Handbook of Optical Constants of Solids, Five-Volume Set: Handbook of Thermo-Optic Coefficients of Optical Materials with Applications 9780080523750, 9780125444156

Citation preview

Handbook of Optical Constants of Solids

Handbook of Optical Constants of Solids

Handbook of Optical Constants of Solids

Edited by EDWARD D. PALIK Institute for Physical Science and Technology University of Maryland College Park, Maryland

A c a d e m i c Press San Diego London Boston New York Sydney Tokyo Toronto

This book is printed on acid-free paper.

@

Copyright 9 1998, 1985 by Academic Press All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. ACADEMIC PRESS 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA 1300 Boylston Street, Chestnut Hill, MA 02167, USA http://www.apnet.com Academic Press Limited 24-28 Oval Road, London NW1 7DX, UK http://www.hbuk.co.uk/ap/

Library of Congress Cataloging-in-Publication Data Handbook of optical constants of solids / edited by Edward D. Palik. p. cm. Includes bibliographical references. ISBN 0-12-544420-6 1. Solids. 2. Optical properties. 3. Handbooks, manuals, etc. I. Palik, Edward D. QC176.8.06H36 1985 530.4'1 84b15870 CIP

Printed in the United States of America 97 98 99 00 01 IC 9 8 7 6

5

4

Before using the tables and figures of the critiques in Part II of this handbook, the reader should refer to the editor's introduction to these materials (Chapter 1, pp. 6-7).

Contents List of Contributors

XV

Part l DETERMINATION OF OPTICAL CONSTANTS Chapter 1 Introductory Remarks EDWARD D. PALIK I. II. III. IV. V. VI.

Introduction The Chapters The Critiques The Tables The Figures of the Tables General Remarks References

Chapter 2 Basic Parameters for Measuring Optical Properties ROY F. POTTER Io II. III.

IV. V. VI. Appendix A. Appendix B. Appendix C. Appendix D. Appendix E.

Introduction Intrinsic Material Parameters in Terms of Optical Constants Reflectance, Transmittance, and Absorptance of Layered Structures The General Lamelliform - - Phase Coherency Throughout The General Lamelliform - - Phase Incoherency in Substrate Summary. Basic Formulas for Fresnel Coefficients General Formulas for the Case of a Parallel-Sided Slab Reflectance, Rjk at j - k Interface Reflectance of Single Layer on Each Side of a Slab and Single Layer on Either Side of a Slab Critical Angle of Incidence Definition of Terms References

Chapter 3 Dispersion Theory, Sum Rules, and Their Application to the Analysis of Optical Data D. Y. SMITH I.

II.

Introduction Optical Sum Rules and Their Physical Interpretation

11 16 18

19 21 24 24 25 26 26 30 33 34

35

36 36 vii

viii

Contents III. IV. V. VI.

Chapter 4

I~

II. III.

Chapter 5

I~

II. III. IV. V. VI. VII.

Finite-Energy Sum Rules Sum Rules for Reflection Spectroscopy Analysis of Optical Data and Sum-Rule Applications Summary References

45 51 55 64 64

Measurement of Optical Constants in the Vacuum Ultraviolet Spectral Region W. R. HUNTER

69

Introduction General Discussion of Reflectance Methods Reflectance Method for Two Media References

69 70 85 87

The Accurate Determination of Optical Properties by Ellipsometry D. E. ASPNES

89

Reflection Techniques; Background and Overview Measurement Configurations Accurate Determination of Optical Properties: Overlayer Effects Living with Overlayers Eliminating Overlayers Bulk and Thin-Film Effects; Effective-Medium Theory Conclusion References

Chapter 6 Interferometric Methods for the Determination of Thin-Film Parameters JOSEPH SHAMIR I~

II. III. IV. V. VI. VII.

Introduction Basic Principles Nonlaser Interferometers K6sters-Prism Interferometers A Self-Calibrating Method Surface Effects Conclusions References

Chapter 7 Thin-Film Absorptance Measurements Using Laser Calorimetry P. A. TEMPLE I~

II. III. IV. V.

Introduction Single-Layer Films Wedged-Film Laser Calorimetry Electric-Field Considerations in Laser Calorimetry Entrance versus Exit Surface Films

89 92 96 99 102 104 108 110

113

113 114 117 123 126 131 132 133

135

135 138 139 143 147

Contents

ix VI.

Chapter 8

I~

II. III. IV. V. VI. VII. VIII.

Chapter 9

I~

II. III. IV.

Chapter 10 I~

II. III. IV. V.

Chapter 11

I~

II. III. IV. V~

VI.

Experimental Determination of o~f, aaf, and afs References

149 153

Complex Index of Refraction Measurements at Near-Millimeter Wavelengths GEORGE J. SIMONIS

155

Introduction Fourier Transform Spectroscopy Free-Space Resonant Cavity Mach- Zehnder Interferometer Direct Birefringence Measurement Overmoded Nonresonant Cavity Crystal Quartz as Index Reference Conclusion References

155 156 161 163 164 165 165 167 167

The Quantum Extension of the Drude-Zener Theory in Polar Semiconductors B, JENSEN

169

Introduction Quantum Theory of Free-Carrier Absorption Theoretical Results Comparison with Experimental Data Appendix References

169 172 174 176 187 188

Interband Absorption--Mechanisms and Interpretation DAVID W, LYNCH

189

Introduction One-Electron Model Electron-Hole Interaction, Excitons Local Field Effects Examples References General References

189 190 198 203 204 210 211

Optical Properties of Nonmetallic Solids for Photon Energies below the Fundamental Band Gap SHASHANKA S. MITRA

213

Introduction Infrared Dispersion by Polar Crystals Kramers-Kronig Dispersion Relations Determination of Absorption Coefficient in the Intermediate Region Absorption Coefficient in the Transparent Regime Multiphonon Absorption

213 215 227 229 230 232

Contents

x

VII. VIII.

Infrared Absorption by Defects and Disorders Infrared Dispersion by Plasmons References

254 263 267

Part II CRITIQUES Subpart 1 Metals Comments on the Optical Constants of Metals

I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII.

I. II. III.

Subpart 2

and an Introduction to the Data for Several Metals DAVID W. LYNCH AND W. R. HUNTER

275

Introduction Anomalous Skin Effect References Copper (Cu) References Gold (Au) References Iridium (Ir) References Molybdenum (Mo) References Nickel (Ni) References Osmium (Os) References Platinum (Pt) References Rhodium (Rh) References Silver (Ag) References Tungsten (W) References

275 277 279 280 280 286 287 296 296 303 304 313 314 324 324 333 334 342 342 350 351 357 358

The Optical Properties of Metallic Aluminum D, Y. SMITH, E, SHILES, AND MITI9 IN9

369

General Features Optical Measurements and Sample Conditions Tabulated'Data References

369 372 377 383

Semiconductors Cadmium Telluride (CdTe) EDWARD D. PALIK

409

References

413

Contents

xi

Gallium Arsenide (GaAs) EDWARD

429

D. PALIK

References

432

Gallium Phosphide (GAP)

445

A, BORGHESI A N D G. GUIZZETTI References

449

Germanium (Ge)

465

R O Y F. POTTER References

469

Indium Arsenide (InAs)

479

EDWARD D. PALIK AND 12. T. HOLM References

481

Indium Antimonide (InSb)

491

R, T. HOLM References

494

Indium Phosphide (InP)

503

O. J. GLEMBOCKI AND H. PILLER References

506

Lead Selenide (PbSe)

517

G. BAUER AND H. KRENN References

518

Lead Sulfide (PbS)

525

G. GUIZZETTI AND A. BORGHESI References

528

Lead Telluride(PbTe)

535

G, BAUER A N D H. KRENN References

538

Silicon (Si)

547

DAVID F. EDWARDS References

552

xii

Contents

Silicon (Amorphous) (a-Si)

571

H, PILLER References

573

Silicon Carbide (SIC)

587

W. J. CHOYKE AND EDWARD D, PALIK References

589

Zinc Sulfide (ZnS)

597

EDWARD D. PALIK AND A, ADDAMIANO References

Subpart 3

602

Insulators Arsenic Selenide (As2Se3) D, J, TREACY

623

References

625

Arsenic Sulfide (As2S3)

641

D, J, TREACY References

644

Cubic Carbon (Diamond)

665

DAVID F. EDWARDS AND H. R, PHILIPP References

668

Lithium Fluoride (kiF)

675

EDWARD D. PALIK AND W. R. HUNTER References

678

Lithium Niobote (LiNb03)

695

EDWARD D. PALIK References

697

Potassium Chloride (KCI)

703

EDWARD D, PALIK References

706

Contents

xiii

Silicon Dioxide (Si02), Type cx (Crystalline)

719

H. R. PHILIPP References

721

Silicon Dioxide (Si02) (Glass)

749

H. R. PHILIPP References

752

Silicon Monoxide (SiO) (Noncrystalline)

765

H. R. PHILIPP References

766

Silicon Nitride (SiaN4) (Noncrystalline)

771

H. R. PHILIPP References

772

Sodium Chloride (NaCI) J. E. ELDRIDGE AND EDWARD D. PALIK

775

References

779

Titanium Dioxide (Ti02) (Rutile)

795

M. W. RIBARSKY References

798

List of Contributors Numbers in parentheses indicate the pageson which the authors' contributions begin. A, ADDAMIANO (597), Naval Research Laboratory, Washington, D.C.

20375 D. E. ASPNES (89), Bell Communications Research, Inc., Murray Hill, New Jersey 07974 G. BAUER (517, 535), Institut for Physik, Montanuniversit/it Leoben, A-8700 Leoben, Austria A. BORGHESI (445, 525), Dipartimento di Fisica "A. Volta" and Gruppo Nazionale di Struttura della Materia del CNR, Universit& di Pavia, Pavia, Italy W. J. CH9 (587), Westinghouse Research and Development Center, Pittsburgh, Pennsylvania 15235, and University of Pittsburgh, Pittsburgh, Pennsylvania 15260 DAVID F. EDWARDS* (547, 665), University of California, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 J. E. ELDRIDGE(775), Department of Physics, University of British Columbia, Vancouver V6T 1W5, British Columbia, Canada O. J. GLEMBOCKI (503), Naval Research Laboratory, Washington, D.C. 20375 G. GUIZZETTI(445, 525), Dipartimento di Fisica "A. Volta" and Gruppo Nazionale di Struttura della Materia del CNR, Universit& di Pavia, Pavia, Italy R. T. HOLM (479, 491), Naval Research Laboratory, Washington, D.C. 20375 W. R. HUNTERt (69, 275, 675), Naval Research Laboratory, Washington, D.C. 20375 MITIO INOKUTI (369), Argonne National Laboratory, Argonne, Illinois 60439 B. JENSEN (169), Department of Physics, Boston University, Boston, Massachusetts 02215 H. KRENN (517, 535), Institut for Physik, Montanuniversit/it Leoben, A-8700 Leoben, Austria * Present address: LawrenceLivermoreNational Laboratory, Livermore,California94550. I"Present address: Sachs/FreemanAssociates, Inc., Bowie,Maryland20715. XV

xvi

List of Contributors

DAVID W. LYNCH (189, 275), Department of Physics and Ames Laboratory,

U.S. Department of Energy, Iowa State University, Ames, Iowa 50011 SHASHANKAS. MITRA(213), Department of Electrical Engineering, University of Rhode Island, Kingston, Rhode Island 02881 EDWARD D. PALIK(3, 409, 429, 479, 587, 597, 675, 695,703, 775), Naval Research Laboratory, Washington, D.C. 20375 H. R. PHILIPP(665, 719,749,765, 771), General Electric Research and Development Center, Schenectady, New York 12301 H. PILLER(503, 571), Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803 ROY F. POTTER(11,465), Department of Physics and Astronomy, Western Washington University, Bellingham, Washington 98225 M. W. RIBARSKY(795), School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332 JOSEPH SHAMIR(113), Faculty of Electrical Engineering, Technion ~ Israel Institute of Technology, Haifa 32000, Israel E. SHILES*(369), Argonne National Laboratory, Argonne, Illinois 60439 GEORGE J. SIMONIS (155), Electronics Research and Development Command, Harry Diamond Laboratories, Adelphi, Maryland 20783 D. Y. SMITH (35, 369), Argonne National Laboratory, Argonne, Illinois 60439, and Max-Planck-Institut for Festk/~rperforschung, 7 Stuttgart 80, Federal Republic of Germany P. A. TEMPLE (135), Michelson Laboratory, Physics Division, Naval Weapons Center, China Lake, California 93555 D. J. TREACY (623, 641), Physics Department, U.S. Naval Academy, Annapolis, Maryland 21402

* Present address: GulfResearch and DevelopmentCompany,Houston, Texas 37048.

Handbook of Optical Constants of Solids

Part [ Determination of Optical Constants

Chapter t Introductory Remarks EDWARD D. PALIK Naval Research Laboratory Washington, D.C.

I. II. III. IV. V. VI.

Introduction The Chapters The Critiques The Tables The Figures of the Tables General Remarks References

3 4 5 6 7 8 9

INTRODUCTION

During a professional career covering molecular spectroscopy, magnetooptical properties of semiconductors, internal-reflection spectroscopy, and general optical properties of electronic materials, I rarely measured refractive index n or extinction coefficient k to better than two significant figures and then only when the experiment really demanded it. However, there was always a need for n and k values for estimating an optical effect, internal reflection in a prism with multilayers, band-edge transmission of a thin semiconductor film on a substrate, or free-carrier reflection effects, for example. Such numbers were not very handy, especially for a variety of metals into the infrared to the far infrared and for semiconductors with various freecarrier concentrations. The present handbook is an attempt to collect references and tabulate n and k for a reasonably large number of materials of technological and physics interest, so that a person having the need could easily find the numbers over the widest spectral range, ideally x-ray to microwave, in a grid fine enough to yield the number by inspection or by a simple interpolation, and provide the pertinent references if more details were needed. For each material a critique is provided to indicate how the numbers were obtained from experiment and how good they are. 3 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

ISBN 0-12-544420-6

I

4

II

Edward D. Palik

THE CHAPTERS

To preface and supplement the critiques for each material, there are 11 chapters (including this one) that discuss various aspects of optical constants with some emphasis on experimental techniques and sample preparation. While n and k are oblivious to the units used in Maxwell's equations, deriving these numbers both theoretically and experimentally can be done in various units that the individual contributors have indicated when necessary (primarily cgs and mks). In Chapter 2 Roy F. Potter develops R, T, and A for a general sample of multiple films on both sides of a thick substrate. This simplifies to the single slab and the single film on a slab substrate, the two most prevalent kinds of samples. In Chapter 3 D. Y. Smith reviews the world of sum rules over infinite and finite frequency ranges and stresses how such analysis can spot experimental errors. In Chapter 4 W. R. Hunter discusses polarized nonnormal incidence reflection techniques especially as they are practiced in the ultraviolet-visible (UV-vis) spectral region, with emphasis on how errors in R are propagated to errors in n and k and the effects of oxide layers. In Chapter 5 D. E. Aspnes outlines wavelength ellipsometry with emphasis on the effects of surface preparation and the properties of a deposited or grown film on a substrate. In Chapter 6 Joseph Shamir reviews standard interferometric techniques for transparent samples and then discusses his more recent laser interferometric technique for measuring both n and k for a film on a substrate. In Chapter 7 P. A. Temple deals with calorimetry for bulk material and for a film on a substrate to determine small absorption constants. In Chapter 8 George J. Simonis discusses the more-specialized techniques used in the far-IR and submillimeter-wave region to determine optical constants. These are of considerable interest now with the advent of gyratron sources. In Chapter 9 B. Jensen summarizes the quantum free-carrier-absorption theory for semiconductors and gives numerical results for the contributions to n and k for GaAs, InP, and InAs for various free-carrier densities. Thus, the reader can calculate from scratch (or use an abbreviated table) to determine the freecarrier contribution to e~ and e2 and thus n and k. The regular critiques for GaAs, InP, and InAs were done for "pure" material, and these values of n and k can be combined with the free-carrier contributions to estimate the optical constants for any sample of arbitrary free-carrier concentration. Finally in Chapter 10 David W. Lynch discusses the absorption mechanisms above the fundamental band gap with emphasis on the variety of interband, plasma, core-level, exciton, and Urbach-tail absorption processes. In Chapter 11 Shashanka S. Mitra describes the absorption mechanisms below the fundamental band gap, such as single- and multiphonon processes, free-carrier absorption, and impurity absorption.

1. Introductory Remarks

5

Special attention has been given to aluminum by D. Y. Smith in a long critique since it is the most-studied metal and its optical constants have been determined over a wide spectral range with the aid of dispersion relations. It would be hoped that the data given for some of the other materials might spur further such analysis. We have stressed the film-on-a-substrate sample because this is becoming more and more the sample confronting us in materials studies since the upsurge in (1) epitaxial film growth, (2) use of insulator films on semiconductors, (3) surface studies of adsorbed layers under ultrahigh-vacuum (UHV) conditions, and (4) reflection and antireflection coatings.

THE CRITIQUES

Thirty-seven materials have been examined with some crystalline and amorphous forms included for a few materials. The critiques have been done by 21 people. Thus, the styles vary somewhat. We wanted a common thread of discussion of experimental techniques, reasons for choices of numbers used in the table, the most important literature references, and some idea of accuracy and precision. This last requirement has been met head-on by some of the contributors (and original workers) who quote plus/minus variations in experimental scatter for a Sellmeier-type fit to express the data, who discuss limits of systematic and random errors, or who (more vaguely) discuss which decimal place is becoming unreliable or uncertain. Because of the wide spectral range along with the number of different experimental techniques used for a given material, this becomes a task impossible to do in detail, and we can only refer the reader back to the original references. In most tables we have overlapped n and k values from different experiments to give the reader an idea of the mismatch. Sometimes, we can figure out the reasons for this mismatch and suggest which is the better set of data. For example, Kramers-Kronig (KK) analysis deteriorates when k becomes smaller than n and at the edges of a particular spectral region; also, the purity and structure of the material can vary among laboratories. Unfortunately, the variations in n and k are such that for nontransmission experiments generally, only one or two figures should be used in quoting absolute values (accuracy), while more are often carried along to show spectral structure as a function of wavelength (precision). Only in the case of measurement of refractive index by minimum deviation or fringes are we able to set down values to four decimal places with some confidence. While we can give examples of ellipsometric measurements and KK analysis of reflectivity in an overlap region (1.5-6.0 eV) that agree to two figures, we also have examples

III

6

Edward D. Palik

in which two laboratories measuring R and doing KK analysis hardly agree to two figures among themselves. Much of the work cited was done by physicists interested in absorption peaks so as to be able to identify phonon absorption and interband absorption transitions in detail. Their precision is generally high in order to distinguish peaks and shoulders, but their accuracy is not necessarily high, as a comparison of two experiments measuring multiphonon absorption in silicon or gallium arsenide will show, for example. We do think that a single set of values of n and k is desirable in contrast to a collection of all the work ever done since the dawn of time with no selectivity or hints as to what is what. To some extent, we feel this handbook accomplishes this goal. There have been a number of previous collections of optical data that have been helpful. The most useful is "The Refractive Index of Alkali Halides," by Li [1], in which Sellmeier-type fits have been given to many materials. The work of Haelbich et al. [2], "Optical Properties of Some Insulators in the Vacuum Ultraviolet Region," was especially useful because it catalogued all the x-ray to UV work of a number of insulators and proved to be a good starting point. Another older collection was published by Moses [3], "Refractive Index of Optical Materials in the Infrared Region." My copy of Ballard et al. [4], "Optical Materials for Infrared Instrumentation," is dogeared from use. Another more recent starting point was Wolfe's [-5] "Properties of Optical Materials." The table of Hass and Hadley [6], "Optical Properties of Metals," was useful, as was the more recent compilation of Weaver et al. [7] based on synchrotron data for metals.

IV

THE TABLES

A few general comments about the format of the tables are in order. We have used photon energy in electron volts, wave number in reciprocal centimeters, and wavelength in micrometers for each material to aid the reader. It is natural to think in electron volts and angstroms in the x-ray-UV region, while in the infrared reciprocal centimeters and micrometers are more comfortable. We will never get used to a number like ! 10,000 cm- ~, although this usage appears in older atomic spectroscopy books [8]. Therefore, we introduce reciprocal centimeters usually near 50,000 cm-x ( ~ 6 eV), keeping them to all lower energies. It is usually obvious what the units were for the original data because of the systematic trends of the simple numbers given to one, two, or three figures. The other two units are usually given to four figures just for consistency and to help identify the original units. To convert electron volts to reciprocal

1. Introductory Remarks

7

centimeters we use 8065.48 c m - l / e V rounded off from the most recently determined [9] fundamental constants e/hc. The real and imaginary parts of the dielectric function e are given by E = el +_ ie2 = (n +_ ik) 2. One obtains n +_ ik by choosing the form of the plane-wave solution to Maxwell's equations as exp[i(-Y-wt _+ q . r)]. This choice also determines signs in the Lorentz-oscillator formula for lattice vibrations. We have elected to tabulate n and k because it is then an easy matter to obtain el = F/2 - k 2 and e 2 = 2nk as well as reflectivity R = [(n - 1)2 + k2]/[(n + 1)2 + k2]. The extinction coefficient is often given in powers of 10. This power is understood to be the same down the column until this power repeats itself just before a new power of 10 appears or the number becomes a pure decimal at 10 -~. In most instances, the values for n and k have been obtained from tables and graphs in the original references. In some cases, the original workers have kindly supplied detailed tables to the critiquers. The tables are acknowledged as private communications. To obtain n and k from a graph produces all sorts of eyestrain problems. Many authors do not provide fiduciary marks on all four sides of the graph or omit the marks for 2 - 9 on log plots. Tables that are too small to read must be expanded. For some reason, artists and/or experimenters cannot draw a linear scale with evenly spaced fiduciary marks. Therefore, some of these numbers probably have an uncertainty larger than the experimental uncertainty quoted in the original paper. When absorption is due to extrinsic mechanisms such as impurities and surface quality, the critiquer has mentioned this and either omitted the data or noted its extrinsic character. The references are given in brackets next to the beginning n and k and are understood to apply down the column until a new reference appears. This may get a little confusing when a significant overlap of two sets of data occurs.

THE FIGURES OF THE TABLES

The tables are plotted for each critique, usually as Fig. 1, on a log-log scale, so that the reader can see at a glance what is going on. Admittedly, the details are lost. The solid line is for 11; the dashed line is for k. After plotting over 30 such tables, 1 find the optical constants of solids to be a boring subject with the same old reststrahlen bands, the same old multiphonon bands, the same old flee-carrier absorption, the same fundamental band gap and higher-lying band gaps and the same K, L, M atomic absorption edges.

V

8 Vl

Edward D. Palik

GENERAL REMARKS

A few interesting observations have been made while I was doing several of the critiques and reading all of them. Room temperature was anything from 292 to 300 K, and it was surprising how many authors neglected to report this number at all. This is no problem except for the refractive-index variation typically in the fourth decimal place for a 1-K change in temperature. Some critiques do quote temperature dependences. Description of the sample surface preparation was often neglected, as was the free-carrier concentration in semiconductors. Mechanical (nonchemical) polishing of the surface should have been outlawed 20 years ago, although whether it affects an optical measurement often depends on the wavelengthto-damage size and the kind of material~metal, semiconductor, or insulator. We sometimes run into trouble with surface roughness at reststrahlen minima at which refleetion is low (R ~ 0.02) and at maxima at which surfaceoptical-phonon features weakly occur because roughness can couple light to the surface-phonon modes. Deposited films were often undercharacterized as to state of crystallinity. The great majority of far-IR determinations of n and k are indirect via the Lorentz-oscillator-model and Drude-model fits to R. The former is notoriously poor in the wings, but it is all we have in many cases, and many of the materials in the handbook are analyzed in the IR by this technique. In a few cases, KK analysis for R was also done, with some agreement with the results of an oscillator-model fit. Fortunately, asymmetric Fourier transform spectroscopy can now give both the real and imaginary parts of R, e~, and e2, and n and k directly, thus avoiding the oscillator model and KK analysis. The other, more direct, technique of polarized nonnormal incidence reflection is not so widespread in the IR yet. The nomenclature for optics gets a little mixed up on occasion in the original references and in the present chapters and critiques. While reflection and transmission are general processes, the measurement one makes on a laboratory sample yields numbers, the reflectance and transmittance. This, of course, includes multiple-reflection effects. As the sample becomes opaque or as the sample is wedged, the reflectance approaches the single-surface reflectivity, and this is the quantity that is used in KK analysis. Since ~ = 4~rk/2 defines the absorption constant, most people measure directly and do not convert to k. In converting some ~ data to k data over a considerable range of 2, I noticed that detailed structure was often washed out somewhat in the final plot of k in contrast to the plot of ~. Also, a peak in ~ did not necessarily fall at the same wavelength as a peak in k. Since power absorption is proportional to ~, measurement of ~ is a better approach to locating resonances.

1. Introductory Remarks

9

Very few of the measurements were done to determine n and k for their own sake, except when the material was new, such as in the case of InSb in 1953. The interest in laser-window materials spurred such measurements in selected transparent materials such as NaC1 and KC1. The availability of synchrotron-radiation sources is now producing physics interests in material structure and will yield better values. Some general reviews of how to calculate and measure optical constants are also listed in the references [10-14]. REFERENCES

1. H. H. Li, J. Phys. Chem. Ref. Data 5, 329 (1976). 2. R.-P. Haelbich, M. Iwan, and E. E. Koch, "Physik Daten, Physics Data," Fach-information zentrum, Karlsruhe, 1977. 3. A. J. Moses, Refractive Index of Optical Materials in the Infrared Region, Report on Contract F33615-68-C-1225, Air Force Systems Command, Hughes Aircraft Company, Culver City, California, 1970. 4. S. S. Ballard, K. A. McCarthy, and W. L. Wolfe, Optical Materials for Infrared Instrumentation, Report from Infrared Information and Analysis Center, University of Michigan, Willow Run Laboratories, Ann Arbor, Michigan, 1959. 5. W. L. Wolfe, in "Handbook of Optics OSA," Section 7, McGraw-Hill, New York, 1978. 6. G. Hass and L. Hadley, in "AIP Handbook," Section 6, pp. 118-169, McGraw-Hill, New York, 1972. 7. J. H. Weaver, C. Krafka, D. W. Lynch, and E. E. Koch, "Physik Daten, Physics Data," "Optical Properties of Metals," Vol. 18-1, Fach-information zentrum, Karlsruhe, 1981. 8. H. E. White, "Introduction to Atomic Spectra," McGraw-Hill, New York, 1934. 9. M. Aguilar-Benitez, R. L. Crawford, F. Frosch, G. P. Gopal, R. E. Hendrick, R. L. Kelly, M. J. Losty, L. Montanet, F. C. Porter, A. Rittenberg, M. Roos, L. D. Roper, T. Shimada, R. E. Shrock, T. G. Trippe, C. Walek, C. G. Wohl, and G. P. Yost, Phys. Lett. lllB, 1 (1982). 10. E. E. Bell, in "Handbuch der Physik," Vol. 25, Part 2a, p. 1 (S. Flugge, ed.), Springer-Verlag, Berlin, 1967. 11. F. Abeles, in "Progress in Optics," Vol. 11, p. 251 (E. Wolf, ed.), North-Holland Publ., Amsterdam, 1963. 12. H. E. Bennett and J. M. Bennett, in "Physics of Thin Films," Vol. 4, p. 1 (G. Hass and R. E. Thun, eds.), Academic Press, New York, 1967. 13. O. S. Heavens, "Optical Properties of Thin Solid Films," Dover, New York, 1965. 14. P. H. Berning, in "Physics of Thin Films," Vol. 1, p. 69 (G. Hass, ed.), Academic Press, New York, 1963.

Chapter 2 Basic Parameters for Measuring Optical Properties ROY F. POTTER Department of Physics and Astronomy Western Washington University Bellingham, Washington

I. II. II1. IV. V.

Introduction Intrinsic Material Parameters in Terms of Optical Constants Reflectance, Transmittance, and Absorptance of Layered Structures The General LamelliformmPhase Coherency Throughout The General Lamelliform--Phase Incoherency in Substrate A. Absorptance of a General Lamelliform B. Reflectance of Single Film on Nontransmitting Substrate VI. Summary Appendix A. Basic Formulas for Fresnei Coefficients Appendix B. General Formulas for the Case of a Parallel-Sided Slab Appendic C. Reflectance R~k at j-k Interface Appendix D. Reflectance of Single Layer on Each Side of a Slab and Single Layer on Either Side of a Slab Appendix E. Critical Angle of Incidence Definitions of Terms References

11 16 18 19 21 22 23 24 24 25 26 26 30 33 34

INTRODUCTION

The interaction of electromagnetic (em) radiation with matter has provided over the years an effective and powerful tool for understanding and probing the electronic and vibrational states of condensed materials. Maxwell's equations describing the propagation of em waves are so well established and accepted that they serve as the underlying bases for optical studies. Based on Faraday's law of induction and Maxwell's extension of Ampere's 11 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright 9 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-544420-6

I

12

Roy F. Potter

law, they are given, in partial differential form, as V x E = -OB/Ot,

V x H = J + OD/Ot,

(1)

along with the divergence relations for induction and displacement

V.B=0,

V.D =0

(assuming that no free charge exists), where E is the electric field, H the magnetic field, B the magnetic induction, D the electric displacement, and J the electric current flux [1]. (Expressions are in mks units.) However, these expressions are not sufficient in themselves to completely describe em propagation through a medium; i.e., no "optical constants" are contained in Maxwell's equations. A full description must include the trio of "material equations" D = e%E,

B = ##oH,

J = o'E

(Ohm's law),

(2)

where the coefficients eeo, ##o, and ~ are restricted for present purposes to be time- and field- independent scalar quantities. (See Born and Wolf [2] for more general tensor treatments.) These restrictions follow the assumption of isotropic media and linear responses. An additional important assumption is that these quantities are continuous in the immediate vicinity of the points where the fields are considered or measured. The coefficients eeo, ##o, and cr are known as the electric permittivity, the magnetic permeability, and the electrical conductivity, respectively; while e0 and #0 are the free-space permittivity and permeability, respectively. The so-called optical constants are contained in the coefficients e, #, and cr. Used for describing the macroscopic optical properties of matter, these coefficients can be related to polarizabilities on the atomic scale and provide details on the physics of the condensed state. The expressions used in this section are based on the rationalized mks system of units. A consequence is that the propagation velocity in a given medium is given by (#e#oeo)- 1/2 = C(#~)- 1/2 where c = (#o%)- 1/2 is the freespace velocity of light. However, when Gaussian units are used, the propagation velocity is stated explicitly in Maxwell's equations, and the freespace velocity defined by (#o%)-1/2. At the risk of belaboring the point, the refractive index for a given medium c/v is defined by the dimensionless quantity (#~)1/2. The Fresnel coefficients, reflectivity and transmittivity, are also defined by dimensionless ratios, namely, of the field strengths. Thus the complex refractive indices (as well as the corresponding complex dielectric functions k) carry no reference to any system of units. When they are used to determine field strength values relative to given input values, the fields must be stated in a consistent set of units (see Table I). An additional point to keep in mind is that the dielectric susceptibility term in the polarization vector P in the rationalized system of units is 4~r times the susceptibility term in an unrationalized system.

2. Basic Parameters for Measuring Optical Properties

13

TABLE I

Conversion of Units Quantity

mks

P e r m i t t i v i t y of free space eo

P e r m e a b i l i t y of free space Po

Speed of light c = (Poeo)-1/2

Practical cgs

10 -9 F 4ha 2 m

10 -11

F

1

4ha 2 cm

H 4n x 1 0 - 7 _ m m a • 108-(a sec

4n H

4n

cm

c2

4n x 10 . 9 _

= 2.9979)

a • 101~

Gaussian

cm

a x 10 l ~

sec

cm sec

I m p e d a n c e of free space Z 0 = (]2o/s

40ha f~

1/2

Electric intensity E

Electric d i s p l a c e m e n t D

4 0 h a f~

4n/c

V

V

m

cm

300 cm

C 4ncm 2

C 12n x 10 9 _ cm 2

Oe

Oe

G

G

C m2

1

V

ampere-turn M a g n e t i c intensity H m W m2b (10 4 G)

Magnetic induction B

Table I lists several parameters in terms of three systems of units in common use.

Introducing the material equations permits Maxwell's equations to be expressed as a modified set of partial differential equations involving only two of the four field parameters, namely, E and H. Assuming that the fields have time dependence proportional to e ~ t , the modified Maxwell's equations become 2n# V x E =

- #po(-I =

- ip#ocoH

=

- i --j-

ZoH ,

(3) Vx

H=J+I)=J+eeoEco=i

e

e

eoooE . . . . E, 2 Zo

where = e -(ia/e,

oCO) = e -

ia2Zo/2n

= e'-

ie,"

and c = ( #0%)-1/2,

Z o = ( # 0 / % ) 1/2,

2 = 2nc/og.

The complex function g has a real term equal to the ratio of the medium permittivity to that of free space, i.e., e, and an imaginary term proportional to the frequency-dependent conductivity a. The wave velocity, impedance, and wavelength for free space are given by c, Zo, and 2, respectively.

14

Roy F. Potter

For the infrared, visible, and ultraviolet spectral regions, the em radiation is considered to be propagating as plane waves. When such waves are incident upon a specimen composed of one or more homogeneous phases separated by specular surfaces, the amount and degree of transmission, reflection, and refraction are direct measures of the material parameters. Many of the usual and useful experimental arrangements have the em waves incident upon a set of parallel-sided slabs, with zero separation between the slabs (lamelliforms). The two field components E and H are orthogonal to each other and are contained in a plane orthogonal to the propagation direction. One needs to consider two cases only: (1) the electric field E normal to the plane of incidence (POI) (TE mode) and (2) the magnetic field component H normal to the POI (TM mode). (The POI is defined as the plane containing the propagation vector and the normal to the surface upon which the wave is incident.) Consider that the normal to the incident surfaces lies parallel to the z axis and that the x axis is normal to the POI. The y axis lies in the POI and is parallel to the surface. Both the TE (also known as s or perpendicular polarization) and TM (p or parallel polarization) modes are treated in similar manners by using separation of variables techniques. The following are taken as solutions for each polarization: TE polarization

TM polarization

y)

H x = U ( z ) X ( t , y)

H~ = V(z)X(t, y)

~ = V(z)X(t, y)

U z = W ( z ) X ( t , y)

Ez = W ( z ) X ( t , y)

F~. = V(z)X(t,

(4)

where X = exp i(cot - xSy), in which x = co/c = 27r/2 = (/./o2o)1/2(.o,

S = (#01/2 sin ~b,

and ~b is the angle of incidence in the medium. Operating on these with the modified Maxwell's equations, V x E = -##o

~H/Ot,

V x H = e%o OE/c3t,

yields for either polarization the expressions u' = -irhv,

V' = - ~ 2 / r h u ,

SU = -rhW,

(5)

where ~ = :r - ifl, 392 = #~ - S 2, and #~ = 112 = (n - ik) 2. The rh is defined for each mode as rh = ~

#Z~ = fi2 = -fi/2

-e/Zo

(TE), (TM).

(6)

2. Basic Parameters for Measuring Optical Properties

15

It should be noted that in this section the prime denotes the use of the operator (1/~c)8/8z. Eliminating W yields two second-order ordinary differential equations in U and V: U" + ~2(z)U - f ' ( z ) U ' = O,

(7) V" + ~ ( z ) V + O'(z)V' = 0,

where f = ln(nS)= ln(~/~/), g = ln(rh/4/2) = ln(~/~/), and where the material parameters m and ~ are still to be considered to vary along z; also:

= _th = { 2fi/~ = Zo#/~ 7 -fi/Z~ = - g / ~ Z o

(TE), (TM).

(8)

The pair of differential equations (7) can be solved analytically for only a few special cases. (See Jacobsson [3] for a comprehensive discussion of this problem.) Two cases are considered here, the first of which is that for an abrupt interface; i.e., fi' is infinite. This is expressed in matrix form as

E ]--['o Uo

Vf]

(9)

where the subscripts refer to the initial side (0) and the opposite side (f) of the slab or interface. This matrix expression is the statement that tangential fields are equal across the interface. The second case is that for a medium characterized as having a uniform refractive index n, i.e., f ' and g' vanish. This results in the usual homogeneous wave equation whose solutions are expressed in the "characteristic matrix" [Uo]=

cos(fz) [[i sin(~z)]/~

i~ sin(fz)] cos(~z)j[U~]=/~/[U~],

,10)

where U, V, and ~ are appropriately defined for either polarization mode. Note that z is the dimensionless product of the distance between points 0 and f with ~c = co/c. The determinants for both of these 2 x 2 matrices are constants and the particular solutions have been chosen to make both equal to unity. The characteristic matrix for a uniform layer is the basis for calculating the resultant matrix MR for a series of parallel layers. Applying matrix multiplication to the matrices for the individual layers gives

. . . .

M 0 x M 1 x'"x (see Macleod [4]).

Mf = M R =

dR

~

DR

(,,,

16

Roy F. Potter

An alternative approach is to apply Eqs. (5) and express the matrix for each interface and layer in terms of the forward and reverse field components:

U--U + + U - = ( V +-v-)~,

V = V + + V - = ( U +-U-)~'-'

(12)

(note that ~V + = + U+). The superscripts + and - refer to forward and reverse components, respectively. For the ith abrupt interface, the matrix equations are

Uo] =

l[ f f~f][U~ LV,-]: es':'-'VU~ ) ]' LU?

fOf = ~f .+. ~0' Cs(V) = tof

tot

=

1 -

-- rof roe =

tOf = 1 + foe--" ~f _~_~0' -lf~ o

(13)

(~s(W)= --mo ]~/f Cs(U)'

f)tof,

while that for the medium with a uniform index is

[u~l=[eSa Oi~a][U~l r U~] Uf J = ~l UoJ

e

LUf

(14)

with d = z t - Zo. Each interface and intervening homogeneous layer can be described by its appropriate matrix. Matrix multiplication must be applied in the proper sequence, beginning with the initial interface matrix (sO) and ending with that for the final interface (sf) and alternating uniform layer-interfaceuniform layer, etc., matrices. The result, CR, corresponding to Eq. (11) is

r

=

r

x Cl, x Cs, x ClS x Css x " ' x

Clf x Csf.

(15)

The matrices expressed in Eqs. (13) and (14) are given in terms of the Fresnel terms f and f as well as the phase factor ~ and are the bases for ellipsometry. The reader is referred to Heavens for a discussion of this approach [5].

INTRINSIC MATERIAL PARAMETERS IN TERMS OF THE OPTICAL CONSTANTS

This section outlines the connections between the macroscopic optical constants and the electric-dipole excitations on the atomic scale. Thus, the accurate determination of the optical parameters of a given medium provide intrinsic information about that medium. At optical frequencies these parameters exhibit rich and varied features unique to the particular composition

2. Basic Parameters for Measuring Optical Properties

17

under study. Even the presence of a small amount of an impurity can significantly alter portions of the spectral behavior. In addition, these parameters can be modified by temperature variations and by additional spectral features introduced by the application of external fields such as electric, magnetic, and stress fields, which can bring about symmetry changes, injection of charges, and so on. It should be understood that the expanded Ampere's law is a generalized statement, that the motion of charges in an oscillating electric field has two components: (1) 90 ~ out of phase with the field with no absorption of power and (2) in phase with energy from the field being dissipated in the material, corresponding to the usual Ohm's law. The optical conductivity is not to be confused with the dc conductivity a0, however. From Ohm's law [-Eq. (2)] the mean power P dissipated per unit volume is given by P = o e~, there exists a critical angle of incidence ~br at which =

where ~b~ =

arcsin

(g,)/n~) 1/2.

(N.B." Many texts define ~b~ as arcsin(nj/no), but this is the special case for a nonabsorbing medium with %(~b~) =/~j(4~) = 0.) Although the Fresnel relations for ~b > ~b~ can differ markedly from those for ~b < ~br the formalism developed in this chapter is directly applicable. Two examples for the three-medium expression of Eq. (36) will illustrate the point.

Example 1. Frustrated Total Reflectance. Consider that the third medium is optically identical to the incident medium and that the intervening medium is completely nonabsorbing, i.e., n2 = no, e'~' < 0, and nt < no. The term Ro: in Eq. (36) is zero for all angles of incidence. Also note that R 0 1 = R 1 2 = g21 + h21, go1 = - g 1 2 , and h o l = - h i 2 . Also ~1 = ~ and W = e x p ( - 2/~ldl). Rearranging Eq. (36) with Ro2 = 0 gives Ro~ [4W sin 2 6~ + (1 - W) 2] Rf = (1 - Ro~ W) 2 + 4W[h21 + (g~l - ho2~)sin2 6~ - 2go~ho~ sin 6~ cos 6~]" (36") It is evident that below the critical angle the reflectance varies between zero

2. Basic Parameters for Measuring Optical Properties

31

and 4Roa/(1 + ROl) 2 as a function of 0~1dl, while above the critical angle, Rox = 1, a l = 0 and Rf has values between zero (zero thickness) and unity (i.e. thickness d I is sufficiently large that f l l d l is such a large number that W~O). What is termed f i ' u s t r a t e d t o t a l reflection occurs for intermediate thicknesses. For small values of 2(S z - nl)2,1/2, al = 2 f l l d l , (S 2

Rf ~"

2

hol

+

-

2 2 nl)dz

(5 2 -

2

n~)d,

vanishes as d 2 and becomes zero, as expected. On the other hand, as the thickness-to-wavelength ratio dl is increased, W decreases exponentially; and as it becomes small in value, R e can be approximated as Rf ~- 1 - [ 4 W h

21/(1 --

W)2].

Although the front-surface reflectance Rox is unity, it is modified by the factor W. This is the result of penetration of the fields into the rarer medium. The W term is a measure of the damping of these fields, and if the two surfaces are not separated by a suitable amount, the evanescent waves couple into the third m e d i u m - - f r u s t r a t i n g the totality of the reflectance. Equations (13), (14), and (22) contain the information necessary for studying the amplitudes and intensities of the fields present in the rarer medium. The following list of equations gives the parameters needed for the frustrated total-reflectance case: Rf =

R~ (1 - Rol

3)1

--

(X l

--

2

Sin2 6 + (1 -- W ) 2] , -) \

9

,,) c,.

.

.

.

.

+ 4 W [ h 2, + (g~, -- h21) sin 2 6 - 2golho1 sin 6 cos 5]'

i fl l ,

70 = no cos r

p2 _ c~21+/32, X o = $2/7o,

~" fl ~:

(n~ - S:) ~a 0

Yl:

~1

72 = ~2 _ f12 _ 2ia~f11 = n 2 -

S 2,

S 2 = n 2 sin 2 4).

0 (S ~ - n~)'/'

-i31 ~, = - ~

w:

1

~:

7,dl

exp(- 2flldl) 0

"7o - 7~

(TE, s)

70 +71 go1"

y2o+72 n~ + n ~ - 2S 2

< (])1 - - X o ) ( ' ~ l

- - ])0)

(7 ~ + Xo)(7~ + 70)

(TM, p)

~,(Xo + ~o) ~ +(~, + Xo~o) ~

(x~- ~,~)(~o~-~,~) 2

2

4

(y~/yo)no + n4i (continued on next page)

32

Roy F. Potter

(continuedfrom precedingpage) h0x:

2i1,'1Yo

0

(TE, s)

0

(TM, p)

2{(S 2 -

n~)(n~

-

Sz)} t/2

2i)'x(),o + Xo)(72 + s 2) (x~ - 7~)I~ - 7~)

2(s ~ _ n2)'/~(n~/~o)n21

70 + 71J Ro,: J [(71 - Xo)(T, - 7o)-]2

I.L(r,

(TM,p)

+ Xo)(~l + ~o)J

Example 2. Attenuated Total Reflectance. A more interesting experimental situation exists when either medium 1 or 2 absorbs a fraction of the radiation; i.e., e,its # 0. In this case, the radiation is also attenuated by the factors exp(-fljdj), hence the name attenuated total reflectance (ATR). Two cases are considered: (1) the layer is the rarer medium but slightly absorbing and (2) the third medium is the rarer one, slightly absorbing and sufficiently ~tt2! thick that exp(-flzd2) is zero. The weak absorption is determined by ~j / 1. Above the critical angle the following approximations can be made:

(e;- 82)2 e~z, as well as weakly absorbing. An example is a coupling prism spaced by air from a sample surface. The intervening layer, medium 1, is nonabsorbing with a refractive index such that e'l > S 2 for all angles of interest so that/31 -~ 0 and W = 1, so that R f --

R o 2 - - Cs, p

1 - Cs,p

,

Us, p

= 49o 1 sin 6 (g 12 sin 6 - h12

COS O).

Evaluating Ro2 in the same manner as that for Ro~ for case 1 gives R o 2 ,~ 1 -

(s ~

-

2e~7o ~h)~/~(n~

-

~h) (TE, s),

2~7o

Ro2 ~ 1 -

(s ~ -

@'/'(n~

-

(S 2 _

~;~)1/2(n2 _

~'~)(1 -

2~7ono~(2s ~ - ~h) a~)(n2S

2e~(70n2)(2S R f '~ 1 -

Depending modulated total as the as a probe,

cJ

2 _

2 -

82),0), 2

(TM, p).

e'2)

(S 2 - - g2)l/2(n2 -- ~3~)(n2S 2 - - e~yo2)(1 -

Cp)

on the thickness of the intervening layer, the ATR for case 2 is by the factor ( 1 - C)-~. In both cases, the reflectance becomes absorption vanishes. Although the ratio ~ = Rp/Rs can be useful care must be taken in case 2 with terms Cs,p.

DEFINITIONS OF TERMS

PPo Magnetic permeability of a medium Magnetic permeability of free space #o g'g,o Permittivity of a medium

34 go (7

h h2 S Zo Z 2 rif

Roy F. Potter Permittivity of free space Speed of light in free space: c = 1/~/-~-oeo Complex refractive index of medium: h = ~ = n - ik Complex dielectric function: h 2 = (e' - ie") = n 2 - k 2 - 2 i n k Snell's law, S is constant: S = h sin ~b = no sin ~bo = n2 sin ~b2 Complex phase term: ~ = h cos q~ = ~ / / h 2 - - S 2 - - ~x - ifl Impedance of free space: Zo = ,v/-#o/eo Complex impedance of medium: 2 = ~/-~/~Zo Wavelength of light in free space: 2 = c/v Fresnel coefficient for reflectivity: ~if = (~e- ~i)/(~f + ~i) Fresnel coefficient for transmittivity across interface in forward direction: tif = 1 + ~if = 2~f/(~f + ~,) Fresnel coefficient for transmittivity across interface in backward direction: Propagation vector in free space: x = 27t/2 Dimensionless parameter for propagation vector x distance along z axis: z = x x physical distance Dimensionless parameter for fixed distance: d = (zf - z0

REFERENCES

1. J.C. Slater and N. I. T. Frank, "Electromagnetism," McGraw-Hill, New York, 1947, or Dover, New York, 1969. 2. M. Born and E. Wolf, "Principles of Optics," Pergamon, New York, 1975. 3. R. Jacobsson, "Progress in Optics," Vol. 5 (E. Wolf, ed.), North-Hol!and Publ., New York, 1966. 4. I. A. Macleod, "Thin Film Optical Filter," American Elsevier, New York, 1969. 5. O. S. Heavens, "Optical Properties of Thin Films," Butterworth, London, 1955, or Dover, New York, 1965. 6. T. Moss, "Optical Properties of Semi-Conductors," Butterworth, London, 1959. 7. R. F. Potter, S P I E P r o c e e d i n g s 276, 204 (1981). 8. C. Gabriel and A. Nedoluha, Opt. A c t a 18, 415 (1971). 9. Edward D. Palik, N. Ginsburg, H. B. Rosenstock, and R. T. Holm, A p p l . Opt. 17, 3345 (1978). 10. D. L. Stierwalt and R. F. Potter, "Semiconductor and Semimetals," Vol. 3 (R. K. Willardson and A. C. Beer, eds.), p. 71, Academic Press, New York, 1967. 11. Roy F. Potter, "Optical Properties" (S. Nudelman and S. S. Mitra, eds.), Plenum, New York, 1969. 12. J. D. Mclntyre and D. E. Aspnes, S u r f . Sci. 24, 417 (1971).

Chapter 3 Dispersion Theory, Sum Rules, and Their Application to the Analysis of Optical Data* D. Y. Smith Argonne

National Laboratory Argonne,

Illinois

and M a x - P l a n c k - l n s t i t u t for F e s t k ~ r p e r f o r s c h u n g Stuttgart, Federal Republic of Germany

I. Introduction II. Optical Sum Rules and Their Physical Interpretation A. Superconvergence Sum Rules for ~(~) B. Superconvergence Sum Rules for N(~o) C. Superconvergence Sum Rules for Functions of N and D. Superconvergence Relations Employing Weighting Functions E. Static-Limit Sum Rules II1. Finite-Energy Sum Rules A. Defect Absorptions and Smakula's Equation B. Finite Oscillator Strength Sums and nerf(~o) IV. Sum Rules for Reflection Spectroscopy A. Normal-Incidence Spectroscopy B. Reflectance between Media at Nonnormal Incidence, Ellipsometry, and Transmission V. Analysis of Optical Data and Sum-Rule Applications A. Construction of Composite Data Sets B. Sum-Rule Tests VI. Summary References

36 36 38 41 42 44 44 45 47 49 51 51 55 55 56 60 64 64

* Work supported by the U.S. Department of Energy and the Max-Planck Gesellschaft. 35 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright 9 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-544420-6

36

D.Y. Smith

INTRODUCTION

The basic optical properties of solids [1, 2], despite their great diversity, are rigorously limited by nature. These limitations take the form of sum rules and dispersion relations that reflect the physical laws governing the dynamics of matter and its interaction with light. Many of these limitations were enumerated in the first half of this century, but in the early 1970s, the discovery of a large number of new sum rules [3-15] deepened our understanding of these constraints and their application. Thus, it is now known that in addition to the celebrated f sum rule for absorption processes, there are companion sum rules for dispersive processes. For example, the real part of the complex refractive index N(og)= n(o9) + ik(co) satisfies [3, 4] fo ~ [n(o9) -- 1] do9 = 0.

(1)

That is, the refractive index averaged over all frequencies must be unity. Moreover, it may be shown that this restriction arises in part from the inertial property of matter. Related rules~with a similar physical interpretation~ have been shown to apply to the complex dielectric function and its inverse. Formally rules such as these are a consequence [4] of the asymptotic behavior of the optical functions and the Kramers-Kronig dispersion relations [16-18]. They are the optical analog of the superconvergence relations [19-21] of high-energy physics. However, physically they arise simply from causality and the dynamical laws of motion [7] and may be viewed as a restatement of these laws in frequency space. Besides giving insight into the mathematical structure of the optical functions, the various sum rules provide a means of relating different physical properties without model fits to spectra and are valuable as self-consistency tests. This chapter is intended as a brief introduction to this topic t with emphasis on the underlying physics and applications. Throughout, the cgs system of units will be used since this is most common in practice. In addition, isotropic media and scalar dielectric functions are assumed. In the case of anisotropic media or symmetry-breaking external fields, the individual dielectric tensor elements must be treated separately. This is reviewed for magneto-optics [11-13] by Smith [23a] and for natural optical activity by Thomaz and Nussenzveig [23b].

II

OPTICAL SUM RULES AND THEIR PHYSICAL INTERPRETATION

The best-known optical sum rule is the f sum rule [1, 24], which may be written in a variety of forms useful in optical analysis [1, 14]: coez(co) do~ = 2

2 0o ogn(og)k(~o) do~ = -~ cop,

(2)

t For some representative references for a broader treatment of linear response theory and sum rules, see Gross, Macdonald et al., Pines et al., Marten et al., and Kubo et al. [22].

3. Analysis of Optical Data

37

00 ok(o) do = ~-O)p, ~ 2

fo

(3)

and fo ~ o9 Im[e-1(o9)]

&o = - ~~ Op, 2

(4)

where e(o9) = e1(o9) + ie2(o)) is the complex dielectric function, in the case* of nonrelativistic electrons in a solid under consideration here, COpdenotes the plasma frequency [4rtXe2/m]X/2; with ~ the total electron density and e and m the electronic charge [e < 0] and mass, respectively. As will become apparent, the factor of 89difference between the right-hand sides of Eqs. (2) and (3) arises because the complex refractive index is the square root of the dielectric function [cf. Eqs. (8) and (22)]. These rules are occasionally rewritten in terms of the optical oscillator strength density [18] f(o9)l e =

(m/2rc2e 2) (_De2(OJ),

(5)

or the absorption coefficient, ~(o) = 2o9k(~)/c,

(6)

where c is the speed of light in vacuo. Note that Eq. (5) is the traditional optical definition of oscillator strength density and arises from the classical notion of the density of Lorentz oscillators. In other applications~such as charged particle energy loss~e-1 plays the fundamental role, and the oscillator strength density for energy-loss processes is taken as [30] f (og)l~- , = -(m/ZrcZe2)o Im[e-1(o9)].

(7)

The f sum rule was first proved by Thomas and Reiche [31] and by Kuhn [32] for nonrealistic systems in 1925. It has played a crucial part in the development of quantum mechanics [33-1, atomic [24, 34] and nuclear [35] * spectroscopy, and it is fundamental to determining the concentrations of defects and dopants in condensed phases via Smakula's equation [36-38]. ~ In , The f sum rules given here are exact for a system of nonrelativistic electrons and infinitely heavy nuclei interacting via a local, velocity-independent potential. Finite nuclear mass may be taken into account via an effective mass. The formalism required for arbitrary nonrelativistic particles of arbitrary mass and a generalization to all electric multipole orders has been given by Sachs and Austern [25]. However, for practical purposes this effective mass correction is generally negligible in treating the optical properties of solids. Relativistic corrections are important for the inner shells of heavy elements. In first order these corrections arise from the mass-velocity term in the approximate relativistic wave equation describing the electrons. These appear to have been first described by Levinger et al. [26] and later found independently by Dogliani and Bailey [27]. Relativistic oscillator strengths for individual transitions have been given by Jacobsohn [28] and by Payne and Levinger [29]. Chapter 1 of Levinger [35] contains an insightful introduction to these rules. wFor a modern treatment see Dexter [37].

38

D.Y. Smith

addition to the f sum rule, several less familiar sum rules for moments of the absorption spectrum have been given by Vinti [39] and others [40, 41]. These have been widely applied to atomic spectra [34], nuclear photodisintegration [35] and quantum chemistry [40], but only limited applications have been made to the spectroscopy of solids [41-43]. Values of the various moments of the absorption spectrum involved in several sum rules have been shown to be interrelated by the uncertainty principle [35, 44]. The existence of numerous sum rules for the absorptive process suggests analogous rules hold for the dispersive process. However, no hint of them is given by the commonly employed [24] proofs of the absorption rules that involve combining the time-dependent Schr6dinger equation and the fermion commutator. Here, an alternative approach [-4], employing the highfrequency (or short-time) behavior of matter, will be used. This has the advantage of allowing the systematic construction of a large class of sum rules and of emphasizing the underlying physics.

A

Superconvergence Sum Rules for s(a))

1

Formal Development At frequencies higher than any characteristic absorption of the system, the excitation frequency dominates the expression for the polarizability, and the complex dielectric function has the asymptotic form [1, 45] lim e(09) = 1 - (0)2/0)2) -{- " " " . r

(8)

- ' * O0

This series expansion is universal: The term in 09- ~ is missing; the term in (D-2 involves only the electron density through the plasma frequency. These first terms are independent of interactions within the system. Physically, the reason is that at frequencies well above the highest characteristic absorption, inertial effects, not restoring or dissipative forces, dominate the dynamics. The latter forces enter into succeeding terms. [For example, the next term in the expansion of e(09) for a Lorentz oscillator with damping constant ~ is

i0927/093.]

Formally, the physics and mathematics may be combined to get the sum rules by comparing the limiting behavior of e(og), Eq. (8), with the KramersKronig relations [1, 16-18]: el(09)- 1 = 2reP

~0o 09,2 o~'~(co') _ 092 d09',

1 if(0) 2 ~1(09') e2(09) - 4~z 09 - - - n 09P fo~ r _ o92 d09'.

(9) (10)

Here P denotes the principal value integral, and rr(0) is the dc value of the

39

3. Analysis of Optical Data

conductivity. The conductivity term is, of course, not present for insulators. These relations hold for linear, causal systems, and their use guarantees the functions involved are analytic, square-integrable, causal response functions [46, 47]. To proceed, note that Eq. (8) actually makes two statements, one for the real part of ~(co), lim el(co)= 1 r

-(0`)2/0`)2),

(ll)

--* o o

and a second for the imaginary part, lim e2(0`) ) falls off faster than 0) -2.

(12)

(D ---~ O0

The first of these may be compared with the high-frequency limit of the first Kramers-Kronig relation lim el(co) - 1 =

r

2 1 i o co,gz(co,)dco, + " "

~ O)2 du

"

(13)

Equating powers of 0,) - 2 in Eqs. (11) and (13) yields the f sum rule t fo o

7"Ccop. 2 coe2(co) dco = ~-

(14)

Similarly, the high-frequency limit of the second Kramers-Kronig relation is lim ~2(co)= 4rta(0) + - -2 1 foo [~,(o~')- 1] d d + . . . .

~o-~oo

0`)

~ 0`)

(15)

Since by Eq. (12) there is no term in 0`)-1 in/32, equating powers of 0`)-1 in Eqs. (12) and (15) yields [4]. ~'~

fo~ [~,(~o)- 1] dco + 2~2~(0) = O.

(16)

* The first proof of the f sum rule employing this method appears to have been given by R. de Laer Kronig [17] in his original work on the Kramers-Kronig relations. An alternative proof [8-1 of Eq. (16) can be made by simply integrating e(o9) over the contour consisting of the real axis and the semicircle at infinity in the upper half-plane. In the case of metals e(og) has a pole at the origin i&za(0)/~o; this contributes the term in a(0) to Eq. (16), and the remainder of the real axis gives the integral over e(~o), while the semicircle at infinity gives a vanishing contribution. At first this proof seems simpler, but the vanishing of the contribution of the semicircle at infinity does not follow directly from the asymptotic behavior along the real axis [Eq. (8)]. Rather, an argument based on the Phragm6n-Lindel6f theorem is required. For details on this point see Titchmarsh [48]; Sec. 5.61. See also Nussenzveig [19], Theorems 2.5.4 and 7.5.1. ago) has units sec-1 in cgs electrostatic units. To convert cr in ohm-1 cm-1 to a in sec-1 multiply by 8.9876 • 1011 ohm cm sec -1. Generally, sum-rule integrals are done over an energy scale resulting in values of ha. The corresponding multipliers for electron volt and c m - 1 energy scales are 5.9158 • 10 -4 ohm cm eV and 4.7713 ohm cm cm-1, respectively.

40

D.Y. Smith

This is the analog for/31(o9) of Eq. (1). Note that this sum rule provides independent information on the parameters describing free-carrier absorption. Traditionally, these are found by making Drude-model fits to e(co). However, the integral of [e~(co)- 1] provides an additional constraint since tr(0)= ne2z/mo, where n is the carrier concentration, z their lifetime, and mo their optical mass (see Stern [1] for details). In taking the high-frequency limit of the dispersion relations, we have relied on the intuitive notion that at large 09 the denominator of the integrals is dominated by 092 [or, more precisely, that (092 - ~0'2)- ~ may be expanded in a power series in inverse powers of 092]. This will be true provided that the numerator decreases fast enough as 09' ~ oo. The necessary restrictions on the numerator are embodied in the superconvergence theorem [49-50].* For the present, it is sufficient to note that these limits hold provided that /32(0) ) falls off at least as fast as 0) - 2 In -~ co, for ~ > 1, as 09 -~ ~ . This is a mild restriction on/32(0)) since it decreases faster than this in actual systems. For example, the contribution to/32(09) from K-shell electrons that dominate the high-frequency absorption falls off as 09-4.5 (neglecting retardation and relativistic effects) [51]. (For more information see Bethe et al. [24] and Rau et al. [51 ].) In discussing the optical properties of metals and superconductors, it is common to rewrite the rules in terms of the complex frequency-dependent conductivity as tr(og) = -(iog/4rc)e(co). (17) The f sum rule then takes the form fo~

tr(~o) do9

7z ~ e 2 2 m

(18)

This rule and its generalizations [-9, 52] as well as related rules for the surface impedance [53, 54-] have been used extensively to relate the difference in conductivity between the superconducting and the normal states in the region of the superconducting gap to the 6-function conductivity at zero frequency in superconductors [53, 55-57].

2

Physical Picture The physical basis of the f sum rule [Eq. (14)] and the dc conductivity rule [Eq. (16)] lies in the dynamical laws of motion. This is most easily seen using the linear-response-theory I-7] result that the polarization P~(z) induced by a 6-function pulse electric field as a function of the time z following the t The term superconvergent was introduced into particle physics to describe the convergence properties of dispersion relation integrals; the corresponding sum rules are often referred to as superconvergence relations. (For an introduction to this theorem see the appendix of Altarelli et al. [4]. Details are given in de Alfaro et al. [49] and Frey et al. [50].)

3. Analysis of Optical Data

41

pulse is just the Fourier transform of the electric susceptibility [e(co) - 1]/4rc. That is, 1 f ~ (e(co_)_--1) i,o~ P~(z) = ~ - -o0 \ 4~z edo).

(19a)

Taking account of the pole in e(co) at the origin for conductors and by using the fact that el(co) is an even function of co while ~;2(o9) is odd leads to Pa(z) = ~

Re[e(e)) - 1] cos(mr) do), )

+ fo ~ Im e(o9)sin(mr)do9 + 2rtz~r(0)~.

(19b)

The sum rules then follow [7] by using elementary arguments to determine P~(r) at small r. For example, inertia requires that the response cannot be instantaneous; P~(z) for z > 0 must join onto Po(r) for z < 0 without a discontinuity. By causality the latter values are all zero (no response before excitation) so that P~(z = 0) = 0. Setting r = 0 in Eq. (19) then leads to the dc conductivity rule [Eq. (16)]. This rule and, by analogy, the corresponding relation for n(co) [Eq. (1)] are therefore direct consequences of causality and the law of inertia; they are thus often referred to as inertial sum rules. The f sum rule may be similarly derived by calculating the initial rate of change of polarization Po(r) in the impulse approximation and equating this to the time derivative of Eq. (19) at r = 0. In this case the determining factor is the dynamical law of motion--Newton's law in a classical picture. The details are given by Altarelli and Smith [7]. In the commonly quoted quantummechanical derivation employing commutation relations [24], the dynamics enters through the time-dependent Schr6dinger equation.

Superconvergence Sum Rules for N(e)) A parallel development [7] may be given for the refractive index. The only difference lies in that for metals there is no pole in the index at the origin arising from the conductivity term. Rather, there is a square-root singularity

[i4rcff(O)/co] 1/2

=

(1 + i)x/2rtcr(0)co- 1/2

at the origin, and the Kramers-Kronig dispersion relations take the form t

n(co)- 1 = -2Pro~ 09,2 co'k(co') _ c~2 de)',

(20)

, For a derivation of these relations for insulator see Section 1.9 in Nussenzveig [19] and for conductors see Smith [58].

B

42

D.Y. Smith

k(e)) = - 2corcP

loOn(co' )1- do)'. ~ -- ~9~

(21)

Comparing the high-frequency limits of these with the limit for N(co) obtained from the square root of Eq. (8), lim N(co)= 1 - ~(OOp/CO x 2 2) +.-.,

(22)

(D--~ O0

yields the f sum rule for k(og) and the companion inertial sum rule for n(e))

fooo (ok(o) do = ~7~ COp, 2

(23)

and (24)

fo ~ In(co)- 13 do~ = 0.

A physical interpretation [7] of these rules in terms of a response function is similar to that for e(co). In addition to sum rules involving integral equations, a variety of integral inequalities hold for the dielectric function, the refractive index, and the reflectance. These have received some formal treatment recently [6, 59] but have not yet been applied to experimental problems. C

superconvergence Sum Rules for Functions of N and

Since an analytic function of an analytic function is also analytic, it is possible to construct analytic functions of N(co) and e(co) and develop sum rules and dispersion relations for them. For example, starting with N(co), N2(o9) would yield the sum rules for e(co). The number of such rules is limited only by one's imagination and a large number of generalized rules involving powers of co and the real and imaginary parts of IN(co)- 1] m, and so on, have been given in the literature [6-8]. In general the higher the power m, the faster the convergence of the sum rule, so that a given rule is more sensitive to a particular spectral range than to others. A few preliminary applications [7] of these rules have been made to test optical data. Since they are based on the same asymptotic behavior as the rules for N(co) and e(co), they must contain the same physics, but with emphasis on different spectral regions. A number of examples [-7] of these rules for an insulating system are (25)

fo ~ Re{ IN(co') - 1]~} aco' = 0, and

fo~

rcc~

( m = 1), (26) ( m > 1).

3. Analysis of Optical Data

43

For m - 1 these yield the familiar inertial and f sum rules, respectively. For higher powers Eq. (25) yields m = 2:

fo ~ {In(co')- 1] 2 - k2(co')} dco'= 0,

(27)

m = 3" fo ~ In(co')- 1]{In(of)- 1] 2 - 3k2(co')} dco'= 0.

(28)

Similarly, Eq. (26) yields the rules m = 2: m = 3:

fo ~ co'k(co')[n(co')- 1] dco'= 0,

(29)

fo ~ co'k(co'){3[n(co')- 1] 2 -- k/(co')} dco'= 0.

(30)

Equation (29) has been previously noted by Stern [1]. Observe that this rule, together with the inertial sum rule for n(co) [Eq. (24)] states that [n(co)- 1] averages to zero either alone or when weighted by cok(co). Analogous rules may be derived [7] for conductors by taking co[N(co)- 1] as the starting point. The reader is referred to the literature [6-8] for details and related rules involving (N - 1)" multipled by powers of co. Equation (27) deserves further comment. It can be rewritten in the more suggestive form

f:

[n(co')- 1] 2 dco'=

fo

k2(c~

(31)

As such it provides a potentially useful check on Kramers-Kronig inversions. It is a particular case of the general theorem [60, Theorems 90 and 91] that the norm of a function is preserved in a Hilbert transformation. Using the identity el(CO) = n2(co)- k2(co) and the inertial sum rule for n(co), it follows from Eq. (16) that the generalization of Eq. (31) for conductors is fo ~ {In(co)- 1] 2 - k2(co)} dco + 2n2a(O)= O.

(32)

The extra term in a(O) arises from the pole at the origin. (Note: Separately [n(o~) - 1] 2 and k2(~o) have poles at the origin but their difference does not; hence, they have been written together under the same integral to avoid undefined expressions.) Since N(o3) has at most a co-~/2 singularity for conductors, ~oN(~o) is square integrable for both insulators and conductors. It then follows that its real and imaginary parts are related by Hilbert transforms. Thus, in analogy with Eq. (31), the norm of the transform is preserved [6],

f0~ (D2[r/(o))- 1] 2 do.)= fO~~(D2k2(Og)do.).

(33)

The fact that both Eqs. (33) and (31) or, in the case of a conductor Eq. (32), must hold is another remarkable example of the constraints imposed on the optical functions.

44

D.Y. Smith

Before leaving this section, it might be noted that a generalization of the f sum rule for higher powers of ogp has been found by Villani and Zimerman [5], who considered powers of o92[N(o9)- 1] and o92[e(o9)_ 1]. In the case of the refractive index their result is

f~

o92m- 11-N(o9)

B

1-Imdo)

~

(

~

1)m+12-m-_ .2m trll, Ujp

.

(34)

--00

D

Superconvergence Relations Employing Weighting Functions

Superconvergence techniques can be applied to quantities consisting of the optical functions multiplied by an appropriate analytic weighting function. Weighting with powers of o9 mentioned in the preceding section is perhaps the simplest example of this. The technique has been applied by using algebraic, exponential, and Gaussian weighting functions by Villani and Zimerman [6], King [8], and Kimel [61]; a general discussion has been given by Fischer et al. [62]. A major result of Villani and Zimmerman based on the Lin-Okubo [63] weighting function is

f: [n(og) - 1] do9 (a 2 - 092) 1/2 =

fa~

k(og) do) (0) 2 -- a2)1/2"

(35)

This is particularly interesting since it relates the index in the interval [0, a] to the extinction coefficient in the remainder of the spectrum. While this relation does not seem to have been used in practical data analysis to date, it should be a helpful guide to constructing composite sets of data. For example, in semiconductors and insulators n(o9) is often available with high accuracy for co less than the band gap, ogg, while above ogg, k(og) may be imprecisely or only partially known. By using Eq. (35) with a < ogg and requiring that the f sum rule be also satisfied, it should be possible to sharpen the limits on the values of k(og) for 09 > COg.

E

Static-Limit Sum Rules

The sum rules discussed thus far all derive from the high-frequency behavior. The static limit also yields several well-known rules [18] that follow directly from the dispersion relations. For insulators both 5(o9) and N(og) are well behaved at the origin, and the Kramers-Kronig relations yield the dc dielectric constant and index as /31(0 ) ~-

1 + (2/re) fo ~ co-le2(o9) do9,

(36)

3. Analysis of Optical Data

45

and n(0) = 1 + (2/7t) fo~ o9-'k(o9)dog.

(37)

The other dispersion relations give the trivial results ez(0) = 0 and k(0) = 0. In the case of conductors e(~o) has a pole at the origin arising from the term i4rca(O)/o9in the dielectric function. Along the real axis this yields an ~o-~ divergence in gZ((_D), but e~(og) remains finite. The latter finite value of e~(0) can be recovered by considering dispersion relations for the function e(co)- [i4rca(O)/co] - 1. The result is /31(0 ) = 1 + -

2 ~.o (_D'g2((D') -- 4~za(0) J0

CO~2

&o'.

(38)

Note that this limit holds only as co approaches zero along the real axis. Because of the singularity, the limit of Re e(og) depends on the direction in the complex ~ plane along which the limit is taken. In the case of the refractive index of a conductor, both n(~o) and k(co) have CO-1/2 singularities and diverge in the static limit.

FINITE-ENERGY SUM RULES

The sum rules discussed to this point involve the infinite frequency interval to, ~ ] and include all absorptive processes. However, a typical spectrum consists of different regions corresponding primarily to a single class of absorptions such as excitation of phonons, conduction, valence, or core electron. It is tempting to seek a means of treating these individual processes. In those favorable cases in which a particular absorptive process is not overlapped by other absorptions, finite-energy sum rules and dispersion relations may be developed [14, 64]. The key requirement [14] is that the absorption in question be sufficiently isolated that it can be viewed to a good approximation as taking place in a transparent medium with real background dielectric function ~b(~O) arising from the dispersion of absorptive processes in all other spectral ranges. The situation is illustrated in Fig. 1. Examples include phonon absorptions in polar insulators and free-carrier absorption in wide-gap semiconductors. In both instances, the absorptions in question are well separated from interband transitions that provide the polarizable background. On a wider spectral range, the valence electrons of a solid may be regarded as moving in a dielectric "medium" consisting of the polarizable ion cores in those materials for which valence and core absorptions do not overlap. To proceed, consider an energy o3 large compared with that of the isolated absorption~or group of absorptions~but well below the energy of the

III

46

D.Y. Smith

INTERBAND

Z 0 (0 Z

- ~ABSORPTION INTERBAND i v\ DISPERSION"-~[ v\\\. / E 2 I

ISOLATED

LL

c.) __1

Eb

L .

.

.

.

.

I A

co FREQUENCY Fig. 1. Absorption spectra for nonoverlapping bands. (Solid curve, absorption; dashed curve, interband dispersion.)

transitions responsible for the dispersive dielectric background. Such an energy is indicated in Fig. 1. The dielectric function may then be expanded in the neighborhood of 03 as [14] lim e(o9)--- eb(0) ) -- (Dpi/0) 2 2 -+-''',

(39)

where 0)pi is the plasma frequency associated with the isolated absorption. In the limit of large separation between the isolated absorption and the other absorptions of the system, eb(O) may be taken as a constant so that 2 2 +''', lim e(o)) = eb -- COpi/CO (40) lim N(0)) = el/2 - 0)2i/2 ~;~/20)2 + . . . , 2 2 2 -~-'''. lim e - 1(o))= •b 1 -~- 0)pi/eb0)

(41)

(42)

to-~ r

In taking the square root and the inverse of Eq. (40) to obtain Eqs. (41) and (42), it has been assumed that o2 >> 0)vi. Notice particularly that the dielectric constant eb appears in the coefficient of the 09-2 term in the limiting values of N(0)) and e-I((_D). Finite-energy sum rules for the frequency interval containing the isolated absorption then follow either by applying superconvergence methods to the idealized problem of an isolated absorption in a medium with a constant eb at all frequencies or by employing [21] a Cauchy-theorem integration over a semicircular path of radius o3 in the upper half-plane. The resulting finite-energy f sum rules are o

n

2

2 f0 0)'k(0)') do)' ~ 7~ (_Dpi 4/31/2'

(43)

(44)

3. Analysis of Optical Data

47 2

fo~'co'Im[e - l(co,)]do),~

Tc 2 copi e2"

(45)

eb] do)' + 2rc 2if(0) ~ 0,

(46)

0,

(47)

fO' { Re[E(co')- 1] _ ~b 1} dcot ,~ O.

(48)

Similarly, the inertial sum rules are

fo ~' [el(CO')

fo

These rules are exact in the limit of e3 ~ oo. For finite e3 the f sums are accurate to within terms of the order of co2~(7/05), and the inertial rules hold to terms of order co2~/03, where 7 is the effective damping constant for the low-energy absorption. The inertial sum rules are what one would expect intuitively: The dispersion associated with the isolated absorption is superimposed--though not linearly in the case of N(co) or ~ - l ( c o ) - - o n the background eb, 81~/2, or 8b 1 and averages to zero (or - 2 7 c 2 times the dc conductivity) with respect to that background. The finite-energy f sum rules are more noteworthy in that the background dielectric constant appears in the right-hand side of the rules for k(co) and Im[e- ~(co)]. That is, the integral of COS2(co) gives copifor an isolated absorption directly. However, the areas under cok(co) or co Im[8-1(o))] do not; in these f sums it is necessary to include the effect of virtual processes from all other transitions via 8b. This is manifest in the factors 81/2 and 82 in Eqs. (44) and (45), respectively, which may be thought of as correcting for the dielectric shielding of the isolated transition by the remainder of the system. The greater the background dielectric function, the weaker the apparent strength of an absorption as measured by k(co) or Im[e- 1(o))] for a fixed copi (i.e., for a fixed oscillator strength of the isolated absorption).

Defect Absorptions and Smakula's Equation

A particularly striking example [64] of the dependence of the amplitude of an isolated absorption on the remainder of the system occurs for defects or impurities that introduce absorption bands in the region of transparency of insulators. This is illustrated in Fig. 2, which shows the modeled absorption spectra for a series of four different defects. Each exhibits a single absorption band; they all have the same strength as measured by COvi 2 and the same half-width but have different absorption frequencies cod. In the figure the bands are labeled by v = cod/COhost,where cohost is the frequency of the host's fundamental absorption edge. As the defect absorption frequency approaches

A

48

D . Y . Smith

0.9

-7 -6

0.8

3 o t--

-5>,.laJ

0.7-~o r-

_4 z la.J

~

I---

.

..~

U_ I.aJ rr Z

,,,0.5-

n0(~)

u.-

!

i,i 0

I--r~

C~ -1-

i- I -0

o 0.3 -

0.2-

O.I-

00

0.2

0.4 0.6 0.8 RELATIVE FREQUENCY, (~/~o,h

I

I 1.0

Fig. 2. Modeled spectra of a weak defect absorption on the low-energy side of the fundamental crystal absorption. Four possible defect absorptions are shown for energies Wd = VCOo,h,where O~o,his host absorption energy. Although all four absorptions have the same oscillator strengths, the absorption coefficient for bands at higher energies are reduced as a result of the larger values of the host refractive index, no(O~). (After Smith and Graham [64].)

that of the host's fundamental absorption edge, eb(~0) increases sharply with frequency and the apparent strength of the defect absorption as measured by the absorption coefficient ~(co)= 2cok(@/c decreases markedly. In this case, it is necessary to treat the case in which the background dielectric function varies over the band. The result is a generalization of Smakula's equation [36-38] for the integrated oscillator strength f = (m/4ne2)e~2i of a defect or impurity in a transparent medium [64]

pf = ~

mfO~no(O~) ak(o~) do~.

(49)

Here p is the density of defects that is taken to be much less than the density of host atoms and effective-field effects have been omitted for simplicity. The

49

3. Analysis of Optical Data

refractive index of the host is denoted by no(Og)= e~/2(oj) (the change in index induced by the defect is negligible), and Ak(og) is the absorption induced by the defect.

Finite Oscillator Strength Sums and n~ff(r~) In analogy with the f sum rule it has become common practice to define the effective number density of electrons contributing to the optical properties up to an energy o9 by the partial f sums [14, 45] neff(O.))[ e = 27c2e2

o.)'82((.0' ) do.)', d o,

(51)

m 2 fo' o9' Im[e-1(o')] dog'. 27zZe

(52)

=

nefe(~O)] _, =

(50)

Here the subscripts e, k, and e- 1 are used to distinguish the partial f sums involving ez(fO), k(o9), and Im[e- 1(o9)], respectively. The f sum rule guarantees that in the limit o9 ~ ~ all three values of neff(m) approach the number of electrons X in the system. For intermediate values of o9 the three values generally do not coincide. This is illustrated in Fig. 3a for the simple classical Drude-model of a metal in which core-electron x-ray transitions are neglected. Starting at zero for ~o = 0, the three values increase monotonically, but on different paths until in the vicinity of the plasma frequency they draw together and asymptotically approach X at infinity. In discussing the spectra of materials with valence or conduction electron absorptions that are well separated from those of core transitions, it is common to calculate neff(e3) for the valence or conduction electrons. Then o3 is chosen to exhaust the valence- or conduction-electron oscillator strength but to lie below the lowest-energy core transitions. A comparison of this situation with the finite-energy f sum rules [Eqs. (43)-(45)] shows that the various n~ff(oS) values for conduction or valence electrons should not be equal when the polarizability of the core electrons is accounted for. Rather [14],

noffi )[

= 4

noffI )l

=

(53)

where eb is the background dielectric constant of the fictitious medium consisting of the polarizable atomic cores alone. This is illustrated in Fig. 3b. Again a Drude model has been employed, but a background dielectric constant eb = 1.05 has been included to take

B

50

D.Y. Smith

1.04(a) 0.8 0.6 I

0.4 0.2 I

0o

2.0

l

3.0

,,.o

~l~p

(b) 1.0 L

E ; 112

0.8 o.6

O.4 0.2 0 0

1.0

I

I

2.0

3.0

4.0

Fig. 3. The functions neff(O) ) for a Drude metal (a) with unpolarizable cores, i.e., eb = 1, and (b) embedded in a dielectric medium with eb = 1.05 corresponding to the core polarizability for a light element such as aluminum. (From Smith and Shiles [14].)

account of the core transitions which lie at higher energies. This value is typical for a light element such as aluminum. A plot of the actual noff(co) for both conduction and core transitions of aluminum is discussed in Section V. It should be further noted that the asymptotic limit of n~ff(o3)l~is generally n o t the actual number of valence or conduction electrons. It is their oscillator strength. To start with the distinction between valence and conduction electrons is only approximate. The separation is never complete because of Cou-

3. Analysis of Optical Data

51

lomb and exchange interactions, and in the one-electron approximation this leads to an "exchange" of oscillator strength between core and valence electrons [2, 38, 65]: The major contribution to this effect can be understood by observing that the Pauli principle prohibits transitions of core electrons to the occupied valence or conduction states, so that the core-electron oscillator strength is less than their number. Similarly, electrons in higher-lying states cannot make transitions to the occupied core states, and they consequently exhibit an oscillator strength greater than their number (Note: Emissive processes have negative oscillator strength.) Typically this Pauli principle effect increases the valence-electron oscillator strength by a few percentages for light elements and from 10 to 20?/o for heavy elements, t In addition a small correction arises from the exchange part of the potential that occurs when electrons are treated in the one-electron approximation. Briefly, the f sum rule for dipole matrix elements applies strictly only for systems in which the interaction operator and spatial coordinates commute [24]. This does not hold for the (nonlocal) exchange part of one-electron Hamiltonian that leads to a small correction in the oneelectron model oscillator strength. This has been treated by Fock [66], who found the effect to be less than 20?/o of the Pauli principle correction. However, note that what is a prohibited absorption for one particular electron is a prohibited emission for another, so that the f sum taken over transitions to all levels still totals to the electronic number of the whole system in accord with the f sum rule.

SUM RULES FOR REFLECTION SPECTROSCOPY

IV

Reflection spectroscopy has become a widely used method for determining optical constants of highly absorbing materials, since it was shown that reflectance spectra could be analyzed for the unknown phase by dispersion methods [67, 68].

Normal-Incidence Spectroscopy

The simplest situation occurs for normal incidence from vacuum or, to a good approximation, air. In this case the complex reflectivity (for the magnetic field vector)is given by [1] ~(co) = r(co)e i~

= IN(co) - 1]/[N(o)) + 1].

t See, for example, Table 1 of Smith and Dexter [38].

(54)

A

52

D.Y. Smith

Since N is analytic, 7(09) is also analytic, and its real and imaginary parts satisfy sum rules and dispersion relations similar to those for N(og) [69-71]. For example,

fo~rm(fo)cos[m0(fo)] dfO = 0

(m = 1, 2, 3 , . . . )

(55)

and

fo~fOrm(fO)sin[m0(fO)] do) = t !

fOp 2

(m = 1),

(56)

(m > 2). These and similar superconvergence relations for powers of f(fo), and so on, all involve a mixture of reflectivity amplitude and phase. They are, therefore, currently of little practical value since the phase is generally not measured. The phase and amplitude may be separated by considering the analytic function In ~(fO) = In r(fO) +

iO(fO).

(57)

There is, however, a major difference between this and the dielectric function, refractive index, and so on. The latter approach unity as co ~ oo, whereas In f(fo) diverges logarithmically since lim,,_,oo r(fO)= 0. Thus, the dispersion relations and sum rules for In f(o) are not of the same form as those for e(o~) or U(fO). The dispersion relations for In Y(co) may be obtained by a number of methods* that yield [68, 72, 73]. * O(fo) = _~2~ P foO coIn'2 r(fo') _ 002 dfo'

(58)

and its inverse [76, 77] In r(fo) - In r(foo) = _2 P 7r

foo fo'0(fo') (co,z m1 (.o2

1 )

fo,2 _ fo~ dfo',

(59)

where it is necessary to assume that the reflectance is known at some frequency 090. The equation for 0(09) [Eq. (58)] has the usual Kramers-Kronig form, while the second is a subtracted dispersion relation. Note that, while the second appears to simply involve the difference between two Kramers-Kronig integrals--one for r(co) and one for r(oo0)~such an interpretation is invalid. The reason is that 0(co) ~ n as 09 ~ ~ , leading to integrals that are divergent when taken separately. Equation (59) displays a further point. The phase does not determine the t For a detailed discussion of the phase retrieval problem see Toll [46], Roman and Marathay [74], and Burge e l al. [75]; a brief, but physical, discussion is given by Stern [1]. : The sign convention used here assumes a time dependence exp[-iogt]. If exp[i~ot] is used, the index must be written as N = n - i k and Eq. (58) for 0(09) changes sign.

3. Analysis of Optical Data

53

reflectance uniquely [77]. One must know the reflectivity amplitude at one point co = coo, to fix the ratio r(co)/r(coo). In other words, two systems with reflectances differing by a constant multiplier have the same phase. [This is clear from Eq. (58) since multiplying the reflectance by a constant c contributes In c ~(co,2 _ co2)-1 do)' to the phase which is identically zero. t] Sum rules are similarly limited [71]. Several methods have been proposed for circumventing this. They are (1) use of convergence factors [13, 70], (2) consideration of the normalized function [71] r(o))/r(O) or of the ratio of the reflectances of two materials [13], and (3) finite-energy sum rules [71]. In general, the reflectance sum rules are too new to have been widely studied or utilized, but a number of them appear promising and will be mentioned briefly. For insulators a sum rule for the derivative of the phase in the static limit that involves the ratio lnlr(co)/r(O)] has been developed [71].

fo~ co-lFr'(co)-] dco = fo~ co-21n r(co) dco r-N [_r(co)_] n12do dO

=o

_--

....n ldodk]

n2(U)

--

,

(60)

o~ = o

where in the first integral r'(co) denotes the derivative. The two integrals are related by integration by parts, and dO/dco]o=o has been written in terms of dk/dco]~,=o by employing Eq. (54). In most instances, dk/dco[,o=o is small so that there is a large degree of cancellation within the integrals of Eq. (60). This has been demonstrated in an application [71] to the reststrahl absorption of NaC1. The negative contribution to co-l[r'(co)/co(r)] exceeds the positive contribution by roughly 10~, the remainder giving dk/dco],,=o to within experimental accuracy. The derivative form of the normalized reflectance sum rule is of potential interest in connection with modulation spectroscopy [78-80], where the measured quantity is the relative change AR/R induced in the intens@ reflection coefficient R = Ill 2 by an oscillatory external perturbation or wavelength modulation. Finite-energy sum rules are particularly useful in reflection spectroscopy of systems with well-isolated absorptions such as polar insulators or wideband-gap semiconductors in which the infrared vibrational modes or freecarrier absorption is separated from band-to-band electronic transitions by a wide region of transparency. In this region of transparency the reflectance arising from the band-to-band transition r bis, to a good approximation, constant except as the fundamental edge is approached, and the infrared reflectance spectra appears superimposed on this constant "optical" reflectance as a background. This superposition is complicated and nonlinear, but one can * N o t e t h a t P I ~ [ d x / ( x 2 - a2)] = 89

f~_~[dx/(x

- a)] = 8 9

~ - o o ( d y / y ) - O.

54

D.Y. Smith

proceed in analogy with the development of finite-energy rules for N(o9), etc., in Section III. Finite-energy dispersion relations and sum rules are then found for the quantity f(Oo)/rb over the finite energy internal [0, e3], where, as in Section III, o3 is a frequency in the region of transparency below the point at which dispersion in rb becomes significant but well above the highest of the isolated absorption bands in the infrared. The two sum rules that appear to have the greatest utility are [71] (1)

A reflectance conservation rule for isolated reflection bands

fo o~[r'(o~)/r(o~)] do =

fo lnlr(~

do~ ~

o.

(61)

rb --(el/2 1)/(~3~/2 q.. 1) is the background reflectance with 13b the "optical" dielectric constant. The integrals are related by integration by parts as in Eq. (60). (2) A phase f sum rule

Here

2 fO

2 w h e r e (Dpi

(62)

~(-'Opi ~' 090(09) do9 '~'~ 2t~bl/2(eb _ 1)'

is the plasma frequency of the infrared absorption.

I.O

Rb 0 0433

O.Ol

o.ool

_

_

0.OOOlI

0

1

I

I

2

I

3

I

4

I

5

I

6

I

7

I

8

1

9

1

I0

iii

1 12

I 13

14

ENERGY (102 crn-~)

Fig. 4. The reflectance of crystalline NaC1 in the infrared plotted on a logarithmic scale versus energy. Note the equality of areas between the curve above and below the background reflectance line at Rb = 0.0433, the reflectance in the visible. (From Smith and Manogue [71].)

3. Analysis of Optical Data

55

The content of the reflectance conservation rule [Eq. (61)] is that, regardless of the processes involved, the logarithm of the reflectance of the lowenergy process averages to the logarithm of the background reflectance. This is illustrated [71] for the NaC1 reststrahl band in Fig. 4. Notice the equality of the areas above and below the dashed line, which denotes the reflectance in the visible or background reflectance Rb -- r 2. The second rule [Eq. (62)] allows the direct calculation of the oscillator strength of the low-energy absorption without the usual intermediate calculation of e2(o~) from the phase and reflectance. It must be stressed that this only applies for a finite frequency interval.

Reflectance between Media at Nonnormal Incidence, Ellipsometry, and Transmission

B

The case of normal incidence already discussed leads to particularly simple dispersion relations and sum rules. In the case of reflection in a medium other than vacuum [81-84] or at nonnormal incidence [81, 84-86], the dispersion relations must be modified to account for more complex terms in the phase shift [46]. The interested reader is referred to the literature [8-86] for details. While this more complex situation has led to discussions of the validity of dispersion analysis [87, 88], sum rules have not been investigated for these more general cases except in the case of ellipsometry. In this procedure the ratio of the reflectivities for light polarized parallel and perpendicular to the plane of incidence is considered. Dispersion relations [89] and sum rules [90] for the real and imaginary parts of the ellipsometric function have been developed. Their use has been limited by the small wavelength range over which ellipsometry is now feasible. However, applications to highly localized absorptions in a finite energy interval is an attractive possibility. Dispersion relations have also been developed for the analysis of transmission measurements [91-93].

ANALYSIS OF OPTICAL DATA AND SUM-RULE APPLICATIONS

Wide-spectral-range composites of optical data have been constructed with increasing frequency as sufficient measurements become available. Some examples of this work are given in Ehrenreich et al. [94], Hagemann et al. [95], Inagaki et al. [96], Shiles et al. [97], and Cardona [98]. The various sum rules provide a means of testing such composites against both theoretical

V

56

D.Y. Smith

constraints and independently measured quantities. We briefly consider several aspects of dispersion analysis and the use of sum rules to eliminate systematic errors. A

Construction of Composite Data Sets

In principle, exact knowledge of both the real and imaginary parts of a single-valued analytic function over a finite interval is sufficient to determine the function for all values of the argument by analytic continuation. However, in practice optical data are subject to experimental errors, so any such continuation outside the measured interval has virtually no meaning. It is thus necessary to have optical measurements at all significant frequencies to determine optical properties over the complete spectral range. While this strict requirement is generally not satisfied, good results over a limited range can be had in favorable cases. We consider this situation first. 1

Dispersion Analysis over a Limited Range

For argument, suppose a single optical function is known (or can be estimated) over all frequencies. Then the others are given by a Kramers-Kronig analysis. As a specific example, if the reflectance for normal incidence from vacuum is known, the phase can be found by using the Simon-RobinsonPrice method [67, 68] by the phase dispersion relation Eq. (58): 0(0)) =

0) Pfo o In R(0)')

--'~-

(,0'2 -- 0)2 do)',

(63)

where the expression has been rewritten in terms of the intensity reflection coefficient R(~o)= the quantity usually measured. Jahoda's proof [72] of this dispersion relation for the optical case was completed by Velick~, [73] using Toll's general theory [46]. The other optical functions then follow from Fresners equations [1]. In practice dispersion integrals are usually integrated numerically after eliminating [1] the principle value by subtracting an integral that is identically zero t to get 0(09) =

oJ fo o In R(0)') - In R(0)) -~0),2 _ 0)2 do)'.

(63')

Graphical [68, 99] and, more recently, Fourier-integral [100] and Fourierseries [101-103]* methods have also been developed to evaluate Eq. (63). Usually only reflection measurements over a limited frequency interval oJ1 < oJ' < ~o2 are available and without further information 0(o9) cannot be * See footnote to the discussion of Eq. (59) in Section IV. * Similar methods have been applied to phase determination in electron microscopy [104, lo5].

3. Analysis of Optical Data

57

determined since the contributions to Eq. (63) from 0 < 0)' < 0)1 and 0)2 < 0)' < Go are unknown. However, "physically reasonable" estimates and extrapolations of R(0)') can often be made and, for 0) in the measured interval, considerable information about the structure of 0(0))--though not its absolute value--can be found. The important point is that the denominator of Eq. (63) tends to emphasize values of R(0)') near 0) more than those far away. This is particularly evident if Eq. (63) is integrated by parts [72, 106, 107_] 0(0)) =

1 foo0 d In]R(0)')] In ]0)' + 0 ) do)'. 2~z do)' 0 ) ' - 0)

(64)

The factor ln[(0)' + 0))/(0)'-0))] is sharply peaked at 0) and weights R(0)') near o~ strongly. Thus, for o~ within the measured range, the portion of the dispersion integral from 0)1 to 0)2 contributes the structure to 0(0)). The values of R(0)') outside this measured interval typically supply a slowly varying "background" contribution A0(0))*. Only under special conditions is this background negligible [111, 112], and tests indicate that errors may be significant particularly in regions of small k [113]. The presence of a slowly varying background component in 0(0)) is crucial to estimating 0(0)) since if 0(0)) is known even at a single point, the transform Eq. (63) or (64) can be "anchored" there. The more such points, the better the estimate of A0(0)). The additional information required can be obtained by ellipsometry in reflecting regions and by absorption or index measurements in transparent regions. Traditionally, such additional data have been used either (1) to establish an extrapolation for R(0)) or (2) to determine the slowly varying component A0(0)) of the phase directly. The two methods are equivalent but have been compared in only a few cases [114]. In the first approach methods used include trial-and-error searches [-115] for an extrapolation of R(0)) that reproduces the additional data, and the determination of free parameters in a simple reflectance extrapolation function. Some common examples of high-energy parametric extrapolations are the power law [116, 117]* R(0))= R((_D2)(0)2/0)) p, the exponential [119] /{(0)) =/{(0)2) exp[B(0)2 - 0))], and a closely related power-law development [120] of In/{(0)). These are useful when reflectance measurements extend to energies high enough so that the strength of conduction or valence electrons is largely exhausted. The parameters p and B are generally chosen to make 0(~o) zero below the fundamental edge in nonmetals. In infrared studies of molecular and lattice modes, constant reflectance extrapolations R(0)') = R L, 0 _< 0)' < 0)1, and R(0)') = RH, 0)2 < 0)' < oe, for the reflectance above and below the measured interval are common [76, 121]. The two constants RL and Rn are chosen to reproduce the auxiliary data. *Explicit examples of this are given in Roessler [108], Velick~ [109], and MacRae et al. [110]. * For correction to Cardona and Greenaway [117] see Scouler [118].

58

D.Y. Smith

A closely related method employs a mean-value approximation to the dispersion integrals in the unmeasured regions [107, 108]. More detailed schemes [122] using parameters to define harmonic-oscillator extrapolations have been developed for cases in which sufficient data is at hand. Arkatova et al. [123] have critically reviewed several of these extrapolation methods. The second approach to the finite measurement interval problem involves finding the slowly varying component of the phase. Methods include calculating A0(og) at specific points at which additional data are available and then fitting the result with a simple function [109, 124] or determining a powerseries expansion of A0(09) [125-127]. The latter approach is based on Velick~,'s observation [109] that since In ~(09) is analytic, the difference between the actual phase 0(09) and the phase calculated from Eq. (63) by using measured data plus a "smooth" extrapolation can be expressed in a power series. The coefficients of the series are determined from the auxiliary measurements of 0(09). [Since 0(09) is an odd function, only odd powers of 09 are present.] The remarkable success of these methods in treating reflectance measurements over a finite interval plus one or more auxiliary data points to anchor the phase can be best understood in terms of subtracted dispersion relations, a technique widely used in high-energy physics [19, 20]. If 0(09) is known at a frequency o) = a, then by Eq. (63) f ~ In R(o)') a2 do)'.

O(a) = __a P Jo 09'2

(65)

Subtracting this from Eq. (63) yields [128]

O(09)=(a)O(a )

09(092 aZ)Pfo~

-~-

In R(09') (09,2 _ 092)(09,2 _ a 2) d09'.

(66)

This new expression manifestly gives the correct value of 0(o9) at 09 = a and is odd in 09 as required. By way of qualitative comparison with the Velick~, [109] series expansion of 0(09), Eq. (66) has the form of a modified dispersion relation plus a term linear in o9. Most important, the modified dispersion integral converges more rapidly than Eq. (63); at large 09' the denominator of the integrand behaves as 09,4 compared with 09,2 for Eq. (63). Thus, the knowledge of additional data in the form of the phase at o9 = a has anchored 0(~o) at the point O(a) and has significantly reduced the dependence of the dispersion integral on the unknown details of R(og) far from the measured interval. The procedure may be carried as far as one has extra data points. For example, with two points the doubly subtracted dispersion relation becomes 0(09)

09 0)2 - -

b2

09 092 - - a 2

= a a 2 - b 20(a)+-bb2_a20(b) O9

rt

(092 _ a2)(092 _

b2)p I,,~

In R(09') do)' . (67) j o (09,2 _ 092)(09,2 _ a2)(09,2 _ b 2)

3. Analysis of Optical Data

59

Now 0(co) is anchored at two points a and b; moreover, the dispersion integral is even less dependent on the high-energy extrapolation. The method of subtracted dispersion relations has had only limited applications [ 128, 129] in optics, but it should provide a systematic scheme utilizing all available data and offering significant savings in effort particularly over trial-and-error extrapolation.

Dispersion Analysis over a Wide Spectral Range For an increasing number of substances some optical data are available in all significant spectral ranges. However, the same type of data are not available at all frequencies. Reflectance or ellipsometric measurements are practical in opaque spectral regions, while the absorption coefficient or refractive index can be measured in regions of near transparency. As a consequence of this mix of data, the Kramers-Kronig method cannot be applied directly since it requires knowledge of a single optical function over all frequencies. However, the combination of Kramers-Kronig and Fresnel equations can be solved as coupled equations. In practice, one employs a method of successive approximations in which a trial value of one of the optical functions is estimated as best as can be from the available data. A Kramers-Kronig analysis is performed on this trial function. A new trial function is then constructed (generally by substitution of measured values in place of the calculated ones) and the procedure repeated until measurements and calculations agree [97, 132]. In principle, any optical function could be chosen as the starting point for an iteration. For materials with large regions of transparencymsuch as insulators--the extinction coefficient has been found to be convenient in practice [96]. However, transmission data can also be treated directly by using a dispersion relation for the phase shift in terms of the transmittance [91-93] that is analogous to Eq. (63). In the case of metals [97], the reflectance is a particularly favorable starting point since in opaque regions reflectance data are the most common, while the reflectance can often be approximated from Fresnel's equations in transparent regions. In the latter, both absorption coefficient and refractive index data can be measured and, even if only absorption data is at hand in the region of transparency, simple extrapolations or model fits give reasonable starting values for the index. This procedure, or at least one cycle of the procedure, has been carried out in a large number of studies by using the reflectance (for example, Ehrenreich et al. [94]) or a combination of reflectance and transmittance data (for example, Hagemann et al. [95]). In principle, there should be no difference in the final result whatever starting function is chosen, provided that the calculation is iterated to self-consistency, although this is not often done.

2

60

D.Y. Smith

A full self-consistent iterative analysis has been applied to polystyrene by Inagaki et al. [96] and to aluminum by Shiles et al. [97] and preliminary results have been reported for silicon [130] and gallium arsenide [131]. Aluminum is particularly favorable for this procedure since one form or another of optical data are now available from approximately 0.04 to 10,000 eV. In addition, in the region near the plasma frequency where optical experiments are difficult, there are accurate electron energy-loss measurements that give the shape of Im[e-~(co)] (though not its absolute magnitude). Specific results are discussed in the section of this handbook devoted to the optical properties of aluminum.

B

Sum-Rule Tests

Sum rules provide significant guidance in the construction of composite sets of data, particularly in selecting the most probable values from the available optical measurements and in pinpointing systematic errors. In the case of metallic aluminum mentioned earlier [97], it has proven possible to generate a comprehensive set of optical functions that satisfies all principal sum rules and is compatible with electron-energy-loss and stopping-power data.

1

The Inertial Sum Rule for n(~)

The rule ;o~ In(co)- 1] dco = 0

(68)

must be satisfied for any n(co)corresponding to a physically acceptable absorption spectrum. Provided that k(co) has an acceptable high-frequency behavior (no co-1 component), the rule tests the acceptability of n(co) as part of an analytic function. Hence, it tests the accuracy of the Kramer-Kronig transform procedure and any high- or low-frequency extrapolation of n(co) that may have been used. In practice, it is convenient to define a verification parameter [7] ~-- f ; [n(co)- 1] d c o /f; In(co)- 1] dco

(69)

to assess the extent to which a spectrum satisfies the rule. Shiles et al. [97] found that values of the order of 2 x 10-3 were obtained for self-consistent solutions with ordinary numerical procedures. Application of the rule to several composite data sets in the literature disclosed large values of ~ (some as large as 0.2) [7, 97]. These unacceptable values appear to arise primarily from nonanalytic modifications of the results

3. Analysis of Optical Data

61

of a Kramers-Kronig transformation. Such modifications include fitting inappropriate extrapolations or otherwise forcing n(o~) to pass through data points inconsistent with the absorption spectrum. The DC-Conductivity Sum Rule

2

A second consequence of causality and inertia is the expression for the dc conductivity f0 ~ [el(co) - 1] dco = - 27c2a(0).

(70)

In the case of insulators, there are only interband transitions, and the righthand side is zero. In metals both inter- and intraband transitions occur. To a good approximation the two processes contribute to e(o~) by linear superposition, and the nonzero term on the right-hand side of Eq. (70) arises primarily from the intraband part. Thus, the dc-conductivity rule measures the conduction-electron contribution to el(~o). Generally, this dispersive contribution extends over a wider and experimentally more accessible range than the corresponding absorptive contribution to e2(o~). Hence, Eq. (70) can be expected to give a more reliable optically derived value of a(0) for comparison with the measured dc conductivity than a Drude-model fit to e2(co) in the far infrared. In comparing values of a(0) derived from optical measurements with the bulk dc conductivity, it must be borne in mind that optical measurements are sensitive to the state of the sample at the surface. Moreover, many measurements are made on evaporated films that may have properties differing significantly from those of the bulk. In applying this rule to aluminum Shiles et al. [97] found that the optical and electrical values were equal within experimental error for unoxidized evaporated samples prepared in ultrahigh vacuum. This would generally not have been the case for contaminated films. The f Sum Rule

The various f sum rules test the conformity of optical data with the known electron density. Further, to the extent that absorptions can be attributed to a particular group of levels, the finite-energy rules test the distribution of oscillator strength within broad spectral regions. The concept of the effective number of electrons contributing to processes up to energy co [14, 45] neff(co)l~_ 2~2e m 2 foO CO,Cp(of) dof ,

(71)

where tI)(co') may be e2(co'), 2k(co'), or -Im[e-x(co')], is particularly useful in applying the f sum rule. A plot of these three quantities for aluminum is

3

62

D.Y. Smith 14 12

-I

.....

I0

,,,i,-. cl)

=

6

-

4 2 0 lO-I

/ I

JI '"',',

"'"

,

, 3.ill

10

10z 103 104 105 ENERGY, 15co(eV) Fig. 5. The functions neff(O ) for metallic aluminum. The dashed curve shows the results for experimental data exhibiting a total oscillator strength over 14 rather than 13 e/at. The solid curve shows neff(O9) after correction of the magnitude of the L-shelldata. (From Shiles et al. [97].)

shown in Fig. 5. Notice that the three quantities differ significantly as a function of energy. They describe different processes: The e2(o)) function concerns the dissipation of energy from an electromagnetic wave, the k(o)) function the attenuation of the wave's amplitude, and Im 5-1(o9) the energy loss of a charged particle. The requirement that the three forms of neff(O.) ) be equal at co = oo, while trivial, is a test of the arithmetic consistency of e, N, and e -~. The further requirement that neff(oo) is the observed electron density has more substance and serves as a test for net systematic error. In applying it, one must remember that reflection data measures the state of matter near the surface and that transmission measurements on thin films may not be representative of the bulk. In both instances, surface contamination and impurities introduced in preparation, or differences in density between thin-film and bulk samples may lead to spurious effects. An example of an apparent systematic error can be seen in the aluminum data of Fig. 5. The dashed curve shows neff(O)) for what initially appeared to be the best available experimental data. However, the total f sum approaches a limit of approximately 14.1 electrons/atom (e/at.) rather than the actual density of 13 e/at. This indicates a significant systematic error in an important spectral region. The partial f sum rules serve as a guide to the location of such errors. Aluminum is an especially favorable case for this since the conduction, L-shell, and K-shell absorptions are well separated and the oscillator strength of one absorption is virtually exhausted before the onset of the next. The partial f sums for the conduction electrons show a plateau region from the plasmon frequency ( ~ 15.0 eV) to the L,.m edge at ~ 72.7 eV. As explained in Section

3. Analysis of Optical Data

63 TABLE I

The "Effective" Number of Electrons per Atom for the Various Energy Levels in Aluminum"

Electronic shell Conduction L shell K shell

Atomic parentage Occupation (3s2, 3p)

(2S2, 2p 6) (ls 2)

Total

Raw data

/leff(O))]e modifieddata

HFS theory

3 8 2

3.11 9.35 1.61

3.11 8.27 1.61

3.12 8.33 1.55

13

14.08

12.99

13.00

" After Shiles et al. [97].

III, the three forms of neff(O ) differ in this asymptotic region because of virtual processes in the polarizable cores. A fit of Eq. (53) to the plateau yields the core background dielectric constant/~b ~'~ 1.035 .The total conduction electron oscillator strength is given by the limiting value of/%ff(o))le; extrapolating the plateau gives a total oscillator strength for the three conduction electrons of 3.1 ~ e/at. Above the Lu,n~ edge the effects of shielding decrease sharply, and the three forms of neff(O) ) draw together. The oscillator strengths for the various energy levels in metallic aluminum as derived from a self-consistent composite of the reported data are listed in Table I under raw data. Also given is the oscillator strength calculated for the corresponding atomic energy levels in a one-electron H a r t r e e - F o c k Slater (HFS) model that takes into account the Pauli principle redistribution of oscillator strength. In going from the atom to the solid, the valence-electron energies and wave functions are vastly altered, leading to a completely different absorption spectrum of the outermost electrons. However, the core levels are not changed significantly, and the net oscillator strength of each group of levels in the metal and the atom should be qualitatively comparable. Making this comparison in Table I, one sees that the L-shell strength of the reported data is too strong by approximately 1 e/at. corresponding to a systematic overestimate of 1 2 - 1 3 ~ averaged over the whole L absorption. In the high-absorption region between the L edge and 500 eV, virtually all measurements have been made on thin evaporated films prepared in conventional vacuum and subsequently exposed to the atmosphere. It is known that such films are highly reactive and form surface layers. Moreover, their absorption shows a surface as well as a bulk component [133, 134]. It was found by trial and error that an ad hoc reduction of the reported thin-film absorption data by 14~ from the L edge to 300 eV followed by a smooth interpolation to the rolled-foil data above 500 eV gave satisfactory agreement with the partial f sum rules. Independent experiments by Balzarotti et al. [134] provide additional evidence for this correction. Using films of

64

D.Y. Smith

various thickness, these authors separated surface and bulk effects and found absorption coefficients some 15% below those generally reported in this region. Unfortunately, their measurements extend only 10 eV above the L edge, but as far as they go, they are in agreement with the values inferred by demanding the f sum rule be satisfied.

VI

SUMMARY

The optical constants obey a large number of integral constraints in the form of sum rules. We have shown that some of the simpler of these rules arise directly from the requirements of causality, inertia, and the dynamical laws. These optical rules may therefore be viewed as the co-space equivalent of the dynamical laws of motion in time space. The sum rules provide a useful means of testing optical measurements, particularly wide-range composite data, both against theoretical constraints and independently measured quantities. ACKNOWLEDGMENTS

The author wishes to express his thanks to Professor Dr. H. Bilz for his hospitality and encouragement and to the Max-Planck-Institut fiir Festk6rperforschung for support during the early stages of preparation of this review. Thanks are also due to Dr. M. Inokuti for stimulating discussions and Mrs. Janice Grant for her aid in preparing the manuscript.

REFERENCES

1. F. Stern, in "Solid State Physics" (F. Seitz and D. Turnbull eds.), Vol. 15, p. 299, Academic Press, New York, 1963. 2. F. Wooten, "Optical Properties of Solids," Academic Press, New York, 1972. 3. W. M. Saslow, Phys. Lett. 33A, 157 (1970). 4. M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, Phys. Rev. B 6, 4502 (1972). 5. A. Villani and A. H. Zimerman, Phys. Lett. 44A, 295 (1973). 6. A. Villani and A. H. Zimerman, Phys. Rev. B 8, 3914 (1973). 7. M. Altarelli and D. Y. Smith, Phys. Rev. B 9, 1290 (1974); M. Altarelli and D. Y. Smith, Phys. Rev. B 12, 3511 (1975). 8. F. W. King, J. Math. Phys. 17, 1509 (1976). 9. K. Furuya, A. H. Zimerman, and A. Villani, Phys. Rev. B 13, 1357 (1976). 10. D. Y. Smith, J. Opt. Soc. Am. 66, 547 (1976). 11. D. Y. Smith, Phys. Rev. B 13, 5303 (1976). 12. K. Furuya, A. H. Zimerman, and A. Villani, J. Phys. C 9, 4329 (1976). 13. K. Furuya, A. Villani, and A. H. Zimerman, J. Phys. C 10, 3189 (1977). 14. D. Y. Smith and E. Shiles, Phys. Rev. B 17, 4689 (1978). 15. C. K. Mukhtarov, Dok. Akad. Nauk SSSR 249, 851 (1979); C. K. Mukhtarov, Soy. Phys. Dokl. 24, 991 (1979). 16. H. A. Kramers, Nature 117, 775 (1926) [abstract]; H. A. Kramers, Atti Congr. Intern. Fisici, Como-Pavia-Roma (N. Zanichelli, Bologna, 1928) 2, 545-557 [reprinted in H. A.

3. Analysis of Optical Data

65

Kramers, "Collected Scientific Papers," North-Holland Publ., Amsterdam, 1956]; H. A. Kramers, Phys. Z. 30, 522 (1929). 17. R. de L. Kronig, J. Opt. Soc. Am. 12, 547 (1926). 18. L. D. Landau and E. M. Lifshitz, "Electrodynamics of Continuous Media," Pergamon, Oxford, 1960. 19. H. M. Nussenzveig, "Causality and Dispersion Relations," Academic Press, New York, 1972. 20. H. Burkhardt, "Dispersion Relation Dynamics," North Holland Publ., Amsterdam, 1969. 21. C. Ferro Font/m, N. M. Queen, and G. Violini, Riv. Nuovo Cimento 2, 357 (1972). 22. B. Gross, Nuovo Cimento Suppl. 3, 235 (1956); J. R. Macdonald and M. K. Brachman, Rev. Mod. Phys. 28, 393 (1956); D. Pines and P. Nozi6res, "The Theory of Quantum Liquids," W. A. Benjamin, New York, 1966; P. C. Martin, Phys. Rev. 161, 143 (1967); J. des Cloizeaux, in "Theory of Condensed Matter," p. 325 Int. At. Energy Ag., Vienna, 1968; R. Kubo and M. Ichimura, J. Math. Phys. 13, 1454 (1972). 23a. D. Y. Smith, in "Theoretical Aspects and New Developments in Magnetooptics" (J. T. Devreese ed.), p. 133, Plenum, New York, 1980. 23b. M. T. Thomaz and H. M. Nussenzveig, Ann. Phys. (N.Y.) 139, 14 (1982). 24. H. A. Bethe and E. E. Salpeter, "Quantum Mechanics of One- and Two-Electron Atoms," Sec. 61 and 62, pp. 357 and 358, Springer-Verlag, Berlin, 1957. 25. R. G. Sachs and N. Austern, Phys. Rev. 81, 705 (1951). 26. J. S. Levinger, M. L. Rustgi, and K. Okamoto, Phys. Rev. 106, 1191 (1957). 27. H. O. Dogliani and W. F. Bailey, J. Quant. Spectrosc. Radiat. Transfer 9, 1643 (1969). 28. B. Jacobsohn, Ph.D. dissertation, University of Chicago, Chicago, 1947 (unpublished). [Available in part through DOE Technical Information Center, Oak Ridge, Tennessee, as ANL-HDY-423 (Del).] 29. W. B. Payne and J. S. Levinger, Phys. Rev. 101, 1020 (1956). 30. U. Fano, in "Annual Review of Nuclear Science" (E. Segr6, G. Friedlander, and H. P. Noyes eds.) Vol. 13, p. 1, Annual Reviews, Palo Alto, California, 1963. 31. W. Thomas, Naturwiss. 28, 627 (1925); and F. Reiche and W. Thomas, Z. Phys. 34, 510 (1925). 32. W. Kuhn, Z. Phys. 33, 408 (1925). 33. B. L. van der Waerden, "Sources of Quantum Mechanics," North-Holland Publ., Amsterdam, 1967. 34. U. Fano and J. W. Cooper, Rev. Mod. Phys. 40, 441 (1968); U. Fano and J. W. Cooper, Rev. Mod. Phys. 41, 724 (1969). 35. J. S. Levinger, "Nuclear Photo-Disintegration," Oxford Univ. Press, London, 1960. 36. A. Smakula, Z. Phys. 59, 603 (1930). 37. D. L. Dexter, in "Solid State Physics" (F~ Seitz and D. Turnbull eds.), Vol. 6, p. 353, Academic, New York, 1958. 38. D. Y. Smith and D. L. Dexter, in "Progress in Optics" (E. Wolf, ed.), Vol. 10, Sec. 3.1, North-Holland Publ., Amsterdam, 1972. 39. J. P. Vinti, Phys. Rev. 41, 432 (1932). 40. J. O. Hirschfelder, W. B. Brown, and S. T. Epstein, in "Advances in Quantum Chemistry" (P.-O. L6wdin, ed.), Vol. 1, Sec. 10. Academic Press, New York, 1964. 41. J. J. Hopfield, Phys. Rev. B 2, 973 (1970). 42. M. Brauwers, R. Evrard, and E. Kartheuser, Phys. Rev. B 12, 5864 (1975). 43. C.-H. Wu, G. Mahler, and J. L. Birman, Phys. Rev. B 18, 4221 (1978). 44. J. Gardavsk), Coll. Phen. 1, 193 (1974). 45. H. Ehrenreich, in "The Optical Properties of Solids" (J. Tauc, ed.), p. 106, Academic Press, New York, 1966. 46. J. S. Toll, Ph.D. thesis Princeton University, Princeton, New Jersey, 1952; J. S. Toll, Phys. Rev. 104, 1760 (1956). 47. N. G. van Kampen, J. Phys, Rad. 22, 179 (1961).

66 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68.

69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88.

D.Y. Smith E. C. Titchmarsh, "The Theory of Functions," Oxford Univ. Press, London, 1968. V. de Alfaro, S. Fubini, G. Rossetti, and G. Furlan, Phys. Lett. 21, 576 (1966). G. Frey and R. L. Warnock, Phys. Rev. 130, 478 (1963). A. R. P. Rau and U. Fano, Phys. Rev. 162, 68 (1967). R. A. Ferrell and R. E. Glover, Phys. Rev. 109, 1398 (1958). G. Br/indli, Phys. Rev. Lett. 28, 159 (1972); G. Br/indli and A. J. Sievers, Heir. Phys. Acta 45, 847 (1972); G. Brandli and A. J. Sievers, Phys. Rev. B 5, 3550 (1972). G. Briindli, Phys. Rev. B 9, 342 (1974). M. Tinkham and R. A. Ferrell, Phys. Rev. Lett. 2, 331 (1959). D. M. Ginsberg and M. Tinkham, Phys. Rev. 118, 990 (1960). L. H. Palmer and M. Tinkham, Phys. Rev. 165, 588 (1968). D. Y. Smith, J. Opt. Soc. Am. 66, 454 (1976). F. W. King, J. Math. Phys. 22, 1321 (1981); F. W. King, Phys. Rev. B 25, 1381 (1982). E. C. Titchmarsh, "Introduction to the Theory of Fourier Integrals," Oxford Univ. Press, London, 1962. I. Kimel, Phys. Rev. B 25, 6561 (1982). J. Fischer, J. Pi~fit, P. Pre~najder, and J. ~ebesta, Czech. J. Phys. B 19, 1486 (1969). Y. C. Liu and S. Okubo, Phys. Rev. Lett. 19, 190 (1967). D. Y. Smith and G. Graham, J. Phys. (Paris) 41, Colloque C6, C6-80 (1980). R. de L. Kronig and H. A. Kramers, Z. Phys. 48, 174 (1928). V. Fock, Z. Phys. 89, 744 (1934). I. ~imon, J. Opt. Soc. Am. 41, 336 (1951). T. S. Robinson, Proc. Phys. Soc. (London) B 65, 910(1952); T. S. Robinson and W. C. Price, Proc. Phys. Soc. (London) B 66, 969 (1953); T. S. Robinson and W. C. Price, in "Molecular Spectroscopy" (G. Sell ed.), p. 211, Inst. Petroleum, London, 1955; J. K. Wilmshurst and S. Senderoff, J. Chem. Phys. 35, 1078 (1961). T. Inagaki, A. Ueta, and H. Kuwata, Phys. Lett. 66A, 329 (1978). F. W. King, J. Chem. Phys. 71, 4726 (1979); I. Kimel, Phys. Lett. 88A, 62 (1982). D. Y. Smith and C. A. Manogue, J. Opt. Soc. Am. 71, 935 (1981). F. C. Jahoda, Ph.D. thesis, Cornell University, Ithaca, New York, 1957; F. C. Jahoda, Phys. Rev. 107, 1261 (1957). B. Velick~r Czech. J. Phys. B 11, 541 (1961). P. Roman and A. S. Marathay, Nuovo Cimento 30, 1452 (1963). R. E. Burge, M. A. Fiddy, A. H. Greenaway, and G. Ross, J. Phys. D 7, L65 (1974). M. Gottlieb, Ph.D. thesis, University of Pennsylvania, Phildelphia, Pennsylvania 1959; M. Gottlieb, J. Opt. Soc. Am. 50, 343 (1960); M. Gottlieb, J. Opt. Soc. Am. 50, 350 (1960). D. Y. Smith, J. Opt. Soc. Am. 67, 570 (1977). G. Bonfiglioli and P. Brovetto, Phys. Lett. 5, 248 (1963); G. Bonfiglioli and P. Brovetto, Appl. Opt. 3, 1417 (1964). B. O. Seraphin, in "Optical Properties of Solids" (F. Abel6s, ed.), p. 163, North-Holland Publ. Amsterdam, 1972. A Balzarotti, E. Colavita, S. Gentile, and R. Rosei, Appl. Opt. 14, 2412 (1975). J. S. Plaskett and P. N. Schatz, J. Chem. Phys. 38, 612 (1963). E. A. Lupashko, V. K. Miloslavskii, and I. N. Shklyarevskii, Opt. Spectrosk. 29, 789 (1970); E. A. Lupashko, V. K. Miloslavskii, and I. N. Shklyarevskii, Opt. Spectrosc. 29, 419 (1970). E. A. Lupashko, V. K. Miloslavskii, and I. N. Shklyarevskii, Opt. Spectrosk. 24, 257 (1968); Lupashko, Miloslavskii, and Shklyarevskii, Opt. Spectrosc. 24, 132 (1968). A. A. Clifford, M. I. Duckels, and B. Walker, J. Chem. Soc. (London), Faraday Transactions II, 68, 407 (1972). D. M. Roessler, Brit. J. Appl. Phys. 16, 1359 (1965). D. W. Berreman, Appl. Opt. 6, 1519 (1967). W. G. Chambers, Infrared Phys. 15, 139 (1975). R. H. Young, J. Opt. Soc. Am. 67, 520 (1977).

3. Analysis of Optical Data

67

89. G. M. Hale, W. E. Holland, and M. R. Querry, Appl. Opt. 12, 48 (1973); M. R. Querry and W. E. Holland Appl. Opt., 13, 595 (1974). 90. T. Inagaki, H. Kuwata, and A. Ueda, Phys. Rev. B 19, 2400 (1979); T. Inagaki, H. Kuwata, and A. Ueda, Surf. Sci. 96, 54 (1980). 91. S. Maeda, G. Thyagarajan, and P. N. Schatz, J. Chem. Phys. 39, 3474 (1963); K. Kozima, W. Su~taka, and P. N. Schatz, J. Opt. Soc. Am. 56, 181 (1966). 92. P.-O. Nilsson, Appl. Opt. 7, 435 (1968). 93. J. D. Neufeld and G. Andermann, J. Opt. Soc. Am. 62, 1156 (1972). 94. H. Ehrenreich, H. R. Philipp, and B. Segall, Phys. Rev. 132, 1918 (1963); H. R. Philipp and H. Ehrenreich, J. Appl. Phys. 35, 1416 (1964). 95. H.-J. Hagemann, W. Gudat, and C. Kunz, J. Opt. Soc. Am. 65, 742 (1975); H.-J. Hagemann, W. Gudat, and C. Kunz, DESY Report SR 74/7, "Optical Constants from the Far Infrared to the X-Ray Region," Deutsches Elektronen-Synchrotron, Hamburg, 1974. 96. T. Inagaki, E. T. Arakawa, R. N. Hamm, and M. W. Williams, Phys. Rev. B 15, 3243 (1977). 97. E. Shiles, T. Sasaki, M. Inokuti, and D. Y. Smith, Phys. Rev. B 22, 1612 (1980). 98. M. Cardona, in "Optical Properties of Solids," (S. Nudelman and S. S. Mitra eds.), p. 137 Plenum, New York, 1969. 99. L. D. Kislovskii, Opt. Spektrosk. 1, 672 (1956); L. D. Kislovskii, Opt. Spektrosk. 2, 186 (1957); L. D. Kislovskii, Opt. Spektrosk. 5, 66 (1958); L. D. Kislovskii, Opt. Spektrosk. 6, 810 (1959); L. D. Kislovskii, Opt. Spectrosc. 6, 529 (1959); L. D. Kislovskii, Opt. Spektrosk. 7, 311 (1959); L. D. Kislovskii, Opt. Spectrosc. 7, 201 (1959). 100. C. W. Peterson and B. W. Knight, J. Opt. Soc. Am. 63, 1238 (1973). 101. D.W. Johnson, J. Phys. A 8, 490 (1975). 102. F. W. King, J. Phys. C 10, 3199 (1977). 103. F. W. King, J. Opt. Soc. Am. 68, 994 (1978). 104. W. O. Saxton, J. Phys. D 7, L63 (1974). 105. D. L. Misell, J. Phys. D 7, L69 (1974). 106. H. W. Bode, "Network Analysis and Feedback Amplifier Design," D. Van Nostrand, New York, 1945. 107. D. M. Roessler, Brit. J. Appl. Phys. 16, 1119 (1965); D. M. Roessler, Brit. J. Appl. Phys. 16, 1777 (1965). 108. D. M. Roessler, Brit. J. Appl. Phys. 17, 1313 (1966). 109. B. Velick~,, Czech. J. Phys. B 11, 787 (1961). 110. R. A. MacRae, E. T. Arakawa, and M. W. Williams, Phys. Rev. 162, 615 (1967). 111. G. R. Anderson and W. B. Person, J. Chem. Phys. 36, 62 (1962). 112. H. J. Bowlden and J. K. Wilmshurst, J. Opt. Soc. Am. 53, 1073 (1963). 113. W. G. Spitzer and D. A. Kleinman, Phys. Rev. 121, 1324 (1961). 114. O. Castafio, M. de Dios, and R. P6rez, comunicaciones (Havana) 17, 65 (1975); O. D. Castafio, M. de Dios Leyva, and R. P+rez Alvarez, Phys. Stat. Solidi B 71, 111 (1975). 115. B. W. Veal in "The Actinides: Electronic Structure and Related Properties, Vol. 2, pp. 87-88, Academic Press, New York, 1974. 116. H. R. Philipp and E. A. Taft, Phys. Rev. 113, 1002 (1959); H. R. Philipp and E. A. Taft, Phys. Rev. 136, A1445 (1964). 117. M. Cardona and D. L. Greenaway, Phys. Rev. 133, A1685 (1964). 118. W. J. Scouler, Phys. Rev. 178, 1353 (1969). 119. M. P. Rimmer and D. L. Dexter, J. Appl. Phys. 31, 775 (1960); M. G. Doane, M.S. thesis, University of Rochester, Rochester, New York, 1961. 120. V. B. Tulvinskii and N. I. Terentev, Opt. Spektrosk. 28, 894 (1970); V. B. Tulvinskii and N. I. Terentev, Opt. Spectrosc. 28, 484 (1970). 121. P. N. Schatz, S. Maeda, and K. Kozima, J. Chem. Phys. 38, 2658 (1963); D. W. Barnes and P. N. Schatz, J. Chem. Phys. 38, 2662 (1963). 122. G. Andermann, A. Caron, and D. A. Dows, J. Opt. Soc. Am. 55, 1210(1965); G. Andermann

68

123.

124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134.

D.Y. Smith and D. A. Dows, J. Phys. Chem. Sol. 28, 1307 (1967); C.-K. Wu and G. Andermann, J. Opt. Soc. Am. 58, 519 (1968). T. G. Arkatova, N. M. Gopshtein, E. G. Makarova, and B. A. Mikhailov, Opt. Mekh. Promst. 48, 44 (1981); T. G. Arkatova, N. M. Gopshtein, E. G. Makatova, and B. A. Mikhailov, Soy. J. Opt. Technol. 48, 552 (1981). D. G. Thomas and J. J. Hopfield, Phys. Rev. ll6, 573 (1959). V. K. Miloslavskii, Opt. Spektrosk. 21, 343 (1966); V. K. Miloslavskii, Opt. Spectrosc. 21, 193 (1966). P.-O. Nilsson and L. Munkby, Phys. Kondens. Materie 10, 290 (1969). V. K. Zaitsev and M. I. Fedorov, Opt. Spektrosk. 44, 1186 (1978); V. K. Zaitsev and M. I. Fedorov, Opt. Spectrosc. 44, 691 (1978). R. K. Ahrenkiel, J. Opt. Soc. Am. 61, 1651 (1971); R. K. Ahrenkiel, J. Opt. Soc. Am. 62, 1009 (1972). R. Z. Bachrach and F. C. Brown, Phys. Rev. B l, 818 (1970); N. J. Carrera and F. C. Brown, Phys. Rev. B 4, 3651 (1971). E. Shiles and D. Y. Smith, Bull. Am. Phys. Soc. 23, 226 (1978). E. Shiles and D. Y. Smith, Bull. Am. Phys. Soc. 24, 335 (1979). G. Leveque, J. Phys. C 10, 4877 (1977). D. H. Tomboulian and E. M. Pell, Phys. Rev. 83, 1196 (1951). A. Balzarotti, A. Bianconi, and E. Burattini, Phys. Rev. B 9, 5003 (1974).

Chapter

4

Measurement of Optical Constants in the Vacuum

Ultraviolet Spectral Region* W, R, HUNTERt Naval Research

Laboratory

Washington,

I. Introduction II. General Discussion of Reflectance Methods A. Reflection at Normal Incidence--Measurement of Index of Refraction B. Reflection at Oblique Incidence C. Effect of Layer Thickness D. Critical-Angle Method E. Measurement of Extinction Coefficients III. Reflectance Method for Two Media References

D.C.

69 70 70 71 78 79 81 85 87

INTRODUCTION

The optical constants of an isotropic material are the index of refraction n and the extinction coefficient k. They are the real and imaginary components of the complex index of refraction N = n ~ ik. They can be measured at a given wavelength by direct methods or inferred from photometric or polarimetric measurements. For wavelengths less than 2000 A, the vacuum ultraviolet (VUV) spectral region, polarimetric methods are seldom useful because the phase-discriminating components of the system are strongly absorbing, consequently, reflectance methods are almost exclusively used for obtaining * This chapter is a condensed version of a paper, "Measurement of Optical Properties of Materials in the Vacuum Ultraviolet Spectral Region," published in Applied Optics 21, 2103 (1982). t Present address: Sachs/Freeman Associates, Inc., Bowie, Maryland. 69 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright 9 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-544420-6

I

70

W.R. Hunter

n and k. A number of methods exist for extracting n and k from specularreflectance measurements at both normal and oblique incidence and for semi-infinite media as well as layers on substrates. This chapter will discuss the most useful methods for obtaining n and k in the VUV and the problems encountered in applying these methods. The K r a m e r s - K r o n i g method will not be discussed here (see Chapter 3) because it cannot be used to deduce n and k from measurements at a single wavelength.

II

GENERAL DISCUSSION OF REFLECTANCE METHODS

The specular reflectance of a substance at an angle of incidence ~, measured from the surface normal, is related to the complex index of refraction by the generalized Fresnel reflection coefficients, R~ = [(a - cos ~b)2 +

b2]/[(a + cos ~b)2 + b 2]

R p - R~[(a- sin ~b tan (I))2 _[_ b2]/[(a q_ sin ~b tan r R~ - 89

..[_b 2]

+ p) + Rs(1 - p)],

where R~ is the component of specular reflection perpendicular to the plane of incidence, Rp the parallel component, and R~ the reflectance for nonpolarized radiation. The polarization of the incident radiation is given by p = (lp-

I~)/(Ip + I~),

and a 2 = l{[(n2 - k 2 - sin 2 t~) + 4nZk2] 2 + (n 2 - k 2 - sin 2 (h)}, b 2 = 89

2 - k 2 - sin 2 +) + 4nZk2] 2 - (n 2 - k 2 - sin 2 th)}.

These equations cannot be solved explicitly for n and k; hence one must resort to graphical or numerical means to solve them [ 1 - ! 1]. Since the equations have two unknowns, two independent measurements are necessary if a solution is to be obtained.

A

Reflection at Normal Incidence--Measurement of Index of Refraction

At normal incidence the two components of reflection are indistinguishable and, for nonabsorbing media, the equations may be reduced to R=(n-1)

2 / ( n + l) 2.

4. Measurement of Optical Constants in the VUV Spectral Region

71

Unlike the generalized coefficients, this equation can be solved for n as n = (1 + R ~

-

R~

The values of n of transparent media in the VUV usually range from 1.3 to 2.0. Over this range the absolute error in determining n from reflectance measurements is approximately 10 times the absolute error in measuring R. Thus, n is sensitive to small errors in R and the accuracy of the method makes it suitable only for determining provisional values to two significant figures at best. This simple method may be used for any material that is transparent in the VUV and can be polished or cleaved to give a specularly reflecting surface. Precautions must be taken to measure reflectance from the first surface only. Reflections from the second surface can be eliminated by making the sample wedge shaped or can be suppressed by grinding the second surface and blackening it.

Reflection at Oblique Incidence

Humphreys-Owen [6] lists nine methods by which n and k can be deduced from reflectance measurements at oblique incidence. These methods are divided into two classes: (1) two reflectance measurements at one angle of incidence or one reflectance measurement at each of two angles of incidence and (2) one reflectance measurement at any angle of incidence and measurement of a special angle of incidence capable of supplying the necessary second measurement. There are only two special angles: (1) the principal angle of incidence, which must be determined by polarimetric methods and so it will not be discussed, and (2) the Brewster, or pseudo-Brewster (pB), angle. The Brewster angle is defined only in terms of dielectric media and is given by n = tan ~bB. At this angle Rp -- 0. As k increases, Rp always has a minimum value, but it is not zero. The angle at which this minimum occurs in the pB angle and, if k > 0, q~pB> ~bB. Humphreys-Owen's list has been rearranged as follows: Class 1 Method 1

Method 2 Method 3

Reflectance at two angles of incidence using natural or polarized radiation; sometimes referred to as the reflectanceversus-angle of incidence method The ratio Rp/R s at two angles of incidence Rs and R v at one angle of incidence

Class 2 Method 4 Method 5

Pseudo-Brewster angle and Rs or Rp at that angle Pseudo-Brewster angle and Rp/Rs at that angle

B

72

W.R. Hunter

Method 6

Pseudo-Brewster angle and Rs, Rp, or Rp/R s at any other angle of incidence

In principle, only two measurements are required, but, because of possible errors in the measurements, a redundancy of measurements is more useful and will give an indication of the errors involved in determining n and k. Although these methods can be used at any wavelength, their sensitivities to errors in measurements of reflectance, of the angles involved, or of the state of polarization are functions of both n and k, as well as the angle of incidence and the state of polarization. Thus, there may be parts of the n, k plane, angles of incidence, and states of polarization, for which the lack of sensitivity reduces their accuracy to unacceptable values. Since the sensitivities of the methods are dependent on n and k, hence on the shape of the R-versus-~b curves, a collection of these curves is shown in Fig. 1 [11] for reference during the ensuing discussion. The large numbers give the values of n and k used in calculating a particular set of curves. These values cover the range of n and k most likely to be encountered in the VUV. The small numbers along the abscissa and ordinate show the angle of incidence and percent reflectance, respectively. The upper curve is always Rs, the lower curve Rp, and the curve between R,. Generally, for a given n, as k increases I00

3.3

5o o IOO w Z

2.5

~ o

50

w _1 1.1_ w

n- I00

0

IZ w

1.3 ~: so w 13_

0 I00

k =0.:5

5o o

o

5o

60

n = 0.3

90 o

o 5o 60 5o 60 90 ANGLE OF INCIDENCE 0.8

1.3

90 o

5o

60 1.8

o 90

5o

60

90

2.3

Fig. 1. R-versus-~ curves calculated using the optical constants shown by the large numbers. The small numbers shown on the abscissa and ordinate are the values of ~b and R, respectively.

4.

Measurement of Optical Constants in the VUV Spectral Region

73

the reflectance at normal incidence increases, the change in R with 4~ becomes small, especially for R,, except at the pB angle, and the pB angle becomes larger. For a given k, as n increases the reflectance at normal incidence decreases, the change in R with 0 becomes more pronounced, especially for Rp, and again the pB angle becomes larger. The value of Rp at the pB angle appears to be zero for n > 1.3 and k = 0.3. This is not so, but the values are extremely small. Method 1

The sensitivity of this method is most easily investigated by means of isoreflectance curves. An isoreflectance curve is the locus of points in the n, k plane corresponding to a given value of R for a specific value of 4~. If isoreflectance curves are plotted for perfect data, i.e., reflectance values calculated using the Fresnel formula, an insight into the sensitivity of the method can be obtained. Figure 2 [11] shows such a set of curves for the parallel component covering that part of the n, k plane 0.3 < n < 2.3 and 0.3 < k < 3.3.

3.3

3.5 3.4 3.3 5.2 3.1

2.5

2.3

2.4 2.3 2.2 2.1

il i

J

].5= 1.41

' i i

I.I

k=O.3

.....

.....

0.5 0.4 0.3 0.2 0.1 0.1

0.3

n = 0.3

0.5

0.6

0.8

0.8

1.0

I.I

1.3

1.3

I.,5

1.6

1.8

1.8

2.0

2.1

2.3

2.5

2.3

Fig. 2. Isoreflectance curves for Rp calculated using the optical constants shown by the larger numbers. The small numbers used for the abscissa and ordinate show the scale in the n, k plane. The numeral 8 designates the isoreflectance curve corresponding to 80 ~ The other curves are for angles of incidence 70, 60, through 10 ~ The curves usually occur in descending order; when they do not, each curve is labeled with a single digit to avoid confusion (method 1).

74

W.R. Hunter

An indication of the sensitivity is the angle of intersection of the isoreflectance curves. If there is an error in the reflectance, the isoreflectance curve will be shifted parallel to itself by an amount depending on the magnitude of the error and in a direction depending on its sign. Thus, if two curves intersect at a small angle, a slight displacement of one with respect to the other may shift the intersection point by a large amount. Such a displacement could be due either to an error in measuring R or an error in measuring ~b. The figure indicates that for n = 0.3, the sensitivity with respect to k decreases as k increases. Thus, for small n and k > n, the R-versus-~b method can provide reasonable values for n but may result in large errors for k. Metals with large reflectance values, e.g., A1, Mg, etc., usually have small n and k > n. For example, at 1216 A, A1 has n ~ 0.06 and k ~ 1.0. For n > 0.3, the isoreflectance curves for small angles of incidence are more nearly parallel than those for the larger angles. Thus, the sensitivity is greatest if measurements are made at the larger angles of incidence. For n = 0.3, however, the maximum sensitivity for small k is obtained using small angles and as n increases, maximum sensitivity is shifted only slowly to larger angles. Generally, the angles with which maximum sensitivity is obtained are those angles at which the curvature in R is maximum, which is not necessarily the angles at which R changes most rapidly with ~b. Similar sets of isoreflectance curves have been calculated for Rs and Ra [11]. Although the curves in Fig. 2 furnish an idea of the sensitivity of the method, the accuracy of the method can only be found by using false data. By adding positive and negative errors to perfect reflectance data, the magnitude of the displacement of the isoreflectance curves can be obtained and the accuracy of the method determined. Hunter [11] has investigated the accuracy of method 1 using angles of incidence of 20 and 70 ~ His results are shown in Fig. 3 for both Rp and Ra. The blackened parallelopipeds represent the error in determining n and k for _+ 19/o errors in measuring the reflectance. The parallelopipeds correspond to the values of n and k shown in large numbers. Lines connect the extremities of these parallelopipeds to indicate the magnitude of the errors at intermediate points. The magnitude of the errors is in keeping with the sensitivity as discussed in connection with Fig. 2. For example, for small n (0.3), the error in n as k increases grows comparatively slowly, but the corresponding error in k becomes large quite rapidly because the parallelopipeds are almost parallel to the k axis. As n increases and for large k, the long axis of the parallelopipeds rotates in a clockwise direction so that the error in determining n increases while that for k decreases. It is evident that Rp is more tolerant of errors than Ra. In using the R-versus-~b method it is not necessary to know the actual R values [11]. The fact that reflectance is defined as the ratio of the reflected intensity at angle ~b to the incident intensity is equivalent to normalizing the reflected intensity at angle ~bto the reflected intensity at 90 ~ angle of incidence, which is defined as unity. For use in the R-versus-4~ method, normalization

4. Measurement of Optical Constants in the VUV Spectral Region

,

5.0 '

RO

,

,

,

75

,

Rp

'

4.0

3.3--

_

i

--r

3.0

I

2.3--

i

2.0

1.3-1.0

k:o.3-I

0 0

n:o.3

I

3.(

0.8

1.3

1.8 :;::).3

3.0

0.3 0.8

1.3

1.8 2.3

Fig. 3. The effect of _+1~ error in the measurement of reflectance on the determination of n and k using method 1. The angles of incidence are 20 and 70~ can be done at any angle of incidence and is referred to as oblique normalization. Field and Murphy [12] have published an analysis of the R-versus-~b method for oblique normalization. For oblique normalization the angle between the isoreflectance curves may not be a good indicator of sensitivity, and one should plot known errors in the obliquely normalized reflectance values to get a correct indication of sensitivity. Oblique normalization has no advantages over the usual mode of normalization and is only used when the physical arrangement of the reflectometer makes measurement of the incident intensity impractical. Method 2

Figure 4 shows two sets of curves for the ratio Rp/Rs. To the left is a set for small angles of incidence and to the right a set for large angles. According to Humphreys-Owen, an indication of the sensitivity is the spacing between contours--if the spacing is large, the sensitivity is good and vice versa. Assuming that his assertion is correct, the large-angle contours are obviously not useful if n and k are large but might be used to obtain good values if n and k are small. The small-angle contours are only useful if n and k are small. H u m p h r e y s - O w e n showed a limited set of contours of Rp/Rs for 80 and 60 ~ and a more extended, calculated set indicates that these two angles may be more useful than those shown here for larger values of n and k.

76

W.R. Hunter I~

'

'

'

'

I

'

'

'

~

Io

f

. . . .

(a)

---

%

. . . .

i

(b)

~,~

I ~ e ~ ~ 2

.-.,

'

'

'

'

'

'

'

Rp/R s (50 ~

'

'

n-4

,'o

Rp/R s (70 ~

Fig. 4. Curves of constant n and k as a function of Rp/R s for (a) small (30-20 ~ and (b) large (70-50 ~ angles of incidence. An and Ak are each 0.2. Curves for integer n and k values are dashed (method 2).

Method 3 Figure 5 shows two sets of curves for Rs and Rp at both 20 and 70 ~ angles of incidence. Judging from sets of curves obtained for other angles of incidence, these two angles are the most useful. F o r the 20 ~ angle of incidence, the curves of constant k are spaced 0.01 apart from k = 0 to k = 0.1. The next k c o n t o u r is for k = 0.2 and thereafter Ak = 0.2. The n contours have An = 0.1. 1.0

.

.

.

.

I

.

.

.

.

0

.

,

,

,

,

I

. . . .

=

Rs

R;

/ "~--...~k = 0 _

o Oo

I.O Rp

~

/

- =0.2

'l--I.O

,

,

,

I

o

,

,

,

(,b) I.O

Rp

Fig. 5. Curves of constant n and k as a function of Rpand Rs for (a) small (20 ~ and (b) large (70 ~ angles of incidence. For (a) Ak is 0.01 for 0 _< k _< 0.1; otherwise Ak is 0.2. An is 0.1 and the curve for n = 0.5 is dashed. The dotted line corresponds to the value of n for the Brewster angle, n - 0.364. For (b) Ak = An = 0.2. Curves for integer n and k values are dashed (method 3).

4. Measurement of Optical Constants in the VUV Spectral Region The dotted line is the contour of,, - '~ " " " be useful for n < 0.5 and k < 0. The set of curves of 70 ~ is k for larger values. For this set R, Ak are 0.2 for this set. The envelope of the 20 ~ cot mately, the curve for large n v two curves for k = 0, which cc for any n and k, the envelope a parabola.

. . . .

'-:~" ~

77 ,~f curves might

Figure 6 shows sets of curv the pB angle. The Rp/R s sel favors somewhat larger n a values. For example, if the [ n could be 0.2 or 0.3. Thus, the l c~;,~,. . . . . . Contours for Rp-versus-pB angle are similar to those for Rp/Ks. According to H u m p h r e y s - O w e n , method 6 gives no advantages over the other methods. In order to obtain quantitative data on the accuracy of methods 2-5, it is necessary to plot the contours for R + AR, or pB _+ ApB, which has not been done. The existence of large spacings between the contours suggests but does not guarantee small displacements of the contours when errors are introduced H u m p h r e y s - O w e n points out that R p , a t the pB angle, can be quite

i~ k

'

o'.s

'-

i.o'

1.5

'--

'

'

n

l

I.O

1

. . . .

0.5

~-----I.5

o 0

el)

ot

90

PSEUDO-BREWSTER

,

,

0

,

,(b 90

ANGLE (deg.)

Fig. 6. Curves of constant n and k as a function (a) of Rp/R~and the pseudo-Brewster (pB) angle and (b) of R~ and the pB angle (methods 4 and 5, respectively). An = Ak = 0.1 and curves for integer and half-integer; n and k values are dashed.

78

W.R. Hunter

small; thus, there may be a large error in measuring it. Furthermore, the minimum in Rp at the pB angle is usually broad, so accurate location of the angle may be difficult. If method 2 or 5 is to be used, normalization to get actual reflectance values is unnecessary. The equation for Rp can be manipulated to obtain the pB angle in terms of n and k [6]. It is given below, 2(p2 + q)v 3 + p2(p2 _ 3)v2 _ 2p4v + p4= O, where p = n 2 nt- k 2,

q = n 2 - - k 2,

and

v = sin(pB).

The experimental values for n and k obtained from the curves can be substituted in the formula to help verify the correctness of the results.

C

Effect of Layer Thickness Preparing a material for reflectance measurements from which n and k are to be obtained is often done by vacuum evaporation of the material onto a highly polished substrate. Since methods 1-5 are intended for use with semi-infinite media, the layer must be thick enough to qualify in this respect. Generally, as the thickness of a vacuum-deposited layer increases, so does its surface roughness. If the layer is not too thick, the error introduced by scattering due to the surface roughness will be small compared to errors from other causes. If, however, the thickness is too small, radiation will penetrate to the substrate/layer boundary, be reflected back to the layer/vacuum boundary, and interfere with the radiation reflected at the layer/vacuum boundary. If this interference is significant, the measured reflectances will not be representative of the reflectances of a semi-infinite medium and the reflectance methods 1-5 will not be useful. Hunter and Hass [13] have investigated the effects of interference on method 1 and found that the thickness required to reduced errors in n and k to approximately 1~o depends on the n, k values of the layer. Their Table I shows that for n = k = 0.3, the geometrical thickness of the layer must be approximately one wavelength, and as n and k increase, the required thickness decreases. At n - 2.3 and k = 3.3, for example, a geometrical thickness of approximately 0.1 wavelength is required. The effect of layer thickness on methods 2 - 5 has not been investigated but is expected to be generally the same as for method 1. The general behavior of R-versus-4~ and isoreflectance curves indicates that for large k/n, the method is not very accurate. Since metals with large reflectance values in the visible and infrared usually have large k/n, reflectance methods appear not to be suitable for obtaining n and k of metals in these spectral regions.

4.

Measurement

of O p t i c a l C o n s t a n t s

in t h e V U V S p e c t r a l

79

Region

Critical-Angle Method There are wavelength ranges in the VUV in which the n of some substances is less than unity and k < 0.1. This occurs for A1, Mg, Si, for example [14], and the alkali metals [15] at wavelengths less than their critical wavelength. For such values of n and k, the R-versus-~b curves exhibit the behavior shown in Fig. 7 [16]. This figure shows curves for a substance with n - 0.707, corresponding to a critical angle 4)~ of 45 ~ and for different values of k. For k = 0, 4>c is well defined and sin 4)c = n. As k increases, however, ~b~ is no longer well defined because the R-versus-~b curves become rounded at angles larger than 4)~. The point of maximum slope of the curve, however, occurs quite close to 4~ and can be used as a good approximation to ~b~. Since only the angle of incidence at which the maximum slope occurs is required, the reflectance values need not be known. To get correct values of n, k must be known and

lOOi

,

'"

, k =0.0

90

80

7O ta.i o z

I-- 6 0

0 i.iJ _J b_

0.5

"'50 I--Z

"0 ' r i,i

40

30

20

IO

i

o

,'o

1

I

20

30 ANGLE

1

5'o

8'o

90

OF INCIDENCE

Fig. 7. Calculated R-versus-~b curves for n = 0.707, ~bc = 45 ~ and different values of k showing the behavior of the reflectance for nonpolarized light in the vicinity of the critical angle.

D

80

W.R. Hunter

a correction made. The figure shows that as k increases, the correction required to obtain the true n also increases. Corrections must also be made for the state of polarization of the incident radiation. These corrections decrease with increasing n, becoming small for n =0.7. Hunter [16] has used this technique to measure n of evaporated layers of quasi-free electron metals such as A1, Mg, Si, and others at wavelengths less than their critical wavelengths. In order to correct the measured angle of maximum slope for the effect of k, he used extinction coefficients obtained from transmittance measurements through thin, unbacked films of the metal under investigation. The presence of the oxide layer that forms readily on the chemically active metals can shift the position of maximum slope. Hunter found that the natural oxide thickness for A1 (30-40 A) tends to shift the i.o 0.8 r r)

0.6

"'=;c ~

,

0.4 0.2 0.4 o

L (145A)

o

2,a(17OA) 0.I

[ 9

,

9

.

~:'II, ~TOMBOULIAN 8 B E D O 9

O'OI

I~~--t~$~.L*

'~. ':

I

I,.,-I.~1"%"

9

"

j ~9

. . > 7 o ~ ~ ~

t

.,/ o-

Y/_

~

jILAVILLA

a MENDLOWITZ'

.~/

J~, /

0 ;

200

~,

" ~

~ I I

!i . . ' oe . Z ~ , ; 9 - 99

.~.., ','j~s' !

o.ool I00

~ 1O

O O

. ~

J J 300

DRUDE THEORY (k c=837/~, r =1.1 xlO "lhsec)

400 500 WAVELENGTH (/~)

600

700

Fig. 8. The curve at the top shows n values for A1 obtained using the critical angle method. The curve at the bottom shows the extinction coefficient of A1 deduced from transmittance measurements. The scatter of points indicates that the critical angle method gives fairly accurate values for n, but that the transmittance method for finding k can give quite variable results.

800

4. Measurement of Optical Constants in the VUV Spectral Region

81

maximum slope to larger angles, especially for small values of n, but that the magnitude of the shift is small. For this method the layer thickness is also important because interference effects can shift the position of maximum slope. Usually, if the thickness exceeds 2000 A, interference effects are negligible. This method is also advantageous because of its relative insensitiveness to surface roughness frequently encountered for layer thicknesses in excess of 2000-3000 A. Whang et al. [ 17] have extended the method to obtain k from the maximum slope of the R-versus-~b curve. Because the value of the slope is required when using their method, actual reflectance values must be measured. In addition, the polarization of the incident radiation must be known. They estimate that the critical-angle method, in spectral regions in which it can be used, gives more accurate results than using method 1. Hunter [ 18] found that the scatter of data points is quite small when using this method, as illustrated by the top part of Fig. 8.

Measurement of Extinction Coefficients

The extinction coefficient can be determined by measuring the transmittance of a known thickness of material and calculating k using Lambert's law after correcting the transmittance measurements for reflection losses at the surfaces. If the light is not coherent, these corrections involve only reflected intensities, and if two different thicknesses of the material are used, the corrections usually can be eliminated. Lambert's law relates transmittance T, thickness t, and k through the equation T = I / I o -- e-47rkt/2.

For two thicknesses, k is obtained from k = 2 ln(T1/Tz)/[4rc(t 2

-

tl)],

where 2 is the wavelength in the same units as t. If the film has surface layers, such as oxide layers, they are frequently much more absorbing than the material to be measured. If the oxide layers are always the same thickness, however, their absorptance does not affect the ratio T 1 / T 2 and their reflectance values are independent of the test-material film thickness for all practical film thicknesses. Therefore, transmittance values obtained using two different test-material thicknesses should give reasonable values for k. This method for determining k can be applied to thin, unbacked films or thin films on a transparent substrate if the value of k is small enough so that the transmitted signal has a reasonable signal/noise ratio. Hass et al. [19]

E

82

W.R. Hunter

obtained k values for unbacked films of A1 at 736 and 584 ~, and Hunter [18, 20] has made extensive measurements on unbacked films of A1 and other materials. Despite the apparent simplicity of the method, there are two possible sources of error: (1) anomalous dependence of transmittance on thickness and wavelength caused by interference effects and (2) methods of preparation of the sample. For most materials in the VUV, the absorptance is so large that transmittances can only be obtained through thin films a few thousand angstroms thick. The coherence length of the light is usually larger than the film thickness, so interference effects will be present, and the simple intensity correction for reflection losses may not be valid. Figure 9 shows the calculated transmittance of an A1 film as a function of the A1 thickness with a 40-A-thick oxide coating on either surface. The k value used for A1 is 0.0233. As the A1 film thickness increases, the interference effects decrease in magnitude. At a thickness of 2000 A the amplitude of the interference effect is about 10% of the transmittance, and at 3000 ,& about 4?/0. By choosing thicknesses corresponding to interference maxima and minima, k values as large as 0.0698 can be obtained, and even negative k values

I

I00

I

I

I

I

I

l

I

I

1

W c_) Z

~p-

\

IO

Or) Z n~

I

I

I

i

",,.. \

I

0

7000 THICKNESS (A)

Fig. 9. Calculated transmittance versus thickness at 584/~, and at normal incidence, for an unbacked A1 film with 40 ,& of oxide on both surfaces. The curve shows how interference effects alter the transmittance as the A1 thickness changes.

4. Measurement of Optical Constants in the VUV Spectral Region

83

can be calculated. Calculated from transmittance values at thicknesses of 6000 and 7000 N, k is 0.0216, about a 7?/o error, and for thicknesses of 9000 and 10,000 ~, not shown in Fig. 9, a value of 0.0299 was obtained, an error of 1.77/o. Thus, in principle, more accurate values of k are obtained as the thickness is increased. In practice, the transmittance values become too small for accurate measurement, so thinner films must be used. Although interference effects are always present in the transmittance values, the experimental error in measuring the transmittance can mask the error caused by interference. For example, if the experimental error is + 1 of full scale, a transmittance of 16.5?/o (t = 2000 ~) can be in error by _+6~, but the error introduced by interference would be 1.6~ (10~ of the transmittance value). Thus, for this example, the error in transmittance caused by interference would contribute little to the error in determining k for film thicknesses in excess of 2000 ~. The transmittance of a thin film can also show interference effects as a function of wavelength. An example is given in Fig. 10 [21] which shows the measured transmittance of an unbacked film of A1 800 A thick as a function of wavelength. Also shown for comparison are the calculated transmittances of an A1 film of the same thickness with and without oxide layers 40 ~ thick on either surface. The oxide is much more strongly absorbing than the A1, which decreases the transmittance of the oxidized film over that of the nonoxidized film, and its presence accentuates the interference effects. If interference-modulated transmittance spectra are used to provide transmittance values for the equation for k, k will also have a wavelength-dependent modulation. Reduction of this modulation is achieved by using thick films. The transmittance method for finding k using thin films is perhaps more susceptible to errors arising from the method of preparation of the sample than from oscillatory terms in the transmittance equation or interferencemodulated transmittance spectra. Unbacked metal films are made most simply by evaporating the metal onto a water-soluble layer [22]. Both the act of removal of the evaporated film from the substrate and the evaporation conditions under which it is made can strongly affect its optical properties. After the layer is dissolved, the films are allowed to float free on the water surface until picked up on a suitable frame for transmittance measurements. No two exposures to water will be alike; therefore, there may be considerable variability in the thicknesses of the oxide layers on either side of the metal film and also in k deduced from transmittance measurements made on such films. Figure 8 [18] shows measurements of k for A1. The scatter of points is quite large, primarily because of differences in the thicknesses of the oxide layers on either side of a metal film and because of differences in thicknesses of the oxide films on the different samples. Some of the scatter, however, is caused by interference effects of the type shown in Figs. 9 and 10. Evaporation conditions are most apt to affect chemically active metals, such as A1, which adsorb oxygen readily. The amount of residual oxygen

84

W.R. Hunter I00 [.... "---..__._.____ :

t ~

-

t

q

W Z

I-. I-. if)

z

,~10

? I Ioo

zoo

3o0

4oo

5o0

6oo

700

800

WAVELENGTH (/~) Fig. 10. The measured transmittance of an unbacked A1 film 800 • thick as a function of wavelength. Included for comparison are calculated curves for A1 with and without 40-A oxide layers on both surfaces. Interference effects cause the fluctuations in transmittance that are accentuated by the presence of the oxide layers.

captured in the film during evaporation must be reduced as much as possible. This can be done by evaporating in an ultrahigh-vacuum evaporator, or by doing a very fast evaporation in a conventional evaporator. If the residual oxygen is not removed or reduced sufficiently, A1 films tend to have transmittances almost independent of thickness.* A more specialized technique exists for measuring k of a material, if it is small (k < 0.1) and if n is known. At a given wavelength the reflectance at near-normal incidence of a substrate/layer combination, wherein the material to be measured forms the layer, is determined as a function of layer thickness. t D. N. Steele, Luxel Corporation, 515 Tucker Avenue, Friday Harbor, Washington 98250, and C. Kunz, DESY HASYLAB, Notkestieg 1, Hamburg 52, Federal Republic of Germany; private communications.

4. Measurement of Optical Constants in the VUV Spectral Region

85

The reflectance of the combination is then calculated as a function of layer thickness, with different k values, until a best fit to the experimental points is found. For this technique n and k of the substrate must be known. Canfield et al. [23] have used this method to determine k of MgF2 layers on A1 at 1216 A, and Cox et al. [24] have used it to determine k of LiF layers on A1 at 1026 A.

REFLECTANCE METHOD FOR TWO MEDIA

In principle, reflectance methods can be used to measure n and k of either component of a layer-substrate combination. One must know n and k of either layer or substrate, the layer thickness, and the layer-substrate transition region must be small compared to the wavelength used to make the measurement. Using such measurements, one could determine the optical constants of the natural oxide that forms on aluminum or of iron or cobalt films that cannot be made opaque because of their large internal strains. Juencker [10] has discussed using method 1 for substrate-layers but did not discuss the conditions under which the method is valid. It is difficult to generalize for two media because of the large number of parameters involved, but there are some avenues of approach as well as some problems that will be discussed in the following paragraphs. In contrast with single-medium reflectance methods, the two-medium reflectance method requires interference between the light reflected from the vacuum-layer and layer-substrate interfaces. Therefore, the layer must be thin enough for this interference to take place. The amount of light available for interference depends on the optical constants of both the layer and substrate, on the layer thickness, and on the angle of incidence. An example will be given to illustrate the method. Assume a substrate with N = 0.4 + i0.1 and a layer with N - 2.0 + i0.1. The upper left panel of Fig. 11 shows how the reflectance of the substratelayer varies with layer thickness. Choosing the first half-wave thickness, 0.13t/2, which may not be the optimum thickness, one calculates the Rversus-~b curves shown in 1lb. Then, isoreflectance curves are plotted. Studies showed that for this particular layer-substrate combination, the parallel component gave the best isoreflectance curves for both layer and substrate. They are shown in 1l c and d, respectively. Note that if only the large angles of incidence are used, there appear to be two solutions. This ambiguity can be resolved by using some reflectance measurements at smaller angles of incidence or by using a second, different thickness. In the latter case the correct intersection remains unmoved, but the spurious intersection shifts to a new location in the n, k plane.

III

86

W.R. Hunter

,oo

,l 2~176 . . . . I

. . . .

II#l#lllll[#l#lllllll

I

'

'~176 I

. . . . . .

I1[[

"~

0.4 -I- iO.I W

n.,

n~

0

'

0

0.5 ~/X

O,0

1.0

90 ANGLE OF INCIDENCE (deg)

(o) 0.4

,

,

,

(b) ,

0.4

4

!

i

I

I

,

85

4O 0

1.7

2.3

0 0.1

I

7

~

r ~

0.7

n

n

(c)

(d)

Fig. 11. Illustration of the procedure Ior obtaining isoreflectance curves for determining n and k using two media. (a) Reflectance versus thickness is shown at normal incidence; choosing t/2 = 0.13 (arrow), (b) R-versus-g5 curves are calculated; isoreflectance curves are obtained (c) for the layer assuming that the substrate is known and (d) for the substrate assuming the layer is known.

There are a number of problems associated with the two-medium reflectance method. First, there is the possibility of interdiffusion between substrate and layer, giving rise to a transition-region dimension comparable to, or greater than, the 'wavelength. If glass or fused silica is used for the substrate, the probability of interdiffusion is remote. There may be situations, however, when a substrate with a large reflectance adds to the contrast of the interference effects; for example, the effective substrate could be an opaque layer of A1 on glass. Unoxidized A1 interdiffuses readily with some other metals, notably Au [25]. Thus, some advance research must be done to ascertain the compatibility of the substrate and layer vis-&-vis interdiffusion. Another problem is nonuniformity of the layer. The two-medium reflectance method tacitly assumes that the layer has a constant n and k as a function of distance into the layer. Such an assumption is not necessarily true. Reflectance measurements made on aged A1 + MgF2 mirrors at 1216

4. Measurement of Optical Constants in the VUV Spectral Region

87

with polarized radiation indicate that n or k or both may not be constant throughout the layer. Magnesium fluoride is known to adsorb water vapor which can modify the optical constants of the MgF2 layer. Fabre et al. [26] have devised a method for measuring n and k of slightly absorbing films in the VUV and have applied it to a study of M g F 2 films [27]. The film is vacuum deposited onto a suitable substrate (glass) such that its thickness varies along the long dimension of the coated area. After the actual thickness and its variation have been determined, the reflectance is measured as a function of thickness by movivg the substrate/film uniformly past the radiation beam. The resultant reflectance curve resembles the curve shown in Fig. 1 l a. Using the enveloping curves for the reflectance maxima and minima, and the position of these extremes, Fabre et al. were able to calculate values for n and k. Daude et al. [28] have used a technique similar to that employed by Canfield et al. and Cox et al. to determine both n and k of M g F 2 films on A1 but that they consider applicable to any thin film with small absorptance. They measure the reflectance of the substrate/film combination as a function of film thickness and fit a curve to the measurements by calculating the reflectance as a function of layer thickness while varying n and k to achieve best fit.

REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

C. Boeckner, J. Opt. Soc. Am. 19, 7 (1919). R. Tousey, J. Opt. Soc. Am. 29, 235 (1939). J. R. Collins and R. O. Bock, Rev. Sci. Inst. 14, 135 (1943). I. Simon, J. Opt. Soc. Am. 41, 336 (1951). D. G. Avery, Proc. Phys. Soc. (London) 65, 425 (1952). S. P. F. Humphreys-Owen, Proc. Phys. Soc. (London) 77, 949 (1961). T. Sasaki and K. Ishiguro, Japan J. Appl. Phys. 2, 289 (1963). A. P. Prishivalko, "Otrazhenie Sveta ot Poglishchaoiuchikh Sred" ["Reflection of Light from Absorbing Media"], Izdate'lstvo, Akademii Nauk, Beloruskoi SSR Minsk, 1963. A Vasicek, "Tables of Determination of Optical Constants from the Intensities of Reflected Light," Nakladatelstvi Ceskoslovenske Akademie Vec, Prague, 1964. D. W. Juenker, J. Opt. Soc. Am. 55, 295 (1965). W. R. Hunter, J. Opt. Soc. Am. 55, 1197 (1965). G. R. Field and E. Murphy, Appl. Opt. 10, 1402 (1971). W. R. Hunter and G. Hass, J. Opt. Soc. Am. 64, 429 (1974). D. Pines, Rev. Mod. Phys. 28, 184 (1956). C. Zener, Nature, 132, 968 (1933). W. R. Hunter, J. Opt. Soc. Am. 54, 15 (1964). U. S. Whang, R. N. Hamm, E. T. Arakawa, and M. W. Williams, J. Opt. Soc. Am. 63, 305 (1973). W. R. Hunter, J. de Physique 25, 154 (1964). G. Hass, W. R. Hunter, and R. Tousey, J. Opt. Soc. Am. 47, 120A (1957). W. R. Hunter, "Proceedings of the International Colloquium on Optical Properties and Electronic Structure of Metals and Alloys" (F. Ab616s, ed.), p. 136, North-Holland Publ., Amsterdam, 1966.

88

W.R. Hunter

21. W. R. Hunter and R. Tousey, J. de Physique 25, 148 (1964). 22. W. R. Hunter, in "Physics of Thin Films" (G. Hass, M. Francombe, and R. W. Hoffman, eds.), p. 43, Academic Press, New York, 1973. 23. L. R. Canfield, G. Hass, and J. E. Waylonis, Appl. Opt. 5, 45 (1966). 24. J. T. Cox, G. Hass, and Ji E. Waylonis, Appl Opt. 7, 1535 (1968). 25. W. R. Hunter, T. L. Mikes, and G. Hass, Appl. Opt. 11, 1594 (1972). 26. D. Fabre, J. Romand, and B. Vodar, J. Phys. Radium 21, 263 (1960). 27. D. Fabre and J. Romand, J. Phys. Radium 22, 324 (1961). 28. A. Daude, A. Savary, A. Seignac, and S. Robin, Opt. Acta 20, 353 (1973).

Chapter 5 The Accurate Determination of Optical Properties

by Ellipsometry D. E. ASPNES Bell Communications Research, Inc. Murray Hill, New Jersey

I. I1. II1. IV. V. VI. VII.

Reflection Techniques; Background and Overview Measurement Configurations Accurate Determination of Optical Properties: Overlayer Effects Living with Overlayers Eliminating Overlayers Bulk and Thin-Film Effects; Effective-Medium Theory Conclusion References

89 92 96 99 102 104 108 110

REFLECTION TECHNIQUES; BACKGROUND AND OVERVIEW

In this section, we give some background in electromagnetic theory to aid in understanding the different techniques used to determine optical properties of materials. An incident plane wave propagating in the z-direction of a local orthogonal coordinate system can be represented as [1, 2] Ei(r, t ) = Re[(2Ex +

.gEy)eik=-i'~

(1)

where Ex and Ey are the complex field coefficients describing the amplitude and phase dependences of the projections of Ei(r, t) along the x and y axes. If the electromagnetic wave is reflected by a smooth surface, the outgoing wave can be represented in the absence of anisotropic effects in another local coordinate system as Er(r, t ) =

Re[(2rpEx + ;rsey)eik='-i~

(2) 89

HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright (~) 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-544420-6

I

90

D.E. Aspnes

where in both local coordinate systems the x axes are in the plane of incidence, the y axis is perpendicular to the plane of incidence, and the z axes define the plane of incidence. In this approximation the effect of the surface is described by two coefficients rp and rs. These complex reflectances describe the action of the sample on the field components parallel (p) and perpendicular (s = German senkrecht) to the plane of incidence. Thus, four parameters~two amplitudes and two p h a s e s ~ a r e necessary to describe completely the incoming wave and four more to describe the sample. The properties of the sample therefore are obtainable if the properties of both incident and reflected waves are known. Reflectometry and ellipsometry are two techniques for obtaining this information. Neither extracts all the available data, but each is an incomplete form of a still more general technique, polarimetry [3]. To appreciate the different types of information obtained in reflectometry, ellipsometry, and polarimetry, it is useful to recast the four parameters describing the plane waves of Eqs. (I) and (2) into four new parameters by means of the concept of the polarization ellipse. The polarization ellipse is the locus traced out in time by the real (observable) electric field vector E(r, t) at any fixed plane z = z~. The polarization ellipse has two attributes that can be termed size and shape. The size E = ([Ex[ 2 -k- [Eyl2)1/2 is a scalar whose square is proportional to the power flow or intensity I. The intensity is the only quantity of interest in a reflectance or transmittance measurement and one of the quantities of interest in a polarimetric measurement. The shape is an intensity-independent quantity that clearly requires two parameters to specify, such as the minor/major axis ratio and the azimuth angle of the major axis of the polarization ellipse. One convenient representation of the shape is the polarization state Z = Ex/Er [2]. Since E x and E r are complex, X is also complex and therefore contains the required two parameters. For example, if E~ and E r a r e in phase, then g is real, both field components are strictly proportional at all times, and the light is linearly polarized with an azimuth angle ~ = tan-~ X. If E~ and E r a r e 90 ~ out of phase, then Z is purely imaginary and one field component is at a maximum while the other is at zero and vice versa; if, in addition, IExl = IE, I, then Z = + i, and the light is circularly polarized. The polarization state is the only quantity of interest in ellipsometry and another of the quantities of interest in polarimetry. In addition, polarimetry is also concerned with depolarized or unpolarized light, the superposition of electromagnetic waves in uncorrelated states of polarization. Given either intensities or polarization states for the incident and reflected beams, the sample properties can be calculated by taking appropriate ratios. In reflectometry, we distinguish between p- and s-polarized light. By Eqs. (1) and (2), the intensity ratios are (IJIi)p = r

E l /Ig 12

=

=

(3a)

5. Determination of Optical Properties by Ellipsometry

(I,./I,)~- Ir~E, JZ/lE, IZ = prs]z-- R~.

91

(3b)

The reflectances Rp and Rs are the absolute squares of the respective complex reflectance coefficients. At normal incidence both polarizations are equivalent and R p - - R s. In ellipsometry, the ratio of polarization states yields a somewhat different perspective of the sample. By Eqs. (1) and (2)

Z~/Zi = (rpE~/rsEr)/(Ex/Er) = rp/rs = p = (tan ~)e ia.

(4a) (4b)

The complex reflectance ratio p is often expressed as an amplitude (tan ~) and a phase A. Depolarization or cross-polarization effects determined by polarimetry are important in more complex sample configurations and in the presence of scattering [3], topics that we shall not discuss here. We can now make some general remarks about the relative advantages and disadvantages of reflectometry, ellipsometry, and polarimetry. For isotropic samples ellipsometry is strictly a nonnormal-incidence technique, while reflectometry and polarimetry can be performed at either normal or nonnormal incidence. Reflectometry deals with intensities and is therefore a power measurement, while ellipsometry deals with intensity-independent complex quantities and is therefore more nearly analogous to an impedance measurement. Because more complicated quantities are involved, an ellipsometer is necessarily more complicated than a reflectometer and a polarimeter is more complicated than an ellipsometer. The most serious drawback of ellipsometers and polarimeters is that transmission through optical elements (e.g., polarizers) is generally required. Therefore, such measurments are limited to those wavelength ranges in which good-quality transmitting elements are available. Leaving polarimetry as a topic not yet sufficiently developed for detailed discussion, we now consider the relative advantages and disadvantages of reflectometry and ellipsometry. Ellipsometry is unquestionably the more powerful for a number of reasons. First, two parameters instead of one are independently determined in any single-measurement operation. Consequently, both real and imaginary parts of the complex dielectric function e of a homogeneous material can be obtained directly on a wavelength-bywavelength basis without having to resort to multiple measurements or to Kramers-Kronig analysis. Two independent parameters also place tighter constraints on models representing more complicated, e.g., laminar microstructures. While two independent parameters Rs and Rp are also available in nonnormal-incidence reflectance measurements, these parameters can be obtained separately only after additional adjustments of system components. Second, ellipsometric measurements are relatively insensitive to intensity fluctuations of the source, temperature drifts of electronic components, and macroscopic roughness. Macroscopic roughness causes light loss by scattering the incident radiation out of the field of the instrument, which can be

92

D.E. Aspnes

a serious problem in reflectometry but not in ellipsometry, for which absolute intensity measurements are not required. Third, accurate reflectometric measurements are difficult, in general requiring double-beam methods. In contrast, ellipsometry is intrinsically a double-beam method in which one polarization component serves as amplitude and phase reference for the other. Finally, p explicitly contains phase information that makes ellipsometry generally more sensitive to surface conditions. Insensitivity to surface conditions is often considered to be an advantage of reflectometry, but this is not correct if the objective is to obtain accurate values of the intrinsic dielectric responses of bulk materials. Because small differences in R can result in large differences in e, reflectances in general must be measured more accurately. Surface artifacts affect ellipsometric results at the measurement level at which they can be identified and often corrected on the spot, whereas they affect reflectometric results at the data-reduction level at which it may be too late to do anything about them.

II

MEASUREMENT CONFIGURATIONS

Despite the intrinsic advantages of ellipsometry, reflectometry historically has been a far more significant factor in advancing our general knowledge of the optical properties of materials. Reflectometers are well suited for spectral measurements, while the classical null ellipsometer is by nature a single-wavelength instrument. The development of photometric ellipsometers has now changed the picture, although spectroscopic instruments must still be built by the user. For this reason, we review in this section some configurations of interest (as of July 1982). More detailed discussions have been given by Azzam and Bashara [-2], Hauge [3], Muller [4], and Rzhanov and Svitashev [5]. A schematic diagram of the general optical configuration for reflectometry, ellipsometry, or polarimetry is shown in Fig. 1. The source provides collimated

SOURCE

DETECTOR

ENTRANCE ,,//~PTICS

Ii,

EXIT OPTIC~ ~ , * ~ k / ~

Xi " % 1 / I~ X~ F//////A SAMPLE

Fig. 1. General configuration for determining optical properties by reflectance techniques.

5. Determination of Optical Properties by Ellipsometry

93

monochromatic light. The entrance and exit optics establish and analyze, respectively, the intensities, and/or polarization states. The detector may determine absolute or relative intensities or their time dependences. The simplest configuration is that of a normal-incidence reflectometer. Here, there are no entrance or exit optics. The detector may be moved to intercept either entrance or exit beam or a rotating light pipe may be used. The movable-detector approach is particularly well suited for synchrotron radiation sources because they provide light in spectral ranges in which transmission elements are not available. Because reflectometry is discussed in detail elsewhere in this volume [6], we now restrict the discussion to ellipsometry. As already implied, the most important instrument until recently was the classical null ellipsometer. Here, the entrance optics consist of a polarizer and compensator, or quarter-wave plate, and the exit optics consist of a second polarizer, or analyzer. The polarizer-compensator combination functions as a general elliptical polarizer. In normal operation the ellipticity is adjusted so that it is exactly cancelled by reflection. The reflected beam can then be blocked by properly adjusting the azimuth of the analyzer. The null ellipsometer is the optical analog of the ac impedance bridge. The twin advantages of the null ellipsometer are mechanical simplicity and high accuracy. If the light source is sufficiently intense, the accuracy limit is determined solely by the mechanical components and their alignment and is not affected by detector nonlinearity. However, these advantages are relatively minor compared to its disadvantages. The null condition cannot be achieved by independent adjustments because the polarizer and analyzer azimuth settings interact. The intensity near null depends quadratically on both azimuths, so the null is not well defined. While Faraday-cell modulation of the plane of polarization can be used to improve null sensitively [-7, 8], the detector still operates at minimum light levels so the rate of information flow is small and dark-current and shot noise may be significant. The traditional approach toward dealing with these problems has been to increase the source intensity, which means that in practice most null ellipsometric measurements have been performed at the Hg green line 2 = 5461 A or more recently at the HeNe laser line 2 = 6328 A. Consequently, the use of a compensator, a component with a relatively narrow useful wavelength range, has not proved to be the significant handicap. The modern era in ellipsometry can essentially be traced back to the automatic photometric ellipsometer described by Cahan and Spanier in 1969 [9]. Photometric instruments operate on entirely different principles than do null ellipsometers. One of the components is a modulator that is used to produce a time dependence on the transmitted intensity that is later decoded for the properties of interest. Because most of the light is transmitted, photometric systems are compatible with the relatively weak continuum sources suitable for infrared-visible-near-UV spectroscopy. In the Cahan-Spanier design

94

D.E. Aspnes

[9], the modulator consisted simply of the final analyzer prism, which was rotated mechanically to produce a sinusoidal variation of the transmitted intensity (see the next paragraph). In addition, the data were obtained automatically, with operator feedback not required. A number of configurations [2-5] have also been developed involving photoelastic modulators [10-13] and Pockels cells [14] as well as mechanically rotating elements [15-20] and featuring also digital encoding and analysis of the transmitted intensity. The more elaborate designs [3, 17, 18] also can distinguish between polarized and unpolarized light. Photometric ellipsometers now totally dominate the field as far as research and spectroscopic applications are concerned, with null ellipsometers now mainly relegated to routine measurements on simple systems, such as determining oxide thicknesses on semiconductor wafers. Only a few of the many photometric designs have been considered sufficiently practical to warrant the time and effort of actual development. The simplest of these are the rotating-analyzer ellipsometer (RAE) [15, 16] and its complement, the rotating-polarizer ellipsometer (RPE) [20]. In these configurations, the entrance and exit optics consist of single polarizing elements, the polarizer and analyzer, respectively. As the names imply, one of these is rotated mechanically while the other is held fixed. The advantages are simplicity and the absence of a compensator; the only wavelengthdependent element is the sample itself. A second advantage is that the transmitted intensity has a very simple Fourier spectrum consisting of a single ac component on a dc background. Disadvantages include the requirement of either a rigorously unpolarized source for an RPE or a rigorously polarization-insensitive detector for an RAE. In addition, as with all photometric systems, the detector must be rigorously linear. Finally, these configurations cannot distinguish between circularly polarized or unpolarized light or the handedness of circularly polarized light, and a significant loss of accuracy occurs for e2 if e2 ~ 0. Because photodiodes and end-on photomultipliers are relatively polarization insensitive while equivalently unpolarized sources do not exist, the RAE is preferred over the RPE unless the samples or their environments are themselves sources of significant amounts of radiation. Stray light suppression is much better in an RPE because the monochromator can be placed between sample and detector. With respect to the other limitations, depolarization is not usually a factor unless sample quality is poor. Handedness can be determined from spectral dependences. The inability to measure ~2 accurately for ~2 ~ 0 can be circumvented by using a compensator, but this introduces another wavelength-dependent element into the system. A combination of a rotating analyzer followed by a fixed analyzer has been found useful in the infrared [19], where the principal objective is to determine threshold energies of line structures rather than to determine accurate values of dielectric functions.

5. Determination of Optical Properties by Ellipsometry

95

If a compensator must be used, then the rotating-compensator ellipsometer (RCE) [18] is a more practical alternative. In this configuration, the entrance optics consist of a fixed polarizer and the exit optics of a rotating compensator followed by a fixed analyzer. The fixed polarizers at either end completely eliminate polarization sensitivity artifacts due to source or detector. Moreover, the RCE is capable of measuring handedness and can also distinguish between circularly polarized and unpolarized light [18]. The only reason that RCEs have not completely replaced RAEs and RPEs is that suitable achromatic compensators do not yet exist. With currently available elements, the usable wavelength range of an RCE is about a factor of 2. For fixed-wavelength operation the RCE is superior to an RAE, an RPE, or a null ellipsometer. Photometric systems that rely on mechanically rotating components are relatively slow, with complete optical cycles requiring about 10 msec. Much higher speeds can be obtained with photoelastic modulators or with Pockels cells. Ellipsometers using these elements are identical to the null ellipsometer except that the compensator is replaced with the modulating element. Modulation frequencies are of the order of 100 kHz for photoelastic units [10-13] and upwards of 1 MHz for Pockels cells [ 14]. Disadvantages of birefringentmodulation systems include the introduction of another degree of freedom, the modulation amplitude, which in principle should also be measured on a wavelength-by-wavelength basis and should also be correlated with the wavelength for optimum sensitivity. In principle, the modulation amplitude can be obtained from an analysis of the higher-frequency harmonics. Because the operational frequency is much higher, digital analysis may be impractical and several lock-in amplifiers may be required. This introduces the usual calibration and gain uncertainties associated with several independent demodulators. Higher speeds also require higher source intensities. Other configurations have also been developed, but those already discussed have accounted for most published work and appear entirely adequate when used within their limitations. Future developments will take advantage of the capabilities of photometric configurations to measure absolute intensities as well as polarization states and therefore to determine reflectances as well as complex reflectance ratios. At present, this capability has only been exploited differentially in thin-film measurements in which changes in the reflectance, rather than the reflectance itself, are measured [21]. The development of a truly achromatic compensator would be significant because an RCE using this element could perform spectroscopic measurements over a wide energy range without the limitations of an RAE or RPE. But for most purposes, at least in the near IR-visible-near UV, instrumentation is a solved problem. The most significant factor responsible for differences among e spectra is now sample preparation and maintenance, not measurement method or instrumentation. We discuss sample effects in the next section.

96 III

D.E. Aspnes

ACCURATE DETERMINATION OF OPTICAL PROPERTIES: OVERLAYER EFFECTS

The accurate determination of the intrinsic optical properties of a material in its pure bulk form is a double challenge, requiring not only accurate instrumentation but also well-characterized samples. Indeed, all reflection techniques for determining e are model dependent, and the key to success is to prepare and maintain the sample in such a way that it can be described by fewer variables than independent experimental quantities. This is not a trivial matter: visible-near-UV optical properties are so strongly affected by microstructure and surface conditions that optical measurements are a convenient means of characterizing both microstructure [22] and surface quality [23]. As a particular case, the optical properties of evaporated metal films may not be accurately described by intrinsic bulk properties of the constituent material because the samples generally contain a significant volume fraction of voids [24]. We shall return to bulk problems after first discussing difficulties associated with imperfect surfaces. It is useful to begin with the ideal case, in which the substrate is homogeneous and isotropic and the surface is mathematically sharp. Then e and the normal incidence reflectance R are given by the two-phase (substrateambient) model [2]: 8/8a

=

sin 2 q~ + sin 2 q~ tan 2 ~b [(1 - p)/(1 +/9)] 2,

R = [ ( n - na)/(n + na)[2,

(5a) (5b)

where ~b is the angle of incidence and n - 81/2 and n ~ - e1/2 are the complex and ordinary indices of refraction of substrate and ambient, respectively. (We assume that the ambient is transparent.) Although 8 and p may be transformed interchangeably by Eq. (5a), there are no corresponding expressions for 8 and R, R~, and Rp. The analogous equations for these quantities must be solved numerically for 8 [25]. For real samples, the assumption of a mathematically sharp interface leads to immediate practical difficulties. Transition-region widths are finite even if the lattice termination is atomically perfect. Under typical laboratory conditions, surfaces are covered with oxides or adsorbed contaminants. Surfaces are usually microscopically rough, often as a result of cleaning processes used to remove the other overlayers. If such effects are not taken into account, the optical parameters calculated from Eqs. (5) may be considerably in error [26]. However, in the analysis of ellipsometric data it is often useful to assume perfection anyway and to convert the measured quantity p into a derived quantity, the pseudodielectric function (8), by Eq. (5a). The pseudodielectric function necessarily is an average of the dielectric responses of substrate and overlayer, and the accuracy by which it actually represents e depends on the accuracy by which the sample configuration approximates

5. Determination of Optical Properties by Ellipsometry

97

that of the ideal two-phase model. The pseudodielectric function is useful simply because it expresses p in a form related to the fundamental quantity of interest. The complex reflectance ratio can always be recovered from (e) by inverting Eq. (5a). The importance of surface conditions can be appreciated by noting that penetration depths of light in opaque materials are typically 100-500 A in the near UV, whence a 1-A surface film can affect p, (~), or R as much as several percentage points. This can be shown quantitatively by expanding p, (e), and R to first order in d/2, where d is the film thickness. We have [26,27]

[ P ~P0

4rcidn,cos dp 1+

(~} -~ ~ + ~4hid

2

e(t-

~(t- %)(%- ea)

%)(%- ~a) (•

ea sin 2 ~)1/2;

~- t + (4nid/2)e 3/2", 1+

(6a)

(6b)

t o ( t - ea)

,~

R~Ro

].

to(t-- ta)(tcot 2 ~b-~;a) '

o

A

(6c)

Im

.

ta

(6d)

In Eqs. (6), to is the effective dielectric function of the overlayer and Po and Ro are the corresponding values for the ideal system. The quantity to may be anisotropic because eo can be defined in terms of spatial averages that are different for s- and p-polarized light [28]. However, effects due to anisotropy are usually small, and we ignore them here. Equation (6c) follows from (6b) in the limit that It] >> leo[ >> lea[. Note that eo has dropped out of Eq. (6c) completely, so that relative changes in d can be investigated through (e) without having to identify the nature of the overlayer materials [29]. Effects of oxide overlayers and microscopic roughness on (e) and R are shown in Figs. 2 and 3 for a typical semiconductor, GaAs. These calculations were done by using the exact three-phase (substrate-overlayer-ambient) model [-2] instead of Eqs. (6). The dielectric function of the oxide was assumed to be that of an electrochemically grown anodic film [30]. Microscopic roughness was modeled in the Bruggeman effective medium approximation [31] as a layer consisting of 60% substrate material and 40% voids, values typical of microscopically rough surface on semiconductors [29, 32]. The effect on (e) of a microscopically rough layer 14 A thick is identical to that of a 10-A oxide on the scale of Fig. 2. Figures 2 and 3 show that at higher energies (e) is about three times more sensitive than R to the presence of overlayers; the quantitative ratio of the relative sensitivities of R and ( t ) is seen by Eqs. (6c) and (6b) to be

~(~}/(~}

1~13/=

Im

'

ta

,

(7)

98

D.E. Aspnes ~ T

r

T

'T

T

r

El

I

~

r

! I

E2

2O

^

40

v

',L__J -10 2

4

5

5

6

E (ev) Fig. 2. Effect of a IO-A oxide on (e) for GaAs. The results for a 14-A-thick microscopically rough surface are identical to this scale. [ ~ , abrupt; . . . . , oxidized (d = 10/~).]

0.7

I

'

1

i

i

'

1

'

i

EZ 0.6

=

o.5-

_

-

t7

',

-

7/ 0.4

0.5

I 8

z

1

i.

3 E

[

1 4

I

1 5

i 6

(ev)

Fig. 3. As in Fig. 2 but for R. Note that the film thicknesses are larger than those in Fig. 2. .... , a b r u p t ; - - - - - , rough (d = 30 A); . . . . , oxidized (d-- 30 A).]

5. Determination of Optical Properties by Ellipsometry

99

about 89at the 4.78-eV E 2 peak in 82 for an oxide on GaAs. At low energies where e and eo are real, the first-order dependence of R on d vanishes. Figure 3 shows that the effect of microscopic roughness on R, as with (s), is also qualitatively similar to that of an oxide overlayer. The spectroscopic dependence of the effect of an overlayer on ( s ) can be understood from Eq. (6c). At low energies ~;3/2 is real, so overlayers affect only /(E2> Fig. 4. Configuration for calculating the absorptance of a sample as per Eq. (3).

III

140

P.A. Temple

where ~(z) and n(z) are the absorption coefficient and the real part of the index of refraction, respectively, of the sample at the position z and no is the index of the material outside the sample, which is normally air. Here ~(z) is related to the extinction coefficient by c~(z) = 4rck(z)/2. The relative energy density, which contains the incident beam normalization factor, is p(z). The purpose of wedged-film laser calorimetry is to determine ~(z) for the film and the film interfaces by using Eq. (3b). This equation relates the measured absorptance A and the calculable quantity p(z) to the material property c~(z). Equation (3a) makes it clear that the contribution that any region in the sample makes to sample heating is proportional to ~(z) in that region times the electric-field strength in that same region. In order to make use of Eq. (3b), one assumes that the bulk absorption coefficient of the film ~e is constant throughout the film and is independent of film thickness. We also assume that all of the air-film interface absorption takes place in a thin sheet of material of absorption coefficient ~af and thickness A,f, and the film-substrate interface absorption takes place in a thin sheet of thickness Afs, with absorption coefficient ~fs. Then the air-film and film-substrate interface absorption are characterized by the specific absorpt a n c e s aaf -- ~afAaf and afs -- ~fsAfs . Finally, in calculating p(z), which will be described in Section IV, we assume no absorptive attenuation of the illuminating beam in passing through the sample. In samples where A < 10-2, this approximation is certainly acceptable. Without this simplification, knowledge of ~(z) is necessary to calculate p(z), and the entire problem becomes more difficult. Under these approximations, the absorptance of a single-layer film on a transparent substrate can be written as* A = Pafaaf + /~fo~flf+ Pfsafs +/~Ao,

(4)

where i0f and/~s are the spatially averaged relative energy densities within the film and substrate and Ao is the absorptance measured on the uncoated substrate. Here Paf and pfs are the relative energy density at the air-film and film-substrate interfaces, respectively. As the film thickness is varied, aaf , ~fafs , and Ao remain constant and A, If, Paf, fif, Pfs, and i0s change. Of these A, Ao, and If are the experimentally measured absorptances and physical film thickness. And Paf, fif, Pfs, and/~ are calculated quantities. Values of Pae, iof, Pfs, and p~ for 2/4 and 2/2 thickness films are given in graphical form at the end of this chapter. By measuring the absorptances of an appropriate set of film thicknesses, we obtain a set of linearly independent equations that can be solved for ~f, aae, and afs, the three coefficients that characterize a single film on a transparent substrate. In order to use Eq. (4) to determine the bulk absorption of the film ~r, one measures the absorptance of a pair of film thicknesses that have the same * The last term in this expression has a further approximation, whichis discussed in Section V.

7. Thin-Film Absorptance Measurements Using Laser Calorimetry

141

relative power densities at the interfaces, in the bulk of the film, and in the substrate. Such a pair consists of any two films that differ by 2/2 in optical thickness. Differentiating Eq. (4) and using absorptances from this pair of film thicknesses, we obtain af = (At+ 4/2 -

A,)IPd~/2,

(5)

where pf is calculated for film thickness t and l;~/2 , the physical thickness of a film of 2 / 2 0 T . Like Eq. (2), two measurements are required; but in contrast to Eq. (2), this expression accounts for interface absorption and electric-field strength in the film and gives the bulk absorption coefficient of the thinfilm coating. A convenient sample configuration for this measurement is a sample-onehalf of which has been masked for part of the deposition. In this way, one side of a diameter has a coating of OT t while the other side has a coating of OT t + 2/2. While any thickness t will suffice, values of/~f for the 2/2, 2 pair (or the 2/4, 32/4 pair) are given here. When measuring the absorptances of these two coatings, the beam is aligned slightly off-center so that it passes through one or the other of the coating halves. The use of one sample eliminates substrate-to-substrate and deposition-to-deposition variations that are present when two substrates and two depositions are needed to obtain the t and t + 2/2 samples. With additional measurements on the bare substrate and on a 2 / 4 0 T film, we can determine aaf and afs. The relative power densities at both interfaces, within the film, and within the substrate all change in going from a 2 / 4 0 T film to a 2 / 2 0 T film. This fact is utilized in the determination of aaf and afs. By rewriting Eq. (4) for these two cases (unprimed for 2 / 2 0 T and primed for 2 / 4 0 T ) with all measured and/or calculated quantities on the right, including ~f, which is determined by using Eq. (5), we have aafPaf-~- afsPfs = [ A - iOfO~f/f-/~sAo] = [F], t

t

-t

t

aafPaf -k- af~Pfs-- [A'-- pf0~f/f- ffsao] = [F'],

(6a) (6b)

which have the solutions aaf =

[F]p~s- [F']pfs

,

(7a)

.

(7b)

PafPfs - - PafPfs

afs =

[ F ' ] p a f - [F]p'af PafPfs - - PafPfs

Figure 5 shows a convenient sample configuration for making this type of measurement. The maximum film thickness is ,~ 12 OT, allowing measurement at the four regions required for obtaining A o , A1/4, A1/2, and A1. A small-diameter laser beam is used with the sample tilted slightly. The required thickness is found by moving the sample until the transmittance minimum

Io

[~

DIAMETER SCANNED

Fig. 5. Wedged-film sample used to obtain film bulk and interface absorption for single-layer films. The film varies from zero to ~ 12 OT. (From Temple [12].)

I

I

I

I

I

I 0

I 0.25

I 0.50 THICKNESS, /am

I 0.75

1.0

14

12

10 o x klJ

8

z ,< F" 6 0 4

0

Fig. 6. The absorptance of an entrance surface film on As~Se 3 on a CaF 2 substrate measured at two wavelengths and at 2/4, 2/2, 32/4, and 12 OT. The two data points at 0.0-#m thickness are bare substrate absorptances Ao. The two lines have been drawn through 2/2 and 12 OT data points. The 2/4 and 32/4 data points have lower absorptances because of reduced Pfs for these thicknesses. (V, 2.87 #m; (3, 2.72/~m; for the upper curve e - 0.4cm -1, a,f ~ 0.0, and afs = 4.2 x 10-4; for the lower curve, e = 0.0 c m - 1, a,f ~ 0.0, and afs -- 2.3 x 10-4.) (From Temple [12].)

I

7. Thin-Film Absorptance Measurements Using Laser Calorimetry

143

or maximum is found. This type of sample is particularly useful when absorptances are to be measured at a variety of wavelengths, since precise 2/4, 2/2, and 12 OTs can be found for any wavelength. As in the case in which two film thicknesses are present on the same substrate, this type of sample eliminates substrate-to-substrate and deposition-to-deposition variation present when three separate depositions are made to obtain the 2/4, 2/2, and 2 0 T films. Figure 6, which is an example of this type of measurement [12, 13], shows the absorptance measured on the bare CaF 2 substrate (shown at 0.0 film thickness) and at 2/4, 2/2, 32/4, and 2 0 T positions of an AszSe 3 wedged film. Measurements were made at 2.72- and 2.87-#m wavelength by using a line-tuned HF cw gas laser. The substrate index is 1.42 and the film index is 2.6, which causes a 2 / 2 0 T film to be ~0.5 pm thick. The values of ef were calculated from the 2/2 and 2 0 T data (solid lines through data). The presence of water in the filmed substrate is evidenced by the higher absorptance as measured at 2.87 rather than at 2.72 #m. However, the bulk absorption coefficient of the film is quite low. The analysis of interface absorption indicates that nearly all the absorption is taking place at the film-substrate interface. Analysis of the bare substrate and 2 / 2 0 T data and Eq. (2) gives a fl of 14.60 cm-1, which is ~ 35 times greater than ef calculated by the wedgedfilm technique.

ELECTRIC-FIELD CONSIDERATIONS IN LASER CALORIMETRY

The technique and considerations in calculating p(z) for use in Eqs. (7a) and (7b) will now be described. The accurate determination of p(z) must take into consideration (1) the coherent interaction of a single beam with a dielectric interface and (2) the effect of the multiple (but incoherent) bounces the beam undergoes between the two faces of the sample. A convenient method of making the first calculation was first given by Leurgans [14] and was more recently detailed by DeBell [15]. As originally described, the technique uses the Smith chart, but it is easily adapted to a computer or even a hand calculator. Leurgans's method, which we will briefly describe, is restricted to normal-incidence plane waves and lossless films. While only single films are of concern here, the general multilayer description will be given. Figure 7 shows the layer-labeling system. The substrate is of index no, and the incident medium is of index rig. The layers are labeled 1 through k - 1 and are of physical thickness 11 to lk-1. A right-traveling, normally incident plane wave of wavelength 2 is represented by E+ = E +e i(ot-6), where 6 = 2rcnz/2. At a dielectric boundary, the

IV

144

P.A. Temple ,,/ ",/

/

\ n k ' , //

\

ni +i \ / n i

nk_ 1

,?

x

,; ",/ \

\

N /

/

9

9

9

/

\

N //

/

// '

\

/

\/

/ /

N

k

/ /

/

,, /

/

\ / ni_ 1 ,,/ \

/

x

/

X

/O

9

9

,, / \/

/ / / " ,// n 2 / n I NN / no / "/ / / ,, / / / \ / / /' ", / " /

\

',/ 9

\ // \

,/

\

,?

\

/

/ /

",

/

\

/

/

\

/

N

\ -Z \

/'

+Z ,

/

\

3

i+i

2

1

Fig. 7. The layer-labeling system for a thin-film stack.

partially reflected left-going field is given by ff~- = E + 16e i(~ +'~),

where the reflection coefficient/i is complex. Now, in general,

~ ( z ) - I~(z)l ei(2a++) within any given layer. In the lossless case, It~(z)l is constant within a layer and fi only changes in phase through the layer. To calculate fi across a boundary, we use the boundary condition requiring continuity of Eta. and Hta n. This is done by using the reduced admittance .9(z): A

A

.9(z) = H(z)/nE(z).

The boundary condition then requires that ;9i+l(Zb)ni +l = .gi(Zb)ni, where Zb is the position of the boundary between the ith and (i + 1)th layers. Finally, .9(z) and fi(z) are related by t~,(z) =

1 - ~,(z) 9 1 + ;~(z)'

~,(z) =

1 - ~,(z)

1 + ~,(z)

These expressions, along with the condition that there is no beam incident on the exit surface from the right, are sufficient to uniquely determine fi(z). The calculation begins within the substrate and proceeds from right tO left. The procedure is the following:

(1) Realize that Po = 0 and )30 = 1; (2) (3) (4) (5) (6) (7)

use )31(0) = ~ono/nx to cross the substrare-film (1) interface; use ~(0) - [-1 - )31(0)-]/[ 1 + )3~(0)] t6~find fi~(0); use/91(z)--I/~l(0)[e i2'~ to find ~l(z)in the first film; use )~(z~)= [1 - ~(z~)]/[1 + fil(z~)] and )~2(zl) = ; l ( z l ) n l / n 2 to find )32(zl); now proceed as at step (3) above to find fi2(z), and so on.

7. Thin-Film Absorptance Measurements Using Laser Calorimetry

145

In the work by Leurgans [14] and DeBell [15], the calculation was used to determine the reflectance R of the stack, which is given by R =/~k/3~'. Having found p(z) throughout the system by the method of Leurgans [14], we will now calculate the electric-field strengths throughout the layered system. At any point z in the ith layer, the net field is

E,(z) =/~+ (z) +/~? (z) = [ 1 4-/3,(z)]/~ +(z). The time average of the square of the field is given by

1 ~, [/~,(z)

= -T Jo

4- s

2

2

dt

+2

=

Eimax[-1 4-/3,(z)]]-i 2

+/3*(z)l

or, finally

> 1. The transition to the low-frequency regime ~o~ < 1 is given by

a "- 0"0/092Z2/~ =

ao/(1 + co2z2) ~

0.0, (0.0/092~2 '

~ov > 1,

(44)

where a o = nee2z/mn = e n e # ,

# = eT/mn.

(45)

For degenerate materials, m, in Eq. (45) should be replaced by m* = m,(1 + 2Ef/G),

(46)

which is the effective mass at the Fermi surface from the Kane theory. Detailed calculations of the scattering rates are given by Jensen [12, 14, 21]. Numerical results for various compounds are also given by Jensen [18-20] for 1/z as a function of no and co, from which the complex dielectric constant in Eq. (8) can be immediately calculated. The remaining quantities, including the complex refractive index, absorption coefficient, optical conductivity, and skin depth, then follow as discussed in Section I. Values of n, k, and R versus wavelength, calculated by using the method discussed for GaAs, InP, and InAs, are tabulated in Section IV.

IV

COMPARISON WITH EXPERIMENTAL DATA

The theoretical results are calculated by using experimental values for the various quantities given in Table I. For polar-optical mode scattering, no adjustable parameters are involved. Theoretical results are obtained for the scattering rate 1/T in Eq. (43). This enables one to calculate el and e2 in Eqs. (7) and (8), from which n and k are obtained by using Eqs. (11-13). The reflectivity R is found as a function of n and k from Eq. (14). This is the quantity measured experimentally. A calculation of ~a appropriate to electrons in polar semiconductors with the band structure of the Kane theory and based on the quantum densitymatrix equation of motion yields a high-frequency modification to Eq. (7). For coz >> 1 and X < 0.1, one obtains [18, 22] = [eoo/(1 - X)][1 [coo/(1

-

x)](1

2/0 2] -

&2/~02)

x E o - kO,

ho9 > E o + kO.

The phonon has energy kO and wave vector q, the separation of the band edges in reciprocal space. In general, there are several phonons, one from each branch, which may scatter, so the absorption edge is then a sum of terms like the equations above with different values of 0 and different scattering matrix elements (hidden in the constant prefactor). Symmetry considerations sometimes lead to a vanishing scattering probability for some phonons. The resultant absorption coefficients have characteristic shapes, as shown in Fig. 2. (The foregoing assumed the electron excitation by the photon was allowed by the electric-dipole selection rules. If not, then the wave functions are expanded in a series in k. The term in k I gives the first nonvanishing absorption. The exponent 2 in the preceding equation then becomes 3, for these indirect forbidden transitions.) In small-band-gap semiconductors and in heavily doped semiconductors, the foregoing absorption-edge description is incorrect because the Fermi level may lie inside the conduction or valence band. Then one gets a sharp turn-on of absorption with a shape dominated by the Fermi-Dirac distribution function. In this case, the position of the Fermi level rises with increasing doping for n-type material, giving a shift in the edge energy with doping level, the Burstein-Moss shift. This is as far as we shall go with the one-electron picture. It provides a qualitatively good picture for metals; indeed, often a quantitatively good one. (One must first subtract the intraband contribution to e2 from the measured e2 spectrum.) For semiconductors the electron-hole interaction, discussed in Section III, alters the expected spectrum appreciably near threshold but apparently not strongly in regions far above threshold. As the band gap increases, such effects become stronger and, in some cases, the band picture may not be the best starting point.

III

ELECTRON-HOLE INTERACTION, EXClTONS

The excited electron and the hole are created at the same lattice site, so their interaction cannot be neglected. This leads to excitons, and to other effects as well. We first consider only the lowest energy M0 critical point transition with gap energy Eo. There are two extreme models for the electronhole interaction. The Wannier-Mott model assumes that the electron-hole

10. Interband AbsorptionmMechanisms and Interpretation

199

interaction is weak. The effective-mass model is employed and the hydrogen atom Hamiltonian results, but with the factors e 2 and m replaced by e2/e and/z* = m~mh/(me* * * + m~), respectively. The static dielectric constant is e, and the m* are the electron and hole effective masses, here assumed to be isotropic. This Hamiltonian leads to discrete levels and a continuum. The former, nonconducting excited states, have energies of E o - Rr 2 with Rex = 13.6 eV (l~*/m)~-2 and n an integer. The radii of the exciton "orbits" are r, = 0.53 A {emn2/12*), and if the model is to be self-consistent, the smallest radius must be at least several interatomic spacings in order to make the picture of dielectric screening meaningful. This is usually the case for smallband-gap materials that usually have small effective masses and high dielectric constants, but it may be less valid for the lowest energy exciton than for the others. In such cases a central-cell correction is needed to account for the departure of the potential from the simple -eZ/~r assumed when the electron and hole are nearby. This may be done with a nonlocal dielectric function e(r). The other extreme model is the Frenkel exciton, basically an excited atom or ion in the crystal. The electron and hole are localized on one site or a small number of sites. The solid is polarized about this site, so free-atom or free-ion energy levels are not expected. Such a model was formerly believed to be the appropriate one for all large-band-gap insulators, e.g., the solid rare gases and the alkali halides, but now it seems that the Wannier model is quite suitable for all of them, except for the n = 1 exciton, and perhaps for a few higher levels in solid Ne. The Frenkel model is very useful for describing excitons in organic crystals. An electron-hole interaction of intermediate strength requires the use of the Frenkel model with the excitation "localized" on a large number of sites or the Wannier model with Bloch functions employed with many wave vectors, not just those near the critical point wave vector. Thus, as the electron-hole interaction increases in strength, more and more of the band wave functions are used in the Wannier model, and the exciton is no longer associated with a particular critical point; possibly more than one conduction band may have to be used. In such a case, the band picture may not be the best starting point for an understanding of all the optical properties of insulators. In both of these models the excitation is not permanently localized. It propagates through the crystal with a wave vector K and, in the case of the Wannier exciton, with a kinetic energy hZKZ/2(m * 4-m'~), giving rise to a band. The selection rule on wave vector, however, limits the excitons that can be directly photoexcited to those with K = 0. For indirect transitions, one can excite part of the exciton band because the phonon participating in the transition supplies the proper wave vector to allow this to happen. The hydrogen-atom problem also has unbound continuum states that differ from free-electron states, especially near threshold. In the crystal, the

200

David W. Lynch

Ez Eo

E

I I

I

f

/

/

I

Eo-R

Eo

Eo+R

Eo+5R ENERGY

Fig. 3. An Mo absorption-edge spectrum (dashed line) and how it is modified by the electron-hole interaction (solid line). Only the n = 1, 2, and 3 excitons are shown, and these are shown unbroadened as delta functions, with heights proportional to their oscillator strengths. An observed spectrum resembles the spectrum in inset.

electron-hole interaction modifies the absorption above the critical point as well. Elliott [-9] has given a description of the effects of the electron-hole interaction for an Mo critical point for a direct-band-gap material and for an indirect-band-gap absorption threshold. As shown schematically in Fig. 3, the series of excitons merges into a continuum. The enhanced absorption of the Coulomb states above Eo meets them, the result being a series of lines merging into a continuum with no apparent structure at E o. If there is any broadening present, as there must be, the continuum begins further below Eo, possibly leaving only the n = 1 exciton resolved. For Mo critical points in two-dimensional crystals, the exciton spectrum is different, with E = Eo R~,J(2n- 1)2, and the dependence of absorption strength on n is different from the three-dimensional case [ 10, 11]. For an indirect gap, the edge is also distorted, and its analysis into two terms according to Fig. 2 is then invalid over an energy range from about R,~ below to several times R~ above E o. For direct-band-gap materials with large values of R~, the entire valenceto-conduction-band transitions can be distorted by the electron-hole interaction [12]. The electron-hole interaction is present for electron-hole pairs produced by exciting any electron to any excited state, not just at Mo thresholds. At an M1 critical point any discrete bound electron-hole pair we try to make will be degenerate with other excited states into which it may decay, for the

10. Interband Absorption,--., Mechanisms and Interpretation

201

M1 critical point transitions overlap other transitions. Such an excited state is called a hyperbolic exciton, after the nature of the Eif(k) surface, and if the decay rate is not too great, it may be observable. Such excitons are believed to have been observed in several semiconductors, although the subject is controversial [13-15]. The electron-hole interaction decreases the absorption strength below that expected for the one-electron picture for M 2 and M3 critical points, making them less prominent in spectra than expected by the one-electron model [16]. At energies above the band-gap energy, it is possible to think of other types of discrete excited states overlapping the continuum of excitations above the fundamental edge. Such discrete excitations could arise from the excitation of excitons below another Mo critical point involving a second, higher, conduction band or a deeper valence band, e.g., the spin-orbit "partner" of the highest one. Core-electron excitation may also give rise to such a situation. In such cases, the two excitations may not be independent, and the absorption spectrum is not the sum of the two individual spectra. The final excited state of the system is a superposition of both the discrete and continuum excited states. When this superposition is used to evaluate the electric-dipole matrix element, interference effects occur upon squaring, and the strength of the coupling of two types of excitations has a strong effect on the appearance of the resultant absorption spectrum. There can be shifts of the absorption peak, which now is broadened, and a dip may appear on either the high- or low-energy side. Fano [17-19] first described such effects, and the resultant line shapes are called Fano line shapes. (Hyperbolic excitons give rise to Fano line shapes.) The electron-hole interaction is not just the Coulomb attraction. This is also an exchange term that is important, and the net interaction may be attractive or repulsive. Toyozawa et al. [20] have considered the electron-hole interaction at each type of critical point and have summarized the results in a figure, reproduced here as Fig. 4. It shows how the interaction mixes the line shape expected for a given critical point with the line shape for an "adjacent" critical point, the mixing increasing with increasing strength of the electron-hole interaction. The result is that one cannot identify properly a critical point in the observed spectrum of an insulator without consideration of the effect of the electron-hole interaction. The inclusion of exchange also can alter considerably the apparent spin-orbit features in the spectrum [213. Electrons and holes each interact with the vibrating lattice, and despite the apparent electrical neutrality of an exciton, the exciton-phonon interaction may be rather strong. The strength is described by the ratio of the average coupling energy to the bandwidth of the excitons. The coupling is usually to the longitudinal optical phonons or to zone-boundary acoustic phonons [22], and the bandwidth is a measure of exciton mobility. The exciton line shape is an asymmetric Lorentzian for weak coupling and

202

David W. Lynch

\

M3

f

/

V\

\

\

/ J

/

j MI

Fig. 4. Metamorphosis of critical points by an attractive electron-hole interaction. A threedimensional critical point of type M r (j = 0, 1, 2, 3) is as shown with no electron-hole interaction. As this interaction grows, the shape distorts according to the arrow until at very large interaction the M r has become Mr+ 1 (J = 4 --, j = 0). A repulsive interaction acts similarly but with the arrows pointing counterclockwise. (From Toyozawa et al. [20].)

Gaussian for strong coupling [23, 24]; both shapes are temperature dependent. As a result of the phonon coupling, there is a shift in the energy of the peak in the exciton from that calculated for the electronic system alone [23, 24]. On the low-energy side of the first exciton absorption peak, the absorption coefficient is found to be exponential with energy over a range of up to four decades for a variety of insulating and semiconducting materials. This is the "Urbach rule" [25], according to which #(E) = #o e x p [ - b(Eo - E)]. In some materials, e.g., highly doped semiconductors, b is a constant, while for many other materials, e.g., alkali halides, b = bo(2k T/hf9o) tanh(hogo/2kT) with hco0 an effective phonon energy. An Urbach tail is found in amorphous semiconductors, as well. At low temperatures in some crystals, e.g., CdTe [26], the exponential edge shape changes, and phonon sidebands become resolved. There are two models that give the observed exponential absorption edge. One is based on the electric fields arising from impurities, defects, and

10. Interband Absorption--Mechanisms and Interpretation

203

phonons [27]. There is convincing evidence that this model applies to heavily doped semiconductors [28] and disordered materials [29]. The other model is the standard exciton-phonon coupling model [23, 24], which can yield an exponential edge [30-34] or phonon sidebands [32], depending on the coupling strength to the phonons. There is convincing evidence that this mechanism is operative in GeS, for the phonons involved have been identified as phonons with no accompanying electric field [37]. In many other materials with exponential absorption edges, the two models cannot be so clearly distinguished. A phenomenological model [35, 36] also has been used for a number of materials.

LOCAL FIELD EFFECTS

We have just described what appears to be a two-body effect, but, in fact, all the valence electrons participate in screening the electron-hole interaction. Their roles are subsumed in e(r). There are other many-body effects that play a real role. We consider only the local field effect, an old classical problem. The electric field acting on an electron, in our case causing it to become excited, is the local field caused by the incident electromagnetic wave and by the polarization of the medium around the electron in response to that wave and to the polarization in the medium. A self-consistent local field must be found [38-45]. The resultant local field depends on the degree of localization of the electron under consideration. For a spatially delocalized electron, we can use just the average field in the medium. For a highly localized electron, one confined to a unit cell, the local field "correction" becomes the largest. Classically it is given by Eloc(r) = (e + 2/3)E(r) for a cubic or isotropic crystal, and e ~ g in regions of absorption. As the electron becomes more delocalized, the size the correction decreases to unity. Also, at high energies g ~ 1, and the correction is negligible, no matter how localized the electron may be. The use of a local field correction depending on e alters the shape of the/32 spectrum obtained from experiment [46, 47]. In Fig. 5 we show some data for an alkali halide. The dielectric function has been derived from a reflectance measurement by Kramers-Kronig analysis, with two different assumptions about the local field, the most extreme assumptions. The differences in the spectra are very large, and it is not at all clear which may be the more correct in a given spectral region, for we do not really know how much local field correction should be applied to a given transition. This is clearly an area in which more understanding is needed.

IV

204

David W. Lynch !

I

[

i

t

i

i

i

i

4 W

^_A

.....

I

I

I ""

i

I

l

I

I

T

!

I"

r

T

I

1

T

I"

2 .

-I

6

.

.

.

.

_1

L

~

t

8

10

12

14

.

.

.I

~

16

18

_L

I

J.

20

22

24

26

ENERGY (eV) Fig. 5. Dielectric function spectra for CsC1 obtained by Kramers-Kronig analysis of reflectance spectra; solid line represents no local field correction used and dashed line full Lorentz-Lorenz local field correction used. (From Bergstresser and Rubloff [46].)

V

EXAMPLES

We now give some examples of interband spectra for some simple materials. Figure 6 shows the interband optical conductivity of an A1 single crystal at 4.2 K, obtained by a Kramers-Kronig analysis of absorptance and reflectance [48]. The theoretical result presented is a fit to the parallel-band model for a polyvalent, nearly free-electron metal [49]. The model is based on the pseudopotential formalism, and in the case of Fig. 6, four parameters have been used to fit both the shape and the magnitude, with two of the parameters consonant with Fermi-surface data on A1. Moreover, a more elaborate pseudopotential band calculation has been carried out, and the optical properties calculated from it [50]. The results are in good agreement with the measured 1.5-eV peak and in reasonably good agreement for the 0.5-eV peak, but the calculated results are uncertain at low energy because they result from small differences in nearly equal energy eigenvalues. Reasonably good fits have been achieved for other similar metalsmIn, Ga, Pb, and Tl--although in Pb and T1, spin-orbit splitting of certain bands contributes features in the experimental spectra not considered in the model. Other polyvalent metalsmBe, Zn, C d m a n d the alkaline earths do not conform to this simple (local) pseudopotential model. Figure 7 shows the measured ]-51] g2 for Cu and the results of an empirical, nonlocal pseudopotential calculation of the band structure and the dielectric

10. Interband AbsorptionmMechanisms and Interpretation

205

TABLE I Important Interband Transitions in Cu

Energy (eV)

Bands involved

2.1 3.2 3.7 3.7 3.96 4.25 4.33 4.5 5.0

4, 5 ---,6 4, 5 --, 6 3 -* 6 3, 4, 5 ~ 6 4, 5 --*6 6 ~ 7' 1~ 6 3, 4, 5 ~ 6 1, 2 ~ 6

Location in Brillouinzone

Type of transition

A5 ~ AI(EF) Near X E 1-* ]~I(EF) Near X X 5 ~ X~ L~ ---,L~ ~1 ~ ZI(EF)

Fermi edge Volume Fermi edge Volume M1 Mo Fermi edge Volume Volume

function [52]. This calculation also used optical and photoemission data to fix a total of eight parameters in the band calculation. Although onsets of structure may be attributed to transitions to the Fermi level from the relatively fiat 3d bands and to a few critical points, much of the large background giving the magnitude of ~2 comes from transitions occurring in large volumes of the Brillouin zone. The contributing transitions are shown in Table I. The optical properties of Cu have also been calculated quite accurately from first principles. Janak et al. [53] have calculated the (self-consistent) energy bands, dielectric function, and photoemission spectra from a potential for copper adjusted only to give the correct Fermi surface. The only other adjustable parameter characterized the electron-electron interaction, and it was adjusted to improve agreement with e2. The agreement is about the same as shown in Fig. 7, but only one parameter was adjusted to produce the fit. Finally, L/isser et al. 1-5] have calculated the dielectric function for the noble metals by using an interpolation scheme, evaluating the dipole matrix elements from E(k) by simple differentiation [5]. The fit, although not quite as good as that shown in Fig. 7, extends to an energy of 24 eV, and the departures from the experimental spectra seem understandable on the basis of the model. Figure 8 shows the measured interband e2 for Mo obtained by K r a m e r s Kronig analysis of reflectance spectra [54, 55], and one calculated ~2 [56]. The calculation began with energy bands but ignored electric-dipole matrix elements. Thus, the calculated e2 is in arbitrary units. By using only specific regions of the Brillouin zone, the origin of structures or large featureless contributions to e2 could be identified. However, another group [57] used a different set of energy bands as a starting point and a different technique for locating the origin of structures by using energy, rather than momentum, windows, and obtained comparably good agreement with experiment (again without dipole matrix elements) but different assignments for a number of the

I

03 uJ

I

I

I

I

I

I

I00

0 >-I.>

8o

_~\\

-

,I,

/.~

^

5D a z o ~ a z

60

-

40

-

~ +

20

m

i,i I-z

',.

"~

I

_

"

_

+

/ o

A %\ ~~

/:/.

~

I

-

-/ 0,0

I

I 0,8

I

I

I 1,6 ENERGY (eV)

I 2,4

I

Fig. 6. Interband contribution to the optical conductivity of A1 (solid). The crosses and dashed curve represent model calculations of the interband conductivity. (From Benbow and Lynch [48].)

3

'~ w

(a)

t.~

2,~

T'I2 -5 2

Kz ~ X

L

X

4'3

Lu3

F "~/2

t"i

_~

i k

I _

I

_

!

l

!

Cb)

//

l N

od AS"~A I 0

I

0

1.0

)~ 2.0

Z i ,.,.~Z I (I.,.--6")

-

n

I

I

I

3.0

4.0

5.0

6.0

ENERGY

(e V)

Fig. 7. (a) Band structure of Cu calculated by an empirical pseudopotential method, (b) interband contribution to e2 for Cu (dashed) ([51]) and an empirical pseudopotential calculation of e2 (solid). Some important transitions are labeled. (See also Table I and Fig. 7a.) (From Fong et al. [52].)

10. Interband Absorption--Mechanisms and Interpretation I

i

1

I

I

207 !

I

7

/!

25

/ \\

I

\

20

E2 15

I0

0

I

I

I

2

I

3 ENERGY

I

4 (eV)

I

I

I

5

6

?

Fig. 8. e2 of Mo. Solid line represents values measured by Weaver et al. [54]; dashed line values calculated by Pickett and Allen [56].

transitions. It would be most useful to evaluate 82 for a transition metal by using dipole matrix elements to see if/32 can be calculated well for a material with unfilled d bands. In both the transition metals and the noble metals, it now is clear that some of the structure and much of the magnitude in the ~2 spectra arise from transitions throughout large volumes of the Brillouin zone, and such transitions are not easily accounted for in any way other than a numerical integration over the Brillouin zone. Simpler models, such as are appropriate for simple metals, do not suffice. Calcium is an example of a simple metal in which the Fermi surface and critical points play no role in the optical properties [58]. Figure 9 shows the e 2 spectrum for Si, a covalently bonded semiconductor [59]. Although the one-electron calculation of e 2 was quite successful for Ge, a similar material it is clear that the peak at 3.5 eV in Si appears as only a shoulder at 3.4 eV in the one-electron calculation, whose integrated area is too small by about 30~ [42]. A number of critical points have been identified in derivative spectra, so the problem seems not to be grossly inadequate energy bands in the calculation. The bonding electrons, those being excited in this spectrum, are highly localized in Si, and several calculations of local field effects have shown that they have a large effect, although some of the early calculations did not lead

208

David W. Lynch

40 (

9

§

50

N

§ 9

2O

§

I0

2..I.

§

2

4

I

6

ENERGY (eV) Fig. 9. /32 of Si. Solid line represents experiment by Philipp and Ehrenreich [59]; filled circles, values calculated by a one-electron empirical pseudopotential method (Louie et al. [41]; crosses) calculated by using a local field correction. (From Hanke and Sham [45].)

to an effect that produced better agreement with experiment. The recent calculations [45] show that proper treatment of the local field and other many-electron effects is a sine qua non to the quantitative understanding of the optical spectrum of silicon and of diamond. Such calculations have not been carried out on other solids for which local field effects are believed to be important. The absorption-edge region is shown in Fig. 10 [60, 61]. These data were taken with high resolution at a number of temperatures. There is considerable resemblance to the spectrum shown schematically in Fig. 2, indicating the occurrence of indirect transitions with the participation of a phonon. Detailed analysis of all the spectra shows that the band-gap energy is temperature dependent, that the slopes near threshold are not as expected because of the electron-hole interaction, and that there are several overlapping structures at higher temperatures because phonons with four different energies (i.e., from four different branches) participate. A second indirect edge, at which the electrons are excited into a higher-lying set of conduction-band minima, has been observed [62] with great difficulty, for it is weak and lies at 1.65 eV, where the absorption shown in Fig. 10 is very large. As an example of an insulator we can use CsC1, shown in Fig. 5. The sharp structures near 8 and 13 eV are clearly excitons, those near 8 eV arising from the valence-to-conduction-band transitions, and those at 13.5 eV from the excitation of Cs 5p core levels. Considerable analysis has been carried out on the valence and core excitons in alkali halides and solid rare gases, but far less is known about the other structures. (Many weaker exciton features,

10. Interband A b s o r p t i o n m M e c h a n i s m s and Interpretation I

1

i

i

i

i

I

i

i

209 I

I

6.0 o) o

>.

77K 4.0

,

mo

5.0

-

4.2

3.0 2.0 ::L I.O

.00

1.04 1.08

1.12

1.16

1.20

1.24 1.28

ENERGY (eV) I

I

!

4 -(b)

3-

N

>

I

m i

E r

B

v

/ /

Cki

~

tl I J~

:L ---

J,

I

/ (fl

1.16

-

: I

1.18

r

I

1.20 1.22 ENERGY (eV)

1.124

1.2

Fig. 10. (a) Absorption-coefficient spectra for Si at the fundamental (indirect) edge at several temperatures (from Macfarlane et al. [60]) and (b) decomposition of the 4.2 K spectrum into two components, phonon emission (dotted line beginning at higher energy) and phonon absorption (dashed line lying under the solid line, then leaving it). The solid line is the spectrum from both components together.

not apparent in Fig. 2, are known in many alkali halides.) These have been associated with structures expected in the joint density of states for interband transitions, but the large electron-hole interaction leads one not to expect a one-to-one correspondence between the one-electron spectrum and the experimental spectrum.

210

David W. Lynch

ACKNOWLEDGMENT

The Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under contract No. W-7405-Eng-82. This research was supported by the Director for Energy Research, Office of Basic Energy Sciences.

REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

W. Eberhardt and F. D. Himpsel, Phys. Rev. B 21, 5572 (1980). G. Borstel, M. Neumann and M. Wohlecke, Phys. Rev. B 23, 3121 (1981). R. L. Benbow, Phys. Rev. B 22, 3225 (1980). N. V. Smith, Phys. Rev. B 19, 5019 (1979). R. L~isser, N. V. Smith, and R. L. Benbow, Phys. Rev. B 24, 1895 (1981). K. Nakao, J. Phys. Soc. Jpn. 25, 1343 (1968). Y. Sasaki, C. Hamaguchi, A. Moritani, and J. Nakai, J. Phys. Soc. Jpn. 36, 177 (1974). M. T. Okuyana, T. Nishino, and Y. Hamakawa, J. Phys. Soc. Jpn. 37, 431 (1974). R. J. Elliott, Phys. Rev. 108, 1384 (1957). H. I. Ralph, Solid State Commun. 3, 303 (1965). M. Shinada and S. Sugano, J. Phys. Soc. Jpn. 20, 1274 (1965). U. RSssler and O. Schiitz, Phys. Stat. Solidi B 56, 483 (1973). C. B. Duke and B. Segall, Phys. Rev. Lett. 17, 19 (1966). J. Hermanson, Phys. Rev. Lett. 18, 170 (1967). J. E. Rowe, F. H. Pollak, and M. Cardona, Phys. Rev. Lett. 22, 933 (1969). B. Velick~, and J. Sak, Phys. Stat. Solidi 16, 147 (1966). U. Fano, Phys. Rev. 124, 1866 (1961). K. P. Jain, Phys. Rev. 139, A544 (1965). Y. Onodera, Phys. Rev. B 4, 2751 (1971). Y. Toyozawa, M. Inoue, T. Inui, and E. Hanamura, J. Phys. Soc. Jpn. 22, 1337, 1349 (1967). Y. Onodera and Y. Toyozawa, J. Phys. Soc. Jpn. 22, 833 (1967); Y. Onodera and Y. Toyozawa, J. Phys. Soc. Jpn. 22, 1337, 1349 (1967). Y. Toyozawa and M. Inoue, J. Phys. Soc. Jpn. 21, 1663 (1966). Y. Toyozawa, Prog. Theor. Phys. 20, 53 (1958). H. Sumi, J. Phys. Soc. Jpn. 32, 616 (1972). F. Urbach, Phys. Rev. 92, 1325 (1953). D. T. F. Marple, Phys. Rev. 150, 728 (1966). J. D. Dow and D. Redfield, Phys. Rev. B 5, 594 (1972). D. Redfield and M. A. Aframovitz, Appl. Phys. Lett. 11, 138 (1967). J. Szczyrbowski, Phys. Stat. Solidi B 105, 515 (1981). K. Cho and Y. Toyozawa, J. Phys. Soc. Jpn. 30, 1555 (1971). H. Sumi and Y. Toyozawa, J. Phys. Soc. Jpn. 31, 342 (1971). H. Miyazaki and E. Hanamura, J. Phys. Soc. Jpn. 50, 1310 (1981). S. Schmitt-Rink, H. Haug, and E. Mohler, Phys. Rev. B 24, 6043 (1981). M. Schreiber and Y. Toyozawa, J. Phys. Soc, Jpn. 51, 1544 (1982). T. Skettrup, Phys. Rev. B 18, 2622 (1978). D. J. Dunstan, J. Phys. C 30, L419 (1982). J. D. Wiley, E. Schonherr, and A. Breitschwerdt, Solid State Commun. 34, 891 (1980). S. L. Adler, Phys. Rev. 126, 413 (1962). N. Wiser, Phys. Rev. 129, 62 (1963). Y. Onodera, Prog. Theor. Phys. 49, 37 (1973). S. G. Louie, J. R. Chelikowski, and M. L. Cohen, Phys. Rev. Lett. 34, 155 (1975). W. Hanke and L. J. Sham, Phys. Rev. B 12, 4501 (1975). R. D. Turner and J. C. Inkson, J. Phys. C 9, 3585 (1976).

10. Interband Absorption--Mechanisms and Interpretation 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62.

211

R. Bonneville, Phys. Rev. B 21, 368 (1980). W. Hanke and L. J. Sham, Phys. Rev. B 21, 4656 (1980). T. K. Bergstresser and G. W. Rubloff, Phys. Rev. Lett. 30, 794 (1973). S. R. Nagel and T. A. Witten, Phys. Rev. B 11, 1623 (1975). R. L. Benbow and D. W. Lynch, Phys. Rev. B 12, 5615 (1975). N. W. Ashcroft and K. Sturm, Phys. Rev. B 3, 1898 (1971). D. Brust, Phys. Rev. B 2, 818 (1970). U. Gerhardt, Phys. Rev. 172, 651 (1968). C. Y. Fong, M. L. Cohen, R. R. L. Zucca, J. Stokes, and Y. R. Shen, Phys. Rev. Lett. 25, 1486 (1970). J. F. Janak, A. R. Williams, and V. L. Moruzzi, Phys. Rev. B 11, 1522 (1971). J. H. Weaver, D. W. Lynch, and C. G. Olson, Phys. Rev. B 10, 501 (1974). B. W. Veal and A. P. Paulikas, Phys. Rev. B 10, 1280 (1974). W. E. Pickett and P. B. Allen, Phys. Rev. B 11, 3599 (1975). D. D. Koelling, F. M. Mueller, and B. W. Veal, Phys. Rev. B 10, 1290 (1974). P. O. Nilsson and G. Forssell, J. Phys. F 5, L159 (1975). H. R. Philipp and H. Ehrenreich, Phys. Rev. 129, 1550 (1963). G. G. Macfarlane, T. P. McLean, J. E. Quarrington, and V. Roberts, Phys. Rev. 111, 1245 (1958). G. G. Macfarlane, T. P. McLean, J. E. Quarrington, and V. Roberts, J. Phys. Chem. Solids 8, 388 (1959). R. A. Forman, W. R. Thurber, and D. E. Aspnes, Sol. State Commun. 14, 1007 (1974).

GENERAL REFERENCES

Abel,s, F., in "Optical Properties of Solids" (F. Abel,s, ed.), p. 93, North-Holland Publ., Amsterdam, 1972. Altarelli, M., J. de Phys. 39, C4-95 (1978). Aspnes, D. E., in "Handbook of Semiconductors" (M. Balkanski, ed.), Vol. 2, p. 109, NorthHolland Publ., Amsterdam, 1980. Bassani, F., Appl. Opt. 19, 4093 (1980). Bassani, F., and Pastori Parravicini, G., "Electronic States and Optical Transitions in Solids," Pergamon, Oxford, 1975. Brown, F. C., in "Synchrotron Radiation Research" (H. Winick and S. Doniach, eds.), p. 61, Plenum, New York, 1980. Brown, F. C., Solid State Phys. 29, 1 (1974). Brown, G. S., and Doniach, S., in "Synchrotron Radiation Research" (H. Winick and S. Doniach, eds.), p. 353, Plenum, New York, 1980. Cardona, M., in "Semiconductors and Semimetals" (R. K. Willardson and A. C. Beer, eds.), Vol. 3, p. 125, Academic Press, New York, 1967. Cardona, M., "Modulation Spectroscopy," Solid State Phys. Supp. 11 (1969). Cohen, M. H., in "Optical Properties of Solids" (J. Tauc, ed.), p. 1, Academic Press, New York, 1966. Dexter, D. L., and Knox, R. S., "Excitons," Wiley, New York, 1963. Dow, J. D., in "Optical Properties of Solids--New Developments" (B. O. Seraphin, ed.), p. 3, North-Holland Publ., Amsterdam, 1976. Ehrenreich, H., in "Optical Properties of Solids" (J. Tauc, ed.), p. 106, Academic Press, New York, 1966. Evans, B. L., in "Optical and Electrical Properties" (P. A. Lee, ed), p. 1, Reidel, Dordrecht, 1976. Greenaway, D. L., and Harbeke, G., "Optical Properties and Band Structure of Semiconductors," Pergamon, New York, 1968.

212

David W. Lynch

Hamakawa, Y., and Nishino, T., in "Optical Properties of Solids--New Developments" (B. O. Seraphin, ed.), p. 255, North-Holland Publ., Amsterdam, 1976. Hanke, W., in "Festkorper Probleme XIX--Advances in Solid State Physics" (J. Treusch, ed.), p. 43, Vieweg, Braunschweig, 1979. Harbeke, G., in "Optical Properties of Solids" (F. Ab61~s, ed), p. 21, North-Holland Publ., Amsterdam, 1972. Johnson, E. J., in "Semiconductors and Semimetals" (R. K. Willardson and A. C. Beer, eds.), Vol. 3, p. 154, Academic Press, New York, 1967. Koch, E. E., Kunz, C., and Sonntag, B., Phys. Rep. 28C, 154 (1977). Kotani, A., and Toyozowa, Y., in "Synchrotron Radiation Techniques and Applications" (C. Kunz, ed.), p. 169, Springer-Verlag, Heidelberg, 1979. Knox, R. S., "Theory of Excitons," Solid State Phys. Supp. 5 (A63). Kunz, C., J. de Phys. 39, C4-112 (1978). Kunz, C., in "Optical Properties of SolidsmNew Developments" (B. O. Seraphin, ed.), p. 423, North-Holland Publ., Amsterdam, 1976. Lynch, D. W., in "Synchrotron Radiation, Techniques and Applications" (C. Kunz, ed.), p. 357, Springer-Verlag, Heidelberg, 1979. Moss, T. S., "Optical Properties of Semiconductors," Butterworth, London, 1959. Nilsson, P. 0., Solid State Phys. 29, 139 (1974). Pankove, J. I., "Optical Processes in Semiconductors," Prentice-Hall, Englewood Cliffs, New Jersey, 1971. Petroff, Y., in "Handbook of Semiconductors" (M. Balkanski, ed.), Vol. 2, p. 1, North-Holland Publ., Amsterdam, 1980. Philipp, H. R., and Ehrenreich, H., in "Semiconductors and Semimetals" (R. K. Willardson and A. C. Beer, eds.), Vol. 3, p. 93, Academic Press, New York, 1967. Phillips, J. C., in "Optical Properties of Solids" (J. Tauc, ed.), p. 155, Academic Press, New York, 1966. Phillips, J. C. Solid State Phys. 18, 55 (1966). Reynolds, D. C., and Collins, T. C., "Excitons, Their Properties and Uses," Academic Press, New York, 1981. Schwentner, N., Appl. Opt. 19, 4104 (1980). Skibowski, M., Sprussel, G., and Saile, V., Appl. Opt. 19, 3978 (1980). Tauc, J., in "Optical Properties of Solids" (J. Tauc, ed.), p. 277, Academic Press, New York, 1966. Toyozawa, Y., Appl. Opt. 19, 4101 (1980). Wooten, F., "Optical Properties of Solids," Academic Press, New York, 1972. Zavetova, M., and Velicky, B., in "Optical Properties of Solids--New Developments" (B. O. Seraphin, ed.), p. 379, North-Holland, Publ., Amsterdam, 1976.

Chapter 1 1 Optical Properties of Nonmetallic Solids for Photon Energies below the Fundamental Band Gap SHASHANKA S, MITRA Department of Electrical Engineering University of Rhode Island Kingston, Rhode Island

I. II. II1. IV. V. VI.

Introduction Infrared Dispersion by Polar Crystals Kramers-Kronig Dispersion Relations Determination of Absorption Coefficient in the Intermediate Region Absorption Coefficient in the Transparent Regime Multiphonon Absorption A. General Remarks B. Absorption in the Region ~o > ~OLO C. Absorption in the Region co < COTO VII. Infrared Absorption by Defects and Disorders A. General Remarks B. Crystals with Substitutional Impurities C. Amorphous and Glassy Systems VIII. Infrared Dispersion by Plasmons References

213 215 227 229 230 232 232 236 249 254 254 255 259 263 267

INTRODUCTION There are two chief sources of absorption in a pure solid, viz., lattice vibrations and electronic transitions. For most materials, a sufficiently wide spectral window exists between these two limits where the material is transparent (see Fig. 1). However, this transparent regime displays residual absorption 213 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright (~) 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-544420-6

I

214

Shashanka S. Mitra

I t-z uJ 0,=,= LL LL ILl

/~A /

"L

Intrinsic .lattice tail

Urbach / tail

o o z

o I-Q. nO

co rn
8 #m. Refractive-index data in the region of the fundamental absorption edge and above (0.1-0.5 eV) were obtained by Zemel et al. [10], who used epitaxially grown films on NaC1 and optical techniques described in the critique on PbS. At T = 300 K, around 0.3 eV the well-known peak in n (see Table IX) is present. The precision of the n measurements was estimated to be _+39/0. Further information on the temperature dependence of n in this region is found in Preier [11]. Absorption in this region has been measured by Scanlon [12], and Moss [13] has cited unpublished work by Avery [as cited by Moss [13], p. 187] in the 0.5-5-#m range. Avery measured the reflectance for incident light polarized in as well as perpendicular to the plane of incidence, choosing three angles of incidence per wavelength. Both data sets are given in Table IX. In the region above the fundamental absorption edge, several authors have studied reflectivity in the 1-6-eV [ 14-17] and 1-22-eV [ 15, 17] ranges. Synchrotron radiation was used for a study in the 14-26-eV region [18]. Whereas up to 6 eV the results of a K K analysis of reflectivity data obtained with cleaved bulk samples can be compared with optical constants derived from reflectivity and transmission of thin films (d -~ 300 A), from 6 to 22 eV only reflectivity data could be analyzed. For the K K analysis and for the extrapolation to energies higher than 25 eV, it was assumed that the reflectivity is given by R oc Coo -4, where C is calculated so as to have the reflectivity continuous at 25 eV. Actually n and k values were quoted up to 15 eV [17] and are tabulated in Table IX. The structure in the optical constants in the region is due to interband transitions and the effect of the d electrons of lead. For review papers on the optical and electronic properties of PbSe, we refer the reader to Dalven [19, 20], Ravich et al. [21], and Nimtz [22]. REFERENCES

1. E. Burstein, R. Wheeler, and J. Zemel, Proc. Int. Conf. Phys. Semicond. (M. Hulin, ed.), p. 1065. Dunod, Paris, 1964. 2. H. Burkhard, R. Geick, P. K/istner, and K. H. Unkelbach, Phys. Star. Solidi B 63, 89 (1974). 3. P. M. Amirtharaj, B. L. Bean, and S. Perkowitz, J. Opt. Soc. Am. 67, 939 (1977). 4. A. Aziza, E. Amzallag, and M. Balkanski, Solid State Commun. 8, 873 (1970). 5. A. Mycielski, A. Aziza, M. Balkanski, M. Y. Moulin, and J. Mycielski, Phys. Star. Solidi B 52, 187 (1972). 6. I.V. Kucherenko, Y. A. Mityagin, L. K. Vodopyanov, and A. P. Shotov, Soy. Phys. Semicond. 11, 282 (1977); I. V. Kucherenko, Y. A. Mityagin, L. K. Vodopyanov, and A. P. Shotov, Fiz. Tekh. Poluprovodn 11, 488 (1977). 7. K. V. Vyatkin and A. P. Shotov, Soy. Phys. Semicond. 14, 785 (1980); K. V. Vyatkin and A. P. Shotov, Fiz. Tekh. Poluprovodn. 14, 1331 (1980).

Lead Selenide (PbSe)

519

8. A. Maitre, R. Le Toullec, and M. Balkanski, Proc. Int. Conf. Phys. Semicond. (M. Miasek, ed.), p. 826. PWN-Polish Scientific Publishers, Warsaw, 1972. 9. R. N. Hall and J. H. Racette, J. Appl. Phys. (Suppl.) 32, 2078 (1961). 10. J. N. Zemel, J. D. Jensen, and R. B. Schoolar, Phys. Rev. A 140, 330 (1965). 11. H. Preier, Appl. Phys. 20, 189 (1979). 12. W. W. Scanlon, J. Phys. Chem Solids, 8, 423 (1959). 13. T. S. Moss, "Optical Properties of Semiconductors," p. 189. Butterworth, London, 1959. 14. M. L. Belle, Soy. Phys. Solid State, 5, 2401 (1964); M. L. Belle, Fiz. Tverd. Tela 5, 3282 (1963). 15. S. E. Kohn, P. Y. Yu, Y. Petroff, Y. R. Shen, Y. Tsang, and M. L. Cohen, Phys. Rev. B 8, 1477 (1973). 16. F. I. Bogacki, A. K. Sood, C. Y. Yang, S. Rabii, and J. E. Fischer, Surf Sci. 37, 494 (1973). 17. M. Cardona and D. L. Greenaway, Phys. Rev. A 133, 1685 (1964). 18. M. Cardona, C. M. Penchina, E. E. Koch, and P. Y. Yu, Phys. Stat. Solidi B 53, 327 (1972). 19. R. Dalven, Infrared Phys. 9, 141 (1969). 20. R. Dalven, Solid State Phys. 28, 179 (1973). 21. Y. I. Ravich, B. A. Efimova, and I. A. Smirnov, in "Semiconducting Lead Chalcogenides." Plenum, New York, 1979. 22. G. Nimtz, in "Landolt-B6rnstein, Numerical Data and Functional Relationships in Science and Technology" (K.-H. Hellwege and O. Madelung, eds.), Group III, Vol. 17, Subvolume f, pp. 168, 110. Springer-Verlag, Berlin, 1983.

10 2

i0 ~

,o~

'-- .......

~-I~ bl

d

,h

,.-..,

r

i//

,o4

i //' 'l i i I I S ~

i

O i0 -i

i0 o

I01

WAVELENGTH

-

II

,o-'I F-

t

il !i

[

! -

10 2

IO s

(,um)

Fig. 8. Log-log plot of n ( - - ) and k ( - - - ) versus wavelength in micrometers for lead selenide. Curves marked (2) and (3) are for samples of two different free-carrier concentrations from Burkbard et al. [2] and Amirtharaj et al. [3], respectively.

520

G. Bauer and H. Krenn TABLE IX Values of n and k for Lead Selenide Obtained from Various References ~

eV 14.5 14.0 13 12 11.5 11 10.5 10 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3.0 2.0 1.65 1.5 1.24 1.0 0.99 0.83 0.80 0.75 0.71 0.70 0.65 0.62 0.60 0.55 0.50

cm- 1

/~m

48,390 44,360 40,330 36,290 32,260 28,230 24,200 16,130 13,310 12,100 10,000 8,065 7,985 6,694 6,452 6,049 5,726 5,646 5,243 5,001 4,839 4,436 4,033

0.08551 0.08856 0.09537 0.1033 0.1078 O.1127 0.1181 0.1240 0.1305 0.1378 0.1459 0.1550 0.1653 0.1771 0.1907 0.2066 0.2254 0.2480 0.2755 0.3100 0.3542 0.4133 0.6199 0.7514 0.8266 1.000 1.240 1.252 1.494 1.550 1.653 1.746 1.771 1.907 2.000 2.066 2.254 2.480

n

0.72 [17] 0.70 0.70 0.70 0.70 0.71 0.70 0.68 0.68 0.7 0.725 0.715 0.71 0.70 0.68 0.64 0.58 0.54 0.56 0.64 0.74 1.25 [17] 3.65

0.20 [17] 0.25 0.35 0.38 0.40 0.44 0.46 0.50 0.56 0.66 0.63 0.62 0.66 0.71 0.78 0.86 1.0 1.2 1.3 1.9 2.4 3.20 [17] 2.9

4.64

2.64

4.65

1.1

3,992 3,871 3,710 3,629 3,549 3,388 3,307

2.505 2.583 2.695 2.755 2.818 2.952 3.024

n

k

3.00113] 3.90 4.51

3.50113] 3.20 1.73

4.67

1.54

4.69 4.64

1.14 0.950

4.61

0.830

4.59

0.770

4.59

0.740

4.58

0.740

4.57

0.740

4.57

0.740

0.284 [12] 0.269 0.251 0.240

4.70 0.495 0.48 0.46 0.45 0.44 0.42 0.41

k

0.254 0.238 0.235 0.450 [17]

4.90 [ 1O] 4.88 0.220 [12] 4.885 4.895

Lead Selenide (PbSe)

521 TABLE IX

(Continued)

Lead Selenide eV

cm- ~

pm

0.40 0.39 0.38 0.36 0.354 0.35 0.34 0.33 0.32 0.31 0.30

3,226 3,146 3,065 2,904 2,855 2,823 2,742 2,662 2,581 2,500 2,420

3.100 3.179 3.263 3.444 3.502 3.542 3.647 3.757 3.875 4.000 4.133

0.292 0.29 0.28

2,355 2,339 2,258

4.246 4.275 4.428

0.275 0.27 0.26 0.25 0.248 0.24 0.23 0.22 0J0 0.18 0.17 0.16 0.15 0.14 0.13 0.124 0.1215 0.1190 0.1165 0.1141 0.1116 0.1091 0.1066 0.1041 0.1017 0.09919 0.09671 0.09423 0.09175

2,218 2,178 2,097 2,016 2,000 1,936 1,855 1,774 1,613 1,452 1,371 1,290 1,210 1,129 1,049 1,000 980 960 940 920 900 880 860 840 820 800 780 760 740

4.509 4.592 4.769 4.959 5.000 5.166 5.391 5.636 6.199 6.888 7.293 7.749 8.266 8.856 9.537 10.00 10.20 10.42 10.64 10.87 11.11 11.36 11.63 11.90 12.20 12.50 12.82 13.16 13.51

n 4.91 4.85 4.93 4.95

4.96 4.98

4.94

k

0.219 0.225

0.199 0.194 0.189 0.173 0.127 0.104

4.57

0.740

4.56

0.756

4.54

0.782

4.52 5.05 [12] 5.29 [7] 4.48113]

0.796

5.13 [7] 4.48 [13] 4.94

4.90 4.83 4.86 4.82

4.80 4.78 4.77 4.75 4.75 [2] 4.75 4.74 4.74 4.74 4.74 4.73 4.73 4.73 4.73 4.72 4.72 4.71 4.71

0.836

5.72 x 10- 2 3.53 x 10 -2 [10]

9.17 • 10 -3 [12] 2.85 1.54 1.38 1.28

4.47 5.02 [-7] 4.45 [13]

0.890

4.91 [7] 4.90 4.87 4.86 4.85

1.06 • 10 -3 [2] 1.13 1.20 1.28 1.37 1.46 1.57 1.68 1.80 1.94 2.09 2.26 2.44 2.65

0.836

4.82 4.72 [3] 4.71 4.71 4.70 4.69 4.69 4.68 4.67 4.66 4.66 4.65 4.64 4.62 4.61

2.22 2.48 2.69 3.4 4.10 4.36 4.64 4.95 5.29 5.66 6.06 6.51 6.99 7.53 8.13 8.79 9.52 1.03

x 10 -2 [7]

x 10 -3 [3]

x 10-3 x 10-2

(continued)

522

G. Bauer and H. Krenn TABLE IX

(Continued)

Lead Selenide eV 0.08927 0.08679 0.08431 0.08183 0.07935 0.07687 0.07439 0.07191 0.06943 0.06699 0.06447 0.06199 0.05951 0.05703 0.05455 0.05207 0.04959 0.04711 0.04463 0.04215 0.03968 0.03720 0.03472 0.03224 0.02976 0.02728 0.02480 0.02232 0.01984 0.01736 0.01488 0.01240 0.01215 0.01190 0.01165 0.01141 0.01116 0.01091 0.01066 0.01041 0.01017 0.009919 0.009671 0.009423 0.009175 0.008927

cm- 1

720 700 680 660 640 620 600 580 560 540 520 500 480 460 440 420 400 380 360 340 320 300 280 260 240 220 200 180 160 140 120 100 98 96 94 92 90 88 86 84 82 80 78 76 74 72

~m 13.89 14.29 14.71 15.15 15.63 16.13 16.67 17.24 17.86 18.52 19.23 20.00 20.83 21.74 22.73 23.81 25.00 26.32 27.78 29.41 31.25 33.33 35.71 38.46 41.67 45.45 50.00 55.56 62.50 71.43 83.33 100.0 102.0 104.2 106.4 108.7 111.1 113.6 116.3 119.0 122.0 125.0 128.2 131.6 135.1 138.9

n 4.70 4.70 4.69 4.69 4.68 4.67 4.66 4.65 4.64 4.63 4.62 4.60 4.59 4.57 4.55 4.52 4.49 4.46 4.42 4.37 4.31 4.24 4.15 4.04 3.89 3.69 3.42 3.01 2.34 1.34 1.02 1.18 1.21 1.25 1.29 1.34 1.39 1.44 1.51 1.57 1.65 1.73 1.83 1.93 2.05 2.18

k 2.88 3.14 3.43 3.75 4.12 4.54 5.02 5.58 6.21 6.95 7.80 8.81 x 10- 3 1.00 x 10- 2 1.14 1.31 1.52 1.77 2.08 2.47 2.97 3.62 4.48 5.64 7.26 9.61 x 10 -2 0.13 0.19 0.30 0.56 1.49 3.19 5.02 5.22 5.43 5.64 5.87 6.09 6.33 6.58 6.83 7.10 7.38 7.67 7.97 8.29 8.62

n 4.60 4.59 4.57 4.55 4.53 4.51 4.49 4.47 4.44 4.41 4.37 4.33 4.29 4.23 4.17 4.10 4.02 3.92 3.80 3.66 3.48 3.26 2.96 2.55 1.97 1.27 0.96 0.91 0.99 1.15 1.45 1.97 2.04 2.12 2.21 2.30 2.39 2.50 2.61 2.73 2.87 3.02 3.18 3.36 3.56 3.79

k 1.13 1.23 1.34 1.48 1.63 1.80 1.99 2.22 2.48 2.78 3.14 3.56 4.07 4.68 5.43 6.34 7.49 8.95 x 10- 2 0.11 0.13 0.17 0.22 0.30 0.43 0.70 1.41 2.49 3.58 4.71 6.00 7.57 9.64 9.89 10.1 10.4 10.7 11.0 11.3 11.6 11.9 12.3 12.7 13.1 13.5 13.9 14.4

Lead Selenide (PbSe)

523 T A B L E IX

(Continued)

Lead Selenide eV

cm- a

pm

n

k

0.008679 0.008431 0.008183 0.007935 0.007687 0.007439 0.007191 0.006943 0.006695 0.006447 0.006199 0.005951 0.005703 0.005455 0.005207 0.004959 0.004711 0.004463 0.004215 0.003968 0.003720 0.003472 0.003224 0.002976 0.002728 0.002480 0.002232 0.001984 0.001736 0.001488 0.001240

70 68 66 64 62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10

142.9 147.1 151.5 156.3 161.3 166.7 172.4 178.6 185.2 192.3 200.0 208.3 217.4 227.3 238.1 250.0 263.2 277.8 294.1 312.5 333.3 357.1 384.6 416.7 454.5 500.0 555.6 625.0 714.3 833.3 1000

2.33 2.50 2.69 2.91 3.16 3.46 3.80 4.20 4.68 5.25 5.93 6.7 7.7 8.8 10.0 11.2 12.3 13.2 13.8 14.1

8.97 9.34 9.73 10.1 10.6 11.0 11.5 12.0 12.5 13.1 13.6 14.1 14.5 14.7 14.8 14.6 14.1 13.4 12.5 11.7

14.0 13.8 13.4 13.0 12.7 12.6 12.8 13.2 14.1 15.5 17.4

10.6 10.5 10.1 11.3 12.2 13.4 14.9 16.6 18.1 21.1

a References are indicated in brackets.

11.0

4.04 4.33 4.67 5.05 5.51 6.10 6.71 7.51 8.49 9.68 11.1 12.7 14.4 15.9 16.9 17.4 17.4 17.1 16.8 16.5 16.3 16.4 16.6 17.1 17.8 18.7 19.9 21.4 23.3 25.6 28.6

14.9 15.4 16.0 16.6 17.2 17.9 18.6 19.3 20.0 20.6 21.1 21.2 20.9 20.1 19.0 17.8 16.9 16.4 16.3 16.5 17.0 17.7 18.1 19.7 20.9 22.3 23.1 25.4 27.3 29.6 32.5

Lead Sulfide (PbS) G. GUIZZEITI a n d A. BORGHESI Dipartimento di Fisica "A. Volta" and Gruppo Nazionale di Struttura della Materia del CNR Universita di Pavia Pavia, Italy

The room-temperature values of n and k tabulated here were obtained from the following works and references therein: Riedl and Schoolar [1], Geick [2], Schoolar and Zemel [3], Schoolar and Dixon [4], Zemel et al. [5], Dixon and Riedl [6], and Vakulenko et al. [7] for the medium and far infrared; Schoolar and Dixon [4], Avery [8], Scanlon [9], Wessel [-10], and Semenov and Shileika [11] for the near infrared and visible; Wessel [10], Cardona and Greenaway [12], Rossi and Paul [13], Cardona and Haensel [14], Cardona et al. [15], and Heckelmann et al. [16] for the visibleultraviolet and vaCuum-ultraviolet. For the review papers on the optical and electronic properties of PbS we refer to Moss [17], Zemel [18], Dalven [19, 21], and Ravich et al. [20]. A composite smooth-curve plot is given in Fig. 9. Numerical values are given in Table X with appropriate references listed in brackets. The optical constants were obtained generally by reflectance R and/or transmittance T measurement on single-crystal bulk or film samples. After 1963 almost everyone producing PbS films adopted the technique described by Schoolar and Zemel [3]: single-crystal n-type films were grown in vacuum ( ~ 10 -6 torr) from bulk PbS on heated rocksalt substrates (generally cleaved from synthetic single-crystal of NaC1 or KC1). The good crystalline quality of such films, which have very uniform thickness, has been well established [3, 5]. The film's surfaces require on further treatment to make them suitable for either optical T or R measurement. Furthermore, experiments dealing with the electrical, optical, and mechanical properties indicate that the films behave like the best available bulk crystals. No significant thickness dependence of the optical constants has been detected [3-5, 7]. In the far-infrared (2 > 30 #m) bulk reflectance measurements at near normal incidence (8-10 ~) [2, 6] and transmission on films [2, 18] were performed. We have disregarded earlier R spectra [22, 23] that were in qualitative agreement with the findings of Geick and Dixon and Riedl [2, 6] but had 525 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright (~) 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-544420-6

526

G. Guizzetti and A. Borghesi

values too low. The n and k values were obtained by fitting R in the 302000-#m range by a single classical dispersion oscillator (calculations including a Drude term suggested that the influence of free carriers on the optical constants is negligible for N < 1016 cm-3, at least out to 1000/~m). The R samples were natural or synthetic single crystals with different carrier concentrations N ranging from ~ 1016 t o ' ~ 1019 cm-3. The optical surfaces were ground mechanically and polished chemically, and it was checked that the damage caused by mechanical polishing does not significantly affect the R spectrum in this spectral region. Experimental and calculated spectra from Geick [2] and Dixon and Riedl [6] were in agreement to within the experimental uncertainties. In Table X n and k values obtained through the fit have been labeled according to Geick [2]. The parameters for the sample with the lowest carrier concentration were [2] eo = 150 for the static relative dielectric constant; eoo = 16.8 for the optical dielectric constant; vv = 71 cm- 1 and vL = 212 cm-1 for the transverse and longitudinal optical phonon frequencies, respectively; and 7 = 15 cm- 1 for the damping. The fit values for eo and vT agree within 7~ with those derived directly (eo from R value at 2 = 2 mm and v~ from the minimum in T spectra [2, 18]). Besides, the fit values are in good agreement with those deduced from phonon-assisted tunneling by Hall and Racette [24], microwave measurements by Sokoloski and Fang [25], inelastic neutron scattering by Cochran et al. [26], and n data in the medium infrared by plotting n 2 versus 22 and extrapolating the linear region to zero wavelength [1, 5, 6]. The sight dependence of e~ on carrier density 1-6] is discussed in the PbTe critique. As for the 0.95-26-#m range, Table X contains experimental n values deduced from the position of interference peaks in R and/or T for films [3-5]. The thicknesses, ranging from ~0.4 to ~ 5 #m, were determined by using the interferometric and weighing techniques described in Riedl and Schoolar [1], with an estimated error less than _+3~. The typical carrier concentration was N = 2 • 1018 e/cm 3. At such high carrier density n is a slight function of N and is not expected to join smoothly to the calculated n [2] with no free-carrier contribution. The values of n [2] have been listed to 10 #m and can represent n for pure PbS. Although Geick [2] also lists k values, they are only a qualitative description of the lattice absorption of pure PbS. Phase shifts at the interfaces due to the nonzero values of the extinction coefficients and the optical properties of the substrate were estimated and found to be negligible. The basic precision of the n measurements was estimated to be __+3~. The n values for the films are in excellent agreement with those obtained from bulk-reflectivity data [1]; their differences are within the experimental uncertainty. The same agreement exists among the data from [3-5] and the data obtained by other authors [1, 7, 11] with the same technique on films, in narrower spectral ranges. The refractive index in the 2-15-#m region was determined also at 77 and 373 K [5]. We observe that the sharp peak in the n spectrum at ~ 3 #m results from the rapid change in the absorption at the fundamental absorption edge.

Lead Sulfide (PbS)

527

The absorption coefficient c~between 3 and 15 #m was measured by using a single crystal of n-type galena 0.22 mm thick, with N = 2 x 1017 cm- 3 [7]. This absorption coefficient, typical of free carriers, was found to vary as 22.8 +o.2 giving k to vary as 238. We would not expect this k to join smoothly on the k calculated for a purer sample in Dixon and Riedl [6]. Multiplereflection effects were taken into account. Values for c~were determined with an error not exceeding 20?/o and were in agreement, at short wavelengths, with those from Schoolar and Dixon [4] Scanlon [9], while the values from Semiletov et al. [27] were systematically too high (about one order of magnitude over the entire 3-15-#m range). In the spectral region of the fundamental absorption edge, c~ has been measured at room temperature several times. The most recent and accurate measurements have been performed by Schoolar and Dixon [4], who, moreover, have discussed the previous results and the large discrepancies. Reflectance (at an angle of incidence less than 15~) and transmittance measurements were made in [4] on n-type films, with N _~ 3 x 10 TM cm-3 and thicknesses from 0.37 to 2.36/tm. The experimental error in R and T was estimated to be less than __+4~o, and the spectral energy resolution was less than 0.01 eV in the vicinity of the edge. Extinction coefficient k was derived from T and R, including multiple-reflection effects and the influence of the film backing. The results were estimated to be uncertain to _+7%. In particular, the results approach those of Avery [8] in the higher energy region; in addition, they agree with those of Scanlon [9] in the steep portion of the edge but are greater by amounts up to 30% out to 1.2/~m. This fact and the long absorption plateau reported by Scanlon 1-9] were probably due to the presence of pinholes in the samples or scattered light in the monochromator, as suggested by Cardona and Greenaway [12]. The roomtemperature Burstein-Moss effect is probably negligible at the fundamental band gap [28]. For all shorter wavelengths, the free-carrier effects are probably negligible. The optical constants from 1.2 #m to about 0.2/~m were determined either by Kramers-Kronig (KK) analysis of near-normal-incidence reflectance data of bulk samples [12, 14] or from systematic and accurate measurement of R and T on thin films 1-10, 13]. The R values of more than 25 films and bulk agree in both magnitude and structure within about 4% [13]. The same agreement exists between R data from [10] and [13], which are higher than those reported in [12] and [16]. In Wessel [10] the error in R was taken to be _+2% and from 3 to 10% in T. In Rossi and Paul [13] R was repeatable to within + 3% absolute, and the total error was estimated _+5% absolute; the total error in T was estimated to be between 10 and 25%, including errors caused by scattered light from other wavelengths in the thicker samples and errors caused by imperfections in the films and film substrates (NaC1 or KC1). No attempt was made to determine the impurity levels, but it was found [12] that the R spectra were not influenced by normally encountered variations in impurity concentrations. The films' thicknesses, ranging from

528

G. Guizzetti and A. Borghesi

200 to ~ 800 ,~, were determined by assuming the carefully obtained optical constants of Schoolar and Dixon [4] near 1 #m and adding several methods of cross-checking. The thickness was considered to be accurate to approximately _ 15~o. The optical constants obtained from R and T in Wessel [10] and Rossi and Paul [13] agree quite well (within 10~) with those obtained from KK analysis [-13], and their error was about the same as the one involved in the R and T measurements, i.e., + 5?/0 or less, if the thickness can be obtained accurately. This is true except in the energy range in which n and k values are such that a small uncertainty in R can lead to a large uncertainty in n [29]; this occurs in PbS near 0.4 #m. In Table X we have reported n and k values derived from thin-film optical data, which are in several respects more reliable than those derived from dispersion relations. In the region between 0.2 and 0.05 #m only R measurements [12, 15, 16] were performed, either on expitaxial layers or on cleaved single crystals. The optical constants have been obtained from the K K analysis on R spectra from [12] and [-15], extended to longer wavelengths with the data from [2] and [6] for the infrared and from [10] and [13] for the visible-ultraviolet. The R values of [12] and [15] were multiplied by a constant factor to make them agree with results from [13] at 0.2 #m and were extrapolated as R = R o ( 2 o / 2 ) - P for 2 < 20 = 0.05 #m. Value P was adjusted to give values of k nearly equal to zero below the energy gap and agreeing with the values of [4, 10, 13] above the gap. The final value of P = 2.77 was used. Transmission measurements on films in the 0.033-0.008-#m range were performed by Cardona and Haensel [14] with synchrotron radiation and with a typical resolution of 2 ,~ over the whole spectral range. The T technique was chosen over that of reflectance because of the small R or PbS (< 1~) in this region, which increases the error due to scattered light of long wavelengths. The thin carbon substrates of the films made it impossible to extend the measurements to longer wavelengths. The thicknesses were determined with a Tolansky interferometer and also during evaporation with a calibrated quartz-crystal monitor. The absorption coefficient values were affected by a scaling error of about 20?/0 because of uncertainties in the thicknesses. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

H. R. Riedl and R. B. Schoolar, Phys. Rev. 131, 2082 (1963). R. Geick, Phys. Lett. 10, 51 (1964). R. B. Schoolar and J. N. Zemel, J. Appl. Phys. 35, 1848 (1964). R. B. Schoolar and J. R. Dixon, Phys. Rev. A 137, 667 (1965). J. N. Zemel, J. D. Jensen, and R. B. Schoolar, Phys. Rev. A 140, 330 (1965). J. R. Dixon and H. R. Riedl, Phys. Rev. A 140, 1283 (1965). O. V. Vakulenko, M. P. Lisitsa, and Ya. F. Kononets, Soy. Phys. Solid State 8, 1356 (1966). D. G. Avery, Proc. Phys. Soc. London Ser. B 67, 2 (1954). W. W. Scanlon, Solid State Phys. 9. 109 (1959). P. R. Wessel, Phys. Rev. 153, 836 (1967).

Lead Sulfide (PbS) 11. 12. 13. 14. 15. 16. i7. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

529

Ya. A. Semenov and A. Yu. Shileika, Soy. Phys. Solid State 1419 (1967). M. Cardona and D. L. Greenaway, Phys. Rev. A 133, 1685 (1964). C. E. Rossi and W. Paul, J. Appl. Phys. 38, 1803 (1967). M. Cardona and R. Haensel, Phys. Rev. B 1, 2605 (1970). M. Cardona, C. M. Penchina, E. E. Koch, and P. Y. Yu, Phys. Stat. Solidi B 53, 327 (1972). G. H. Heckelmann, H. H. Landfermann, U. Nielsen, and T. S. Wagner, Phys. Stat. Solidi B 69, K99 (1975). T. S. Moss, in "Optical Properties of Semiconductors," p. 183. Butterworth, London, 1959. J. N. Zemel, Proc. Int. Conf. Phys. Semicond., Paris, 1964, p. 1061. Dunod, Paris, 1964. R. Dalven, Infrared Phys. 9, 141 (1969). Yu. I. Ravich, B. A. Efimova, and I. A. Smirnov, in "Semiconducting Lead Chalcogenides," p. 43. Plenum, New York, 1970. R. Dalven, Solid State Phys. 28, 179 (1973). J. Strong, Phys. Rev. 38, 1818 (1931). H. Yoshinaga, Phys. Rev. 100, 753 (1955). R. N. Hall and J. H. Racette, J. Appl. Phys., Suppl. 32, 2078 (1961). M. M. Sokoloski and P. H. Fang, Phys. Lett. 16, 222 (1965). W. Cochran, R. Cowley, and G. Dolling, Proc. R. Soc. London Ser. A 293, 433 (1966). S. A. Semiletov, I. P. Voronina, and E. I. Kortukova, Soviet Phys. Crystallogr. (Eng. Trans.) 10, 429 (1966). Edward D. Palik, D. L. Mitchell, and J. N. Zemel, Phys. Rev. A 135, 763 (1964). P. M. Grant, Bull. Am. Phys. Soc., Ser. II 10, 546 (1965).

I

I

I I IIII

I

I

I I 11111 I

I

I

I IIIII

i01

I

I

1 I IIII1

l

//// \\

iIJ

i0 o

K j-/,./-

~\\

ll/

kt,\\

',,

I

I

iO-I

/

/

\

I /

"',, \ \ 2 \\-

_

f-.iI

10-2

iI II iI

I iI I

I iI

i0-~

I

I _

il

10-2

I

I

I

Illill

l

io-i

I

I

lillll

I

I

I

IIIIll

I0 o WAVELENGTH

1

I01

I

I

liilll

1

102

I

i

llilll

103

(,u.m)

Fig. 9. Log-log plot of n ( - - ) and k (. . . . ) versus wavelength in micrometers for lead sulfide. Data from various references do not join together in overlap regions on occasion.

530

G. Guizzetti and A. Borghesi TABLE X Values of n and k for Lead Sulfide from Various References a

eV 150 145 140 135 130 125 120 115 110 105 100 95 90 85 80 75 70 65 60 58 55 50 45 40 38 25 24 23.3 22.7 21.9 21.5 21.3 20.6 19.8 19.2 18.6 18.0 17.5 17.0 16.5 16.0 15.6 15.2 14.8 14.4 14.0 13.5

cm- 1

pm 0.008266 0.008551 0.008856 0.009184 0.009537 0.009919 0.01033 0.01078 0.01127 0.01181 0.01240 0.01305 0.01378 0.01459 0.01550 0.01653 0.01771 0.01907 0.02066 0.02138 0.02254 0.02480 0.02755 0.03100 0.03263 0.04959 0.05166 0.05321 0.05462 0.05661 0.05767 0.05821 0.06019 0.06262 0.06457 0.06667 0.06888 0.07084 0.07293 0.07514 0.07749 0.07948 0.08157 0.08377 0.08610 0.08856 0.09183

n

0.845 [12, 15] 0.800 0.772 0.760 0.831 0.870 0.858 0.847 0.881 0.911 0.881 0.846 0.814 0.778 0.734 0.687 0.651 0.628 0.619 0.627 0.651 0.687

k 3.86 x 10 - 3 [14] 4.09 4.37 4.66 5.10 5.59 6.40 8.96 x 10 - 3 1.05 x 10 -z 1.26 1.54 1.78 2.06 2.38 2.88 3.72 4.64 5.48 6.17 6.44 6.73 7.05 5.43 4.28 4.23 x 10 -2 0.171 [12, 15] 0.196 0.238 0.291 0.372 0.339 0.324 0.339 0.370 0.332 0.300 0.294 0.298 0.307 0.327 0.364 0.412 0.472 0.538 0.606 0.665 0.725

Lead Sulfide (PbS)

531 TABLE X

(Continued)

Lead Sulfide

eV 13.0 12.5 12.0 11.5 11.0 10.5 10.0 9.5 9.0 8.6 8.2 7.7 7.2 6.8 6.4 6.2 5.9 5.65 5.4 5.15 4.95 4.75 4.6 4.45 4.30 4.15 4.0 3.85 3.75 3.65 3.55 3.45 3.35 3.25 3.15 3.00 2.90 2.75 2.55 2.35 2.20 2.00 1.80 1.60 1.40 1.30

cm-

50,010 47,590 45,570 43,550 41,540 39,920 38,310 37,100 35,890 34,680 33,470 32,260 31,050 30,250 29,440 28,630 27,830 27,020 26,210 25,410 24,200 23,390 22,180 20,570 18,950 17,740 16,130 14,520 12,910 11,290 10,490

1

~m 0.09537 0.09918 0.1033 0.1078 0.1127 0.1181 0.1240 0.1305 0.1378 0.1442 0.1512 0.1610 0.1722 0.1823 0.1937 0.2000 0.2101 0.2194 0.2296 0.2407 0.2505 0.2610 0.2695 0.2786 0.2883 0.2988 0.3100 0.3220 0.3306 0.3397 0.3493 0.3594 0.3701 0.3815 0.3936 0.4133 0.4275 0.4509 0.4862 0.5276 0.5636 0.6199 0.6888 0.7749 0.8856 0.9537

0.725 0.758 0.789 0.811 0.821 0.836 0.879 0.968 1.007 0.997 0.921 0.999 1.037 1.024 0.951 1.13113] 1.22 1.22 1.33 1.43 1.52 1.53 1.55 1.53 1.57 1.67 1.73 1.90 2.07 2.33 2.58 2.98 3.17 3.32 3.62 3.88 4.12 4.25 4.35 4.33 4.30 4.29 [10, 13] 4.53110] 4.62 4.50 4.48 [4, 10]

0.769 0.807 0.843 0.872 0.912 0.974 1.050 1.096 1.066 1.060 1.140 1.273 1.299 1.361 1.489 1.70113] 1.78 1.85 1.95 2.05 2.10 2.20 2.30 2.45 2.53 2.70 2.83 3.00 3.17 3.32 3.37 3.37 3.37 3.30 3.17 3.00 2.70 2.33 2.00 1.67 1.55 1.48110,13] 1.30 [ 10] 0.94 0.83 0.627 [-4, 10]

(continued)

532

G. Guizzetti and A. Borghesi TABLE X (Continued) Lead Sulfide

eV 1.24 1.18 1.13 1.03 0.953 0.885 0.825 0.775 0.730 0.690 0.650 0.620 0.590 0.563 0.539 0.516 0.496 0.477 0.459 0.443 0.432 0.413 0.40O 0.393 0.387 0.381 0.3542 0.3306 0.3100 0.2917 0.2755 0.2480 0.2066 0.1771 0.1550 0.1378 0.1240 0.1178 0.1116 0.1054 0.1033 0.09919 0.09299 0.08856 0.08679 0.08059

cm- 1

10,000 9,524 9,091 8,333 7,692 7,143 6,667 6,250 5,882 5,556 5,263 5,000 4,762 4,545 4,348 4,167 4,000 3,846 3,704 3,571 3,484 3,333 3,226 3,175 3,125 3,077 2,857 2,667 2,500 2,353 2,222 2,000 1,667 1,429 1,250

1,111 1,000 950 900 850 833.3 800 750 714.3 700 650

1.00 1.05 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.87 3.0 3.1 3.15 3.2 3.25 3.5 3.75 4.0 4.25 4.5 5 6 7 8 9 10 10.53 11.11 11.76 12 12.50 13.33 14 14.29 15.38

4.43 4.385 4.355 4.30 4.275 [4] 4.26 4.25 4.24 4.235 4.235 4.24 4.25 [4, 5"] 4.245 [5] 4.255 4.26 4.28 4.30 4.335 4.38 4.42 4.435 4.345 4.30

4.26 4.215 4.185 4.16 4.14 4.125 4.115 4.10 4.09 4.07 4.04 4.01 4.02 [2] 4.01 4.00 3.98 3.90 [3] 3.97 [2] 3.95 3.795 [3] 3.93 [2-1 3.90

0.597 [4] 0.543 0.508 0.458 0.403 0.390 0.370 0.357 0.341 0.334 0.318 0.310 0.304 0.285 0.271 0.255 0.235 0.212 0.182 0.144 9.03 x 10- 2 4.59 [9] 2.27 1.43 7.9x 10 -3 3.46 6.48 x 10 -4 6.22 6.38 6.51 7.47 9.25 x 10 -4 1.38 x 1 0 - 3

2.06 3.06 4.42 6.32 x 10-3

1.14 x 10-2

1.83

Lead Sulfide (PbS)

533 TABLE X

(Continued)

Lead Sulfide eV

cm- 1

#m

0.07749 0.07439 0.06888 0.06819 0.06199

625.0 600 555.6 550 500.0

16 16.67 18 18.18 20

0.05636 0.05579 0.05166 0.04959 0.04339 0.03720 0.03444 0.03263 0.03100 0.02952 0.02818 0.02695 0.02583 0.02480 0.02254 0.02066 0.01771 0.01550 0.01378 0.01240 0.01033 0.008856 0.007749 0.006888 0.006199 0.004959 0.004133 0.003542 0.003100 0.002480 0.001653 0.001240 0.0008266 0.0006199

454.5 450 416.7 400 350 300 277.8 263.2 250.0 238.1 227.3 217.4 208.3 200.0 181.8 166.7 142.9 125.0 111.1 100.0 83.33 71.43 62.50 55.56 50.00 40.00 33.33 28.57 25.00 20.00 13.33 10.00 6.667 5.000

22 22.22 24 25.00 28.57 33.33 36 38 40 42 44 46 48 50 55 60 70 80 90 100 120 140 160 180 200 250 300 350 400 500 750 1,000 1,500 2,000

n 3.68 1-3] 3.86 1-2] 3.545 [3] 3.81 [2] 3.39 [3] 3.75 [2] 3.205 1-3] 3.66 [2] 2.995 [3] 3.53 [2-] 3.33 2.99 2.74 2.53 2.28 1.98 1.61 1.13 0.688 0.514 0.433 0.452 0.582 0.809 1.175 1.79 5.06 17.41 20.47 17.87 16.27 14.40 13.62 13.21 12.96 12.69 12.44 12.35 12.29 12.27

9.77 x 10-2 [2] 0.127 0.167 0.226 0.326 0.540 1.03 1.59 2.67 3.55 5.14 6.74 8.48 10.51 15.98 17.94 7.10 3.47 2.20 1.15 0.785 0.604 0.495 0.367 0.228 0.167 0.109 0.0815

a Values obtained from the various references are indicated in brackets.

Lead Telluride (PbTe) G. BAUER a n d H, KRENN Institut f[~r Physik Montanuniversit6t Leoben Leoben, Austria

Lead telluride is a member of the IV-VI compound semiconductor family, crystallizing in the cubic rocksalt structure with 10 electrons in the unit cell. The optical direct energy gap is very small, located at the L point of the Brillouin zone, which is the case for all three lead salts: PbS, PbSe, and PbTe [ 1-3]. The dielectric properties are characterized by unusually high dielectric constants e~ and es, where in PbTe strong temperature dependence of es indicates the inherent tendency of the lattice to undergo a structural phase transition [4]. The room-temperature values of n and k tabulated in Table XI and plotted in Fig. 10 were obtained from the following works and references cited therein: the far-infrared results from 10cm -1 (1000/~m) to 1000cm -1 (10 #m) are from Buss and Kinch [5], Burkhard et al. [6], Perkowitz [7], and Lowney and Senturia [8] (2.5-40 ~tm). Values of n and k in the midinfrared region are from Zemel et at. [9] (0.1-0.6 eV), Piccioli et al. [10] (1500-3500 cm-a), Globus et al. [11] (0.15-0.6 eV), McCarthy et al. [21] (1000-2600 cm-1), Walz [31] (0.25-0.35 eV); in the visible and UV region, from Cardona and Greenaway [12] (0.1-17 eV), Korn and Braunstein [13] (0.1-21 eV), and Cardona and Haensel [14] (36-150 eV). In the low-energy part of the spectrum, the far-infrared results were usually obtained by reflectivity measurements [4-7, 15, 17]. Single-crystal bulk material as well as epitaxially grown films were used. The latter method yield high-quality material with relatively low free-carrier concentration and high carrier mobilities. For more than 10 years the hot wall technique [16] has been used to deposit films of uniform thickness on alkali halide (NaC1, KC1) or BaF2 substrates. Due to the differences in the lattice constants and the thermal expansion coefficients, stresses appear that influence the electrical and optical properties [29], especially at low temperatures. The PbTeBaF 2 system is the one in which stresses play the smallest role at room temperature. 535 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright 9 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-544420-6

536

G. Bauer and H. Krenn

Epitaxially grown films usually do not require any surface treatment such as grinding, polishing, or etching after growth; bulk materials require electropolishing after surface treatment. The n and k values in the far infrared were obtained by fitting the reflectivity data [4-8, 15, 17], taken at near normal incidence (~ 10~ with a classical dispersion oscillator for the infrared active phonon and Drude term for the free carriers as given by e = (n -

ik) 2 = e oo +

%2 C02T _ O92 + iFfo

co(fo + iv)

Here, units of inverse centimeters are used instead of radians per second. The values given in Table I were obtained by using the following fitting parameters: e~ = 32.8 for the high-frequency dielectric constant [8, 19, 20], es = 400 + 20 for the static dielectric constant [4, 6], COT= 32 cm -~ for the transverse optic phonon frequency (which is also in agreement with inelastic neutron scattering data [18]), and F = 10cm -~ for the phonon damping parameter. The room-temperature electron damping parameter in the Drude term was taken to be 30 cm-1. The plasma frequency given by fOp2 = N e 2 / e o m p in mks units is determined by the electron concentration N and effective mass mp. (e0 = 8.9 x 10-12 C / N m 2 is the permittivity of free space.) We obtain fOp = 1320 cm -1 from the fit with N = 1018 cm -3. The relaxation time r is the reciprocal of y if we stay in radians per second and ? (cm-1) = 7 ( r a d / s e c ) / 2 r c c . Experimental and calculated spectra are usually in agreement within experimental and calculated errors. A detailed discussion on improvements of classical oscillator fits has been given by Perkowitz [7]. At frequencies much greater than the longitudinal optic phonon frequency coL and the plasma frequency but less than that of the fundamental absorption edge, n can be expressed by

[n(co)]

+ A (co)- ~%- +

1-

~3s

fo2 9

(1)

Here, Ae is a function that describes additional dispersion due to bound carriers at frequencies near the fundamental absorption edge, and Eq. (1) is valid for (for)2 > 1 [8, 19]. Lowney and Senturia [8] describe in detail the determination of the dispersion of n with the aid of thin-film transmission and reflectance data exhibiting interference fringes. A carrier-concentration dependence of e~ is related to Im e(fo) by Dionne and Woolley [19] and Moss [30]. e oo = 1 + -

2 [.o Im e(fO) JO

O9

do).

(2)

It turns out that for carrier densities below 1019 cm-3, the values of e~ are also independent of electron concentration N. In the region below and around the fundamental absorption edge, optical constants have been determined and discussed in a number of works [9-11, 21-28]. Early precise measurements on the refractive index in the 0.1-0.5-eV

Lead Telluride (PbTe)

537

range were performed by Zemel et al. [9] on epitaxial films by using an interferometric and weighing technique with an experimental error in n of about _+3~. There exists a peak in the refractive index close to the fundamental band edge (see Table XI). Piccioli et al. [10] repeated the experiments of Zemel et al. [9] but freed the PbTe films (p ~ 101 s cm-3) from the substrate and gave data on absorption as well. The error in refractive index is estimated to be _+ 1~ and in absorption index _+5~. The maximum values of n fit Zemel et al.'s [9] results, apart from a small shift in wavelength. McCarthy et al. [21] investigated PbTe films on BaF2 substrates in the 5-12-#m range and showed that a significant reduction in absorption occurs below the band gap as compared with PbTe-NaC1 films, presumably due to chlorine surface contamination. Whereas the overall agreement of the refractive-index data with those of Piccioli et al. [10] and Zemel et al. [9] are satisfactory, the absorption coefficient data are smaller than those of Piccioli et al. [10] and of Riedl [22] for samples with carrier concentrations of the order of 101 v cm-3 by about a factor of 10 to 50. The most recent data on PbTe-BaF2 films are those by Globus et al. [11]. Reflectivity and transmission were measured by using PbTe films (thickness: 0.6 #m) grown on BaF2 substrates in the wavelength region 2 = 2-8 #m. Reflectivity has been determined with an 8 ~ angle of incidence by using samples of comparatively low carrier concentrations (1016 cm-3). With expressions for interference and multiple reflection of the complete film-substrate system, a careful analysis of their data and a comparison with previously published data [9, 10, 24, 27] were performed. The refractive-index data compare favorably with those of Piccioli et al. [10], Zemel et al. [9], and McCarthy et al. [21]; only in the region near the absorption edge are there some differences (see Table XI). Scanlon's data [24] for k are substantially lower (about 40~) than those of Globus et al. [11]. The results by Globus et al. [11] are not influenced by a Burstein-Moss shift at room temperature. However, there is some fine structure in the absorption spectrum, probably due to phonon-assisted transitions. In addition, the authors suggest that apart from transitions at the L point (~;gL ~ 0.32 eV), transitions originate from other close-lying, valence-band extrema presumably in the direction of A. The influence of free carriers on the absorption constant is discussed in the framework of several band models (Kane, twoband, six-band, Cohen models) in [11, 27, 28]. Optical constants of the important Pbl_xSnxTe system can be found [31, 32] in the range of the fundamental absorption edge. The optical constants above the fundamental absorption gap and in the visible and UV region from 0.5 to 25 eV were determined by KramersKronig (KK) analysis of near-normal-incidence reflectance data of bulk [12, 13, 24, 30] and thin-film samples [12] and from measurements of reflectivity and transmission (below 6 eV) of thin films [12] with thicknesses of the order of 300 A. Wavelength modulation spectroscopy [13, 35] and thermoreflectance [33, 34] have also been used to detect fine structure in the

538

G. Bauer and H. Krenn

reflectance spectra, at low temperatures [35] that are related to critical points in the joint density of states. Cardona et al. [36] used synchrotron radiation to measure reflectivity in the 14-26-eV range. Since there are systematic differences between the data of Cardona and Greenaway [12], who used cleaved bulk material and epitaxial films, and those obtained by Korn and Braunstein [13], who worked on polished and electrochemically etched samples, we decided to show both sets of data in Table XI and Fig. 10. Korn and Braunstein [13] cite a maximum absolute reflectance error of about 2~ throughout the spectral range 0.1-21 eV. Despite some differences in the extrapolations in the KK analysis between the two papers [12, 13], the discrepancy probably results from the different sample preparation. In the 36-150-eV region, transmission experiments have been performed with thin PbTe films by Cardona and Haensel [14], who used synchrotron radiation. Whereas below 20 eV the structures are due to valence-band to conduction-band transitions, above 20 eV structures appear to be due to transitions from the outermost d electrons [37]. The measurements were performed on films deposited on carbon substrates, and reflectivity measurements could not be performed because the small reflectivity in the 36-150-eV region (R < 0.01) would increase the error due to scattered light. Measurements of the thickness were made by using the Tolansky interferometric method with films deposited on glass substrates next to the carbon substrate used for the transmission experiments in the far-UV range. From uncertainties in the thickness evaluation, the absorption index Cardona and Haensel's [14] data are supposed to be affected by an error of about 20~. Absorption constant and reflectivity data below and above the fundamental gap were recently compiled by Nimtz and Schlicht [38], and Nimtz [39]. REFERENCES

1. R. Dalven, Infrared Phys. 9, 141 (1969). 2. R. Dalven, Solid State Phys. 28, 179 (1973). 3. Yu. I. Ravich, B. A. Efimova, and I. A. Smirnov, in "Semiconducting Lead Chalcogenides." Plenum, New York, 1970. 4. G. Bauer, H. Burkhard, W. Jantsch, F. Unterleitner, A. Lopex-Otero, and G. Schleussner, in "Lattice Dynamics" (M. Balkanski ed.), p. 669. Flammarion, Paris, 1978. 5. D. D. Buss and M. A. Kinch, J. Nonmet. 1, 111 (1973). 6. H. Burkhard, G. Bauer, and A. Lopez-Otero, J. Opt. Soc. Am. 67, 943 (1979). 7. S. Perkowitz, Phys. Rev. B 12, 3210 (1975). 8. J. R. Lowney and S. D. Senturia, J. Appl. Phys. 47, 1771 (1976). 9. J. N. Zemel, J. D. Jensen, and R. B. Schoolar, Phys. Rev. 140, A330 (1965). 10. N. Piccioli, J. M. Beson, and M. Balkanski, J. Phys. Chem. Solids, 35, 971 (1974). 11. T. R. Globus, B. L. Gelmont, K. I. Geiman, V. A. Kondrashov, and A. V. Matveenko, Soy. Phys. JETP 53, 1000 (1981); T. R. Globus, B. L. Gelmont, K. I. Geiman, V. A. Kondrashov, and A. V. Matveenko, Zh. Eksp. Teor. Fiz. 80, 1926 (1981).

Lead Telluride (PbTe) 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

539

M. Cardona and D. L. Greenaway, Phys. Rev. 133, A1685 (1964). D. M. Korn and R. Braunstein, Phys. Rev. B 5, 4837 (1972). M. Cardona and R. Haensel, Phys. Rev. B 1, 2605 (1970). W. E. Tennant and J. A. Cape, Phys. Rev. B 13, 2540 (1976). A. Lopez-Otero, Thin Solid Films, 49, 1 (1978). E. G. Bylander and M. Hass, Solid State Commun. 4, 51 (1966). W. Cochran, R. A. Cowley, G. Dolling, and M. M. Elcombe, Proc. R. Soc. London 293, 433 (1966). G. Dionne and J. C. Woolley, Phys. Rev. B 6, 3898 (1972). E. Burstein, S. Perkowitz, M. H. Brodsky, J. Phys. C4 29, 78 (1968). S. L. McCarthy, W. H. Weber, and M. Mikkor, J. Appl. Phys. 45, 4907 (1974). H. R. Riedl, Phys. Rev. 127, 162 (1962). J. R. Dixon and H. R. Riedl, Phys. Rev. 138, A873 (1965). W. W. Scanlon, J. Phys. Chem. Solids 8, 423 (1959). R. N. Tauber, A. A. Machonis, and I. B. Cadoff, J. Appl. Phys. 37, 4855 (1966). F. Stern, Phys. Rev. 133, A1653 (1964). D. Genzow, A. G. Mironov, and O. Ziep, Phys. Status Solidi B 90, 535 (1978). W. W. Anderson, Infrared Phys. 20, 363 (1980). E. D. Palik, D. L. Mitchell, and J. N. Zemel, Phys. Rev. 135, A763 (1964). T. S. Moss, "Optical Properties of Solids," p. 181. Butterworth, London, 1959. V. M. Walz, thesis, Naval Postgraduate School, Monterey, California (1972); available from NTIS, Springfield, Virginia, order No. AD 767-677. W. G. Opyd, thesis, Naval Postgrade School, Monterey, California (1973); available from NTIS, Springfield, Virginia, order No. AD 767-677. M. Baleva, Phys. Status Solidi B 88, 335 (1978). R. I. Bogacki, A. K. Sood, C. Y. Yang, S. Rabii, and J. E. Fischer, Surf. Sci. 37, 494 (1973). S. E. Kohn, P. Y. Yu, Y. Petroff, Y. R. Shen, Y. Tsang, and M. L. Cohen, Phys. Rev. B 8, 1477 (1973). M. Cardona, C. M. Penchina, E. E. Koch, and P. Y. Yu, Phys. Status Solidi B 53, 327 (1972). D. E. Aspnes and M. Cardona, Phys. Rev. 173, 714 (1968). G. Nimtz and B. Schlicht, in "Springer Tracts in Modern Physics" (G. H6hler, ed.), Vol. 98, p. 1. Springer-Verlag, Berlin, 1983. G. Nimtz, in "Landolt-B6rnstein, Numerical Data and Functional Relationships in Science and Technology" (K.-H. Hellwege and O. Madelung, eds.), Group III, Vol. 17, Subvolume f, pp. 177 and 473. Springer-Verlag, Berlin, 1983.

540

G. Bauer and H. Krenn I0 a

I0 ~

I0 o ",c

;;//"

-

"-

/

7 / t

',. ....

I"

/

r-

/ /

io-i

f---J

-i

I I

I

I I

10-2 -I I t i0-3

,,I

10-2

!

!

t

I i

i

I

I

]

I

i

I -

I

i ,I,,,,I

I

lO-I

I , I,I,,I

i

I0o

WAVELENGTH

, ,I,,,,I

,

I0 i

i

, ,I,,,,I

I0 2

i i I IIII

103

(jam)

Fig. I0. Log-log plot of n (--) and k (----) versus wavelength in micrometers for lead telluride. Data from various references do not join together in overlap regions on occasion.

541

Lead Telluride (PbTe) T A B L E XI

Values of n and k for Lead Telluride Obtained from Various References a eV 150 145 140 135 130 125 120 115 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 17 16.5 16 15.5 15 14.5 14 13.5 13 12.5 12 11.5 11 10.5 10 9.5 9 8.5 8 7.5

cm- 1

/tm 0.008266 0.008551 0.008856 0.009184 0.009537 0.009919 0.01033 0.01078 0.01127 0.01181 0.01240 0.01305 0.01378 0.01459 0.01550 0.01653 0.01771 0.01907 0.02066 0.02254 0.02480 0.02755 0.03100 0.03542 0.04133 0.07293 0.07514 0.07749 0.07999 0.08266 0.08551 0.08856 0.09184 0.09537 0.09919 0.1033 0.1078 0.1127 0.1181 0.1240 0.1305 0.1378 0.1459 0.1550 0.1653

n

0.8 [12] 0.78 0.75 0.73 0.72 0.70 0.67 0.65 0.63 0.63 0.62 0.60 0.62 0.65 0.66 0.68 0.68 0.7 0.73 0.8

k

n

k

2.37 • 10-3 [14] 3.06 4.37 5.85 7.59 9.71 x 10 -3 1.31 • 10- 2 1.79 2.43 3.43 4.39 4.94 5.36 5.73 6.09 6.43 6.62 6.76 6.84 6.80 6.87 6.78 6.81 7.10 7.77 • 10-2 0.1 [12] 0.1 0.1 0.15 0.17 0.2 0.22 0.26 0.3 0.35 0.37 0.45 0.50 0.58 0.60 0.62 0.7 0.78 0.82 0.92

(continued)

542

G. Bauer and H. Krenn T A B L E XI

(Continued)

Lead Telluride eV

cm-

1

k

#m

7 6.5 6 5.5 5.0 4.8 4.6 4.5 4.4 4.2 4 3.8 3.6 3.5 3.4 3.2 3.0 2.8 2.6 2.5 2.4 2.2 2.0 1.8 1.6 1.5 1.4 1.2 1.0 0.90 0.85 0.80 0.75 0.70 0.65 0.62 0.61 0.60

48,390 44,360 40,330 38,710 37,100 36,290 35,490 33,880 32,260 30,650 29,040 28,230 27,420 25,810 24,200 22,580 20,970 20,160 19,360 17,740 16,130 14,520 12,900 12,100 11,290 9,679 8,065 7,259 6,856 6,452 6,049 5,646 5,243 5,001 4,920 4,839

0.1771 0.1907 0.2066 0.2254 0.2480 0.2583 0.2695 0.2755 0.2818 0.2952 0.3100 0.3263 0.3444 0.3542 0.3647 0.3875 0.4133 0.4428 0.4769 0.4959 0.5166 0.5636 0.6199 0.6888 0.7749 0.8266 0.8856 1.033 1.240 1.378 1.459 1.550 1.653 1.771 1.907 2.000 2.033 2.066

0.59 0.58 0.57 0.56 0.55

4,759 4,678 4,597 4,517 4,436

2.101 2.138 2.175 2.214 2.254

0.9 0.9 0.8 0.75 0.72

0.92 0.94 0.92 0.87 1.0

0.65

1.22

0.7

1.55

0.86

1.85

1.0

2.2

1.35

2.86

3.8

4.55

n

k

0.90 [13] 0.9 0.9

1.88 [13] 2.02 2.16

0.9 0.91 0.96 1.05 1.2

2.34 2.54 2.75 2.96 3.18

1.4 1.55 1.65 1.8 2.14

3.35 3.54 3.8 4.2 5.0

3.3 5.4 6.4 6.16 6.0

5.8 5.9 4.3 3.1 2.85

6.4 7 6.5

2.74 1.8 1.04

6.25

0.71

6.10111] 6.10113] 6.11111] 6.10 6.10 6.10 6.11

0.527111] 0.529 0.521 0.46113] 0.516111] 0.508 0.492 0.496 0.505

3.1

2.2 0.413 [24] 0.367 0.348 0.311 0.289 0.271

0.261

Lead Telluride (PbTe)

543 TABLE XI

(Continued)

Lead Telluride eV

cm- 1

,um

0.54 0.53 0.52 0.51 0.50

4,355 4,275 4,194 4,113 4,033

2.296 2.339 2.384 2.431 2.480

0.49 0.48 0.47 0.46 0.45 0.44 0.43 0.42 0.41 0.40

3,952 3,871 3,791 3,710 3,629 3,549 3,468 3,388 3,307 3,226

2.530 2.583 2.638 2.695 2.755 2.818 2.883 2.952 3.024 3.100

0.39 0.38

3,146 3,065

3.179 3.263

0.37 0.36

2,984 2,904

3.351 3.444

0.35

2,823

3.542

0.34

2,742

3.647

0.335 0.33 0.322

2,702 2,662 2,597

3.701 3.757 3.850

0.31

2,500

4.000

0.30

2,420

4.133

0.29

2,339

4.275

0.28

2,258

4.428

0.27

2,178

4.592

0.26

2,097

4.769

n

5.76

k

1.4 [12] 0.228 [24]

6.04 [9] 6.04 0.220 6.05 6.08 6.06 [10] 6.075 6.12 [9] 6.15 6.13 [10] 6.09 6.1619] 6.12 [10] 6.23 6.05 [-9] 6.17 rio] 6.16 5.96 [-9] 6.02 [ 10]

5.95 5.89 [9]

5.90 [ 10] 5.82 [9] 5.88 [10] 5.85 5.82 [9]

0.370 [10] 0.374 0.345 0.331 0.220 [24] 0.285 0.195 0.284 0.261 0.178 0.227 0.188 0.145

[10] [24] [10] [243 [ 10] [24]

0.177 [10] 0.119 [24] 7.81-'10 -2 [10] 7.32 [24] 5.61 [10] 3.78 [24] 9.55 • 10 -3 [22] 3.55 x 10 -2 [10] 1.23 [24] 5.2 • 10 -3 5.11 [22] 2.49 x 10 -2 [10] 4.20 x 10-3 [24] 2.17 • 10 -2 [10] 4.35 x 10-3 [24] 4.44 [22]

6.10 6.07 6.10 6.10 6.07

0.522 0.523 0.504 0.509 0.507

6.06 6.05 6.06 6.09 6.10 6.11 6.14 6.07 6.07 6.09 6.00 [13] 6.10111] 6.13

0.505 0.493 0.465 0.447 0.438 0.427 0.407 0.399 0.381 0.382 0.24113] 0.349111] 0.345

6.14 6.15

0.320 0.307

6.17

0.245 0.175131] 0.183111] 0.151131] 0.134111] 0.149131] 5.50 x 10 -1 [11] 2.75 [31] 1.13111] 1.13 [31]

6.17

6.12111] 6.08 6.02

5.97

7.95 • 10- 3

5.91

7.14

5.89 5.92 [8] 5.86111]

1.04121] 7.05131] 6.94

5.85

7.02

(continued)

544

G. Bauer and H. Krenn T A B L E XI

(Continued)

Lead Telluride eV

cm- 1

/~m

0.25

2,016

4.959

0.24 0.235 0.23 0.226 0.22

1,936 1,895 1,855 1,823 1,774

5.166 5.276 5.391 5.486 5.636

0.21 0.20

1,694

1,613

5.904 6.199

0.19 0.185

1,532 1,492

6.526 6.702

0.18 0.175 0.17 0.16 0.155 0.15 0.14 0.124

1,452 1,411 1,371 1,290 1,250 1,210 1,129 1,000

6.888 7.085 7.293 7.749 7.999 8.266 8.856 10.00

0.1215 0.1190 0.1165 0.1141 0.1116 0.1103 0.1091 0.1066 0.1041 0.1017 0.09199 0.09671 0.09423

980 960 940 920 900 890 880 860 840 820 800 780 760

10.20 10.42 10.64 10.87 11.11 11.24 11.36 11.63 11.90 12.20 12.50 12.82 13.16

0.09299 0.09175 0.08927 0.08679 0.08431 0.08183

750 740 720 700 680 660

13.33 13.51 13.89 14.29 14.71 15.15

n

k

5.84 [10-] 5.79 [9-] 5.79 5.82 [103

5.84

7.18

5.83

7.32

6.72 x 10 -4 [21]

3.9

x

10 -3

5.74 5.76110] 5.72 [9]

5.8 5.80 5.70[8] 5.79111]

5.77 5.69 [8] 5.77111] 5.76 5.68 [8] 5.76111]

5 68 [9"1 5.66 [9, 10"] 5.56 [4-7] 5.55 5.54 5.54 5.53 5.52 5.66 [8-1 5.51 [4-7] 5.49 5.48 5.47 5.45 5.44 5.64 [8] 5.51 [9, 10-] 5.42 [4-7] 5.40 5.38 5.36 5.34 5.31 5.57 [8-]

k

5.82 5.83 [8] 5.82111]

5.80 5.76[9] 5.78 [10] 5.77 5.74 [9-]

5.70 5.72 [9, 10]

n

5.67 [8] 4.97115,17] 5.29 5.63 6.01 6.42 6.87 7.37 7.91 8.5 9.16 9.9 x 10- a 1.07 x 10- 2

1.16 1.26 1.37 1.50 1.64 1.81

1.1 x 10 -a [21]

8.45 x 10 -4

Lead Telluride (PbTe)

545 T A B L E XI

(Continued)

Lead Telluride eV

cm- 1

pm

0.07935 0.07687 0.07439 0.07191 0.06943 0.06695

640 620 600 580 560 540

15.63 16.13 16.67 17.24 17.86 18.52

0.06447 0.06199 0.05951 0.05703 0.05455 0.05207 0.04959 0.04711 0.04463 0.04215 0.03968 0.03720 0.03472 0.03224 0.02976 0.02728 0.02480 0.02232 0.01984 0.1736 0.01488 0.01240 0.01215 0.01190 0.01165 0.01141 0.01116 0.01091 0.01066 0.01041 0.01017 0.009919 0.009671 0.009423 0.009175 0.008927 0.008679

520 500 480 460 440 420 400 380 360 340 320 300 280 260 240 220 200 180 160 140 120 100 98 96 94 92 90 88 86 84 82 80 78 76 74 72 70

19.23 20.00 20.83 21.74 22.73 23.81 25.00 26.23 27.78 29.41 31.25 33.33 35.71 38.46 41.67 45.45 50.00 55.56 62.50 71.43 83.33 100.0 102.0 104.2 106.4 108.7 111.1 113.6 116.3 119.0 122.0 125.0 128.2 131.6 135.1 138.9 142.9

n 5.28 [4-7] 5.25 5.22 5.18 5.13 5.09 5.48 !-8] 5.03 [4-7] 4.97 4.90 4.82 4.72 4.61 4.50 4.32 4.13 3.89 3.58 3.18 2.62 1.79 1.01 0.80 0.78 0.84 0.95 1.15 1.47 2.00 2.07 2.14 2.22 2.31 2.40 2.50 2.60 2.71 2.83 2.95 3.1 3.2 3.4 3.6 3.7

k

n

k

1.99 2.2 2.45 2.73 3.06 3.45 3.90 4.44 5.09 5.88 6.85 8.1 9.6 x 10- 2 0.12 0.14 0.18 0.23 0.32 0.48 0.87 1.9 3.2 4.4 5.6 6.9 8.5 10.4 13.0 13.3 13.6 13.9 14.3 14.6 15.0 15.4 15.8 16.2 16.6 17.1 17.6 18.1 18.6 19.2

(continued)

546

G. Bauer and H. Krenn T A B L E XI

(Continued)

Lead Telluride eV 0.008431 0.008183 0.007935 0.007687 0.007439 0.007191 0.006695 0.006447 0.006199 0.005951 0.005703 0.005455 0.005207 0.004959 0.004711 0.004463 0.004215 0.003968 0.003720 0.003472 0.003224 0.002976 0.002728 0.002480 0.002232 0.001984 0.001736 0.001488 0.001240

cm- 1 68 66 64 62 60 58 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10

n

/~m 147.1 151.5 156.3 161.3 166.7 172.4 185.2 192.3 200.0 208.3 217.4 227.3 238.1 250.0 263.2 277.8 294.1 312.5 333.3 357.1 384.6 416.7 454.5 500.0 555.6 625.0 714.3 833.3 1000

"References are indicated in brackets.

3.9 4.2 4.4 4.7 4.9 5.3 6.0 6.4 6.9 7.5 8.2 9.1 10.2 11.6 13.6 16.6 20.8 25.3 27.7 27.5 26.5 26.0 26.4 27.6 29.4 32.0 35.4 39.6 45.1

k 19.8 20.4 21.1 21.8 22.5 23.3 25.1 26.1 27.2 28.4 29.8 31.3 32.9 34.8 36.9 38.8 39.9 38.8 35.7 33.4 33.2 34.5 36.6 39.1 42.0 45.1 48.7 52.8 57.8

n

k

Silicon (Si) DAVID F, EDWARDS t University of California Los Alamos National Laboratory Los Alamos, New Mexico

Index of refraction values have been reported for the transparent region (2 > 1.12 ~tm) of silicon by a number of investigators. For these reports the samples have ranged from single cystals to evaporated thin films. The measuring techniques with their inherent differences in accuracy have varied from investigator to investigator as have the measured wavelength ranges. For a given group of investigators using equivalent measurement techniques on samples of about the same quality, the index values disagree by amounts greater than their stated precisions [1-3]. Rather than attempting to resolve this disagreement, we have selected those data by using the criteria given below and present them in Table XII as representative of average values. The data extend from 1.12 to 588 #m and are the results of six investigative groups. Essentially two different measuring techniques were used; angle of minimum deviation for the near infrared and channel spectra for the far infrared. The data-selection criteria are as follows" The samples were single crystal of good or optical quality. The reported refractive-index values were either tabulated or given in a figure which was readable to at least a few places in the third decimal place. We also required agreement in overlap areas to less than five in the third place between different data sets. Meeting these criteria were the following investigators: Primak [1], who determined the index from 1.12 to 2.16/tm from the angle of minimum deviation measured by an autocollimation technique; Salzberg and Villa [2], who measured the index from 1.3570 to 11.04 #m by a similar autocollimation technique; Edwards and Ochoa [3], who used a channel-spectra technique to measure the index from 2.4373 to 25 #m; Icenogle et al. [4], who used the autocollimator technique to measure the index from 2.554 to 10.270 ~tm; Loewenstein et al. [5], who determined the far-IR index (28.57-333 ~tm) from

* W o r k performed under the auspices of the U.S. D e p a r t m e n t of Energy. t Present address: Lawrence Livermore National Laboratory, Livermore, California. 547 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright (g) 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-544420-6

548

David F. Edwards

the channel spectra; and Randall and Rawcliffe [6], who also used the channelspectra technique to determine the index from 67.8 to 588/~m. Each of these investigators report the index values for room-temperature samples. We have fit these data to both a Herzberger-type and a Sellmeier-type dispersion formula. A total of 184 data points was used. The Herzberger-type dispersion formula [7] is n = A + B L + C L 2 -F D22 + E24,

where L = 1 / ( 2 2 - 0.028) with the wavelength 2 in micrometers and 0.028 is the square of the mean asymptote for the short-wavelength abrupt rise in index for 14 materials (silicon included) [7]. For the 184 data points, from 1.12 to 588 #m, the coefficients are A = 3.41906, B = 1.23172 x 10-1, C = 2.65456 x 1 0 - 2 , D - -2.66511 x 10-8, a n d E = 5.45852 x 10-~4. The quality of the fit of reported indices to the dispersion formula is good with differences in the third and fourth decimal places. The pertinent temperature is 26~ The Sellmeier-type dispersion formula [8] is n 2 = ~ + A / 2 2 + B22/(22 _ 22),

where 2~ = 1.1071/~m for silicon [8]. For the same 184 data points the coefficients are e = 1.16858 x 101, A = 9.39816 x 10 -1, and B = 8.10461 • 10- 3. The quality of the fit is similar to that for the Herzberger-type formula. In addition to their room-temperature measurements, Icenogle et al. [4] have measured the index for a number of temperatures between about 100 and 296 K for the range from 2.554 to 10.270 pm. For a given wavelength their temperature-dependent indices were fit to a polynomial from which we have evaluated d n / d T . These d n / d T are given in Table XIII for four wavelengths. Loewenstein et al. [-5] report the far-infrared index at 1.5 K in addition to 300 K. Channel spectra used for far-infrared index determination has been described by Randall and Rawcliffe [6] and by Loewenstein et al. [5]. This technique is particularly well suited for the far infrared because of the low order of the interference, typically 20 or 30. For the near infrared, special care must be exercised in using the channel spectra because of the very large fringe order and the difficulty in its determination [9]. A method suggested by Baumeister [10], and Edwards and Ochoa [3] was used to avoid this problem. The Baumeister method has a broad application, and the details are given here. For the channel-spectra technique, the samples must have faces that are flat to 2/4 (visible) or better and parallel to a second of arc or better. The flatness and wedge can be the determining factors in the accuracy of the measurement. Fringes of equal chromatic order [11] are produced when radiation undergoes multiple reflections between the parallel surfaces of the sample. The spacing between the fringes depends only on the sample thickness

Silicon (Si)

549

and the real part of the refractive index. Knowing the sample thickness and the wavelength of the fringe, one can, in principle, determine the index. A limitation is knowing the order of the fringe [9]. To overcome this [10], an arbitrary order number is assigned to a fringe and the entire channel spectra is fit to the polynomial m = ao + ale r + a2 0"2 + a3 0"3,

(1)

where the a~ coefficients are determined by a least-squares fit and m is the order number of the fringes at wave number a. The slope for this curve at any wave number is dm/dtr = a l + 2a2tr + 3a302.

(2)

In terms of the sample parameters, the order of the fringe is given by m = 2n(tr)htr,

(3)

where n(a) is the refractive index at wave number a and h the metric thickness. For most materials, such as silicon, with broad transparent regions; the index is a slowly varying function of tr and can accurately be represented as a polynomial

(4)

n = no + n~cr + n2 0-2,

where the n/are coefficients that can be evaluated from the ai coefficients and the metric thickness no = al/(2h),

nl = az/(2h),

n2 = a3/(2h).

(5)

The channel spectrum is usually taken in a beam of finite angle, and thus the ni coefficients must be corrected [6]. In Eq. (5) this beam-angle correction is accomplished by replacing h by h(1 + cos tim)/2, where flrn is the maximum internal angle of a ray with the normal to the surface. The index values evaluated by this Baumeister method are insensitive to the choice of the initial order. The accuracy of the index values is set by the sample thickness, flatness, and parallelism. Good optical practices will ensure sufficient flatness and parallelism. In a metrology laboratory the metric thickness can be determined to a few parts in 105, and thus the uncertainty in the index will be determined to first order by the uncertainty in the sample thickness. For index 2 this means an index uncertainty of about 1 - 5 x 10 -4. Optical-quality silicon is highly transparent from the band edge at 1.1071 #m to the far infrared with the exception of the 5- to 25-/~m lattice absorption band. Outside this lattice band and for 2 > 1.1071 #rn, the absorption coefficient has been measured at a few select laser wavelengths. Using a photoacoustic technique, Hordvik and Skolnik [12] have measured the absorption coefficient for silicon at the H F and D F laser wavelengths, 2.7 and 3.8 #m, respectively. Their results are given in Table XII.

550

David F. Edwards

Multiphonon absorption limits the transparency in the lattice band of silicon and other wide-gap semiconductors. Because of crystal symmetry, the fundamental vibration in silicon has no dipole moment and is infrared inactive. Lax and Burstein [13] have shown that the phonons can interact directly with the radiation field through terms in the electric moment of second order (or higher) in the atomic displacement. Johnson [14] analyzed the absorption in the 6.7- to 25-/~m region for oxygen-free silicon. He verified the absorption to be two phonon. Bendow et al. [15] extended these measurements to wider frequency and temperature ranges and found the multiphonon absorption structure to persist to low absorption levels and high temperatures. The work of Johnson [14] and Bendow et al. [15] confirms multiphonon interpretation of the 5- to 25-~m band. The absorption coefficient values of Johnson [ 14] and Bendow et al. [15] are given in Table XII for the major absorption peaks and shoulders. These correspond to combinations of four characteristic phonon energies. At wavelengths longer than 28/~m, the values for ~ of Loewenstein et al. [5] are approximately an order of magnitude larger than the extrapolation of the work of Lax and Burstein [13] and Johnson [14]. Therefore, we must omit this work as being anomalous due to surface damage or some other mechanism, as the authors suggest. Several recent Fourier-transform measurements of n and k (not using Kramers-Kronig analysis) have given smaller values of k [16-19]. The Si used was 10 ~ cm [16-18] or 1000 ~ cm [19] resistivity with carrier type not specified. These showed free-carrier effects below 100 cm-~ with n decreasing and k increasing. The plasma frequency for 10 fl cm n-type material is ~ 4 cm-1, and the carrier density is ~ 5 x 1014 cm-3. The data in the lattice-band region near 600 cm-1 in Honijk et al. [16] and Passchier et al. [17] appear to be in error both in peak value and in shape of bands compared to Lax and Burstein [13] and Johnson [14]. For example, the peak at 606 cm-1 is not 17 Np/cm but 9.5 cm-1. Also, there is no band of Si at 650 cm-1. The neper is usually a unit of amplitude attenuation and is ~/2 in optics language. Therefore, we do not use these data to lower frequency. Since the k values below 100 cm- 1 varied among the reports of Loewenstein et al. [5], Randall and Rawcliffe [6], Honijk et al. [16], Passchier et al. [17], Birch [18], and Afsar and Hasted [19], we conclude that the free-carrier densities were different. We choose the data of Afsar and Hasted [19] for k below 160 cm-~ (measured at 303 K) because the resistivity was the largest, which should minimize the free-carrier effects on k. Wavelength ellipsometry measurements have been carried out by Aspnes and Theeten [20] to determine n and k in the 1.5-6.0-eV region. Temperature was not stated but was probably 298 K. Samples of resistivity ~26 fl cm with {100}, {111}, and {110} surface orientations were prepared by Syton polishing; and dry thermal oxides were grown, stripped, and regrown so that a study of the transition layer between S i O 2 and Si could be made. The analysis involved mathematical removal of the oxide and its 6-A transition

Silicon (Si)

551

layer to obtain the final optical constants. The data were combined with e2 data of Hulth6n [21] below 2.5 eV to generate the final values we have tabulated. The Hulth6n work was transmission measurements of epitaxial films of silicon on sapphire substrates, so that surface damage was not a problem as might be the case for Dash and Newman [22], who used mechanically polished thin samples. Temperature was not stated but was probably 298 K. This is fairly important, since the band gap is quite sensitive to temperature [23]. It would also be useful to know whether the entire optical image was focused on the sample (possibly raising the temperature) or if the radiation was predispersed. Strain in the film might also cause a shift in band gap. We obtained n and k from reading graphs [21] and list them so as to get some overlap with data of Aspnes and Theeten [20] near the band edge. To slightly lower energy (1.0-1.3 eV), the absorption has been measured by McFarlane and Roberts [23] at 290 K for 100-f~-cm material, mechanically polished. The absorption coefficient is dependent on optical-phonon emission and absorption processes for the indirect transitions, thus being temperature sensitive. Since the data of Hulth6n [21] and McFarlane and Roberts [23] were at 298 and 290 K, respectively, the discrepancy at 1.2 eV is not too surprising (bearing in mind the graph-reading process, also). However, a factor of 5 difference at 1.2 eV is puzzling and leads us to believe that sample surface preparation, the type and magnitude of impurity doping, and possibly strain must produce significant absorption effects near the band edge. It has been pointed out by Russo [24] through reflection measurements of unpolarized incident radiation that n and k above the fundamental band gap do depend on resistivity (free-carrier density). However, sample resistivity was in the range 10-2 ~ cm (n ~ 1 x 1019 cm-3) compared to 95 f~ cm (~ 5 x 1013 cm -3) for the effect to be large. For example, at 0.5750 ~m k increased by a factor of 3 while n increased by 10%. This is probably a combination of free-carrier absorption coming in from the infrared and the Burstein-Moss effect at the fundamental and other higher-lying band gaps. Several investigators have measured reflectivity in the visible-ultraviolet region and then determined n and k from a Kramers-Kronig analysis [25-28]. The effect of native oxide on reflectivity measurements in this spectral region is discussed by Philipp [26]. It appears that the samples of Sasaki and Ishiguro [28] were mechanically polished, which would make them suspect at high energies. Philipp [25-27] used chemically prepared surfaces with minimum time spent in air before insertion into the vacuum spectrometer. We have therefore elected to tabulate the Philipp data from 5 to 25 eV overlapping the ellipsometric data for comparison. Interestingly, the data of Philipp [26] and Sasaki and Ishiguro [28] are in reasonable agreement at higher energies, typically < 30% for n but < 10% for k as well as at lower energy, despite the apparently different surface preparation. The

552

David F. Edwards

data of Aspnes and Theeten [20] and Philipp [26] are typically within < 59/o for k and < 109/o for n in the overlap region 5-6 eV. For the range 20-95 eV we found data of Hunter [29]. Samples were evaporated Si on glass plates for measurement of n by a critical-angle method; they were freestanding for measurement of k by transmission. While the samples were probably amorphous, we find that the data fit reasonably well between the results of Henke [30] and Philipp [26]. An interpolation has been done between 95 and 99 eV to keep the continuity. The absorption coefficient near the L2, 3 absorption edge of Si at ~ 100 eV has been measured [31, 32] with chemically etched and polished samples and with films flash evaporated onto NaC1 substrates and floated off; the latter were polycrystalline. A synchroton source was used. Some of the extended fine structure is apparent. From 99 to 2000 eV we have calculated n and k from a model of Henke et al. [30] as discussed by Lynch and Hunter in the metal section of this handbook. While the calculation cannot account for the fine structure, the magnitude of k is in reasonable agreement with experiment [31, 32]. Very recently, new measurements [33] of e' and e" have been made on n-type Si wafers at 23~ in the millimeter-wave region at 107.3 GHz (2 = 2.796 mm). The resistivity varied from 5 to 50 f~ cm (N = 1 x 1015 to 1 x 10 ~4 cm- 3). A transmission-reflection technique With the sample in free space was used. It was shown that a Drude model fit the data well with ~' - e o o

1

(1 + o92v 2) '

o9(1 + o92~2) '

2 = NeZ/eo ~ o~m* is the plasma frequency, N is the free-electron where ~Op density, e oo = 11.7 is the high-frequency dielectric constant, m* = 0.26 is the effective mass, T = 1 x 10-~3 sec is the constant scattering time, and mks units are used. Since the data fit well over the order-of-magnitude change in N, it is probably safe to extend the calculation of ~' and e" at least another order of magnitude in N in both directions to estimate how transparent a sample will be. Also, it is probably safe to calculate the reflectance and transmittance an order of magnitude to either side of 107.3 GHz.

REFERENCES

1. 2. 3. 4. 5. 6. 7.

W. Primak, App. Opt. 10, 759 (1971). C. D. Salzberg and J. J. Villa, J. Opt. Soc. Am., 47, 244 (1957). D. F. Edwards and E. Ochoa, Appl. Opt. 19, 4130 (1980). H. W. Icenogle, B. C. Platt, and W. L. Wolfe, Appl. Opt. 15, 2348 (1976). E. V. Loewenstein, D. R. Smith, and R. L. Morgan, Appl. Opt. 12, 398 (1973). C. M. Randall and R. D. Rawcliffe, Appl. Opt. 6, 1889 (1967). M. Herzberger and C. D. Salzberg, J. Opt. Soc. Am. 52, 420 (1962); M. Herzberger, Opt. Acta. 6, 197 (1959).

Silicon (Si) 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

31. 32. 33.

553

H. H. Li, J. Phys. Chem. Ref. Data 9, 561 (1980). T. S. Moss, "Optical Properties of Semi-Conductors." Butterworth, London, 1959. P. Baumeister, Optical Coating Laboratory, Inc., private communication (1979). S. Tolansky, Philos. Mag. 36, 225 (1945). A. Hordvik and L. Skolnik, Appl. Opt. 16, 2919 (1977). M. Lax and E. Burstein, Phys. Rev. 97, 39 (1955). F. A. Johnson, Proc. Phys. Soc. London 73, 265 (1959). B. Bendow, H. G. Lipson, and S. P. Yukon, Appl. Opt. 16, 2909 (1977). D. D. Honijk, W. F. Passchier, M. Mandel, and M. N. Afsar, Infrared Phys. 17, 9 (1977). W. F. Passchier, D. D. Honijk, M. Mandel, and M. N. Afsar, J. Phys. D 10, 509 (1977). J. R. Birch, Infrared Phys. 18, 613 (1978). M. N. Afsar and J. B. Hasted, Infrared Phys. 18, 835 (1978). D. E. Aspnes and J. B. Thee'ten, J. Electrochem. Soc. 127, 1359 (1980); D. E. Aspnes and J. B. Theeten, private communication (1980). R. Hulth~n, Phys. Scr. 12, 342 (1975). W. C. Dash and R. Newman, Phys. Rev. 99, 1151 (1955). G. G. McFarlane and V. Roberts, Phys. Rev. 98, 1865 (1955). O. I. Russo, J. Electrochem. Soc. 127, 953 (1980). H. R. Philipp and E. A. Taft, Phys. Rev. 120, 37 (1960). H. R. Philipp, J. Appl. Phys. 43, 2836 (1972); H. R. Philipp, private communication (1982). H. R. Philipp and H. Ehrenreich, in "Semiconductors and Semimetals" (R. K. Willardson and A. C. Beer, eds.), Vol. 2, p. 93. Academic Press, New York, 1967. T. Sasaki and K. Ishiguro, Phys. Rev. 127, 1091 (1962). W. R. Hunter, private communication of unpublished data (1982). B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, and B. K. Fujikawa, "Low Energy X-Ray Diagnostics--1981" (D. T. Attwood and B. L. Henke, eds.), AlP Conference Proc., No. 75, p. 340. American Institute of Physics, New York, 1981. C. G~ihwiller and F. C. Brown, Proc. Int. Conf. Phys. Semicond., lOth, Cambridge, 1970, p. 213. U.S. Atomic Energy Commission, Oak Ridge, Tennessee, 1970. F. C. Brown, R. Z. Bachrach, and M. Skibowski, Phys. Rev. B 15, 478 (1977). H. T. Kinasewitz and D. Senitzky, J. Appl. Phys. 54, 3394 (1983).

554

D a v i d F. E d w a r d s

I0 i -I ~ ' " I

f

'

'1

~'"l

'

'

'1

''"1

iii IJ I III

i(3o -

I

I,%/ L I

i(3-I _ /

#

/ /

/1

//'/

i(3.2 -

1 l 1 t

/ ;

c"

/

i9

~I

_

/

1I

/

I I I

/

I

o

I I

_ _

iI /

/

-

/

t~

I I

I

/ / /

I

11

;

i{3-4 _

/ / /

/

LI L2,3

/

[/ -

i i

-

I;

-

V

_ I

10-6- KI

I

I _

/

/

I I

10-7, [ ,,,,I i0-5

I

J

I I,,Itl

I

I

I I tltll

10-2

t

10-1

t

t I Jllll

WAVELENGTH F i g . 11. L o g - l o g plot of n ( ~ )

~

J

~ I,il,I

i0 o

,

I01

,

I I tJlll

l

t

~ I JlJ

10 2

(~m)

and k (. . . . ) versus wavelength in micrometers for silicon. D a t a from various references sometimes do not join together in overlap regions.

10 3

Silicon (Si)

555 T A B L E XII

Values of n and k for Silicon Obtained from Various References a eV 2,000 1,952 1,905 1,860 1,815 1,772 1,730 1,688 1,648

1,609 1,570 1,533 1,496 1,460 1,426 1,392 1,358 1,326 1,294 1,263 1,233 1,204 1,175 1,147 1,119 1,093 1,067 1,041 1,016 992 968 945 923 901 879 858 838 818 798 779 760 742 725 707 690 674

cm- 1

A

n

6.199 6.351 6.508 6.665 6.831 6.997 7.166 7.345 7.523 7.705 7.897 8.087 8.287 8.492 8.694 8.906 9.129 9.350 9.581 9.816 10.05 10.30 10.55 10.81 11.08 11.34 11.62 11.91 12.20 12.50 12.81 13.12 13.43 13.76 14.10 14.45 14.79 15.16 15.54 15.92 16.31 16.71 17.10 17.54 17.97 18.39

0.9999048 [30] 0.9999131 0.9999222 0.9999318 0.9999113 0.999885 0.999874 0.999863 0.999854 0.999844 0.999834 0.999824 0.999814 0.999803 0.999792 0.999781 0.999769 0.999756 0.999743 0.999729 0.999715 0.999700 0.999684 0.999667 0.999649 0.999631 0.999612 0.999591 0.999570 0.999548 0.999524 0.999500 0.999474 0.999447 0.999418 0.999388 0.999358 0.999325 0.999289 0.999253 0.999214 0.999174 0.999134 0.999088 0.999041 0.99899

k 3.19 3.46 3.76 4.08 3.93 4.32 4.74 5.20 5.70 6.24 6.85 7.49 8.22 8.99 9.81 1.07 1.17 1.28 1.40 1.53 1.67 1.83 2.00 2.18 2.39 2.60 2.84 3.10 3.39 3.70 4.04 4.41 4.80 5.24 5.73 6.25 6.80 7.42 8.11 8.84 9.66 1.05 1.14 1.25 1.36 1.48

n

k

• 10 -5 [30]

• 10 -5 x 10 -6

x 10 -6 • 10 -5

x 10 -5 x 10 -4

(continued)

556

David F. Edwards T A B L E XII

(Continued)

Silicon eV 658 642 627 612 597 583 569 555 542 529 516 504 492 480 469 458 447 436 426 415 406 396 386 377 368 359 351 342 334 326 318 311 303 296 289 282 275 269 262 256 250 244 238 232 227 221 216

cm- 1

A 18.84 19.31 19.77 20.26 20.77 21.27 21.79 22.34 22.87 23.44 24.03 24.60 25.20 25.83 26.43 27.07 27.74 28.44 29.10 29.87 30.54 31.31 32.12 32.89 33.69 34.53 35.32 36.25 37.12 38.03 38.99 39.86 40.92 41.88 42.90 43.96 45.08 46.09 47.32 48.43 49.59 50.81 52.09 53.44 54.62 56.10 57.40

k 0.99894 0.99889 0.99883 0.99878 0.99871 0.99865 0.99858 0.99851 0.99843 0.99836 0.99827 0.99819 0.99810 0.99800 0.99791 0.99781 0.99770 0.99759 0.99748 0.99735 0.99723 0.99710 0.99695 0.99682 0.99667 0.99651 0.99636 0.99619 0.99602 0.99585 0.99566 0.99549 0.99528 0.99509 0.99489 0.99468 0.99446 0.99427 0.99403 0.99382 0.99361 0.99339 0.99318 0.99296 0.99277 0.99254 0.99236

1.61 1.76 1.91 2.08 2.27 2.47 2.69 2.94 3.19 3.47 3.79 4.11 4.47 4.87 5.27 5.72 6.22 6.77 7.34 8.01 8.63 9.39 • 10 -4 1.02 • 10-3 1.11 1.20 1.31 1.41 1.53 1.66 1.80 1.95 2.10 2.29 2.47 2.67 2.88 3.13 3.36 3.65 3.93 4.23 4.57 4.94 5.34 5.70 6.19 6.63

n

k

Silicon (Si)

557 T A B L E XII

(Continued)

Silicon eV 211 210 206 205 201 200 196 195 191 190 187 185 182 180 178 175 174 170 166 165 162 160 158 155 154 152.5 150 147.5 147 145 143 140 136 135 133 130 127.5 127 125 124 121 120 118 115 112 110

cm- ~

A 58.76 59.04 60.18 60.48 61.68 61.99 63.25 63.58 64.91 65.26 66.30 67.02 68.12 68.88 69.65 70.85 71.25 72.93 74.69 75.14 76.53 77.49 78.47 79.99 80.51 81.30 82.66 84.05 84.34 85.51 86.70 88.56 91.16 91.84 93.22 95.37 97.24 97.62 99.19 99.98 102.5 103.3 105.1 107.8 110.7 112.7

n

k

0.99219

7.12

0.99206

7.63

0.99192

8.18

0.99179

8.79

0.99169

9.46 • 10 -3

0.99167

1.00 x 10 -2

0.99164

1.07

0.99164

1.14

0.99167 0.99176 0.99235

1.21 1.28 1.45

0.99235

1.45

0.99287

1.52

0.99340

1.57

0.99384

1.62

0.99415

1.66

0.99460 0.99496 0.99549

1.72 1.77 1.83

0.99594 0.99645

1.88 1.94

0.99705

2.00

0.99773 0.99856

2.06 2.12

0.99958 1.0008 1.0024 1.0039

2.19 2.26 2.33 2.39

6.30 x 10 -3 [31] 6.59 6.96 7.34 7.84 8.48 9.21 x lO -a 1.01 x 10- 2 1.14 1.21

1.29 1.38 1.42 1.40 1.37 1.43 1.54 1.66 1.81 1.88 1.91

1.85

1.63 1.38

(continued)

558

David F. Edwards T A B L E XII

(Continued)

Silicon eV 107 105 104.8 104.6 104.4 104.2 104 103.8 103.6 103.4 103.2 103 102.8 102.6 102.4 102.2 102 101.8 101.6 101.4 101.2 101 100.8 100.6 100.4 100.2 100 99.8 99.6 99.4 99.2 99.19 99.0 98.40 96.86 95.37 95 93.93 91.17 88.56 82.66 77.49 72.93 68.88 65.26 61.99 59.04

cm- 1

A 115.9 118.1 118.3 118.5 118.8 119.0 119.2 119.4 119.7 119.9 120.1 120.4 120.6 120.8 121.1 121.3 121.6 121.8 122.0 122.3 122.5 122.8 123.0 123.2 123.5 123.7 124.0 124.2 124.5 124.7 125.0 125 125.2 126 128 130 130.5 132 136 140 150 160 170 180 190 200 210

n 1.0070

k

n

k

2.47

~.081-32]

1.0132

2.56

1.0241

2.62

1.025

2.4 • 10- 2

1.030

5.0 • 10-3

1.032

1.54

1.034 [29]

1.0 [29]

1.034 1.034 1.03

1.02 1.09 1.13

1.022 1.008 1.000 0.993 0.991 0.988 0.985 0.982 0.978 0.976

1.2 1.3 1.43 1.78 2.15 2.54 2.97 3.43 3.93 4.43

1.09 1.08 1.08 1.09 1.09 1.07 1.09 1.14 1.14 1.12 1.16 1.17 1.13 1.13 1.16 1.22 1.29 1.36 1.37 1.40 1.40 1.34 1.12 1.OOx 10 -2 9.7 x 10-a 4.9 3.6 3.4 3.4 3.4

4.9131]

Silicon (Si)

559 T A B L E XII

(Continued)

Silicon eV

cm i

A

56.36 53.91 51.66 49.59 47.69 45.92 44.28 42.75 41.33 40.00 38.75 37.57 36.47 35.42 34.44 33.51 32.63 31.79 3 i.00 30.24 29.52 28.83 28.18 27.55 26.95 26.38 25.83 25.30 24.80 24.31 23.84 23.39 22.96 22.54 22.14 21.75 21.38 21.01 20.66 20.33 20.00

220 230 240 250 260 270 280 290 300 310 320 33O 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 52O 530 540 550 560 570 580 590 600 610 620

19.68 19.5 10.37 19.07

630 635.8 640 650

n 0.972 0.968 0.964 O.960 0.956 0.952 0.947 0.942 0.937 0.930 0.925 0.918 0.913 0.906 0.899 0.893 0.885 0.877 0.869 0.860 0.853 0.843 0.834 0.824 0.814 0.803 0.792 0.778 0.766 0.752 0.737 0.722 0.706 0.691 0.675 0.659 0.644 0.627 0.61 0.59 0.567 0.569 [26] 0.549 [29] 0.542 [26] 0.53 [29] 0.513

k 5.0 5.53 6.1 6.7 7.3 7.85 8.4 9.0 9.5 x 10 -3 1.0• 10 -2 1.04 1.O 1.13 1.17 1.21 1.24 1.28 1.32 1.35 1.38 1.42 1.47 1.52 1.58 1.68 1.78 1.92 2.05 2.23 2.43 2.64 2.92 3.25 3.65 4.05 4.55 5.1 5.8 6.5 7.4 8.35 • lO-Z 0.122 [26] 9.3 • 10-2 O.140 [26] 0.10129] 0.113

(continued)

560

David F. Edwards TABLE XII

(Continued)

Silicon eV 19.0 18.5 18.0 17.75 17.5 17.25 17.0 16.80 16.7 16.6 16.5 16.25 16.0 15.75 15.5 15.25 15.0 14.75 14.5 14.25 14.0 13.75 13.5 13.25 13.0 12.75 12.5 12.25 12.0 11.75 11.5 11.25

11.0 10.75 10.5 10.25 10.0 9.75 9.5 9.25 9.0 8.75 8.5 8.25 8.0 7.75 7.5 7.25

cm- 1

pm 0.06526 0.06702 0.06888 0.06985 0.07085 0.07187 0.07293 0.07380 0.07424 0.07469 0.07514 0.07630 0.07749 0.07872 0.07999 0.08130 0.08266 0.08406 0.08551 0.08700 0.08856 0.09017 0.09184 0.09357 0.09537 0.09724 0.09919 0.10120 0.1033 0.1055 0.1078 0.1102 0.1127 0.1153 0.1181 0.1210 0.1240 0.1272 0.1305 0.1340 0.1378 0.1417 0.1459 0.1503 0.1550 0.1600 0.1653 0.1710

n

0.514 [26] 0.485 0.455 0.440 0.426 0.411 0.397 0.386 0.379 0.374 0.369 0.357 0.345 0.333 0.323 0.313 0.304 0.296 0.288 0.281 0.275 0.269 0.265 0.261 0.258 0.256 0.255 0.256 0.257 0.259 0.263 0.267 0.272 0.278 0.286 0.295 0.306 0.318 0.332 0.348 0.367 0.389 0.414 0.444 0.478 0.517 0.563 0.618

0.163 [26] 0.189 0.219 0.237 0.255 0.275 0.296 0.314 0.323 0.333 0.342 0.367 0.394 0.421 0.450 0.479 0.510 0.541 0.573 0.607 0.641 0.677 0.714 0.752 0.792 0.833 0.875 0.918 0.963 1.01 1.06 1.11 1.16 1.21 1.26 1.32 1.38 1.45 1.51 1.58 1.66 1.73 1.82 1.90 2.00 2.10 2.21 2.32

k

Silicon (Si)

561 TABLE XII

(Continued)

Silicon eV 7.0 6.75 6.5 6.25 6.0 5.98 5.96 5.94 5.92 5.9 5.88 5.86 5.84 5.82 5.8 5.78 5.76 5.74 5.72 5.7 5.68 5.66 5.64 5.62 5.6 5.58 5.56 5.54 5.52 5.5 5.48 5.46 5.44 5.42 5.4 5.38 5.36 5.34 5.32 5.3 5.28 5.26 5.24 5.22 5.2 5.18 5.16

cm t

48,390 48,230 48,070 47,910 47,750 47,590 47,430 47,260 47,100 46,940 46,780 46,620 46,460 46,300 46,130 45,970 45,810 45,640 45,490 45,330 45,170 45,010 44,840 44,680 44,520 44,360 44,200 44,040 43,880 43,710 43,550 43,390 43,230 43,070 42,910 42,750 42,590 42,420 42,260 42,100 41,940 41,780 41,620

~m 0.1771 0.1837 0.1907 0.1984 0.2066 0.2073 0.2,080 0.2087 0.2094 0.2101 0.2109 0.2116 0.2123 0.2130 0.2138 0.2145 0.2153 0.2160 0.2168 0.2175 0.2183 0.2191 0.2198 0.2206 0.2214 0.2222 0.2230 0.2238 0.2246 0.2254 0.2263 0.2271 0.2279 0.2288 0.2296 0.2305 0.2313 0.2322 0.2331 0.2339 0.2348 0.2357 0.2366 0.2375 0.2384 0.2394 0.2403

n

k

0.682 0.756 0.847 0.968 1.11

2.45 2.58 2.73 2.89 3.05

1.24

3.18

1.40

3.33

1.51

3.40

1.64

3.44

1.75

3.42

1.78

3.36

1.010 [20] 1.036 1.046 1.066 1.070 1.083 1.088 1.102 1.109 1.119 1.133 1.139 1.155 1.164 1.175 1.180 1.195 1.211 1.222 1.235 1.247 1.265 1.280 1.299 1.319 1.340 1.362 1.389 1.416 1.445 1.471 1.502 1.526 1.548 1.566 1.579 1.585 1.590 1.591 1.592 1.589 1.586 1.582

2.909 [20] 2.928 2.944 2.937 2.963 2.982 2.987 3.005 3.015 3.025 3.045 3.061 3.073 3.086 3.102 3.112 3.135 3.150 3.169 3.190 3.206 3.228 3.245 3.267 3.285 3.302 3.319 3.334 3.350 3.359 3.366 3.368 3.368 3.364 3.358 3.353 3.346 3.344 3.344 3.347 3.354 3.363 3.376

(continued)

562

David F. Edwards TABLE XII

(Continued)

Silicon

eV 5.14 5.12 5.1 5.08 5.06 5.04 5.02 5.0 4.98 4.96 4.94 4.92 4.9 4.88 4.86 4.84 4.82 4.8 4.78 4.76 4.74 4.72 4.7 4.68 4.66 4.64 4.62 4.6 4.58 4.56 4.54 4.52 4.5 4.48 4.46 4.44 4.42 4.4 4.38 4.36 4.34 4.32 4.3 4.28 4.26 4.24 4.22 4.2

cm- 1 41,460 41,300 41,130 40,970 40,810 40,650 40,490 40,330 40,170 40,000 39,840 39,680 39,520 39,360 39,200 39,040 38,880 38,710 38,550 38,390 38,230 38,070 37,910 37,750 37,590 37,420 37,260 37,100 36,940 36,780 36,620 36,460 36,290 36,130 35,970 35,810 35,650 35,490 35,330 35,170 35,000 34,840 34,680 34,520 34,360 34,200 34,040 33,880

,um 0.2412 0.2422 0.2431 0.2441 0.2450 0.2460 0.2470 0.2480 0.2490 0.2500 0.2510 0.2520 0.2530 0.2541 0.2551 0.2562 0.2572 0.2583 0.2594 0.2605 0.2616 0.2627 0.2638 0.2649 0.2661 0.2672 0.2684 0.2695 0.2707 0.2719 0.2731 0.2743 0.2755 0.2768 0.2780 0.2792 0.2805 0.2818 0.2831 0.2844 0.2857 0.2870 0.2883 0.2897 0.2910 0.2924 0.2938 0.2952

n

k

1.72

3.42

1.68

3.58

n

1.579 1.573 1.571 1.570 1.569 1.568 1.569 1.570 1.575 1.580 1.584 1.591 1.597 1.608 1.618 1.629 1.643 1.658 1.673 1.692 1.713 1.737 1.764 1.794 1.831 1.874 1.927 1.988 2.059 2.140 2.234 2.339 2.451 2.572 2.700 2.833 2.974 3.120 3.277 3.444 3.634 3.849 4.086 4.318 4.525 4.686 4.805 4.888

k

3.389 3.408 3.429 3.451 3.477 3.504 3.533 3.565 3.598 3.632 3.670 3.709 3.749 3.789 3.835 3.880 3.928 3.979 4.031 4.088 4.149 4.211 4.278 4.350 4.426 4.506 4.590 4.678 4.764 4.849 4.933 5.011 5.082 5.148 5.206 5.257 5.304 5.344 5.381 5.414 5.435 5.439 5.395 5.301 5.158 4.989 4.812 4.639

Silicon (Si)

563 TABLE XII

(Continued)

Silicon eV 4.18 4.16 4.14 4.12 4.1 4.08 4.06 4.04 4.02 4.0 3.98 3.96 3.94 3.92 3.9 3.88 3.86 3.84 3.82 3.8 3.78 3.76 3.74 3.72 3.7 3.68 3.66 3.64 3.62 3.6 3.58 3.56 3.54 3.52 3.5 3.48 3.46 3.44 3.42 3.4 3.38 3.36 3.34 3.32 3.3 3.28 3.26

cm-

1

33,710 33,550 33,390 33,230 33,070 32,910 32,750 32,580 32~420 32,260 32,100 31,940 31,780 31,620 31,460 31,290 31,130 30,970 30,810 30,650 30,490 30,330 30;160 30,000 29,840 29,680 29,520 29,360 29,200 29,040 28,870 28,710 28,550 28,390 28,230 28,070 27,910 27,750 27,580 27,420 27,260 27,100 26,940 26,780 26,620 26,450 26,290

#m 0.2966 0.2980 0.2995 0.3009 0.3024 0.3039 0.3054 0.3069 0.3084 0.3100 0.311.5 0.3131 0.3147 0.3163 0.3179 0.3195 0.3212 0.3229 0.3246 0.3263 0.3280 0.3297 0.3315 0.3333 0.3351 0.3369 0.3388 0.3406 0.3425 0.3444 0.3463 0.3483 0.3502 0.3522 0.3542 0.3563 0.3583 0.3604 0.3625 0.3647 0.3668 0.3690 0.3712 0.3734 0.3757 0.3780 0.3803

n

k

n 4.941 4.977 4.999 5.012 5.020 5.021 5.020 5.018 5.015 5.010 5.009 5.010 5.009 5.012 5.016 5.021 5.029 5.040 5.052 5.065 5.079 5.095 5.115 5.134 5.156 5.179 5.204 5.231 5.261 5.296 5.336 5.383 5.442 5.515 5.610 5.733 5.894 6.089 6.308 6.522 6.695 6.796 6.829 6.799 6.709 6.585 6.452

4.480 4.335 4.204 4.086 3.979 3.885 3.798 3.720 3.650 3.587 3.529 3.477 3.429 3.386 3.346 3.310 3.275 3.242 3.211 3.182 3.154 3.128 3.103 3.079 3.058 3.039 3.021 3.007 2.995 2.987 2.983 2.984 2.989 2.999 3.014 3.026 3.023 2.982 2.881 2.705 2.456 2.169 1.870 1.577 1.321 1.110 0.945

(continued)

564

David F. Edwards TABLE XlI

(Continued)

Silicon

eV 3.24 3.22 3.2 3.18 3.16 3.14 3.12 3.1 3.08 3.06 3.04 3.02 3.0 2.98 2.96 2.94 2.92 2.9 2.88 2.86 2.84 2.82 2.8 2.78 2.76 2.74 2.72 2.7 2.68 2.66 2.64 2.62 2.6 2.58 2.56 2.54 2.52 2.5 2.48 2.46 2.44 2.42 2.4 2.38 2.36 2.34 2.32

cm- 1 26,130 25,970 25,810 25,650 25,490 25,330 25,160 25,000 24,840 24,680 24,520 24,360 24,200 24,040 23,870 23,710 23,550 23,390 23,230 23,070 22,910 22,740 22,580 22,420 22,260 22,100 21,940 21,780 21,660 21,450 21,290 21,130 20,970 20,810 20,650 20,490 20,330 20,160 20,000 19,840 19,680 19,520 19,360 19,200 19,030 18,870 18,710

~m 0.3827 0.3850 0.3875 0.3899 0.3924 0.3949 0.3974 0.4000 0.4025 0.4052 0.4078 0.4105 0.4133 0.4161 0.4189 0.4217 0.4246 0.4275 0.4305 0.4335 0.4366 0.4397 0.4428 0.4460 0.4492 0.4525 0.4558 0.4592 0.4626 0.4661 0.4696 0.4732 0.4769 0.4806 0.4843 0.4881 0.4920 0.4959 0.4999 0.5040 0.5081 0.5123 0.5166 0.5209 0.5254 0.5299 0.5344

n

k

n

6.316 6.185 6.062 5.948 5.842 5.744 5.654 5.570 5.493 5.420 5.349 5.284 5.222 5.164 5.109 5.058 5.009 4.961 4.916 4.872 4.831 4.791 4.753 4.718 4.682 4.648 4.615 4.583 4.553 4.522 4.495 4.466 4.442 4.416 4.391 4.367 4.343 4.320 4.298 4.277 4.255 4.235 4.215 4.196 4.177 4.159 4.140

k

0.815 0.714 0.630 0.561 0.505 0.456 0.416 0.387 0.355 0.329 0.313 0.291 0.269 0.255 0.244 0.228 0.211 0.203 0.194 0.185 0.185 0.170 0.163 0.149 0.149 0.133 0.131 0.130 0.131 0.134 0.120 0.120 0.090 0.094 0.083 0.079 0.077 0.073 0.073 0.066 0.072 0.060 0.060 0.056 0.053 0.043 0.045

Silicon (Si)

565 TABLE XII

(Continued)

Silicon eV 2.3 2.28 2.26 2.24 2.22 2.2 2.18 2.16 2.14 2.12 2.1 2.08 2.06 2.04 2.02 2.0 1.98 1.96 1.94 1.92

1.9 1.88 1.86 1.84 1.82

1.8 1.78 1.76 1.74 1.72

1.7 1.68 1.66 1.64 1.62

1.6 1.58 1.56 1.54

1.52 1.5 1.4 1.3 1.28 1.26 1.24

cm

-1

18,550 18,390 18,230 18,070 17,910 17,740 17,580 17,420 17,260 17,100 16,940 16,780 16,610 16,450 16,290 16,130 15,970 15,810 15,650 15,490 15,320 15,160 15,000 14,840 14,680 14,520 14,360 14,200 14,030 13,870 13,710 13,550 13,390 13,230 13,070 12,900 12,740 12,580 12,420 12,260 12,100 11,290 10,490 10,320 10,160 10,000

#m 0.5391 0.5438 0.5486 0.5535 0.5585 0.5636 0.5687 0.5740 0.5794 0.5848 0.5904 0.5961 0.6019 0.6078 0.6138 0.6199 0.6262 0.6326 0.6391 0.6458 0.6526 0.6595 0.6666 0.6738 0.6812 0.6888 0.6965 0.7045 0.7126 0.7208 0.7293 0.7380 0.7469 0.7560 0.7653 0.7749 0.7847 0.7948 0.8051 0.8157 0.8266 0.8856 0.9537 0.9686 0.9840 0.9999

2.96 • lO-Z [21]

2.4

2.25

1.97

1.42

1.25 x 10- 2

4.123 4.106 4.089 4.073 4.057 4.042 4.026 4.012 3.997 3.983 3.969 3.956 3.943 3.931 3.918 3.906 3.893 3.882 3.870 3.858 3.847 3.837 3.826 3.815 3.805 3.796 3.787 3.778 3.768 3.761 3.752 3.745 3.736 3.728 3.721 3.714 3.705 3.697 3.688 3.681 3.673

0.048 0.044 0.044 0.032 0.038 0.032 0.034 0.030 0.027 0.030 0.030 0.027 0.025 0.025 0.024 0.022 0.022 0.019 0.018 0.017 0.016 0.016 0.015 0.014 0.013 0.013 0.013 0.012 0.011 0.011 0.010 0.010 0.009 0.009 0.008 0.008 0.007 0.007 0.006 0.006 0.005

7.75 x 10-3 2.26 6.4 • 10 -4 [23] 5.1 4.0

(continued)

566

David F. Edwards T A B L E Xll

(Continued)

Silicon eV

cm- '

l~m

1.22 1.2 1.18 1.16 1.14 1.12 1.107 1.10 1.084 1.08 1.06 1.033 0.9037 O.8856 0.8093 0.7749 0.7310 0.6888 0.6199 O.5087 0.4959 0.4568 0.4133 0.3753 0.3626 0.3542 0.3263 0.3100 0.2480 0.2418 0.2356 0.2294 0.2232 0.2170 0.2108 0.2066 0.2046 0.1984 0.1922 0.1860 0.1798 0.1771 0.1736

9,840 9,679 9,518 9,356 9,195 9,034 8,929 8,872 8,741 8,711 8,550 8,333 7,289 7,143 6,527 6,250 5,896 5,556 5,000 4,103 4,000 3,684 3,333 3,027 2,925 2,857 2,632 2,500 2,000

1,400

1.016 1.033 1.051 1.069 1.088 1.107 1.12 1.127 1.144 1.148 1.170 1.20 1.372 1.4 1.532 1.6 1.696 1.8 2.0 2.4373 2.50 2.7144 3.00 3.3033 3.4188 3.50 3.80 4.00 5.00 5.128 5.263 5.405 5.556 5.714 5.882 6.00 6.061 6.250 6.452 6.667 6.897 7.00 7.143

0.1674 0.1612

1,350 1,300

7.407 7.692

1,950 1,900 1,850 1,800 1,750

1,700 1,667 1,650 1,600 1,550 1,500 1,450 1,428

n

k

n

k 2.8 1.8 1.2 x 10 -4 6.7 x 10 -5 4.2 2.5

1.07

3.5361 [1, 2, 3, 8] 1.3 x 10 -5 3.5295 5.8 x 10 -6 1.5 3.5193 3.5007 3.4876 3.4784 3.4710 3.4644 3.4578 3.4490 3.4434 3.4424 3.4393 3.4361 3.4335 3.4327 3.4321

2.5 x 10 -9 [12]

1.3 x 10 -8 3.4294 3.4261

1.99 x 10-7 [15] 2.82 x 10-'1 2.97 3.26 3.94 4.05 4.17

3.4242 5.64 8.46 1.74 2.66 2.45

x x x x

10 -7 10 -6 10 -6 10- 5

3.4231 1.68 1.87114] 1.62 [15] 2.49 2.82114]

Silicon (Si)

567 T A B L E XII

(Continued)

Silicon eV

cm- t

#m

n 3.4224

0.1550 0.1488

1,250 1,200

8.000 8.333

0.1426 0.1401 0.1395 0.1389 0.1378 0.1376 0.1364

1,150 1,130 1,125 1,120 1,111 1,110 1,100

8.696 8.850 8.889 8.929 9.00 9.009 9.091

0.1339 0.1314 0.1289 0.1265 0.1240 0.1215 0.1203 0.1190 0.1178 0.1165 0.1141 0.1116 0.1103 0.1091 0.1079 0.1066 0.1054 0.1041 0.1029 0.1017 0.1004 0.09919 0.09795 0.09671 0.09547 0.09423 0.09299 0.09184 0.09175 0.09051 0.08927 0.08803 0.08679 0.08555 0.08550

1,080 1,060 1,040 1,020 1,000 980 970 960 950 940 920 900 890 880 870 860 850 840 830 820 810 800 790 780 770 760 750 740.7 740 730 720 710 700 690 689.7

9.259 9.434 9.615 9.804 10.00 10.20 10.31 10.42 10.53 10.64 10.87 11.11 11.24 11.36 11.49 11.63 11.76 11.90 12.05 12.20 12.35 12.50 12.66 12.82 12.99 13.16 13.33 13.50 13.51 13.70 13.89 14.08 14.29 14.49 14.50

k 1.53115] 2.41 2.52114] 5.13115] 7.11 [14] 6.44115] 8.38 [14]

3.4219

3.4215

8.46 5.49 7.38 4.72 3.75 3.67 4.84 6.76 1.09 1.22 1.24 1.22 1.27 1.66 2.02 2.08 2.02 2.01 1.73 1.53 1.44 1.77 2.06 1.97 1.77 2.07 2.14 2.27 2.39 2.59

[15] [ 14]

• 10-5 • 10 -4

3.4209 3.12 3.03 2.21 1.57 1.02 1.12 3.4208

(continued)

568

David F. Edwards T A B L E XII

(Continued)

Silicon eV 0.08431 0.08307 0.08265 0.08183 0.08059 0.07935 0.07811 0.07687 0.07563 0.07513 0.07439 0.07315 0.07191 0.07067 0.06943 0.06695 0.06447 0.06199 0.05951 0.05703 0.05455 0.05207 0.04959 0.04835 0.04711 0.04587 0.04463 0.04339 0.04215 0.04092 0.03720 0.03100 0.02480 0.01984 0.01736 0.01488 0.01364 0.01240 0.01116 0.009919 0.008679 0.007430 0.006199 0.004959 0.003720

cm - 1 680 670 666.7 660 650 640 630 620 610 608 600 590 580 570 560 540 520 500 480 460 440 420 400 390 380 370 360 350 340 330 300 250 200 160 140 120 110 100 90 80 70 60 50 40 30

#rrt 14.71 14.93 15.00 15.15 15.38 15.63 15.87 16.13 16.39 16.50 16.67 16.95 17.24 17.54 17.86 18.52 19.23 20.00 20.83 21.74 22.73 23.81 25.00 25.64 26.32 27.03 27.78 28.57 29.41 30.30 33.33 40.00 50.00 62.50 71.43 83.33 90.91 100.0 111.1 125.0 142.9 166.7 200.0 250.0 333.3

a References "are indicated in brackets.

n

k

n

1.25 1.24 3.4207

3.4204

3.4201

1.41 1.52 1.64 5.05 9.63 1.11 1.25 7.16 4.18 3.65 4.15 3.84 3.40 3.18 2.86 2.22 1.56 1.27 1.06 9.15 9.39 1.05 1.12 1.08

x x x x

10 -4 10 -3 10- 3 10 -4

x 10 - 4

x 10 -5 x 10-5 x 10 -4

1.08 3.4200 [5, 6] 3.4199 3.4197 3.4195 3.4192 3.4190 3.4188 3.4185 3.4180 3.4170 3.4165 3.4160 3.4155

1.7 [19] 1.9 2.3 2.9 3.5 4.3 5.5 7.2 x 10 -4 1.Ox 10 - 3 1.4

k

Silicon (Si)

569 TABLE XIII Temperature Dependence of Index of Refraction a 2 (pm)

B1

B2

B3

2.554

-1.1127

x 10 - 4

2.0722 • 10 - 6

2.732

-1.0255

• 10 - 5

9.6748 • 10 - 7

-1.2145

5.190

-4.0089

• 10 - 5

1.3368 • 10 - 6

- - 2 . 3 4 5 8 • 10 - 9

10.270

-6.5198

x 10 -5,

1.6358 • 10 - 6

-3.2457

a

- - 4 . 0 • 10 - 9

dn/dT for T = 100 to 296 K; dn/dT - B 1 Jr- B 2 T -F B3 T2.

• 10 - 9 x 10 - 9

Silicon (Amorphous) (a-Si) H. PILLER Depaffment of Physics and Astronomy Louisiana State University Baton Rouge, Louisiana

The optical properties of amorphous silicon (a-Si) are sensitive to preparation conditions and to doping with hydrogen. Surface conditions and oxide films on the surface affect the optical measurements strongly, especially in the ultraviolet (UV). The optical properties of a-Si are also affected by the amount of disorder in the samples. Disorder is determined by substrate temperature, deposition rate, deposition method, impurities, vacuum conditions, annealing conditions, and environmental conditions. The mismatch of the expansion coefficients of the sample and the substrate introduces stress that can cause grain boundaries and voids in the sample. Annealing of the samples will reduce the strain but will generally increase the surface roughness. The a-Si samples should never be exposed to air before making the optical measurements. A measurement of the surface roughness should be performed, since specular reflectance measurements need to be corrected for imperfect (rough) surfaces. X-ray and low-energy electron-diffraction (LEED) measurements should be made to investigate the amorphous and surface properties of the samples. Pierce and Spicer [ 1] measured reflectance R from 0.4 to 11.8 eV. KramersKronig (KK) analysis was used to calculate the optical constants. The samples were evaporated in a vacuum of less than 5 x 10 - 6 torr by using low evaporation rates of 2 to 5 A/sec and large evaporator-to-substrate distance (~ 40 cm). Silicon single crystals (1000 ~ cm) were used as substrates. The R measurements were made in situ on a 600-A-thick film. The scattering in the R data indicated a relative measurement accuracy of _+1~. The optical constants above 12 eV were derived from an extrapolation of the measured R and transmittance T and are less accurate. A zero-frequency refractive index n of 3.4, identical to that in crystalline Si (c-Si), was obtained. The maximum of the extinction coefficient k occurs at 4 eV compared to a maximum in k at 4.3 eV in c-Si. The data are listed in Table XIV and shown in Fig. 12 for both 571 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright (~) 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-544420-6

572

H. Piller

n and k. Weiser et al. [2] measured R of amorphous glow-discharge Si (g-Si) in the energy range of up to l0 eV. The g-Si films were deposited on quartz substrates in a rf discharge in pure silane at 0.2 mbar. The deposition rate was 2 A/sec. The H'Si ratio in the samples varied as a function of deposition temperature (room temperature to 400~ between 50 and 4~. The optical constants quoted in Table XIV and shown in Fig. 12 were determined from the measurements on a sample with a substrate temperature of 400~ Here, R of g-Si shows a strong dependence on deposition temperature but a small dependence on annealing. Brodsky et al. [3] measured the optical constants in a-Si films deposited on single-crystal sapphire substrates. The films were deposited by rf sputtering of c-Si onto a room-temperature substrate in an Ar atmosphere at a pressure of 0.01 torr. The deposition rates varied between 200 and 600 A/min. Thicknesses of the films varied from 0.3 to 10 #m and were measured to within _+ 10~. The films were stored in air. The data quoted in the table and shown in the figure were determined on a room-temperature sample. The k values agree very well with the data of Pierce et al. [1]. Zanzucchi et al. [4] measured the absorption in g-Si. Films prepared from silane contain up to an estimated 15-20 at. ~ of H. The films were deposited by either dc or rf glow discharge in silane at a pressure of 0.5 to 2.0 torr of silane and at deposition rates in the range 0.1-0.5 #m/min. It was found that n of g-Si is within 10 to 20~ of the c-Si values. Vibration-band absorption due to bonded hydrogen was observed in g-Si associated with two modes, one at 2050 cm-1 and the other one at 635 cm-~. These modes are associated with the Si-H stretching mode and the low-energy wagging mode, respectively. The data given in Table XIV and shown in Fig. 12 are for rf g-Si deposited at 215 and 300~ respectively. The k values of rf g-Si, with substrate temperature of 420~ agree very well with the values of Weiser et al. [2] in the band-edge region. Here, the absorption coefficient was determined from the amplitude of interference fringes in 1- to 3-~tm-thick films. The measurement accuracy was _+(100-200)cm -1, which at 0.6/tm translates into _+(4.7-9.4) • 10 - 4 for k. The experimental data show that the type of discharge is an important factor in determining the optical properties of g-Si. The Si-H bond was shown to be stable to thermal treatments. Moddel et al. [5] determined the low-energy absorption in a-Si'H. The measurements were made on a 10-/tm-thick sputtered film. The range of photon energy hv was extended to lower energy by making use of photoconductivity (PC) data. The samples were made by rf sputtering of c-Si onto glass substrates in an Ar-H atmosphere. Interference fringes in the PC resulting from front and back surface reflections were not observed because of the low resolution of the spectrometer. Optical constants are shown in Fig. 12 and are listed in Table XIV. Evangelisti et al. [6] determined the absorption coefficient ~ of a-Si'H from the spectral dependence of the PC over four decades. The samples were

Silicon (Amorphous) (a-Si)

573

grown by rf glow discharge in a capacitive reactor in pure silane or in a silane-argon mixture. Film thicknesses ranged between 0.5 and 1 #m; the hydrogen content varied between 9 and 17%. The exponential dependence of the PC on photon flux and hv was studied, and it was found that the exponent/~ varies with hr. Value ~ was then evaluated by means of a recurrent expression. The optical constants presented in Table XIV and shown in Fig. 12 were determined by this procedure. Transmission data were used to normalize ~ deduced from PC. Freeman and Paul l-7] studied the transmission spectra of a-Si'H as a function of H content and substrate temperature. The samples were prepared by rf sputtering of a c-Si target in an Ar and H plasma. Corning 7059 glass was used as a substrate. The deposition rate was 1 A/sec and the Ar pressure was 5 x 10-3 torr. The hydrogenated sample (22 at. %) shows a lower n for hv between 0.04 and 0.50 eV; n decreases with increasing H pressure; n of a-Si and a-Si'H show very little hv dependence for hv < 0.5 eV. Resonance effects due to Si-H vibrational modes are observed in n of a-Si'H. Here, n of a-Si'H is about 25% less than n of a-Si. The error in n is ___4%. An increase in the H content in the film shifts the absorption edge to larger hv and also causes a decrease of the density. Refractive indices are shown in Fig. 12 and are listed in Table XIV. Brodsky and Lurio [8] measured the IR vibrational spectra of a-Si between 35 and 700 cm-1. The absorption spectrum is interpreted in terms of a disorder-induced breakdown of the selection rules for vibronic transitions. Wedged c-Si substrates were used to suppress internal multiple-reflection effects in the substrate. Interference effects within the thin a-Si films are negligibly small because of the low reflection coefficient at the a-Si to c-Si interface (approximately 0.03). Extinction coefficients are shown in Fig. 12 and are listed in Table XIV. Brown and Rustgi [9] studied the soft x-ray absorption spectra of a-Si. The spectrum for a-Si shows none of the detailed structure observed in c-Si. Extinction coefficients are shown in Fig. 12 and are listed in Table XIV. Jackson and Amer [ 10] measured ~ at the absorption edge in a-Si'H down to hv = 0.6 eV by using the photothermal deflection spectroscopy (PDS), which measures directly the optical absorption and is insensitive to scattering. No optical constants are listed here, however. Additional work is discussed by Chaleravertz and Kaplan [11]. REFERENCES

1. D. T. Pierce and W. E. Spicer, Phys. Rev. B 5, 3017 (1972). 2. G. Weiser, D. Ewald, and M. Milleville, J. Non-Cryst. Solids 35/36, 447 (1980). 3. M. H. Brodsky, R. S. Title, K. Weiser, and G. D. Pettit, Phys. Rev. B 1, 2632 (1970). 4. P. J. Zanzucchi, C. R. Wronski, and D. E. Carlson, J. Appl. Phys. 48, 5227 (1977). 5. G. Moddel, D. A. Anderson, and W. Paul, Phys. Rev. B 22, 1918 (1980). 6. F. Evangelisti, P. Fiorini, G. Fortunato, A. Frova, C. Giovannella, and R. Peruzzi, J. NonCryst. Solids 55, 191 (1983).

574

7. 8. 9. 10. 11.

H. Piller

E. C. Freeman and W. Paul, Phys. Rev. B 20, 716 (1979). M. H. Brodsky and A. Lurio, Phys. Rev. B 9, 1646 (1974). F. C. Brown and O. P. Rustgi, Phys. Rev. Lett. 28, 497 (1972). B. Jackson and N. M. Amer, Phys. Rev. B 25, 5559 (1982). B. K. Chakravertz and D. Kaplan, J. de Phys. 42, Colloque C-4 (1981).

'

'

' I''"I

'

'

' I''"I

'

'

' I''"I

'

'

[2]

../-~

' I~'~,

I

[7]

}'I \

[z]

i0 o-

i0-I

/

//

L lll

I

/x

l

\

' I

/ /

//D]

/

1

io-Z _~ I

'

1'i[4]

\

I[9]

c-

[411

i!

[4]\ \

I

"

i

I \\\

f~

\ [3]

[8]

io-3

II

io-4

~[5] 10-5 J

Io-Z

~

, I,,~I

[

i

I lllil

I0-I i0 0 WAVELENGTH

I

i

I ]liill

I

I01

i

i

J,tli

102

(~Lrn)

Fig. 12. Log-log plot of n ( - - ) and k (. . . . ) versus wavelength in micrometers for amorphous silicon. Various references are indicated in brackets.

Silicon (Amorphous) (a-Si)

575

.2

r--"l

r•"-I

/

0 X ,-~




._i >

0

=

A

ii!

9

l

i

i

i

Iiii

o 0

i

I

i

i

illll

i

I

i

i

i

lllll

I

I

I

I

IIIII

I

I

I

I

I

0

o~ ,o u 0

A r s e n i c S e l e n i d e (As2Se3) illl

627

I

lOo

1I

J!

I

iO-I

fi

I

\J \\ \

\ \\ \ \\ \ \\ \\

Io-z

\ \\ \ \\\ \

II;

\

\ \\ \

~J

io-3 I

I

I I I I I

10-4 t.-'

l

I I i

10-5 ,,,

I

I 1 [ lllll

0o

l]

1

I

I

II111

I0 ~ i0 2 WAVELENGTH

I

I

[

I iiill

i

I

I I 1t111

103

i

i

i

I ill

104

05

(p,m)

Fig. 2. Log-log plot of n ( - - ) and k(. . . . ) versus wavelength in micrometers for vitreous AszSe3. Two significant figures are sometimes not enough to show a smooth variation in n and k, and so the curves have been drawn through these points to smooth out steps. TABLE I Values of n and k for Crystalline As2Se 3 from Various References"

eV 2.194 2.168 2.141 2.123 2.098 2.094 2.091 2.073 2.060 2.049 2.036 2.023 2.013 2.009

cm- 1 17,700 17,480 17,270 17,120 16,920 16,890 16,860 16,720 16,610 16,530 16,420 16,310 16,230 16,210

#m 0.565 0.572 0.579 0.584 0.591 0.592 0.593 0.598 0.602 0.605 0.609 0.613 0.616 O.617

n,,

nc

k.

kc

0.30 [-2] 0.25 0.20 0.17 O.13

0.10 0.079

0.26 [-2] 0.26 0.23 0.20 0.17 0.15 0.12

0.050 0.097

(continued)

628

D.J. Treacy TABLE I

(Continued)

Crystalline As2Se 3

eV

cm- 1

2.000 1.987 1.977 1.974 1.962 1.953 1.949 1.937 1.925 1.922 1.905 1.893 1.881 1.859 1.848 1.845 1.842 1.831 1.826 1.821 1.818 1.815 1.807 1.802 0.06199 0.05904 0.05636 0.05391 0.05166 0.04959 0.04769 0.04592 0.04428 0.04275 0.04133 0.04065 0.04000 0.03936 0.03875 0.03815 0.03757 0.03701 0.03647 0.03594 0.03542 0.03493 0.03444

16,130 16,030 15,940 15,920 15,820 15,750 15,720 15,630 15,530 15,500 15,360 15,270 15,170 14,990 14,900 14,880 14,860 14,770 14,730 14,680 14,660 14,640 14,580 14,530 500.0 476.2 454.5 434.8 416.7 400.0 384.6 370.4 357.1 344.8 333.3 327.9 322.6 317.5 312.5 307.7 303.0 298.6 294.1 289.9 285.7 281.7 277.8

lam

0.620 0.624 0.627 0.628 0.632 0.635 0.636 0.640 0.644 0.645 0.651 0.655 0.659 0.667 0.671 0.672 0.673 0.677 0.679 0.681 0.682 0.683 0.686 0.688 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 30.5 31.0 31.5 32.0 32.5 33.0 33.5 34.0 34.5 35.0 35.5 36.0

na

ka

nc

0.082 0.063 0.031 0.051 0.038 0.030 0.020 0.022 0.017 0.012 8.6x 10 -3 6.4 5.2 3.1 1.7 x 10 -3 2.0 1.3 x 10 - 3

1.2 x 10 - 3 9.0 x 10 -4 6.4 4.7

8.6 x 10 -4 3.4

3.211] 3.1 3.1 3.1 3.1 3.1 3.1 3.0 3.O 3.0 3.0 2.9 2.9 2.9 2.9 2.9 2.8 2.8 2.8 2.7 2.7 2.7 2.6

2.911] 2.9 2.9 2.9 2.8 2.8 2.8 2.8 2.8 2.8 2.7 2.7 2.7 2.7 2.7 2.6 2.6 2.6 2.6 2.6 2.5 2.5 2.4

5.5 4.1 1.7 x 10 -3 [1] 2.1 2.5 3.0 3.6 4.3 5.2 6.3 7.6 9.2 x 10 - 3 0.011 0.012 0.014 0.015 0.017 0.019 0.021 0.024 0.027 0.031 0.037 0.040 0.046

1.8 x 10 -3 [1] 2.2 2.6 3.1 3.7 4.5 5.3 6.4 7.7 9.3 x 10 - 3 0.011 0.012 0.014 0.015 0.017 0.019 0.021 0.023 0.027 0.030 0.034 0.040 0.046

Arsenic Selenide (As2Se3)

629 TABLE I

(Continued)

Crystalline As2Se 3 eV 0.03397 0.03351 0.03306 0.03263 0.03220 0.03179 0.03139 0.03100 0.03061 0.03024 0.02988 0.02952 0.02917 0.01883 0.02850 0.02818 0.02786 0.02755 0.02725 0.02695 0.02666 0.02638 0.02610 0.02583 0.02556 0.02530 0.02505 0.02480 0.02455 0.02431 0.02407 0.02384 0.02362 0.02339 0.02317 0.02296 0.02275 0.02254 0.02234 0.02214 0.02194 0.02175 0.02156 0.02138 0.02119 0.02101

cm- 1

#m

274.0 270.3 266.7 263.2 259.7 256.4 253.2 250.0 247.0 244.0 241.0 238.1 235.3 232.6 229.9 227.3 224.7 222.2 219.8 217.4 215.1 212.8 210.5 208.3 206.2 204.1 202.0 200.0 198.0 196.1 194.2 192.3 190.5 188.7 186.9 185.2 183.5 181.8 180.2 178.6 177.0 175.4 173.9 172.4 170.9 169.5

36.5 37.0 37.5 38.0 38.5 39.0 39.5 40.0 40.5 41.0 41.5 42.0 42.5 43.0 43.5 44.0 44.5 45.0 45.5 46.0 46.5 47.0 47.5 48.0 48.5 49.0 49.5 50.0 50.5 51.0 51.5 52.0 52.5 53.0 53.5 54.0 54.5 55.0 55.5 56.0 56.5 57.0 57.5 58.0 58.5 59.0

na 2.6 2.5 2.4 2.4 2.3 2.2 2.1 1.9 2.0 1.7 1.4 1.2 1.1 1.2 1.6 2.3 3.4 4.2 4.1 3.7 3.3 2.8 2.3 2.0 2.2 3.0 4.8 6.5 6.6 6.1 5.7 5.4 5.2 5.0 4.8 4.7 4.6 4.5 4.5 4.4 4.3 4.3 4.2 4.2 4.1 4.1

nc 2.4 2.3 2.3 2.2 2.1 2.0 1.7 1.7 2.6 2.4 2.1 1.9 1.6 1.3 1.2 1.2 1.5 2.0 3.0 4.4 5.2 5.3 5.0 4.7 4.5 4.3 4.2 4.0 3.9 3.9 3.8 3.7 3.7 3.6 3.6 3.5 3.5 3.5 3.4 3.4 3.4 3.4 3.3 3.3 3.3 3.3

ka 0.053 0.063 0.083 0.089 0.11 0.14 0.19 0.38 0.33 0.41 0.62 1.0 1.6 2.2 2.8 3.3 3.3 2.5 1.6 1.2 1.0 1.1 1.5 2.2 3.2 4.3 4.8 3.6 2.0 1.2 0.76 0.54 0.41 0.33 0.27 0.23 0.20 0.17 0.15 0.14 0.13 0.12 0.11 0.10 0.098 0.094

0.054 0.066 0.081 0.10 0.14 0.21 0.41 1.0 0.95 0.46 0.38 0.44 0.61 0.94 1.4 2.0 2.7 3.3 3.9 3.7 2.6 1.6 1.0 0.71 0.52 0.40 0.32 0.26 0.22 0.19 0.17 0.15 0.14 0.13 0.12 0.11 0.11 0.10 0.10 0.099 0.098 0.097 0.097 0.097 0.098 0.10

(continued)

630

D.J. Treacy TABLE I (Continued) Crystalline As2Se 3

eV

cm- 1

I~m

na

?tc

ka

0.02084 0.02066 0.02049 0.02033 0.02016 0.02000 0.01984 0.01968 0.01953 0.01937 0.01922 0.01907 0.01893 0.01879 0.01864 0.01851 0.01837 0.01823 0.01810 0.01797 0.01784 0.01771 0.01759 0.01746 0.01734 0.01722 0.01710 0.01698 0.01687 0.01675 0.01664 0.01653 0.01642 0.01631 0.01621 0.01610 0.01600 0.01590 0.01579 0.01569 0.01560 0.01550 0.01540 0.01531 0.01521 0.01512 0.01503

168.1 166.7 165.3 163.9 162.6 161.3 160.0 158.7 157.5 156.3 155.0 153.8 152.7 151.5 150.4 149.3 148.1 147.1 146.0 144.9 143.9 142.9 141.8 140.8 139.9 138.9 137.9 137.0 136.1 135.1 134.2 133.3 132.4 131.6 130.7 129.9 129.0 128.2 127.4 126.6 125.8 125.0 124.2 123.4 122.7 122.0 121.2

59.5 60.0 60.5 61.0 61.5 62.0 62.5 63.0 63.5 64.0 64.5 65.0 65.5 66.0 66.5 67.0 67.5 68.0 68.5 69.0 69.5 70.0 70.5 71.0 71.5 72.0 72.5 73.0 73.5 74.0 74.5 75.0 75.5 76.0 76.5 77.0 77.5 78.0 78.5 79.0 79.5 80.0 80.5 81.0 81.5 82.0 82.5

4.1 4.0 4.0 4.0 4.0 3.9 3.9 3.9 3.9 3.8 3.8 3.8 3.8 3.8 3.7 3.7 3.7 3.7 3.7 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.6 3.7 3.7 3.7 3.7 3.7 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 [1] 3.7 3.7 3.7

3.3 3.2 3.2 3.2 3.2 3.2 3.2 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 [1] 3.1 3.1 3.1 3.0

0.091 0.089 0.087 0.086 0.085 0.085 0.086 0.087 0.088 0.091 0.093 0.097 0.10 0.11 0.11 0.12 0.13 0.14 0.15 0.16 0.18 0.19 0.21 0.23 0.26 0.28 0.31 0.33 0.36 0.38 0.40 0.41 0.42 0.42 0.41 0.40 0.38 0.37 0.35 0.33 0.31 0.29 0.27 0.26 0.24 0.23 0.22

0.10 0.10 0.11 0.11 0.12 0.12 0.13 0.13 0.14 0.15 0.15 0.16 0.17 0.18 0.20 0.21 0.22 0.24 0.25 0.27 0.28 0.30 0.32 0.33 0.35 0.37 0.38 0.40 0.41 0.42 0.43 0.44 0.44 0.45 0.45 0.44 0.44 0.43 0.43 0.42 0.41 0.40 0.39 0.38 0.37 0.37 0.36

Arsenic Selenide (As2Se3)

631 TABLE I

(Continued)

Crystalline AszSe 3 eV

cm- 1

#m

na

0.01494 0.01485 0.01476 0.01467 0.01459 0.01450 0.01442 0.01433 0.01425 0.01417 0.01409 0.01401 0.01393 0.01384 0.01378 0.01370 0.01362 0.01355 0.01348 0.01340 0.01333 0.01326 0.01319 0.01312 0.01305 0.01298 0.01292 0.01284 0.01278 0.01272 0.01265 0.01259 0.01252 0.01246 0.01240 0.01228 0.01216 0.01204 0.01192 0.01181 0.01170 0.01159 0.01148 0.01137 0.01127 0.01117

120.5 119.8 119.0 118.3 117.6 117.0 116.3 115.6 114.9 114.3 113.6 113.0 112.3 111.7 111.1 110.5 109.9 109.3 108.7 108.1 107.5 107.0 106.4 105.8 105.3 104.7 104.2 103.6 103.1 102.6 102.0 101.5 101.0 100.5 100.0 99.01 98.04 97.09 96.15 95.24 94.34 93.46 92.59 91.74 90.91 90.09

83.0 83.5 84.0 84.5 85.0 85.5 86.0 86.5 87.0 87.5 88.0 88.5 89.0 89.5 90.0 90.5 91.0 91.5 92.0 92.5 93.0 93.5 94.0 94.5 95.0 95.5 96.0 96.5 97.0 97.5 98.0 98.5 99.0 99.5 100.0 101.0 102.0 103.0 104.0 105.0 106.0 107.0 108.0 109.0 110.0 111.0

3.7 3.7 3.7 3.6 3.6 3.6 3.5 3.5 3.5 3.4 3.4 3.4 3.3 3.2 3.2 3.1 3.0 3.0 3.0 3.0 3.2 3.5 4.0 4.4 4.7 4.8 4.8 4.8 4.7 4.7 4.6 4.5 4.5 4.5 4.4 4.3 4.3 4.2 4.2 4.2 4.1 4.1 4.1 4.1 4.1 4.0

3.0 3.0 3.0 3.0 2.9 2.9 2.9 2.9 2.8 2.8 2.8 2.7 2.7 2.6 2.6 2.5 2.4 2.4 2.3 2.3 2.4 2.4 2.6 2.8 3.0 3.2 3.3 3.3 3.3 3.3 3.2 3.1 3.0 2.8 2.7 2.4 2.1 1.9 2.2 3.0 4.7 6.0 6.1 5.7 5.3 5.1

0.21 0.21 0.20 0.20 0.20 0.20 0.21 0.21 0.22 0.24 0.26 0.28 0.32 0.37 0.43 0.52 0.64 0.81 1.0 1.3 1.6 1.8 1.9 1.8 1.5 1.2 0.94 0.74 0.59 0.48 0.40 0.34 0.29 0.25 0.22 0.18 0.15 0.12 0.11 0.094 0.084 0.076 0.069 0.064 0.059 0.055

0.35 0.35 0.34 0.34 0.34 0.34 0.34 0.34 0.35 0.36 0.37 0.39 0.42 0.45 0.49 0.55 0.62 0.71 0.83 0.97 1.1 1.3 1.5 1.5 1.5 1.5 1.3 1.2 1.0 0.92 0.84 0.79 0.77 0.78 0.81 1.0 1.4 2.1 3.0 3.9 4.1 3.0 1.7 1.0 0.70 0.51

(continued)

632

D.J. Treacy TABLE I (Continued) Crystalline As2Se 3

eV

cm- ~

#m

n.

nc

ka

k~

0.01107 0.01097 0.01088 0.01078 0.01069 0.01060 0.01051 0.01042 0.01033 0.01025 0.01016 0.01008 0.009999 0.009919 0.009840 0.009763 0.009686 0.009611 0.009537 0.009465 0.009393 0.009322 0.009253 0.009184 0.009117 0.009050 0.008984 0.008920 0.008856 0.008551 0.008266 0.007999 0.007749 0.007514 0.007293 0.007085 0.006888 0.006702 0.006526 0.006358 0.006199

89.29 88.50 87.72 86.96 86.21 85.47 84.75 84.03 83.33 82.64 81.97 81.30 80.65 80.00 79.37 78.74 78.12 77.52 76.92 76.34 75.76 75.19 74.63 74.07 73.53 72.99 72.46 71.94 71.43 68.97 66.67 64.52 62.50 60.61 58.82 57.14 55.55 54.05 52.63 51.28 50.00

112.0 113.0 114.0 115.0 116.0 117.0 118.0 119.0 120.0 121.0 122.0 123.0 124.0 125.0 126.0 127.0 128.0 129.0 130.0 131.0 132.0 133.0 134.0 135.0 136.0 137.0 138.0 139.0 140.0 145.0 150.0 155.0 160.0 165.0 170.0 175.0 180.0 185.0 190.0 195.0 200.0

4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.8

4.9 4.8 4.7 4.6 4.5 4.4 4.3 4.3 4.2 4.2 4.2 4.1 4.1 4.1 4.0 4.0 4.0 4.0 4.0 4.0 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.9 3.8 3.8 3.8 3.7 3.7 3.7 3.7 3.7 3.7 3.7 3.6 3.6 3.6

0.051 0.048 0.045 0.043 0.041 0.039 0.037 0.036 0.034 0.033 0.032 0.030 0.029 0.028 0.028 0.027 0.026 0.025 0.024 0.024 0.023 0.023 0.022 0.022 0.021 0.021 0.020 0.020 0.019 0.018 0.016 0.015 0.014 0.013 0.012 0.012 0.011 0.010 0.010 9.5 x 10 -3 9.1

0.39 0.31 0.26 0.22 0.19 0.17 0.15 0.14 0.13 0.12 0.11 0.10 0.094 0.088 0.083 0.079 0.075 0.072 0.069 0.066 0.063 0.061 0.059 0.057 0.055 0.053 0.051 0.050 0.048 0.043 0.038 0.034 0.032 0.029 0.027 0.025 0.024 0.022 0.021 0.020 0.019

a The radiation electric field is parallel to the a and c axes of the crystal. References are indicated in brackets and a reference applies down the column until a new bracket appears. Usually, two significant figures are given.

Arsenic Selenide (As2Se3)

633 TABLE II

Values of n and k for Vitreous As2Se3 a

eV 2.056 2.039 2.026 2.006 1.990 1.965 1.946 1.925 1.905 1.876 1,848 1.826 1.810 1.794 1.771 1.749 1.729 1.715 1.701 1.647 1.629 1.596 1.579 1.562 1.544 1.529 1.512 1.494 1.476 1.442 1.409 1.378 1.348 1.319 1.292 1.265 1.240 1.216 1.192 1.170 1.148 1.127 1.107 1.088 1.069 1.051

cm- 1 16,580 16,450 16,340 16,180 16,050 15,850 15,700 15,530 15,360 15,130 14,900 14,730 14,600 14,470 14,290 14,100 13,950 13,830 13,720 13,280 13,140 12,870 12,740 12,590 12,450 12,330 12,200 12,050 11,910 11,630 11,360 11,110 10,870 10,640 10,420 10,200 10,000 9,804 9,615 9,434 9,259 9,091 8,929 8,772 8,621 8,475

#m 0.603 0.608 0.612 0.618 0.623 0.631 0.637 0.644 0.651 0.661 0.671 0.679 0.685 0.691 0.700 0.709 0.717 0.723 0.729 0.753 0.761 0.777 0.785 0.794 0.803 0.811 0.820 0.830 0.840 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18

n

3.07 [19] 3.06 3.05 3.05 3.04 3.03 3.03 3.02 3.01 3.00 2.99 2.98 2.97 2.96 2.95 2.94 2.93 2.93 2.92 2.92 2.92 2.90 2.90 2.89 2.89 2.89

k 0.12 [2] 0.11 0.11 9.9 x 10 -2 9.0 7.6 6.8 5.6 4.5 3.1 2.1 1.4 1.2x 10 -2 8.9 x 10 -3 6.2 4.5 3.3 2.6 2.2 x 10 -3 4.6 x 10 -4 4.0 2.7 1.9 1.3 x 10 -4 9.4 x 10 .5 6.3 4.2 2.8 1.8

(continued)

634

D.J. Treacy TABLE II

(Continued)

Vitreous eV 1.033 0.2555 0.2380 0.2344 0.1345 0.1339 0.1333 0.1308 0.1302 0.1289 0.1277 0.1227 0.1215 0.1203 0.1196 0.1178 0.1170 0.1165 0.1159 0.1147 0.1141 0.1116 0.1103 0.1091 0.1079 0.1066 0.1054 0.1041 0.1029 0.1017 0.1004 0.09919 0.09795 0.09671 0.09547 0.09423 0.09299 0.09175 0.09051 0.08927 0.08803 0.08679 0.08555 0.08431 0.08307 0.08183 0.08059

cm

-1

8,333 1,980 1,919 1,890 1,085 1,080 1,075 1,055 1,050 1,040 1,030 990 980 970 965 950 943.4 940 935 925 920 900 890 880 870 860 850 840 830 820 810 800 790 780 770 760 750 740 730 720 710 700 690 680 670 660 650

As2Se 3

#m 1.20 5.05 5.21 5.29 9.22 9.26 9.30 9.48 9.52 9.62 9.71 10.10 10.20 10.31 10.36 10.53 10.60 10.64 10.70 10.81 10.87 11.11 11.24 11.36 11.49 11.63 11.76 11.90 12.05 12.20 12.35 12.50 12.66 12.82 12.99 13.16 13.33 13.51 13.70 13.89 14.08 14.29 14.49 14.71 14.93 15.15 15.38

n 2.88 1.6 x 10 -7 [15] 9.9 x 10 -8 1.1 x 10 - 7 4.4 3.7 4.4 4.5 4.9 6.1 6.6 8.4115] 8.9 9 . 9 x 10 - 7 1.0 x 10 - 6 1.1 1.3 1.1 1.1 1.2 1.1 1.8 1.8 1.8 1.8 1.9 1.9 2.3 2.9 3.9 4.9 7.0 • 10 - 6 1.Ox 10 -5 1.5 2.2 2.9 3.7 4.3 5.5 6.1 6.7 6.8 6.9 5.9 5.9 6.0 6.1

Arsenic Selenide (As2Se3)

635 TABLE II

(Continued)

Vitreous As2Se3 eV 0.07935 0.07811 0.07687 0.07563 0.07439 0.07315 0.07191 0.07067 0.06943 0.06819 0.06695 0.06633 0.06571 0.O6509 0.06447 0.O6385 0.06323 0.O6261 0.06199 0.06137 0.06075 0.06024 0.06013 0.05951 0.05889 0.05827 0.05765 0.05703 0.05641 0.05579 0.05517 0.05455 0.O5393 0.05331 0.05269 0.05207 0.05145 0.05083 0.05021 0.04959 0.04862 0.O4769 0.04679 0.04592 0.04509

cm- 1

#m

640 630 620 610 600 59O 58O 570 560 55O 540 535 530 525 52O 515 510 5O5 500 495 490 485.9 485 48O 475 470 465 460 455 45O 445 44O 435 430 425 420 415 410 4O5 4O0 392.2 384.6 377.4 370.4 363.6

15.63 15.87 16.13 16.39 16.67 16.95 17.24 17.54 17.86 18.18 18.52 18.69 18.87 19.05 19.23 19.42 19.61 19.80 20.00 20.20 20.41 2O.58 20.62 20.83 21.05 21.28 21.51 21.74 21.98 22.22 22.47 22.73 22.99 23.26 23.53 23.81 24.10 24.39 24.69 25.0 25.5 26.0 26.5 27.0 27.5

2.8116] 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7111] 2.6 2.6 2.6 2.6 2.6

6.2 6.3 7.7 7.8 9.3 x 1.2 • 1.4 1.8 2.8 3.3 4.3 5.2 7.2 • 1.2 • 1.7 2.2 2.8 3.3 4.0 4.3 4.9 5.2 5.1 5.1 5.2 4.9 4.6 4.2 3.7 3.2 2.5 2.2 1.7 1.4 1.1 x 8.5 x 7.3 8.3 9.4 x 1.2 • 1.6 3.9 5.0 8.0x 1.2 •

10 -5 [16] 10 -4

10 -4 10 -3

10 - 3

10 -4

10 -4 10 -3 [11]

10 -3 10 -2

(continued)

636

D.J. Treacy TABLE II

(Continued)

Vitreous As2Se 3

eV 0.04428 0.04350 0.04275 0.04203 0.04133 0.04065 0.04000 0.03936 0.03875 0.03815 0.03757 0.03701 0.03647 0.03594 0.03542 0.03493 0.03444 0.03397 0.03351 0.03306 0.03263 0.03220 0.03179 0.03139 0.03100 0.03061 0.03024 0.02988 0.02952 0.02917 0.02883 0.02850 0.02818 0.02786 0.02755 0.02725 0.02695 0.02666 0.02638 0.02610 0.02583 0.02556 0.02530 0.02505 0.02480 0.02455 0.02431

cm- 1

#m

n

357.1 350.9 344.8 339.0 333.3 327.9 322.6 317.5 312.5 307.7 303.0 298.5 294.1 289.9 285.7 281.7 277.8 274.0 270.3 266.7 263.2 259.7 256.4 253.2 250.0 246.9 243.9 241.0 238.1 235.3 232.6 229.9 227.3 224.7 222.2 219.8 217.4 215.1 212.8 210.5 208.3 206.2 204.1 202.0 200.0 198.0 196.1

28.0 28.5 29.0 29.5 30.0 30.5 31.0 31.5 32.0 32.5 33.0 33.5 34.0 34.5 35.0 35.5 36.0 36.5 37.0 37.5 38.0 38.5 39.0 39.5 40.0 40.5 41.0 41.5 42.0 42.5 43.0 43.5 44.0 44.5 45.0 45.5 46.0 46.5 47.0 47.5 48.0 48.5 49.0 49.5 50.0 50.5 51.0

2.6 2.6 2.6 2.6 2.5 2.5 2.5 2.5 2.5 2.5 2.4 2.4 2.4 2.4 2.3 2.3 2.3 2.3 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.3 2.5 2.7 2.9 3.2 3.5 3.7 3.8 3.8 3.9 3.8 3.8 3.8 3.7 3.7 3.7

1.7 2.2 2.8 3.6 4.4 5.3 6.3 7.3 8.2 9.3 x 10 -2 0.11 0.12 0.14 0.16 0.18 0.21 0.24 0.28 0.32 0.37 0.42 0.48 0.54 0.60 0.67 0.73 0.81 0.89 1.0 1.1 1.3 1.4 1.6 1.7 1.8 1.8 1.7 1.5 1.3 1.1 0.94 0.80 0.68 0.59 0.52 0.45 0.40

Arsenic Selenide (As2Se3)

637 T A B L E II

(Continued)

Vitreous As2Se 3

eV 0.02407 0.02384 0.02362 0.02339 0.02317 0.02296 0.02275 0.02254 0.02234 0.02214 0.02194 0.02175 0.02156 0.02138 0.02119 0.02101 0.02184 0.02066 0.02049 0.02033 0.02016 0.02000 0.01984 0.01968 0.01953 0.01937 0.01922 0.01907 0.01893 0.01879 0.01864 0.01851 0.01837 0.01823 0.01810 0.01797 0.01784 0.01771 0.01759 0.01746 0.01734 0.01722 0.01710 0.01698 0.01687

cm-

#m

194.2 192.3 190.5 188.7 186.9 185.2 183.5 181.8 180.2 178.6 177.0 175.4 173.9 172.4 170.9 169.5 168.1 166.7 165.3 163.9 162.6 161.3 160.0 158.7 157.5 156.2 155.0 153.8 152.7 151.5 150.4 149.2 148.1 147.1 146.0 144.9 143.9 142.9 141.8 140.8 139.9 138.9 137.9 137.0 136.1

51.5 52.0 52.5 53.0 53.5 54.0 54.5 55.0 55.5 56.0 56.5 57.0 57.5 58.0 58.5 59.0 59.5 60.0 60.5 61.0 61.5 62.0 62.5 63.0 63.5 64.0 64.5 65.0 65.5 66.0 66.5 67.0 67.5 68.0 68.5 69.0 69.5 70.0 70.5 71.0 71.5 72.0 72.5 73.0 73.5

n

3.6 3.6 3.6 3.5 3.5 3.4 3.5 3.4 3.4 3.4 3.4 3.4 3.4 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1

0.36 0.33 0.30 0.27 0.25 0.23 0.22 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.14 0.13 0.13 0.12 0.12 0.11 0.11 0.10 0.10 0.10 0.10 O.10 9.6 x 10- 2 9.4 9.3 9.1 9.0 8.9 8.8 8.8 8.7 8.7 8.7 8.7 8.7 8.7 8.7 8.8 8.8 8.9 9.0

(continued)

638

D.J. Treacy TABLE II

(Continued)

Vitreous AszSe 3

eV 0.01675 0.01664 0.01653 0.01642 0.01631 0.01621 0.01610 0.01600 0.01590 0.01579 0.01569 0.01560 0.01550 0.01540 0.01531 0.01521 0.01512 0.01503 0.01494 0.01485 0.01476 0.01467 0.01459 0.01450 0.01442 0.01433 0.01425 0.01417 0.01409 0.01401 0.01393 0.01384 0.01378 0.01370 0.01362 0.01355 0.01348 0.01340 0.01333 0.01326 0.01319 0.01312 0.01305 0.01298 0.01292 0.01285 0.01278

cm- 1

#m

135.1 134.2 133.3 132.5 131.6 130.7 129.9 129.0 128.2 127.4 126.6 125.8 125.0 124.2 123.5 122.7 121.9 121.2 120.5 119.8 119.0 118.3 117.6 117.0 116.3 115.6 114.9 114.3 113.6 113.0 112.4 111.7 111.1 110.5 109.9 109.3 108.7 108.1 107.5 106.9 106.4 105.8 105.3 104.7 104.2 103.6 103.1

74.0 74.5 75.0 75.5 76.0 76.5 77.0 77.5 78.0 78.5 79.0 79.5 80.0 80.5 81.0 81.5 82.0 82.5 83.0 83.5 84.0 84.5 85.0 85.5 86.0 86.5 87.0 87.5 88.0 88.5 89.0 89.5 90.0 90.5 91.0 91.5 92.0 92.5 93.0 93.5 94.0 94.5 95.0 95.5 96.0 96.5 97.0

n 3.1 3.1 3.1 3.1 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 2.9 3.0 3.0 3.0 3.0 3.1

9.1 9.2 9.4 9.6x 10 -2 0.10 0.10 0.10 0.10 0.11 0.11 0.11 0.12 0.12 0.13 0.13 0.14 0.14 0.15 0.15 0.16 0.17 0.18 0.19 0.19 0.21 0.22 0.23 0.24 0.25 0.27 0.29 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.47 0.49 0.51 0.53 0.55 0.56 0.58 0.59

A r s e n i c S e l e n i d e (As2Se3)

639 TABLE II

(Continued)

Vitreous As2Se 3

eV 0.01272 0.01265 0.01259 0.01252 0.01246 0.01240 0.01228 0.01216 0.01204 0.01192 0.01181 0.01170 0.01159 0.01148 0.01137 0.01127 0.01117 0.01107 0.01097 0.01088 0.01078 0.01025 0.007606 0.006199 0.004592 0.002799 0.001826 0.001273 0.0006491 0.0004376 0.0002903 0.0001716 0.00009047 0.00005621 0.00002774 0.00001439

cm- 1 102.6 102.0 101.5 101.0 100.5 100.0 99.01 98.04 97.09 96.15 95.24 94.34 93.46 92.59 91.74 90.91 90.09 89.28 88.49 87.72 86.96 82.64 61.35 50.00 37.04 22.57 14.73 10.27 5.236 3.530 2.341 1.384 0.7297 0.4534 0.2237 0.1161

/~m 97.5 98.0 98.5 99.0 99.5 100.0 101.0 102.0 103.0 104.0 105.0 106.0 107.0 108.0 109.0 110.0 111.0 112.0 113.0 114.0 115.0 121.0 163.0 200.0 270.0 443.0 679.0 974.0 1910.0 2833.0 4271.0 7224.0 13704 22056 44699 86153

n 3.1 3.1 3.2 3.2 3.2 3.2 3.3 3.3 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.3 3.2 3.1 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0

k 0.60 0.60 0.60 0.60 0.60 0.59 0.57 0.54 0.51 0.47 0.43 0.40 0.37 0.34 0.31 0.28 0.26 0.24 0.23 0.21 0.20 0.14 0.12 7.2 4.5 2.8 2.1 1.1 7.5 5.0 3.1 1.6 9.9 5.2 2.6

x 10-2 [7]

x 10 -2 x 10-3

x 10- 3 x 10-*

a The material is now isotropic with one set of n and k. At 400 cm-1, data of Lucovsky [11] and Yreacy and Taylor [16] were matched by averaging. At 163 and 200/~m, data of Strom and Taylor [7] and Lucovsky [11] were matched by averaging. Two significant figures are given in most cases.

Arsenic Sulfide (As,S ) D. J. TREACY Physics Department U.S. Naval Academy Annapolis, Maryland

The values of n and k listed in Tables III and IV are room-temperature values. The discussion will cover both the crystalline and vitreous phases.

Crystalline As2Sa The crystalline phase has been investigated, primarily, as a comparison with the vitreous phase, and there are substantial gaps in the data. The crystalline form of As2S3, orpiment, is a naturally occurring mineral, but it is difficult to obtain large crystals of good optical quality. The crystal exhibits micaceous cleavage, and this makes it difficult to obtain optical data perpendicular to its layers. The crystalline form is monoclinic [13; hence the results will be tabulated as n,, k, (E II a); no, ko (E II b); and no, kc (E [I c) in Table Ill. The data are plotted in Fig. 3. In the spectral region from 100 to 25 #m, the spectra are reflection spectra [2, 3] with some transmission measurements at the position of weaker bands. The spectra were fit employing a superposition of several oscillators. In Zallen et al. [2-1 the criterion used to determine the oscillator fit was to minimize the root mean square of the deviation of the theoretical fit from the observed reflectivity spectra. The two studies agree essentialiy but differ systematically with n from Zallen et al. [2] in general being greater than Treacy and Taylor [3]. The systematic difference in n is due to the choice of the high-frequency dielectric constant. In Zallen et al. [2] higher reflectivities were observed at short wavelength, leading to a higher dielectric constant. Both choices of the dielectric constant, however, are probably in error because one overestimates and the other underestimates the index of refraction as determined from accurate fringe measurements by Evans and Young [4] in the near infrared and the visible. This is a fairly common problem in attempting to determine optical constants from reflectivity data, for the reflectivity can vary over a substantial range, particularly in crystals showing micaceous cleavage. For the spectral region from 100 to 25/~m, the data presented for E I[ a and E IIc are from Zallen 641 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright (~) 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-544420-6

642

D.J. Treacy

et al. [2] and are probably accurate to about 10~ in n and k. The data for E 1[b are from Treacy and Taylor [3] and are probably good within the same limits. A point was added to the table at 35.8/.tm to account for a weak mode that was observed in transmission studies. In the spectral range from 2 = 21.7 to 9.35 #m (the multiphonon region) the data for E [Ia, E [1c, are taken from Klein et al. [5]. These are transmission measurements and are probably good to about 10~ with a somewhat better accuracy in the stronger absorption bonds. The values of k in the relatively transparent region were limited by the quality of the natural crystals and probably are representative of impurity absorption and scattering. The values of k observed in this region decrease with wavelength near 25 #m faster than the oscillator fits used from 100 to 25 #m. In the spectral range from 2 to 0.42/~m, the data for n are taken from Evans and Young [4]. The measurements for Ell a and Ell c were very accurate fringe measurements that involved an absolute determination of the order of interference. The results for na and n~ are probably accurate to better than 1~ and are in substantial agreement with an earlier determination [6]. The measurements for E ]] b were again fringe measurements, but an intermediate step involving the birefringence was necessary. The values of k for E il a and E ]]c are a mixture from Evans and Young [4] and Zallen et al. [7]. The data at small values of k were taken from Evans and Young [4]; for the moderate values of k Evans and Young [-4] and Zallen et al. [7] agree, and for large values of k the data are from Zallen et al. [7]. Values of k for low temperatures may be obtained from Evans and Young [4] (T = 77 K) and from Zallen et al. [7] (T = 10 K). It is difficult to evaluate what limits of confidence can be placed on these data and what effect, if any, impurities have at low absorption levels. Both measurements agree for moderate values of k and are probably good to about 10~. At high values of k the data from Zallen et al. were used, but these values of k are probably good only within about 20Vo. Since the reflectivity data for 2 < 0.4 #m [7] have not been analyzed but rather presented for their qualitative features, no further constants will be listed.

Vitreous As2S3 The glassy (amorphous) form of As2S 3 has been investigated in substantially greater depth than the crystalline form due to its possible technological applications. The values for n and k are listed in Table IV and plotted in Fig. 4. There is one very accurate low-frequency measurement of the dielectric constant and the loss factor [8 3. The loss is sufficiently low that it may be considered negligible, and the index of refraction is determined directly from the real part of the dielectric constant. At this point it should be emphasized that there are substantial differences between commercially available forms of a-AsES 3. It should also be noted

Arsenic Sulfide

(As2S3)

643

that the commercially available forms of a-As2S 3 differ from a-As2S 3 that was prepared specifically for its purity. It is worthwhile when using this set of tables to make allowance for the specific type of a-AsES 3 in use. The measurements reported by Fontanella [8] are accurate to substantially better than 1~ within one batch of commercial material but are not better than 3~o when different melts and suppliers are considered. This measurement [8] was taken at 300 K. In the region from 2 = 1.2 x 104 to 116 #m, the data are taken from Strom and Taylor [9]. The first point in this study was taken from microwave perturbation techniques and is probably good within a factor of 2. The remainder of the points were taken in transmission by using a laser source. These points are probably good to within 20~o. From 2 = 100 #m to 2 = 25 #m, the data are taken from Lucovsky [10]. These data were taken as a reflectance spectrum and fit with oscillator parameters. The caveat given earlier concerning reflectance spectra applies here, but since the glass can be polished better than the crystal, the values of n and k are probably somewhat better than 15~. The values of n from 24.69 to 12.5 #m were taken from Young [15]. When these values are compared with the extrapolation of the oscillator fit in Lucovsky [ 10], it is seen that the values given by the oscillator fit do not rise as quickly as the measured points as wavelength decreases. These measured values of n were determined from fringe measurements in transmission. The absolute value of the order of the fringes was obtained, and the measurements are probably good to about 1~. The values of k for 2 = 25 to 12.5 #m were taken from Klein et al. I l l ] . This was a sample in-sample out transmission spectrum, and the values of k listed are within 10~. The values of n for 12.2 to 0.5600 #m were taken from Rodney et al. [12]. These are accurate prism measurements and the original values were listed to six significant figures. The data were fit with a five-term Sellmeier equation, and the fit was typically less than one part in 104. The authors note, however, that this is for a particular sample and that the difference between samples is of the order of 4~o in the visible and of the order of 0.1~ near 12 #m. (The values from Rodney et al. [12] and Young [15] agree to better than l~o at 12 #m.) This underscores the earlier comment concerning the difference between commercial suppliers and different melts of glass from any single commercial source. The values of k for 2 = 12.2 to 6.6 #m are taken from Maklad et al. [16]. At 12.5 #m Klein et al. [11] and Maklad et al. [16] agree. Most points were taken in transmission on a double-beam instrument and are good to within 20~. The points between 9.2 and 10.9 #m were taken by using laser calorimetry. For the points taken on the double-beam instrument, it is difficult to estimate the absolute magnitude of scattering losses, and the values of k listed can be interpreted as an upper bound.

644

D.J. Treacy

The value of k at 2 = 1.0/~m is taken from Wood and Tauc [13] and is probably good to 20%. The values of k from 2 = 0.8 to 0.56 #m were taken from Zallen et al. [7]. These were done in transmission with a foreprism monochrometer instrument and are probably good to within 20%. In the region of overlap between ZaUen et al. and Wood and Tauc [13], the values quoted in Wood and Jauc [13] are slightly higher (about 25% higher at 0.5 pm). The values of n and k from 2 = 0.5585 to 0.22/~m were taken from Young [15]. For the values of n, 0.05 was added to the data presented by Young [15]. This was done because (a) this is well within the uncertainty between melts quoted in Rodney et al. [12], (b) there is greater uncertainty in the values of thickness for the thin samples quoted in Young [15], and (c) the curves run parallel to each other in the region of overlap. It must be emphasized that the differences in the index of refraction are representative differences between melts. The values of k listed here were taken on a prism-grating double monochrometer and should be accurate to better than 20%. In the region in which Zallen et al. [7] was used, the values from Young [15] agree quite well, but Zallen et al. [7] is consistently quite a bit lower for values of 2 less than 0.56 #m. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16.

N. Morimoto, Mineral J. Sapporo 1, 160 (1954). R. Zallen, M. L. Slade, and A. T. Ward, Phys. Rev. B 3, 4257 (1971). D. J. Treacy and P. C. Taylor, Phys. Rev. B 11, 2941 (1975). B. L. Evans and P. A. Young, Proc. R. Soc. Ser. A 297, 230 (1967). P. G. Klein, P. C. Taylor, and D. J. Treacy, Phys. Rev. B 16, 4501 (1977). E. S. Larsen and H. Berman, U. S. Geol. Surv. Bull. 848, 213 (1934). R. Zallen, R. E. Drew, R. L. Emerald, and M. L. Slade, Phys. Rev. Lett. 26, 1564 (1971). J. J. Fontanella, private communication (1975). U. Strom and P. C. Taylor, in "Amorphous and Liquid Semiconductors" (J. Stuke and W. Brenig, eds.), p. 375, Taylor and Francis, London, 1974; U. Strom and P. C. Taylor, Phys. Rev. B. 16, 5512 (1977). G. Lucovsky, Phys. Rev. B 6, 1480 (1972). P. B. Klein, P. C. Taylor, and D. J. Treacy, Phys. Rev. B 16, 4511 (1977). W. S. Rodney, I. H. Malitson, and T. A. King, d. Opt. Soc. Am. 48, 633 (1958). D. L. Wood and J. Tauc, Phys. Rev. B 5, 3144 (1972). E. L. Zorina, Soy. Phys. Solid State 7, 269 (1965). P. A. Young, J. Phys. C 4, 93 (1971). M. S. Maklad, R. K. Mohr, R. E. Howard, P. B. Macedo, and C. T. Moynihan, Solid State Commun. 15, 855 (1974).

Arsenic Sulfide (As2S3)

645 r.~

I

""l

'

'

'

I""I

'

'

l ~'''

'

I '

'

'

I '~'

I '

'

'

I ''~'

I '

'

'

r

I~

I

~

'

'

I

I ~'

_~

0

"i"

-

I

_

-

-~

~_

-

(.9

-

~ % -

.

.

.

~c

.

r~

~

"~ ~ o

-

.

=

T ~

~

-i" oI,~ -

i

iii

I

]

I

l l l l l J

7~

0

_

1

I

0

M

]

T~

o _

'~

'i,

'I:,

~o 9

~

%I'Du '

'

'

1""1

~

'

'

I""1

'

'

'

1""1'

'

'

1""1

1

-

~,~..

_

~

'

'

J

_

-

\\

__

..~

= _

- # , ~

'

'

I " " 1 '

'

'

1 " " I '

~ T

~1,,,~1,

t

a

,

InalllJ

i

q~'

"1'

~

'

'

l""l

'

'

'

l " " l '

i

[aln,I

I

L

~

I,,,,IL

K

qu

'

'

l " " l '

'

'

.--...1

[11 T T ] o _

I

_

_

~g

_ _

"~

-

~._~

(_9

-

)

_.

~

--

/

0

.1 1,,, --

0

-

=

.,-,

i I

0

~':'1' ~u

._~ -~

646

D.J.

i01

'

~

1 ' ' " 1

~

I

I

I ~ " 1

'

'

f

I ' ' " 1

w

~

~

Treacy

I ' ' " 1

,,....p__ I

I

\

i0 o

L

I

l,i v

i0-~

\ \

_ _

\

\ \\

_

Y

10-2_-Z _

I/

_

IJ ii tl II

10-:5__ ,,

/ /

0

I

+-'

i

o

j::

o

u.

Inl

0

N

"r 0

o~ ,o u

~:

w

I

Lithium Niobate (LiNbO3)

699

TABLE VII Values of n and k for Lithium Niobate Obtained from Various References" eV

cm-1

,urn

10.0 9.75 9.5 9.25 9.0 8.75 8.5 8.25 8.0 7.5 7.0 6.5 6.0 5.75 5.5 5.25 5.0 4.75 4.5 4.25

80,650 78,640 76,620 74,610 72,590 70,570 68,560 66,540 64,520 60,490 56,460 52,430 48,390 46,380 44,360 42,340 40,330 38,310 36,290 34,280

0.1240 0.1272 0.1305 0.1340 0.1378 0.1417 0.1459 0.1503 O.1550 0.1653 O.1771 0.1907 0.2066 0.2156 0.2254 0.2362 0.2480 0.2610 0.2755 0.2917

4.2 4.15 4.1 4.05 4.0 3.95 3.9 3.85 3.8 3.75 3.7

33,880 33,470 33,070 32,670 32,260 31,860 31,460 31,050 30,650 30,250 29,840

0.2952 0.2988 0.3024 0.3061 0.3100 0.3139 O.3179 0.3220 0.3263 0.3306 0.3351

eV 3.0642 2.952 2.8447 2.755 2.6503 2.5831 2.480 2.4379 2.2705 2.254 2.1489 2.1415 2.1102

cm - ' 24,714 23,810 22,944 22,220 21,376 20,834 20,000 19,663 18,313 18,180 17,332 17,272 17,019

pm 0.40463 0.42 0.43584 0.45 0.46782 0.47999 0.50 0.50858 0.54607 0.55 0.57696 0.57897 0.58756

no(l) 1.67 [-1] 1.88 2.04 2.19 2.32 2.12 1.82 1.58 1.43 1.38 1.54 1.82 2.16 2.36 2.68 3.22 3.62 3.52 3.19 2.91

ko(-[-) 1.22 [1] 1.28 1.17 1.09 0.75 0.45 0.36 0.46 0.59 0.98 1.30 1.48 1.60 1.67 1.75 1.60 1.27 0.43 0.11 0.00

2.76

no(l)

ne(l) 1.73 [-1] 1.97 2.14 2.32 2.42 2.17 1.89 1.63 1.48 1.48 1.75 2.07 2.30 2.46 2.76 2.98 3.10 2.99 2.79 2.59

2.49

he(l)

2.4317 [6]

2.3260 [6]

2.3928

2.2932

2.3634 2.3541

2.2683 2.2605

2.3356 2.3165

2.2448 2.2285

2.3040 2.3032 2.3002

2.2178 2.2171 2.2147

no(_L)

go(I) 1.1811] 1.22 1.12 1.03 0.72 0.34 0.29 0.40 0.57 0.94 1.24 1.26 1.22 1.22 1.16 1.00 0.64 0.25 0.06 0.00 1.02 x 10-2 [3] 5.87 • 10- 3 3.09 1.71 x 10 -3 9.01 • 10 -4 4.19 2.12 • 10 -4 7.84 • 10-5 3.33 1.69 • 10-5 8.16 x I0 -6 5.07

,,o~11

2.4089 [5]

2.3025 [5]

2.3780

2.2772

2.3410

2.2457

2.3132

2.2237

(continued)

700

Edward D. Palik TABLE VII

(Continued)

Lithium Niobate

eV

cm- 1

2.066 1.9257 1.8566 1.771 1.7549 1.550 1.5321 1.4224 1.378 1.3251 1.2915 1.240 1.2227 1.1352 1.0745 1.0707 1.033 0.96284 0.8856 0.86103 0.7749 0.75683 0.6888 0.64871 0.6199 0.56762 0.5636 0.51662 0.5166 0.4769 0.47412 0.45410 0.4428 0.42793 0.4133 0.40631 0.3875 0.3647 0.3444 0.3263 0.3100

16,670 15,532 14,974 14,290 14,154 12,500 12,357 11,472 11,110 10,688 10,417 10,000 9,961.9 9,156.34 8,666.11 8,636.03 8,333 7,765.78 7,143 6,944.59 6,250 6,104.22 5,556 5,232.18 5,000 4,578.17 4,545 4,166.75 4,167 3,846 3,824.03 3,662.53 3,571 3,451.45 3,333 3,277.10 3,125 2,941 2,778 2,632 2,500

eV

cm- 1

0.2480 0.2066 0.1771 0.1550 0.1378 0.1240

2,000 1,667 1,429 1,250 1,111 1,000

um 0.60 0.64385 0.66782 0.70 0.70652 0.80 0.80926 0.87168 0.90 0.93564 0.95998 1.00 1.0140 1.09214 1.15392 1.15794 1.20 1.28770 1.40 1.43997 1.60 1.63821 1.80 1.91125 2.00 2.18428 2.20 2.39995 2.40 2,60 2.61504 2.73035 2.80 2.89733 3.00 3.05148 3.20 3.40 3.60 3.80 4.00 jum 5.0 6.0 7.0 8.0 9.0 10

.o(•

.o(11)

2.2835 2.2778

2.2002 2.1953

2.2699

2.1886

2.2541 2.2471

2.1749 2.1688

2.2412 2.2393

2.1639 2.1622

2.2351 2.2304 2.2271 2.2269

2.1584 2.1545 2.1517 2.1515

2.2211

2.1464

2.2151

2.1413

2.2083

2.1356

2.1994

2.1280

2.1912

2.1211

2.1840

2.1151

2.1765 2.1724

2.1087 2.1053

2.1657

2.0999

2.1594

2.0946

no(Z) 2.05 [7] 1.97 1.84 1.71 1.52 1.20

ko(-L)

.o(•

.,,(11)

2.2967

2.2082

2.2716

2.1874

2.2571

2.1745

2.2448

2.1641

2.2370

2.1567

2.2269

2.1478

2.2184

2.1417

2.2113

2.1361

2.2049

2.1306

2.1974

2.1250

2.1909

2.1183

2.1850 2.1778

2.1129 2.1071

2.1703

2.1009

2.1625

2.0945

2.1543 2.1456 2.1363 2.1263 2.1155

2.0871 2.0804 2.0725 2.0642 2.0553

.o(11)

2.oo ['7-1 1.93 1.83 1.72 1.55 1.30

k~(ll)

Lithium Niobate (LiNb03)

701 T A B L E VII

(Continued)

Lithium Niobate eV

cm- 1

ltm

no(L)

O.1127 0.09919 0.09671 0.09423 0.09175 0.08927 0.08679 0.08431 0.08307 0.08183 0.07935 0.07811 0.07687 0.07563 0.07439 0.07315 0.07191 0.07067 0.06943 0.06819 0.06695 0.06447 0.06199 0.05951 0.05703 0.05455 0.05331 0.05207 0.04959 0.04711 0.04587 0.04463 0.04339 0.04215 0.04092 0.03968 0.03844 0.03720 0.03596 0.03472 0.03348 0.03286 0.03224 0.03100 0.02976 0.02852 0.02728

909.1 800 780 760 740 720 700 680 670 660 640 630 620 610 600 590 580 570 560 550 540 520 500 480 460 440 430 420 400 380 370 360 350 340 330 320 310 300 290 280 270 265 260 250 240 230 220

11 12.50 12.82 13.16 13.51 13.89 14.29 14.71 14.93 15.15 15.63 15.87 16.13 16.39 16.67 16.95 17.24 17.54 17.86 18.18 18.52 19.23 20.00 20.83 21.74 22.73 23.26 23.81 25.00 26.32 27.03 27.78 28.57 29.41 30.30 31.25 32.26 33.33 34.48 35.71 37.04 37.74 38.46 40.00 41.67 43.48 45.45

0.60 0.20 [8] 0.21 0.22 0.23 0.26 0.54 0.90 0.90 0.99 0.95

1.24 [8] 1.43 1.56 1.74 1.94 2.30 2.51 2.60 2.52 2.52

0.90

2.88

1.10

3.63

1.83 3.01 4.77 4.88 4.51 4.00 3.27 2.52 1.78 1.03 1.20 1.30 1.01 1.33 1.73 2.61 2.64 1.34 1.89 3.39 4.09 2.80

4.62 5.30 4.55 2.97 2.10 1.24 0.84 0.59 0.62 1.44 2.11 1.92 2.45 3.36 3.74 4.10 2.83 2.96 4.75 5.52 3.04 1.96

0.92 2.19 4.19 5.71 6.79 5.76 5.82 4.94

4.34 6.84 8.34 7.91 2.94 2.68 2.31 1.21

ko(•

ne(ll)

k~(l[)

0.81 0.46 [8] 0.44 0.45 0.46 0.53 0.56 0.75

1.1018] 1.48 1.78 2.17 2.34 2.67 3.01

1.05 1.44 1.85 2.52 3.42 4.02 4.24 4.47

3.33 3.83 4.05 4.16 4.09 3.48 2.83 2.12

4.17

1.20

3.67 3.20 2.67 2.28 2.06 1.82

0.68 0.47 0.37 0.44 0.48 0.55

1.27 0.83 0.48

0.79 1.09 1.65

0.32

2.47

0.37

3.34

0.70 1.16 1.20 0.75 0.80 0.94

4.30 4.51 4.35 4.64 5.62 6.91

3.04 6.58 7.48 8.55 8.32

9.86 10.4 7.34 4.91 3.36

(continued)

702

Edward D. Palik T A B L E VII (Continued)

Lithium Niobate eV

cm- 1

pm

0.02604 0.02480 0.02356 0.02232 0.02108 0.01984 0.01860 0.01736 0.01612 0.01488 0.01240 0.01054 0.009919

210 200 190 180 170 160 150 140 130 120 100 85 80

47.62 50.00 52.63 55.56 58.82 62.50 66.67 71.43 76.92 83.33 100 117.6 125

0.009299 0.008679 0.007439

75 70 60

133.3 142.9 166.7

0.006199 0.004959

50 40

200 250

0.003720 0.002480 0.001240

30 20 10

333.3 500 1000

no(Z )

ko(-L)

~(11)

2.54

1.57

0.98 2.04 5.43 7.96 10.1 10.6 9.83 8.57

5.09 7.36 10.1 9.92 5.68 3.06 1.67 0.70

7.66 6.78 7.08 7.86 8.03 6.93 6.33 6.17 6.01 5.84 5.66

7.87

0.13

5.52

7.34 7.06111] 6.90 6.85 [8-] 6.80111] 6.70 6.64 6.61

3.5 • 10 -2 [10] 3.0 7 [8] 1.9 x 10 -2 [10] 9.5 X 1 0 - 3

5.38 5.22111] 5.165

6.0 4.5 3.2 2.0

5.125 5.10 5.07 5.06

k~(ll) 2.60 2.35 2.85 2.4t 0.75 0.36 0.38 0.40 0.43 0.43 0.35 2.3 x 10 -2 [10] 0.18 [8] 1.8 • 10 -2 [10] 1.1 9 x 10 -2 [8] 8.0 x 10-3 [10] 4.8 3.6 2.9 2.0 1.6

a The reference is given in brackets at the beginning of the tabulation of n and k and is understood to refer to all n's or all k's below it until a new reference appears. When an exponent of 10 is given, all numbers below it have the same exponent until the power of 10 changes or the next number has 10 -1, which is usually written as a decimal. Boyd et al. [5] n values are for stoichiometric material and Nelson and Mikulyak [6] n values are for congruent material. We specify no(l) and ne(ll) for redundancy.

Potassium Chloride (KCl) EDWARD D. PALIK Naval Research Laboratory Washington, D,C.

The optical constants n and k are collected in Table VIII and plotted in Fig. 8. The spectral region 3000-300 #m primarily contains absorption due to two- and three-phonon absorption processes. Here, k has been determined by transmission measurements of slabs. The data of Genzel et al. [1] are tabulated in Table VIII. Measurement was with a grating monochromator utilizing an Hg lamp and the harmonics of a 12-ram Klystron as sources. The tabulated numbers were read off a wavelength graph with a smooth line drawn through the data points, which scattered by < 10~. The more-limited results of Stolen and Dransfeld [2] at 300 K were in good agreement, with the temperature dependence studied in detail. The spectral region 200-30 #m has beeen studied by Johnson and Bell [3] by asymmetric Fourier-transform spectroscopy. This technique yields the phase and amplitude of the reflectivity experimentally, thus circumventing Kramers-Kronig (KK) analysis of R or oscillator-model fits. At the reststrahlen minimum, a power reflectance of 0.0004 was measured, which suggests little polishing damage. Resolution was 2 cm-~. An accuracy of 0.005 in amplitude reflectance was stated. Shoulders on the high-frequency side of Vv are due to multiple-phonon absorption processes. We have read graphs to obtain n and k. Bravo to workers who put fiduciary marks on all four sides of a graph and include all the digits from 2 to 9 on the log scales! We typically give three figures, but our reading ability is probably somewhat less than this. For the region 50-12.5 #m, Deutsch [4] has collected values of absorption coefficient ~ from several sources all more or less at room temperature. Transmission of slab samples was used. On semilog paper these form a straight line to within the scatter, and we have read the values on the line. Boyer et al. [5] treat the temperature dependence of such absorption in numerous materials including KC1. 703 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

ISBN 0-12-544420-6

704

Edward D. Palik

Calorimetry measurements at single laser wavelengths have been carried out in the transparent region [5-7] and the k values are listed. These represent bulk values with uncertainties of factors of 2 from sample to sample and lab to lab. Surface preparation included chemical polishing. In the transparent region 35-0.18 #m, Li 1-8] has assembled and analyzed a great deal of n data from many references. Minimum deviation was the experimental method in most cases. He has fitted a Sellmeier-type equation with nine adjustable parameters to represent the data to the fifth decimal place. The dispersion formula has the form n 2 = 1.26486 +

0.3052322 0.4162022 + 22 _ (0.100) 2 22 _ (0.131) 2

0.1887022 2.620022 + 2 2 _ (0.162)2 + 2 2 _ (70.42)2

(1)

at 293 K. This indicates that 2T = 70.42 #m (VT = 142.0 cm-X). Li [8] has provided a temperature-dependence formula of the form dn 3.39324 142.5624 2n d--T = - 11.13(n 2 - 1) + 0.19 + (22 _ 0.02624) 2 + (22 _ 4958.98) 2.

(2)

where d n / d T is in units of 10 -5 K -~. Thus, a 3-K difference in room temperature between labs begins to affect the fourth decimal place. For the spectral region 0.24-21 /zm, Li [8] states that the fourth decimal place is meaningful, with estimated uncertainties in the range 0.0001-0.0005. To either side, the uncertainty moves to the third decimal place. At and above the fundamental band gap, a variety of n and k data have been assembled. Some overlap is provided to show discrepancies and variations. For 295 K, Tomiki [9] has measured transmission of thin slabs to determine k in the region 6.8-7.9 eV, including the Urbach tail and exciton effects. Two discharge lamps (hydrogen and CO2) were utilized. Much attention was given to the growth and preparation of pure samples to minimize impurity absorption effects. For the region 6.4-11.6 eV, reflectivity of cleaved samples has been fitted by K K analysis to yield n and k as listed [10]. A great deal of temperature data was obtained by Tomiki et al. [11]. In this region absorption is due primarily to a number of interband transitions between lower-lying valence and higher-lying conduction bands. In the spectral region 5-26 eV, R was measured by Roessler and Walker [12] with K K analysis. More recent measurements have been performed [13] that we choose to emphasize. As usual, differences of 20~ in n and k appear in overlap regions, although the features seem the same. Antinori et al. [13] used a continuum He discharge as a source. Reflectivity was

Potassium Chloride (KCI)

705

reproducible among various cleaved samples to within 53/o. As Table II in Antinori et al. [13] emphasizes, the reflectivity at 18.4 eV varied by a factor of 2 among seven different references. As Fig. 5 in Antinori et al. [13] indicates, spectra in this region are also affected by resolution; two examples of spectra of 1.5 and 3 A resolution are shown to differ significantly. More structure was resolved in room-temperature spectra than previously had been seen. Absorption in this region is due primarily to plasma oscillations of the 6p electrons of C1- filling the valence-band states (,,~ 14 eV) and to transitions of 3p electrons of K + to the bottom of the conduction band (~ 20 eV). Balzarotti et al. [14] have measured reflectivity in the region from 20 to 44 eV by utilizing synchrotron radiation. Kramers-Kronig analysis values of n and k are listed. Peaks in this spectral region are primarily due to electron transitions from 3p levels of K + and 3s levels of C1- into the conduction band. No doubt, cleaved samples were used. Above this energy, absolute values of k become sparse, although a great deal of work has been done at the various L and K absorption edges. The work is collected by Haelbich et al. [15], and it proved very useful as a starting point for the present work. They give a figure summarizing absorption coefficient e in inverse centimeters for the range 50-280 eV. However, some of the original references contain only relative values. Therefore, we have found some other references to supplement these. For the spectral region 50-200 eV, the data of Lukirskii and Zimkina [ 16] as given in Haelbich et al. [15] indicated a featureless decrease in k to high energy. Lukirskii et al. [17] report measurements of n and k at two x-ray wavelengths. The k values are about a factor of 2 smaller than those given in Haelbich et al. [15], but we list them along with the n values since they seem to join the high-energy data better. For the spectral region 200-240 eV, the data of Brown et al. ]-18], obtained with synchrotron radiation, are only relative in the original paper. We use the work of Aita et al. [19], who used an x-ray tube as a continuous source to study the L2, 3 edge of C1- in evaporated films. Iguchi et al. [20] used synchrotron radiation to measure relative absorption in the 200-280 eV region. Finally, the K absorption edge of C1- near 2823 eV has been studied by Parratt et al. [21] with an x-ray source spectrometer, and k is tabulated. They pointed out a resolution (spectral window) problem that distorted the measurement of c~ as a function of sample thickness near the absorption edge. Kiyono and Sugiura [22] also measured relative absorption in this region. The K absorption edge of potassium near 3610 eV has been studied by Sugiura [23]. It is not clear that the L edges of potassium have been measured in the 290-300 eV region.

706

Edward D. Palik

REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

18. 19. 20. 21. 22. 23.

L. Genzel, H. Happ, and R. Weber, Z. Physik 154, 13 (1959). R. Stolen and K. Dransfeld, Phys. Rev. 139A, 1295 (1965). K. W. Johnson and E. E. Bell, Phys. Rev. 187, 1044 (1969). T. F. Deutsch, J. Phys. Chem. Solids 34, 209 (1973). L. L. Boyer, J. A. Harrington, M. Hass, and H. R. Rosenstock, Phys. Rev. liB, 1665 (1975). S. D. Allen and J. A. Harrington, Appl. Opt. 17, 1679 (1978). J. A. Harrington, B. L. Bobbs, M. Braunstein, R. K. Kim, R. Stearns, and R. Braunstein, Appl. Opt. 171, 541 (1978). H. H. Li, J. Phys. Chem. Ref. Data 5, 329 (1976). T. Tomiki, J. Phys. Soc. Jpn. 23, 1280 (1967). T. Tomiki, J. Phys. Soc. Jpn. 22, 463 (1967). T. Tomiki, T. Miyata, and H. Tsukamoto, J. Phys. Soc. Jpn. 35, 495 (1973). D. M. Roessler and W. C. Walker, J. Opt. Soc. Am. 58, 279 (1968). M. Antinori, A. Balzarotti, and M. Piacentini, Phys. Rev. B 7, 1541 (1973); M. Antinori, A. Balzarotti, and M. Piacentini, private communication (1982). A Balzarotti, A. Bianconi, E. Burattini, and G. Strinati, Solid State Commun. 15, 1431 (1974). R. P. Haelbich, M. Iwan, and E. E. Koch, "Physik Daten, Physics Data." Fach-information Zentrum, Karlsruhe, 1977. A. P. Lukirskii and T. M. Zimkina, in "Rontgen Spektren und Chemische Bindung," p. 187. Karl-Marx Universitat, Leipzig, 1966. A. P. Lukirskii, E. P. Savinov, O. A. Ershov, and Y. F. Shepelev, Opt. Spectrosc. 16, 168 (1964); A. P. Lukirskii, E. P. Savinov, O. A. Erskov, and Y. F. Shepelev, Opt. Specktrosk. 16, 310(1964). F. C. Brown, C. Gahwiller, H. Fujita, A. B. Kunz, W. Scheifley, and N. Carrera, Phys. Rev. B 2, 2126 (1970). O. Aita, I. Nagakura, and T. Sagawa, J. Phys. Soc. Jpn. 30, 1414 (1971). Y. Iguchi, T. Sagawa, S. Sato, M. Watanabe, H. Yamashita, A. Ejiri, M. Sasanuma, S. Nakai, M. Nakamura, S. Yamaguchi, Y. Nakai, and Y. Oshio, Solid State Commun. 6, 575 (1968). L. G. Parratt, C. F. Hempstead, and E. L. Jossem, Phys. Rev. 105, 1228 (1957). S. Kiyono and C. Sugiura, Tech. Repts. Tohoku Univ. 30, 9 (1965). C. Sugiura, Sci. Repo. Tohoku Univ. 45, 248 (1961).

Potassium Chloride (KCI) I0 i

'

'

,

I ''"I

'

l

l~,,,l

707 ,

i

i

l,i,~

I

~

i

i

IT,111

i

,

~I~,~i

,

,

i

l1~q,

|

1

I

I ] 1 liT!

\

_

/

I0 o v.

Y! IIII I I IIII

Id, ~H. t~

I

,~

!1111

~ ~1 I

J

\ \

!

lO-i

\

_

\ \\

\\

/

I0-2

7 \

I

\N

\

I

I

/

E

/

I//

I I

I Mj(CI)

I J

i(3-3 I I M2, 3(K) MI

i(3.4

-

I I L2,5 (CI) Li

I

-_ K (CI) _ _

I

II

K(K)

L t L2, 3 (K)

i(3.5

lO-e

10-5

10-2

i0-1

i0 0

WAVELENGTH Fig. 8. Log-log plot of n ( - - ) sium chloride.

and k (. . . .

I01

102

03

(,u.m)

) versus wavelength in micrometers for potas-

TABLE VIII Values of n and k for Potassium Chloride Obtained from Various References a eV 2860.3 2859.3 2858.3 2857.3 2856.3 2855.3

A 4.3347 4.3362 4.3377 4.3392 4.3408 4.3423

n

k 3.93 3.83 3.73 3.63 3.52 3.39

x 10 - 6

[21]

(continued)

708

Edward D. Palik TABLE VIII

(Continued)

Potassium Chloride eV 2854.3 2853.3 2852.3 2851.3 2850.3 2849.3 2848.3 2847.3 2846.3 2845.3 2844.3 2843.3 2842.8 2842.3 2841.8 2841.3 2840.8 2840.3 2839.8 2839.3 2838.8 2838.3 2837.8 2837.3 2836.8 2836.3 2835.8 2835.3 2834.8 2834.3 2833.8 2833.3 2832.8 2832.3 2831.8 2831.3 2830.8 2830.3 2829.8 2829.3 2828.8 2828.3 219 218.5 218 217.5 217 216.5 216 215.5

4.3438 4.3453 4.3468 4.3484 4.3499 4.3514 4.3530 4.3545 4.3560 4.3575 4.3590 4.3606 4.3614 4.3621 4.3629 4.3637 4.3644 4.3652 4.3660 4.3668 4.3675 4.3683 4.3691 4.3698 4.3706 4.3714 4.3721 4.3729 4.3737 4.3745 4.3752 4.3760 4.3768 4.3775 4.3783 4.3791 4.3799 4.3806 4.3814 4.3822 4.3830 4.3837 56.61 56.74 56.87 57.00 57.13 57.27 57.40 57.53

3.32 3.60 3.91 4.36 4.74 4.61 4.16 4.09 4.13 4.20 4.06 3.82 3.71 3.71 3.82 3.89 3.54 2.54 2.08 1.95 2.09 2.57 3.65 4.80 4.94 4.87 5.85 5.95 5.85 4.53 3.55 3.66 4.53 5.85 7.49 9.34 9.06 6.45 1.57 1.05 • 10 - 6 6.98 • 10 - 7 4.19 • 10-7 1.82 x 10 -3 [19] 1.82 1.85 1.88 1.87 1.89 1.92 1.91

Potassium Chloride (KCI)

709 TABLE VIII

(Continued)

Potassium Chloride eV 57.67 57.80 57.94 58.07 58.21 58.34 58.48 58.62 58.76 58.9O 59.04 59.18 59.32 59.46 59.61 59.75 59.89 60.04 60.19 60.33 60.48 60.63 60.78 60.92 61.07 61.23 61.38 61.53 61.68 61.84 61.99 67 113

215 214.5 214 213.5 213 212.5 212 211.5 211 210.5 210 209.5 209 208.5 208 207.5 207 206.5 206 205.5 205 204.5 204 203.5 203 202.5 202 201.5 201 200.5 200 185.1 109.7 eV 43 42 41 40 39 38 37 36.5 36 35.5 35 34.5 34

cm-

1

0.99874 [ 17] 0.99578

1.84 1.84 1.84 1.85 2.02 2.19 2.14 1.87 1.82 1.91 1.71 1.48 1.45 1.42 1.46 1.55 1.56 1.51 1.55 1.56 1.52 1.40 1.23 1.31 1.46 1.15 1.13 1.22 1.04 1.03 1.03 1.01 [17] 4.22 x 10- 3

0.96113,14] 0.955 0.945 0.925 0.91 0.89 0.87 0.845 0.86 0.830 0.87 0.851 0.89

3.0 x 10 -2 [13, 14] 2.5 2.0 1.8 2.0 3.0 5.0 7.04 8.0 9.16 x 10 -2 0.115 0.105 0.12

pm 0.02883 0.02952 0.03024 0.03099 0.03179 0.03263 0.03351 0.03397 0.03444 0.03493 0.03542 0.03594 0.03646

(continued)

710

Edward D. Palik TABLE VIII

(Continued)

Potassium Chloride eV 33.5 33.1 32.7 32.3 31.9 31.5 31.1 30.7 30.3 29.9 29.5 29.1 28.9 28.7 28.5 28.3 28.1 27.9 27.5 27.1 26.7 26.3 26.1 25.9 25.7 25.5 25.3 25.1 24.7 24.3 23.9 23.5 23.1 22.9 22.7 22.5 22.3 21.9 21.7 21.5 21.3 21.1 20.9 20.7 20.5 20.3 20.1

cm- 1

~um 0.03701 0.03746 0.03792 0.03839 0.03887 0.03936 0.03987 0.04039 0.04092 0.04147 0.04203 0.04261 0.04290 0.04320 0.04350 0.04381 0.04412 0.04444 0.04509 0.04575 0.04644 0.04714 0.04750 0.04787 0.04824 0.04862 0.04901 0.04940 0.05020 0.05102 O.O5188 0.05276 0.05367 0.05414 0.05462 0.05510 0.05560 0.05661 0.05714 0.05767 0.05821 0.05876 0.05932 0.05990 0.06048 0.06108 0.06168

0.854 0.821 0.800 0.797 0.795 0.797 0.799 0.795 0.779 0.756 0.743 0.730 0.729 0.743 0.785 0.805 0.816 0.814 0.789 0.775 0.755 0.745 0.750 0.779 0.805 0.824 0.836 0.851 0.849 0.813 0.806 0.819 0.834 0.849 0.860 0.868 0.872 0.814 0.790 0.820 0.918 1.01 1.06 1.07 1.02 0.926 0.910

7.83 • 10- 2 6.77 9.24 • 10 - 2 0.111 0.121 0.131 0.134 0.134 0.134 0.145 0.175 0.207 0.234 0.265 0.275 0.261 0.245 0.228 0.230 0.241 0.263 0.302 0.335 0.359 0.362 0.358 0.357 0.346 0.317 0.328 0.363 0.388 0.413 0.418 0.420 0.420 0.418 0.416 0.488 0.569 0.612 0.587 0.530 0.449 0.355 0.411 0.495

Potassium Chloride (KCI)

711 TABLE VIII

(Continued)

Potassium Chloride

eV

cm- 1

#m

19.9 19.7 19.5 19.1 18.7 18.3 17.9 17.7 17.5 17.3 17.1 16.9 16.7 16.3 15.9 15.5 15.1 14.7 14.3 13.9 13.5 13.1 12.9 12.7 12.5 12.3 12.1 11.9 11.7 11.6 11.5

0.06230 0.06294 0.06358 0.06491 0.06630 0.06775 0.06927 0.07005 0.07085 0.07167 0.07251 0.07336 0.07424 0.07606 0.07798 0.07999 0.08211 O.O8434 0.08670 0.08920 0.09184 0.09465 0.09611 0.09763 0.09919 0.1008 0.1025 0.1042 0.1060 0.1069 0.1078

11.4 11.3

0.1088 0.1097

11.2 11.1

0.1107 0.1117

11.0 10.9

0.1127 0.1137

10.8 10.7

0.1148 0.1159

10.6 10.55

0.1170 0.1175

0.982 1.00 1.01 1.09 1.17 1.15 1.11 1.07 1.07 1.09 1.13 1.14 1.13 1.09 1.05 1.01 0.965 0.931 0.883 0.791 0.690 0.841 0.914 1.04 1.34 1.57 1.67 1.74 1.73 1.67 [10] 1.66 1.69 1.65 1.63 1.62 1.60 1.57 1.55 1.54 1.50 1.50 1.47 1.42 1.40 1.38 1.36

0.584 0.497 0.533 0.525 0.489 0.387 0.367 0.382 0.411 0.428 0.420 0.376 0.357 0.322 0.317 0.328 0.344 0.367 0.415 0.441 0.731 0.95O 1.04 1.22 1.29 1.12 0.959 0.767 0.640 0.58 [lO] 0.53 0.520113] 0.49 [10] 0.45 0.425 [13] 0.41 [10] 0.38 0.374 [13]

o.3511o] 0.325 0.363113]

o.31 [~o] 0.315 0.336113] 0.36 [ 10] 0.38

(continued)

712

Edward D. Palik TABLE VIII

(Continued)

Potassium Chloride eV

cm

-1

/~m

10.5 10.4 10.3 10.2 10.1 10.0 9.9 9.8 9.75 9.7 9.65 9.6 9.55 9.5 9.45 9.4 9.35 9.3 9.2 9.1 9.0 8.9 8.8 8.7 8.6 8.5 8.4 8.3 8.2 8.1 8.0 7.95 7.9 7.85 7.8 7.75 7.7

0.1181 0.1192 0.1204 0.1216 0.1228 0.1240 0.1252 0.1265 0.1272 0.1278 0.1285 0.1291 0.1298 0.1305 0.1312 0.1319 0.1326 0.1333 0.1348 0.1362 0.1378 0.1393 0.1409 0.1425 0.1442 0.1459 0.1476 0.1494 0.1512 0.1531 0.1550 0.1560 0.1569 0.1579 0.1589 0.1600 0.1610

7.65 7.6

0.1621 0.1631

2.63

7.55 7.5

0.1642 0.1653

2.92 3.0

7.45 7.4

0.1664 0.1675

2.8 2.61

1.37 1.395 1.35 1.30 1.24 1.16 1.05 0.87 0.80 0.94 1.38 1.81 2.07 2.18 2.12 1.99 1.90 1.85 1.81 1.79 1.75 1.73 1.68 1.54 1.36 1.15 1.13 1.30 1.92

0.385 0.38 0.37 0.35 0.355 0.38 0.41 0.58 0.71 0.90 1.1 1.27 1.41 1.495 1.48 1.38 1.28 1.13 0.85 0.64 0.50 0.43 0.40 0.39 0.37 0.345 0.34 0.265 0.215 0.25 0.46 0.67 0.90 1.12 1.36 1.54 1.64 1.67 I-9] 1.6110] 1.34 1.43 1-9] 1.1 [10] 0.51

0.494 [9] o.211o] 6 x 10 - 2 5.60 x 10-2 [9]

Potassium Chloride (KCI)

713 TABLE VIII

(Continued)

Potassium Chloride eV 7.3 7.2 7.1 7.0 6.9 6.888 6.812 6.8 6.738 6.7 6.666 6.6 6.595 6.525 6.5 6.4 6.199 5.904 5.636 5.391 5.166 4.959 4.769 4.592 4.428 4.275 4.133 3.999 3.874 3.757 3.647 3.542 3.473 3.444 3.351 3.263 3.179 3.100 2.952 2.818 2.695 2.616 2.583 2.480 2.384 2.296

cm- 1

50,000 47,620 45,450 43,480 41,670 40,000 38,460 37,040 35,710 34,480 33,330 32,260 31,250 30,300 29,410 28,570 28,010 27,780 27,030 26,320 25,640 25,000 23,810 22,730 21,740 21,100 20,830 20,0OO 19,230 18,520

~m 0.1698 0.1722 0.1746 0.1771 0.1797 0.180 0.182 0.1823 0.184 0.1850 0.186 0.1879 0.1888 0.190 O.1907 0.1937 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.357 0.36 0.37 0.38 0.39 0.40 0.42 0.44 0.46 0.474 0.48 0.50 0.52 0.54

n 2.34 2.19 2.08 2.0 1.94 1.89324 [8] 1.86392 1.89 [10] 1.83883 [-8] 1.85 [-10] 1.81705 [8] 1.82 [10] 1.79789 [-8] 1.78087 1.80 [10] 1.78 1.71739 [-8] 1.67539 1.64517 1.62226 1.60426 1.58972 1.57775 1.56772 1.55921 1.55191 1.54558 1.54005 1.53518 1.53087 1.52703 1.52358

3.11 1.78 7.96 8.46 2.29

• • x • •

10 -3 10 -4 10 -6 10 -7 10- 7

1.45 • 10-7

5.1 x 10 - i ~ [7] 1.52049 1.51769 1.51515 1.51283 1.51072 1.50701 1.50386 1.50115 7.6 x 10-11 1.49882 1.49679 1.49501 1.49344

(continued)

714

Edward D. Palik TABLE VIII

(Continued)

Potassium Chloride eV

cm- 1

pm

2.214 2.138 2.066 2.000 1.937 1.879 1.823 1.771 1.722 1.675 1.631 1.590 1.550 1.512 1.476 1.442 1.409 1.378 1.348 1.319 1.291 1.265 1.240 1.127 1.033 0.9537 0.8856 0.8265 0.7749 0.7293 0.6888 0.6525 0.6199 0.5636 0.5166 0.4769 0.4428 0.4133 0.3874 0.3647 0.3444 0.3263 0.3100 0.2952 0.2818 0.2695 0.2583

17,860 17,240 16,670 16,130 15,630 15,150 14,710 14,290 13,890 13,510 13,160 12,820 12,500 12,200 11,900 11,630 11,360 11,110 10,870 10,640 10,420 10,200 10,000 9,091 8,333 7,692 7,143 6,667 6,250 5,882 5,556 5,263 5,000 4,545 4,167 3,046 3,571 3,333 3,125 2,941 2,778 2,632 2,500 2,381 2,273 2,174 2,083

0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8

1.49205 1.49081 1.48969 1.48869 1.48779 1.48697 1.48623 1.48555 1.48493 1.48436 1.48384 1.48336 1.48291 1.48250 1.48212 1.48176 1.48143 1.48111 1.48082 1.48055 1.48030 1.48006 1.47983 1.47887 1.47813 1.47755 1.47708 1.47668 1.47634 1.47605 1.47580 1.47557 1.47536 1.47498 1.47464 1.47433 1.47403 1.47374 1.47345 1.47315 1.47285 1.47254 1.47223 1.47190 1.47156 1.47122 1.47085

1.27 x 10- lo [6]

1.97 x 10- lO

Potassium Chloride (KCI)

715 TABLE VIII

(Continued)

Potassium Chloride eV

cm- 1

~um

n

0.2480 0.2384 0.2339 0.2296 0.2214 0.2138 0.2066 0.2000 0.1937 0.1879 0.1823 0.1771 0.1722 0.1675 0.1631 0.1590 0.1550 0.1512 0.1476 0.1442 0.1409 0.1378 0.1348 0.1319 0.1291 0.1265 0.1240 0.1216 0.1192 0.1170 0.1148 0.1127 0.1107 0.1088 0.1069 0.1051 0.1033 0.1016 0.09999 0.09919 0.09840 0.09686 0.09537 0.09393 0.09252 0.09116

2,000 1,923 1,887 1,852 1,786 1,724 1,667 1,613 1,563 1,515 1,471 1,429 1,389 1,351 1,316 1,282 1,250 1,220 1,190 1,163 1,136 1,111 1,087 1,064 1,042 1,020 1,000 980.4 961.5 943.4 925.9 909.1 892.9 877.2 862.1 847.5 833.3 819.7 806.5 800 793.7 781.3 769.2 757.6 746.3 735.3

5.0 5.2 5.3 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 9.6 9.8 10.0 10.2 10.4 10.6 10.8 11.0 11.2 11.4 11.6 11.8 12.0 12.2 12.4 12.50 12.6 12.8 13.0 13.2 13.4 13.6

1.47048 1.47010 2.11 • 10 -11 1.46970 1.46928 1.46886 1.46842 1.46796 1.46749 1.46701 1.46651 1.46600 1.46547 1.46493 1.46437 1.46379 1.46290 1.46260 1.46198 1.46134 1.46069 1.46002 1.45934 1.45864 1.45792 1.45719 1.45644 1.45567 1.45489 1.45409 1.45327 1.45244 1.45159 1.45072 1.44984 1.44893 1.44801 1.44707 1.44611

2.53 x 10 -s [5]

1.49 x 10 -7 [4] 1.44514 1.44415 1.44313 1.44210 1.44105 1.43999

(continued)

716

Edward D. Palik TABLE VIII

(Continued)

Potassium Chloride eV 0.08984 0.08856 0.08731 0.08679 0.08610 0.08492 0.08377 0.08265 0.08157 0.08051 0.07948 0.07847 0.07749 0.07653 0.07560 0.07469 0.07439 0.07380 0.07293 0.07208 0.07125 0.07044 0.06965 0.06888 0.06812 0.06738 0.06666 O.O6595 0.06525 0.06457 0.06391 0.06326 0.06262 0.06199 0.06048 0.05904 0.05767 0.05636 0.05510 0.05391 0.05276 0.05166 0.05060 0.04959 0.04862 0.04769 0.04679

cm- 1

pm

724.6 714.3 704.2 700 694.4 684.9 675.7 666.7 657.9 649.4 641.0 632.9 625.0 617.3 609.8 602.4 600 595.2 588.2 581.4 574.7 568.2 561.8 555.6 549.4 543.5 537.6 531.9 526.3 520.8 515.5 510.2 505.1 500.0 487.8 476.2 465.1 454.5 444.4 434.8 425.5 416.7 408.2 400.0 392.2 384.6 377.4

13.8 14.0 14.2 14.29 14.4 14.6 14.8 15.0 15.2 15.4 15.6 15.8 16.0 16.2 16.4 16.6 16.67 16.8 17.0 17.2 17.4 17.6 17.8 18.0 18.2 18.4 18.6 18.8 19.0 19.2 19.4 19.6 19.8 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0 25.5 26.0 26.5

1.43890 1.43779 1.43667 1.14

•

10 - 6

8.76

•

10 - 6

1.43552 1.43343 1.43317 1.43197 1.43074 1.42950 1.42823 1.42694 1.42563 1.42 1.42295 1.42158 1.42018 1.41877 1.41733 1.41587 1.41438 1.41287 1.41134 1.40979 1.40821 1.40661 1.40498 1.40333 1.40165 1.39995 1.39822 1.39647 1.39469 1.39012 1.38538 1.38047 1.37537 1.37009 1.36461 1.35892 1.35303 1.34692 1.34059 1.33402 1.32721 1.32014

7.64 • 10 -5

6.57 x 10-'*

P o t a s s i u m C h l o r i d e (KCI)

717

TABLE VIII

(Continued)

Potassium Chloride eV 0.04592 0.04508 0.04428 0.04350 0.04275 0.04215 0.04203 0.04133 0.04065 0.03999 0.03967 0.03936 0.03874 0.03843 0.03815 0.03757 0.03719 0.03701 0.03647 0.03595 0.03594 0.03542 0.03471 0.03409 0.03348 0.03268 0.03224 0.03100 0.02976 0.02914 0.02852 0.02790 0.02728 0.02666 0.02604 0.02542 0.02480 0.02356 0.02232 0.02170 0.02108 0.02046 0.01984 0.01922 0.01860 0.01798

cm- x

/~m

n

370.4 363.6 357.1 350.9 344.8 340 339.0 333.3 327.9 322.6 320 317.5 312.5 310 307.7 303.0 300 298.5 294.1 290 289.9 285.7 280 275 270 265 260 250 240 235 230 225 220 215 210 205 200 190 180 175 170 165 160 155 150 ! 45

27.0 27.5 28.0 28.5 29.0 29.41 29.5 30.0 30.5 31.0 31.25 31.5 32.0 32.26 32.5 33.0 33.33 33.5 34.0 34.48 34.5 35.0 35.71 36.36 37.04 37.74 38.46 40.00 41.67 42.55 43.48 44.44 45.45 46.51 47.62 48.78 50.00 52.63 55.56 57.14 58.82 60.61 62.50 64.52 66.67 68.97

1.31281 1.30520 1.29731 1.28911 1.28060 1.30 [3] 1.27175 [8] 1.2626 1.2530 1.2431 1.28 [-3] 1.2327 [8] 1.2220 1.25 [3] 1.2108 [8] 1.1991 1.20 [3] 1.8669 [8] 1.1741 1.15 [3-] 1.1608 [8] 1.1469 1.10 [3] 1.05 0.99 0.93 0.85 0.86 0.80 0.70 0.53 0.61 0.70 0.57 0.41 0.31 0.23 0.17 0.15 0.17 0.24 0.44 1.1

6.63 x 10- 3

2.3 x 10-2 [-3] 2.5 2.8 3.2 4.2 8.3 x 10 -2 0.16 0.17 0.18 0.20 0.35 0.58 0.52 0.50 0.53 0.77 1.05 1.5 2.2 2.9 4.0 5.5

(continued)

718

Edward D. Palik TABLE VIII

(Continued)

Potassium Chloride eV

cm- ~

0.01736 0.01674 0.01612 0.01550 0.01488 0.01364 0.01240 0.01178 0.01116 0.01054 0.009919 0.009299 0.008679 0.007439 0.006199 0.004133 0.003874 0.003100 0.002480

140 135 130 125 120 110 100 95 90 85 80 75 70 60 50 33.33 31.25 25.00 20.00

0.002066 0.001771 0.001550 0.001378

16.67 14.29 12.50 11.11

0.001240 0.0008265 0.0006199 0.0004133

10.00 6.667 5.000 3.333

jum 71.43 74.07 76.92 80.00 83.33 90.91 100.0 105.3 111.1 117.6 125.0 133.3 142.9 166.7 200.0 300 320 400 500 600 700 800 900 1,000 1,500 2,000 3,000

5.3 5.0 4.1 3.5 3.2 2.9 2.7 2.6 2.5 2.4 2.3 2.2

4.0 0.80 0.3.2 0.23 0.20 0.15 0.11 9 . 0 x 10 -1 6.6• 10 -2 0.11 0.11 9.2 • 10 -2 5.3 3.1 x 10 -a [-1] 2.65 [2] 2.511] 1.9 1.83 [23 1.611] 1.3 1.2 1.0 1.07 ]-2] 9.0 • 10 -3 [1] 5.5 3.7 2.0

" T h e original references for the n and k data are indicated in brackets. A given reference applies down the column until a new one appears. The smaller values of k are given in powers of 10 until 10-2, whereupon we revert to decimals only.

Silicon Dioxide (SiO2), Type (Crystalline) H, R. PHILIPP General Electric Research and Development Center Schenectady, New York

The room-temperature optical properties of type e, crystalline S i O 2 have been the subject of numerous studies. In the infrared spectral region, this information is reasonably detailed. In the vacuum ultraviolet, however, few attempts have been made to obtain a self-consistent set of optical constants n and k for this material. These parameters are generally obtained by KramersKronig (KK) analysis of reflectance data, which are difficult to measure with high accuracy. In addition, the analysis in the case of S i O 2 requires extrapolation of the reflectance into spectral regions in which no data exist. This introduces considerable uncertainty in the derived n and k values. In the region of low absorption between the infrared and vacuum ultraviolet bands, the index of refraction can be evaluated from prism data, and this has been accomplished with great precision. The measured k values, on the other hand, can be strongly influenced by the presence of impurity and defect absorption, and for this reason no k values are given in the wavelength range 6.0 #m > 2 > 0.1476 #m, although some data are available for the users' consideration and will be referenced. A set of optical parameters n and k are given in Table IX and plotted in Fig. 9 for the wavelength region 333.3/tm > 2 > 0.04768/~m. They were obtained from a variety of sources. The wavelength A, index of refraction n and extinction coefficient k are given to four places in most of the table entries. The accuracy of these values does not, in general, warrant this precision. This is done to show the possible presence of weak structure in the optical properties that can often be discerned by comparison of precise, but not necessarily accurate, values over small wavelength ranges. The no and ko values refer to the ordinary (o) ray (electric field perpendicular to the optic axis), and n e and k~ refer to the extraoridinary (e) ray (electric field parallel to the optic axis). 719 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright 9 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-544420-6

720

H.R. Philipp

In the sub-millimeter-wave region from 1000 to 200 #m the no results of several workers have been combined and fitted by Simonis* with a frequencysquared dependence. This fit gives values in agreement with all these experiments to _+0.0015, except for the results of Loewenstein et al. [1], which are consistently 0.005 higher. Most of these results were obtained by analysis of channel spectra. In the far infrared 333 > 2 > 50 #m the optical constants determined by Loewenstein et al. [-1] are used. The estimated experimental error in n is _+0.001, but at least in the region 333-200 pm it is considerably more. The error in k is considerably larger than the error in n. Their data extend to shorter wavelength (2 = 30.3 pm) for temperature T = 1.5 K but were not evaluated at room temperature. Because of the strong temperature dependence of the absorption in this region and the presence of a resonance absorption at 2 ~ 37.88 pm, considerable caution should be taken in using the low-temperature data to estimate room-temperature values. In the infrared, 32.00 #m > 2 >_ 2.000 #m, the values were obtained from the analysis of Spitzer and Kleinman [2]. They carefully measured the reflectance and fitted the results to classical dispersion theory. Their n and k values reproduce the measured reflectance and independently obtained transmission curves in detail over most of the spectral range. They also compare their results with values derived by KK analysis of the same reflectance data. They find that the agreement in k is quite good for k > 0.1 but is poor for k < 0.1. We believe that this inconsistency is due, at least in part, to the lack of adequate precision in the reflectance data that is necessary to evaluate accurately small k values by KK analysis. We note in this connection that in the regions of very low reflectance, the measured values are generally appreciably higher (on a relative basis) than those calculated from the n and k values. This could result from a small scattered-light component in the measuring beam. The n and k values given in the table were computed from the dispersion parameters of Spitzer and Kleinman [2] that do not consider absorption processes (resonances) above 2 ~ 35 #m. The results for the e ray join, more or less smoothly, in the region 32 pm < 2 < 50 ktm with the results of Loewenstein et al. [1], while those for the o ray, which show the presence of a resonance near 38 #m, do not. At the shorter wavelengths, the results of Spitzer and Kleinman [2] agree very well with index of refraction data for SiO2 obtained from prism measurements [3]. For the o ray, the precise prism measurements extend as high as 4 #m, while for the e ray, they extend only to 2 #m. It should be noted here that less precise index of refraction values given in the American Institute of Physics Handbook [3] for the range 4.2 pm < 2 < 7.0 #m do not agree very closely with the dispersion analysis results [-2] that are given in the table. No k values are given in the table for infrared wavelengths below 6.00 #m. Both Saksena [4] and Drummond [5] have examined the absorption spectra * See Chapter 8 of this handbook.

Silicon Dioxide (SiO2), Type ~ (Crystalline)

721

of ~ - S i O 2 for 2 < 6.00 #m, and the results show a variety of features. It is difficult to determine which of these features are intrinsic to SiO2, and, hence, these results are not considered further here, although they do contain a wealth of information. In the wavelength region 2.00/~m > 2 > 0.185/~m, the precise index of refraction values of the American Institute of Physics Handbook [3] are given in table. Some time ago Sosman [6] prepared a list of most probable values of the refraction indices of SiO2, and these values are also given in the table, in which they are denoted by brackets. These latter values are generally more precise than those of the American Institute of Physic Handbook [3]; however, I cannot comment on their accuracy. For 2 < 0.185 #m, the optical constants given in the table were obtained from the reflectance measurements of Philipp [7] augmented by the transmission measurements of Appleton et at. [8] and Philipp [9] in the tail of the fundamental absorption edge. Both Lamy [10] and Philipp [7] have measured the reflectance and determined n and k by K K analysis. The analysis of Lamy [10] uses the reflectance values of Philipp [-7] for 2 < 0.10/~m. The n values from these two sources agree very closely for wavelengths down to ~0.11/~m. The k values show very good agreement for 2 < 0.08/~m. The differences found below these wavelengths of ~ 103/o must be attributed to the extrapolation procedures used in the K K analysis. The choice of using Philipp's [7] values over those of Lamy [10] was an arbitrary one. Care should be exercised in using the table results, especially for 2 < 0.08 ~m, until data are available for 2 < 0.05/~m, which more specifically details the reflectance extrapolation. The k values determined from the K K analysis of Phillipp [7] are tabulated for wavelengths of 0.1278/~m and below. Above this wavelength, where they become small, they are unreliable as determined by this technique. In the range 0.1476/~m > 2 > 0.1409/~m, they are obtained from the absorption measurements of Appleton et al. [8] and Philipp [9], which agree reasonably closely. A smooth curve was drawn between the absorption data and the K K results, and the values so determined are given in the table, in which they are delineated by brackets; they are not reliable. No k values are given for 2 > 0.1476 ~m because they may have a component associated with a defect or impurity absorption. It is recommended that anyone who uses the table carefully examine the material and references cited in the above-named references [1-10] as well as other references [ 11 - 15]. REFERENCES

1. E. V. Loewenstein, D. R. Smith, and R. L. Morgan, Appl. Opt. 12, 398 (1973). 2. W. G. Spitzer and D. A. Kleinman, Phys. Rev. 121, 1324 (1961). 3. "American Institute of Physics Handbook," 3rd ed., Chapter 6, p. 27. McGraw-Hill, New York, 1972.

722

H.R. Philipp

4. B. D. Saksena, Proc. Phys. Soc. London 72, 9 (1958). 5. D. G. Drummond, Proc. R. Soc. London 153, 328 (1935). 6. R. B. Sosman, "The Properties of Silica," p. 591. Chem. Catalog Co. (Tudor), New York, 1927. 7. H. R. Philipp, Solid State Commun. 4, 73 (1966). 8. A. Appleton, T. Chiranjivi, and M. Jafaripour-Ghazvini, in "The Physics of SiO 2 and Its Interfaces" (S. T. Pantelides, ed.), p. 94. Pergamon, New York, 1978. 9. H. R. Philipp, J. Phys. Chem. Solids 32, 1935 (1971). 10. P. L. Lamy, Appl. Opt. 16, 2212 (1977). 11. I. Simon and H. O. McMahon, J. Chem. Phys. 21, 23 (1953). 12. J. Reitzel, J. Chem. Phys. 23, 2407 (1955). 13. E. Loh, Solid State Commun. 2, 269 (1964). 14. S. Roberts and E. Coon, J. Opt. Soc. Am. 53, 1023 (1962). 15. D. E. McCarthy, Appl. Opt. 2, 591 (1963).

Silicon D i o x i d e (SiO2), T y p e ~ ( C r y s t a l l i n e )

.

.

.

.

. . . ... . . . . ... . . . . .. .

723

.

~.- - ~

.

.

.

.

0

.

.

_ ~3

.

o

z ~

b

N

..Q ,~,,

_o_

0 I ~

,

,

I,,,,

I

~

I

,

I~,~I

oo

,

,

~o .

,

I,,,,I

"b

.

i

,

,.,

I,,,.I

,

,

,

l,.,,I

~o .

~

.

~C~)

--

.

~,

a~ 1 '~u

I"~'I

' '

'

I""I

' '

o ~

I""'

'

1 I" ,

,

I"'"-

/ .... /

I i

I J -

o_

~

/

___~

0

~

T.~ 0

b

T~

o

o

T

o

%1 '~

e~

b

co

b

~

4 ~

9

I

724

H.R. Philipp

o~

r

o~

~.

~

~

~

~

~

~.

~

~

~

~

O 0

~

~

~.

~

~

~

w

w

~

~

~

w

~

~

~

C~

~

~

~

~

~

~

~

0

~

~

.

. ~.

o~

o., m o~ I_,

~

I

~ ' ~

~

~

. ~.

725

Silicon Dioxide (SiO~), Type = (Crystalline)

o o o o o o c ~

~--.--~.~.~.~.~.~.~.

~

~ 5 o o o o ~ o o

.~.

.~.~.~.

o-;

~r ~ . ~ .

"-;~

"

"oo

~ . ~ . ~ . ~ . ~ ~

o o o o o

~

~

~

~.

726

H.R. Philipp

o~

m

.~

r~

[.,,, m

o~,

I

Silicon Dioxide (SiO2), Type ~ (Crystalline)

X

X

X

X

X

X

727

728

H.R. Philipp

Q~ ~ c

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

~ ..~

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 ~ 6 ~ 6 6 6 6 6 6 6 6

I

Silicon Dioxide (SiO2), Type 0~ (Crystalline)

729

r-"-1 r-m O

t~l

~

~

t~

t'~

s

t~

~

~j[,~l

r"-"l

t__j

i..~1

r-"'3

i..._J

i__1

~ _ 1

1_._1

l__J

,~,

730

H.R. Philipp I-'-! C'.,I I_.=1

e X

I=--I I=_,..I

oO

oO

~

{'~

c'.~1 ("~

M'~ I.t'3 ' ~ "

(%1 ~.~

oO

~'~

oO

~

I-'-I C'.l I_,..=1

o==

r

J

X

Jo..., r.~

o

I

w

,~.,-~ ["~

('~

~1~ oO

~"~

~''~.

'~=

Silicon Dioxide (SiO2), Type ~ (Crystalline)

X

X

X

X

~'~

731

~:::) "~"

X

~

X

,-~

~'~'~ ~'~

~"I

~,~ ~

~

~'I

b"-

,,'~

~

,"'~ ~

I"~

0'~ '~:I" ,-~

~

,..~

C',l ~ "

~r~ ~C) I ' ~

~

(~'~

732

H.R. Philipp

I

.,.i

~

m

o.-, r~

I

Silicon Dioxide (SiO2), Type ~ (Crystalline)

733

.,..~

7 X

734

H.R.

I

X

I ~ m m

c X

9S

r~

~

~ m ~

I

Philipp

Silicon Dioxide (SiO2), Type ~ (Crystalline)

735

I

X

I

O

O

O

O

OO

~ O

OO

O O O O

OO

O O O

O

~

O

O

O

O O O

O

O

OO

736

H.R. Philipp

I

~

c

J J

X

~

~

0,~

,..~ ~

,~t. ~C)

~'~

~:~

~'~ O~

X

C

[,. ~

T

~

O~

C~

P~- ["".

P'-'. I""-

~::~ ~

~:)

~1~ ~r ~

~

~t'~ ~1- ~ -

Silicon Dioxide (SiO2), Type 0~ (Crystalline)

737

.,,.~

O X

X

X

X

738

H.R. Philipp

I

X

.,.i m

o

[,,,

~ J~

r--

I

r--: r--: r--

r--

m.

r--- r.-- r-.: r--

r.-- r.-- r.-- ~

r--- r.-- r--. r'-- r-.- ~

~

~

~

~

~

~

~

Silicon Dioxide (SiO/), Type ~ (Crystalline)

739

~j

I

X

740

H.R. Philipp

J oO

m I

I

o

['-' .,.i

d d d d e ~ $ ~ e e d d d d d d d d d d d d d d d d

i~

,..~

t...,i t-..i

~',~. ~

t.q

t.q

t.,q

t-...i t , q

t.-.,I ~,,..q r.q

t-q

t.q

t,q

~

oo

',,0

'~1- ~

oo

oo

oo

~,'h t",'h 0

oo

',,0

',~"

0',,

r

0",,

r

,,.,,

0",,

r

r~,l

t..q

oO

t~,l

,...-

t,q

~,1

t.,.,I t.,.,i ~

~

~

~

~

0., oo

oo

~1::::, ~,"~ r,"~

cq

,,--,

~

~

oO

t'--.

I"'--

~

~

~

~

I'-..- I'-.-

Silicon Dioxide (Si02), Type ~ (Crystalline)

741

I

•

~:~ ~

C~ ~t"

~0

~

t~l

~

,--~ r'-- ~

~

~

,-~

,--~

0"~ ~

[~-

~::~ ~

~

~'~ ~"~ ~

~t'~ ~rh ~

0"~ ~

'~t"

H.R. Philipp

742

O 0

eq

. 0

{N

~4 ~

eq

0

0

0

0

0

0

0

~-~ --~ --~ ,-~ ,-~ ,-~ --~

0

. 0

~4 eq e4 c4 e4 ~4 e4 eq ~

--~ ,-~ ,-~ ~ 4

r

~

~

~

~r

~

i'~ o o

0

0

(~

~

~

(~

1.0). The optical properties of SiO and SiO~ are different, and the values given here should only be applied to materials of stoichiometry SiO. For completeness, references will be given that also describe the optical behavior of SiO~ materials. Although SiO has some important applications, it is a rather dull material from an optical point of view. For this and other reasons, there is not very much information available on its optical properties. However, in the visible and near-ultraviolet spectral regions in which data from several sources can be compared, the results agree with one another quite well. This would indicate that SiO is a reasonably well defined material, when prepared in the absence of oxygen, with definitive optical properties. A set of optical parameters n and k for noncrystalline SiO are given in Table XI and plotted in Fig. 11 for the wavelength region 14.0/~m > 2 > 0.050 ~m. The wavelength ~, index of refraction n, and extinction coefficient k are given to four places in most of the table entries. The accuracy of the values does not warrant this precision. This is done to show the possible presence of weak structure in the optical properties that can often be discerned by comparison of precise, but not necessarily accurate, values over small wavelength ranges. In the infrared, 14.0 ~m > 2 > 1.00 #m, the values given in the table were obtained from Hass and Salzberg [ 1]. The values were derived through analysis of transmittance and reflectance measurements on a series of SiO layers 765 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright (~) 1985 by Academic Press, inc. All rights of reproduction in any form reserved. ISBN 0-12-544420-6

766

H.R. Philipp

of various thicknesses. Although the overall accuracy of these values cannot be simply specified, they fit the measured transmittance and reflectance to within _+2~. In the region 1.00/~m > 2 > 0.049/~m, the values are taken from Philipp [2]. They were derived from Kramers-Kronig (KK) analysis of reflectance data augmented by transmission measurements in the region of low absorption. These results agree reasonably well with those of Hass and Salzberg [ 1], whose measurements extend to 0.24 #m, as well as with the absorption data of Cremer et al. [3]. The results of Philipp are given in the table because they cover a wider wavelength range and more explicitly evaluate the extinction coefficient in the region of low absorption compared to the other measurements. They are not necessarily more accurate than those of Hass and Salzberg [1] and Cremer [3]. The differences that are found in these measurements [1-3], _+5~ or less in n and slightly larger in k, may be due in part to sample preparation (evapoaration rate, vacuum conditions, substrate temperature, etc.), and as stated earlier, it is encouraging that they agree as well as they do. It should also be pointed out that the table values at the shorter wavelengths, especially those below 0.100 ~m, depend somewhat on the extrapolation used in the KK analysis and should be treated with caution until more definitive measurements are made for 2 < 0.05 ~m, which more clearly delineate the proper reflectance extrapolation. It is recommended that anyone who uses the table carefully examine the measurements and references cited in the references discussed [1-3] as well as other references [4-7] that also describe the optical behavior of materials of stoichiometry SiOx (x > 1). REFERENCES

1. 2. 3. 4. 5. 6. 7.

G. Hass and C. D. Salzberg, J. Opt. Soc. Am. 44, 181 (1954). H. R. Philipp, J. Phys. Chem. Solids 32, 1935 (197]). E. Cremer, T. Kraus, and E. Ritter, Z. Elektrochem. 62, 939 (1958). E. Ritter, Opt. Acta 9, 197 (1962). A. P. Bradford and G. Hass, J. Opt. Soc. Am. 53, 1096 (1963). G. Hass, J. Am. Ceramic Soc. 33, 353 (1950). A. P. Bradford, G. Hass, M. McFarland, and E. Ritter, Appl. Opt. 4, 97! (1965).

Silicon Monoxide (SiO) (Noncrystalline) I0 i _,

767

~ ....

I0 o /

/

/

f

\

\

/

I i I/ iO-I

=10-2

10~,

i

lt,,,I

,

n

,

t

i

llllll

IO0

WAVF/E-N@TH Fig. 11. L o g - l o g plot n ( - - ) monoxide (noncrystalline).

,

l,lt,l

lO-I and k (. . . .

IO I

(,L.tm)

) versus wavelength in micrometers for silicon

TABLE XI Values of n and k for Silicon Monoxide (Noncrystalline) from Various References" eV 25 24 23 22.5 22 21.5 21 20.5 20 19.5 19 18.5 18 17.5 17 16.5 16 15.5

cm- 1

pm 0.04959 0.05166 0.05391 0.05510 0.05636 0.05767 0.95904 0.06048 0.06199 0.06358 0.06526 0.06702 0.06888 0.07085 0.07293 0.07514 0.07749 0.07999

n 0.8690 [2] 0.8444 0.8371 0.8391 0.8454 0.8519 0.8610 0.8721 0.8853 0.9007 0.9178 0.9376 0.9596 0.9825 1.008 1.036 1.066 1.098

k 0.2717 [2] 0.3060 0.3505 0.3761 0.3987 0.4222 0.4456 0.4688 0.4919 0.5140 0.5362 0.5578 0.5771 0.5961 0.6147 0.6309 0.6453 0.6566

(continued)

768

H.R. Philipp TABLE XI

(Continued)

Silicon Monoxide (Noncrystalline) eV 15 14.5 14 13.5 13 12.5 12 11.5 11 10.5 10 9.5 9 8.5 8 7.5 7 6.5 6 5.75 5.5 5.25 5 4.8 4.6 4.4 4.2 4 3.8 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.240 0.6199 0.4133 0.3100 0.2480

cm- x

/~m

48,390 46,380 44,360 42,340 40,330 38,710 37,100 35,490 33,880 32,260 30,650 29,040 27,420 25,810 24,200 22,580 20,970 19,360 17,740 16,130 14,520 12,900 11,290 10,000 5,000 3,333 2,500 2,000

0.08266 0.08551 0.08856 0.09184 0.09537 0.09919 0.1033 0.1078 0.1127 0.1181 0.1240 0.1305 0.1378 0.1459 0.1550 0.1653 0.1771 0.1907 0.2066 0.2156 0.2254 0.2362 0.2480 0.2583 0.2695 0.2818 0.2952 0.3100 0.3263 0.3444 0.3647 0.3875 0.4133 0.4428 0.4769 0.5166 0.5636 0.6199 0.6888 0.7749 0.8856 1.000 2.000 3.000 4.000 5.000

n

k

1.132 1.166 1.199 1.231 1.259 1.283 1.307 1.311 1.320 1.345 1.378 1.412 1.445 1.482 1.530 1.593 1.667 1.746 1.829 1.871 1.914 1.957 2.001 2.034 2.066 2.094 2.119 2.141 2.157 2.162 2.160 2.144 2.116 2.085 2.053 2.021 1.994 1.969 1.948 1.929 1.913 1.87 [1] 1.84 1.82 1.80 1.75

0.6651 0.6692 0.6698 0.6666 0.6602 0.6523 0.6464 0.6293 0.6529 0.6701 0.6843 0.6920 0.7002 0.7153 0.7333 0.7473 0.7479 0.7348 0.7084 0.6890 0.6663 0.6383 0.6052 0.5723 0.5364 0.4948 0.4499 0.4006 0.3453 0.2872 0.2287 0.1706 0.1211 0.08374 0.05544 0.03533 0.02153 0.01175 0.00523 0.00151

Silicon Monoxide (SiO) (Noncrystalline)

769

TABLE XI (Continued) Silicon Monoxide (Noncrystalline) eV 0.2066 0.1771 0.1653 0.1550 0.1459 0.1378 0.1305 0.1240 0.1181 0.1153 0.1127 0.1078 0.1033 0.09537 0.08856

cm- ~ 1,667 1,492 1,333 1,250 1,176 1,111 1,053 1,000 952.4 930.2 909.1 869.6 833.3 769.2 714.3

/~m 6.000 7.000 7.500 8.000 8.500 9.000 9.500 10.00 10.50 10.75 11.00 11.50 12.00 13.00 14.00

" References are indicated in brackets.

n 1.70 1.60 1.42 1.15 0.90 0.91 1.20 2.00 2.85 2.86 2.82 2.50 2.13 2.04 2.01

k

0.1811] 0.75 1.20 1.38 0.90 0.58 0.40 0.20 0.14 0.20 0.30

Silicon Nitride (Si3N4) (Noncrystalline) H. R, PHILIPP General Electric Research and Development Center Schenectady, New York

Noncrystalline silicon nitride (Si3N4) is an important material in integratedcircuit technology. In this application it is used in thin-film form and can be prepared by a variety of deposition techniques, including (a) pyrolytic decomposition of a mixture of gases containing silicon and nitrogen (for example, Sill 4 and NH3), (b) sputtering, and (c) rf glow-discharge techniques. Silicon oxynitride (SiOxNr) films can also be readily formed by adding small amounts of NO or 02 to the reactive gases. The optical properties of these films are strongly dependent on the deposition temperature and the Si-to-N- and Si-to-O-atom ratios. Other studies also indicate the presence of chemically bound hydrogen in the film. Thus, the stoichiometry of an arbitrary film can be more accurately indicated by SiOxNyH~, and the refractive index and other optical properties will be a function of x, y, and z. Samples prepared by hightemperature pyrolysis are generally considered to have, or come close to, the ideal stoichiometry, Si3N4 (exclusive of hydrogen content). While there have been a number of studies on the optical properties of silicon nitride materials, the data have generally been limited to the evaluation of the index of refraction in the visible region of the spectrum [1-14] and to infrared measurements of the lattice absorption bands near 11 #m and other bands, particularly those which show up N - H and Si-H bonding [1-7, 9, 12, 15-20]. The infrared measurements have been used mainly for structural and chemical evaluation purposes. While Taft [12] measured the absorption coefficient of silicon nitride films prepared by several different techniques in the region of 11-#m wavelength, no attempt has been made to determine a self-consistent set of optical parameters for this material in the infrared. In the region of strong electronic absorption, both Philipp [21] and Bauer [22] have evaluated n and k from the visible into the vacuum ultraviolet region. The results of Philipp [21], which extend to 24 eV, were determined by Kramers-Kronig analysis of reflectance and absorption data. The films 771 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright (~) 1985 by Academic Press, Inc. All rights of reproduction in any form reserved, ISBN 0-12-544420-6

772

H.R. Philipp

he used were prepared by pyrolytic decomposition at 1000~ of a mixture of Sill4 and NH3 gases in the ratio 1 to 40,000. Bauer's [22] measurements extend to only 7.5 eV but were obtained on samples prepared by a variety of deposition techniques including pyrolysis of a mixture of SiO4 and NH3 gases at substrate temperatures in the range 720 to 1000~ His n and k values were evaluated by error-function analysis of reflectance and transmission data. For pyrolytic silicon nitride films, the results of Philipp [21] and Bauer [22] show reasonable but not precise agreement. A set of optical parameters n and k for noncrystalline silicon nitride are given in Table XII and are plotted in Fig. 12 for the energy range 1-24 eV (1.24-0.0517-/~m wavelength). They are taken entirely from the work of Philipp [21]. These results are not necessarily more accurate than other data but were primarily chosen because they cover the widest energy range. In addition, Theeten et al. [23] in a study of Si3N 4 in the region from 1.5 to 5.8 eV using spectroscopic ellipsometry found good agreement with the values of Philipp [21] (and it is in this energy range that the largest differences occur between the results of Philipp [21] and Bauer [22]). In most of the table entries, the index of refraction n values are given to four places and the extinction coefficient k values are given to three places. The accuracy of these values does not warrant this precision. This is done to show the possible presence of weak structure in the optical properties that can often be discerned by comparison of precise, but not necessarily accurate, values over small energy ranges. It is also recommended that anyone who uses the table carefully examine the material and references cited in the references. Since the optical properties of silicon nitride depend on the method and temperature of deposition, care must be taken in using the table values for materials prepared by using techniques other than high-temperature pyrolysis of gases with a high N-to-Si ratio. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

V. Doo, D. Nichols, and G. Silvey, J. Electrochem. Soc. 133, 1279 (1966). K. Bean, P. Gleim, R. Yeakley, and W. Runyan, J. Electrochem. Soc. 114, 733 (1967). T. Chu, C. Lee, and G. Gruber, J. Electrochem. Soc. 114, 717 (1967). V. Doo, D. Kerr, and D. Nichols, J. Electroehem. Soc. 115, 61 (1968). D. Brown, P. Gray, F. Heumann, H. Philipp, and E. Taft, J. Electrochem. Soc. 115, 311 (1968). M. Grieco, F. Worthing, and B. Schwartz, J. Electrochem. Soc. 115, 525 (1968). R. Levitt and W. Zwicker, J. Electrochem. Soc. 114, 1192 (1967). B. Deal, P. Fleming, and P. Castro, J. Electrochem. Soc. 115, 300 (1968). G. Brown, W. Robinette, Jr., and H. Carlson, J. Electrochem. Soc. 115, 948 (1968). E. V. Shitova, I. A. Yasneva, and N. A. Genkina, Opt. Spectrosc. 43, 140(1977); E. V. Shitova, I. A. Yasneva, and N. A. Genkina, Opt. Spektrosk. 43, 244 (1977). 11. Y. N. Volgin, O. P. Borisov, Y. I. Ukhanov, and N. I. Sukhanova, J. Appl. Spectrose. 24, 115 (1976); Y. N. Volgin, O. P. Borisov, Y. I. Ukhanov and N. I. Sukhanova, Zh. Priklad. Spektrosk. 24, 164 (1976). 12. E. A. Taft, J. Electrochem. Soc. 118, 1341 (1971). 13. M. J. Rand and D. R. Wonsidler, J. Electrochem. Soc. 125, 99 (1978).

Silicon Nitride (Si3N4) (Noncrystalline)

773

14. T. Wittberg, J. Hoenigman, W. Moddeman, C. Cothern, and M. Gulett, J. Vac. Sci. Technol. 15, 348 (1978). 15. S. Yoshioka and S. Takayanagi, J. Electrochem. Soc. 114, 962 (1967). 16. H. J. Stein, Appl. Phys. Lett. 32, 379 (1978). 17. S. M. Hu, J. Electrochem. Soc. 113, 693 (1966). 18. H. J. Stein and H. A. R. Wagener, J. Electrochem. Soc. 124, 908 (1977). 19. P. S. Peercy, H. J. Stein, B. L. Doyle, and S. T. Picraux, J. Electron. Mater. 8, 11 (1979). 20. L. F. Cordes, Appl. Phys. Lett. 11, 383 (1967). 21. H. R. Philipp, J. Electrochem. Soc. 120, 295 (1973). 22. J. Bauer, Phys. Status. Solidi A 39, 411 (1977). 23. J. B. Theeten, D. E. Aspnes, F. Simondet, M. Errman, and P. C. Miirau, J. Appl. Phys. 52, 6788 (1981).

I

l illll

l

l

i

..... I

i l llll

I

I0 o

I I I I I I I I I I I

lO-i _

c--

I I

I I I I

10-2 _

I

I I I I I I

I I I i I,,,,I

i

Ii

i l lill[

10-I WAVELENGTH

Fig. 12. Log-log plot of n ( - - ) nitride (noncrystalline).

and k (. . . .

I i0 0

(,LLm)

) versus wavelength in micrometers for silicon

774

H.R. Philipp TABLE XII Values of n and k for Silicon Nitride (Noncrystalline) from Various References" eV 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10.5 10 9.5 9 8.5 8 7.75 7.5 7.25 7 6.75 6.5 6.25 6 5.75 5.5 5.25 5 4.75 4.5 4.25 4 3.5 3 2.5 2 1.5 1

cm- 1

~m

n

96,790 88,720 84,690 80,650 76,620 72,590 68,560 64,520 62,510 60,490 58,470 56,460 54,440 52,430 50,410 48,390 46,380 44,360 42,340 40,330 38,310 36,290 34,280 32,260 28,230 24,200 20,160 16,130 12,100 8,065

0.05166 0.05391 0.05636 0.05904 0.06199 0.06526 0.06888 0.07293 0.07749 0.08266 0.08856 0.09537 0.1033 0.1127 0.1181 0.1240 0.1305 0.1378 0.1459 0.1550 0.1600 0.1653 0.1710 0.1771 0.1837 0.1907 0.1984 0.2066 0.2156 0.2254 0.2362 0.2480 0.2610 0.2755 0.2917 0.3100 0.3542 0.4133 0.4959 0.6199 0.8266 1.240

0.655 [-213 0.625 0.611 0.617 0.635 0.676 0.735 0.810 0.902 1.001 1.111 1.247 1.417 1.657 1.827 2.000 2.162 2.326 2.492 2.651 2.711 2.753 2.766 2.752 2.724 2.682 2.620 2.541 2.464 2.393 2.331 2.278 2.234 2.198 2.167 2.141 2.099 2.066 2.041 2.022 2.008 1.998

" Data obtained from Philipp [21].

k 0.420 0.481 0.560 0.647 0.743 0.841 0.936 1.03 1.11 1.18 1.26 1.35 1.43 1.52 1.53 1.49 1.44 1.32 1.16 0.962 0.866 0.750 0.612 0.493 0.380 0.273 0.174 0.102 5.7 x 2.9 1.1 x 4.9 x 1.2 x 2.2 x

[213

10 -2 10 -2 10-3 10 -3 10 -4

Sodium Chloride (NaCl) J. E. ELDRIDGE Department of Physics University of British Columbia Vancouver, British Columbia, Canada EDWARD D. PALIK Naval Research Laboratory Washington, D.C.

The room-temperature values of n and k tabulated here were obtained from the following works and references therein: The centimeter-millimeterwave-region results, 0.33 cm-1 (30 mm) to 5 cm-1 (2 ram), are from Owens [ 1]; the 3.1-cm- 1 (3200-/~m) to 10-cm- ~ (1000-#m) results are from Stolen and Dransfeld [2], Genzel et al. [3], and Dianov and Irisova [4]; the farinfrared reststrahlen results, 10 cm-a (1000/tm) to 500 cm-1 (20/~m), are from Eldridge and Staal [5]; the multiphonon-tail data, 500 cm- ~ (20 #m) to 900 cm-~ (11.1 fire), are from Harrington et al. [6]; values of k in the transparent region, 943 cm -I (10.6/tm) to 3571 cm -1 (2.8/~m), are from Allen and Harrington [7]; while those in the visible, 19,430 cm-~ (0.5145/tm) to 28,489 cm-1 (0.3511/~m), are from Harrington et al. [8]; values of n from 500 cm-~ (20/~m) to 50,000 cm-1 (0.2/tin), also in the transparent region, are from Li [9], who consolidated various data [10-13]; the visible and ultraviolet values are from Miyata and Tomiki [14] and Roessler and Walker [15]; L2. 3 absorption-edge data for C1- are from Aita et al. [16]. A composite smooth-curve plot of log n and log k versus log wavelength is given in Fig. 13. Numerical values are given in Table XIII. In the centimeter-millimeter-wave region a filled resonant-cavity method was used [1] to measure the real part of the dielectric function e' and the loss tangent, tan c5 = c,"/e,'. From these, n and k could be extracted. Other results in the millimeter-wave region were obtained with transmission measurements made on slab samples of different thicknesses to determine n and k [2-4]. While not listed in the table, Breckenridge [17] gives 85~ data over a wide range of frequencies, 102-101~ Hz by measuring the capacitance and 775 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright @ 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-544420-6

776

J.E. Eldridge and Edward D. Palik

susceptance of blocks of NaC1. Generally, k is less than the last value given in the table, as was noted by Owens [1]. Above 10 cm-1 the far-infrared results were obtained mainly by the technique of dispersive-reflection spectroscopy [5]. In this technique a flatpolished (to within 0.3 #m) crystal of the material under investigation replaces the mirror in one of the arms of a Michelson interferometer. The temperature was 290 K. The resulting interferogram is asymmetric and contains information on both the reflectance amplitude r and phase change 0 where = r e i~ for the crystal. When ratioed with a background spectrum, obtained when the usual mirror is in place, both r and 0, and consequently a l l of the real and imaginary optical properties, can be obtained simultaneously and directly. This obviates the need for a Kramers-Kronig (KK) analysis of power reflectivity measurements, which has serious and well-known limitations [18]. For instance, the KK analysis will give poor values of n or k if either one is much smaller than the other. It works best when the values are comparable, since they both then have an effect of the power reflectivity. Of course, one also has to measure this power reflectivity over a large range for the integral to be at all valid. Furthermore, the power reflectivity becomes extremely small following a strong resonance such as the transverse-optic lattice mode in alkali halides. This occurs in the region of the longitudinaloptic frequency. All of the fine structure in the optical properties at this frequency is therefore usually lost. Another difficult region is the reststrahlen peak when the reflectivity can be very close to unity and an accurate value for k difficult to obtain. One has then to resort to measurements of transmission through thin films, with their concomitant dependence on interference effects and the quality and properties of the surfaces. The dispersive-reflection technique is therefore obviously superior, but it is not without its difficulties. The main one of these is the physical interchange of reference mirror and sample in exactly the same position (either that or the exact relative position must be known). In Eldridge and Staal [5] this problem was overcome by aluminizing a portion of the large NaC1 crystal, which would then act as the reference surface. (A correction must be made to the reflectivity and phase change of aluminum, depending on the film thickness.) In order to avoid the other main problem of ratioing spectra taken subsequent to one another, with the associated errors due to micrometer mirror-drive backlash, thermal drift, and component drift, a mask was placed in front of the aluminized sample and switched from background to sample at each step of the moving mirror. Dual interferograms were therefore simultaneously obtained. The eight-sector design of the mask and aluminized sample also relaxed the condition of flatness required of the polished sample, since both sample and reference were equally affected. The results obtained cover the reststrahlen region between 100cm - 1 (100 #m) and 340 cm-1 (29.4/~m). Below this, and up to 500 cm-1 (20 #m),

Sodium Chloride (NaCI)

777

the results were supplemented with transmission measurements through thin, wedge-shaped, polished single crystals. One can see in Fig. 13 that there is structure after the resonance, between 200 and 300 cm- 1 (50 to 33/tm), and a distinct shoulder at 20 cm- 1 (500 #m). In the past a multioscillator model has been used to fit reflection and transmission data [3]. However, there is only one oscillator, the TO mode, but it has an extremely frequency-dependent damping. The primary component of this damping is the coupling of the TO mode, through cubic anharmonicity, to two phonons. Energy and crystal momentum are conserved. This accounts for most of the structure in the damping. Three-phonon and higherorder processes also contribute strongly, depending on the temperature and the frequency, but they contain little structure, especially at room temperature. There is also some damping around the resonance due to the isotopic disorder of the chlorine ion, although this is extremely small at room temperature. All of the damping contributions mentioned can now be calculated with high accuracy, by use of the theories that have been developed, in conjunction with good lattice-dynamical data, which are obtained by inelastic neutron scattering (see Eldridge and Staal [5]). The agreement between experiment and theory is excellent. The similarities in the optical-phonon spectra of many alkali halides are discussed by Bilz [19]. Beyond the reststrahlen peak, in the "transparent" region (20-0.2 #m), the refractive index n has been determined by Li [9], who has fitted a multitude of experimental data [10-13] with a dispersion formula of the Sellmeier type, assuming k = 0. Temperature no doubt varied from 293 to 298 K for various laboratories. The formula for 293 K is n 2 -- 1.00055 +

0.19822 0.4839822 0.3869322 + -k22 _ (0.05) 2 22 _ (0.1) 2 22 _ (0.128) 2

0.2599822 0.0879622 3.1706422 0.3002822 + 22 - (0.158)2 + 22 - (40.5) 2 + 22 - (60.98)2 + 22 - (120.34) 2.

(1)

The 15 adjustable parameters were first chosen as input parameters for least-square fitting based on previous fits of more restricted data. The first 5 terms constitute the dispersion due to all interband oscillators above the band gap. Note that Li [9] used three far-IR classical oscillators to account for dispersion well above the TO phonon frequency (60.98 #m). This choice is based on earlier work of Genzel et al. [3] and Geick [18], who measured and collected n and k data, interpreting them as indicating oscillators at these frequencies. Li's [9] fit agrees with the experiment generally to a few units in the fourth decimal place (+0.03%). We have therefore used this fit in the table for the region from 0.2 to 20/tm as representative of experimental data. Typical measurements of n were done with a prism utilizing the angle of minimum deviation [-10-13, 20, 21] with four to five decimal places quoted.

778

J.E. Eldridge and Edward D. Palik

Interestingly, even present-day determinations of n for laser-window materials use this technique and achieve an accuracy of several parts in 106, quoting five decimal places [22]. Since the temperature dependence of n is -3.27 x 10- 5 K - ~ at 2000 cm- ~, one sees that control of room temperature (293 K) is important, since a variation of 1 K from laboratory to laboratory leads to changes of three units in the fifth decimal place [9]. The values of k between 550 cm-1 (18.2 pm) and 900 era- ~ (11.1 #m) in Table XIII were taken from Harrington et al. [6] and are the average transmission results through single crystals, taken at two laboratories with different spectrometers. Corrections for the low-temperature emittance of the sample were also employed. Sample temperature was 300 K. These agree at the 500-cm-1 end with the infrared results of Eldridge and Staal [5] and at the high-energy end with the laser-calorimetry results of Allen and Harrington [7]. These authors obtained values at five distinct laser wavelengths between 10.6 ,and 2.8/tm. Here the values of k are so small that the calorimetry method is essential. Up to 1000 cm- ~ or so (i.e., above 10 pro), the absorption is multiphonon damping of the TO resonance as previously mentioned, and agreement between experiment and theory is good. Beyond 1000 cm-1, the absorption is greater than predicted by the theory, and one must assume that much of it is extrinsic and due to impurity levels in the gap [7, 8]. Certainly, as materials are purified, this absorption decreases. One also has a sizable contribution from surface absorption, which can, however, be dynamically separated from the bulk absorption in the lasercalorimetric technique. Surface polish is still very important. The bulk values are quoted in Table XIII. In the UV region near the band gap, there are two detailed measurements [14, 15]. Thin transmission samples at 298 K have been used to determine n and k in the Urbach tail [14], while in the exciton region (0.158/~m) and above the band gap, near-normal-incidence reflectance measurements of cleaved samples have been made followed by KK analysis. In Miyata and Tomiki [14] the reflectance at wavelengths shorter than 0.092 #m has been approximated by extrapolation to zero, while the contribution of the reststrahlen region at long wavelength has been added in performing the KK integrals. The experimentally determined values of k in the Urbach tail are fitted smoothly to the values of k above the band gap, since KK values determined below the gap are erratic. There is no reliable oscillator model for interband absorption to give the detailed values and structure shown in Fig. 13, although in Eq. (1) lossless oscillators are assumed above the fundamental band gap to account for the dispersion below the band gap. The second set of data [15] was measured over a wider range from 0.048 to 0.247 #m (at a temperature of 300 K) and a method of KK analysis used that utilized a restricted data range (as justified in Roessler [23]). Farinfrared contributions were neglected. The two sets of data differ by as much as 20% at peaks and valleys in n and k. Perusal of the reflectance data

Sodium Chloride (NaCI)

779

indicates that the reflectance of Miyata and Tomiki [14] was higher at peaks than the Roessler and Walker [15] data, although not always at the valleys. Each work quotes experimental uncertainties in R to +_3~ in the range 0.2479-0.1033 #m with Roessler and Walker [15] giving _+ 10~ for 0.10330.0476 ~tm. Such variations are enough to account for the differences in n and k. At low temperature adsorbed gases were a serious problem affecting R; this did not seem to be a problem at room temperature, although pressures of about 3 x 10- ~ torr were reported in Miyata and Tomiki [14]. This suggests that monolayers of absorbed molecules such as water and hydrocarbons form within minutes after cleaving. Each work is presumably an absolute reflectance measurement involving comparison of the direct intensity with the reflected intensity with no additional mirror reflection involved. The use of refractive-index data in the transparent region by Miyata and Tomiki [14] as a secondary standard seems to be reasonable. In the interband spectral region from 0.04 to 0.16/tm, the structure in k is intrinsic due to a series of excitions associated with various interband transitions [15, 24]. The Urbach tail at wavelengths slightly longer than 0.16 #m is due to intrinsic rather than impurity excitons, as temperature dependence shows [ 14]. Extrapolation of this tail to longer wavelength yields values of k much smaller than observed experimentally [8], which suggests absorption due to color centers or metal-ion impurities. Since we have a classic case of two reflectance measurements differing slightly with the subsequent differences in n and k upon KK analysis, we elect to display both sets of data for this one material. Above 25 eV there are few data giving any sort of absolute values for n and k. The available data have been reviewed by Haelbich et al. [25]. Relative absorption in the 25-72 eV region has been measured by Le Comte et al. [26] for samples at 80 K. The room-temperature, L2, 3 absorption edge of C1- in NaC1 has been measured on an absolute scale by Aita et al. [16] in the 200-210-eV region at 300 K and is tabulated. The samples were evaporated polycrystalline films on transparent substrate films. The source was an x-ray tube emitting continuous radiation. Brown et al. [27], using synchrotron radiation, have made measurements in a somewhat wider spectral region but only in arbitrary units. The absorption in arbitrary units has been measured in the region 200280 eV by Iguchi et al. [28]. The K absorption edge of Na + near 1076 eV has been measured by Rule [29], and the K absorption edge of C1- near 2825 eV has been measured by Sugiura [30]. REFERENCES

1. J. C. Owens, Phys. Rev. 181, 1228 (1969). 2. R. Stolen and K. Dransfeld, Phys. Rev. 139A, 1295 (1965). 3. L. Genzel, H. Happ, and R. Weber, Z. Physik 154, 13 (1959).

780

J.E. Eldridge and Edward D. Palik

4. E. M. Dianov and N. A. Irisova, J. Appl. Spectros. 5, 187 (1966); E. M. Dianov and N. A. Irisova, Zhur. Prikl. Spektrosk. 5, 251 (1966). 5. J. E. Eldridge and P. R. Staal, Phys. Rev. B 16, 4608 (1977). 6. J. A. Harrington, C. J. Duthler, F. W. Patten, and M. Hass, Solid State Commun. 18, 1043 (1976). 7. S. Allen and J. A. Harrington, Appl. Opt. 17, 1679 (1978). 8. J. A. Harrington, B. L. Bobbs, M. Braunstein, R. K. Kim, R. Stearns, and R. Braunstein, Appl. Opt. 17, 1541 (1978). 9. H. H. Li, J. Phys. Chem. Refer. Data 5, 329 (1976). 10. H. Rubens and E. F. Nichols, Ann. Phys. Chem. 60, 418 (1897). 11. H. Rubens and A. Trowbridge, Ann. Phys. Chem. 60, 724 (1897). 12. F. Paschen, Ann. Physik 26, 120 (1908). 13. F. F. Martens, Ann. Physik 6, 603 (1901); F. F. Martens, Ann. Physik 8, 459 (1902). 14. T. Miyata and T. Tomiki, J. Phys. Soc. Jpn. 24, 1286 (1968); T. Miyata and T. Tomiki, J. Phys. Soc. Jpn. 22, 209 (1967); T. Miyata and T. Tomiki, private communication (1981). 15. D. M. Roessler and W. C. Walker, J. Opt. Soc. Am. 58, 279 (1968); D. M. Roessler and W. C. Walker, Phys. Rev. 166, 599 (1968). 16. O. Aita, I. Nagakura, and T. Sagawa, J. Phys. Soc. Jpn. 30, 1414 (1971). 17. R. G. Breckenridge, J. Chem. Phys. 16, 959 (1948). 18. R. Geick, Z. Physik 166, 122 (1962). 19. H. Bilz, in "Correlation Functions and Quasiparticle Interactions in Condensed Matter" (J. W. Halley ed.), p. 531. Plenum, New York, 1978. 20. S. P. Langley, Ann. Astrophy. Obs. Smithsonian Inst. 1, 219 (1902). 21. W. W. Coblentz, J. Opt. Soc. Am. 4, 443 (1920). 22. A. Feldman, D. Horowitz, R. M. Waxler, and M. J. Dodge, NBA Technical Note 933, Optical Materials Characterization. U.S. Department of Commerce, National Bureau of Standards, Washington, D.C., February 1979. 23. D. M. Roessler, Brit. J. Appl. Phys. 16, 1119 (1965); D. M. Roessler, Brit. J. Appl. Phys. 17, 1313 (1966). 24. T. Tomiki, T. Miyata, and H. Tsukamoto, Z. Naturforsch, 29, 145 (1974). 25. R. P. Haelbich, M. Iwan, and E. E. Koch, "Physik Daten-Physics Data", p. 57. Fachinformation Zentrum, Karlsruhe, 1977. 26. A. Le Comte, A. Savary, M. Morlais, and S. Robin, Opt. Commun. 4, 296 (1971). 27. F. C. Brown, C. G/ihwiller, H. Fujita, A. B. Kunz, W. Scheifley, and N. Carrera, Phys. Rev. B 2, 2126 (1970). 28. Y. Iguchi, T. Sagawa, S. Sato, M. Watanabe, H. Yamashita, A. Ejiri, M. Sasanuma, S. Nakai, M. Nakamura, S. Yamaguchi, Y. Nakai, and T. Oshio, Solid State Commun. 6, 575 (1968). 29. K. C. Rule, Phys. Rev. 66, 199 (1944). 30. C. Sugiura, Phys. Rev. B 6, 170 (1972).

Sodium Chloride (NaCI) I01

,,,,i

I

I

I

lilil

I

781 i

I

i

ltl,i

I

i

I

i

IIIII

I

I

!

I

lilll

I

i

i

i lliil

I

I

t

I

III

--

I0o

#

/

Jvl \

lO-i

\\ ,,,,\ \

\

\ \

IU z

\

\ \

-

___,d_

I

d

1

\\

/

\

\

i0-3

/

I I L I L 3 (Na)

\

I

\-

?

I L2,3 (CI) 10-4

10-5

10-6

Jllli

I

10-2

I I llllll

I

I I Illttl

iO-I

I

I

t llllll

i

i0 o

I01

WAVELENGTH

(~m)

Fig. 13. L o g - l o g plot of n ( - - ) sodium chloride.

and k ( . . . .

1

J llllll

I

102

I 1 lilJll

i

103

J t lill

104

) versus wavelength in micrometers for

T A B L E XIII Values of n and k for Sodium Chloride from Various References" eV 209.5 209 208.5 208 207.5 207 206.5 206

cm-l

/~ 59.18 59.32 59.46 59.61 59.75 59.89 60.04 60.19

n

k 2.54 2.56 2.58 2.55 2.56 2.59 2.64 2.62

x

10 -3

n

k

[16]

(continued)

782

J.E. Eldridge and Edward D. Palik TABLE XIII

(Continued)

Sodium Chloride eV

cm- 1

205.75 205.5 205.25 205 204.75 204.5 204.25 204 203.75 203.5 203.25 203 202.75 202.5 202.25 202 201.5 201 200.5 200 eV 26.0 25.6 25.4 25.2 25.0 24.9 24.7 24.3 24.1 24.0 23.9 23.8 23.4 23.3 23.2 23.1 23.0 22.8 22.6 22.5 22.4 22.3 22.2 22.0

/~

n

60.26 60.33 60.41 60.48 60.55 60.63 60.70 60.78 60.85 60.92 61.00 61.07 61.15 61.23 61.30 61.38 61.53 61.68 61.84 61.99 cm- ~

/~m 0.04769 0.04842 0.04880 0.04919 0.04959 0.04979 0.05019 0.05101 0.05144 0.05166 0.05187 0.05209 0.05298 0.05320 0.05343 0.05366 0.05391 0.05437 0.05485 0.05510 0.05534 0.05559 0.05584 0.05636

k 2.57 2.45 2.48 2.53 2.60 2.68 2.65 2.64 2.52 2.38 2.14 2.08 2.04 2.06 2.24 2.22 1.96 1.91 1.91 1.92

n 0.83 [15] 0.82 0.83 0.83 0.83 0.82 0.82 0.81 0.81 0.82 0.82 0.83 0.85 0.85 0.85 0.83 0.82 0.80 0.79 0.80 0.80 0.81 0.82 0.83

k 0.15 [15] O.17 O.17 0.18 0.18 0.18 0.18 0.21 0.22 0.23 0.23 0.24 0.23 0.22 0.21 0.21 0.21 0.23 0.26 0.27 0.28 0.29 0.30 0.31

n

k

Sodium Chloride (NaCI)

783 TABLE XIII

(Continued)

Sodium Chloride eV 21.9 21.8 21.5 21.1 20.9 20.8 20.6 20.5 20.2 20.1 20.0 19.9 19.8 19.6 19.5 19.4 19.3 19.2 19.1 18.9 18.8 18.7 18.6 18.5 18.4 18.3 18.2 18.0 17.5 17.3 17.1 17.0 16.9 16.8 16.7 16.5 16.4 16.3 16.2 16.1 15.9 15.7 15.5 15.4 15.3 15.2

cm- ~

~um 0.05661 0.05687 0.05766 0.05875 0.05931 0.05960 0.06018 0.06048 0.06137 0.06168 0.06199 0.06230 0.06261 0.06325 0.06358 0.06390 0.06423 0.06457 0.06490 0.06559 0.06594 0.06629 0.06666 0.06702 0.06737 0.06774 0.06811 0.06888 0.07085 0.07166 0.07250 0.07293 0.07335 0.07379 0.07423 0.07514 0.07559 0.07601 0.07652 0.07700 0.07797 0.07896 0.07999 0.08050 0.08103 0.08156

n 0.84 0.85 0.87 0.88 0.88 0.88 0.88 0.87 0.87 0.88 0.88 0.89 o.91 0.92 0.92 0.91 0.90 0.88 0.87 0.86 0.86 0.87 0.88 0.89 0.90 0.90 0.90 0.89 0.85 0.84 0.82 0.82 0.82 0.82 0.82 0.80 0.79 0.77 0.74 0.74 0.74 0.74 0.75 0.76 0.76 0.77

k

n

k

0.32 0.32 0.32 0.31 0.30 0.30 0.30 0.31 0.33 0.33 0.34 0.34 0.34 0.32 0.31 0.30 0.29 0.29 0.30 0.33 0.34 0.35 0.35 0.35 0.35 0.34 0.33 0.33 0.33 0.34 0.36 0.37 0.38 0.38 0.38 0.38 0.39 0.40 0.43 0.45 0.49 0.53 0.57 0.59 0.60 0.62

(continued)

784

J.E. Eldridge and Edward D. Palik TABLE XlII

(Continued)

Sodium Chloride eV 14.9 14.8 14.7 14.65 14.6 14.55 14.5 14.45 14.4 14.3 14.25 14.2 14.1 14.0 13.9 13.8 13.7 13.5 13.40 13.30 13.20 13.1 13.05 13.0 12.95 12.93 12.90 12.88 12.85 12.8 12.76 12.7 12.65 12.60 12.55 12.51 12.45 12.40 12.36 12.35 12.31 12.30 12.20 12.10 12.09 12.0 11.99

cm- ~

pm 0.08320 0.08376 0.08433 0.08460 0.08491 0.08520 0.08551 0.08570 0.08609 0.08669 0.08700 0.08730 0.08792 0.08856 0.08919 0.08983 0.09049 0.09184 0.09255 0.09322 0.09390 0.09464 0.09500 0.09537 0.09574 0.09587 0.09611 0.09626 0.09648 0.09686 0.09712 0.09762 0.09802 0.09840 0.09879 0.09907 0.09958 0.09998 0.1003 0.1004 0.1007 0.1008 0.1016 0.1025 0.1025 0.1033 O.1034

n

k

0.77 0.76 0.78 0.79 0.81 0.82 0.84 0.85 0.86 0.88 0.89 0.91 0.94 0.98 1.02 1.06 1.10 1.16

0.67 0.71 0.75 0.77 0.78 0.79 0.80 0.81 0.82 0.84 0.85 0.86 0.88 0.89 0.90 0.90 0.89 0.85

1.20 [15]

0.81 [15]

1.21

0.74

1.19 1.17

0.71 0.70

1.16

0.70

1.14 1.12

0.71 0.71

1.10

0.75

1.10 1.14

0.81 0.83

1.15 1.17

0.84 0.85

1.19

0.85

1.21 1.24 1.24

0.85 0.83 0.79

1.22

0.79

n

k

1.096114]

0.916114]

1.197

0.868

1.169

0.759

1.104

0.765

1.088

0.794

1.083

0.834

1.082

0.896

1.119

0.967

1.197

1.02

1.230

1.02

1.304

0.999

1.320

0.940

1.290

0.921

Sodium Chloride (NaCI)

785 TABLE

XIII

(Continued)

Sodium Chloride eV 11.92 11.90 11.84 11.80 11.78 11.71 11.70 11.65 11.64 11.60 11.58 11.55 11.51 11.50 11.46 11.45 11.43 11.40 11.38 11.35 11.32 11.30 11.25 11.23 11.20 11.10 11.08 11.03 11.00 10.94 10.90 10.82 10.77 10.68 10.63 10.60 10.55 10.50 10.45 10.42 10.40 10.35 10.29 10.20 10.15 10.13

cm- 1

gm O. 1040 0.1042 O. 1047 0.1051 0.1052 0.1058 0.1060 0.1064 0.1065 0.1069 0.1071 O. 1073 0.1077 0.1078 0.1082 0.1083 O. 1085 0.1088 0.1090 0.1092 0.1095 O. 1097 0.1102 0.1104 0.1107 0.1117 0.1119 O. 1124 O. 1127 0.1133 O. 1137 O. 1146 O. 1151 0.1161 0.1166 0.1170 O.1175 0.1181 0.1186 O.1189 0.1192 0.1198 O. 1205 O. 1216 0.1221 O. 1224

n

k

1.20

0.79

1.16

0.80

1.12 1.10

0.86 0.89

1.07

0.96

1.09

1.06

1.15

1.14

1.24

1.19

1.32

1.21

1.40

1.23

1.49 1.57

1.22 1.19

1.64 1.75

1.15 1.04

1.82

0.93

1.82

0.78

1.67

0.56

1.58 1.54

0.53 0.54

1.51 1.49

0.56 0.59

1.50

0.70

n

k

1.272

0.931

1.234

0.937

1.209 1.204

0.987 1.05

1.219

1.14

1.255

1.205

1.338

1.28

1.422

1.34

1.491

1.36

1.588

1.35

1.682

1.34

1.809 1.864 1.918

1.29 1.27 1.21

2.002 2.003

1.02 0.938

1.990

0.834

1.975 1.947 1.873 1.821

0.706 0.650 0.569 0.543

1.730 1.707

0.546 0.563

1.654

0.575

1.637 1.637 1.640

0.623 0.645 0.673

1.664

0.694

(continued)

786

J.E. Eldridge and Edward D. Palik T A B L E XIII

(Continued)

Sodium Chloride eV 10.10 10.08 10.06 10.00 9.908 9.825 9.80 9.757 9.662 9.60 9.586 9.557 9.539 9.444 9.40 9.356 9.214 9.20 9.093 9.00 8.947 8.90 8.894 8.840 8.80 8.775 8.75 8.70 8.652 8.60 8.550 8.50 8.490 8.45 8.409 8.40 8.367 8.35 8.325 8.30 8.290 8.267 8.25 8.238 8.20 8.169 8.15

cm -1

~m 0.1227 0.1230 0.1232 0.1240 0.1251 0.1262 0.1265 0.1271 0.1283 0.1291 0.1293 0.1297 0.1300 0.1313 0.1319 0.1325 0.1345 0.1348 0.1363 0.1378 0.1386 0.1393 0.1394 0.1402 0.1409 0.1413 0.1417 0.1425 0.1433 0.1442 0.1450 0.1459 0.1460 0.1467 0.1474 0.1476 0.1482 0.1485 0.1489 0.1494 0.1496 0.1500 0.1503 0.1505 0.1512 0.1517 0.1521

n

k

1.52

0.71

1.54 1.55

0.71 0.71

1.59

0.70

1.63

1.65

n

k

1.673

0.687

1.678 1.711 1.742

0.697 0.715 0.700

1.751 1.764

0.681 0.670

1.770 1.779 1.776 1.787

0.658 0.655 0.647 0.623

1.780 1.792

0.617 0.597

1.791

0.585

1.804

0.564

1.792 1.789

0.549 0.552

1.789

0.546

1.776 1.765 1.749

0.529 0.521 0.534

1.745 1.746 1.735

0.539 0.539 0.531

1.719

0.531

1.695

0.534

1.682 1.665

0.542 0.542

1.638 1.604 1.547

0.550 0.567 0.598

0.68

0.65

1.66

0.63

1.67

0.60

1.66

0.59

1.66

0.60

1.66 1.66

0.59 0.59

1.65

0.58

1.64

0.58

1.64

0.58

1.63

0.58

1.61

0.57

1.59

0.58

1.55

0.58

1.50

0.60

1.44

0.65

Sodium Chloride (NaCI)

787 TABLE XIII

(Continued)

Sodium Chloride eV 8.127 8.10 8.091 8.08 8.062 8.05 8.040 8.03 8.024 8.00 7.98 7.95 7.93 7.918 7.90 7.885 7.88 7.859 7.85 7.83 7.802 7.78 7.767 7.75 7.738 7.73 7.710 7.70 7.684 7.65 7.613 7.60 7.588 7.567 7.55 7.50 7.45 7.419 7.40 7.353 7.30 7.276 7.27 7.205 7.20 7.112

cm- ~

/~m 0.1525 0.1531 0.1532 0.1534 0.1538 0.1540 0.1542 0.1544 0.1545 0.1550 0.1554 0.1560 0.1563 O.1566 0.1569 0.1572 0.1573 0.1577 0.1579 0.1583 0.1589 0.1594 0.1596 0.1600 0.1602 0.1604 0.1608 0.1610 0.1613 0.1621 0.1628 0.1631 0.1634 0.1638 0.1642 0.1653 0.1664 0.1671 0.1675 0.1686 0.1698 0.1704 0.1704 0.1721 0.1722 0.1743

n

k

1.38

0.74

1.36

0.80

1.33

0.89

1.34

0.99

1.38 1.42 1.51 1.60

1.10 1.18 1.29 1.36

1.74

1.43

1.84

1.46

2.00 2.12

1.49 1.49

2.42

1.41

2.59

1.33

2.72

1.26

2.91

1.05

2.90

0.59

2.69

2.55 2.46 2.40

n

k

1.483

0.691

1.461 1.451 1.446

0.784 0.829 0.878

1.470

0.974

1.490

1.01

1.550

1.20

1.727 1.809 1.929 2.025

1.40 1.45 1.51 1.53

2.206

1.52

2.563 2.695 2.778 2.911 3.019 3.078 3.161

1.45 1.39 1.37 1.31 1.21 1.12 0.991

3.252 3.180 3.019 2.974 2.902 2.827

0.737 0.383 0.151 0.126 8.67 x 2.76 X

10 - 2 10 - 2

2.636 5.67 x 10 -4

2.34 1.11 x 10 -7 2.25

2.17 2.081

1.37 1.37 2.02 2.02 4.43

x x x x x

10-5 10 -5 10 -6 10 -6 10 -7

(continued)

788

J.E. Eldridge and Edward D. Palik T A B L E XIII

(Continued)

Sodium Chloride eV 7.00 6.996 6.75 6.50 6.23 6.199 6.00 5.904 5.636 5.50 5.391 5.166 5.00 4.959 4.769 4.592 4.428 4.275 4.133 3.875 3.647 3.531 3.444 3.408 3.263 3.100 2.952 2.818 2.708 2.695 2.583 2.541 2.480 2.410 2.384 2.296 2.214 2.138 2.066 2.000 1.937 1.879 1.823 1.771 1.722 1.675 1.631

cm-

50,000 48,390 47,620 45,450 44,360 43,480 41,670 40,330 40,000 38,460 37,040 35,710 34,480 33,330 31,250 29,410 28,480 27,780 27,490 26,320 25,000 23,810 22,730 21,840 21,740 20,830 20,490 2O,000 19,440 19,230 18,520 17,860 17,240 16,670 16,130 15,630 15,150 14,710 14,290 13,890 13,510 13,160

~m

0.1771 0.1772 0.1837 0.1907 0.1990 0.20 0.2066 0.21 0.22 0.2254 0.23 0.24 0.2480 0.25 0.26 0.27 0.28 0.29 0.30 0.32 0.34 0.3511 0.36 0.3638 0.38 0.40 0.42 0.44 0.4579 0.46 0.48 0.4880 0.50 0.5145 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76

2.03 1.83

1.91 1.84 1.792 1.7899 [9]

175[~5] 1.74676 [9] 1.71499 1.69115] 1.6905619] 1.67118 1.65 [15] 1.65537 [9] 1.64227 1.63124 1.62184 1.61374 1.60669 1.59506 1.58590 6.4 x 10-~o [8] 1.57852 4.3 1.57248 1.56746 1.56324 1.55965 2.9 1.55657 1.55390 1.1 x 10 - l~ 1.55157 4.9 x 10 T M 1.54953 1.54773 1.54613 1.54471 1.54343 1.54228 1.54124 1.54030 1.53944 1.53865 1.53794 1.53728 1.53667

Sodium Chloride (NaCI)

789 TABLE XIII

(Continued)

Sodium Chloride eV 1.590 1.550 1.512 1.476 1.442 1.409 1.378 1.348 1.319 1.292 1.265 1.240 1.127 1.033 0.9537 0.8856 0.8266 0.7749 0.7293 0.6888 0.6526 0.6199 0.5904 0.5636 0.5391 0.5166 0.4959 0.4769 0.4592 0.4428 0.4275 0.4133 0.4000 0.3875 0.3757 0.3647 0.3542 0.3444 0.3351 0.3263 0.3179 0.3100 0.3024 0.2952 0.2883 0.2818

cm- 1 12,820 12,500 12,200 11,900 11,630 11,360 11,110 10,870 10,640 10,420 10,200 10,000 9,091 8,333 7,692 7,143 6,667 6,250 5,882 5,556 5,263 5,000 4,762 4,545 4,348 4,167 4,000 3,846 3,704 3,571 3,448 3,333 3,226 3,125 3,030 2,941 2,857 2,778 2,703 2,632 2,564 2,500 2,439 2,381 2,326 2,273

~m 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4

n 1.53611 1.53560 1.53512 1.53467 1.53426 1.53387 1.53351 1.53317 1.53285 1.53255 1.53227 1.53200 1.53088 1.53000 1.52930 1.52873 1.52824 1.52782 1.52745 1.52712 1.52682 1.52654 1.52627 1.52602 1.52578 1.52554 1.52531 1.52509 1.52486 1.52463 1.52441 1.52418 1.52395 1.52372 1.52349 1.52325 1.52301 1.52277 1.52252 1.52226 1.52201 1.52174 1.52148 1.52121 1.52093 1.52065

k

n

k

(1.3 + 0.1) • 10 -9 [7]

(1.8 + 0.2) x 10-9

(continued)

790

J.E. Eldridge and Edward D. Palik TABLE XlII

(Continued)

Sodium Chloride eV 0.2755 0.2695 0.2638 0.2583 0.2530 0.2480 0.2431 0.2384 0.2339 0.2296 0.2254 0.2214 0.2175 0.2138 0.2101 0.2066 0.2000 0.1937 0.1879 0.1823 0.1771 0.1722 0.1675 0.1631 0.1590 0.1550 0.1512 0.1476 0.1442 0.1409 O.1378 0.1348 0.1337 0.1319 0.1292 0.1265 0.1240 0.1216 O.1192 0.1170 0.1148 0.1127 0.1116 O.1107 O.1088 O.1069 0.1054

cm- ~ 2,222 2,174 2,128 2,083 2,041 2,000 1,961 1,923 1,887 1,852 1,818 1,786 1,754 1,724 1,695 1,667 1,613 1,563 1,515 1,471 1,429 1,389 1,351 1,316 1,282 1,250 1,220 1,190 1,163 1,136 1,111 1,087 1,079 1,064 1,042 1,020 1,000 980.4 961.5 943.4 925.9 909.1 900.0 892.9 877.2 862.1 850.0

pm 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.27 9.4 9.6 9.8 10.0 10.2 10.4 10.6 10.8 11.0 11.11 11.2 11.4 11.6 11.76

n 1.52036 1.52006 1.51977 1.51946 1.51915 1.51883 1.51851 1.51818 1.51785 1.51751 1.51716 1.51681 1.51645 1.51609 1.51572 1.51534 1.51457 1.51377 1.51294 1.51209 1.51122 1.51031 1.50938 1.50842 1.50744 1.50643 1.50539 1.50432 1.50322 1.50210 1.50094 1.49996

k

(1.7 -I- 0.5) • 10 -9

(2 + 1.5) • 10 -8 1.49855 1.49731 1.49604 1.49473 1.49340 1.49204 1.49065 1.48922 1.48776

(8 + 0.8) • 10 -8

1.5 • 10 -7 [6] 1.48628 1.48476 1.48320 4.5 • 10- v

n

k

Sodium Chloride (NaCI)

791 TABLE XIII

(Continued)

Sodium Chloride eV o.1051 0.1033 0.1016 0.09999 0.09919 0.09840 0.09686 0.09537 0.09393 0.09299 0.09252 0.09116 0.08984 0.08856 0.08731 0.08679 0.08670 0.08610 0.08492 0.08377 0.08266 0.08157 0.08059 0.08051 0.07948 0.07847 0.07749 0.07653 0.07560 0.07469 0.07439 0.07380 0.07293 0.07208 0.07125 0.07044 0.06965 0.06888 0.06819 0.06812 0.06738 0.06666 0.06595 0.06526 0.06457 0.06391

cm- ~ 847.5 833.3 819.7 806.5 800.0 793.7 781.3 769.2 757.6 750.0 746.3 735.3 724.6 714.3 704.2 700.0 699.3 694.4 684.9 675.7 666.7 657.9 650.0 649.4 641.0 632.9 625.0 617.3 609.8 602.4 600.0 595.2 588.2 581.4 574.7 568.2 561.8 555.5 550.0 549.4 543.5 537.6 531.9 526.3 520.8 515.5

/~m 11.8 12.0 12.2 12.4 12.50 12.6 12.8 13.0 13.2 13.3.3 13.4 13.6 13.8 14.0 14.2 14.29 14.3 14.4 14.6 14.8 15.0 15.2 15.38 15.4 15.6 15.8 16.0 16.2 16.4 16.6 16.67 16.8 17.0 17.2 17.4 17.6 17.8 18.0 18.18 18.2 18.4 18.6 18.8 19.0 19.2 19.4

n

k

n

k

1.48162 1.48000 1.47834 1.47665 1.4

X 10 -6

3.8

X 10 -6

1.47493 1.47318 1.47138 1.46956 1.46769 1.46579 1.46385 1.46188 1.45986 1.1 x 10 -5 1.45781 1.45572 1.45359 1.4514 1.4492 3.6 x 10- 5 1.4469 1.4446 1.4423 1.4399 1.4375 1.4351 1.4325 8.6 x 10-5 1.4300 1.4274 1.4247 1.4220 !.4193 1.41649 1.41364 2.2 • 10 -4 1.41074 1.40779 1.40478 1.40173 1.39861 1.39544 1.39222

(continued)

792

J.E. Eldridge and Edward D. Palik TABLE XIII

(Continued)

Sodium Chloride eV

cm- i

l~m

0.06326 0.06262 0.06199

510.2 505.0 500.0

19.6 19.8 20.0

0.05951 0.05703 0.05455 0.05207 0.04959 0.04711 0.04463 0.04215 0.03967 0.03720 0.03658 0.03596 0.03534 0.03472 0.03410 0.03348 0.03286 0.03224 0.03162 0.03124 0.031 O0 0.03038 0.02976 0.02938 0.02914 0.02889 0.02852 0.02790 0.02728 0.02666 0.02604 0.02542 0.02480 0.02418 0.02356 0.02294 0.02232 0.02170 0.02108 0.02046 0.02021 0.01984 0.01922

480 460 440 420 400 380 360 340 320 300 295 290 285 280 275 270 265 260 255 252 250 245 240 237 235 233 230 225 220 215 210 205 200 195 190 185 180 175 170 165 163 160 155

20.83 21.74 22.73 23.81 25.00 26.32 27.78 29.41 31.25 33.33 33.90 34.48 35.09 35.71 36.36 37.04 37.74 38.46 39.22 39.68 40.00 40.82 41.67 42.19 42.55 42.92 43.48 44.44 45.45 46.51 47.62 48.78 50.00 51.28 52.63 54.05 55.56 57.14 58.82 60.61 61.35 62.50 64.52

n 1.38893 1.38559 1.3822 1.41 [5] 1.38 1.35 1.33 1.32 1.27 1.23 1.17 1.12 1.01 0.85 0.83 0.80 0.78 0.69 0.59 0.49 0.42 0.45 0.57 0.59 0.58 0.51 0.46 0.44 0.46 0.49 0.43 0.30 0.25 0.20 0.18 O.16 O.14 O.15 O.19 0.24 0.35 0.59 1.35 6.17 7.76 6.92 5.50

k

6.2 x 8.7 • 1.1 x 1.5 2.3 3.5 5.3 9.8 1.7 • 4.5 x 0.11 0.13 O.13 O.14 O.17 0.22 0.31 0.50 0.66 0.71 0.67 0.66 0.69 0.81 0.87 0.87 0.87 0.87 1.00 1.15 1.36 1.56 1.74 1.99 2.29 2.63 3.02 3.72 4.57 6.03 7.41 6.03 2.14 0.87

10 -'~ [5] 10 -'~ 10 - 3

10 -2 10- 2

n

k

Sodium Chloride (NaCI)

793 T A B L E XIII

(Continued)

Sodium Chloride eV

cm- ~

#m

0.01860 0.01798 0.01736 0.01674 0.01612 0.01488 0.01364 0.01240 0.01116 0.009919 0.008679 0.007439 0.006199 0.005579 0.004959 0.004339 0.004132 0.003720 0.003100 0.002480

150 145 140 135 130 120 110 100 90 80 70 60 50 45 40 35 33.3 30 25 20

66.67 68.97 71.43 74.07 76.92 83.33 90.91 100.0 111.1 125.0 142.9 166.7 200.0 222.2 250.0 285.7 300 333.3 400.0 500.0

0.001860 0.001240

15 10

n 4.52 4.07 3.72 3.47 3.31 3.02 2.85 2.74 2.63 2.57 2.52 2.48 2.45 2.44 2.44 2.43 2.43 2.43 2.43

k

n

k

0.380 0.309 0.219 0.145 0.135 0.110 0.095 0.087 0.078 0.077 0.068 0.055 0.047 0.045 0.041 0.036 0.030 [2] 0.032 1-5] 0.027 0.024

0.024 E2]

0.001215 0.001127 0.001033 0.0008856 0.0007872 0.0006888 0.0006199 0.0005904 0.0004959 0.0004797 0.0003875 0.0001464 0.00004053

9.8 9.091 8.333 7.143 6.349 5.556 5.000 4.762 4.000 3.869 3.125 1.181 0.3269

666.7 1000 1020 1100 1200 1400 1575 1800 2000 2100 2500 2584 3200 8469 30590

2.43 2.43

0.014 [5] 0.006 [5] 0.01 [3] 8.1 x 10 -3 [2] 9.5 [3] 8.8 6.5 5.76 5.4

2.43 1-4]

2.43 [1] 2.43 2.43

4.3 4.4 2.1 [1] 3.3 x 10 - 3 [3] 5.8 x 10 .4 [1] 2.5

" T a b u l a t i o n of n and k as a function of p h o t o n energy (electron volts), wave n u m b e r (inverse centimeters), and wavelength (micrometers). References for the d a t a are given in brackets a n d a p p l y to all d a t a below the brackets until a new reference appears; k is often very small a n d e x p o n e n t s often are given for n u m b e r s smaller t h a n 10- 2. The power of 10 applies for all d a t a below the e x p o n e n t until a new e x p o n e n t appears or the e x p o n e n t becomes 10-1, w h e r e u p o n the n u m b e r is given in decimal form.

Titanium Dioxide (TiO 2) (Rutile) M. W. RIBARSKY School of Physics Georgia Institute of Technology Atlanta, Georgia

Titanium dioxide exists in three crystal structures: rutile, brookite, and anatase. The work reported here is all on rutile (tetragonal) single crystals. Rutile is easily reduced, and the crystal will have some oxygen vacancies or titanium interstitials. Since the structure cannot be treated as a slightly perturbed cubic crystal, there are pronounced polarization effects. It is therefore necessary to measure the optical constants for E [[ c axis and E _1_c axis. Compiled in Table XIV and plotted in Fig. 14 are room-temperature values of n and k for wavelengths between 36 and 0.11 #m. There are evidently very few reports of optical constants for single-crystal TiO2 in the ultraviolet above 20 eV. In the far infrared, Smith and Loewenstein [1] measured no (2_) at 1.5 K from channeled spectra in the range 400-100/~m. Meriakri and Ushatkin [-2] measured n at 1030 #m at an unspecified room temperature. The technique involved the use of a millimeterwave source whose frequency could be varied periodically over a short range and a Michelson interferometer in which the path length could be varied. A beating fringe pattern was obtained from which an average n could be obtained. Parker [3] measured the susceptability of oriented rutile in a capacitor at low frequency and determined dc values of n~ = 170 and nZ~ = 86 at 300 K. The infrared results of 36.4 to 11.1 #m are from Gervais and Piriou [4]; the results in the near infrared and visible 1.5 to 0.425 ~m are from Devore [5], and overlapping sets of results for the visible to ultraviolet regions are from Cardona and Harbeke [6] (1.2-0.11 #m) and from Vos and Krusemeyer [7] (0.413-0.207 gm). Cardona and Harbeke [6] have also taken the reflectivity from 0.11 to 0.056 #m with unpolarized light. All the results were obtained from power reflectivity measurements. In the case of Cardona and Harbeke [6] and Vos and Krusemeyer [7] a K r a m e r s 795 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS

Copyright (~) 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-544420-6

796

M.W. Ribarsky

Kronig (KK) analysis was used to obtain the optical constants. The uncertainty due to truncation of the reflectivity to a finite frequency range was somewhat relieved by extrapolation to previous measurements at frequencies below the measured region. Above the measured region, a standard powerlaw fit to the reflectivity [7] was used, or a fit of the expansion coefficients in an approximate expression for the phase of the complex reflectivity was made [6]. Neither procedure took account of the higher interband transitions in any direct way. Such uncertainties introduce errors in the magnitudes of the optical constants and in the positions of peaks and other features [8]. These errors are more pronounced near the boundaries of the spectrum. Most of the spectra were derived from reflectivity measurements. All the experiments used single crystals with the orientation of the c axis determined by x-ray diffraction. The infrared reflectivity spectra were taken with a single-beam monochromator on optically polished rutile crystals [4]. Errors were of the order AR/R = 2~o and AE = 2-5 cm-1. The very near-infrared refractive-index results [5] were taken directly from the measured amount of refraction of light from a mercury arc by a single-crystal rutile prism. Cardona and Harbeke [-6] measured the reflectivity spectra below 6 eV with a grating spectrometer and used a vacuum-UV spectrometer at higher energies. Vos and Krusemeyer [7] took the absolute specular reflectance with a relative uncertainty of less than 0.5~. They cut their samples from flame-fusiongrown boules of rutile that were polished and etched in NaOH. The optical constants in the infrared were tabulated by using the following four-parameter semiquantum model [4] for the dielectric function g=

2 m 0) 2 ~"~jLO ~---iTjLOO g l d- ie 2 = ~3 H -oo -J-" ~jTO2 __ 0)2 .~_ iTjTO(_O

(1)

Each TO (LO) phonon mode is assumed to be an oscillator with frequency ~vo (flLO) and damping 77o (TLO). In general, the parameters are frequency and time dependent, but tests of Eq. (1) for realistic cases have shown that the parameters vary slowly with frequency even for wide reflectivity bands [9]. The temperature dependences of 7To and ?LO for E [[ c indicate large anharmonicities due perhaps to sixth-order coupling at higher temperatures. The model dielectric function was used in the expression for the power reflectivity, and the resulting fit to the reflectivity data was within the experimental error. The results tabulated here use this model with the parameters given in Tables I and II of Gervais and Piriou [4]. The data show that there are only one TO mode and one LO mode in the Azu (Eli c) spectrum but that there is also a secondary mode at 592 cm-~. This could be a two-phonon peak since high phonon densities exist at energies about half the secondary oscillator energy [10]. However, it could also be a forbidden mode, such as the A lg mode, activated by substitutional impurities

Titanium Dioxide (TiO2) (Rutile)

797

[4]. The E, (E 2_ c) spectrum has three TO and LO modes and a secondary peak near the same energy as the A2u peak. There are apparently no recent measurements of the optical constants in the range 1.5-11.1 #m. However, in 1921 Liebisch and Rubens [11] measured the index of refraction of a rutile prism in this range. For the region in which their measurements overlap those of Devore [5] (0.7-1.5 #m), Liebisch and Rubens's [11] results are in agreement with Devore's [5] for E 2_ c. However, for E IIc the refractive-index curve of Liebisch and Rubens [11] has the same shape as Devore's [5] but is offset by approximately -0.1. A similar offset for E IIc exists between the curves of Devore [5] and Cronemeyer [12] in the range 0.4-0.6 #m. However, Devore's [5] values are quite close to the original measurements of Baerwald [ 13]. There are no measurements of the extinction coefficient in the range 0.45-1.5/~m, but it is presumably quite small and structureless. Devore has fit his refractive-index results with the equations n 2 = 5.913 + 2.441 x 107/(22- 0.803 x 107), n~ = 7.197 + 3.322 x 107/(22- 0.843 x 107),

(2)

where 2 is in angstroms. These equations give results within 0.001-0.003 of the experimental results, and they are used here to obtain the tabulated results from 0.425 to 1.5 gin. At 0.405 #m Cronemeyer [12] gives k = 0.086. The data of Cardona and Harbeke [6] and of Vos and Krusemeyer [7] are significantly different in the region of overlap (0.21-0.41 #m). Vos and Krusemeyer [7] ascertain that bulk pretreatment of their samples did not affect the reflectivity and that there were no errors in their measuring procedure. Gupta and Ravindra [ 14] point out that the values of e~ obtained by integrating g2 over the spectral range are in better agreement with experiment for the results of Cardona and Harbeke [6] than for those of Vos and Krusemeyer [7]. However, the Vos and Krusemeyer [7] results are over a more limited energy range and the truncated KK analysis can introduce shifts in the optical-constant spectra [-8] that would affect the evaluation of e~. We have therefore elected to present both the results of Cardona and Harbeke [6] and of Vos and Krusemeyer [7]. There is a rapid rise in all the spectra above the fundamental edge at 3.05 eV (0.405 #m). Comparison with a semiempirical band structure [15] indicates that the rise is due to nearly parallel conduction and valence bands. The calculations also show that the structures in ~2 for both directions of polarization are due to transitions to two groupings of conduction bands, one from 3 to 6 eV and the other from 6 to 9 eV above the valence bands. The calculated ~2 for E _1_c is not as close to experiment as for E II c [15]. The shift to lower energy of the main peak in the experimental spectrum may indicate the presence of excitonic effects.

798

M.W. Ribarsky

REFERENCES

1. D. R. Smith and E. V. Loewenstein, Appl. Opt. 15, 859 (1976). 2. V. V. Meriakri and E. F. Ushatkin, Prib. Tekh. Eksp. 16, 143 (1973); V. V. Meriakri and E. F. Ushatkin, Instrum. Exp. Tech. 16, 498 (1973). 3. R. A. Parker, Phys. Rev. 124, 1719 (1961). 4. F. Gervais and B. Piriou, Phys. Rev. B 10, 1642 (1974). 5. J. R. Devore, J. Opt. Soc. Am. 41, 416 (1951). 6. M. Cardona and G. Harbeke, Phys. Rev. 137A, 1467 (1965). 7. K. Vos and H. J. Krusemeyer, J. Phys. C 1O, 3893 (1977). 8. W. E. Wall, M. W. Ribarsky, and J. R. Stevenson, J. Appl. Phys. 51, 661 (1980). 9. F. Gervais and B. Piriou, J. Phys. C 7, 2374 (1974). 10. G. A. Samara and P. S. Peercy, Phys. Rev. B 7, 1131 (1973). 11. T. Liebisch and H. Rubens, Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl. 211 (1921). 12. D. C. Cronemeyer, Phys. Rev. 87, 876 (1952). 13. C. Baerwald, Z. Krist. 7, 167 (1883). 14. V. P. Gupta and N. M. Ravindra, Phys. Chem. Solids 41, 591 (1980). 15. K. Vos, J. Phys. C 1O, 3917 (1977).

IC'~i~.jl

I

'

'

I''"I

'

'

'

i'i,il

,

(a)'

i

I0 i

l_

i

i

!

i i ill

I

i

i

|

i Jill

1

w

(b)

lI~,~I/

,/wZ

/

-

/

i'~ll[

/

/

/

I0 o

c---

c--

I

io-i

iO-I

io-z _

iO-Z i

i0 -f

i

i

i

i iltl

i

i0 o

1

i

iiiiii

i

I0 ~

WAVELENGTH (,t.J.m)

i

l

i

io-i

i

i

i

i iiii

1

i

IO0 WAVELENGTH

I

Iiiiii

I..

!

I

I

IOi (~m)

Fig. 14. (a) Log-log plot ofn_L( ~ ) , k_L(. . . . ) versus wavelength in micrometers for titanium dioxide. (b) Log-log plot of nil ( ~ ) , kll (. . . . ) versus wavelength in micrometers for titanium dioxide. The _L and II symbols refer to radiation electric field E perpendicular and parallel to the c axis of the crystal. The values of n at 1030 #m given in Meriakri and Ushatkin [2] are not shown.

Titanium Dioxide (Ti02) (Rutile)

799 TABLE XIV

Values of n and k for Titanium Dioxide Obtained from Various References"

eV

cm -1

/~m

12.04 11.07 10.25 9.116 8.377 8.051 7.798 7.293 7.167 6.965 6.391 6.048 5.904

48,780 47,620

0.103 0.112 0.121 0.136 0.148 0.154 0.159 0.170 0.173 0.178 0.194 0.205 0.210

5.713 5.610 5.510 5.438 5.344 5.188 4.940 4.862 4.661 4.592 4.558 4.541 4.492 4.350

46,080 45,250 44,440 43,860 43,100 41,840 39,840 39,220 37,590 37,040 36,760 36,630 36,230 35,090

0.217 0.221 0.225 0.228 0.232 0.239 0.251 0.255 0.266 0.270 0.272 0.273 0.276 0.285

4.305 4.217 4.174 4.105 4.052 4.038 3.899 3.827 3.791 3.701 3.604 3.463 3.444 3.195 3.100 3.084

34,720 34,010 33,670 33,110 32,680 32,570 31,450 30,870 30,580 29,850 29,070 27,930 27,780 25,770 25,000 24,870

0.288 0.294 0.297 0.302 0.306 0.307 0.318 0.324 0.327 0.335 0.344 0.358 0.360 0.388 0.400 0.402

.(11)

n(_L)

k(ll)

k(•

0.590 [6] 0.510 0.560 0.490 0.670

0.790 [6] 0.854 0.816 0.740 1.07

0.820 [6] 0.893 0.947 1.31 1.67

0.950 1.04

1.32 1.51

1.31 1.55

1.54 1.46 1.98 [7] 1.46 [6] 2.02 [7]

0.760 [6] 0.908 1.05 1.42 1.74 1.81 1.77 1.94 2.01 2.03 1.77

1.82 [-7]

1.46

1.65

2.31 [6] 2.36 [7]

2.01 [-7] 1.97 [6] 2.53 [7] 2.59 [-6] 2.92

2.49 [6] 2.50 2.59 [-7]

2.83 [6] 3.02 3.00 [7]

3.84 [7] 5.38 [6]

1.35 2.21 [6] 1.17 2.1516]

3.87 [7] 3.49 3.40 [5]

2.29 1.1917] 1.29

3.36 [6] 2.09 [7] 3.47 [6] 3.49 2.21 [-7] 3.56 [-6] 3.55 3.48

1.67 [6] 1.65 [6] 1.65 1.41 [7]

1.82 [-6] 1.97

1.68 [7] 2.1316] 1.95 [7]

3.42 4.00 [6]

2.18 [6] 1.44

3.87 [7] 4.22 [7] 4.36 [-6]

1.38

1.65 1.1017] 1.68 [6] 1.20 [73

1.71 [7] 2.37 2.44 [6]

3.03 [7] 3.56 [6] 4.23

1.50

0.971 [-7] 2.26

1.81 1.45 [6]

1.57

1.61 [-7] 1.79 [5] 1.44 0.810 [7]

0.788 [7] 4.30 [6] 3.38 [7]

0.117 [7] 0.251

2.88 3.00 [5] 3.00 [7]

0.008

(continued)

800

M.W. Ribarsky TABLE XIV (Continued) Titanium Dioxide eV

3.009 2.952 2.818 2.817 2.695 2.583 2.480 2.384 2.300 2.296 2.214 2.138 2.066 2.000 1.937 1.879 1.823 1.771 1.722 1.675 1.631 1.590 1.550 1.512 1.476 1.442 1.409 1.378 1.348 1.319 1.291 1.265 1.240 1.216 1.192 1.170 1.148 1.127 1.107 1.088 1.069 1.051 1.033 1.016 0.9999 0.9840

cm- 1

24,270 23,810 22,730 22,670 21,740 20,830 20,000 19,230 18,550 18,520 17,860 17,240 16,670 16,130 15,630 15,150 14,710 14,290 13,890 13,510 13,160 12,820 12,500 12,200 11,900 11,630 11,360 11,110 10,870 10,640 10,420 10,200 10,000 9,804 9,615 9,434 9,259 9,091 8,929 8,772 8,621 8,475 8,333 8,197 8,065 7,937

ktm 0.412 0.420 0.440 0.441 0.460 0.480 0.500 0.520 0.539 0.54O 0.560 0.580 0.600 0.620 0.640 0.660 0.680 0.700 0.720 0.740 0.760 0.780 0.800 0.820 0.840 0.860 0.880 0.900 0.920 0.940 0.960 0.980 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18 1.20 1.22 1.24 1.26

"(1[)

n(•

3.24 [7] 3.29 [5] 3.20

2.91 [5] 2.84

k(ll) 0.022

0.080 [6] 3.13 3.08 3.03 3.00 2.95 [6] 2.97 ]-5] 2.94 2.92 2.90 2.88 2.87 2.85 2.84 2.83 2.82 2.81 2.81 2.80 2.79 2.79 2.78 2.78 2.77 2.77 2.76 2.76 2.76 2.75 2.75 2.75 2.74 2.74 2.74 2.74 2.74 2.73 2.73 2.73 2.73 2.73 2.72 2.72

k(_L)

2.79 2.75 2.71 2.68 2.66 [5] 2.64 2.62 2.60 2.59 2.58 2.57 2.56 2.55 2.54 2.54 2.53 2.52 2.52 2.52 2.51 2.51 2.50 2.50 2.50 2.49 2.49 2.49 2.49 2.48 2.48 2.48 2.48 2.48 2.47 2.47 2.47 2.47 2.47 2.47 2.47 2.46

0.040 [63

Titanium Dioxide (Ti02) (Rutile)

801

TABLE XIV

(Continued)

Titanium Dioxide

eV 0.9686 0.9537 0.9393 0.9252 0.9116 0.8984 0.8856 0.8731 0.8610 0.8492 0.8377 0.8265 0.5123 0.3668 0.3271 0.2897 0.2535 0.2164 0.1839 0.1729 0.1542 0.1107 0.1088 0.1069 0.1051 0.1033 0.1016 0.09999 0.09840 0.09686 0.09537 0.09393 0.09252 0.09116 0.08984 0.08856 0.08731 0.08610 0.08492 0.08377 0.08265 0.08157 0.08051 0.07948 0.07847

cm-

1

7,813 7,692 7,576 7,463 7,353 7,246 7,143 7,042 6,944 6,849 6,757 6,667 4,132 2,959 2,639 2,336 2,045 1,745 1,484 1,395 1,244 892.9 877.2 862.1 847.5 833.3 819.7 806.5 793.7 781.2 769.2 757.6 746.3 735.3 724.6 714.3 704.2 694.4 684.9 675.7 666.7 657.9 649.4 641.0 632.9

~m 1.28 1.30 1.32 1.34 1.36 1.38 1.40 1.42 1.44 1.46 1.48 1.50 2.42 3.38 3.79 4.28 4.89 5.73 6.74 7.17 8.04 11.2 11.4 11.6 11.8 12.0 12.2 12.4 12.6 12.8 13.0 13.2 13.4 13.6 13.8 14.0 14.2 14.4 14.6 14.8 15.0 15.2 15.4 15.6 15.8

,,~ll)

n(_L)

2.72 2.72 2.72 2.72 2.72 2.72 2.72 2.71 2.71 2.71 2.71 2.71 2.59 [12] 2.58 2.57 2.51 2.49 2.43

2.46 2.46 2.46 2.46 2.46 2.46 2.46 2.46 2.46 2.46 2.46 2.45 2.40 [12] 2.41 2.39 2.34 2.32 2.24 2.11

2.27 2.06 1.30 [4] 1.21 1.11 1.01 0.882 0.743 0.587 0.441 0.345 0.296 0.269 0.254 0.245 0.241 0.239 0.239 0.240 0.243 0.247 0.253 0.259 0.268 0.278 0.290

1.01 [4] 0.905 0.784 0.649 0.513 0.406 0.340 0.301 0.278 0.264 0.255 0.250 0.249 0.250 0.253 0.258 0.265 0.275 0.287 0.303 0.322 0.347 0.378 0.418

k(ll)

k(3_)

0.075 [4] 0.091 0.111 0.136 0.171 0.224 0.310 0.450 0.625 0.791 0.940 1.07 1.20 1.31 1.42 1.52 1.62 1.72 1.81 1.90 1.98 2.07 2.15 2.24

0.176 [4] 0.209 0.256 0.328 0.440 0.591 0.751 0.902 1.04 1.17 1.30 1.42 1.53 1.65 1.76 1.87 1.98 2.09 2.21 2.32 2.44 2.56 2.69 2.81

(continued)

802

M.W. Ribarsky TABLE XIV

(Continued)

Titanium Dioxide

eV 0.07749 0.07653 0.07560 0.07469 0.07380 0.07293 0.07208 0.07125 0.07044 0.06965 0.06888 0.06812 0.06738 0.06666 0.06595 0.06525 0.06457 0.06391 0.06326 0.06262 0.06199 0.06138 0.06078 0.06019 0.05961 0.05904 0.05848 0.05794 0.05740 0.05687 0.05636 0.05585 0.05535 0.05486 0.05438 0.05391 0.05344 0.05298 0.05253 0.05209 0.05166 0.05123 0.05081 0.05040 0.04999 0.04959

cm- 1

625.0 617.3 609.8 602.4 595.2 588.2 581.4 574.7 568.2 561.8 555.6 549.5 543.5 537.6 531.9 526.3 520.8 515.5 510.2 505.1 500.0 495.0 490.2 485.4 480.8 476.2 471.7 467.3 463.0 458.7 454.5 450.5 446.4 442.5 438.6 434.8 431.0 427.4 423.7 420.2 416.7 413.2 409.8 406.5 403.2 400.0

#m 16.0 16.2 16.4 16.6 16.8 17.0 17.2 17.4 17.6 17.8 18.0 18.2 18.4 18.6 18.8 19.0 19.2 19.4 19.6 19.8 20.0 20.2 20.4 20.6 20.8 21.0 21.2 21.4 21.6 21.8 22.0 22.2 22.4 22.6 22.8 23.0 23.2 23.4 23.6 23.8 24.0 24.2 24.4 24.6 24.8 25.0

,,(11) 0.306 0.326 0.351 0.377 0.398 0.406 0.401 0.389 0.378 0.370 0.366 0.364 0.365 0.368 0.372 0.377 0.382 0.389 0.395 0.403 0.410 0.418 0.427 0.435 0.444 0.453 0.463 0.472 0.482 0.492 0.502 0.513 0.524 0.535 0.546 0.557 0.569 0.581 0.593 0.605 0.617 0.630 0.643 0.656 0.670 0.683

n(_L) 0.468 0.533 0.614 0.710 0.814 0.908 0.972 1.00 1.00 1.00 1.02 1.06 1.15 1.31 1.56 1.96 2.62 3.64 4.96 5.88 5.95 5.53 5.01 4.51 4.06 3.66 3.29 2.96 2.64 2.33 2.03 1.73 1.45 1.21 1.03 0.904 0.822 0.772 0.744 0.733 0.737 0.756 0.790 0.841 0.916 1.02

k(ll) 2.32 2.39 2.46 2.52 2.57 2.61 2.66 2.71 2.77 2.84 2.91 2.98 3.05 3.12 3.19 3.25 3.32 3.39 3.45 3.52 3.59 3.65 3.72 3.78 3.84 3.91 3.97 4.04 4.10 4.16 4.23 4.29 4.35 4.42 4.48 4.54 4.60 4.67 4.73 4.79 4.85 4.92 4.98 5.04 5.11 5.17

~• 2.94 3.07 3.19 3.30 3.39 3.45 3.49 3.54 3.62 3.76 3.94 4.16 4.43 4.74 5.10 5.51 5.91 6.12 5.72 4.51 3.17 2.21 1.63 1.27 1.05 0.909 0.826 0.783 0.772 0.792 0.846 0.941 1.09 1.28 1.51 1.75 1.98 2.19 2.40 2.59 2.78 2.96 3.14 3.32 3.51 3.70

Titanium Dioxide (TiOJ (Rutile)

803

TABLE XIV

(Continued)

Titanium Dioxide eV 0.04920 0.04881 0.04843 0.04805 0.04769 0.04732 0.04696 0.04661 0.04626 0.04592 0.04558 0.04525 0.04492 0.04460 0.04428 0.04397 0.04366 0.04335 0.04305 0.04275 0.04246 0.04217 0.04189 0.04160 0.04133 0.04105 0.04078 0.04052 0.04025 0.03999 0.03974 0.03948 0.03923 0.03899 0.03874 0.03850 0.03827 0.03803 0.03780 0.03757 0.03734 0.03712 0.03690 0.03668 0.03647

cm- 1

396.8 393.7 390.6 387.6 384.6 381.7 378.8 375.9 373.1 370.4 367.6 365.0 362.3 359.7 357.1 354.6 352.1 349.7 347.2 344.8 342.5 340.1 337.8 335.6 333.3 331.1 328.9 326.8 324.7 322.6 320.5 318.5 316.5 314.5 312.5 310.6 308.6 396.7 304.9 303.0 301.2 299.4 297.6 295.9 294.1

#m 25.2 25.4 25.6 25.8 26.0 26.2 26.4 26.6 26.8 27.0 27.2 27.4 27.6 27.8 28.0 28.2 28.4 28.6 28.8 29.0 29.2 29.4 29.6 29.8 30.0 30.2 30.4 30.6 30.8 31.0 31.2 31.4 31.6 31.8 32.0 32.2 32.4 32.6 32.8 33.0 33.2 33.4 33.6 33.8 34.0

,,(11) 0.697 0.711 0.726 0.740 0.755 0.771 0.786 0.802 0.818 0.834 0.851 0.867 0.885 0.902 0.920 0.938 0.956 0.975 0.994 1.01 1.03 1.05 1.07 1.09 1.12 1.14 1.16 1.18 1.20 1.23 1.25 1.28 1.30 1.33 1.35 1.38 1.40 1.43 1.46 1.48 1.51 1.54 1.57 1.60 1.63

n(• 1.17 1.39 1.70 2.15 2.70 3.18 3.30 3.02 2.52 1.95 1.40 0.968 0.704 0.553 0.465 0.411 0.377 0.356 0.344 0.336 0.333 0.332 0.334 0.336 0.341 0.346 0.351 0.358 0.365 0.372 0.380 0.389 0.398 0.407 0.416 0.426 0.436 0.446 0.457 0.468 0.480 0.491 0.503 0.516 0.528

k(ll) 5.23 5.29 5.36 5.42 5.48 5.55 5.61 5.67 5.74 5.80 5.87 5.93 5.99 6.06 6.12 6.19 6.25 6.32 6.38 6.45 6.51 6.58 6.64 6.71 6.78 6.84 6.91 6.98 7.04 7.11 7.18 7.25 7.31 7.38 7.45 7.52 7.59 7.66 7.73 7.79 7.86 7.93 8.00 8.08 8.15

k(• 3.90 4.09 4.26 4.33 4.18 3.67 2.93 2.26 1.84 1.66 1.72 1.95 2.24 2.51 2.75 2.95 3.13 3.30 3.44 3.58 3.71 3.83 3.94 4.05 4.15 4.25 4.35 4.45 4.54 4.63 4.72 4.81 4.90 4.98 5.07 5.16 5.24 5.33 5.41 5.49 5.58 5.66 5.75 5.83 5.92

(continued)

804

M.W. Ribarsky TABLE XIV

(Continued)

Titanium Dioxide

eV 0.03625 0.03604 0.03583 0.03563 0.03542 0.03522 0.03502 0.03483 0.03463 0.03444 0.03425 0.03406 0.00120

cm-' 292.4 290.7 289.0 287.4 285.7 284.1 282.5 280.9 279.3 277.8 276.2 274.7 9.71

#m 34.2 34.4 34.6 34.8 35.0 35.2 35.4 35.6 35.8 36.0 36.2 36.4 1,030

n(ll) 1.66 1.69 1.73 1.76 1.79 1.83 1.86 1.90 1.93 1.97 2.01 2.05 12.85

~J-) 0.541 0.555 0.569 0.583 0.598 0.613 0.628 0.644 0.661 0.678 0.695 0.713 9.15 [2]

k~ll) 8.22 8.29 8.36 8.43 8.50 8.58 8.65 8.72 8.79 8.87 8.94 9.01

k(J_) 6.00 6.09 6.17 6.26 6.34 6.43 6.52 6.60 6.69 6.78 6.87 6.96

a References are indicated in brackets. While we use the notation 11, _L, these are the same as e, o and c, a used by others.

Handbook of Optical Constants of Solids II

Handbook of Optical Constants of Solids II

Edited by EDWARD D. PALIK Institute for Physical Science and Technology University of Maryland College Park, Maryland

A c a d e m i c Press San Diego London Boston New York Sydney Tokyo Toronto

This book is printed on acid-free paper.

~)

Copyright 9 1998, 1991 by Academic Press All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. ACADEMIC PRESS 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA 1300 Boylston Street, Chestnut Hill, MA 02167, USA http://www.apnet.com Academic Press Limited 24-28 Oval Road, London NW1 7DX, UK http://www.hbuk.co.uk/ap/

Library of Congress Cataloging-in-Publication Data Handbook of optical constants of solids II / edited by Edward D. Palik. p. cm. Includes bibliographical references. ISBN 0-12-544422-2 1. Solids. 2. Optical properties. 3. Handbooks, manuals, etc. I. Palik, Edward D. QC176.8.06H363 1991 90-826 530.4' 12 CIP

Printed in the United States of America 97 98 99 00 01 IC 9 8 7 6

5

4

3

2

Contents List of Contributors

Part l

XV

DETERMINATION OF OPTICAL CONSTANTS

Chapter 1 Introductory Remarks

3

E D W A R D D. PALIK

1. II. llI. IV. V. VI. VII.

Introduction The Chapters The Critiques The Tables The Figures of the Tables General Remarks Errata References

Chapter 2 Convention Confusions

3 3 5 6 7 7 8 11 21

R. T. H O L M I. II. IiI. IV. V. VI. VII.

Chapter 3

Introduction Units Wave Equation Polarization Fresnel's Amplitude Reflection Coefficients Nomenclature Concluding Remarks Acknowledgments References

21 22 22 26 37 5O 51 52 52

Methods for Determining Optical Parameters of Thin Films

57

E. PELLETIER I.

II. III. IV. V. VI.

Introduction An Example of Comparison between the Different Techniques of Index Determination Principle Methods of Index Determination Sensitivity and Accuracy of the Determination Methods Overlapping and Tests of Validity of the Model Results and Conclusion References

57 58 60 64 67 71 71

vi

Contents

Chapter 4 The Attenuated Total Reflection Method

75

G. J. SPROKEL AND J. D. SWALEN Io II. III. IV. V. VI.

Chapter 5 I,

II. III. IV. V. VI. VII. VIII.

Chapter 6

I~

II. III. IV.

Chapter 7

I~ II.

Introduction Surface-Plasmon Oscillations Optical Analysis of ATR Structures Guided Modes Applications of ATR Conclusion References

75 77 81 84 87 94 94

Optical Properties of Superlattices P. APELL AND O. HUNDERI

97

Introduction The Classical Optical Response of a Multilayer System Semiconductor Superlattices Amorphous Superlattices Collective Excitations Lattice Disorder The Effect of Nonlocal Dielectric Functions Concluding Remarks References

97 98 101 109 112 118 120 122 122

Calculation of the Refractive Index of Compound Semiconductors below the Band Gap B. JENSEN

125

Introduction The Quantum-Density-Matrix Formulation of the Complex Dielectric Constant Comparison with Experimental Results Summary References

Calculation of Optical Constants, n and k, in the Interband Region A. R. FOROUHI AND I. BLOOMER Introduction The Extinction Coefficient,

Chapter 8

Io

II.

128 138 147 148

151

151

k(E),

and the Refractive Index,

n(E) III. IV. V.

125

Discussion Examples Summary References

154 163 166 174 175

Temperature Dependence of the Complex Index of Refraction MICHAEL E. THOMAS

177

Introduction Temperature Dependence of n and k

177 179

.~

Contents

vii III.

Chapter 9

Conclusions References

199 199

Measurement of n and k in the XUV by the Angle-of-Incidence Total-External-Reflectance Method

2o3

MARION k. SCOTT I.

II. III. IV. V.

Introduction General Considerations Aluminum Reflectance versus Angle in UHV Determination of n and k Comparison with Other Results References

Chapter 10 Spectroscopic Ellipsometry in the 6-35 eV Region J. BARTH, R. L. JOHNSON, AND M. CARDONA I.

II. III. IV. V.

Chapter 11

I,

II. III. IV. V. VI. VII. VIII.

Chapter 12

I.

II. III.

Introduction The Bessy VUV Ellipsometer Theory Results and Comparison with Reflectance Spectra Summary Acknowledgment References

Methods Based on Multiple-Slit Fourier Transform Interferometry for Determining Thin-Film Optical Constants in the VUV/Soft X-Ray Range ROMAN TATCHYN Introduction to MSFTI Fundamental Dimensional Constraints and Inequalities Relationship of MSFTI to Alternative Metrological Techniques General Experimental Equations and Sensitivity Analysis A Characterization of the Optical Constants of Au Using MSFTI in the 280-640 eV Range Problems and Prospects in Sample Preparation Prospects for MSFTI in Other Spectral Regimes Summary Acknowledgments References

203 204 206 207 210 212 213

213 217 226 237 244 245 245

247

247 253 254 258 265 270 272 273 274 274

Determination of Optical Constants from Angular-Dependent, Photoelectric-Yield Measurements H.-G. BIRKEN, C. BLESSING, AND C. KUNZ

279

Introduction Theoretical Background Experimental Details

279 280 285

viii

Contents IV. V.

Chapter 13

I.

II. III. IV. V. VI. VII. VIII.

Chapter 14

Io II.

Part II Subpart 1

I~

II. III. IV. V. VI. VII. VIII. IX.

Examples Conclusions and Future Prospects References

286 289 292

Determination of Optical Constants by High-Energy, Electron-Energy-Loss Spectroscopy (EELS) J. PFLOGER AND J. FINK

293

Introduction Description of an EELS Experiment Scattering Cross-Section Exact Determination of the Loss Function Kramers-Kronig Analysis Brief Summary of the Evaluation Procedure Comparison with Reflectivity Measurements Optical Properties of TiC, VC, TiN, and VN References

293 295 297 298 299 300 300 303 310

Optical Parameters for the Materials in HOC I and HOC II EDWARD D. PALIK

313

Introduction The Parameters References

313 313 334

CRITIQUES Metals An Introduction to the Data for Several Metals DAVID W. LYNCH AND W. R. HUNTER

341

Introduction References Alkali Metals References Lithium (Li) References Sodium (Na) References Potassium (K) References Chromium (Cr) References Iron (Fe) References Niobium (Nb) References Tantalum (Ta) References

341 341 342 344 345 347 354 357 364 367 374 376 385 387 396 397 408 409

Contents

ix

Beryllium (Be) E. T. ARAKAWA, T. A. CALLCOTT, AND YUN-CHING CHANG

421

References

425

Cobalt (Co) L. WARD

435

References

440

Graphite (C) A. BORGHESI AND G. GUIZZETTI

449

References

453

Liquid Mercury (Hg) E. T. ARAKAWA AND T. INAGAKI

461

References

465

Palladium (Pd) A. BORGHESI AND A. PIAGGI

469

References

472

Vanadium (V) G. GUIZZETTI AND A. PIAGGI

477

References

480

Subpart 2 Semiconductors Aluminum Arsenide (AlAs) EDWARD D. PALIK, O. J. GLEMBOCKI, AND KENICHI TAKARABE

489

Acknowledgments References

492 492

Aluminum Antimonide (AISb) DAVID F. EDWARDS AND RICHARD H. WHITE

501

References

505

Aluminum Gallium Arsenide (Al.Ga,_~s) O. J. GLEMBOCKI AND K. TAKARABE

513

Acknowledgments References

515 515

Contents

x

Cadmium Selenide (CdSe) H. PILLER

559

References

563

Cadmium Sulphide (CdS) L. WARD

579

References

586

Gallium Antimonide (GaSb) DAVID F. EDWARDS AND RICHARD H. WHITE

597

References

601

Silicon-Germanium Alloys (SixGe~_.) J. HUMLJ(~EK, F. LUKES, AND E. SCHMIDT

607

References

611

Lead Tin Telluride (PbSnTe) H. PILLER

637

References

639

Mercury Cadmium Telluride (H~_xCdxTe) P. M. AMIRTHARAJ

655

Acknowledgments References

666 667

Selenium (Se) EDWARD D. PALIK

691

Acknowledgments References

693 694

Cubic Silicon Carbide (~SiC) SAMUEL A. ALTEROVlTZ AND JOHN A. WOOLLAM

705

References

706

Tellurium (Te) EDWARD D. PALIK

709

Acknowledgments References

713 713

Tin Telluride (SnTe) H. PILLER

725

References

727

Contents

xi

Zinc Selenide (ZnSe) Zinc Telluride (ZnTe) L. WARD

737

References

747

Subpart 3 Insulators Aluminum Oxide (AIz03) FRAN(~OIS GERVAIS

761

References

764

Aluminum Oxynitride (ALON) Spinel WILLIAM J. TROPF AND MICHAEL E. THOMAS

777

References Notes Added in Proof

780 781

Barium Titanate (BaTi03) CHUEN WONG, ,YE YUNG TENG, J. ASHOK, AND P. L. H. VARAPRASAD

789

Acknowledgments References

793 793

Beryllium Oxide (Be0) DAVID F. EDWARDS AND RICHARD H. WHITE

805

References

807

Calcium Fluoride (CaFz) D. F. BEZUIDENHOUT

815

References

825

Amorphous Hydrogenated "Diamondlike" Carbon Films and Arc-Evaporated Carbon Films SAMUEL A. ALTEROVlTZ, N. SAVVlDES, F. W. SMITH, AND JOHN A. WOOLLAM

837

References

840

Cesium Iodide (Csl) JOHN E. ELDRIDGE

853

References

857

Copper Oxides (Cu20, Cu0) CARL G. RIBBING AND ARNE ROOS

875

References

878

xii

Contents Magnesium Aluminium Spinel (MgAIz0,) WILLIAM J. TROPF AND MICHAEL E. THOMAS

883

References

887

Magnesium Fluoride (MgFz) THOMAS M. COTTER, MICHAEL E. THOMAS, AND WILLIAM J. TROPF

899

References

903

Magnesium Oxide (Mg0) DAVID M. ROESSLER AND DONALD R. HUFFMAN

919

Acknowledgments References

939 939

Polyethylene (C2H4).

957

J. ASHOK, P. L. H. YARAPRASAD, AND J. R. BIRCH References

968

Potassium Bromide (KBr) EDWARD D. PALIK

989

References

992

Potassium Dihydrogen Phosphate (KHzP04, KDP) and Three of Its Isomorphs DAVID F. EDWARDS AND RICHARD H. WHITE

1005

References

1008

Sodium Fluoride (NaF) I. OHLJDAL AND K. NAVRATIL

1021

References

1024

Strontium Titanate (SrTi03) FRAN(~OIS GERVAIS

1035

References

1037

Thorium Fluoride (ThF4) I. OHLJDAL AND K. NAVRATIL

1049

References

1054

Contents

xiii

Water (HzO) MARTIN R. QUERRY, DAVID M. WlELICZKA, AND DAVID J. SEGELSTEIN

1059

References

1063

Yttrium Oxide 1Y203) WILLIAM J. TROPF AND MICHAEL E. THOMAS

1079

References Notes Added in Proof

1084 1086

List of Contributors Numbers in parentheses indicate the pages on which the authors' contributions begin.

SAMUEL A. ALTEROVITZ (705,837), Space Electronics Division, NASA Lewis Research Center, Cleveland, Ohio 44135 P. M. AMIRTHARAJ (655), U.S. Army Center for Night Vision and Electro-Optics, Mail Code AMSEL-NV-IT, Ft. Belvoir, Ft. Belvoir, Virginia 22060 P. APELL (97), Institute of Theoretical Physics, Chalmers University of Technology, S-412 96 G6teborg, Sweden E. T. ARAKAWA (421, 461), P.O. Box 2008, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 J. ASHOK (789,957), Department of Applied Physics, Andhra University, Visakhapatnam 530 003, India J. BARTH (213), Elktro-Optik GmbH, F6rdestr. 35, D-2392 Gliicksburg, Federal Republic of Germany D. F. BEZUIDENHOUT (815), Division of Production Technology, P.O. Box 395, Council for Scientific and Industrial Research, Pretoria, South Africa J. R. BIRCH (957), Division of Electrical Science, National Physical Laboratory, Teddington, Middlesex TWll 0LW, United Kingdom H.-G. BIRKEN (279), Institute f~ir Experimentalphysik, Universit~it Hamburg, Luruper Chausee 149, D-2000 Hamburg 50, Federal Republic of Germany C. BLESSING (279), Institut fiir Experimentalphysik, Universitat Hamburg, Luruper Chausee 149, D-2000 Hamburg 50, Federal Republic of Germany I. BLOOMER (151), Department of Physics, San Jos6 University, One Washington Square, San Jos6, California 95192 A. BORGHESI (449,469), Dipartimento di Fisica "A. Volta," Universit& di Pavia, Via A. Bassi, 6, 27100 Pavia, Italy T. A. CALLCOTT (421), Physics Department, The University of Tennessee, Knoxville, Tennessee 37933 M. CARDONA (213), Max-Planck Institut fiir Festk6rperforschung, Heisenbergstrasse 1, D-7000 Stiittgart 80, Federal Republic of Germany YUN-CHING CHANG (421), Physics Department, The University of Tennessee, Knoxville, Tennessee 37933

XV

xvi

List of Contributors

T. M. COTTER (899), Applied Physics Laboratory, The Johns Hopkins

University, Johns Hopkins Road, Laurel, Maryland 20707 DAVID F. EDWARDS (501, 597, 805, 1005), University of California, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, California 94550 J. E. ELDRIDGE (853), Department of Physics, 6224 Agriculture Road, The University of British Columbia, Vancouver, British Columbia V6T 2A6, Canada J. FINK (293), Kernforschungszentrum Karlsruhe, Institute fiir Nukleare Festk6rperphysik, P.O. Box 3640, D-7500 Karlsruhe, Federal Republic of Germany A. R. FOROUHI (151), Loral Fairchild Imaging Sensors, 1801 McCarthy Boulevard, Milpitas, California 95035 FRANCOIS GERVAIS (761, 1035), Centre de Recherches sur la Physique des Hautes Temp6ratures, 1D Avenue de la Recherche Scientifique, 45071 Orl6ans Cedex, France O. J. GLEMBOCKI (489, 513), Code 6833, Naval Research Laboratory, Washington, D.C. 20375 G. GUIZZETTI (449,477), Dipartimento di Fisica "A. Volta," Universit& di Pavia, Via A. Bassi, 6, 27100 Pavia, Italy R. T. HOLM (21), Code 6833, Naval Research Laboratory, Washington, D.C. 20375 DONALD R. HUFFMAN (921), Physics Department, University of Arizona, Tucson, Arizona 85721 J. HUML[CEK (607), Department of Solid State Physics, Masaryk University, Kotl~i~sk~i2, 611 37 Brno, Czechoslovakia O. HUNDERI (97), Division of Physics, NTH, N 7034 Trondheim, Norway W. R. HUNTER (341), S.F.A., Inc., 1401 McCormick Dr., Landover, Maryland 20785 T. INAGAKI (341,461), P.O. Box 2008, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 B. JENSEN (125), Department of Physics and Applied Physics, College of Pure and Applied Science, University of Lowell, Lowell, Massachusetts 01854 R. L. JOHNSON (213), Institut f/Jr Experimentalphysik, Universit~it Hamburg, Luruper Chausee 149, D-2000 Hamburg 50, Federal Republic of Germany C. KUNZ (279), Institut fiir Experimentalphysik, Universit~it Hamburg, Luruper Chausee 149, D-2000 Hamburg 50, Federal Republic of Germany F. LUKES (607), Department of Solid State Physics, Masaryk University, Kotl~i~sk~i 2, 611 37 Brno, Czechoslovakia DAVID W. LYNCH (341), Department of Physics, 12 Physics Building, Iowa State University, Ames, Iowa 50011

List of Contributors

xvii

K. NAVRATIL (1021, 1049), Department of Solid State Physics, Masaryk University, Kotl~skfi 2,611 37 Brno, Czechoslovakia L. OHLfDAL (1021, 1049), Department of Solid State Physics, Masaryk University, Kotlfi~skfi 2,611 37 Brno, Czechoslovakia EDWARD D. PALIK (3, 313, 489, 691, 709, 989), Institute of Physical Sciences and Technology, University of Maryland, College Park, Maryland 20742 E. PELLETIER (57), Laboratoire d'Optique des Surfaces et des Couches Minces, Unit6 Associ6e au CNRS, Ecole Nationale Sup6rieure de Physique de Marseille, Domaine Universitaire de St. J6r6me, 13397 Marseille Cedex 13, France J. PFLUGER (293), Synchrotronstrahlungslabor, HASYLAB at DESY, Notkestrasse 85, D-2000 Hamburg 52, Federal Republic of Germany A. PIAGGI (469, 477), Dipartimento di Fisica "A. Volta," Universit& di Pavia, Via A. Bassi, 6, 27100 Pavia, Italy H. PILLER (559, 637, 725), Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803 MARVIN R. QUERRY (1059), Department of Physics, University of Missouri-Kansas City, Kansas City, Missouri 64110 CARL G. RIBBING (875), School of Engineering, Department of Technology, Uppsala University, Box 534, S-751 21 Uppsala, Sweden DAVID M. ROESSLER (921), Physics Department, General Motors Research Laboratories, 30500 Mound Road, Warren, Michigan 48089 ARNE ROOS (875), School of Engineering, Department of Technology, Uppsala University, Box 534, S-751 21 Uppsala, Sweden N. SAWIDES (837), CSIRO Division of Applied Physics, National Measurements Laboratory, Sydney, Australia 2070 E. SCHMIDT (607), Department of Solid State Physics, Masaryk University, Kotlfi~skfi 2,611 37 Brno, Czechoslovakia MARION L. SCOTT (203), E 549, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 DAVID J. SEGELSTEIN (1059), Bell Communications Research, Red Bank, New Jersey 07701 F.W. SMITH (837), Department of Physics, City College of CUNY, New York, New York 10031 G. J. SPROKEL (75), IBM Almaden Research Center, Department K33, 650 Harry Road, San Jose, California 95120 J. D. SWALEN (75), IBM Almaden Research Center, Department K33, 650 Harry Road, San Jose, California 95120 KENICHI TAKARABE (489,513), Department of Natural Science, Faculty of Science, Okayama University of Science, 1-1 Ridai-cho, Okayama City, Okayama 700, Japan ROMAN TATCHYN (247), Stanford Electronics Laboratory, Stanford University, Stanford, California 94305

xviii

List of Contributors

YE YUNG TENG (789), Department of Physics and Applied Physics,

University of Lowell, 1 University Avenue, Lowell, Massachusetts 01854 MICHAEL E. THOMAS (177, 777, 883, 899, 1079), The Johns Hopkins University, Applied Physics Laboratory, Johns Hopkins Road, Laurel, Maryland 20707 WILLIAM J. TROPF (777, 883,899, 1079), The Johns Hopkins UniverSity, Applied Physics Laboratory, Johns Hopkins Road, Laurel, Maryland 20707 P. L. H. VARAPRASAD (789), Department of Applied Physics, Andhra University, Visakhapatnam 530 003, India L. WARD (435, 579, 737), Department of Applied Physical Science, Coventry (Lanchester) Polytechnic, Coventry CV1 5FB, United Kingdom RICHARD H. WHITE (501, 597, 805, 1005), University of California, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, California 94550 DAVID M. WIELICZKA (1059), Department of Physics, University of Missouri-Kansas City, Kansas City, Missouri 64110 JOHN A. WOOLLAM (705, 837), Center for Microelectronic and Optical Materials Research and Department of Electrical Engineering, University of Nebraska, Lincoln, Nebraska 68588 CHUEN WONG (789), Department of Physics and Applied Physics, University of Lowell, 1 University Avenue, Lowell, Massachusetts 01854

Handbook of Optical Constants of Solids II

Part I Determination of Optical Constants

Chapter 1 Introductory Remarks EDWARD D, PALIK Institute for Physical Science a n d T e c h n o l o g y University of M a r y l a n d C o l l e g e Park, M a r y l a n d

I. Introduction II. The Chapters III. The Critiques IV. The Tables V. The Figures of the Tables Vl. General Remarks VII. Errata References

3 3 5 6 7 7 8 11

INTRODUCTION

I

The original list of materials prepared for the Handbook of Optical Constants of Solids (HOC I) [1] contained about 60 materials, but because of a lack of critiquers, the published handbook contained only 37 materials, more or less evenly divided among metals, semiconductors, and insulators. Useful, important materials had to be omitted. More than three years after the publication of HOC I, I approached Academic Press about a second volume to cover these additional materials. Having now added another 48 materials to the list, I can see another 30 materials that might be of interest also.

THE CHAPTERS

As with HOC I the first part of the Handbook of Optical Constants of Solids I1 (HOC II) consists of short topics dealing with the determination of optical constants. In Chapter 2, R. T. Holm discusses sign and polarization conventions HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS II

3

Copyright (~) 1991 by Academic Press. All rights of reproduction in any form reserved. ISBN 0-12-544422-2

II

4

Edward D. Palik

starting with the form of the electromagnetic wave used to solve Maxwell's equations. It seems that in many optics books and papers I read, the authors have not stated this wave form and used it consistently; they sometimes change wave forms. Although this does not affect the final calculated answers, it surely confuses the student with + and - signs and sines and cosines. In Chapter 3, E. Pelletier discusses the problems involved in determining the optical constants of a film on a substrate (a common sample in this day and age of epitaxial layers, and reflection and antireflection coatings). Also, the improvement in deposition processes is now providing thin-film samples, which look more and more like bulk material (except for strain, perhaps). In Chapter 4, G. J. Sprokel and J. D. Swalen discuss the determination of n and k by the technique of attenuated total reflection. This is a powerful technique of examining deposited, single or multiple thin films; oxide layers on substrates; bulk material; and surface properties such as surface plasmons and phonons. In Chapter 5, P. Apell and O. Hunderi extend thin-film structures to superlattice structures, discussing how fundamental optical properties are changed and how we can treat a superlattice as a new material with new, effective n and k values different from the parent materials. This should be very important as multiple-quantum wells, and so forth, become routine optical samples in the future, and as the field of textured surfaces and textured bulk becomes optically more interesting in the context of photonic band structure. In Chapter 6, Barbara Jensen calculates the index of refraction at and below the fundamental band gap of semiconductors from quantum mechanics principles, stressing the ternary and quaternary materials of widespread interest. This gives a calculational way of determining n for AlxGal_~As for any value of x, for example. In Chapter 7, A. R. Forouhi and I. Bloomer develop a model for electronic transitions across the band gaps, which accounts for n and k in the interband region. They assign the several peaks in k to transitions with strengths and widths, fitting the experimental k spectrum and then Kramers-Kronig analyzing it to get the n spectrum. Thus, one has an analytical form for n and k in the interband region. In Chapter 8, Michael E. Thomas develops a model to account for multiphonon absorption as a function of temperature in the region straddling the fundamental transverse optical phonon resonance. This is useful in determining the intrinsic (versus extrinsic) absorption coefficient, especially in studies of laser-window materials in the IR. The next three chapters concentrate on the determination of n and k in the U V - X - R a y regions. Because of the advent of synchrotron radiation sources, this is a very active spectral region.

1. Introductory Remarks

5

In Chapter 9, Marion L. Scott discusses the use of total external reflection to determine n and k in the U V - X - R a y region. All materials eventually have n < 1 above the interband region where this technique applies, and variable-angle-of-incidence measurements can yield n and k. In Chapter 10, J. Barth, R. L. Johnson, and M. Cardona describe spectroscopic ellipsometry, extending it beyond the usual region of 1.56 eV to as high as 35 eV. Linear polarization is not so good at these higher photon energies, and the data must be analyzed in terms of partially linearly polarized radiation. In Chapter 11 Roman Tatchyn describes the interference effects of a microfabricated Au transmission grating in the soft x-ray region and how the complex index of refraction can be extracted from such interference patterns by comparing relative intensities of interference orders. There now follow two chapters describing "nonoptical" techniques for determining n and k in the interband region. Although in the critiques we do not often use the numbers obtained from such techniques, they are still of interest. In Chapter 12, H.-G. Birken, C. Blessing, and C. Kunz describe the angular-resolved photoemission process as a probe of optical constants in the interband region. Although light is used to produce the electrons, it is the electron currents emitted by a crystalline sample (proportional to k) that are of interest. In Chapter 13, J. Pfl~iger and J. Fink describe how n and k are obtained from a truly nonoptical experiment, electron-energy-loss spectroscopy. This technique has been used for many years as a way of determining Im(-1/e) for unsupported thin films. Finally, in Chapter 14, I have attempted to collect many optical parameters for the 86 materials in H O C 1 and II and display them in a gigantic table. Again, these are parameters that I have often needed but could not find in one place, things like crystal symmetry, lattice constant, irreducible representation, TO and LO resonant frequencies, plasma frequency rap, fundamental band gap Eg, d.c. dielectric constant e0, and so on. In regards to Chapter 2, which discusses conventions, it is interesting to note that in three of the chapters n - i k is used, whereas in four others n + ik is used. The other chapter writers have avoided making the choice. In the critiques both conventions are also used.

THE CRITIQUES

The critiques are meant to be the critiquer's own judgment of the best values of n and k over the widest spectral range. Some spectral-region

III

6

Edward D. Palik

values can overlap slightly t o indicate typical differences among laboratories and methods. When possible, estimates of uncertainties are stated, but a rule of thumb is that n is good to two figures and k is good to a factor of two. Beyond this, one has to examine the experimental methods; n can be good to five figures for minimum-deviation techniques, and k can be good to two figures if one measures transmittance carefully. In a few cases, data such as reflectance R and transmittance T are found in the literature, but no determination of n and k has been made; in such cases we can only cite the work. This happens most often in the UV interband region and in the IR reststrahlen region. A good review and source of references for properties of many materials (including optical) is "Landolt-B6rnstein Numerical Data and Functional Relationships in Science and Technology," (K.-H. Hellwege, ed.), Springer-Verlag, Berlin, 1983. This series consists of 23 volumes, which cover quite a few of the materials discussed in H O C I and II.

IV

THE TABLES

The biggest change from H O C I is in the use of camera-ready tables of n and k. Although uniformity could not be achieved in every case, the tables should be more free of typographical errors. We have given the various critiquers more leeway in the format. The eV, cm-1, and/~m units may not run continously from X-ray to mm-wave regions, but may "back up" for another author's values to overlap, and then continue on. The exponential form of k varies; sometimes x 10 -5 runs down to x 10 -4 line by line. (This is because it is easier for the critiquers than the typesetter to include every single x l0 -5. Other critiquers leave the intermediate lines blank and x 10 -5 is understood until a change takes place at x 10 -4. Although we try to continue exponents down to x 10 -5, then switch to decimals for x 10 -1, x 10~ x 101, this is not done consistently. Sometimes 4.6 E - 03 is used, and sometimes a star indicates the multiplication sign ( 4 . 6 , 1 0 - 2 ) . Some authors have carried cm -~ up into the millions; we usually stop mentioning it above 50,000cm -~. Always keep in mind the conversion factor 8065.48 cm-1/eV. Although we have used this conversion factor to move among the units in most cases, some critiquers have used values differing by one digit in the fourth significant figure. This should cause no problems in an experiment until the fourth figure is reached in the eV, cm -~ or ~m units, which is not frequently. Values of n and k are identified by a reference number in brackets, which is understood to apply in that particular column until a new reference appears. Several alloy semiconductors have been included, such as AlxGal_xAs,

1. Introductory Remarks

7

Pb~_xSn~Te, Hgl_xCdxTe, SixGe~_x. Although it would be nice to list x in tenths from zero to one, these are not necessarily available from UV to IR, so in most instances we have taken what the literature supplies for a few representative values of x. Hopefully, the dependence of band gap and optical-phonon frequencies on x can be obtained from the cited literature. Values of n and k have been read from expanded graphs, from tables in the original papers, original reports, and data kindly supplied by the investigators in private communications, and in at least one or two cases, the critiquers have made additional measurements for their particular critiques.

THE FIGURES OF THE TABLES

V

L o g - l o g plots of n, k versus r have been prepared in a uniform format. The details are often lost in the thickness of the pen line, but the reader can always return to the tables to interpolate values. The figures give an instant bird's-eye view of n and k. We also cannot plot all the data for every x tabulated for an alloy semiconductor, for example, so we have chosen only a few examples, where hopefully there is a wide spectrum of data for one or two values of x.

GENERAL REMARKS

In H O C I, we had concentrated on room-temperature (often not stated in the original papers) values of n and k. Some critiquers have included temperature dependence of n at various wavelengths and commented on what raising or lowering the temperature does to k. Of special interest is the fact that lowering the temperature significantly widens the spectral windows in the optic-phonon regions, and materials that are optical windows on dewars at room temperature invariably become much better windows when used as low-temperature windows inside the dewar (sapphire is a good example). We cannot hope to tabulate every useful optical parameter for these materials, although in Chapter 14 there is a table listing some of the parameters for each material. Of some interest is the effort that has gone into a material like BaTiO3 (for physics and technology), but its UV properties have not been

Vl

8

Edward D. Palik

measured to 30 eV. Also, a material like Cu20 is very prominent in solar research and is used as an example in a book on IR and Raman selection rules for optic phonons, but again no UV data are available. Even a material of great interest for quantum-well studies and superlattices in general, like AI,Ga~_,As, has not been studied from UV to far IR for x = 0, 0.1,0.2 . . . . . 1.0 in detail. Only bits and pieces seem to be done with existing or readily grown materials for a few values of x. The uncertainty in x is also a problem, often being +0.01. Our knowledge of deposited films improves steadily. Ion-beam-assisted layer growth is producing samples that look more and more like bulk materials (either single crystal or polycrystal). Variations in n from the substrate to outer surface, related to the columnar growth process, are now seen; the strain effects in films on substrates are not yet optically clear. I proposed the title of this volume to be H a n d b o o k o f Optical Functions o f Solids because n and k are not constant, and physicists use the term often. However, to relate to H O C I better, we retain the use of optical constants in the title and in most of the critiques and chapters. We have stretched the truth a little in the title, in that two liquids are discussed: Hg (as a classic liquid metal) and water (because H_,O surrounds us). Also, we have included polyethylene because it is a prototype plastic and has been widely used in IR spectroscopy. In addition, it shows scattering in the visible because of its structure and therefore provides a tricky task for the experimentalist to characterize it. Also, a-C:H (amorphous hydrogenated carbon) is of considerable interest (as is a-Si'H), so we touch upon this material. Among the materials offered but not claimed by critiquers are AgCI CsBr, NaBr, KI, BaO, NiO, PbF_,, HgS, HgSe, GaN, Mg, Cd, Pb, Mn, Ti, Teflon, Plexiglas, mica, and liquid nitrogen (the only material not at room temperature). The high-temperature superconductors were considered for inclusion, but at this date, the optical properties have not "stabilized."

VII

ERRATA

In using H O C l, I have found mistakes and typographical errors; a few readers have also pointed out mistakes. p. 6, line 18: insulators p. 7, line 5: for the choice n + ik the wave form is exp[i( - tot + q- r)]; for n - ik the wave form is exp[i(~ot + q . r)]. The + on the term q- r pertains to propagation in the + r direction for n + ik, and to propagation in the -Y-r direction for t l - ik. The variations uscd in the chapters and critiques testify

9

1. Introductory Remarks

to the fact that we will never agree on one convention, and it behooves us to learn how to switch back and forth between them. See Chapter 2, Table 1 in this volume to sort out the _ signs. p. 15, line 7" f = In ( t h ) = In ( ~ ) g = In (rh/~) 2) = In (~/~) p. 19, in denominator in two lines after Eq. (22): 1 + &fP*of + 2 Re (Pof) 1 +/~of + 2 Re (rof)

p. 21, numerator in last line 9 ;';"* "I'I Re(1/r p. 23, denominator of Eq. (36): 1 - { R 0 , R , 2 ( 1 - W 2 ) + 2 ( 1 - W) Re(Po, P,2) + 4Wsin 6[sin 6 Re(P~,~?~:) - c o s 6 Im(P~,~P~2)}/D with D = 1 + RillRi2 + 2 Re(f01?12),

p. 24, denominator of Eq. (38): 1 - 4{fid[Ro, R ,2 + Re(fi0, PI2 )] - dim(P,,, P,2 )}/D

p. 25, denominator of Eq. (31): 1 - R I R h E 2 p. 27, line 9 up from bottom: gjk = [( p 2 p 2 _ S 4

.i *

hi, - 2[" 'e . . . . .

) (P z "~

_ _

p2

~ .i ) - 4 S 2( a ~fi "k - ot ~",fi "/", ) ] l D i,

-

99

p. 28, line 2 below Fig. 1" 77.t= (1 + R02 + 2go2)~//ee~,~2//,

p. 28, line 3" T t' = (1 + R , , 2 - 2 g , , e ) - - ~ R e

-~

9

Th(1-R,,2) Re { ~,,'] p. 28, line 4" 1 + R o 2 - 2&,2 \ ~ j : T,,2

T

p. 2 8 , 1 i n e 7 " T l = ( 1 - R , , ~ )

t-(i-R,,2)

( 1+( i -2

10

Edward D. Palik

p. 28, line 10: Tr=(1 + Roz + 2go2) -~ine

, T}= (1 + R o2 -

w

2go2 )

1(r

~f Re- \ ~2J

p. 28, line 11: TyT'I= TITb p. 31, line 6" 2(S 2 - nZ)1/Zd, = 2fl~dl p. 31, denominator in line 7: h2~ + (S 2-n~)d~22 p. 38, line 14: advantage p. 62, line 7: arithmetic p. 70, lines 20, 21: the term ( n 2 - k 2 - s i n 2 q S ) in brackets [ ] should be squared p. 310, 16.8 eV-0.07380 pm p. 429, last line: is given by Bennett p. 466, line 9: O)],2/(~, 2 - E) p. 470, Fig. 4: k curve should be dashed between 3 • 10 -3 and 6 • 10 -3/zm p. 480, line 13: subtracted p. 480, line 21: footnote b p. 623, ReAs2Se3: Room-temperature data for a and e2 in the 2-12 eV region can be obtained from R. E. Drews, R. L. Emerald, M. L. Slade, and R. Zallen, Solid State Commun. 10, 293 (1972). p. 704, Eq. (2): dn/dT is in units of 10 -5 K -1 p. 704, line 15 up from bottom: Tomiki p. 704, line 8 up from bottom: Tomiki p. 706, Refs. 9, 10, 11: Tomiki p. 719, line 9 up from bottom: The wavelength 2, index p. 731, on 0.1579 e V - 1,274 cm -1 line: k0 = 6.949 p. 778, line 15 up from bottom: Miyata p. 798, Ref. 9: (1974) In the critiques for ZnS (cubic), LiF, and SiO2 (glass) in HOC I, the Henke model was used to calculate n and k in the X-ray region. The densities used in this calculation are: ZnS (cubic), 4.1; LiF, 2.635; SiO2 (glass), 2.648. In the critiques for AlAs and AlxGa~_xAs in HOC H, the densities used were calculated from the density equation g = 5 . 3 6 - 1.6x

1. Introductory Remarks

11

given in the AlxGal_~As critique: AlAs, 3.76; A10.3Ga0.7As, 4.88. Since slightly different densities can be found in various handbooks for the same material, it is important to know the value used. Regarding the LiF, ZnS, and SiO2 critiques in H O C I: Since the publication of H O C I, it has been brought to our attention that the k values for LiF calculated using the scattering factors reported by Henke et al. [2] do not agree with experiment and appear to be too small by approximately a factor of two. This discrepancy was discovered by Powell and Tanuma [3] during an investigation of mean free paths of photoelectrons in LiF. On closer scrutiny of our calculations, and in consultation with Mark Thomas of Lawrence Berkeley Laboratory, we found that indeed the formulas converting scattering factors to n, k had been misinterpreted. The consequence is that the calculated n, k values for LiF, ZnS, and SiO2 reported in H O C I are too small. Using the corrected formulas the difference in n is small, but the new k is larger by about a factor of two for the binary compounds and about 30% larger for the ternary compound. Calculations for the elements are not changed. We take this opportunity to present the newly calculated values for these three materials in Table I at the end of this chapter. These values are calculated from a new set of scattering factors disseminated by Henke et al. [4] and cover a wider range of wavelengths, 1.24-240 A. We regret any inconvenience that may have been caused through use of the older values. REFERENCES

1. Edward D. Palik, ed., "Handbook of Optical Constants of Solids," Academic Press, Orlando (1985). 2. B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, and B. K. Fujikawa, "Low Energy X-ray Diagnostics--1981" (D. T. Attwood and B. L. Henke, eds.), p. 340, AlP Conf. Proc. No. 75, American Institute of Physics, New York, 1981. 3. C. J. Powell, National Institutes of Science and Technology, Gaithersburg, MD 20899, USA, and S. Tanuma, Central Research Laboratories, Nippon Mining Company, Ltd., 3-17-35 Niizo-Minami, Toda, Saitama 335, Japan, private communication. 4. B. L. Henke, J. C. Davis, E. M. Gullikson, and R. C. C. Perera, "A preliminary report on x-ray photoabsorption coefficients and atomic scattering factors for 92 elements in the 1010000eV region," Lawrence Berkeley Laboratory Report 26259, Nov. 1988, Berkeley, CA.

12

Edward D. Palik

(X) OO OO (D (X) OO (X) OO OO I'- t'- I'- ~. I'- ~-- ['- I'- P- ~- I'- ~

~

~

~

~

~

~

~

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I ~

I ~

I ~

I ~

I ~

I I I I C,.] C,-] r.,-] ~

I I I C,-] b-] ~

I I I I I b-] r.,-] r.,-] C,-] ~

I I I I C,-] [,-] C,-] ~

I ~

I I C,-] ~

I I I I r~ C,.] C,.] r.,-]

e,l

o

-,4 CO)

e~

oooooooooooooooooooooooodood 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I

E

I

0

I

~

0

I

I

I

I'-- C~ 0

9

,

9

9

I

I

I

I

I

I

I

I

I

I

O,l U9 l ~ O~ C,~l U") CD (,~I ~.0 0

,

,

,

.

,

.

o

,

,

,

I

I

I.~ 0

o

,

I

I

I

I

I

I

If) ,-I 09 I ~ P") 0

,

o

o

.

,

,

,

I

I

I

I

,-I Od ~

~

~

,

,

,

,

o

.=. 0 "

e,l

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(X) OO (X) (X) (X) OO OO eD (X) OO OO (X) OO (X) OO aO (X) OO (X) O0 ~

~

~

~

~

~

~

~

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I

I

I

I

I

I

~ " ~:) !"- 0", ,-I ~

~

~

~

I

I

I

I

I

I

I

I

I

~:) aO ,-I ' ~ aO ,--I l O 0

~ ~ I C ~

I

I

u9 0

I

I

I

kO e'~ 0

I

I

I

I

I

I

I

,-4 ('~! ~

~

~

~

~0 ~r1~r} ~

~

~

~

~

~

(X) I"- 0

04 ~I ,M ~-4 O 0 ~ h ( ~ O 0 0 0

t~

~

I

m

8 == 0

~h dh dh dh d~ (F~ ~

d~ ~

(Y~ dh dr~ dh dh ~

~

dh (Y~ (Y~ dh ~

~

~

~

~

oooooooooooooooooooooooodood

e

9

9

9

9

9

9

9

9

9

.

9

9

,

9

9

9

9

9

,

9

v--4 v--~ v..~4 v.-~ r-~ ~-~.~ v-.g r-.~ v-.4 r..-~ r-..~ v-.~ ~-~ r-~ v...-~ ~-4 ~..j r..-~ v..-~ v.-~ ~

O~ 0 0

~ ~ ~

~ ~ ~

0 ~ ~

~ ~ ~

~ ~ ~

~ ~ ~

~ ~

~ 0

~ ~ ~

~ ~

0 ~

~

~

0 ~

0 ~

~

9

~

~ ~ ~

9

9

~

9

N

~

~

~ ~

~

~

~

9

9

9

~

~

~

1. Introductory Remarks

~" [" [" ~" C" [" ~O kO ~) ~

13

~,O kO ~O ~,O ~

~

iO '.O ~

~,O kO iO ~

~

~

~

~

~

~

~

~

~

O0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

I I ~r.d

I I I I I I r-,-1 r-,.1 r.,.1 c,.I c,-1 ~

~ r

o

i ~ ~,, @4 o 9

,

I I I r.,.1 r.=l ~

I ~

I I c,.1 ~

I ~

,_i @,i {Y) ~p ic} i ~ oo o

9

.

9

9

,

,

.

,

I ~

I ~

I I r.=l ~

I ~

(Xl ~:r ko ~

.

.

I ~

I I r~ ~

I I I r.,.1 b.1 ~

,_i ~p i ~ ,_i ~I, oo ~

,

,

o

.

I I I I I I r.,.1 r.,.1 r.,.1 r.,.1 r.,.1 ~ ~

~

.

~

~

~

,

~

,

~

~

I ~

I ~

~

o

,

.

Gooooooooooooooooooooooooooooooood 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I ~

I I I r~ b-1 ~

I ~

I I I I f.,.1 r.,-1 r.,-1 ~

9

.

,

,

9

9

.

.

.

.

,

,

.

o

.

9

I I r.,-1 ~

I I I I r.d r.,.1 r.,.1 ~

I I r.,.1 ~

I ~

I I r.,.1 ~

I I I I I b.1 r.,.1 cd r.d ~

I I r.,.1 ~

I ~

I ~

I ~

I ~

I I r.,.1 ~

I ~

,

,

,

,

o

.

,

o

.

,

,

.

.

,

,

,

o

,

,

,

,

,

,

,

o

,

.

,

.

o

.

.

.

,

,

.

.

.

,

,

o

,

,

,

,

,

.

,

.

.

,

o

I

I

o 0 0 0 o 0 0 0 o 0 0 0 0 0 o 0 0 o o o 0 o 0 0 o 0 0 o 0 0 0 0 o o

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I

I

I

~ O~ .

I

~ ~ .

I

I

~ ~ ,

9

I

~ ~ ,

I

I

~ ~ ,

.

I

0 ~ ,

I

I

~ ~ o

,

I

0 ~ .

I

I

~ 0 ,

,

~ ~ ,

I

I

I

~ 0 ,

,

I

~ ~ ,

I

0 ~ o

I

I

~ ~ ,

o

I

0 ~ o

I

I

~ ~ ,

I

~ ~

,

,

I

I

0 0 ,

I

~ ~

,

o

I

I

I

~ ~ ,

o

,

,

,

,

,

0~0"~ O~ O~ O~ 0"~ O~ 0"~ O~ O~ 0"~ O~ O~ O~ O~ O~ O~ O~ O~ O~ O~ O~ ~

~

~

~

~

~

~

~

~

~

~

~

0~0~

0"~ O~ O~ O~ O~ O~ O~ O~ 0"~ O~ O~ O~ O~ O~ O~ O~ 0'~ O~ O~ O~ ~

~

~

~

~

~

~

~

~

~

~

~

O0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14

Edward D. Palik

o

o

o

o

I

I

I

I

I

I

,,::r ,w, i.~ ~o o'~ o4 !.... (v') o oo ( D o ,,:::p r... (x) oo (x) o ~ ,,o , - i oo LO ~ ,'-I O4 ("') "::1' I.~ ! ' " (X) O e,4 ~ I.~ O0 O O 0 0 ,-I O4 ',::~' I ~ ~O glD ( ~ ,-I ~

~ ~

~ ~

~ O

O ~

e,,io ,,:::~ o ,i,o ,-i .r:-. o4 oo o,) o.% i.~ , - i (v.) go (v~ o,) iv) f,.) (.~ o4 ,_1 o ~ ~ ~ o o o~ o'~ oo (x) r ~ i...- ~o ,ko ~o ~.o i ~ ,w, o,~ o4 , - i o ( ~ (x) !...- L.O ,,::p ~ (~1~ O~ gO (X) 00 (D (:D (X) (X) (X) (X) O0 O0 O0 (X) (D gO OO r'-. !"-. ~ ~ ~ ~

~ ~ ~

~ O ~

~ ~ ~

~ ~ ~

o

o

I

o

I

o

I

o

I

o

I

o

o

o

o

I

I

I

I

o

o

I

o

o

I

o

I

o

I

I

o

o

o

I

I

I

o

o

I

o

I

o

I

o

I

o

I

o

0

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

O O O O O O O O O O O O O O O O O O O O O O O O O O O O

I

I

I

I

I

9

9

9

9

9

I

9

I

I

9

9

I

9

l

9

I

9

l

9

I

9

I

9

I

9

I

9

I

9

I

I

I

I

9

9

9

9

9

I

9

I

I

9

9

["... ',,o l.~ ,,::~, , ~ ~ ('~l , - i o ( ~ c o r.- k o l ~ ,,::r e,) (,~i ,-4 o o o l . - ,,.o ~ o o o o o o (x) (x) (x) (x) o o (x) r-. I.- r.. r..- r... r-. r... r... l..-. i.... ,4~ ~ ~ ~

~ ~

I

I

I

40

~ ~

o ~

I

9

~ ~

9

~ ~

oooooooooooooooooooooooooood wnq

O O O O O O O O O O O O O O O O O O O O O O O O O O O O

I

I

I.O "::1' gO I ~ I " O") ',:~' ,--I 0") O4 ~P ~ f") I.O r - O ,W, O ["- u9 ,4D !"- ~ ~ (v) [--. ,-.i ~o , - i I.~ o,~ o I--.. i.~ o ,,_.l r f,..) ~1, l o !..- o,~ o o4 ,,::~, l o o.~ ~

I

I

~ ~

9

~ o

I

9

o o

I

9

9

~ o

I

9

~ ~

I

I

9

I

9

o ~

9

~

I

I

,

,

~ ~

I

,

~ ~

I

,

I

,

~ ~

I

.

~ ~

I

,

I

,

o ~

I.

,

~ o

I

,

~ ~

I

I

,

,,

~ ~

I

o ~

I

,

~ ~

I

,

a,

o

,

I

I

~ ~ ,

I

~ o ,

~ ~ 9

~

~

o

~

9

~

ddddddddddddddddddddddddgddd

OO ("~10'~ r -

O0 O

M

I.-. e.,) O0 i f ) ,_1 r.. (.,.) oo o4 ,i.o (x) ( ; , b o o ~o ~

~

~

~

~

1. Introductory Remarks

15

O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

.

9

.

9

,

9

,

9

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

o

.

,

I

,

I

9

I

,

I

,

I

o

,

oooooodooooooooooooooooooooooodood O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

o4 ,-I ('q (',I u') O

,--IO

O

~I' ed u9 l'- ~

I

I

I

I

I

I

I

I

I

I

9

,

,

,

~

~

~

L'- ,-I ~I' I'-- 00 (~ 00

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O I

I

I

~~-I ,

~ ~ ~

I

~ 9

,

~ ~ ~

,

N

I

I

r

~

.

~ ~ ~

I

~

,

O

~ ~

I

,

I

~ ,

~ ~ ~

I

,

N O ~

I

~

I

I

I

I

~ ,

~ ~ ~

I

I

I

I

I

I

I

I

I

I

I

I

I

I

,

,

,

,

r~r~r.=lr~r~r,=lr~r~r~r.=1~r.=l~~ ,

,

~

,

O

,

O

~ ~

,

~

O ~

,

O ~ ~

~ ~

,

~ O N

,

,

~

,

~ ~ ~

,

O

,

~

~ ~

,

~ ~ ~

~ ~

.

,

O O O

,

~ ~ ~

~ ~

Zddddd66dg6dddddddddds

9

9

9

9

9

9

9

9

9

,

9

9

9

9

o

,

,

,

,

,

,

,

,

,

,

.

,-4,-I ,-I ,-I ,-I ,-I ,-I ,-I ,-I ,-I ,-I ,-I ,-I ,M ,-I ,--I ,-I ,-I ,-I ,-4 ,-I ,-I ~

9

9

9

I~t~O rl

rt

rl

9

9

9

,

,

o

,

,

,

I ~ q:~ r

!~- ~P 0~1 0

~--I r-I v-I r l

r-I ~-I ~.-I ~-I

,

,

,

,

,

,

,

r~- I ~ C~ r-I O~ [~- i.ir~ ~

,

,

,

,

,

9

~

,

r-i O~ I ~ i.c) ~1, ~

9

9

~

,

~

,

0

9

9

~

~

,

~

,

~

9

,

~

9

~

9

~

,

~

,

~

,

~

,

0

~

16

Edward D. Palik

0

0

I

0

I

9

0

,

0

0

I ~

0

0

I ~

0

0

0

0

I

I

I

,

0

0

I

~

0

9

0

0

I

0

I

9

0

0

0

I

,

0

0

,

0

0

0

0

0

0

0

I

I

I

0

I

,

N

0

I

0

~

0

0

,

~

0

0

0

0

o

o

o

o

I

I

I

0

I

0

0

I

I

0

0

0

0

0

0

0

0

I

I

0

~

0

0

0

o

o

o

o

I

I

,

0

0

I

I

,

0

0

0

0

I

~

O

0

0

0

0

I

I

I

,

0

I

0

0

I

,

0

0

I

,

~

0

I

,

N

0

0

0

I

~

0

.

0

I O

0

,

0

0

I

I

,

0

0

I

0

I

,

,

,

0

0

0

0

0

0

0

I

I

I

~

0

0

I

0

~

0

0

I

,

,

0

0

0

0

0

0

o

o

o

o

o

I

I

I

0

I

O

0

0

0

I ~

0

0

0

I

I

,

0

I

~

0

I

,

I

,

0

0

0

0

I

I

0

0

I

I

,

,

I

,

I

I

I

I

~

0

0

~Q ~Q

o

o

o

o

I

I

I

I

I

I

I

I

I

(D (D (D (D (X) (X) (X) (X) (X) (X) D-- r-- P- ~- r-- P- r-- t---to t~ ~

~

~

~

~

~

~

~

~

.

9

I

o

I

o

o

9

o

I

o

9

o

I

o

9

0

9

o

9

o

9

o

o

I

o

o

9

9

o

I

o

I

o

9

I

o

9

o

9

o

9

o

9

o

I

o

9

o

I

o

I

I

o

o

o

o

9

9

9

9

o

o

9

o

9

o

o

9

o

o

9

o

o

9

o

9

o

o

9

9

9

1. Introductory Remarks

O0

17

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

I

I

I

9

,

I

9

9

I

I

I

I

i

9

9

I

,

9

I

I

,

I

,

I

,

I

o

I

,

I

,

I

,

I

o

I

,

I

,

I

,

I

,

I

,

I

,

I

o

I

.

I

,

I

,

I

,

I

o

I

,

I

.

I

,

I

.

9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I

I

I

,

,

9

0

I

9

9

0

I

.

9

0

I

9

9

0

9

0

I

.

9

0

I

9

9

,

.

,

9

0

0

0

0

9

9

0

I

,

0

I

I

I

.

I

9

I

9

.

9

9

0

I

.

9

0

I

.

,

0

I

9

,

0

0

I

9

9

0

I

.

.

0

I

0

I

.

I

.

I

,

,

9

.

.

0

0

0

0

I

.

I

.

.

9

.

0

I

9

9

0

0

I

9

9

0

.

0

0

I

I

9

.

9

.

0

I

.

0

I

I

.

I

9

~

9

0

0

.

9

I

9

9

9

I

I

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I

I

9

I

.

~ ~

I

I

.

~ ~

9

9

~ ~

I

I

I

.

9

9

~ ~

I

I

.

~

.

~

~

I

I

.

,

~

~

0

I

I

.

~

I

9

,

~

0

~

I

I

.

.

~ ~

I

I

9

~

I

,

I

9

0

.

~ ~

~

I

I

9

I

.

0 ~

I

9

9

~ ~

I

9

I

.

~

.

0

~

~

I

I

.

9

~ 0

I

I

.

.

~ ~

0

.

I

9

~

dGddddddgdddddddddddddddgGdddddddd

9

9

9

9

9

9

(",,I O,,i "~1' L~ kO i "

9

,

,

9

9

9

9

9

9

O0 dr~ 0

9

9

9

9

9

9

9

9

9

9

9

9

,--I 0,,I ',::1' L~ kO I'~- 0"~ 0

9

,

9

9

9

.

9

,

,

9

9

.

9

9

.

9

O,,i er) Lr} ~0 O0 Oh, r-I ~

9

9

9

9

.

.

9

9

9

9

9

9

~

,

~

9

~

9

0

~

~

,

9

9

9

9

9

.

9

~

~

9

9

~

18

Edward D. Palik

O4 OQ O4 O4 O4 ('q O4 ('q ('Q O4 O4 O4 O4 ('q O4 O4 eq (N O4 O4 ~ ~ ~ ~ ~ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

I

I

9

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

I

,

~

I

~

I

,

~

I

,

,

~

I

,

I

,

,

,-4 ,-4 ,-4 ,-4 ,-I ,-4 ,-4 ,-4 ,-I ,-I ,--I ,-I ,-4 ,-4 ,-I ,-4 ,-4 ,-4 r4 r4 ~

~

~

~

~

~

~

~

~

CD (X) ~0 (D (X) (X) ~0 (D (D (X) (X) (X) (X) ~0 r-- r-- I'-- I'-- r-- io ~

~

~

~

~

~

~

~

~

o -H

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

oQ ('Q o4 o4 ('q o4 o4 ('Q o~i o4 (,Q (w o4 eq o4 o4 o4 ('q o4 o4 ~ ~ ~ ~ ~ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

,-I ,-4 ,-I ,-I ,-I ,-I ,-I ,-I ,-I rl ,-I ,--IO4 O4 O4 O4 O4 04 r

I

I

Cq ~

I

~

I

~

I

~

~

I

M

~

I

M

~

I

~

~

I

I

M

M

O4 ~

~

L~I O

M

(X) M

~

~' O

ur) O

~r o

~

o

~, oo tw tD o

e,~ tD O'~ ,-4 t,') IJr)I'- O

(X) (X) (X) (X) (X) (X) co o0 (X) (X) (X) (X) cD (X) o0 (X) o0 (X) r-- I-- I'-- I',- I'-- I-,- I-- I--- I'-- r-- r-o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

eq O4 eQ O4 O4 O4 O4 04 ('Q O4 e4 O4 O4 O4 eq O4 O4 ('Q 04 ~ ~ ~ ~ ~ ~ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

!

I

I

~

I

~

I

~

I

~

I

I

T~ .H

O~I O4 ,--IO O'~ I'- U9 O40'~ ;JP ,-ItD ,-4 CD IJr)Oq O (~I ~ ,-I iO CD ~ u9 "~' M O~I O (~h (X) r-- u9 ~" M ,--IO (D I'- tD ur) ~I, e,'~e') O4 rl ~

~ O

~ ~

~ ~

~ ~

~ ~

~ ~

~

~

~

~

~

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

J (~h (D O0 (~h O

O4 If) O0 O4 r,- O4 I'- C,~ O

~--I I--I r-I ~-I r-~ ~1 r l

~1 q~l i~-I ~1 r-I i - I i--I

t- I~ f,~ eq ,-4 ,-4 ,-4 O4 M

~

1. Introductory Remarks

19

0 0 0 0 0 0 0 0 0 0 0 0 0 0 I

I

~ 9

9

I

I

I

I

I

~

~

r,lr=1~ ~

~

~

.

.

.

oo tJr) o~ O

I

I

I

.

,,

9

OO lx'p ('q O

,,

I

I

r=lr~ ~

I

~

9

I

,,

9

9

.

F~. trp c%I c~ t~ ~:l,

0 o 0 0 o o 0 0 o o o o o o

0 0 0 0 0 0 0 0 0 0 0 0 0 0

I

I

I

I

I

I

I

I

I

I

I

I

I

gJ~J, 4 4 4 4 4 4 4 4 J ,

I

J,J,

0 0 ~ CO I'--tO I~0 ~P C'~ e~l ,-I 0 O0 I'-I'-- I'--tO tO tO tD ~P tD tO tO tD tO uO tO

oooooooooooood 0 0 0 0 0 0 0

I

I

~ ~

I

I

0 0 0

I

~ ~ ~ 0 ~ ~ ~ 9

~

I

9

~

9

.

~

.

~

9

0

I

I

0 ~ .

0 ~ 9

0

.

o

0 9

0

0

~

~

9

~

I

I

~

9

.

9

~

oooooooooodooo

D.. oo oo o~ ~ o o ,-l ,--l O~l o~l ~ o*} ~r ,-.l ,-l ,-l ,-l ,-I t~l O~l OQ O4 0~l O~I Oq OQ Oq

Chapter 2 Convention Confusions R. T. HOLM Naval R e s e a r c h L a b o r a t o r y W a s h i n g t o n , D.C.

I. Introduction II. Units III. Wave Equation IV. Polarization A. Linear Polarization B. Circular Polarization V. Fresnel's Amplitude Reflection Coefficients A. s Polarization B. p Polarization C. Phase Change VI. Nomenclature VII. Concluding Remarks Acknowledgments References

21 22 22 26 28 30 37 38 40 41 50 51 52 52

INTRODUCTION As with other scientific fields, optics has its own genre of conventions. The general subject of conventions includes units, symbols, definitions, nomenclature, and sign choices. One of the criticisms of H O C I was a lack of a consistent set of conventions by the contributing authors. This criticism can also be voiced about optics literature in general. Almost everyone, even "old hands," has experienced that frustrating feeling of trying to reconcile differences caused by conventions. However, most authors (for whatever reasons) sidestep the convention issue completely. We do not intend to advocate any particular set of conventions. Instead, we present different conventions that are in general use and discuss conflicts that result because of their differences. We also include reasons that are put forward to justify a particular convention. Here and there, we weave in some history. For a detailed account of the beginnings of the wave theory of light, see the book by Buchwald [1]. HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS II

21

Copyright 9 1991 by Academic Press. All rights of reproduction in any form reserved. ISBN 0-12-544422-2

I

22

R.T. Holm

In this chapter we discuss the plane-wave solutions to the electromagnetic wave equation. The solution of choice dictates which of the two forms of the complex refractive index (n + ik or n - ik) is used. We point out that opposite terms are used to label the same polarization state. The different sign conventions used in Fresnel's reflection equations and the subsequent misunderstandings that this leads to are discussed. The various definitions for the expressions phase change, in phase, and out of phase, and the resulting ambiguities, are presented. Finally, the meanings of reflection, reflectance, and reflectivity are given. We assume the reader has some familiarity with optic and electromagnetic theory. The actual set of conventions that one uses is completely arbitrary~the physics, after all, has to be independent of convention. As far as we know, Muller has provided the most extensive discussion of different conventions that are used in optics [2]. At the Second International Conference on Ellipsometry, held at the University of Nebraska in 1968, he proposed a standard set of conventions, which have become known as the Nebraska conventions. Bennett and Bennett have also commented on the convention issue [3]. Although the comments and recommendations by Muller and by Bennett and Bennett are addressed primarily to the ellipsometric community, they are relevant to anyone working with electromagnetic waves.

II

UNITS

Since 1960, the single convention in optics that should be standard is electromagnetic units. The International System of Units, abbreviated SI units, has been adopted nearly worldwide [4-6]. The SI system is based on the rationalized MKSA (meter, kilogram, second, ampere) system of units. Although most journals request that SI units be used, many authors continue to use one of the other systems (Gaussian, practical [7], and so on). Jackson [8] and Frankl [9] provide enlightening comparisons of the common systems of units and conversion tables between systems. We use SI units in this chapter.

III

W A V E EQUATION

The wave equations for the electric and magnetic fields, as developed in any electromagnetic textbook, are 02E 0E V 2E = 6~'o/~o- - ~ + O'po cot

( 1 a)

2. Convention Confusions

23

02H OH VZH -- ee0~0-~2 + okt00t '

(lb)

where E is the electric field strength in volt/meter (V/m), H is the magnetic field strength in ampere/meter (A/m), e0 is the permittivity of free space in coulomb2/newton meter 2 (C2/N m2),/to is the permeability of free space in weber/ampere meter ( W b / A m ) , e is the dielectric constant (relative permittivity), and a is the conductivity in siemens/meter (S/m). The wave equations shown above are applicable to media that are homogeneous, isotropic, linear, and nonmagnetic. For simplicity, one usually establishes a right-handed Cartesian coordinate system in which the wave propagates along one axis, say the z axis. For layered media, the z axis is chosen to be perpendicular to the interfaces. Whether the wave equation is for an electromagnetic wave, a stretched string, or a sound wave, the same four well-known plane-wave solutions apply [10, 11]. These four solutions for the electric field of an electromagnetic wave may be formally expressed in complex form as E = Eoe ''~i~'2q~-~''l,

(2)

where c~ and c2 are sign coefficients and can take the values of +1; q ( - 2 J r / 2 ) is the wave number [12] and is the magnitude of the wave or propagation vector, q; 2 is the vacuum wavelength; ~o( = 2~f) is the angular frequency; f is the frequency; and E0 is a vector representing the direction and maximum amplitude of E. In Eq. 2 the coefficient c2 determines the direction of propagation. Time, t, is inherently positive and always increasing. When c2= + 1, z must increase so that the phase term q z - (ot remains constant. Thus, the wave propagates toward increasing z ( + z direction). When c2 = - 1, the phase term - q z - o J t remains constant for decreasing z, that is, propagation toward decreasing z ( - z direction). As will be discussed shortly, the choice of cl determines the form in which complex numbers have to be expressed. If ct= + 1, complex numbers take the form a' + ia", whereas if c~ = - 1 , complex numbers take the form a ' - i a " , where prime and double prime denote the real and imaginary parts of a complex quantity, respectively. The effects of the values of cl and c2 are summarized in Table I. We have now reached the point at which the first arbitrary choice must be made. One has to decide to use either c~ = + 1 or cl = - 1, that is, to use either a minus sign or a plus sign in front of ~ot. However, whichever choice is made for c~, both values of c2 are admissible. A general solution consists of a linear superposition of two waves propagating in opposite directions. In the early days of electromagnetic theory, the exp[i(o~t+qz)] solution was commonly used. Today, most opticists still prefer that solution. However, physicists (at least since the advent of quantum mechanics)

24

R.T. Holm

TABLE I Possible Forms of Solution to Wave Equation, with Corresponding Propagation Direction and Form for Complex Numbers

C1

C2

e cli(c2qz-~')

Propagation direction

Complex numbers a' + ia"

1

1

e i(qz-~~

+z

1

- 1

e i(-qz-wt)

-z

a' + ia"

- 1

1

e i(~'t-qz)

+z

a' - ia"

-- 1

-- 1

e i(~~

--z

a ' -- i a "

( n + i k ) e = g = e~

1-

i),--------~+

o~- -Tr,o/

~oi, w

~ 7 - o) y

~

Included is the infrared dielectric function for polar semiconductors expressed in the two complex-number forms.

prefer to use the exp[i(+qz-~ot)] solution. The one-dimensional form of the Schr6dinger wave equation for a free particle of mass m is [13] 0/p

h2

02~

ih Ot-

2m

OZ 2"

(3)

Note that the Schr6dinger equation contains a first derivative with respect to time rather than a second derivative as the electromagnetic wave equation does. As a result, the kinetic energy of the particle is positive [13] only if the exp [i(+qz-~ot)] solution is used. Being comfortable with this form, physicists carry this solution from quantum mechanics to other fields, such as optics. Other reasons have been proffered for picking one choice of c~ over the other. One consideration is whether one is dealing with temporal or spatial phenomena. For temporal effects, it is the damping in time at a fixed point in space that is important. In this situation, the angular frequency, ~o, is taken as complex [14], so that the imaginary term accounts for damping in time. Hence, it is convenient to choose exp(i~ot) and discard the spatial terms. Circuit theory primarily deals with temporal phenomena, so electrical engineers are usually reared on exp(iogt). Then, when they study wave propagation, they continue to use the exp(i~ot) formalism because that is what they are comfortable with. Spatial phenomena are primarily concerned with steady-state effects.

2. Convention Confusions

25

For a wave propagating in the z direction, for example, it is the spatial damping that is important. Thus, q is complex, and it is convenient to use e x p ( i q z ) and to discard the temporal terms. In either case, a positive exponent is chosen, so that one does not continually have to carry a minus sign along when writing equations (laziness). Many textbooks simply state a solution to the wave equation without discussing reasons for their choice or even mentioning that another possibility exists. Several textbooks first give an abstract solution of the form f ( q z - wt) + g ( q z + ~ot). Then without any explanation, however, they say that it is "convenient" to use the e x p [ i ( o g t + q z ) ] solution. An otherwise excellent optics book actually uses both conventions throughout the book. Although each section is consistent within itself, it becomes confusing when comparing sections. Before leaving this section, let us have that talk about complex numbers. In a lossy material the wave number is complex and given by o)

c) = --h,

(4)

c

where h is the complex refractive index and a tilde (-) signifies complex numbers. All complex quantities must be expressed in the form a = a' + icla",

(5)

ft = q ' + ic~q",

(6a)

h = n + iclk,

(6b)

so that

and

where n is the refractive index, k is the extinction coefficient [15], and c~ takes the value chosen in Eq. 2. Note that q', q", n, and k are all positive. The factor c~ in Eq. 5 assures that a wave decays (rather than grows) exponentially as it propagates; see Table I. To reiterate, if one uses the e x p [ i ( + _ q z - ~ o t ) ] solution, the complex refractive index must be written h = n + ik. For the exp[i(~ot+qz)] solution, the h = n - ik form is required. An equation expressed in one form of complex number can be converted to the other form by taking its complex conjugate [16]. As an example, we

26

R.T. Holm

have included in Table I the equation for the classical dielectric function, g, which describes the optical properties of a polar semiconductor in the infrared [17]. Free carriers (second term) and phonons (third term) are formulated in terms of the Drude and Lorentz models, respectively. In the dielectric function e~ is the high-frequency dielectric constant; ~0p is the bulk plasma frequency; 7 is the free-carrier damping constant; ~oL and a~x are the longitudinal and transverse optic-phonon frequencies, respectively; and F is the phonon damping constant. One must exercise caution when programming reflectance (and other) formulas that contain complex variables [18]. When taking the square root of a complex number, whether it be in the (a' + ia") or the (a' - ia") form, most (all?) compilers will return the answer in the correct form~ However, when taking the square root of a negative real number, compilers return a positive imaginary number; for example, ( - 4) 1/2= + 2i. This is the correct answer for the ( a ' + ia") form, but for the (a'-ia") form, the correct answer is - 2i. If one writes one's own programs and uses formulas that require the (a'-ia") form, it is easy to test complex variables before taking a square root. Just insert a line of code that sets (square root) - - ( s q u a r e root) if a complex number has only a negative real part. Such situations arise when calculating internal reflection spectra (IRS) [19], in which terms like [ 1 - ( n / s i n Oi)2/nt2]1/2, with n/sin Oi>nt, a r e encountered. As Spiller [18] points out, commercial computer programs may not offer any provision for setting the form of complex numbers. To obtain the correct answer, either assign negative values to the thicknesses of those films whose extinction coefficients are zero, or set k to some small value that does not significantly affect the results. Before those who use the ( n - i k ) form panic, we point out that reflectance programs that do not test complex numbers yield~the correct results most of the time. We had used an IRS program successfully for over a year before it gave reflectances greater than unity for a particular IRS configuration. It takes a long time to find such a "bug." To be on the safe side, put in a test. Finally, let us give a gentle reminder. The form for fi (and other complex variables) has nothing to do with the direction of propagation. The form is determined solely by the sign of ~0t in Eq. 2.

IV

POLARIZATION

In addition to its amplitude, frequency, and direction of propagation, an electromagnetic wave is characterized by the orientation of its field vectors,

2. Convention Confusions

27

E and H. Since, in an isotropic medium, an electromagnetic wave is transverse rather than longitudinal, its field vectors are normal to its wave vector. The temporal behavior of the orientation of the electric field, E, as observed at a fixed location along the propagation direction, determines the polarization state of the wave [20-22]. The polarization state is represented by the curve traced out by the tip of the instantaneous E vector. The electric field is chosen as the descriptor of the polarization state because it exerts a much stronger force on electrons than the magnetic field [23]. Polarization terminology and even the word polarization itself [24-26] are perplexing, if not downright contradictory. The first reported observation of an effect arising directly from polarization was by Erasmus Bartholinus [27, 28]. In 1670 he described double refraction, in which two images are seen, in calcite (Iceland spar) [29]. It was not until 1808 that Etienne Louis Malus discovered polarization by reflection. While looking through a calcite crystal at the reflection of the setting sun from a window of a nearby building, he saw only one image, rather than two, for certain orientations of the crystal [30, 31]. Malus introduced the term polarization to distinguish the two light beams that emerge from a calcite crystal. We shall now present the various ways that polarization terms are defined and used. When a phase term d is included, four different forms can be found in the literature to express the electric-field strength of a monochromatic wave propagating in the + z direction [2]: E = Eo ei(ot-q~+6)

(7a)

E = Eo ei(~'t-qz-~)

(7b)

E = Eoe i~q~-~'+6)

(7c)

E = Eo ei 0 with the (cot + 6) form and b < 0 with the (cot- 6) form), then 6 represents an increase in time, or a phase advance (leading). Likewise, if the sign of the phase term is opposite that of ~ot (that is, 6 < 0 with the (~ot+ 6) form and 6 > 0 with the (~ot- 6) form), then 6 represents a phase retardation (lagging). An arbitrary wave, E, propagating in the +z direction can be treated as a superposition of two orthogonal waves, E~ and Ey, expressed in the ( c o t - b ) form bv Ex = Exo cos(cot- qz - 6x)

(9a) Ey = Eyo c o s ( ~ t - qz - 6r),

and in the (cot + 6) form by Ex = Ego cos(cot- qz + 6x) Ey = Eyo cos(tot- qz + 6y),

(9b)

such that E =/'E~ +]F~,

(10)

where t and ] are unit vectors in the +x and +y directions, respectively, of an (again arbitrary) right-handed Cartesian coordinate system. However, practical considerations based on the experimental system ultimately dictate the selection of a coordinate system. The actual polarization state of the wave depends on the relative values of Ex0 and Ey0 and of 6~ and 6y. For arbitrary values of E~0, Ey0, 6x, and 6y, the wave is elliptically polarized. Two special cases result: (A) when 6 = 6 y - 6 ~ = 0 or _+Jr; and (B) when Ex0 = Ey0 and in addition 6 = _+Jr/2. (Almost every author defines 6 as by - 6x, rather than as d x - 6y.)

A

Linear Polarization

Even though polarization is referenced to the temporal variation of the orientation of E, many authors provide a spatial visualization of the

2. Convention Confusions

29

component (Ex and E,.) and resultant (E) waves at an instant of time. Such a snapshot [22, 33] is shown in Fig. 1 for 6 = 0 . A sinusoidal curve is an expedient representation of a monochromatic light wave. If we fixed our attention at some point on the z axis, we would see that E oscillates, with time, along a fixed line that is normal to the z axis. E would change direction every half cycle, having maximum extensions of +E0 [ = (E20 + E~0)1/2]. The curve is drawn by connecting the tips of the instantaneous E vectors along the z axis at an instant of time. The resultant wave lies in a plane inclined at an angle a[ = tan-~(Eyo/Exo)] to the x axis. Such a wave is said to be linearly polarized. Sometimes a linearly polarized wave is referred to as being plane polarized because the wave lies in a plane, as in Fig. 1; however, many authors discourage the use of this term [24, 26]. Three points are worth noting. First, both E~ and E,, in Fig. 1 are also linearly polarized. Second, if 6 = _+~r, then Ev inverts, which means that E lies on the other side of the x axis. Third, both the ( ~ t - 6 ) and (~ot + 6) forms (Eq. 9) give the same description of linear polarization. A n o t h e r expression that has long been in disfavor [24, 26] is plane of polarization. The plane of polarization, as originally defined by Malus [27], is the plane that is normal to the crystal surface and contains the light ray polarized by reflection. That plane is nothing more than the plane of incidence. Malus' definition was more closely tied to the procedure of producing polarized light than to the physical nature of light. A b o u t forty years later, people discovered the vector (E and H) nature of light, and they figured out that it is actually the H field, rather than the E field, that lies in Malus' plane of polarization. As a result, some authors have tried to redefine the plane of polarization to be the plane containing E and the wave vector q; such are the ways that ambiguity raises its evil head.

X

Ex

f--

,,

!

Z

Fig. 1. Snapshot of a linearly polarized wave propagating in the + z direction. E, and Ev are the component fields; E is the resultant field inclined at an angle c~to the x axis.

30

R.T. Holm

To make matters worse, the I E E E has defined the plane of polarization [34] to be the plane containing E and H. Thus, the IEEE plane of polarization is perpendicular to that of opticists. Several authors [24, 25] have pointed out that a plane of polarization does not define a unique wave (although a particular wave does have a unique plane of polarization). As an alternative, many authors prefer to prescribe a linearly polarized wave by the direction of its wave vector and the direction of vibration of the E field. Clark [26] suggests that azimuth (rather than direction) of vibration is a better term to describe the orientation of the E field.

B

Circular Polarization

When Exo=Eyo and 6 = +at/2, electromagnetic waves are circularly polarized. The sense of rotation depends on the sign of 6 and which form of Eq. 9 is used. Since it is the difference between the phases of Ex and Ey that is important, we let 6x = 0 so that 6 = by. Initially, circular polarization will be described from a strictly mathematical point of view. Then the contradictory ways in which various disciplines label circular polarization will be presented.

1

(~,t- ~) Form First, consider the (tot-6) form with Exo = Eyo = Eo. The component field equations from Eq. 9a for 6 - - Jr/2 are

Ex = Eo cos(tot- qz)

(11a)

Ey = - E 0 sin(tot- qz), and the equations for 6 = + Jr/2 are

Ex = Eo cos(tot- qz)

(llb)

Ey = Eo sin(tot- qz). Figure 2a shows a plot of the Ex and Ey components as a function of tot for 6 = - n/2, and Fig. 2b shows a similar plot for 6 = + er/2. It can be seen that when 6 = - =r/2, Ey leads Ex by n/2 (Fig. 2a), whereas when 6 = + er/2, Ey lags Ex by n/2 (Fig. 2b).

2. Convention Confusions

31 (a)

Ey

leads

Ex

(b)

Ey lags Ex

E6

a _J W U_ -E6 n" I-" w E6

..J iJJ

wt

-E6 Fig. 2. The orthogonal components of circularly polarized waves as a function of time. In (a) Ey leads E~, and in (b) Ey lags Ex.

Recall that the resultant field, E, is the vectorial sum (Eq. 10) of Ex and Ev and that polarization is related to the time variation of E at a fixed location along z. Figures 3a and 3b show the position (orientation) of E in the x - y plane located at z = 0 , at several values of cot, for 6 = - ~/2 and 6 = + ~/2, respectively. In Fig. 3a, E rotates, with constant a m p l i t u d e and angular f r e q u e n c y co, from the positive x axis t o w a r d the negative y axis. In

(a)

Ey leads E x Y

(b)

Ey lags Ex _~(Tr/21

Elo~t-O)

E(37 T / 4 ) ~ . . . ~

E(?r14)

IE(Tr/2) Fig. 3. Orientation of the field vector, E, of circularly polarized waves at several values of ~t, with the sense of rotation indicated. In (a) Ey leads E, (right-handed, traditional optics), and in (b) Ev lags Ex (left-handed).

32

R.T. Holm

contrast, E rotates from the positive x axis toward the positive y axis in Fig. 3b.

Oot+ #) Form Let us now repeat the above procedure for the (~ot+6) form. The component field equations from Eq. 9b for 6 = - Jr/2 are

Ex = Eo cos(cot- qz)

(12a)

Ey = Eo sin(cot- qz),

and the equations for 6 = + zr/2 are

Ex = E0 cos(cot- qz)

Ey = -Eo sin(cot- qz).

(12b)

We see that Eq. 12a is identical to Eq. 11b and that Eq. 12b is identical to Eq. lla. Thus, it is apparent that Fig. 2b (Ey lags Ex) and Fig. 3b are applicable to the (cot + 6) form when 6 = - Jr/2. Likewise, Fig. 2a (Ey leads Ex) and Fig. 3a are applicable to the (cot + 6) form when 6 = + Jr/2. We suspect that the reader feels befuddled by now. The crux of the analysis can be summarized in two statements. (1) If the phase term has the same (opposite) sign as cot, then Ey leads (lags) Ex (see the discussion between Eqs. 8 and 9). (2) If Ey leads (lags) Ex, then the sense of rotation of E is from the positive x axis toward the negative (positive) y axis [35].

3

Spatial Visualization In addition to the temporal pictures given in Figs. 2 and 3, many authors also present a spatial visualization. Let us do so at cot=0. Rather than depicting in detail the four combinations obtained from the (cot + 6) forms with 6 - _+ Jr/2, it will suffice to consider only one combination, let us say the (~ot-6) form with 6 = - :r/2.

33

2. Convention Confusions X

Y

Fig. 4. Spatial representation (snapshot) of a circularly polarized wave propagating in the + z direction. E, and E~ are the component fields; E is the resultant field, in the form of a right-handed helix.

The component fields are Ex-Eocos(qz)

(13)

E,, = E, sin(qz). The tip of the vector representing the resultant field, E, traces out a helix whose axis coincides with the z axis. The helix, along with the component fields, is shown in Fig. 4. The helix has the same pitch direction as a righthanded screw. The (~0t-6) form with 6-- + ~/2 gives a left-handed helix. As one would expect (by now, we hope), the (~ot + 6) form with 6 = + ~/2 and 6 = - ~ / 2 results in right-handed and left-handed helices, respectively. Note that the handedness of a helix is independent of an observer's viewpoint--a right-handed screw looks like a right-handed screw, no matter from what direction you look at it. The temporal behavior of the helix has been the subject of some confusion. Several authors [36-38] have stated that the helix rotates as time increases, whereas others [39-41] point out that the helix moves forward without rotating. Since both scenarios cannot be correct, which one is? Figure 5 shows six snapshots of a right-handed helix at ~ot- ~ for the first snapshot and incrementing by ~/4 for each succeeding snapshot. The spatial patterns of the component and resultant waves are shown between z = 0 and z = 22. In addition, the resultant E vector lying in the x - y plane at z - ( 9 / 8 ) 2 is shown. If the helix were to rotate as it moved forward, then the resultant E vector at a fixed point (say, at z = (9/8)2) would remain fixed in space. Any E vector in front of the x - y plane at z - (9/8)2 would rotate into the position shown in the top picture in Fig. 5 as it passed through that plane. The helix would appear to be passing through a hole, located at the tip of the E vector, in the x - y plane. Therefore, in order for

34

R.T. Holm

X

X

Z

Y

E

Y

X

g__z

Y X

Z

X

Fig. 5. Snapshots of a propagating circularly polarized wave and its components. The snapshots, taken every Jr/4w of a second, show the wave between z = 0 and z = 22. The field vector, E, at z = 9 2 / 8 rotates clockwise, as viewed by looking toward the source ( - z direction).

2. Convention Confusions

35

the resultant E vector to rotate in a plane, the helix must move forward without rotating, as demonstrated in Fig. 5. As the helix moves forward a distance of 2, it scribes out a complete circle in any x - y plane.

Terminology The terminology used to classify circular polarization varies with country and field of science and has changed with time [42]. To begin with, there are two ways to view a circularly polarized wave: opticists and physicists prefer to look into the oncoming wave (that is, toward the source); and radio and microwave engineers usually look at the receding wave (that is, in the direction of propagation). Thus, opticists and physicists say that E is rotating clockwise in Fig. 3a and counterclockwise in Fig. 3b, whereas engineers say just the opposite. As the reader looks at Fig. 3, the + z axis is normal to the page and pointing toward the reader. Therefore, the view corresponds to the optics convention of looking toward the source. Engineers would be behind the figure looking out at the reader. When explaining the convention you use to define clockwise and counterclockwise, it is important to be specific and to avoid ambiguous phrases, such as "as seen looking down the beam." Which direction is down? The traditional optics convention refers to the clockwise (counterclockwise) rotating field E as being right (left) circularly polarized. One reason for this is that the spatial pattern for a right-handed (left-handed) wave would then be a right-handed (left-handed) helix, even though polarization is supposed to be referenced to the temporal variation. Of course, the helix serves as a good mnemonic, since its sense of inclination is invariant to the point of observation. Engineers usually reverse the definitions for the handedness of circularly polarized waves [34, 43-45]. For them, the E vector of a right-handed wave rotates clockwise (as seen looking in the direction of propagation), but has a left-handed helix. This definition is appropriate for a right-handed, helical-beam antenna, which receives right circular (left-handed helix) polarization [45]. As everyone knows, a light wave has a linear momentum p. In addition, a circularly polarized wave also carries an angular momentum L [46-51]. The direction of angular momentum is obtained from the right-hand rule. When the right hand curls around the direction of propagation with the fingers pointing in the direction that E is rotating (see Fig. 3), the thumb points in the direction of the angular momentum. Thus, the linear and angular momenta of a left circularly polarized wave (traditional optics definition) point in the same direction, whereas they point in opposite directions for a right circularly polarized wave. The direction that the angular momentum of the wave points determines whether electron or

4

36

R.T. Holm

Eih

------..........

Erh

B

Fig. 6. Schematic representation of (a) electron and (b) hole cyclotron resonance. The charge carriers revolve in a plane normal to the applied dc magnetic induction, B. The election and hole angular momenta are L and L÷, respectively. The angular momenta of the left and right circularly polarized waves are L~hand Lrh, respectively, and the linear momenta are Plh and Prh.

hole cyclotron resonance is excited in a semiconductor [52]. As depicted schematically in Fig. 6, left and right circularly polarized waves are propagating in the direction of the applied dc magnetic induction, B. In Fig. 6a, electrons revolve counterclockwise around B (looking against B), and in Fig. 6b, holes revolve clockwise around B. Electron and hole angular m o m e n t a are parallel and antiparallel to B, respectively. In order to satisfy selection rules [52], the angular m o m e n t u m of a circularly polarized wave must point in the same direction as that of the charge carrier. Therefore, electrons absorb left circular polarization, and holes absorb right circular polarization. Modern physics has also seen fit to reverse the meaning of right-handed f r o m that of traditional optics. The convention of particle physics is that both the linear m o m e n t u m and the angular m o m e n t u m of a right-handed

2. Convention Confusions

37

wave point in the same direction. Thus, the wave represented in Fig. 3b is right-handed in modern physics.

FRESNEL'S AMPLITUDE REFLECTION COEFFICIENTS

Augustin Fresnel's derivation in 1823 of the amplitude reflection coefficients of light waves is a classic case of knowing the answer and working backwards [53]. Working nearly fifty years before Maxwell published his electromagnetic theory, Fresnel was fettered by a bogus concept known as the luminiferous ether (aether). Making an analogy between sound and light, early scientists had postulated the existence of the ether. Since sound could not propagate in vacuum, it seemed logical that light could not either, that it needed some medium which filled all space out to the farthermost star [54]. Even though common sense (at that time) dictated the existence of the ether, common sense was noticeably absent when the ether was endowed with some rather remarkable properties [55]: it must be millions of times more rigid than steel to transmit waves having the frequency and velocity of light; it offers no resistance to the passage of the earth and planets around the sun; and it is composed of extremely minute •particles, far smaller than atoms. Working within the elastic-solid theory, Fresnel adjusted his concept of the ether until he was able to derive reflection formulas that agreed with experiment [56]. Fresnel knew that something was vibrating, and whatever it was, the vibration was transverse to the direction of propagation. H e also assumed that the vibration was normal to the plane of polarization (the q - H plane). He finally based his concept of the elastic-solid theory on the following three tenets [57]: (1) Conservation of energy--the energy of the incident wave is equal to the sum of the energies of the reflected and transmitted waves. (2) Uniform elasticity--the densities of the ether in each medium could be different, but not the elasticity (the ether is incompressible). (3) Continuity of displacement--there is no tangential slipping of the ether across the interface. Using these principles and without any knowledge of the electromagnetic character of light waves, Fresnel was able to derive the correct formulas. Since Maxwell's time, however, Fresnel's reflection coefficients have been derived in the more familiar way of using the E and H fields and their associated boundary conditions. In 1890, Otto Wiener demonstrated that a node in the electric field exists at the surface of a mirror, indicating that the electric field reverses phase upon reflection at normal incidence [58, 59].

V

38

R.T. Holm

Thus, the reflection coefficients are usually derived in terms of the electric field. The standard procedure is to resolve the electric field of the incident wave into two orthogonal, linearly polarized components, one normal to the plane of incidence (Es), and the other parallel (Ep). These two components are then treated as independent cases. For both the normal case and the parallel case, three waves are involved--incident, reflected, and transmitted--and each of the three waves can be represented by an E vector that can point in one of two opposite directions. The first step is to pick the direction for the E vector of the incident wave. This choice may depend on many factors, including type of experiment, personal preference, academic heritage, and so on. The second step is to select the direction for the reflected (and transmitted) E vector. Hindsight usually influences this choice. We do not know a priori in which direction the E field of the reflected wave will point; indeed, its direction depends on the relative refractive indices of the two contiguous media at the interface. It does not matter which direction we choose because the reflection coefficient will ultimately provide the correct direction for the reflected E field relative to that of the incident wave. Since each field can point in one of two opposite directions, there are eight possible arbitrary ways to arrange the three electric fields at the interface for both the normal and parallel cases. Fortunately, all eight configurations of each case have not been used; but those used have been enough to cause confusion. Luckily, some are redundant in that they give the same expressions for the reflection coefficients [60]. In general, reflection must be treated as a three-dimensional problem. Thus, an orthographiclike drawing aids the reader in visualizing the field vectors in relation to the plane of incidence and the reflecting surface. Over the past forty years or so, several excellent variations [61-68] have been presented. Figure 7 is typical of the type presented to depict the relative orientations of the field vectors of the incident, reflected, and transmitted waves. The coordinate system has been chosen so that the reflecting surface is in the x - y plane and the plane of incidence is in the y - z plane. The angles of incidence and transmission, related by Snell's law, are 0i and 0,, respectively.

A

s Polarization

In Fig. 7a, the electric fields are normal to the plane of incidence. Such waves are usually denoted as s-polarized, transverse electric (TE), or _1_ waves. Almost every author has chosen Ei~, Er~(FC), and E,s to point in the same direction [69, 70], which is usually the + x direction. The subscripts, is, rs, and ts, designate the incident, reflected, and transmitted waves,

2. Convention Confusions

39

Z

Fig. 7. The conventions used to derive the Fresnel reflection coefficients. The field (E and H) and wave (r vectors of the incident (i), reflected (r), and transmitted (t) waves are shown for (a) s polarization, (b) p polarization, Fresnel convention, and (c) p polarization, Verdet convention.

40

R.T. Holm

respectively, for s polarization. The FC label attached to the reflected wave in Fig. 7a signifies the Fresnel convention. We use this label to emphasize that Er~(FC) is an arbitrarily selected vector from which one derives the Fresnel reflection coefficient. The actual reflected field may point in the same direction as the convention vector in Fig. 7a or in the opposite direction. Once the direction of E is chosen, the direction of H is determined by q x E = ~o~t0H. It must be pointed out that Fig. 7 is just a schematic representation of the actual situation. Although it is the field vectors located at the interface that satisfy the boundary conditions, the field vectors are drawn some distance from the interface for clarity [71]. In addition, the vectors represent peak values, rather than instantaneous values, and the direction each vector points is defined as the positive direction for that vector [72]. The Fresnel amplitude reflection coefficient for s polarization is [73]

Ers(FC)

?1i COS 0 i -- ~t COS 0 t

sin(0i- Ot)

~-- Ei'----------f---=hicosOi+htcosOt=-sin(Oi+Ot)'

(14)

where h~ and h t a r e the complex refractive indices of the incident and transmission media, respectively.

B

p Polarization

For the other case in which the electric field is parallel to the plane of incidence, a wave is denoted as a p-polarized, transverse magnetic (TH), or II wave. One of the oldest (and continuing) controversies in optics is associated with the sign of the reflection coefficient for p polarization. Fresnel felt that the tangential component of the displacement associated with the reflected wave should point in the same direction as that of the incident wave [74]. This means, in electromagnetic terms, that the tangential components of Eip and Erp(FC) point in the same direction, as shown in Fig. 7b. For this situation, the Fresnel amplitude reflection coefficient for p polarization is [75]

E~p(FC)

rp(F)=

E~p

h~ c o s 0 t - / ~ t c o s 0 i

tan(0~- 0t)

=t~ cos 0~+h~ cos 0~

tan(0~+ 0t)

(15)

Fresnel argued for this form because at normal incidence (Oi=O~ r~--rp(F). It was intuitively consistent because at normal incidence there

2. Convention Confusions

41

is no distinction between s and p polarizations. However, and herein lies the root of the controversy, at glancing incidence (0~--+90~ r ~ = - r p ( F ) . When Verdet published Fresnel's work, he changed the sign of rp because he felt that r~ should equal rp at glancing incidence rather than at normal incidence [76]. This is equivalent to reversing the direction of the E vector of the reflected wave, as shown by Erp(VC) in Fig. 7c, where the VC label signifies the Verdet convention. Thus, in this case, the form of the reflection coefficient for p polarization is [73]

Erp(VC) //t COS 0 i --/~i

COS 0 t

tan(O~- Ot)

?p(V) = Ei-----7 = fit cos Oi + t/i cos O, = tan(Oi + 0,)"

(16)

There is nothing wrong with rs = - rp(V) at normal incidence. Figures 7a and 7c do not correspond to the same convention at 0~ = 0 ~ so why should r~ necessarily equal rp? The reasons for choosing either Fig. 7b or 7c have changed over time. Now they are related to whether one relates the field vectors to a coordinate system tied to the sample surface or to a coordinate system localized with the field vectors themselves [77]. Fresnel's reflection formula for p polarization was not universally accepted in the beginning because it did not completely agree with experiment. (Presumably, the data were in error.)

Phase Change C Everyone knows that an electromagnetic wave "suffers a phase change" upon reflection. As expected, the various conventions discussed so far result in different phase changes. However, as we shall show, they all yield the same reflected Er field in the final analysis. There is, in addition, no consensus on the meanings of the expressions in phase and out of phase. We shall first give the pertinent mathematical expressions for reflection phase change. Next, we shall summarize some of the ideas about phase change that are encountered in the literature. Finally, we shall discuss the physical significance of phase change. The Fresnel reflection coefficients may be formally expressed in the form

~ - r~ + ir~'= Irs e io., !

9

f ! _____

~p= r p + Irp

Irp c lap ,

(17a)

(17b)

42

Iq. T. Holm

where

Irs, pl = [(r~,p)2+ (r~'.p)2]1/2

(18a)

/r" \ 68, p = tan-1 / r~,pS'--2P)"

(18b)

The corresponding electric fields for the reflected waves are given by

Ers(F) = ]rsleias(V)Eis

(19a)

grp(F , V) = ]Fp]ei6p(F'v)gip ,

(19b)

where Eis and Eip point as shown in Fig. 7. E~s(F) and Erp(F,V) are the actual reflected fields for s and p polarization, respectively, and are distinguished from the fields Ers(FC) and Erp(FC,VC) used to establish the reflection conventions in Fig. 7. The reflection phase changes (shifts), 6s and 6p, are usually thought of as the difference between the phases of the incident and reflected waves. However, this thought is somewhat misleading. Electric field, being a vector, has a magnitude and a direction. Equations like Eq. 19 account for the magnitude. The direction is implicit through the conventions expressed in Fig. 7. Therefore, it is more apropos to think of 6p as the difference between the phases of Erp(F,V) and Erp(FC,VC), with Erp(FC,VC) related to Eip through Fig. 7. Erp(FC,VC) acts as an intermediary between Erp(F,V) and Eip. In other words, Erp(F,V) is what Erp(FC,VC) looks like after rp(F,V) is taken into account. Similar points of view hold for 6s. Since the phase change also depends on whether one uses (n + ik) or ( n - ik), there are two expressions for dis and four expressions for be. From Eqs. 14-18, one finds n + ik form

Os = tan-1 (--Snum) \ Sden /

n - ik form

tan-X (Snum] \Sdenf

1(Pnu___zm] 6p(V) = tan- \Pden/

i(--num)

alp(F) = tan- \ --Pden,]

tan-

('numl eden /

Pnum tan- 1 --eden/'

(20)

43

2. Convention Confusions

where (21a)

Snum = 2 3 COS 0i Sden = COS2 0 i -

(21b)

O~2 -- 3 2

P.um = 2 COS Oi[2ntkta- (nt -

k )31

Pden = (r/~ + k~) 2 c o s 2 0 i - a 2 - f12 Gt2

(21c) (21d)

= 71 { [ ( n2t - kt2 - s i n 2 0i) 2 + (2ntkt)2] 1/2 + n t2- kt2 - s i n 2 0i}

(21e)

2 - k 2 - sin 2 0i) 2 -t'- (2ntkt)2] 1/2- n 2 + k 2 + sin 2 0i} ,

(21f)

/32-- 89

where num and den refer to numerator and denominator, respectively [78], and assuming n~ = 1. Note that the equation for dp(F, n + ik) has a minus sign in both the numerator and the denominator. As we shall show, one must not inadvertently cancel out these minus signs. There is another, equivalent expression for the numerator of the reflection phase shift for p polarization; this alternative form was derived by Abel6s and is used by some authors [79-81]. Abel6s expressed the reflection coefficients in terms of effective refractive indices. For p polarization these a r e rti eff= 1/cos 0~ and/~t eff'-/~t/COS 0t, with n~ = 1. In the Abel6s method, Pnum (Abel6s), which is equivalent to Pnum in Eq. 21c, is given by

Pnum(Abel6s) = 2 cos 0i3(c~ 2 + f 1 2 - sin 2 0i).

(22)

Figure 8 presents the phase changes as a function of the angle of incidence, 0i, calculated from Eq. 20 for steel (2 = 589.3 nm, nt = 2.485, k t - 3.432) [82, 83]. The (n + ik) forms are in Fig. 8a and the ( n - ik) forms in Fig. 8b. Note that the four dp curves are different, as are the two ds curves. Figures 8a and 8b are mirror images of each other, and the dp curves in Fig. 8a differ by 180 ~ as do those in Fig. 8b. Care must be exercised to place ds and dp in the correct quadrant of the complex r~ and rp planes, especially when calculating phase changes with a computer. By convention, angles are measured positive in a counterclockwise direction starting from the positive real axis [84]. One must test separately the signs of the numerator and denominator to determine proper placement. If both r' and r" are positive (negative), d lies in quadrant I (III). If r' is positive (negative) and r" is negative (positive), d lies in quadrant IV (II). Thus, tan -1 (a/b) and tan -1 ( - a / - b ) differ by

44

R.T. Holm

I

I

I

I

I

I

(a) 300

8p(F,qz-wf) -I00

v

J

UJ

z

200

8s(qz-~t)

T

A -200

LIJ O0

180~ in front of surfoce

node

Fig. 11. Standing wave pattern in front of a surface with r = 1. The dependence of the nodal position on 6 is shown. The values of 6 are for the Fresnel convention with the (n- ik) form.

50

R.T. Holm

the surface, whereas if b were greater than 180 ~ the node would occur in front of the surface. Once again, we point out that Eqs. 26-29 are applicable only for the Fresnel convention and the ( n - ik) form. For the other cases in Fig. 8, Eqs. 26-29 have to be adjusted accordingly. However, all conventions result in the same standing wave pattern. The preceding discussion is mathematically rigorous, but does not completely comply with physical reality. First of all, IrJ never equals unity so that the nodes are minima in the standing wave pattern, rather than zeros. Second, phase shift does not vary over the entire range from 0 to 2~. Thus, in Fig. 10, an antinode occurs at the surface when reflection is from a less optically dense dielectric (k = 0); a node occurs when reflection is from a more optically dense dielectric. When the incident medium is transparent and the transmission medium is lossy (k4:0), the phase shift will lie between 0 and ~ (Fresnel convention, ( n - ik) form, normal incidence). Therefore, the node would appear to be behind the surface. Since phase shift cannot exceed ~, a node will not occur in front of the surface (within a distance of 2/4). Before proceeding to the next section, four points are worth making. First, reflection coefficients are characterized by two parameters: a magnitude and a phase. It is difficult to express meaningfully the values of both these parameters in an orthographic figure of the type shown in Fig. 7. A drawing such as in Fig. 9, in which E~ is drawn along the positive real axis and Er has the appropriate magnitude and phase angle, is more informative [a06]. The second point is that some authors express phase changes as negative values. In other words, they measure phase change clockwise from the positive real axis. Thus, if everything is done correctly, 6(clockwise)= 6(counterclockwise) - 360 ~ The right-hand ordinate in Fig. 8 measures the phase changes as negative quantities. The final Er field is the same--just the intermediate steps are different. The third point concerns phase change when k t - 0 . As might be surmised from Fig. 8, when the angle of incidence is less than the Brewster angle (and for ni < nt) [83,107], 0p(g, n - ik)= 360 ~ not 0 ~ as most books state. Likewise, above the Brewster angle, 0p(F, n + ik)= 360 ~ Finally, an interesting optical morsel. Abel6s [108-110] showed that at 0~= 45 ~ Irp] = Irsl2. In addition, for the Verdet form of r v, 6p = 26s. This is true in general for reflection from any semi-infinite medium.

Vl

NOMENCLATURE

Nomenclature is not a burning issue, but is worthy of mention. The distinctions between reflection, reflectance, and reflectivity are subtle [111-

2. Convention Confusions

51

115] but are favored by many authors. Reflection is the act or process by which the incident wave is redirected back into the incident medium. Reflectance is a measured value. It is the fractional amount of the incident energy or intensity that is reflected from the surface. Reflectivity refers to a fundamental or characteristic property of a material. It is the theoretical reflectance that a semi-infinite medium would have. Applying the -ion, -ance, and -ity suffixes to the words transmit and absorb yields transmission, transmittance, and transmissioity, and absorption, absorptance, and absorptioity. These words have the same usage as the corresponding words derived from reflect.

CONCLUDING REMARKS

Although we did not cover such specific topics as Jones matrices, Poincar6 sphere, qJ and A in ellipsometry, and so forth [2], we have presented the fundamental conventions dealing with electromagnetic wave propagation. It is important to recognize that conventions are a matter of definition or convenience. One must be able to separate physics from conventions and one convention from another. An amusing example of the misunderstandings that can arise is presented by a series of papers (and counter-papers), which discusses the "proper way" to account for reflection phase changes in interferometric measurements [116-122]. Since the various authors involved were using different conventions, they did not agree on the proper way and made scathing comments about each other. Finally, in an effort to end the controversy, the editors of Optics and Spectroscopy stated that they " . . . consider the discussion of phase shifts during reflection of light to be closed [117]." Of course, the controversy continues. Even though we do not advocate a particular set of conventions, we do have one recommendation. Textbook authors should devote more time to the convention issue. They should point out the various conventions that a student is likely to encounter. How can you determine which conventions an author uses if he fails to mention them? Look for obvious clues. Try to find equations, figures, tables, and so on, in which either exp(kot) or exp(-i~ot) is used. Are optical constants listed as n - i k or n + ik? Is rp given in the Fresnel or Verdet convention? Review papers may by their very nature contain mixed conventions, especially when they quote from the original papers. For example, Weinberg [123] states that a metal is characterized by f~ = n - i k but then presents phase changes in an (n + ik) form. Also, he presents dielectric constants in the (a' + ia") form.

VII

52

R.T. Holm

The two watchwords are convention and consistency. When you write a paper, either state your conventions explicitly or refer to a paper that does. Regardless of which set of conventions you choose, use them consistently throughout [3]. ACKNOWLEDGMENTS The author wishes to express his thanks and appreciation to J. M. Bennett, D. M. Roessler, E. D. Palik, A. S. Glass, W. R. Hunter, and P. J. McMarr for valuable comments and suggestions. It is a pleasure to acknowledge the effort by M. E. Dixon in typing the manuscript. REFERENCES 1. J. Z. Buchwald, "The Rise of the Wave Theory of Light," University of Chicago Press, Chicago (1989). 2. R. H. Muller, Surf. Sci. 16, 14 (1969); P. S. Hauge, R. H. Muller, and C. G. Smith, Surf. Sci. 96, 81 (1980). 3. J. M. Bennett and H. E. Bennett in "Handbook of Optics," (W. G. Driscoll, ed.), McGraw-Hill, New York, 10-1 (1978). 4. "The International System of Units (SI)," NBS Special Publication 330 (1986). 5. "Units of Measure," ISO Standards Handbook 2, 1st ed., International Organization for Standardization, Switzerland (1979). 6. R. A. Nelson, "SI: The International System of Units," American Association of Physics Teachers, Stony Brook, New York (1981). 7. M. Abraham, "The Classical Theory of Electricity and Magnetism," 2nd ed., Blackie & Son, London, 158 (1950). 8. J. D. Jackson, "Classical Electrodynamics," 2nd ed., John Wiley and Sons, New York, 811 (1975). 9. D. R. Frankl, "Electromagnetic Theory," Prentice-Hall, Englewood Cliffs, N.J., 147 (1986). 10. J. B. Marion and S. T. Thornton, "Classical Dynamics of Particles and Systems," 3rd ed., Harcourt Brace Jovanovich, New York, 476 (1988). 11. See Ref. 9, p. 260. 12. Many books use k for both the wave number and the imaginary part of the refractive index. Be careful. 13. L. I. Schiff, "Quantum Mechanics," 3rd ed., McGraw-Hill, New York, 20 (1968). 14. A. V. Sokolov, "Optical Properties of Metals," English ed., Elsevier, New York, 20 (1967). 15. Some authors define the refractive index as h = n(1 + be) or n ( 1 - &), where ~c is the index of absorption (not to be confused with the absorption coefficient, a = 2wk/c). Also, n and k are referred to as the optical constants, even though neither is constant. 16. H. C. Chen, "Theory of Electromagnetic Waves," McGraw-Hill, New York, xiii (1983). 17. S. S. Mitra in "Handbook of Optical Constants of Solids," (E. D. Palik, ed.), Academic Press, New York, 213 (1985). 18. E. Spiller, Appl. Opt. 23, 3036 (1984). 19. N. J. Harrick, "Internal Reflection Spectroscopy," John Wiley and Sons, New York, (1967); F. M. Mirabella and N. J. Harrick, "Internal Reflection Spectroscopy: Review and Supplement," Harrick Scientific Corp., Ossining, N.Y. (1985). 20. See Ref. 16, p. 93. 21. P. Yeh, "Optical Waves in Layered Media," John Wiley and Sons, New York, 14 (1988).

2. Convention Confusions

53

22. E. Hecht, "Optics," 2nd ed., Addison-Wesley, Reading, Mass., 270 (1987). 23. M. Born and E. Wolf, "Principles of Optics," 6th ed., Pergamon Press, London, 28 (1980). 24. W. A. Shurcliff, "Polarized Light: Production and Use," Harvard University Press, Cambridge, Mass., 6 (1962). 25. D. Clark and J. F. Grainger, "Polarized Light and Optical Measurement," Pergamon, Oxford, 173 (1971). 26. D. Clark, Appl. Opt. 13, 3 (1974); ibid. 13, 222 (1974). 27. E. Mach, "The Principles of Physical Optics," republication of 1926 English translation, Dover Publications, 186 (1953). 28. T. Preston, "The Theory of Light," 4th ed., Macmillan and Co., London, 25 (1944). 29. See Fig. 8.21 in Ref. 22. 30. See Ref. 27, p. 190. 31. See Ref. 1, p. 41. 32. F. L. Pedrotti and L. S. Pedrotti, "Introduction to Optics," Prentice-Hall, Englewood Cliffs, N.J., 341 (1987). 33. For an algorithm to generate perspective drawings, see R. E. Myers, "Microcomputer Graphics," Addison-Wesley, Reading, Mass., Chapter 6 (1982). 34. "IEEE Standard 145-1983, IEEE Standard Definitions of Terms for Antennas," IEEE Trans. Antenna Propagar AP-31, Part II, 1 (1983). 35. The sense of rotation is determined by rotating the phase-leading component toward the phase-lagging component. See C. A. Balanis, "Advanced Engineering Electromagnetics," John Wiley and Sons, New York, 163 (1989). 36. M. V. Klein, "Optics," John Wiley and Sons, New York, 487 (1970). 37. R. F. Harrington, "Time-Harmonic Electromagnetic Fields," McGraw-Hill, New York, 46 (1961). 38. J. Strong, "Concepts of Classical Optics," W. H. Freeman and Company, San Francisco, 28 (1958). 39. See Ref. 24, p. 5. 40. See Ref. 25, p. 19. 41. J. A. Kong, "Electromagnetic Wave Theory," John Wiley and Sons, New York, 37 (1986). 42. S. Shibuya, "A Basic Atlas of Radio-Wave Propagation," John Wiley and Sons, New York, 14 (1987). 43. J. D. Kraus, "Electromagnetics," 3rd ed., McGraw-Hill, New York, 497 (1984). 44. P. A. Rizzi, "Microwave Engineering," Prentice-Hall, Englewood Cliffs, N.J., 36 (1988). 45. See Ref. 35, p. 158. 46. See Ref. 22, p. 275. 47. R. H. Webb, "Elementary Wave Optics," Academic Press, New York, 80 (1969). 48. R. W. Ditchburn, "Light," 3rd ed., Academic Press, New York, 638 (1976). 49. R. P. Feynman, R. B. Leighton, and M. Sands, "The Feynman Lectures on Physics," Vol. 1, Addison-Wesley, Reading, Mass., 33-10 (1963). 50. See Ref. 36, p. 488. 51. F. H. Read, "Electromagnetic Radiation," John Wiley and Sons, New York, 37 (1980). 52. B. D. McCombe and R. J. Wagner in "Advances in Electronics and Electron Physics," (L. Marton, ed.), Vol. 37, Academic Press, New York, 1 (1975). 53. E. Whittaker, "A History of the Theories of Aether and Electricity," Vol. 1, Philosophical Library, New York, 125 (1951). 54. See Ref. 28, p. 29. 55. R. W. Wood, "Physical Optics," 3rd ed., Macmillan, New York, 4 (1934).

54

R.T. Holm

56. A. Schuster, "An Introduction to the Theory of Optics," 3rd ed., Longmans, Green, and Co., New York, 241 (1924). 57. See Ref. 28, p. 360. 58. F. A. Jenkins and H. E. White, "Fundamentals of Optics," 4th ed., McGraw-Hill, New York, 244 (1976). 59. See Ref. 55, p. 211. 60. G. Friedmann and H. S. Sandhu, Am. J. Phys. 33, 135 (1965). 61. F. W. Sears, "Optics," 3rd ed., Addison-Wesley, Reading, Mass., Fig. 7-3 (1958). 62. B. Rossi, "Optics," Addison-Wesley, Reading, Mass., Figs. 8-9, 8-10 (1957). 63. See Ref. 22, Figs. 4.20 and 4.21. 64. W. N. Hansen, J. Opt. Soc. Am. 58, 380 (1968). 65. D. T. Paris and F. K. Hurd, "Basic Electromagnetic Theory," McGraw-Hill, New York, Fig. 8-5 (1969). 66. See Ref. 32, Figs. 23-1 and 23-2. 67. R. I. Sarbacher and W. A. Edson, "Hyper and Ultrahigh Frequency Engineering," 8th printing, John Wiley and Sons, New York, Figs. 4-4, 4-5 (1950). 68. D. R. Corson and P. Lorrain, "Introduction to Electromagnetic Fields and Waves," W. H. Freeman and Co., San Francisco, Figs. 11-4, 11-5 (1962). 69. Three books (Refs. 61, 67, 70) present figures in which Ers points in the opposite direction. However, the expression for the reflection coefficient given by Refs. 61 and 70 corresponds with the convention of Fig. 7a. 70. A. B. Winterbottom, "Optical Studies of Metal Surfaces," Kgl. Norske Vedenskab. Selskabs Skrifter, Trondheim, Norway, 15 (1955). 71. See how much easier it is to visualize Fig. 7-3 than Fig. 7-4 in Ref. 61. 72. See Ref. 61, p. 167. 73. See Ref. 22, p. 94. 74. See Ref. 28, p. 361. 75. See Ref. 68, p. 363. 76. See Ref. 25, p. 55. 77. See Ref. 14, p. 31. 78. L. Ward, "The Optical Constants of Bulk Materials and Films," Adam Hilger, Bristol, 10 (1988). 79. F. Abel,s, in "Progress in Optics," (E. Wolf, ed.), Vol. 2, North-Holland, Amsterdam, 251 (1963). 80. See Ref. 3, p. 10-10. 81. P. H. Berning, in "Physics of Thin Films," (G. Hass and R. E. Thun, eds.), Vol. 1, Academic Press, New York, 69, (1963). 82. F. A. Jenkins and H. E. White, "Fundamentals of Physical Optics," 1st ed., McGraw-Hill, New York, 402 (1937). Note that Fig. 180 and Eqs. (18n) and (18o) do not appear in later editions. 83. See Ref. 3, p. 10-12. 84. Typically, x is chosen to be the real axis and counterclockwise rotation is from the positive x axis toward the positive y axis. 85. In a calculation, phase angles must be expressed in radians, not degrees. See H. Anders, "Thin Films in Optics," English ed., The Focal Press, London, 24, footnote (1967). 86. R. S. Longhurst, "Geometrical and Physical Optics," 2nd ed., John Wiley and Sons, New York, 471 (1967). 87. Since we are showing the actual reflected field vectors, we have dropped the F and V labels. 88. See Ref. 48, p. 514. 89. R. M. A. Azzam and N. M. Bashara, "Ellipsometry and Polarized Light," North-Holland, Amsterdam, 67 (1977).

2. Convention Confusions

55

90. See Ref. 3, p. 10-20. 91. See Ref. 58, p. 528. 92. H. E. Bennett and J. M. Bennett, in "Physics of Thin Films," (G. Hass and R. E. Thun, eds.), Vol. 4, Academic Press, New York, 66 (1967). 93. A trivial but not necessarily inconsequential point is that no author (that we know of) tells the reader where to put his feet when looking into the oncoming wave. Left and right, as well as up and down, are relative to the body's orientation. Thus, the reader must be able to position (in his mind, at least) his body correctly to a sample surface, plane of incidence, optical element, or whatever. If the viewer stands on a plane parallel to the plane of incidence in Fig. 10 and looks into the wave, either incident or reflected, the y axis points up and the x axis points to the right. 94. See Ref. 22, p. 100. 95. See Ref. 25, p. 44. 96. See Ref. 58, p. 268. 97. See Ref. 35, Fig. 5-22 and Eq. (5-90a). 98. See Ref. 25, p. 48. 99. See Ref. 35, p. 236. 100. See Ref. 43, p. 519. 101. See Ref. 42, pp. 18 and 53. 102. G. W. DeBell, SPIE Proc. 140, 2 (1978). 103. See Ref. 43, p. 439. 104. E. E. Wahlstrom, "Optical Crystallography," 5th ed., John Wiley and Sons, New York, 154 (1979). 105. See Ref. 58, p. 242. 106. J. M. Bennett, J. Opt. Soc. Am. 54, 612 (1964). 107. R. H. Muller, in "Advances in Electrochemistry and Electrochemical Engineering," (R. H. Muller, ed.), Vol. 9, John Wiley and Sons, New York, 167 (1973). 108. F. Abel6s, C.R. Acad. Sci. 230, 1942 (1950). 109. R. M. A. Azzam, Opt. Acta 26, 113 (1979); ibid. 26, 301 (1979). 110. See Ref. 3, p. 10-86. 111. J. F. Snell, in "Handbook of Optics," (W. G. Driscoll, ed.), McGraw-Hill, New York, 1-1 (1978). 112. D. L. MacAdam, J. Opt. Soc. Am. 57,854 (1967). 113. G. Hass, J. B. Heaney, and W. R. Hunter, in "Physics of Thin Films," (G. Hass, M. H. Francombe, and J. L. Vossen, eds.), Vol. 12, Academic Press, New York, 1 (1982). 114. E. D. Palik and J. K. Furdyna, Rept. Prog. Phys. 33, 1193 (1970). 115. J. D. Rancourt, "Optical Thin Films," Macmillan, New York, 12 (1987). 116. I. N. Shklyarevskii and N. A. Nosulenko, Opt. Spec. 14, 127 (1963). 117. I. N. Shklyarevskii and V. K. Miloslavskii, Opt. Spec. 12, 449 (1962). 118. L. Dunajsky, Opt. Spec. 12, 448 (1962). 119. I. N. Shklyarevskii, N. A. Vlasenko, V. K. Miloslavskii, and N. A. Noculenko, Opt. Spec. 9, 337 (1960). 120. N. Y. Gorban and I. A. Shaikevich, Opt. Spec. 11,404 (1961). 121. P. E. Ciddor, Opt. Acta 7, 399 (1960). 122. I. N. Shklyarevskii, Soo. Phys. Tech. Phys. 1,327 (1956). 123. W. H. Weinberg, in "Solid State Physics: Surfaces," (R. L. Park and M. G. Lagally, eds.), Vol. 22, Academic Press, New York, 22, Fig. 26b (1985).

Chapter

3

Methods for Determining Optical Parameters of Thin Films E. PELLETIER Laboratoire d'Optique des Surfaces et des Couches Minces Unite Associee au CNRS Ecole Nationale Superieure de Physique de Marseille Domaine Universitaire de St. Jer6me Marseille, France

I. Introduction II. An Example of Comparison between the Different Techniques of Index Determination II1. Principle Methods of Index Determination IV. Sensitivity and Accuracy of the Determination Methods V. Overlapping and Tests of Validity of the Model VI. Results and Conclusion References

57 58 60 64 67 71 71

INTRODUCTION

The aim of this chapter is to give an overview of the different methods used for determining the parameters that optically characterize a thin film. We limit ourselves to the case of a single layer deposited on a substrate, the properties of which are well known, and the materials involved are exclusively those generally used for optical multilayer coatings. Practically, these coatings can be used in the ultraviolet, visible, or infrared regions of the spectrum (lOOnm to lO0~m). In a filter some wavelengths are transmitted, and others are completely reflected, according to the characteristics of the layers that constitute this optical filter. In a general way, each layer is characterized by its refractive index n, its extinction coefficient k, and its thickness d [1]. For the production of optical filters, we will deal more precisely with the materials in their transparent region, that is, the spectral range in which the extinction coefficient is very weak. HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS II

57

Copyright 9 1991 by Academic Press. All rights of reproduction in any form reserved. ISBN 0-12-544422-2

I

58

E. Pelletier

Concerning the optical characterization of a single film, we can find in the literature numerous references with particularly exhaustive review articles. It is impossible to give an analysis of these works in a short chapter. However, it seems useful to display the evolution of the problems and the techniques used to solve them for some decades. In 1963, Abel6s [2] gave a general review of the work done by different groups in the world. Some years later, with the introduction of computers, data processing greatly improved, and numerous ways were found to ease the acquisition of experimental results and the characterization of thin films. On this subject, we can refer to Liddell's book in which a complete chapter is devoted to the question of index determination [3]. At present, with the development of very high-performing optical filters, a more precise determination of the index values is of great importance. What kinds of methods can meet these requirements? What progress has been made recently and how can we go further? We will try to give answers to these questions.

AN EXAMPLE OF COMPARISON BETWEEN THE DIFFERENT TECHNIQUES OF INDEX DETERMINATION

To have access to the values of refractive indices of thin layers, we will attempt to exploit directly the optical measurements made on these layers. Different techniques can be of potential use according to the transparency or nontransparency of the material. According to each case, measurements of transmission and reflection parameters or ellipsometric measurements of A, tlS angles can be used [4]. Each author has his preferred method, and it is not possible to distinguish clearly the advantages and disadvantages of each without a detailed study, at least from the point of view of accuracy. We are indebted to the Optical Materials and Thin Films technical group of the Optical Society of America, with J. A. Dobrowolski as leader, for an attempt to address this question [5]. Seven laboratories were given films of different thicknesses of ScO2 and Rh. The substrates used for the coatings were commercially polished fused silica (Dynasil). All the samples of each material were prepared in a single deposition run; the aim of the panelists (R. M. Azzam, J. M. Bennett, W. E. Case, C. K. Carniglia, J. A. Dobrowolski, U. J. Gibson, H. A. Macleod, and E. Pelletier) was to compare the optical constants obtained. The results are meaningful: Table I gives the optical constants for scandium oxide. We give here only the values and variances of refractive index for different wavelengths in the 400-750 nm spectral range. Two

3. Methods for Determining Optical Parameters of Thin Films 0"~ C'J

59

~J

1.

+1 +1 c q ~'b r~

c-a cq ~-

~= ~J

-H

-H

+1

-H

+1

-H

+1

-H

I

~a

Cq

~"

Cq Cq

~"

u

L, ~a ~a =. r~ ~a .=

.=. 0

f., ~a

~

U'b Cq C'4 ~ "

~a o..

0 ..

~ 0

-5

..= c'a

~-

0J

9 ej t..

v3 ~a ~a I.

~a

E

0 9 4=

0

0


.i.a

0.025

m

(D r r

. _ . _ .-- .-- . - : - : ~ 0.5

0

. . . . . . . . . . . . . . . . . . . 1 1.5 2

0

2.5

Change in Critical Angle (Degrees)

Fig. 2. Relative change in index of refraction, An~n, versus critical-angle uncertainty, A0c, for three different values of n. reflectance values as one would expect. Calculations with the computer algorithm indicate that a change in k of +0.003 from a nominal value of 0.01 leads to changes in reflectance values that vary from zero to +7% depending on the incidence angle. The fact that the effects due to variations in n and k are essentially separate means that it is very unlikely that ambiguities will occur in determining n and k values with A O I T E R .

100

!

- -

-

'-

....

-

"

....

9

i

"-

-"

'-

i

-

k I

75

/.~5..-"

////

.,-... ....... >, , B

9>--

50

cj

/

25

/ .

.

.

. ,

0

.

i

25

.

.

.

.

l

50

.

L

l

,,

,

i

75

Angle (Degrees)

Fig. 3. Reflectance versus incidence angle for n -0.7 and three values of k, the absorption index.

210

C

Marion L. Scott

Results at 58.4 nm and 30.4 nm

With the above inputs to the reflectance calculation, the n and k of the aluminum thin film can be varied to get the best fit to the measured X U V reflectance data. The error in the n and k determined from reflectance data can be estimated from the +_2% reflectance error in the following way: one adds 2% to each reflectance data point to generate a new (higher) reflectance curve. This new curve is again fit to determine n and k. One then subtracts from the original reflectance data points to generate a new (lower) reflectance curve, which is also fitted to determine n and k. The excursion of n and k from the original data fit is used as the error bar on n and k. The results of this fitting procedure for aluminum are given as the A O I T E R entries in Table I. In situ U H V - A O I T E R measurements on aluminum at other X U V wavelengths have not yet been performed.

V

COMPARISON WITH OTHER RESULTS

Other methods exist to determine the optical constants of materials in the XUV. Typically, these methods utilize reflection a n d / o r transmission values measured at normal incidence or at more than one incident angle, to infer the optical constants of the material.

A

A l u m i n u m at 58.4 n m

Using a combination of transmission and reflection measurements, Ditchburn and Freeman [6] determined the n and k of aluminum at various wavelengths. A linear interpolation of their values gives n = 0 . 6 9 and k = 0 . 0 2 3 at 58.4 nm. Madden, Canfield, and Hass [7] used a reflection versus time method to minimize the effects of oxidation in determining the optical constants of aluminum. The values quoted in this reference are n = 0 . 7 1 and k = 0 . 0 1 8 for 58.4 nm. In early work [8], Hunter used reflection measurements and determination of the critical angle to deduce the optical constants of aluminum to be n =0.72 and k =0.018 at 58.4 nm. In later work [9, 10], Hunter used transmission measurements to find k values between 0.018 and 0.023 with a best-fit curve for multiple-wavelength data passing through k =0.023 at 58.4 nm. Hass and Tousey [11] indicate that interference-fringe analysis gives n = 0 . 7 at 58.4 nm. They state that the k value for aluminum at this wavelength is less than 0.04; however, the aluminum reflectance calculations in this paper were performed with

9. Measurement of n and k in the XUV

211

n = 0.7 and k = 0.018. The extensive compilation and analysis of aluminum optical constant data by Smith et al. [12] indicates that n = 0 . 7 1 5 and k = 0 . 0 2 3 7 at 5 8 . 4 nm.

A l u m i n u m at 30.4 nm

B

Fomichev and Lukirskii [13] measured absorption coefficients of thin aluminum films in the wavelength range from 2.36 nm to 41 nm. The value of k linearly interpolated from this reference at 3 0 . 4 n m is k = 0 . 0 0 9 . Hunter [9, 10] used the critical-angle method to determine n and transmission measurements to determine k at wavelengths between 10 nm and 80nm. The approximate values from these references at 3 0 . 4 n m are n = 0.94 nm and k = 0.009 (these values were taken from the best-fit line on a graphical presentation of this multiple-wavelength data). The aluminum optical-constant data from the compilation by Smith et al. [12] indicates that n = 0.943 and k = 0.0079 at 30.4 nm.

Drude Free-Electron Model

This model utilizes only two parameters, the plasma critical wavelength and the electronic relaxation time, to characterize the optical constants of a TABLE I A Comparison of the Optical Constants of Aluminum a Source of data

n

k

58.4 nm AOITER Ref. [6] Ref. [7] Ref. [8] Ref. [9, 10] Ref. [11] Ref. [ 12] Drude model

0.700 + 0.005 0.69 + 0.02 0.71 + 0 . 0 2 0.72+0.015 0.72 _+0.02 0.70+0.02 0.715 + 0.02 0.72

0.010 + 0.002 0.023 + 0.012 0.018+0.009 0.018+0.009 0.023 + 0.01 0.018_+_0.020 0.0237 + 0.01 0.010

30.4 nm AOITER Ref. [13] Ref. [9, 10] Ref. [12] Drude model

0.944 _+0.002 -0.94 _+0.03 0.943 _+0.03 0.93

0.0055 + 0.0008 0.009+0.004 0.009 +_0.004 0.0079 _+0.004 0.001

" E r r o r s have been estimated by the author.

C

212

Marion L. Scott

given material over a wide range of wavelengths. The values of these two parameters that give the best fit [14] to available data on aluminum (in the U V and visible spectral regions) are 2crit-" 83.7 nm and r = 1.1 • 10 -15 sec. The optical constants can be calculated in the Drude model by means of e 1 . n. 2 . k 2. e2 = 2 n k =

1

2 T 2//(1 + 0) 2 z. 2)], [0)crit

(0)r)- 110)2critz.2/( l

0) crit2___4 : r N e 2 / m = (2JgC/2crit)2

+ 0)2 ,g2)],

(5)

(6) (7)

where c is the speed of light, N is the effective valence-electron density, e is the charge of an electron and m is its mass. The values for n and k calculated with the simple Drude model at 58.4 nm and 30.4 nm using the above values for the parameters are given in Table I along with a comparison of the various measured values discussed above.

D

Discussion The A O I T E R results given in Table I are very similar to previous results, except that the A O I T E R values obtained for k and the error bars on n and k are significantly smaller. The larger previous values of k are presumed to arise from the presence of some oxide deposited in the coating, surface-oxide layers, surface contamination, or impurities in the starting material. REFERENCES

1. J. G. Endriz and W. E. Spicer, Phys. Reo. B 4, 4144 (1971). 2. M. L. Scott, P. N. Arendt, B. J. Cameron, J. M. Saber, and B. E. Newnam, Appl. Opt. 27, 1503 (1988). 3. M. L. Scott (unpublished). 4. M. Born and E. Wolf, "Principles of optics," Macmillan, New York, 57 (1964). 5. D. F. Edwards, in "Handbook of Optical Constants of Solids," (E. D. Palik, ed.), Academic Press, Orlando, 547 (1985). 6. R. W. Ditchburn and G. H. C. Freeman, Proc. R. Soc. London, Ser. a 294, 20 (1966). 7. R. P. Madden, L. R. Canfield, and G. Hass, J. Opt. Soc. Am. 53, 620 (1963). 8. W. R. Hunter, J. Opt. Soc. Am. 54, 15 (1964). 9. W. R. Hunter, Appl. Opt. 21, 2103 (1982). 10. W. R. Hunter, in "Proceedings, International Colloquium on Optical Properties and Electronics Structure of Metals and Alloys," (F. Abeles, ed.), North-Holland, Amsterdam, 136 (1966). 11. G. Hass and R. Tousey, J. Opt. Soc. Am. 49, 593 (1959). 12. D. Y. Smith, E. Shiles, and M. Inokuti, in "Handbook of Optical Constants of Solids," (E. D. Palik, ed.), Academic Press, Orlando, 369 (1985). 13. V. A. Fomichev and A. P. Lukirskii, Opt. Spectrosc. 22, 432 (1967). 14. W. R. Hunter, J. Appl. Phys. 34 1565 (1963).

Chapter 10 Spectroscopic Ellipsometry in the 6-35 eV Region J. BARTH,* R. L. JOHNSON, t and M. CARDONA Max-Planck-lnstitut fQr Festk6rperforschung D-7000 Stuttgart 80 Federal Republic of Germany

I. Introduction II. The BESSY VUV Ellipsometer A. Basic Experimental Considerations B. Optimum Geometry C. Design of the Ellipsometer D. VUV Polarizers E. Photodiodes as VUV-Detectors II1. Theory A. Basic Formulas B. Calibration C. Error Analysis IV. Results and Comparison with Reflectance Spectra A. III-V Semiconductors B. II-VI Compounds C. Calcium Fluoride V. Summary Acknowledgment References

213 217 217 218 221 224 225 226 226 231 232 237 237 238 242 244 245 245

INTRODUCTION

The technique of ellipsometry has experienced a renaissance recently driven by the ever-increasing demand for rapid, nondestructive analysis of surfaces and thin films. Ellipsometry enables the optical constants of * Present address: Elektro-Optik GmbH, F6rdestrage. 35, D-2392 Glticksburg, Federal Republic of Germany. t Present address: II. Institut fiir Experimentalphysik, Universit~it Hamburg, Luruper Chaussee 149, D-2000 Hamburg 50, Federal Republic of Germany. HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS II

213

Copyright 9 1991 by Academic Press. All rights of reproduction in any form reserved. ISBN 0-12-544422-2

I

214

J. Barth et al.

materials to be determined with high accuracy and can therefore help to solve a wide variety of problems in different disciplines. The fact that ellipsometric measurements can be performed without a vacuum environment in the visible and near-UV regions is a definite advantage over other surface-science techniques for industrial applications. Ellipsometry is widely used in assessing the quality of semiconductor materials and oxide coatings and to characterize such diverse systems as polymers, liquid crystals, and Langmuir-Blodgett films. There are now a number of commercial ellipsometers available that exploit recent advances in detection systems and computer technology and allow precise and rapid measurement of the optical properties of thin films and surfaces in the I R - V I S - n e a r - U V spectral region (1.5-6.0 eV). The microscopic-imaging ellipsometer described by Beaglehole [1] is a particularly interesting instrument that enables ellipsometric measurements to be performed with a lateral resolution of about 3/~m. Given these developments and the incontestable advantages of ellipsometry for determining the optical constants of materials, it seems obvious to extend the spectral range for ellipsometry into the vacuum-ultraviolet (VUV) region. This expands the number of systems accessible for ellipsometric studies to include large-band-gap insulators and excitations from deeper valence bands and also from shallow core levels into the q.onduction bands. The best light source for spectroscopic ellipsometry in the VUV region is without doubt monochromatized synchrotron radiation because of its tunability, high intensity, and natural collimation. It is also highly polarized, although the degree of linear polarization is not as high as that which can be achieved in the visible region. The possibility of carrying out ellipsometry with synchrotron radiation was demonstrated by Schledermann and Skibowski in 1970 [2, 3] at the DESY synchrotron in Hamburg. Apparently, since then no further attempts have been made despite recent advances in the development of high brightness storage rings as much more powerful sources of synchrotron radiation. At present, reflectometry is the best-established technique for determining the optical constants in the VUV region. A detailed description of the method and data analysis has been presented by Hunter [4]. To perform such measurements, one simply measures the specularly reflected intensity from the sample using a collimated monochromatic beam of light, which has a well-defined state of polarization. Reflectometry can be performed throughout the entire VUV and X-ray regions, and by scanning the photon energy, a wealth of spectroscopic information can be obtained. The greatest uncertainty in such measurements is usually caused by variations in the intensity and spectral purity of the primary photon flux. At high energies diffuse scattering and geometrical-shadowing effects can introduce significant errors; however, by taking suitable precautions and using highly monochromatic synchrotron radiation, excellent data can be

10. Spectroscopic Ellipsometry in the 6-35 eV Region

215

obtained. In order to calculate the optical constants from reflectance data, it is necessary to make use of the Kramers-Kronig relation, which implies that data over a very wide energy range should be available. It is difficult to estimate the amount of uncertainty in the absolute values of the dielectric function introduced by the unavoidable extrapolation necessary for Kramers-Kronig analysis. A more complete discussion of these aspects can be found in the article by Smith [5]. The alternative method of multiple-angle reflectivity measurements is a time-consuming and laborious exercise [6]. In ellipsometry one measures the change in polarization state of the beam of light after nonnormal reflection from the sample. The polarization state is defined by two parameters, for example, the relative phase and relative amplitude of the orthogonal electric-field components of the polarized light wave. On reflection both electric-field components are modified in a linear way, so a single ellipsometric measurement provides two independent parameters. The name ellipsometry derives from the fact that linearly polarized light after oblique reflection becomes elliptically polarized due to the different reflection coefficients for s- and p-polarized light. By measuring the polarization ellipse, one can determine the complex dielectric function of the sample. The principles of ellipsometry were established about 100 years ago [7], but the technique only became practical with the development of the rotating-analyzer ellipsometer (RAE) [8], which makes use of digital computers to handle the rather unwieldy equations. The RAE measures continuously, so wavelength scanning or spectroscopic ellipsometry can be performed. A conventional ellipsometer consists of a light source and a rotatable polarizer, so that the plane of polarization can be inclined to the plane of incidence. The measurement geometry is shown in Fig. 1. Polarized light falls on the sample at an angle of incidence of about 70 ~ typically, and the reflected light is analyzed using a second rotatable polarizer and a suitable detector. The simplest ellipsometers use a single-wavelength light source and manual determination of the ellipsometric angles A and qJ (where A corresponds to the phase shift and tan ~p= ]~p/r=l,the amplitude ratio of the reflected components). The measurement procedure is to locate the minimum transmission through the optical system by alternately rotating polarizer and analyzer (or equivalently polarizer and compensator). The extinction condition determines both of the ellipsometric angles; thus, a single ellipsometric measurement provides two parameters. Furthermore, since it is the polarization state and not the intensity which is being measured, the results are unaffected by source intensity fluctuations. The fact that ellipsometry simultaneously determines the relative phases and amplitudes accounts for the high sensitivity of this technique. In particular, the direct measurement of the phase changes makes ellipsometry highly surface-sensitive, and it is possible to detect extremely thin adsorbed films

216

J. Barth et al.

9

7~reftected

ht

~

~~

/

surface .--

normal

Cho~ing t

Fig. 1. Measurement geometry for ellipsometry: the angle of incidence is ~0, and the angle between the plane of incidence and the plane of polarization is ~. The polarization ellipse of the reflected light is characterized by the ratio of the axes and the angle a0.

(down to 0.01 monolayer, with a good instrument). By assuming an appropriate model, the measured ellipsometric parameters can be evaluated in terms of a macroscopic refractive index with real and imaginary components. Alternatively, a single ellipsometric measurement can be used to determine the thickness and refractive index of transparent films. The interpretation of the measured changes in polarization in terms of optical functions is a problematic aspect of ellipsometry. The simplest approach is to apply the Fresnel equations and a two-phase model to the vacuum-solid interface. The Fresnel equations are strictly applicable only to an abrupt interface between homogeneous isotropic media. In reality, no materials are perfectly isotropic and, on a microscopic scale, no interfaces are abrupt; however, the two-phase model generally provides the most useful basis for interpreting ellipsometric data. The application of the twophase model to anisotropic media enables a "pseudodielectric function" to be calculated, which allows anisotropic samples to be characterized. A full determination of all the components of the dielectric tensor in anisotropic media is, in principle, possible by performing measurements at multiple angles of incidence for different orientations of the sample. Measurement of optical anisotropies is a challenging field of current research [9]. So far we have outlined some of the salient aspects of ellipsometry, which we can summarize as follows: spectroscopic ellipsometry is now a

10. Spectroscopic Ellipsometry in the 6-35 eV Region

217

well-established technique for determining the optical constants of bulk materials and thin films in the I R - V I S - n e a r - U V spectral region over the photon-energy range 1.5-6.0 eV. It is generally accepted to be a technique with high intrinsic precision and accuracy, which allows both real and imaginary parts of the dielectric function to be measured simultaneously. In the visible, conventional instruments can yield information on samples with absorption coefficients down to typically 10 4 c m -1, and extinction indices --~0.1. More details about conventional ellipsometry can be found in review articles [10, 11] and the proceedings of the ellipsometry conferences [12]. The purpose of this article is to introduce the concepts of VUV ellipsometry and to demonstrate that it is a viable technique to determine the optical constants of solids. There are important differences between ellipsometry in the visible and VUV regions. The fact that there are no perfect polarizers for the VUV means that all four components of the Stokes vector have to be considered in the data analysis. This, in turn, means that the full Mueller matrix notation has to be used in the theoretical description. On the experimental side it was necessary to develop new instrumentation and procedures to calibrate the imperfect polarizers and also to measure the polarization of the incident light. By making use of synchrotron radiation with its high degree of linear polarization and triple-reflection polarizers, we built an automated instrument that extends the spectral range of spectroscopic ellipsometry up to 35 eV [13]. We present data that demonstrate the usefulness of our instrument for measuring the optical constants of solids. By combining the data with theoretical calculations, new insights about the band structure of wideband-gap semiconductors and insulators have been gained.

THE BESSY VUV ELLIPSOMETER

II

Basic Experimental Considerations

A

There are a number of different optical configurations that can be used for ellipsometry. A typical system consists of a collimated unpolarized monochromatic light source, a polarizer, a compensator, the reflecting sample under investigation, the polarization analyzer, and a detector. The modern rotating-analyzer ellipsometer has a great advantage over the older null technique in that it can be readily automated to allow the wavelength to be scanned continuously, yielding dielectric-function spectra. In a rotating-analyzer ellipsometer, the state of polarization of the

218

J. Barth et al.

reflected light (ellipticity and orientation of the ellipse) is measured with a photodetector behind an analyzer, which rotates at a fixed frequency. Thus, the intensity at the detector has, as a function of analyzer rotation angle, a, the form of a sinusoidal curve on a dc background, which can be written as Idet = const(1 + a~ cos 2a + a2 sin 2a).

(1)

Fourier analysis yields the coefficients al and a2, which are used to determine numerically el(h~o) and ez(h(_D). The detected intensity is a function of 2a because the optical cycle of the rotating analyzer corresponds to half a mechanical cycle. The rotating analyzer is advantageous because of its higher speed of data collection, its high precision, and the fact that a compensator is not necessary. Practically all aspects of rotating-analyzer ellipsometry have been described by Aspnes [8, 10, 11, 14, 15], and our present VUV ellipsometer is based on his pioneering work. One obstacle to performing ellipsometric measurements in the VUV is the lack of suitably intense laboratory light sources. The brightest tunable VUV light is provided by specially constructed synchrotron radiation facilities. Synchrotron radiation (SR) has the additional advantage of being highly polarized. SR extends over a broad continuum from the IR to X-ray regions, so a monochromator has to be used to filter out a narrow bandwidth. The present VUV ellipsometer at the dedicated light source BESSY in Berlin is installed on the 2m Seya-Namioka beam line, which nominally covers the photon-energy range from 5-35 eV using two gratings. The photon flux falls off rapidly near the ends of the spectral range, so it is primarily the monochromator that limits the range of the ellipsometer. In spectroscopic ellipsometry the monochromator plays a particularly important role, since alignment has to be maintained while scanning the monochromator. The present system dispenses with a polarizer before the sample and uses the fact that synchrotron radiation is linearly polarized in the plane of orbit and that the optical components in the beam line enhance the degree of polarization.

B OptimumGeometry In ellipsometry, linearly polarized light is incident at an oblique angle on the surface under investigation. The reflected light is in general elliptically polarized due to the differences in the complex ~s and ~p reflection coefficients. The complex reflection ratio can be obtained di-

10. Spectroscopic Ellipsometry in the 6-35 eV Region

219

rectly from the ellipsometric angles ~p and A since, by definition, p = ?p/?s = tan ~p exp(iA). The reflection coefficients depend on the angle of incidence, the photon energy, and the material properties, so it is clear that different geometrical arrangements are required to obtain the best sensitivity in ellipsometry in different spectral regions. Furthermore, different optimum geometries are to be expected for such different samples as wide-band-gap semiconductors or adsorbates on metals. A versatile instrument could be conceived with a variable angle of incidence, but it would be difficult and expensive to attain the necessary precision in ultra-high vacuum. A theoretical analysis of the optimum experimental geometry is, therefore, a prerequisite for establishing a practical design of a VUV ellipsometer. General equations for optimizing the precision of rotating-analyzer ellipsometers have been given by Aspnes [14]. He found that for rotatinganalyzer systems with an ideal detector, the angle of incidence should be as close as possible to the "principal angle." At this angle of incidence, for which A = +:r/2, linearly polarized light is generally reflected as elliptically polarized light, but when the angle between the plane of polarization and the plane of incidence is equal to ~p, the reflected light is circularly polarized. For spectroscopic measurements over a given spectral range, minimum uncertainty is obtained as a compromise between the conditions that yield circularly polarized light and those that maximize the transmitted flux. It is interesting to note that the optimum precision in the idealdetector case is achieved when the measured Fourier coefficients al and a2 are zero, so the complex-reflectance ratio p is determined purely by instrument settings. The angle of incidence is one of the most important parameters in ellipsometry and has to be determined with at least the same accuracy as the ellipsometric angles. In the principal-angle-of-incidence ellipsometer [16, 17], the rotating analyzer is used as a null detector seeking circularly polarized light. The dielectric function of the sample is determined solely by the angle of incidence and the polarizer position. The best accuracy in the determination of the optical constants is obtained at the principal angle of incidence; the linearity of the detector becomes unimportant. Hence, accurate determination of the dielectric function requires high angular precision and good mechanical stability of the ellipsometer. Aspnes [14] applied the theory to the practical cases of silicon and gold surfaces measured with a rotating-analyzer ellipsometer and shot-noiselimited detector without using a compensator. He found that for an angle of incidence of 68 degrees, the optimum polarizer azimuth was 30 degrees over the energy range 2.5-5.5 eV. In general, the optimum sensitivity is obtained for an angle of incidence close to the principal angle (A = +:r/2) and an optimum polarizer azimuth equal to ~p (where tan ~p= I 1). We can make use of the fact that at high photon energies el~-1 and e2---0 to derive the optimum experimental

220

J. Barth et al.

geometry in the X-ray limit. In this case the optimum angle of incidence is 45 ~, and the optimum polarizer azimuth tends toward zero. Thus, we see that the optimum angle of incidence decreases from about 70 ~ in the visible region to 45 ~ in the XUV. Unfortunately, the reflectivity of the sample is reduced at the smaller angle of incidence, and this constitutes a fundamental limitation to performing ellipsometry in the soft X-ray region. The strong dependence of the precision obtainable in ellipsometry on the measurement geometry is interesting and in some ways nonintuitive. It is somewhat paradoxical that it is necessary to know the dielectric function of the sample before one can measure it with the greatest precision by ellipsometry. It is also difficult to appreciate that the same sample will require different geometries in different wavelength regions. It is instructive to plot the variation of el and e2 with energy as shown in Fig. 2 and Fig. 3 for InAs. The curve A-B in Fig. 2 corresponds to photon energies from 1.5-9 eV and the curve in Fig. 3 corresponds to photon energies 14-25 eV. The optimum sensitivity is obtained when the dielectric-function curve lies in the vicinity of the principal-angle curves shown as semicircles for angles of incidence of 67.5 ~, 56.25 ~, and 45 ~. These figures demonstrate clearly that for InAs, the optimum angle of incidence tends rapidly toward 45 ~

E2

I

I

I

0

10

20

30

20

10

0

-10

I

E

Fig. 2. The dielectric function of InAs from 1.5 eV (A) to 9 eV (B) plotted in the complex plane (Argand diagram). The semicircle corresponds to the values of el + ie2 for which A = - 90 ~ at an angle of incidence q~0= 67.5~

10. Spectroscopic Ellipsometry in the 6-35 eV Region I

I

221 I

3

%0o:56.25 ~

1 q::'o=Z'5~

o

1

0

1

2

e:~

Fig. 3. Argand diagram for the dielectric function of InAs from 14 eV (A) to 25 eV (B). The semicircles correspond to the values of el + ie2 for which A = - 90~ at angles of incidence of 56.25~ and 45~.

with increasing photon energy. Although the magnitudes of e I and e2 become smaller at higher energies, it is important to note that good precision can be achieved by measuring with the optimum geometry. As a compromise between optimum precision and mechanical stability, our ellipsometer provides two different angles of incidence. In the energy region 5 - 1 0 e V , the incidence angle is 67.5 ~ and a MgF2 Rochon polarizer is used. Above 10eV, a 45 ~ incidence angle produces better results for most materials and only triple-reflection polarizers can be used at these energies. In practice, the two fixed angles of incidence have been found to be quite adequate, since the optimum conditions do not have to be fulfilled stringently. The two angles also provide complementary information, which is useful when studying thin films and adsorbates.

Design of the Ellipsometer The basic design of the V U V ellipsometer is modular, which enables functions to be separated and individually modified and upgraded as the

C

222

J. Barth et

al.

system evolves. The present ellipsometer comprises of three vacuum chambers: the main UHV chamber (base pressure 10 -l~ mbar) around the sample manipulator, the analyzer chamber, and a preparation chamber with load-lock. In order to tilt the plane of polarization with respect to the plane of incidence, usually accomplished by turning the polarizer, the whole ellipsometer has to be rotated about the axis of the incoming light. As shown in Fig. 4, the sample is mounted on the precision manipulator in the center of the main UHV chamber. Samples are introduced via a rapid load-lock and exchanged with a transfer system and can be heated or cooled on the manipulator (77-900 K). Facilities are provided for cleaving samples and for evaporation in situ. The analyzer chamber can be readily removed and positioned at ports corresponding to angles of incidence of either 67.5 ~ or 45 ~. It contains the most important parts of the ellipsometer, that is, the rotating analyzer and the detector. The prealigned standardized polarizer assemblies can be easily interchanged and fit snugly inside a rotating drum, which is driven via a friction wheel by a tachogenerator-stabilized dc motor. The angular orientation of the analyzer is measured with an angle encoder (1000 pulses per revolution) mounted directly on the drum. The pulses from the angle sampte transfer and preparation

port~ rn._/ _ ~ viewport ~__~ ,/ .'~s. . . . ~ ~~~._L~ anatyzer chamber

~--X~~--

~


515

498 722

725

305B, (R)

TABLE h

KHzPO4; R and IR Values from Three References for Comparison Al (R)

B2 (R)

B1 (R)

(IRI)

(IR _L)

E(R)

[29]

[30]

[29]

[30]

[29]

[30]

[30]

[31]

[29]

[30]

360 514 918

365

156 479 570

155 473 540

80 174 *

100 183

60 178

52 185

75 95 113

96 116

2700

918

1366 1806 2390

386 510 1350

470

190 320 490 530 568 960 1145

[30]

[31]

192

86 102 122 152 206

90 106 123 152 200

533

526

540

#

1325 Above the Curie temperature (123 K), the crystal is paraelectric (tetragonal); below To, it is ferroelectric (orthorhombic).

328

Edward D. Palik

TABLE i

NH4HzPO4; R and IR Values from Two References for Comparison Bl (R)

A, (R)

B2

(IR[)

E(R)

[301

[301

[29]

[30]

[301 51 70

(R)

(IR L )

[291

[30]

[29]

[3ol

337

341

178

182

80

100

76

100

68

474

476

185

204

198

120

125 140 175

520 918 2700

[29]

#

541 1298 1750 2370

922

401

397 547 1320

172 268 390 470 538 568 960 1106

450

150 164 350

542

540

1338 Above the Curie temperature (148K), the crystal is paraelectric (tetragonal)" below To, it is antiferroelectric (orthorhombic).

TABLEj LiNbO3[18] 0=0

90 ~

90 ~

0

E(T)(R, IR J_ )

E(L)(R, IR _L)

AI(T)(R, IRI[)

A,(L)(R, IRI)

155 ~ 238 265

--------~ 198 243 ) 255 276 ~ - - - - - ~ 334 (

371 431 ~

9 275

,295

325

* 371 - - - - - - ~ 428 454 * - ~

582 668 743 - - - - - - - ~

~333

436 ) 633

668 739 880 *-------

876

14. Optical Parameters for the Materials

329

TABLE k A!203 [22]

0=0

90 ~

90 ~

Eu(T)(IR _L)

Eu(L) (IR _L)

A2u(T)(IR] )

A2u(L)(IR )

385 442 569 635

388 480 625 900

400 583

512 871

378 418 432 451 578 645 751

Eg (R) A~g Eg Eg Eg Alg Eg ,,,

TABLE I Se [2, 231

0=0

90 ~

90 ~

E(T)(R, IR _1_)

E(L)(R, IR • )

A2(T)(IR )

A2(L)(IR )

144 225

150 225

102

106

106 1145

90

96

237 A 1 ( e )

Te 92 144 120 A~ (R)

330

Edward D. Palik TABLE m SiO 2 [17] 0=0

90 ~

90 ~

E(T)(R, IR _1_)

E(L)(R, IR • )

A2(T)(IR [)

127 265

129 270

A2(L)(IR[)

364 ~---- ~ ~

394" 450

) 489 510*---) 699

697

547 789

777 ~

809

796 1076 1161

207 356 464 1085

387

403

> 1080 ---~1161 1236 * ~

1240

A~. (R) AI A1 A1

TABLE n As2S3; The Layer Symmetry Properties Dominate the R and IR Spectra [27]

EIIb

Ela

182 246 279 293 348 354 369 384

189

328 354 369 370 390

T

L

186 386

392

140 159 185 201 260 278 307 355 385

148 165 189 203

94 105 132 217 248 268

97 108 136 232 250

326 368 390

As2Se3 106 132 201 224 249

108 134 209 238

14. Optical Parameters for the Materials

331 TABLEo

CuO: Spectra Were Not Necessarily Obtained for E-field Polarization (IR) and/or Wave Vector (R) Parallel to the a, b, c Axes [25]

A~(T)(IR)

(L)

Bu(T)(IR)

(L)

164 355 (421)

165

145 480 603

147 593 624

296 Ag (R) 346 Bg 636 Bg

TABLE p (CH2)n: The Formula for Polyethylene Is Designated by the Molecular Unit CH~, Four of Which Are in the Unit Cell and in the Primitive Cell

(R)

(IR) Translation x Translation y Translation z Translatory Translatory CH 2 rocking CH2 rocking CH2 twisting CH 2 wagging CH 2 scissors CH2scissors C-H sym stretch C-H sym stretch C-H asym stretch C-H asym stretch

0 B3u 0 B2u 0 Blu 86 Blu 109 B2u 721 B2u 736 Blu 1043 B3u 1177 B3u 1472 B2u 1478 Blu 2851 B~u 2853 B2u 2921 B~u 2922 B2u

Translatory CH2 twisting CH2 wagging

55 Au 1041 Au 1181 A.

(o)

Libration Libration Optical skeletal Optical skeletal Optical skeletal Optical skeletal CH2 rocking CH 2 rocking CH2 twisting CH2 twisting CH 2 wagging CH 2 wagging CH 2 scissors CH 2 scissors C-H sym stretch C-H sym stretch C-H asym stretch C-H asym stretch

101 B3g 141 Ag 1066 Big 1070 B2g 1137 B3g 1138 Ag 1175 Ag 1176 B3g 1293 B2g 1297 Big 1371 Big 1373 B2g 1445 Ag 1458 B3g 2850 B3g 2855 Ag 2885 Ag 2889 B3g

These calculated frequencies are usually within 5 cm-~ of the observed frequencies. The translation modes (acoustic phonons) B3, + B2, + Blu are included in this example but not in the irreducible representation. While the acoustic-phonon modes are not IR-active, they are included in the IR column because of the u subscript. They should be active in Brillouin scattering, however. Most of the other observed modes in the IR are combination bands of these fundamentals [32-34].

332

Edward D. Palik

the IR-active modes are not also R-active. In this case we state only the T and L modes omitting the arrows. It is interesting to note the differences in R and IR experiments. Raman measurements are very specific as to which phonon mode is being measured because of the constraints on wave vector and energy. Infrared measurements do not have the constraint of wave-vector conservation. Therefore, for the configuration of Fig. 2b, many T O phonons traveling in many directions in the x , y plane (and for that matter, other directions having a component of dipole moment lIE) contribute to the absorption line seen for EIIc (they may not all have the same frequency). The imaginary part of the IR transverse dielectric function (ee = 2nk) has a peak at toT. The real part of the transverse dielectric function (el = n e - k e) usually has a zero very near here, and it usually has a second zero above toT at to L. This zero is presumably at the frequency of the LO phonon propagating

lie. If the unit cell has a center of symmetry, modes are mutually exclusive; an R-active mode is not also IR-active and vice versa. In the optic frequency Tables a - p , a notation such as E ( L ) ( I R _ L ) , A(L)(IRII) indicates that the E and A modes are IR-active (their transverse members are seen; the longitudinal members are not seen because of dipole-moment, matrix-element considerations, but we can obtain their valt~es from an oscillator fit of the reststrahlen reflectivity by finding where

el =0). (12) and (13) Plasmafrequency.Metalshaveplasmafrequenciestypically in the V I S - U V because of their large free-carrier (usually electron) densities (1022-1023cm-1). Since semiconductors can be doped with controlled amounts of impurities, their free-carrier plasma frequencies can be varied. We therefore omit this p a r a m e t e r for semiconductors. The Drude plasma frequency to2 (rad/s) is given by the formula ~o~,= 4:rNe2/m*e with N in cm-3; e = 4 . 7 7 x 10 -l~ esu; m* is the effective mass compared to the free-electron mass of 9.10 • 10 - 2 8 gm; e is the background lattice dielectric constant at the plasma frequency. The values of top for a given material obtained from different laboratories often vary by tens of percent, so we use the fewest possible number of significant figures, namely two. A polariton composed of a coupled EM wave and the plasmon can exist in the region where e~ is negative (from 0 to oJp) and is measurable by a total internal reflection technique in o~-q space, where q is the polariton wave vector. Three ways of determining ogp are fitting the reflectivity with a Drude model; measuring the energy loss of an electron beam passing through a thin layer; and determining where the real part of the dielectric function is zero (e~ = n 2 - k 2 - 0). This last criterion is easy to determine for the metals in the Tables in HOC 1 and HOC II. When interband absorption effects are small, n crosses k at only one point in the V I S - U V and this point often agrees with a Drude-model fit (which ignores interband effects). However,

14. Optical Parameters for the Materials

333

for some metals, n = k occurs at two, three, or four wavelengths, and the definition is obscured. This is usually due to strong interband transitions giving strong absorption/dispersion effects. For metals, semiconductors, and insulators the valence (bound) electrons also produce a plasma effect in the interband region usually at a frequency higher than the free-electron plasma frequency for a metal. For a metal the first zero should locate the free-electron plasma frequency and the third should locate the valence plasma frequency. Between the second and third zeros, e~ is negative and a valence plasma polariton should be definable just as a free-electron plasma polariton in ~o-q space is definable from 0 to ogr. Ideally for a semiconductor or insulator, this valence plasma frequency is at the second zero (above the band gap). The polariton mode is confined between the first and second zeros in el above the band gap in the spectral region where el is negative. These frequencies are listed in the !1=k column, although with overlapping interband effects, the valence plasma frequency may not be pinpointed by the second zero (higherenergy zero) of each pair of zeros. In Table I, in the n = k column for some of the materials, there is no crossing (dashed line); in other cases there are two sets of data that differ, and the crossing cannot be pinpointed (crosses); in still other cases, no data are available in the interband region, and the space is left blank. In addition to a phonon resonance producing a zero in el very close to OJr for a semiconductor and insulator, the zero somewhat above the resonance allows a longitudinal mode to exist" ~oL, for example. This idea is straightforward for lattice vibrations. It also holds for the free-carriers in metals and doped semiconductors, which have a resonant frequency at 09 = 0 and an ogp (a longitudinal propagating plasmon) at e l = 0 . However, the interband region (especially at the second of each pair of zeros) and the nature of the longitudinal mode are not so clear. (14) The f u n d a m e n t a l band gap Eg o f semiconductors and insulators. The analysis of the absorption coefficient to e x t r a c t Eg is model-dependent for indirect (i) and direct (d) energy gaps. In lieu of a model fit, some investigators just define Eg as the point at which the absorption coefficient a = 47ck/2 = 100-1000 cm- 1. (15) The dc dielectric constant eo. The real part of e0 near w = 0 for insulators is a positive number, which is straightforwardly determined from low-frequency optical or capacity experiments. However, for doped semiconductors and metals e, tends to large, negative values because of the free-carrier dispersion. According to the Drude model, e = el + ie2 contains components e~ __ e~ I1 - ~,/(~2 + w2)] and e2 -eoo ~r,20.Lr/~O(~)2 + (.02) with ~o~(s-1) _ 1/r, where -r(s) is the free-carrier lifetime and w~ is the damping constant. However, e2 approaches infinity as co -+ 0. To escape this "dilemma," we can switch our attention to the real part of the electrical conductivity. Since o- - -ie~o/4rr, then 131"1 82 ~o/47r - e~ w~, Ogr/4"rr(w2 + oo2). The dc conductivity is given by O-o(S-1) - e~ o~2/4Ir0%, a finite num-

-

334

Edward D. Palik

ber. The resistivity po(S) = 1/tro; po(ohm cm) = 9 • 1011 /90(S ). For metals e~ = 1; for semiconductors e~ "-" 4 - 16. We do not include the high-frequency dielectric constant e| The real part of e= for a metal is generally taken as unity in the X-ray region. For a semiconductor or ~insulator it is usually defined in connection with the Lorentz-model fit to reststrahlen reflectivity data; it is the value of e~ below the fundamental band gap where e~ is nearly a constant before the lattice-vibration dispersion begins to decrease e~ significantly.

REFERENCES

1. H. P. R. Frederikse, in "ALP Handbook," (D. E. Gray, ed.), 3rd ed., McGraw-Hill, New York, 9-56 (1972). 2. G. R. Wilkinson, in "Raman Effect," (A. Anderson, ed.), Marcel Dekker, New York, 811 (1973). 3. G. F. Koster, J. O. Dimmock, R. G. Wheeler, and H. Statz, "Properties of the Thirty-Two Point Groups," MIT Press, Cambridge, Mass. (1963). 4. K.-H. Hellewege and D. O. Madelung, eds. "Landolt-B6rnstein Numerical Data and Functional Relationships in Science and Technology," Springer-Verlag, New York (1985). 5. G. Burns and A. M. Glazer, "Space Groups for Solid State Scientists," Academic Press, New York (1978). 6. R. W. G. Wyckoff, "Crystal Structures," Vols. 1--4, Interscience, New York, (1965). 7. R. Loudon, Ado. Phys. 13, 423 (1964). 8. W. G. Fateley, F. R. Dollish, N. T. McDevitt, and F. F. Bentley, "Infrared and Raman Selection Rules for Molecular and Lattice Vibrations: The Correlation Method," Wiley-Interscience, New York (1972). 9. H. Poulet and J. P. Mathieu, "Vibration Spectrum and Symmetry of Crystals," Gordon and Breach, Paris (1976). 10. W. Hayes and R. Loudon, "Scattering of Light by Crystals," Wiley-Interscience, New York (1978). 11. S. S. Mitra and S. Nudelman, eds., "Far-Infrared Properties of Solids," Plenum, New York (1970). 12. D. W. Berreman, Phys. Reo. 130, 2193 (1963). 13. R. Dalven, Infrared Phys. 9, 141 (1969); PbS, PbSe, PbTe. 14. A. Scalabrin, A. S. Chaves, D. S. Shin, and S. P. S. Porto, Phys. Stat. Sol. B 79, 731 (1977); BaTiO3. 15. M. A. Ordal, R. J. Bell, R. W. Alexander, Jr., L. L. Long, and M. R. Querry, Appl. Opt. 26, 744 (1987); several metals. 16. J. L. Servoin, R. Gervais, A. M. Quittet and Y. Luspin, Phys. Reo. B 21, 2038 (1980); Ref. [2], p. 872; SrTiO3. 17. S. M. Shapiro and J. D. Axe, Phys. Reo. B6, 2420 (1972); SiO:. 18. R. Claus, Z. Naturforsch. 27A, 1187 (1972); LiNbO3. 19. Ref. [7], p. 172; Ref. [2], p. 861; wurzite semiconductors. 20. S. P. S. Porto, P. A. Fleury, and T. C. Damen, Phys. Reo. 154, 522 (1967); Ref. [2], p. 878; TiO2, MgF2. 21. F. Tuinstra and J. L. Koenig, J. Chem. Phys. 53, 1126 (1970); graphite. 22. S. P. S. Porto and R. S. Krishnan, J. Chem. Phys. 47, 1009 (1967); Ref. [2], p. 894; A1203. 23. G. Lucovsky, A. Mooradian, W. Taylor, G. B. Wright, and R. C. Keezer, Solid State Commun. 5, 113 (1967); Ref. [2], p. 851; Se.

14. Optical Parameters for the Materials

335

24. P. Dawson, M. M. Hargreave, and G. R. Wilkinson, J. Phys. Chem. Solids 34, 2201 (1973); Cu20. 25. S. Guha, D. Peebles, and T. J. Wieting, Bull. Amer. Phys. Soc. 35, 293, 782 (1990); Phys. Rev. (in press), Bull. Mater. Sci. (in press); CuO. 26. Ref. [2], p. 859; Zinc blende semiconductors. 27. R. Zallen, M. L. Slade, and A. T. Ward, Phys. Rev. B3, 4257 (1971); AszS3, AszSe 3. 28. D. W. Feldman, J. H. Parker, W. J. Choyke, and L. Patrick, Phys. Rev. 170,698 (1968); Phys. Rev. 173, 787 (1968); SiC. 29. D. K. Agrawal and C. H. Perry, in "Proc. 2nd Inter. Conf. Light Scattering in Solids," (M. Balkanski, ed.), Flammarion, Paris, 429 (1971); KDP, ADP. 30. T. Kawamura and A. Mitsuishi, Technology Reports, Osaka University 23, 365 (1973); KDP, ADP. 31. F. Onyango, W. Smith, and J. F. Angress, J. Phys. Chem. Solids 36, 309 (1975); KDP. 32. J. R. Nielsen and A. H. Wollett, J. Chem. Phys. 26, 1391 (1957); (CH2),,. 33. R. Zbinden, "Infrared Spectroscopy of High Polymers," Academic Press, New York (1964); (CH2),,. 34. P. C. Painter, M. M. Coleman, and J. L. Koenig, "The Theory of Vibrational Spectroscopy and its Application to Polymer Materials," Wiley, New York, 1982; (CH2),,. 35. L. M. Fraas and S. P. S. Porto, Solid State Commun. 8, 803 (1970); Be.

Part II Critiques

Metals

An Introduction to the Data for Several Metals DAVID W. LYNCH Department of Physics and Ames Laboratory Institue for Physical Research and Technology Iowa State University Ames, Iowa and

W. R. HUNTER SFA, Inc. Landover, Maryland

INTRODUCTION

We present data below for several metals: the alkali metals Li, Na, and K, and the transition metals Cr, Fe, Nb, and Ta. There are some other data compilations that it may be useful to consult. Weaver et al. [1] present optical data for noble metals, transition metals, lanthanides, actinides, and a few other metals. They present plots that show the agreement, or lack thereof, of data from many investigators. The spectral region covered is usually up to 200 eV or less. Henke et al. [2, 3] present calculated atomic scattering factors, and derive an empirical set of data based on experimental results. The spectra range is 10-10,000eV. Actual data do not exist over most of this range, and some of the data presented were taken on atoms (vapor samples) and some on solids. REFERENCES

1. J. H. Weaver, C. Krafka, D. W. Lynch, and E.-E. Koch, "Optical Properties of Metals," Parts I and II (Physics Data Series No. 18-1 and 18-2, Fachinformationszentrum Energie Physik Mathematik, Karlsruhe, 1981). 2. B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, and B. K. Fujikawa, Atomic Data and Nucl. Data Tables 27, 1 (1982). 3. B. L. Henke, J. C. Davis, E. M. Gullikson, and R. C. C. Perera, Lawrence Berkeley Laboratory Report No. LBL-26259 (1988). HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS II

341

Copyright 9 1991 by Academic Press. All rights of reproduction in any form reserved. ISBN 0-12-544422-2

I

342

II

David W. Lynch and W. R. Hunter

ALKALI METALS

The alkali metals are the simplest of all metals, and their dielectric functions the simplest and best understood. The nearly-free-electron model should be valid. The Fermi level should lie about in the middle of the lowest conduction band, and the gap between the bands at the zone boundaries should be small. The Fermi surface should be spherical, and all interband optical transitions should occur in a central spherical region of the Brillouin zone. The interband optical properties should be accurately calculable by perturbation theory [1-3]. Indeed this might be the case for all alkali metals except Li. Li has only three electrons per atom, with a ls22s configuration. In a pseudopotential approach, there are no core p electrons to which the 2p part of the valence electrons can be orthogonalized. Consequently, the pseudopotential is not as small as for other alkali metals, and the band gap is larger. The Fermi level then comes closer to the zone boundary, and the Fermi surface is distorted from a sphere. Another view is that the screening of the core potential by the single 2s valence electron of the atom is less effective in Li than by the corresponding electrons in the other alkali metals, and the gap is consequently larger. Thus, Li is expected to be somewhat different from the other alkali metals, and this is borne out by experiment. In fact, however, recent photoemission measurements [4, 5] show that the apparent filled region of the valence band in Na and K (and presumably other alkali metals) is narrower than band-structure calculations show. This is an effect of the response of the conduction electrons to the photohole, that is, one should not interpret photoelectron spectra (or optical spectra) on the basis of the one-electron density of states, but rather the spectral density of the interacting system of electrons. An equivalent picture is to augment the calculated one-electron band structure by the (complex) self energy, which shifts bands nonuniformly and broadens them [6-9]. The experimental situation is complicated by several factors. The alkali metals are reactive. They have low melting points, and relatively high diffusion rates at room temperature, leading to surfaces that are possibly rough. A number of measurements were taken not at free metal surfaces, but at quartz-alkali metal interfaces to avoid contamination, even in ultrahigh vacuum, and to yield a smooth interface. The roles of interfacial grain structure, reactions, and strain are unknown. A number of years ago, a significant peak in the optical conductivity was reported for several alkali metals, with potassium being the most studied [10, 11]. The peak occurred at an energy lower than the threshold for interband absorption. Many theoretical explanations were proposed in the ensuing years, but that peak was not found in many subsequent measurements, especially, but not

343

An Introduction to the Data for Several Metals

exclusively, those made on the quartz-metal interface. The best early explanation was that it arose because of roughness at the alkali metalvacuum interface [12]. Some experiments showed that it may be present only when water is adsorbed on the alkali metal surface [13]. This led to a new interpretation, that it arose from charge-density waves, at least in potassium [14, 15]. Subsequent measurements did not confirm the predicted optical structures [16]. A final complication is the fact that at low temperatures both Na and Li undergo a structural phase transition from their normal body-centered cubic structure. There are not many measurements in the infrared on alkali metals. They should behave as free-electron gases in this region, for there is a threshold energy of the order of 1 eV for direct interband absorption. The measurements are difficult, for the reflectance is very high, and one needs great precision and accuracy, of the order of four decimal places in the reflectance. Ellipsometry, using several reflections, has often been used for metals in the infrared, but again the measurement is difficult, for the real part of the ratio of p- to s-polarized electric-field reflection coefficients is very close to one, and the phase shift is very close to 180~. One really needs the .difference between the optical properties of the metal and those of a perfect conductor in such cases, and few measurements have been made at wavelengths longer than 2.5 ~tm for any alkali metal. Intrinsic surface states have been calculated to be present on alkali metals, and to contribute significantly to the infrared optical properties [17-21], in contrast to their contribution in noble metals, for example Ag, where such states contribute effects of only about a part in 10 4. The large predicted effects have not been seen, but there are really no data in the right spectral region on clean samples. Since there may be interest in infrared optical properties, and the only such measurements do not tie in well with those in the visible, the only option is to use a nearly-free-electron (Drude) model. The parameters in this model may be obtained from data in the near infrared. In the discussion of each alkali metal, we give those parameters that have been extracted from experimental data, but there is considerable scatter in them. The real and imaginary parts of the refractive indices can be obtained from the real and imaginary parts of the Drude dielectric function from 2n2-- -+-~1 -}- ( ce2 "-]'-~2)1/2 21, : =

with ~1--

+

+

2 2/(1 + (~ov)2)

1 7I- ~ xx - - (-/) p'g

e2-- ~ v / ~ ( 1 + (~v)2). exx is the contribution of high-energy interband transitions to the lowfrequency dielectric function, ~Opis the plasma frequency, given in cgs units

344

David W. Lynch and W. R. Hunter

by (_Op2 =4:rNeZ/m ,, with N the number density of electrons and m* their effective mass. r is the relaxation time. However, to obtain a good fit to data, r must be frequency-dependent, in the form l / r - - 1/z'0 + f l ~ 2 . /

Thus, to characterize the infrared optical properties, we must know four parameters. Note that ~Opand ro should give the correct static conductivity, ~Opro/4ar. (Sometimes data taken above the lowest interband absorption are fit to a Drude expression. However, in such cases the relaxation time will be different, and the plasma frequency should contain the free-electron mass, or an effective mass closer to it than the infrared value.) The following critiques, tables, and graphs contain what we consider to be the most useful n and k values for the chosen metals. One of the steps in producing the tables containing E (energy), 2 (wavelength), n (index of refraction), and k (extinction coefficient) is to convert E (usually the experimental units in eV) to 2 by means of the formula E (eV)2 (/k)= 12,398.42. This was done and the values subsequently rounded to those found in the table (usually to four figures). After rounding, however, E2 may not give the original conversion factor to four significant figures. Thus, if E from the table is converted to 2 using the above conversion factor, the result may not equal the wavelength value in the table and vice versa. This slight discrepancy happens most often in the data taken from Henke et al. but should produce no practical problems. REFERENCES

1. A. H. Wilson, "The Theory of Metals," Cambridge University Press, Cambridge, 133 (1936). 2. P. N. Butcher, Proc. Phys. Soc. (London) A64, 50 (1951). 3. N. W. Ashcroft and K. Sturm, Phys. Rev. B3, 1898 (1971). 4. E. Jensen and E. W. Plummer, Phys. Reo. Lett. 55, 1912 (1985). 5. I.-W. Lyo and E. W. Plummer, Phys. Rev. Lett. 60, 1558 (1988). 6. J. E. Northrup, M. S. Hybertsen, and S. G. Louie, Phys. Rev. Lett. 59, 819 (1987). 7. K. W. Shung, B. E. Sernelius, and G. D. Mahan, Phys. Rev. B36, 4499 (1987). 8. K. W. Shung and G. D. Mahan, Phys. Rev. B38, 3856 (1988). 9. J. E. Northrup, M. S. Hybertsen, and S. G. Louie, Phys. Reo. B39, 8198 (1989). 10. H. Mayer and M. H. El Naby, Z. Physik 174, 269 (1963). 11. H. Mayer and B. Heitel in "Optical Properties of Solids," (F. Abeles, ed.), North-Holland, Amsterdam, 47 (1972). 12. J. P. Marton, Appl. Phys. Lett. 18, 140 (1971). 13. P. Harms, dissertation, Clausthal, 1972. 14. A. W. Overhauser and N. R. Butler, Phys. Rev. BI4, 3371 (1976). 15. F. E. Fragachan and A. W. Overhauser, Phys. Rev. B31, 4802 (1985). 16. L. H. Dubois and N. V. Smith, J. Phys. F Met. Phys. 17, L7 (1987). 17. R. Garron, Surf. Sci. 79, 489 (1979). 18. R. Garron, Surf. Sci. 91,358 (1980). 19. R. Garron and H. Roubaud, Surf. Sci. 92, 593 (1980).

An Introduction to the Data for Several Metals

345

20. R. Garron and H. Roubaud, Surf. Sci. 146, 527 (1984). 21. R. Garron, Appl. Opt. 20, 1747 (1981).

LITHIUM (Li)

Lithium is a reactive material, and generally only data taken under ultrahigh-vacuum conditions will be mentioned. Li is normally bodycentered cubic, but at low temperatures there is a transition to a phase often reported to be hexagonal, a phase with different optical properties. It is the stable phase below about 80 K. However, the phase transition is sluggish, and a low-temperature sample may be a mixture of phases. In fact, the early reports of this transition stated the low-temperature phase was hexagonal close packed [1, 2], but more recent descriptions [3] are 9R, a closed-packed stacking that is mixed fcc and hcp. We report only data on the bcc phase. We do not consider data taken in arbitrary units, for example, optical density of a film or electron-energy-loss spectra without suitable normalization, unless those are the only data available in a particular wavelength region. In such a case, we give only a reference to the original work. The first modern measurements on Li are those of Hodgson [4]. Although the films were not prepared in ultrahigh vacuum, the ellipsometric measurements were made on an Li-silica interface, which remained stable for about a day. The data, taken in the 0.22-2 [am region and at two temperatures, were fit to a free-electron model in the longer-wavelength region. To obtain a good fit, the relaxation time had to depend on frequency. The interband-absorption threshold was not well-defined. B6senberg [5, 6] made attenuated-total-reflectance measurements on films deposited in better vacuum, using the Li-sapphire interface. His data, covering the range of 2-3.8 eV, are not in good agreement with Hodgson's, but agree better with data taken later by other groups. Myers and coworkers [7-9] made ellipsometric measurements at the vacuum-metal interface in the range 0.6-4 eV. They evaporated films on several substrates at several temperatures, obtaining different results. There were changes in the spectra with time after deposition. Others (see below) also observed time-dependence. For what was believed to be bcc Li, the first interband threshold was at 2.6 eV. At lower energies, the data were fitted to a free-electron model, but to obtain a fit it was necessary that the relaxation time depend on frequency. Table I lists Drude parameters from several references. Table II lists values of n and k. Figure 1 plots n and k versus ~tm. Myers and Sixtensson [9] also reported the optical constants of hexagonal Li. Rasigni and Rasigni [10] measured the transmittance of Li films of varying thickness, from which measurements they

III

346

David W. Lynch and W. R. Hunter

extracted the optical conductivity. They covered the 1-6 eV region and found two interband peaks. The one with a threshold at about 2.6 eV was the expected interband peak, whereas the anomalous one peaked between 1.5 and 2.6 eV. Films giving the latter were found to be grainier, either discontinuous grains on the substrate or a distribution of grains on a smoother Li film on the substrate. They made additional studies of the nature of the surfaces of films evaporated on substrates at different temperatures, correlating them with optical spectra [11-15]. They were able to eliminate the anomalous peak in later work [16], in which they give optical constants in the 0.15-10.7 eV range, obtained by Kramers-Kronig analysis of their reflectance measurements in the range 0.5--6 eV [13], and the reflectance of Callcott and Arakawa [17] from 3-10.7 eV. Inagaki et al. [18] measured the optical properties of Li in the 0.6-3.8eV region by ellipsometry on the Li-silica interface. They obtained the anomalous absorption mentioned earlier after allowing their films to stand for a day in UHV, even though their measurements were at the Li-silica interface. The data that they took before the growth of this peak do not agree well with those of Myers et al., but the reflectance calculated from their data agrees with that of Rasigni and Rasigni [13]. The spectra of k from the papers mentioned earlier ~gree reasonably well, but there are large differences in the n spectra, n has a minimum near 0.5 ~tm, and the values at this minimum range from 0.13 to 0.21. The data of Rasigni and Rasigni exhibit a local maximum in n at about 0.33 tam not apparent in the other spectra (which end at about 0.33 ~tm). There are also discrepancies at both ends of the range of these measurements, either in the magnitudes of n, the slopes dn/d2, or both. There are no data further into the infrared, and the only data at shorter wavelengths are described in the next paragraph. Of the measurements described in the previous paragraph, only those of Rasigni and Rasigni [16] are close to matching the only data set to shorter wavelengths, and this matching was part of their own data analysis. This does not mean these are the best data, but this matching is the only criterion we can use for having selected them for the table. The data sets are too limited in range to test for consistency by sum rules. Moreover, not all authors report the frequency-dependence of the relaxation time necessary for an extrapolation of their data to longer wavelengths. The discrepancies probably result from differences in samples, either the possible occurrence of two phases, differences in surface roughness, or both. (There may be voids at the Li-substrate interface, which would appear as roughness in measurements made on that interface.) Callcott and Arakawa [17] determined the optical constants of Li films in the 3-10.7 eV region by multiple-angle reflectance measurements. They worked on the Li-vacuum interface and the Li-substrate (MgF2) interface, getting agreement in the optical constants only for energies above 6 eV.

An Introduction to the Data for Several Metals

347

Discrepancies below 6 eV were assigned to substrate roughness and structural effects in the film. Below 6 eV, they chose no best set of data, and only results above 6 eV are tabulated. There are no absolute measurements at energies above 10.7 eV. Haensel et al. [19] presented the results of transmission measurements in the 60-149 eV region. They show considerable structure at and above the Li ls absorption edge, structure that will appear prominently in the k spectrum, but very weakly in that of n. Extrapolation to the infrared may be made from the Drude parameters, but in the case of Li, those reported are not in good agreement with each other. We list many of them in Table I. (The fact that the effective-mass values are considerably larger than unity is a consequence of the distortion of the Fermi surface mentioned earlier). The first interband transition begins at about 2.6 eV. The Drude parameters may be used to obtain e2 f o r energies below this, but the interband contribution to t~2 will persist several tenths of an eV below 2.6 eV. Thus, Drude values of n and k would be reliable below about 2 eV, except that a reliable set of parameters cannot be chosen rationally. (Values of the Drude parameters determined from data in the spectral region above the interband absorption are not given in Table I. Such data yield a smaller effective mass, not suitable for extrapolation in the infrared). We have calculated n and k in the 1.24-240 A (10,000-51.5 eV) region using the model of Henke et al. [20]. The atomic absorption edge is K (55 eV) [21]. We used 8065.48 cm-1/eV to convert among the units of [20], whereas the conversion factor varies by a digit or so for [16]. This should give no problems among the units until the fourth figure. REFERENCES

1. C. S. Barrett, Phys. Rev. 72, 245 (1947). 2. C. S. Barrett, Acta. Cryst. 9, 671 (1956). 3. References to the recent structural literature may be found in R. J. Gooding and J. A. Krumhansl, Phys. Rev. B38, 1695 (1988). 4. J. N. Hodgson in "Optical Properties and Electronic Structure of Metals and Alloys," (F. Abeles, ed.), North-Holland, Amsterdam, 60 (1966). 5. J. B6senberg, Phys. Lett. 41A, 185 (1972). 6. J. B6senberg, Z. Physik B 22, 261 (1975). 7. A. G. Mathewson and H. P. Myers, Physica Scripta 4, 291 (1971). 8. A. G. Mathewson and H. P. Myers, Phil. Mag. 25,853 (1972). 9. H. P. Myers and P. Sixtensson, J. Phys. F 6, 2023 (1976). 10. M. Rasigni and G. Rasigni, J. Opt. Soc. Am. 63, 775 (1973). 11. M. Rasigni and G. Rasigni, J. Opt. Soc. Am. 62, 1033 (1972). 12. M. Rasigni, G. Rasigni, and J. P. Palmari, Phil. Mag. 31, 1307 (1975). 13. M. Rasigni and G. Rasigni, J. Opt. Soc. Am. 66, 826 (1976). 14. M. Rasigni, J. P. Gasparini, G. Rasigni, and R. Fraisse, Solid State Commun. 18, 629 (1976).

348

15. 16. 17. 18.

David W. Lynch and W. R. Hunter

M. Rasigni, G. Rasigni, J. P. Gasparini, and R. Fraisse, J. Appl. Phys. 47, 1757 (1976). M. Rasigni and G. Rasigni, J. Opt. Soc. Am. 67, 54 (1977). T. A. Callcott and E. T. Arakawa, J. Opt. Soc. Am. 64, 839 (1974). T. Inagaki, L. C. Emerson, E. T. Arakawa, and M. W. Williams, Phys. Rev. B13, 2305 (1976). 19. R. Haensel, G. Keitel, B. Sonntag, C. Kunz, and P. Schreiber, Phys. Stat. Solidi (a) 2, 85 (1970). 20. B. L. Henke, J. C. Davis, E. M. Gullikson, and R. C. C. Perera, Lawrence Berkeley Laboratory Report No. LBL-26259 (1988). 21. K. Siegbahn, C. Nordling, A. Fahlman, R. Nordberg, K. Hamrin, J. Hedman, G. Johansson, T. Bergmark, S.-E. Karlsson, I. Lindgren, and B. Lindberg, in "Nova Acta Regiae Societatis Scientiarum Upsaliensis," Ser. IV, Vol. 20, Almqvist and Wiksells Boktryckeri, Uppsala, (1967).

A n I n t r o d u c t i o n to t h e Data f o r Several M e t a l s

I0 2 "-

, i ,lillil

I I I llili I

349

I I I IIlll i

I I I Illlll

I I~_

_ m

/N

/-

_

I01 _-_ _ _

/

_

-= -

-

/

_

_

I0 ~

/

/

/

_=--

/

_ _

/

/

-

/

-= =

i _

!

_

io-i

!

=-..

z

_

_

IO-Z _--

/

B

k

/ /

I

_

I

--

I

-

/

10-3_

/

m

/

_

/

_

/

10-4

/

_

_

_

/

_

_

/

_

/

10-5~

_

K

"-2

/

_ _

_

/

/

i0 -6 -

/ /

-

-

-

_

-

/

_

-

/

-

- -

/

10-7

/ --

- -

/

- -

- -

-

/

io-8 _

-

/

-

- -

/ - -

/

-

-

_

10-9

/ I/,,lll,,I

10-4

, lllll,iI

10-3 10-2 WAVELENGTH

I llll,,,I

i0-i (/u.m)

i

,,I,,,,I

,

,,

I0 o

F i g . 1. L o g - l o g p l o t o f n ( s o l i d l i n e ) a n d k ( d a s h e d l i n e ) v e r s u s w a v e l e n g t h for lithium.

in m i c r o m e t e r s

350

David W. Lynch and W. R. Hunter

TABLE I Drude-Model Parameters from Various References

Ref.

E'oo

[4] [51 [6] [8] [91 [16] [18]

1.0 0.8 0.75 50 eV), the vacuum ultraviolet (VUV, 10 eV < ho~ < 50 eV), the infrared-visible-ultraviolet (IR-V-UV, 1 eV50 eV), the photon energy is above the plasmon energy (18.4 eV), and most of the interband oscillator strength is exhausted. Consequently, Be is weakly absorbing so that n ~ X/~1~1 and e2-~2k~1. The only important physical process affecting the optical properties is the onset of absorption at the Be K-edge at 109 eV. The data available in this region are from thin-film transmittances that provide a measure of the linear absorption coefficient,/~ = 4Jrk/20, where 20 is the wavelength of the incident radiation. The available measurements are reported in Refs. [4-7]. The measurements reported by Henke and Elgin [5] and Barstow et al. [6] are consistent with each other. In the VUV spectral range (10-50 eV), the major feature of the spec-

Beryllium (Be)

423

trum is the transition from reflecting to transmitting behavior at the plasma energy (18.4 eV). Reflectance data from Seignac and Robin [8] and Toots et al. [9] are plotted in Fig. 1. Both sets of data were obtained on films with thickness 300-1000 A evaporated on glass substrates and measured in vacua of ] 0 -7 t o 10 - l ~ torr. The reflectance data in the 1-10 eV range from Seignac and Robin [8], Hunderi and Myers [10], Weaver et al. [11],, and Hull [12] are also plotted in Fig. 1. The most striking feature in this region is the wide range of reflectances obtained from careful studies in a spectral region where the procedures of optical measurements are usually most reliable. The two reflectance curves reported by Weaver et al. for the electric-field vector parallel (E[Ic) and perpendicular ( E • to the c-axis direction, were obtained on single-crystal surfaces prepared by mechanical polishing, with a subsequent chemical etch. These values are represented by the symbols (+) and (D), respectively. The reflectance data of Hull [12] were taken on the carefully polished surface of a polycrystalline mirror blank prepared by pressure sintering a low-oxygen-content (200/k using a rhodium-mirror polarization analyzer; for 2~0.2, 0.1 ~

~}

Q)

0

00 o4 o c O 0 0 r.. m~- ~ - m~- r,% r,) e O

o

o

o

o

o

o

o

o

o

L~

o

o

o

o

o

o

4o u

(D U

cU

~)

t_W--

9-~ ,-~ ,-~ ,-~ o.I e o ~

,--4

(D

C

c u') u ~ u'~ u'~ u'~ o 9

9

9

9

9

,-~ r-- 0 0 e 0

9

9

,

9

,

o4

,

o

6

oJ ,

o

,

r-. o ~ o ,

,

,

,-~ o ,

u4o

,

uu ,q-.

k o ~f) ,~- ~,9 o,j ~ ~I- ~x-) r~. o eoo o4 o % o o,~ u-) ,.4 0 ,-~ ,'-~ ,-4 ,-~ ,-~ o c ~ 0 0 I--. I'-. %~9 ~.o r o k o m~- o 4 ,-~ o r,9 ~-) ~,-) r,9 cv~ r,9 r r c,.i r ckl e,j o,~ o,j ~_~ ,_4 ,_4 ,_~ ,.4

c

.c

GJ

c E E ~ ~ o

~ ~ o

o

~ ~ o

~

~ ~

o

o

~ ~

o

o

o ~

o

o

~

~

~ o

o o

o

~ ~

o

o

~ ~

o

o

o'~ u o

o

666666666d666dddddd I-~1

u .o

Aluminum Gallium Arsenide (AIxGal _xAs) O. J. GLEMBOCKI Naval Research Laboratory Washington, D.C. and

KENICHI TAKARABE Department of Natural Science Okayama University of Science Okayama-City Okayama 700, Japan The AlxGal_xAs system is technologically one of the most important alloy systems, especially when combined with GaAs. It forms the basis of quantum-well, superlattice, and single-barrier device structures, which in turn have a significant effect on high-speed electro-optics. Because of this, it would be of the utmost importance to have the optical constants, n and k, over a wide spectral range and over the concentration range of x = 0 to x = 1, in as narrowly spaced x intervals as possible, preferably Ax =0.1. This, however, is not the case. In fact, we could find only bits and pieces of data at almost random values of x over narrow spectral ranges, except for x-0.3. There is, however, a fair amount of information about other physical parameters such as lattice constant, density, band gap, and phonon frequency. Because these parameters are useful in many applications, we list Adachi's values [1] for these most commonly used ones: 1. Lattice constant (,&) a = 5.6533 + 0.0078x; 2. Density (gm/cm 3) g = 5 . 3 6 - 1.6x; 3. Band gap (eV) Eg- 1.424+1.247x ( 0 < x < 0 . 4 5 ) = 1.900+0.125x+0.143x 2 (0.45I000 > 1000

cm-I 368 353 331 320 309 306 304 298 290 277 256 247 244 242 239 238 236 23#+ 233 232 227 217 209 205 202 197 191 179 176 157 145 131 103 83

ne (

II )

3.27 3.18 3.11 3.07

[45]

3.27

[,~6]

3.27 3.00 2.90

[47] [4B]

no(J.) 27.17 28.35 30.21 31.25 32.36 32.68 32.89 33.56 34.48 36.10

1.78 1.68 1.37 1.16 0.96

39.01

0.55 1.12

40,49 40.98 41.33 41.84 42.02 42.37 42.74 42.92 43.10 44.05 46.08

47.85 48.78 49.51 50.76 52.36 55.87 56.82 63.69 68.97 76.34 97.09 120.3

0.82 0.77 0.33 0.14 0.15

2.03 3.42 4.45

6.16 10.0 7.81 7.29 6.84 5.82 4.62 4.10 4.04 4,10 3.73 3.70 3.63 3.49 3.42 3.34 3.29

[43]

no ( _L ) 3.14 3.06 3.00 2.97

3.11 3.11 3.09 2.90

[45]

[46] [47] [48]

k(both) o . o2 0.03 0.05 0.08 0.13

0.16 0.26 0.53 0.82 1.44 3.11 6.45 9.77 13.15 I0.01 6.69 3.38 1.70 1.02 0.69 0.35

0.18 0.23 0.22 0.23 0.24 0.21 0.13 0.09 0.04 0.08

0.06 0.04 0.04

[42]

Cadmium Sulphide (CdS)

595

TABLE I (Continued) Cadmium Sulphide (Thin Films)

eV 4.133 3.543 3.100 2.480 2.067 1.771 1.550 1.378 1.240 1.127 1.033 0.9538 0.8856 0.8266 0.77h9 0.6888 0.6200 0.6200 0.4133 0.3100 0.2067 0.1550 0.1240 0.1033 0.08856

cm-*

)um

33,330 28,580 25,000 20,000 16,670 14,280 12,500

0.30 0.35 0.40 0.50 0.60 0.70 0.80 0.90 1.0 I.I 1.20 1.30 1.40 1.50 1.60 1.80 2.00

II,II0 I0,000 9,091 8,333 7,693 7,143 6,667 6,250 5,556 5,000 5,000 3,333 2,500 1,667 1,250 1,000 833.3 714.3

2.00 3.00 4.00 6.00 8.00 I0.00 12.00 14.00

~' References are indicated in brackets.

n

k

2.68[28] 2.58 2.55 2.72 2.40 2.35 2.32 2.30 2.28 2.28 2.27 2.27 2.26 2.25 2.25 2.24 2.23 2.300 2.284 2.277 2.271 2.267 2.263 2.260 2.255

>0.70 >0.70 >0.70 0.70 0.54 0.47 0.43 0.40 0.37 0.33 0.28 0.15 0.04

[331

[28]

Gallium Antimonide (GaSb) DAVID F. EDWARDS and RICHARD H. WHITE University of California Lawrence Livermore National Laboratory Livermore, California

Few reports can be found in the literature of the fundamental optical properties of GaSb. Research-sized crystals are relatively easy to grow, but GaSb has developed little technical interest. Aspnes and Studna [1] report n and k values calculated from pseudo-dielectric functions measured by spectroscopic ellipsometry. They call this the pseudo-dielectric function because no corrections were made for possible surface overlayers or subsurface damage effects. For surfaces free of these defects or by making the appropriate corrections, the ellipsometric measurements would give the "true" dielectric function. In either case they calculate the n and k values from the ellipsometric measurements. As pointed out by Aspnes and Studna [1], three recent technical advances have been made that allow for more accurate determination of the optical properties of semiconductors, GaSb included. The first and most important is the development of their new etching and cleanup procedures [2]. These procedures minimize the thickness of the undesirable surface layer that forms on semiconductors. As a result, their ellipsometric measurements and calculated pseudo-dielectric functions more accurately represent the intrinsic bulk values. The second advance is the development of the automatic spectroscopic ellipsometer. This instrument determines both components of the complex pseudo-dielectric function over a broad spectral range. No wavelength extrapolations are needed as in the Kramers-Kronig analysis of reflectance data. The third is the development of unambiguous criteria for optically determining the best surfaces. These "best" surfaces are the ones most free of roughness, unintentional overlayers, absorbed contaminants, and subsurface damage. Aspnes and Studna have been leaders in this field of the ellipsometric determination of the optical properties of solids. Several reports have been made of the reflectance in the near-UV and visible regions. Seraphin and Bennett [3] list the reflectance data of Cardona [4] for the 4-4).5 eV spectral range. Tauc and Abraham [5] report reflectance for the same spectral region. The objective of both reports was HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS II

597

Copyright 9 1991 by Academic Press. All rights of reproduction in any form reserved. ISBN 0-12-544422-2

598

David F. Edwards and Richard H. White

an investigation of the band structure of GaSb, and no extraction of n and k values was made. Johnson, Filinski, and Fan [6] measured the absorption coefficient in a ptype undoped GaSb sample at 1.7 K in the region of the intrinsic energy gap. (Eg=0.8100eV at 1.7 K). They observed three sharp bands. The highest energy band (0.8100 eV) they identify as due to an exciton absorption. The two bands of lower energy (0.8049eV and 0.7958eV) they attribute to exciton-impurity complexes. They report their data in a curve of absorption coefficient (cm-1) as a function of photon energy. From their curve we have extracted the k values. The three exciton peaks are designated a, fl, and 7 in Table I. Edwards and Hayne [7], using the angle-of-minimum-deviation technique, measured the near-IR index of refraction from 1.8-2.5 tam (about 0.7-0.5 eV). Oswald and Schade [8] were among the first to report n and k values for the infrared region. They calculated n and k values from the measured reflectance and transmittance for a GaSb sample with a resistivity of only 0.08 f2 cm. This small resistivity and the monotonically increasing k values with wavelength indicate free-carrier absorption. As a result, the Oswald and Schade data are not given in the table that follows. Wynne and Bloembergen [9] calculated the indices of refraction at 5.3 ~tm and 10.6~tm from the measured transmittance and coherence length for second-harmonic generation in GaSb with a CO2 laser. These data are not included in our tabulation because of the inconsistencies with the data of other sources and the lack of precision of their method for extracting index values. Hass and Henvis [10] measured the fundamental lattice reflection bands of several III-V compound semiconductors including GaSb. These measurements were made at liquid-helium temperature using a grating spectrometer for increased spectral resolution. By comparing the observed reflectance spectrum with that calculated using a single classical oscillator [11], they obtained the "best" values of the oscillator parameters. From these parameters we calculated the n and k values for the 29-100 tam spectral region. Pikhtin and Yas'kov [12] report a semi-empirical method for calculating the dispersion of the refractive index using three adjustable parameters associated with the energy-band structure. For those materials where extensive data are available, their predictions accurately fit the indices. For GaSb they use the data of Refs. [7] and [8], which do not have the spectral breadth necessary for the accurate determination of their three parameters. Test calculations [13] for wavelengths longer than those of Ref. [7] indicate the values of the indices are sensitively dependent on the values of these adjustable parameters. It appears that indices over a broad spectral range are needed to improve the accuracy of the Pikhtin-Yas'kov calculated data for GaSb.

Gallium Antimonide (GaSb)

599

The indices of refraction and extinction coefficients for GaSb are given in Table I and plotted in Fig. 1. Most of the data are for room temperature except where noted. The n and k values from 6.0 eV to 2.1 eV are the measured data of Aspnes and Studna [1]. The k values for the absorptionedge region are calculated from the measured absorption coefficients of Johnson et al. [6]. The n values from 2 . 0 e V to 0.0443 eV (0.62-28 ~tm) were calculated from a Herzberger-type dispersion formula fit to the measured data of various references [1, 7, 10]. These same data were also fit to a modified Sellmeier-type dispersion formula. This was done for convenience to the reader, but the Sellmeier values are not given in Table I. A total of 28 data points were used for these fittings. As seen in Fig. 1, the indices between 6.0 eV and 2.1 eV (0.2067 ~m and 0.5905 ~tm) do not lend themselves to Herzberger- or Sellmeier-type, curve-fitting routines. The Herzberger-type dispersion formula [14] is

n = A + B L + C L 2 + D,~ 2 + E24,

where L = 1/(,~2 - 2.8 x 106) with the wavelength 2 in A, and 2.8 x 106 is the square of the mean asymptote for the short-wavelength abrupt rise in index for 14 materials (GaSb not included) [14]. For the 28 points from 0.62 ~tm to 28 ~tm, and room temperature, the coefficients are

A -- 3.850787 B = 2.024481

x 10 7

C = 1.079226 x 1015 D = -4.397689 x 10-~1 E = 3.527494 x 10-19

The RMS error between the reported indices and the values calculated from the dispersion formula is 0.192%. For the reader's convenience, the modified Sellmeier-type dispersion formula [15] is

I+

8) +

n),

600

David F. Edwards and Richard H. White

where the wavelength 2 is in A. For the 28 data points, the coefficients are A = 3.779678 x 10-8 B = 1.558825 x 10 -9 C = 1.266738 x l01 D = - 1.013537 x 10 -9. The RMS error is 0.226%. In Table I the n and k values from 29 ~tm to 100 ~tm were calculated from the lattice band parameters of Ref. [10] and are for liquid-helium temperature. These calculations were extended beyond the range of measurement to 100 ~tm, in order to estimate eo, the low frequency dielectric constant. Hass and Henvis [10] report eo = 15.69 compared with n 2= 15.98 at 100 ~tm. Cardona [16] reports the measured temperature coefficient of the refractive index of GaSb to be 1 dn ---=8.1 ndT

+ 0 . 2 x lO-SK -~

for the 5-10 tam spectral region. This coefficient was determined from changes in the interference patterns produced by multiple internal reflections in single crystals with low impurity concentrations. He reports that the results are independent of flee-carrier effects for the temperature range of about 85-415 K. Ghosh et al. [17] have developed a simple model for predicting the temperature and pressure coefficients of the refractive index of some zincblende semiconductors, GaSh included. They predict the temperature coefficient at constant pressure to be

= 1.47 x 10 . 4 K -1

at 3.7 tam for the range - 173-127 ~ They compare this with experimental values of 3.11-3.05 x 10 -4 K -1. Their predicted pressure coefficient is

-

= 44.5 - 52.8 x 10 .3 GPa-1. T

Gallium Antimonide (GaSb)

601

They have no experimental data for comparison nor do they give a wavelength or pressure range. REFERENCES 1. D. E. Aspnes and A. A. Studna, "Dielectric Functions and Optical Parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV," Phys. Rev. B 27, 985 (1983). 2. D. E. Aspnes and A. A. Studna, SPIE Proc. 276, 227 (1981); Appl. Phys. Lett. 39, 316 (1981). 3. B. O. Seraphin and H. E. Bennett, in "Optical Properties of III-V Compounds," (R. K. Willardson and A. C. Beer, eds.), Academic Press, New York, 499 (1967). 4. M. Cardona, "Optical Investigation of the Band Structure of GaSb," Z. Phys. 161, 99 (1961); "Fundamental Reflectivity Spectrum of Semiconductors with Zinc-Blende Structure," J. Appl. Phys. 32, Suppl., 2151 (1961). 5. J. Tauc and A. Abraham, "Reflection Spectra of Semiconductors with Diamond and Sphalerite Structures," "Proc. Int. Conf. Semicond. Phys., Prague, 1960," (K. Zaveta, ed.), Czech. Acad. Sci., Prague, and Academic Press, New York, 375 (1961). 6. E. J. Johnson, I. Filinski, and H. Y. Fan, "Absorption and Emission of Excitons and Impurities in GaSb," "Proc. Intern. Conf. Phys. Semicond. 6th, Exeter, 1962," The Institute of Physics and the Physical Society, London, 375 (1962). 7. D. F. Edwards and G. S. Hayne, "Optical Properties of Gallium Antimonide," J. Opt. Soc. Am. 49, 414 (1959). 8. F. Oswald and R. Schade, "Uber die Bestimmung der Optischen Konstanten yon Halbleitern des Typus AHZBv in Infraroten," Z. Naturforschg. 9a, 611 (1954). 9. J. J. Wynne and N. Bloembergen, Measurement of the Lowest Order Nonlinear Susceptibility in III-V Semiconductors by Second-Harmonic Generation with a CO2 Laser," Phys. Rev. 188, 1211 (1969); Erratum, Phys. Rev. B2, 4306 (1970). 10. M. Hass and B. W. Henvis, "Infrared Lattice Refraction Spectra of III-V Compound Semiconductors," J. Phys. Chem. Solids 23, 1099 (1962). 11. M. Born and K. Huang, "Dynamical Theory of Crystal Lattices," University Press, Oxford, Chap. 2 (1954). 12. A. N. Pikhtin and A. D. Yas'kov, "Dispersion of the Refractive Index of Semiconductors with Diamond and Zinc-blende Structures," Sov. Phys. Semicond. 12, 622 (1978). 13. D. F. Edwards, 1989 (unpublished). 14. M. Herzberger and C. D. Salzberg, "Refractive Indices of Infrared Optical Materials and Color Correction of Infrared Lens," J. Opt. Soc. Am. 52, (4), 420 (1962); M. Herzberger, "Colour Corrections in Optical Systems and a New Dispersion Formula," Opt. Acta 6, 197 (1959). 15. D. F. Edwards and R. H. White, see KDP in this volume. 16. M. Cardona, "Temperature Dependence of the Refractive Index and the Polarizability oT Free Carriers in Some III,V Semiconductors, "Proc. Intern. Conf. Semicond. Phys., Prague, 1960," Czech. Acad. Sci., Prague, and Academic Press, New York, 388 (1961). 17. D. K. Ghosh, L. K. Samanta, and G. C. Bhar, "The Temperature and Pressure Dependence of Refractive Indices of Some III-V and II-VI Binary Semiconductors," Infrared Physics 26, 111 (1986).

602

David F. Edwards and Richard H. White

I01~ '

' 'I'"'I

'

'

' 'I'"'I

' ' I""I

/\ I0 ~ -

\

\ lO-I_

E 10-2 _

10-3 _

10.4 i0-1

I0 ~ WAVELENGTH

IO t

I0 z

(#m)

Fig. 1. Log-log plot of n (solid line) and k (dashed line) versus wavelength in micrometers for gallium antimonide.

Gallium Antimonide (GaSb)

603

TABLE I V a l u e s of n and k for Gallium A n t i m o n i d e O b t a i n e d from Various R e f e r e n c e s a eV

O0 5. 9O 5. 8O 5. 7O 5. 60 5. 5O 5. 40 5. 30 5. 20 5. 10 5. O0 4. 90 4. 8O 4. 70 4. 6O 4. 5O 4. 4O 4. 30 4. 20 4. I0 4. O0 3. 90 3. 80 3. 70 3. 60 3. 50 3. 40 3. 30 3. 20 3. 10 3. O0 2. 90 2. 80 2. 70 2. 60 2. 50 2. 40 2. 30 2. 20 2. 10 2. 000 1. 938 I. 900 I. 879 I. 824 I. 800 1. 771 .

cm -I

48,387 47,581 46,774 45,968 45,161 44,355 43,548 42,742 41,935 41,129 40,323 39,516 38,710 37,903 37,097 36,290 35,484 34,677 33,871 33,065 32,258 31,452 30,645 29,839 29,032 28,226 27,419 26,613 25,806 25,000 24,194 23,387 22,581 21,774 20,968 20,161 19,355 18,548 17,742 16,935 16,129 15,625 15,323 15,152 14,706 14,516 14,286

pm

0.2067 0.2102 0.2138 0.2175 0.2214 0.2255 0.2296 0.2340 0.2385 0.2431 0.2480 0.2531 0.2583 O. 2638 0.2696 0.2756 0.2818 0.2884 0.2952 0.3024 0.3100 0.3179 0.3263 0.3351 0.3444 0.3543 0.3647 0.3758 0.3875 O. 4000 0.4133 0.4276 0.4429 0.4593 0.4769 0.4960 0.5167 0.5391 0.5636 0.5905 0.620 0.640 0.6526 0.660 0.680 O.6889 O.700

n

nc

0.935 [1] 0.985 1.022 1.062 1.127 1.212 1.299 1.345 1.356 1.358 1.369 1.387 1.408 I. 444 1.503 1.586 1.723 1.989 2.522 3.100 3.450 3.620 3.701 3.748 3.774 3.785 3.794 3.808 3.800 3.766 3.732 3.728 3.760 3.836 3.984 4.312 4.513 4.492 4.521 4.705 5.2496 5.1058

k

2.416 [ I ] 2.444 2.479 2.535 2.602 2.645 2.653 2.638 2.645 2.685 2.751 2.829 2.928 3.055 3.208 3.392 3.628 3.923 4.130 3.976 3.643 3.323 3.069 2.862 2.690 2.545 2.430 2.319 2.210 2.134 2.109 2.121 2.157 2.211 2.280 2.285 1.962 1.789 1.747 1.803 1.378 0.829

4.9812 4.8724 0.611 4.7771

604

David F. Edwards and Richard H. White

TABLE I (Continued) Gallium Antimonide

eV 1.722 I. 700 1.676 1.632 1.600 1.590 1.550 1.512

cm-I

1.442 I. 409 1.378 1.348 1.319 1.292 1.265 1.240 0.6199 0.4133 0.3100 O. 2480 0.2066 0.1771 0.1550 0.1378 0.1240 0.1127 0.1033 0.09537 0.08856 0.08266 0.07749 0.07293 0.06889 0.06526 0.06199 0.05904 0.05636 0.05391 0.05166 0.04959 0.04769 0.04592 0.04428

i3,889 13,710 13,514 13,158 12,903 12,821 12,500 12,195 12,096 11,905 11,628 11,364 11,111 10,870 10,638 10,417 10,204 10,000 5,000 3,333 2,500 2,000 1,667 1,429 1,250 1,111 1,000 909.1 833.3 769.2 714.3 666.7 625.0 588.2 555.6 526.3 500.0 476.2 454.5 434.8 416.7 400.0 384.6 370.4 357.1

0.04275 0.04133

344.8 333.3

1.500 1.476

~m 0.720 0.7294 0.740 0.760 0.7750 O.780 0.800 0.820 0.8267 0.840 0.860 O.880 O.900 0.920 0.940 O.960 0.980 1.000 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0

12.0 13.0 14.0 15.0 16.0 17.G 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0

n

nc 4.6929 O.485 4.6184 4.5521 0.416 4.4928 4.4396 4.3916 0.344 4.3483 4.3090 4.2733 4.2406 4.2108 4.1834 4.1583 4.1351 4.1137 3.7883 3.7421 3.7285 3.7241 3.7236 3.7251 3.7278 3.7313 3.7352 3.7395 3.7438 3.7481 3.7521 3.7557 3.7586 3.7608 3.7620 3.7620 3.7605 3.7574 3.7524 3.7453 3.7357 3.7235 3.7083 3.6898 3.6677 3.66 [10] 3.65

1.2xi0 -3 [103 1 5xi0 -3

Gallium Antimonide (GaSb)

605

TABLE I (Continued) Gallium Antimonide

eV 0.04000 0.03875 0.03757 0.03647 0.03542 0.03444 0.03351 0.03263 0.03179 0.03100 0.03024 0.02952 0.02945 0.02938 0.02931 0.02924 0.02917 0.02910 0.02904 0.02897 0.02890 O.02883 0.02877 0.02870 0.02863 0.02857 0.02850 0.02844 0.02837 0.02831 0.02824 0.02818 0.02755 0.02695 0.02638 0.02583 0.02530 0.02480 0.02066 0.01771 0.01550 0.01378 0.01240 0.8166 0.8145 0.8128 0.8120 0.8112

cm-1 322.6 312.5 303.0 294.1 285.7 277.8 270.3 263.2 256.4 250.0 243.9 238. I 237.5 237.0 236.4 235.8 235.3 234.7 234.2 233.6 233.1 232.6 232.0 231.5 230.9 230.4 229.9 229.4 228.8 228.3 227.8 227.3 222.2 217.4 212.8 208.3 204.1 200.00 166.67 142.86 125.00 111.11 100.00 6,585.5 6,568.5 6,554.8 6,548.4 6,541.9

pm

31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 39.0 40.0 41.0 42.0 42.1 42.2 42.3 42.4 42.5 42.6 42.7 42.8 42.9 43.0 43.1 43.2 43.3 43.4 43.5 43.6 43.7 43.8 43.9 44.0 45.0 46.0 47.0 48.0 49.0 50.0 60.0 70.0 80.0 90.0 100.0 1.5185 1.5224 1.5256 1.5271 I. 5286

n

nc 3.62 3.60 3.57 3.53 3.48 3.42 3.33 3.21 3.03 2.71 2.02 O.484 0.485 0.502 0.531 0.575 0.637 0.723 0.844 1.02 1.28 1.70 2.43 3.84 6.65 10.3 11.2 10.4 9.39 8.62 8.03 7.57 5.66 5.07 4.78 4.61 4.50 4.42 4.13 4.058 4.026 4.008 3.997

k -~ ~.8xI0 3xi0 -3 30x10 -3 40x10 -3 5.4xi0 -3 7 5xi0-3 1.09x10 -2 1 . 6 9 x 1 0 -2 2.85xio-z

5.61xi0-:;' 0.160 2.06 2.39 2.73 3.08 3.45 3.86 4.31 4.82 5.41 6.12 6.98 8.07 9.36 10.3 8.56 5.04 2.87 1.81 1.24 0.907 0.694 0.148 6 . 6 4 x 1 0 -2 3 . B S x l o -z

2.56xI0 -Z_ 1.84xiO -z 1.41x10 -7 3.4x10 -3 1.8x10 -3

1.2xi0 -3 ~.9xI0 -4 .2xi0 -4

B.OBx10-2[6] 4.85 4.86 5.10 5.11

606

David F. Edwards and Richard H. White

TABLE I

(Continued)

Gallium Antimonide

eV 0,8109 O.8 1 0 5 0.8100 O.8 0 9 5 O.8 0 9 4 0.8087 0.8085 O.8 0 8 0 0.8075 O. 8 0 6 8 O.8 0 6 1 O.8 0 5 5 0.8053 0.8049 O.8 0 4 5 O.8 0 4 0 O.8 0 3 6 O.8 0 3 4 0.8030 O.8 0 2 2 O.8 0 1 6 0.8012 0.8005 O.7 9 9 6 0.7990 O.7 9 8 0 O.7 9 7 0 O.7 9 6 5 O. 7 9 6 4 0.7962 0.7958 O. 7 9 5 7 O.7 9 5 4 0.7950 O.7946 0.7940 O.7 9 3 2 O.7 9 2 2 0.7906

cm-1

pm

6,539.5 6,536.3 6,532.3 6,528.2 6,527.4 6,521.8 6,520.2 6,516.1 6,512.1 6,506.5 6,500.8 6,496.0 6,494.4 6,491.1 6,487.9 6,483.9 6,480.6 6,479.0 6,475.8 6,469.4 6,464.5 6,461.3 6,455.6 6,448.4 6,443.5 6,435.5 6,427.4 6,423.4 6,422.6 6,421.0 6,417.7 6,416.9 6,414.5 6,411.3 6,408 1 6,403.2 6,396.8 6,388.7 6,375.8

1.5292 I. 5299 1.5309 I. 5318 I. 5320 1.5333 1.5337 I. 5347 1.5356 I. 5369 I. 5383 I. 5394 1.5398 I. 5406 I. 5413 I. 5423 1. 5431 I. 5434 1. 5442 1.5457 1.5469 1.5477 1.5490 1.5508 1.5519 1.5339 1.5558 I. 5568 1.5570 I. 5574 1.5582 I. 5584 I. 5590 1.5597 I. 5605 1.5617 I. 5633 I. 5653 1.5684

n

nc

k 5.35 5.84 e 6.33 5.73 4.88 2.5 1.35 I. 07xi0 -2 9 65xi0 -3 9.05 9.06 9.80x 10-3 1 I0xi0 -Z B I. 23 1.03x 10-2 8. lOx 10-3 5.16 3.93 2.70 1.66 1.23 ~ .05x I0 -3 .63xi0 -4 6.54 5.31 3.59 3.10 3.34 4.09 4.71 y 4.96 4.34 2.73 2.23 I. 74 1.49 I. 49 I. 5 1.4xi0 -4

aThe column headed n~ contains refractive-index values calculated with the Herzberger-type formula. The bracketed numbers indicate the reference to the data. Near 1.54 pm excitonic peaks are designated m, B, Y.

Silicon-Germanium Alloys (SixGel-x) J. HUMLi(~EK, F. LUKE,S, and E. SCHMIDT Department of Solid State Physics Masaryk University Brno, Czechoslovakia

Silicon and germanium form a continuous series of crystalline alloys, SixGel_x, from x = 0 to x = 1. Optical constants of both constituents have been critiqued extensively by Potter [1] and Edwards [2] in H O C I. Optical response of the alloy varies continuously over the entire range of compositions x, providing a suitable degree of freedom in tailoring the material for possible optical applications. On the other hand, the alloy is a very good object for physical studies of disorder-induced effects in electron and phonon structure of crystalline solids. It has also been studied in order to understand better the electron band structure of the pure constituents. In spite of numerous studies devoted to this alloy system, the number of papers allowing reasonable extraction of the optical constants is somewhat limited. In the present critique, we compile the optical data for three compositions, representative of Si-rich (x~0.80), Ge-rich (x=0.25), and nearly-equal-content (x~0.50) alloys. This particular choice was influenced by the requirement of nearly coinciding compositions for data available from various sources. With one exception, we did not interpolate the composition dependence at fixed wavelengths to match the chosen values of x; instead, we have tabulated the original data obtained for slightly different compositions. It should be pointed out that the three spectra are a rather limited data base with respect to possible accurate interpolation for other alloy compositions. Data are listed in Table I and plotted in Fig. 1. Early measurements of the energy position of the intrinsic optical absorption edge were reported in 1954 by Johnson and Christian [3]. The near-infrared absorption spectra were subsequently studied in detail by Braunstein et al. [4] in wide ranges of composition and temperature. Bulk samples were single crystals in the range of 0-20% and 90-100% Si; HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS II

607

Copyright @ 1991 by Academic Press. All rights of reproduction in any form reserved. ISBN 0-12-544422-2

608

J. Humli6ek e t al.

intermediate compositions were polycrystals. The alloy composition was determined by spectrographic techniques to +5% of the content of silicon (for 0--70% Si) or germanium (for 70-100% Si). Fluctuations in composition between slices from the same ingot are mentioned; consequently, the spectra of absorption coefficient a were taken on the same sample thinned progressively from 500 to 5 rtm. Higher values of a might be influenced by the inferior quality reported for some of the thinnest alloy samples. Special care was taken to ensure that all the light transmitted through the sample was collected on the detector. This indicates problems with scattered light, which we also mention below in the discussion of phonon spectra. Room temperature was specified to be 296 K; this is fairly important, since the band edge shifts strongly with temperature. The refractive index needed for calculations of a from measured transmittances was obtained from the transmission at long wavelengths. The extensive work of Braunstein et al. [4] provides no spectra for the representative compositions of 25%, 50%, and 80% Si; therefore, we used their results in the following interpolation scheme. The absorption coefficient was fitted in Ref. [4] to the Macfarlane-Roberts expression: ( E - E g - kO)2 ( E - Eg + kO)2] e-)- i j ' a(E, T) = A 1 - e x p ( - 0 / T ) +

(i)

where E is the photon energy, kO the phonon energy (k is the Boltzmann constant, not to be confused with the extinction coefficient), Eg the energy gap, T the temperature, and A a temperature-independent factor. The first term in square brackets of Eq. 1 is to be included for E>Eg + kO, the second for E>Eg-kO; a = 0 below Eg-kO. A detailed compositional dependence of the best-fit values of Eg(x) and the phonon equivalent temperature O(x) is given by Braunstein et al. [4]. We have read their room-temperature spectra from graphs and determined the remaining parameter A that fits them best with appropriate values of Eg and 0. Interpolated values of all of the three parameters were inserted in Eq. 1 to generate the a(E) spectra for T= 296 K. We have limited the spectral range so as to obtain the a values from 1 to 1000 cm -x. The corresponding extinction coefficient is shown in the tables. The results of the interpolation procedure agree to about +20% with values obtained by interpolating the experimental spectra either in compositions for fixed photon energies or vice versa. Because of the smoothing action of Eq. 1, the precision of the tabular data is limited only by the three decimals shown. The accuracy, however, is probably much lower. A factor-of-two difference, which is rather typical of various sources for pure Si and Ge [1, 2], is probably hard to achieve in the case of an alloy.

Silicon-Germanium Alloys (SixGel_x)

609

Reflectance measurements in the spectral range of strong absorption above the band gap, aiming at the identification of several interband thresholds, were performed by Tauc and Abraham [5]. Schmidt [6] measured the reflectance in the energy region from 1-13 eV on a number of alloy samples. Three of them had compositions suitable for our tabulation. The compositions were determined by X-ray diffraction; the mean error of x was _+0.004. The polycrystalline samples were mechanically polished and chemically etched. The thickness of surface oxides was estimated ellipsometrically to be between 2 and 4 nm. No correction was performed for the overlayers. The optical constants were computed from Kramers-Kronig (K-K) relations, using a combination of exponential (1318 eV) and constant (above 18 eV) extrapolation of the reflectance. The K-K analysis of these data was repeated with a modified extrapolation, and the accuracy of the dielectric functions was tested for pure Si and Ge by Humli6ek et al. [7]. We tabulate the room-temperature (300 K) results of the latter K-K computations for selected photon energies. Since the low values of the extinction coefficient k at the red side of the spectral region are very unreliable, only values where k>0.4 are given. Very recently, ellipsometry has been used to measure the optical response from 1.66-5.7 eV by Humli6ek et al. [8]. Two series of samples were studied. Bulk polycrystals were Syton-polished; no polishing was used for single-crystal epitaxial layers. The epilayers were strain-free; the lattice mismatch was relaxed by misfit dislocations extending less than about 100 nm into the layers of total thickness of several micrometers. All samples were measured under flowing dry nitrogen, after a wet chemical treatment was used to remove the surface overlayers. The alloy compositions were determined from the energy position of the strong E, interband threshold; the mean error of x was about +_0.003. The accuracy of the dielectric functions was estimated by examining trends in the data at fixed wavelengths as a function of alloy composition for 20 different samples. In spite of the care taken to avoid both surface damage and overlayers, the surface quality was found to be the limiting factor in the accuracy. In the nine spectra selected for tabulation in Ref. [8], the errors of both real and imaginary parts of the dielectric function are estimated to be a few tenths throughout the whole composition and spectral range. We have not tabulated the uncertain values of k less than 0.3 on the low-energy side of the ellipsometric spectra, because of the poor accuracy and precision of the method in this case [8]. There is a fairly good agreement of the reflectance and ellipsometric results in the energy position of all spectral structures. In particular, the E, absorption edge shifts from 2.1 eV in Ge to 3.4 eV in Si; this offers a convenient way to determine accurately the alloy composition, provided the bowing of the concentration dependence is taken into account [8]. A slight blue shift of the reflectance spectrum of Table I b with respect to the

610

J. Humli6ek et al.

ellipsometric one in the region around 2.7 eV is due to the difference of the compositions. The systematic trends in the differences of both sets of n and k are obviously due to the thicker surface films in the reflectance measurements. The refractive indices agree to two figures in the near infrared. The agreement of extinction coefficients above 5 eV is also satisfactory. The most pronounced failure of the reflectance data is the systematically lower values of n on the high-energy side of the overlap with the ellipsometric spectra. The discrepancy becomes more pronounced for Ge-rich alloys, in accordance with increasing optical thickness of the overlayers with increasing Ge content. For both Ge and Si, refractive indices in the transparent region below the intrinsic absorption gap were measured with high precision by angle of minimum-deviation and channel-spectra techniques [1, 2]. We have not found analogous data for the alloys, obviously because of the problems in obtaining sufficiently homogeneous crystals of high optical quality. We have chosen the values based on the reflectance measured by Schmidt [6]. The R values were fitted with refractive coefficients given by the polynomial n ( E ) = no + n , E 2

(2)

for the photon energy E from 1-1.5 eV, neglecting extinction indices k. The two coefficients no and n l were determined from the least-squares fit for each alloy composition. The extrapolation of Eq. 2 below 1 eV is tabulated here. The values are probably accurate to a few hundredths, as indicated by a reasonable agreement of the results for pure Si and Ge. Infrared lattice absorption in the alloys has been measured by Braunstein [9] throughout the whole range of compositions. However, most of his spectra for a silicon content of 22-81% showed a strong background absorption due to free carriers. Cosand and Spitzer [10] studied the lattice modes for Ge-rich (x ~0.88) alloys in the wavenumber region from 100-700 cm -1. They identified two-phonon bands analogous to those of pure Si and Ge. In addition, several temperature-independent bands are interpreted as being due to singlephonon processes. The latter are not seen in perfect homopolar lattices of Si and Ge, since the single-phonon excitations produce no dipole moment. Unpublished spectra of Humli~ek [11] were measured in the 180-1500cm -~ range at 300 K. The samples were of p-type conductivity with about 1015 free holes per c m 3. No Drude-like background is observable for this low level of doping. The alloy compositions (determined from spectral positions of the E~ interband thresholds [8]) are fairly close to the three target values of the present tabulation. The observed transmission in the transparent region above 1500 cm -1 was found to be lower than the

Silicon-Germanium Alloys (SixGel_x)

611

value predicted by reflection losses of the thick slab of nonabsorbing material"

TIk--~ 1 + R '

R=

+,,

,

(3)

where n is the refractive index. This is obviously due to the scattering of the probing beam by inhomogeneities in the composition, which prevents a part of the transmitted light from reaching the detector. The tabular k values were obtained from the transmittances normalized to the maximum value of Eq. 3 in the region above 1500 cm-1. For x = 0.8, two strong peaks near 400 and 480 cm -1 emerge; they are related to the analogous bands observed for Si-rich alloys by Cosand and Spitzer [10]. For higher Ge content, the band near 400 cm -1 dominates the absorption spectrum with peak values of the absorption coefficient, a = 4~k/2, above 10 cm -1. The phonon absorption is related to inevitable spectral variations of the refractive index. However, the magnitude of the changes from any smooth dispersion curve of the Eq. 2 type is expected to be less than a few thousandths, as found for silicon by Humli~ek and Vojt6chovsk~, [12]. Because of the rich spectral structure changing strongly with the composition, this spectral region is most difficult for dealing with the wavenumber- and composition-dependent extinction coefficients. REFERENCES

1. R. F. Potter, in "Handbook of Optical Constants of Solids," (E. D. Palik, ed.), Academic Press, Orlando, 465 (1985). 2. D. F. Edwards, ibid., p. 547. 3. E. R. Johnson and S. M. Christian, Phys. Rev. 95,560 (1954). 4. R. Braunstein, A. R. Moore, and F. Herman, Phys. Rev. 109, 695 (1958). 5. J. Tauc and A. Abrahfim, J. Phys. Chem. Solids 20, 190 (1961). 6. E. Schmidt, Phys. Status Solidi 27, 57 (1968). 7. J. Humli6ek, F. Lukeg, E. Schmidt, M. G. Kekoua, and E. Khoutsishvili, Phys. Rev. B33, 1092 (1986). 8. J. Humlf~ek, M. Garriga, M. I. Alonso, and M. Cardona, J. Appl. Phys. 65, 2827 (1989). 9. R. Braunstein, Phys. Rev. 130, 879 (1963). 10. A. E. Cosand and W. G. Spitzer, J. Appl. Phys. 42, 5241 (1971). 11. J. Humlf~ek, unpublished data (1989). 12. J. Humli~,ek and K. Vojt6chovsk~,, Phys. Status Solidi (a) 92, 249 (1985).

612

J. HumliSek et al.

'"~lwl

'

' 'I'"'I

i0 o

'

' 'I'"'I

'

' 'I'"'I

'

''I

''+-

!I

s.~

I

iO-I

IO-Z I I I I

iO-S

.

I

'~

-

I

{

10-4 -

I

I

I

I

-

iO-5

iO-Z

I01

'

''I

-(b) i0 o

.... I

'

''I'"'I

'

,/~~-,

!

I1

!11 I I

I I

, ,,I,,,,I , ,I ,,.I , ,,I,,,,II , ,,I,,, I0 -I I0 o I0 i 102 WAVELENGTH (~m)

''I'"'I

'

''I"

I01

'

"

~'

~1'~"1

''

JlJ'"I

~ ''I""1

' ''i"~

.

. (c)

/,~'~

I0 o

_

\

-

!

.=

I0-'-

i0 -I

iO-2~ =_

sO-2.~ _,~

-

I

i0-~ _

IO-a --

I

I

J!

I I I I

i"v I

i

#

I I I

I I I

10"3 I

t

10-%-

II

4 ',I

'II

i

J I

-.

I

I0-52110- i ~l .... I , , , I,,.I I , , ,I .... I , , , I, i l l iO-~ tO o IO ~ t0 WAVELENGTH ()u.m)

"

ral

,

10-2

tO-t

rl

,,I,,,,11

,

,,1,,,,I

I0 o WAVELENGTH

,

I0 ~

I

,,I,

102

(~m)

F i g . 1. L o g - l o g plot of n (solid line) and k ( d a s h e d line) versus w a v e l e n g t h in m i c r o m e ters. (a) F o r S i x G e ] _ x, x = 0.25. (b) F o r S i x G e l _ ~, x = 0.50. (c) F o r S i e G e l _ ~, x - 0.80.

Silicon-Germanium Alloys (SixGel_,,)

613

T A B L E l(a) Values of n and k for Si,Ge,_x, x = 0.25, from Various References* eV 12.8 12.6 12.2 12.0 11.8 11.,I 11.0 10.6 10.2 10.0 9.80 9.40 9.00 8.80 8.60 8.40 8.20 8.00 7.90 7.80 7.70 7.60 7.50 7.40 7.30 7.20 7.10 7.00 6.90 6.80 6.70 6.60 6.50 6.40 6.30 6.20 6.10 6.00 5.90 5.80 5.70 5.60 5.50 5.40 5.30 5.20 5.18

c m -1

49,200 48,390 47,580 46,780 45,970 45,160 .14,360 ,13,550 42 7 4 0 41,940 41,780

pm

n

n.

k

0.09686 o.o~4o

0.8.5 [6] 0.85

0.84 [6] o.ss

0.1016 O. 1033 O. 1051 O. 1088 0.1127 0.1170 0. i 216 0.1240 0.1265 0.1319 0.1378 0.1409 0.1442 0.1476 0.1512 0.1550 O. 1569 O. 1590 0.1610 0.1631 O. 1653 0.1675 0.1698 O. 1722 0.1746 0.1771 0.1797 0.1823 0.1851 0.1879 0.1907 0.1937 0.1968 0.2000

0.85 0.85 0.85 0.85 0.84 0.83 0.81 0.80 0.79 0.79 0.80 0.81 0.82 0.82 0.82 0.83 0.83 0.84 0.85 0.87 0.89 0.91 0.93 0.96 0.99 1.03 1.08 1.11 1.12 1.12 1.12 1.13 1.15 1.17

0.88 0.90 0.92 0.95 0.99 1.02 1.09 1.13 1.17 1.26 1.36 1.40 1.45 1.50 1.57 1.64 1.68 1.72 1.77 1.81 1.85 1.90 1.95 1.99 2.03 2.08 2.10 2.11 2.12 2.15 2.20 2.26 2.32 2.39

0.2033

,.20

2..-,8

0.2o66 0.2101 0.2138 0.2175 0.'2214 0.2254 0.2296 0.2.339 0.'2384 0.239:1

1.27 1.36 1.48 1.58 1.64 1.6.5 1.65 1.64 1.66

2.58 2.66 2.70 2.69 2.66 2.66 2.68 2.74 2.83

1.28 [8] 1.33 1.34 1.33 1.33 1.32

k

2.89 [8] 2.87 2.87 2.90 2.96 2.98

614

J. Humli5ek et al. TABLE l(a) (Continued) SixGel-x, x = 0.25

eV 5.16 5.14 5.12 5.10 5.08 5.06 5.04 5.02 5.00 4.98 4.96 4.94 4.92 4.90 4.88 4.86 4.84 4.82 4.80 4.78 4.76 4.75 4.74 4.72 4.70 4.68 4.66 4.65 4.64 4.62 4.60 4.58 4.56 4.54 4.52 4.50 4.48 4.46 4.44 4.42 4.40 4.38 4.36 4.34 4.32 4.30 4.28

cm -1 41,610 41,,450 41,290 41,130 40,970 40,810 40,650 40,480 40,320 40,160 40,000 39,&'~0 39,680 39,520 39,360 39,190 39,t330 38,870 38,710 38,550 38,390 38,310 38,230 38,060 37,900 37,740 37,580 37,500 37,420 37,260 37,100 36,940 36,770 36,610 36,450 36,290 36,130 35,970 35,810 35,640 35,480 35,320 35,160 35,000 34,840 34,680 34,520

~m 0.2403 0.2412 0.2422 0.2431 0.2441 0.2450 0.2460 0.2470 0.2480 0.2490 0.2500 0.2510 0.2520 0.2530 0.2541 0.2551 0.2562 0.2572 0.2583 0.2594 0.2605 0.2610 0.2616 0.2627 0.2638 0.2649 0.2661 0.2666 0.2672 0.2684 0.2695 0.2707 0.2719 0.2731 0.2743 0.2755 0.2768 0.2780 0.2792 0.2805 0.2818 0.2831 0.2844 0.2857 0.2870 0.2883 0.2897

n

1.69

1.80

1.89

n 1.33 1.33 1.33 1.33 1.33 1.34 1.34 1.35 1.35 1.36 1.37 1.37 1.38 1.40 1.40 1.42 1.43 1.44 1.45 1.47 1.48

1.95

2.02

2.93

3.19

3.34

k 3.00 3.01 3.04 3.06 3.08 3.10 3.12 3.15 3.17 3.20 3.22 3.25 3.28 3.31 3.34 3.37 3.40 3.44 3.47 3.50 3.54

3.43 1.50 1.52 1.54 1.56 1.58

2.09

2.19 2.23 2.28 2.33 2.39 2.47 2.55 2.64 2.74 2.85 2.97 3.11 3.24 3.38 3.52 3.64 3.75

k

3.51

3.58 3.62 3.66 3.70 3.74

3.61 1.61 1.64 1.67 1.71 1.74 1.79 1.84 1.89 1.95 2.02 2.10 2.19 2.30 2.42 2.54 2.68 2.81 2.95 3.08

3.72 3.77 3.82 3.87 3.92 3.97 4.03 4.07 4.12 4.15 4.18 4.18 4.17 4.14 4.09 4.02 3.93

3.79 3.84 3.89 3.94 3.99 4.05 4.10 4.16 4.23 4.29 4.35 4.41 4.46 4.51 4.54 4.55 4.54 4.51 4.48

Silicon-Germanium Alloys (SixGel_x)

615

TABLE I(a) (Continued) SixGe,_x, x = 0.25

eV 4.26 4.25 4.24 4.22 4.20 4.19 4.18 4.16 4.14 4.13 4.12 4.10 4.08 4.06 4.05

4.04 4.02 4.00 3.98 3.96 3.95 3.94 3.92 3.90 3.88 3.86 3.85 3.84 3.82 3.80 3.78 3.76 3.75 3.74 3.72 3.70 3.68 3.66 3.65 3.64 3.62 3.60 3.58 3.56 3.55 3.54 3.52

cm -1 34,350 34,270 34,190 34,030 33,870 33,790 33,710 33,550 33,390 33,310 33,230 33,060 32,900 32,740 32,660 32,580 32r 32,260 32,100 31,930 31,850 31,770 31,610 31,450 31,290 31,130 31,050 30,970 30,810 30,640 30,480 30,320 30,240 30,160 30,000 29,840 29,680 29,520 29,430 29,350 29,190 29,0,30 28,870 28,710 28,6,30 28,550 28,390

pm 0.2910 0.2917 0.2924 0.2938 0.2952 0.2959 0.2966 0.2980 0.2995 0.3002 0.3009 0.3024 0.3039 0.3054 0.3061 0.3069 0.3084 0.3100 0.3115 0.3131 0.3139 0.3147 0.3163 0.3179 0.3195 0.3212 0.3220 0.3229 0.3246 0.3263 0.3280 0.3297 0.3306 0.3315 0.3333 0.3351 0.3369 0.3388 0.3397 0.3406 0.3425 0.3444 0.3463 0.3483 0.3493 0.3502 0.3522

n

n

k

3.19 3.87 3.96

3.29 3.38 3.46

3.65

3.52 3.58 3.62

4.21

3.38

3.17

3.79 3.80 3.82 3.83

4.09

2.58

3.27 3.24 3.22 3.19 3.16

2.53 3.96 3.96 3.98 3.99 4.00

4.09

4.10

2.72

3.45 3.41 3.37 3.34 3.30

2.64 3.90 3.92 3.92 3.93 3.95

4.08

4.08

2.91

3.68 3.63 3.58 3.53 3.49

2.81 3.84 3.86 3.87 3.88 3.89

4.08

4.08

3.92 3.85 3.79 3.73

3.02

3.77

4.09

4.14 4.06 3.99

3.27 3.67 3.70 3.72 3.75

4.09

4.09

4.35 4.29

3.51

4.07 4.09

4.42

3.79

4.02 4.05

k

2.48

3.13 3.11 3.09 3.06 3.04

2.43 4.01 4.02 4.03 4.04 ,1.05

4.11

2.39

3.02 3.00 2.98 2.96 2.94

2.35 4.06 4.08

2.93 2.91

616

J. Humli6ek et al. TABLE I(a) (Continued) SixGel-x, x = 0.25

eV 3.50 3.48 3.46 3.45 3.44 3.42 3.40 3.38 3.36 3.35 3.34 3.32 3.30 3.28 3.26 3.25 3.24 3.22 3.20 3.18 3.16 3.15 3.14 3.12 3.10 3.08 3.06 3.05 3.04 3.02 3.00 2.98 2.96 2.95 2.94 2.92 2.90 2.88 2.86 2.85 2.84 2.82 2.80 2.78 2.76 2.75 2.74

cm -1 28,220 28,060 27,900 27,820 27,740 27,580 27,420 27,260 27,100 27,010 26,930 26,770 26,610 26,450 26,290 26,210 26,130 25,970 25,810 25,640 25,480 25,4OO 25,320 25,160 25,000 2:1,840 24,680 24,600 24,510 24,350 24,190 24,030 23,870 23,790 23,710 23,550 23,390 23,220 23,060 22,980 22,900 22,740 22,580 22,420 22,260 22,180 22,090

um 0.3542 0.3563 0.3583 0.3594 0.3604 0.3625 0.3647 0.3668 0.3690 0.3701 0.3712 0.3734 0.3757 0.3780 0.3803 0.3815 0.3827 0.3850 0.3875 0.3899 0.3924 0.3936 0.3949 0.3974 0.4000 0.4025 0.4052 0.4065 0.4078 0.4105 0.4133 0.4161 0.4189 0.4203 0.4217 0.4246 0.4275 0.4.305 0.4335 0.4350 0.4366 0.4397 0.,1428 0.4460 0.4492 0.4509 0.4525

n

n

4.13

4.09 4.11 4.13

4.14

4.16

1.94

1.91

2.40 2.39 2.38 2.38 2.38

1.92 4.38 4.38 4.40 4.42 4.43

4.19

1.94

2.38 2.38 2.39 2.39 2.40

1.97 4.45

4.48 4.23

2.49 2.47 2.45 2.43 2.41

1.92 4.36 4.36 4.36 4.36 4.37

4.16

4.17

2.03

2.62 2.60 2.57 2.55 2.52

1.98 4.36 4.36 4.36 4.36 4.36

4.19

4.17

2.14

2.75 2.72 2.70 2.68 2.65

2.09 4.31 4.32 4.34 4.35 4.35

4.23

4.21

2.24

2.84 2.82 2.80 2.78 2.76

2.19 4.23 4.25 4.27 4.28 4.30

4.23

4.24

2.89 2.8 7 2.85

2.27 4.14 4.16 4.18 4.20 4.21

4.19

4.21

2.31

4.50 4.54 4.57

4.30

2.01

2.41 2.41 2.42 2.42 2.43

2.04 4.62

2.44

Silicon-Germanium Alloys (SixGel_x)

617

T A B L E I(a) (Continued) SixGe,_x, x = 0.25

eV 2.72 2.70 2.68 2.66 2.65 2.64 2.62 2.60 2.58 2.56 2.54 2.52 2.50 2.48 2.46 2.44 2.42 2.40 2.38 2.36 2.34 2.32 2.30 2.28 2.26 2.24 2.22 2.20 2.18 2.16 2.15 2.14 2.12 2.10 2.08 2.06 2.05 2.04 2.02 2.00 1.98 1.96 1.95 1.94 1.92 1.90 1.88

c m -1 21,930 21~770 21,610 21,450 21,370 21,290 21,130 20,970 20,800 20,640 20,480 20,320 20,160 20,000 19,810 19,680 19,510 19,350 19,190 19,030 18,870 18,710 18,550 18,380 18,220 18,060 17,900 17,740 17,580 17,420 17,.~|0 17,260 17,090 16,930 16,770 16,610 16,530 16,450 16,290 16,130 15,970 15,800 15,720 15,640 15,480 15,320 15,160

ltm 0.4558 0.4592 0.4626 0.4661 0.4679 0.4696 0.4732 0.4769 0.4806 0.4843 0.4881 0.4920 0.4959 0.4999 0.5010 0.5081 0.5123 0.5166 0.5209 0.5254 0.5299 0.53-14 0.5391 0.5438 0.5486 0.5535 0.5585 0.5636 0.5687 0.5740 0.5767 0.579:1 0..5848 0.590,1 0.5~1 0.6019 0.6048 0.6078 0.6138 0.6199 0.6262 0.6326 0.6358 0.6391 0.6458 0.6526 0.6595

n

4.38

n 4.66 4.72 4.78 4.85

4.50

k

2.07

2.44

5.00

5.11

2.09

2.39

5.15

2.08

2.35

5.23 5.30 5.37 5.45 5.53 5.59 5.65 5.70 5.72 5.74 5.72 5.69 5.65 5.60 5.55 5.49 5.44 5.39 5.33 5.29 5.23

2.05 2.02 1.97 1.92 1.87 1.83 1.77 1.71 1.64 1.53 1.41 1.30 1.18 1.07 0.98 0.91 0.84

2.30 2.24 2.18 2.11 2.03 1.93 1.82 1.70 1.57 1.43 1.30 1.17 1.08 0.98 0.89 0.81 0.75 0.67 0.64 0.58 0.54

0.74

0.65 5.19 5.15 5.11 5.07 5.03

4.82

4.77

2.42

5.07

5.00

4.91

0.58

0.49 0.46 0.43 0.38 0.37

0.53 5.00 4.96 4.93 4.90 4.87

4.72

0.35 0.33 0.50 0.30 0.46

4.83 4.81

4.67

2.45 2.45 2.45 2.45

2.09

4.92 4.64 4.72 4.79 4.86 4.92 4.97 5.03 5.08 5.14 5.20 5.28 5.33 5.37 5.37 5.37 5.34 5.30 5.25 5.20

k

4.78 4.75

0.41

618

J. Humli6ek et al. TABLE I(a) (Continued) SixGel-x, x = 0.25

eV 1.86 1.85

1.84 1.82 1.80

1.78 1.76 1.74

1.72 1.70 1.68 1.66 1.60 1.50 1.40 1.30 1.22 1.21 1.20 1.19

1.18 1.17 1.16 1.15 1.14 1.13 1.12 1.11 1.10 1.09 1.08 1.07 1.06

1.05 1.04 1.03

1.02 1.01 1.00

0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92

cm -1 15,000 14,920 14,840 14,670 14,510 14,350 14,190 14,030 13,870 13,710 13,550 13,380 12,900 12,090 11,290 10,480 9,840 9,759 9,679 9,598 9,517 9,437 9,356 9,275 9,195 9,114 9,033 8,953 8,872 8,791 8,711 8,630 8,549 8,469 8,388 8,307 8,227 8,146 8.065 7,985 7,904 7,824 7,743 7,662 7,582 7,501 7,420

mn 0.6666 0.6702 0.6738 0.6812 0.6888 0.6965 0.7O45 0.7126 0.7208 0.7293 0.7380 0.7469 0.7749 0.8266 0.8856 0.9537 1.016 1.024 1.033

n

4.58

4.49

4.17

4.12

1.148

1.158 1.169 1.180

1.305 1.318

1.333 1.347

4.69 4.67 4.65 4.63 4.60 4.58 4.56 4.54 4.51 4.49

4.42 4.36 4.30 4.23

1.107

1.291

k

4.72

1.050 1.059 1.068 1.078 1.087 1.097

1.192 1.203 1.215 1.227 1.239 1.252 1.265 1.278

k

4.62

1.041

1.116 1.127 1.137

n

7.69E-03 [4] 7.29E-03 6.91E-03 6.53E-03 6.16E-03 5.8OE-03 5.45E-03 5.11E-03 4.77E-03 4.44E-o3 4.13E-03 3.82E-o3 3.52E-03 3.23E-03 2.95E-O3 2.68E-o3 2.43E-o3 2.18E-O3 1.95E-03 1.72E-03 1.51E-03 1.31E-O3

4.06

1.13E-O3 9.55E-O4 7.96E-04 6.51 E-o,I 5.21E-O4 4.O6E-O4 3.O6E-O4 2.22E-O4 1.54E-04

Silicon-Germanium Alloys (SixGel_x)

619

TABLE I(a) (Continued) SixGem-x, x = 0.25 eV 0.91 0.90 0.89 0.88 0.87 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.1239 0.1178 0.1147 0.1116 0.1091 0.1066 0.1041 0.1017 0.1000 0.09919 0.09671 0.09423 0.09175 0.08927 0.08679 0.08431 0.08307 0.08183 0.07935 0.07687 0.07439 0.07191 0.06943 0.06695 0.06,147 0.06323 0.06199 0.06075 0.05951 0.05703 0.05455 0.05331 0.05207 0.05083 0.04959

cm -1

pm

7,340 7,259 7,178 7,098 7,017 6,452 5,646 4,839 4,033 3,226 2,420 1,613 1,000 950 925 900 880 860 840 820 807 800 780 760 740 720 700 680 670 660 640 620 600 580 560 540 520 510 500 490 480 460

1.362 1.377 1.393 1.408 1.425 1.549 1.771 2.066 2.479 3.099 4.132 6.199 10.00 10.52 10.81 11.11 11.36 11.62 11.90 12.19 12.39 12.49 12.82 13.15 13.51 13.88 14.28 14.70 14.92 15.15 15.62 16.12 16.66 17.24 17.85 18.51 19.23 19.60 20.00 20.40 20.83 21.74

440

22.72

430 420 410 400

23.25 23.81 24.39 25.00

n

n

k

k

1.04E-04 4.02

7.04E-05 4.75E-05 2.89E-05

1.48E-05 3.97 3.94 3.90 3.88 3.86 3.84 3.83 8.35E-06 [1 l] 1.10E-05

1.83E-05 3.34E-05 4.96E-05 7.28E-05 7.74E-05 4.21E-05 3.82 4.88E-O5 7.41E-O5 5.75E-o5 3.46E-05 3.56E-o5 8.94E-05 2.o9E-o4 2.21E-04 2.o7E-o4 1.78E-04 1.75E-O4 2.27E-O4 2.55E-o4 2.64E-o4 3.66E-o4 6.90E-o4 8.96E-o4 9.58E-o4 8.9OE-O4 8.05E-O4 7.04 E-0-4 4.34E-04 3.37E-04 3.81E-04 5.14E-04 1.62E-03

620

J. HumliEek et al. TABLE l(a) (Continued) SixGel_x, x = 0.25

eV 0.04835 0.04711 0.04587 0.04463 0.04339 0.04215 0.04092 0.03968 0.03906 0.03844 0.03782 0.03720 0.03596 0.03472 0.03348 0.03224 0.02976 0.02728 0.02480 0.02232

c m -1

lzm

390 380 370 360 350 340 330 320 315 310 305 300 290 280 270 260 240 220 200 180

25.64 26.31 27.02 27.78 28.57 29.41 30.29 31.24 31.74 32.25 32.78 33.33 34.47 35.70 37.03 38.45 41.66 45.44 50.00 55.54

n

n

k

k 2.25E-03 2.34E-03 2.35E-03 2.11E-03 1.72E-03 1.55E-03 1.35E-03 1.46F_,-03 1.63E-03 1.62E-03 1.46E-03 1.92E-03 2.50E-03 2.11E-03 1.54E-03 1.18E-03 9.36E-04 9.28E-04 7.10E-04 3.48E-04

a T h e reference from which the values wel~e e x t r a c t e d is given in brackets n e x t to the first value. T h e n o m i n a l alloy c o m p o s i t i o n s are: x = 0 . 2 4 6 [6], x = 0 . 2 5 0 [8], x = 0 . 2 5 0 o b t a i n e d by i n t e r p o l a t i n g the d a t a of B r a u n s t e i n et al. [4], x = 0 . 2 6 0 [11].

Silicon-Germanium Alloys (Si,,Gel_,,)

621 T A B L E l(b)

Values of n and k for Si.Ge~_x, x = 0 . 5 0 , from V a r i o u s R e f e r e n c e s ~

eV

cm -1

12.8 12.4 12.0 11.6 11.2 10.8 10.4 10.0 9.80 9.60 9.20 8.80

8.40 8.00 7.80 7.60 7.40 7.20 7.00 6.90 6.80 6.70 6.60 6.5O 6.40 6.30 6.20 6.10 6.00 5.90 5.80 5.70 5.60 5.50 5.40 5.30 5.28 5.26 5.24 5.22 5.20 5.18 5.16 5.14 5.12 5.10 5.08

/~m

n

k

o.o9686

o.78 [61

o.81 [6]

0.09999

0.75

0.85

0.1033 0.,069

o.rs 0.78

o.ss

0.1107 O. 1148 0.1192 0.1240 0.1265 0.1292 0.1348 0.1409 0.1476 0.1550 0.1590 0.1631 0.1675 0.1722 0.1771 0.1797 0.1823 0.1851 O. 1879 0.1907 0.1937 0.1968

0.74 O. 73 0.72 0.70 0.70 0.70 0.71 0.72 0.74 0.76 0.75 0.75 0.76 0.77 0.82 0.86 0.90 0.95 0.99 1.02 1.05 1.08 1.11 1.16 1.22 1.29 1.37 1.47 1 ..56 1.64 1.70 1.71

0.96 1.01 1.07 1.15 1.19 1.23 1.32 1.42 1.53 1.63 1.70 1.78 1.88 2.00 2.13 2.19 2.24 2.29 2.33 2.38 2.42 2.48

0.2000 49,200 48,390 47,580 46,780 45,970 45,160 44,360 43,550 42,740 42,580 42,420 42,9.60 42,100 41,940 41,780 41,610 41,450 41,290 41,130 40,970

n

0.2033 0.2066 0.2101 0.2138 0.2175 0.2214 0.22.54 0.2296 0.2339 0.2348 0.23.57 0.2366 0.2375 0.2384 0.2394 0.2403 0.2412 0.2422 0.2431 0.24,11

1.72

1.75

0.92

2.55

1.32 [8] 1.37 1.39 1.39 1.39 1.39 1.39 1.40 1 .,10 1.40 1.40 1.39 1.40 1.40 1 .,l I

2.62 2.70 2.77 2.83 2.88 2.91 2.91 2.91 2.93

3.00

3.11

2.8r [8] 2.88 2.88 2.91 2.91 2.93 2.93 2.95 2.96 2.98 2.99 3.01 3.03 3.05 3.07

622

J. Humli6ek et at. TABLE I(b) (Continued) SixGel-x, x = 0 . 5 0

eV 5.06 5.04 5.02 5.00 4.98 4.96 4.94 4.92 4.90 4.88 4.86 4.84 4.82 4.80 4.78 4.76 4.75 4.74 4.72 4.70 4.68 4.66 4.65 4.64 4.62 4.60 4.58 4.56 4.54 4.52 4.50 4.48 4.46

4.44 4.42 4.40 4.38 4.36 4.34 4.32 4.30 4.28 4.26 4.24 4.22 4.20 4.18

cm- 1 40,810 40;650 40,480 40,320 40,160 40,000 39,840 39,680 39,520 39,360 39,190 39,a30 38,870 38,710 38,550 38,390 38,310 38,230 38,060 37,900 37,740 37,580 37,500 37,420 37,260 37,100 36,940 36,770 36.610 36,450 36,290 36,130 35,970 35,810 35,640 35,480 35,320 35,160 35,000 34,840 34,680 34,520 34,350 34,190 34,0.30 33,870 33,710

p.m 0.2450 0.2460 0.2470 0.2480 0.2490 0.2500 0.2510 0.2520 0.2530 0.2541 0.2551 0.2562 0.2572 0.2583 0.2594 0.2605 0.2610 0.2616 0.2627 0.2638 0.26,19 0.2661 0.2666 0.2672 0.2684 0.2695 0.2707 0.2719 0.2731 0.2743 0.2755 0.2768 0.2780 0.2792 0.2805 0.2818 0.2831 0.2844 0.2857 0.2870 0.2883 0.2897 0.2910 0.2924 0.2938 0.2952 0.2966

n

n

1.81

1.41 1.41 1.42 1.43

k

3.24

1.43 1.44

1.90

2.00

1.45 1.46 1.47 1.48

3.36

1.51 1 ..52 1 ..54 1 ..5.5 1 ..57

3.39 3.42 3.46 3.49 3.52

3..51

3.59 1.59 1.60 1.63

3.67

1.66

2.22

3.56 3.60 3.64 3.68

1.68

2.33 2.38 2.44 2.50 2.57 2.64 2.72 2.81 2.91 3.02 3.14 3.27 3.41 3..56 3.70 3.83 3.94 4.03 4.11 4.17 4.2.5 4.33

3.09 3.11 3.13 3.16 3.19 3.21 3.24 3.26 3.30 3.32

1.49

2.07

2.14

3.36

k

3.72 3.78

1.71 1.74 1.77 1.81 1.85 1.89 1.95 2.00 2.06 2.13 2.21 2.30 2.40 2.51 2.63 2.76 2.90 3.02 3.1.5 3.26 3.36 3.4.5 3.53 3.59

3.89 3.93 3.98 4.03 4.07 4.12 4.16 4.20 4.24 4.27 4.29 4.30 4.31 4.27 4.22 4.15 4.05 3.96 3.87 3.78 3.70 3..58

3.77 3.82 3.87 3.91 3.97 4.02 4.07 4.13 ,I.18 4.23 4.29 4.34 4.39 4.43 4.46 4.47 4.46 4.44 4.40 4.34 4.28 4.21 4.14 4.06

Silicon-Germanium Alloys (SixGel_x) TABLE I(b)

623 (Continued)

SixGel-x, x = 0.50

eV 4.16 4.14 4.12 4.10 4.08 4.06 4.05 4.04 4.02 4.00 3.98 3.96 3.95 3.94 3.92 3.90 3.88 3.86 3.85 3.84 3.82 3.80 3.78 3.76 3.75 3.74 3.72 3.70 3.68 3.66 3.65 3.64 3.62 3.60 3.58 3.56 3.55 3.54 3.52 3.50 3.48 3.46 3.45 3.44 3.42 3.40 3.38

cm -1 33,550 33~390 33,230 33,060 32,900 32,740 32,660 32,580 32,420 32,260 32,100 31,930 31,850 31.770 31,610 31,450 31,290 31,130 31,050 30,970 30,810 30,6'10 30,480 30,320 30,240 30,160 30,000 29,840 29,680 29,520 29,430 29,350 29,190 29,030 28,870 28,710 28,630 28,550 28,390 28,220 28,060 27,900 27,820 27,740 27,580 27,420 27,260

pm 0.2980 0.2995 0.3009 0.3024 0.3039 0.3054 0.3061 0.3069 0.308-'1 0.3100 0.3115 0.3131 0.3139 0.3147 0.3163 0.3179 0.3195 0.3212 0.3220 0.3229 0.32,16 0.3263 0.3280 0.3297 0.3306 0.3315 0.3333 0.3351 0.3369 0.3388 0.3397 0.3406 0.3425 0.3444 0.3463 0.3483 0.3493 0.3502 0.3522 0.3542 0.3563 0.3583 0.3.594 0.3(')04 0.3625 0.36,17 0.3668

rt

rt

4.34 4.29 4.30 4.33

3.65 3.70 3.74 3.77 3.8O 3.82

4.34

4.34

2.37

2.98 2.96 2.94 2.93 2.91

2.34 4.16 4.19 4.20 4.22 4.24

4.37

4.40

2.46

3.09 3.06 3.04 3.02 3.00

2.41 4.10 4.11 4.12 4.14 4.1.5

4.32

4.34

2.57

3.22 3.19 3.16 3.14 3.11

2.51 4.04 4.04 4.06 4.07 4.08

4.30

4.31

2.72

3.39 3.35 3.31 3.28 3.2.5

2.64 3.98 3.99 4.00 4.01 :1.02

4.29

4.30

2.92

3.61 3.56 3.52 3.47 3.42

2.81 3.92 3.93 3.94 3.95 3.97

4.30

4.30

3.99 3.92 3.85 3.79 3.73 3.67

3.06 3.84 3.86 3.87 3.89 3.90

4.32

4.31

3.41 3.32 3.31 3.24

2.30

2.89 2.88 2.87 2.85 2.83

2.27 4.26 4.28 4.31 4.33

2.24

2.82 2.81 2.79 2.77

624

J. Humlk3ek et al. TABLE I(b)

(Continued)

SixGel-x, x = 0.50

eV 3.36 3.35 3.34 3.32 3.30 3.28 3.26 3.25 3.24 3.22 3.20 3.18 3.16 3.15 3.14 3.12 3.10 3.08 3.06 3.04 3.02 3.00 2.98 2.96 2.94 2.92 2.90 2.88 2.86 2.84 2.82 2.80 2.78 2.76 2.75 2.74 2.72 2.70 2.68 2.66 2.65 2.64 2.62 2.60 2.58

2.56 2.55

cIil - I 27,100 27,010 26,930 26,770 26,610 26,450 26,290 26,210 26,130 25,970 25,810 25,640 25,480 25,400 25,320 25,160 25,000 24,840 24,680 24,510 24,350 24,190 24,030 23,870 23,710 23,5.50 23,390 23,220 23,060 22,900 22,740 22,580 22,420 22,260 22,180 22,090 21,930 21,770 21,610 21,450 21,370 21,290 21,130 20,970 20,800 20,640 20,560

pm 0.3690 0.3701 0.3712 0.3734 0.3757 0.3780 0.3803 0.3815 0.3827 0.3850 0.3875 0.3899 0.3924 0.3936 0.3949 0.3974 0.4000 0.4025 0.4052 0.4078 0.4105 0.4133 0.4161 0.4189 0.4217 0.4246 0.4275 0.,1305 0.4335 0.4366 0.4397 0.4428 0.4460 0.4492 0.4509 0.4525 0.45.58 0.4592 0.4626 0.4661 0.4679 0.4696 0.4732 0.4769 0.4806 0.4843 0.4862

n

n 4.35

4.44

4.48

4.38 4.41 4.43 4.46 4.48

4.52

4.55

5.09 5.13 5.16

4.82

1.97 1.96 1.95 1.94 1.94 1.93 1.91 1.89 1.87 1.84 1.80 1.75 1.62 1.55 1.46

2.50 2.48 2.46 2.45 2.44 2.43 2.42 2.42 2.41 2.41 2.40 2.39 2.38 2.35 2.32 2.28 2.23 2.16 2.09 2.00

1.24 5.55 5.60 5.64 5.65 5.65

5.01

4.91

2.05

2.62 2.60 2.57 2.55 2,52

2.00 4.60 4.61 4.63 4.64 4.66 4.68 4.71 4.74 4.78 4.82 4.87 4.92 4.98 5.05 5.12 5.19 5.27 5.34 5.42 5.48

5.17

5.10

2.15

2.74 2.72 2.69 2.67 2.65

2.10 4.51 4.53 4.55 4.56 4.58

4.57

4.61 4.62 4.64 4.66 4.69 4.73 4.77 4.80 4.85 4.90 4.95 5.01

2.76 2.20

1.04

1.90 1.79 1.66 1.53 1.40

0.89 5.63 5.59 5.55 5.50 5.45

0.79

0.72

1.28 1.17 1.07 0.99 0.91

Silicon-Germanium Alloys (SixGel_,,)

625

TABLE l(b) (Continued) SixGel_x, x = 0.50 eV 2.54 2.52 2.50 2.48 2.46 2.45 2.44 2.42 2.40 2.38 2.36 2.35 2.34 2.32 2.30 2.28 2.26 2.25 2.24 2.22 2.20 2.18 2.16 2.15 2.14 2.12 2.10 2.08 2.06 2.05 2.04 2.02 2.00 1.98 1.96 1.95 1.94 1.92 1.90 1.88 1.86 1.85 1.84 1.82 1.80 1.78 1.76

cm-

1

20,480 20,320 20,160 20,000 19,840 19,760 19,680 19,510 19,350 19,190 19,030 18,950 18,870 18,710 18,550 18,380 18,220 18,140 18,060 17,900 17,740 17,580 1.7,420 17,340 17,260 17,090 16,930 1.6,770 16,610 16,530 16,450 16,290 16,130 15,970 15,800 15,720 15,640 15,480 1.5,320 1.5,160 1.5,000 14,920 1.4,840 1.4,670 1.4,510 1.4,350 1.4,190

.

~um

0.4881 0.4920 0.4959 0.4999 0.5040 0.5061 0.5081 0.5123 0.5166 0.5209 0.5254 0.5276 0.5299 0.5344 0.5391 0.5438 0.5486 0.5510 0.5535 0.5585 0.5636 0.5687 0.5740 0.5767 0.5794 0.5848 0.5904 0.5961 0.6019 0.60:18 0.6078 0.6138 0.6199 0.6262 0.6326 0.6358 0.6391 0.6458 0.6526 0.6595 0.6666 0.6702 0.6738 0.6812 0.6888 0.6965 0.70:15

n

n

4.74

5.40 5.34 5.29 5.24 5.19 5.15 5.10 5.06 5.01 4.97 4.93 4.89 4.85 4.81 4.77

4.47

4.44

4.45 4.42 4.40 4.38 4.36

4.29

4.26

4.34 4.32 4.30 4.28 4.26

4.24

4.21

0.52

0.41 0.40 0.35 0.36 0.34 0.30

0.46

0.44 4.58 4.55 4.52 4.50 4.47

4.35

4.32

0.58

0.58 0.52 0.49 0.46 0.44

0.49 4.74 4.71 4.67 4.64 4.61

4.41

4.38

0.66

0.83 0.77 0.72 0.66 0.61

0.55

4.56

4.52

k

0.62

4.67

4.61

k

4.25 4.23 4.21 4.20 4.18

0.41

626

J. Humli6ek et al. TABLE l(b)

(Continued)

SixGel_x, x = 0.50 eV 1.74 1.72 1.70 1.68 1.66 1.60 1.50 1.40 1.30 1.29 1.28 1.27 1.26 1.25 1.24 1.23 1.22 1.21 1.20 1.19 1.18 1.17 1.16 1.15 1.14 1.13 1.12 1.11 1.10 1.09 1.08 1.07 1.06 1 .O5 1.04 1.03 1.02 1.01 1.00 0.99 0.98 0.97 0.96 O.95 0.94 0.93 0.92

cm -1

~Lm

14,030 13,870 13,710 13,550 13,380 12,900 12,090 11,290 10,480 10,400 10,320 10,240 10,160 10,080 10,000 9,921 9,840 9,759 9,679 9,598 9,517 9,437 9,356 9,275 9,195 9,114 9,033 8,953 8,872 8,791 8,711 8,630 8,549 8,469 8,388 8,307 8,227 8,146 8,065 7,985 7,904 7,824 7,743 7,662 7,582 7,501 7,420

0.7126 0.7208 0.7293 0.7380 0.7469 0.7749 0.8266 0.8856 0.9537 0.9611 0.9686 0.9763 0.9840 0.9919 0.9999 1.008 1.016 1.024 1.033 1.041 1.0.50 1.0.59 1.068 1.078 1.087 1.097 1.107 1.116 1.127 1.137 1.148 1.158 1.169 1.180 1.192 1.203 1.215 1.227 1.239 1.252 1.265 1.278 1.291 1.305 1.318 1.333 1.347

n

n

4.16

4.16 4.14 4.13 4.11 4.09

4.11 4.06 4.01 3.96

3.92

3.88

3.84

7.24E-03 [41 6.91E-03 6.59E-03 6.27E-03 5.96E-03 5.66E-03 5.36E-03 5.07E-03 4.78E-03 4.5oE-o3 4.23E-o3 3.96E-03 3.7oE-o3 3.44E-o3 3.2oE-o3 2.96E-o3 2.73E-o3 2.51 E-03 2.29E-03 2.08E-03 1.88E-03 1.69E-03 1.51E-03 1.3,1 E-03 1.18E-03 1.02E-03 8.81E-04 7.48E-04 6.26E-04 5.14E-04 4.14E-04 3.25E-04 2.47E-04 1.82E-04 1.30E-04 9.07E-05 6.48E-05 4.67E-05 3.14E-05

Silicon-Germanium Alloys (SixGel_x) TABLE I(b)

627 (Continued)

SixGel-x, x = 0.50 eV 0.91 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.1363 0.1301 0.1239 0.1178 0.1116 0.1091 0.1066 0.1041 0.1017 0.1000 0.09919 0.09671 0.09423 0.09175 0.08927 0.08679 0.08431 0.08183 0.07935 0.07687 0.07439 0.07191 0.06943 0.06695 0.06447 0.06323 0.06199 0.06075 0.05951 0.05827 0.05703 0.05455 0.05207 0.050&3 0.04959 0.04835 0.04711 0.04587

cm -1

pm

7,340 7,259 6,452 5,646 4,839 4 033 3,226 2,,120 1,613 1,100 1,050

1.362 1.377 1.549

1,000

950 9O0 880 86O 840 82O 8O7 8OO 78O 760 740 720 7OO 68O 660 640 620 6OO 580 56O 54O 52O 510 5OO 490 48O 470 46O 440 42O 410 400 390 380 37O

1.771

2.066 2.479 3.099 4.132 6.199 9.091 9.524 10.00 10.52 11.11

n

3.80 3.77 3.75 3.72 3.70 3.69 3.67 3.67

1.90E-05 9.66E-06

3.92E-05 [11 ] 4.84E-05 5.87E-05 8.49E-05 1.29E-04 1.54E-04

11.36 11.62 11.9O

12.19 12.39 12.49 12.82 13.15 13.51 13.88 14.28 14.70 15.15 15.62 16.12 16.66 17.24 17.85 18.51 19.23 19.60 20.00 20.40 2O .83 21.27 21.74 22.72 23.81 24.39 25.00 25.64 26.31 27.02

n

1.89E-04

1.97E-04 1.81E-04

3.66 1.77E-04 1.73E-04 1.45E-04

1.32E-04 1.61E-04 2.32E-04 3.26E-04 3.49E-04 3.28E-04 3.33E-04 3.84E-04 4.55E-04 5.20E-04 6.31E-04 9.62E-04 1.13E-03 1.16E-03 1.02E-03 1.24E-03 1.30E-03 1.15E-03 9.01E-04 8.89E-04 1.12E-03 2.30E-03 2.62E-03 2.26E-03 1.87E-03

628

J. Humlff:ek et al. TABLE I(b) (Continued) SixGel-x, x = 0.50

eV 0.04463 0.04215 0.03968 0.03720 0.03596 0.03472 0.03348 0.03224 0.02976 0.02728 0.02480 0.02232

cm -1

l~m

360 340 320 300 290 280 270 260 240 220 200 180

27.78 29.41 31.24 33.33 34.47 35.70 37.03 38.45 41.66 45.44 50.00 55.54

rt

n

k

k 1.47E-03 1.43E-03 1.39E-03 1.57E-03 2.11E-03 2.13E-03 1.80E-03 1.71E-03 1.70E-03 1.70E-03 1.81E-03 1.95E-03

a T h e reference from which the values were e x l r a c t e d is given in brackets n e x t to t h e first value. T h e nominal alloy c o m p o s i t i o n s are: x=0.551 [6], x = 0 . 4 8 7 [8], x = 0 . 5 0 0 o b t a i n e d b y i n t e r p o l a t i n g the d a t a of Braul~stein et al. [4], x=0.,150 [11].

Silicon-Germanium Alloys (SixGel_x)

629

T A B L E I(c) Values of n and k for SixGe~_x, x = 0.80, from Various References" cV

12.8 12.6 12.2 11.8 11.4 11.0 10.6 10.2 9.80 9.40 9.00 8.60 8.20 7.90 7.80 7.40 7.00 6.90 6.80 6.70 6.60 6.50 6.40 6.30 6.20 6.10 6.00 5.90 5.70 5.60 5.50 5.40 5.30 5.28 5.26 5.24 5.22 5.20 5.18 5.16 5.14 5.12 5.10 5.08 5.06 5.04 5.02

cm -1

49,200 48,390 47,580 45,970 45,160 44,360 43,550 42,740 42,580 42,420 42,260 42,100 41,940 41,780 41,610 41,450 41,290 41,130 ,10,970 dO,810 ,10,650 40,480

pm 0.09686 0.09840 0.1016 0.1051 0.1088 0.1127 0.1170 0.1216 0.1265 0.1319 0.1378 0.1442 0.1512 0.1569 0.1590 0.1675 0.1771 0.1797 0.1823 0.1851 0.1879 0.1907 0.1937 0.1968 0.2000 0.2033 0.2066 0.2101 0.2175 0.2214 0.2254 0.2296 0.2.339 0.23-'18 0.2357 0.2366 0.2375 0.2384 0.2394 0.2403 0.2412 0.2422 0.2431 0.2441 0.2450 0.2460 0.2470

n 0.64 0.65 0.65 0.64 0.63 0.62 0.61 0.60 0.60 0.60 0.60 0.63 0.65 0.67 0.67 0.69 0.77 0.82 0.87 0.91 0.95 0.99 1.02 1.05 1.08 1.12 1.1 8 1.24 1.40 1.48 1.56 1.62 1.66

1.67

1.67

n

k

[6]

0.78 0.80 0.84 0.88 0.92 0.98 1.04 1.11 1.2O 1.29 1.4O 1.51 1.63 1.73 1.76 1.95 2.18 2.25 2.30 2.34

k

[6]

2.38

1.28 1.35 1.43 1.47 1.47 1 .,18 1.48 1.48 1 .,t8 1.48 1.48 1.48 1.48 1.45 1.48 1.48 1.49 1 .,19

[8]

2.42 2.45 2.51 2.57 2.64 2.71 2.78 2.89 2.92 2.95

2,96 2,98

3.01

3.06 [8] 3.11

3.13 3.14 3,15 3.15 3.16 3.17 3.18

3.20

3.08

3.21 3.23 3.25 3.27 3.29 3.31 3.34 3.37

630

J. HumliEek et al. TABLE I(c) (Continued) SixGel-x, x = 0.80

eV 5.00 4.98 4.96 4.94 4.92 4.90 4.88 4.86 4.84 4.82 4.80 4.78 4.76 4.75 4.74 4.72 4.7O 4.68 4.66 4.65

4.64 4.62 4.60 4.58 4.56 4.54 4.52 4.50 4.48 4.46 4.44 4.42 4.40 4.38 4.36 4.34 4.32 4.30 4.28 4.26 4.24 4.22 4.20 4.18 4.16 4.14 4.12

cm -1 40,320 40,160 40,000 39,840 39,680 39,520 39,360 39,190 39,030 38,870 38,710 38,550 38,390 38,310 38,230 38,060 37,900 37,740 37,580 37,500 37,420 37,260 37,100 36,940 36,770 36,610 36,450 36,290 36,130 35,970 35,810 35,640 35,480 35,320 35,160 35,000 34,840 34,680 34,520 34,350 34,190 34,030 2,3,870 33,710 33,550 33,390 33,230

pm 0.2480 0.2490 0.2500 0.2510 0.2520 0.2530 0.2541 0.2551 0.2562 0.2572 0.2583 0.2594 0.2605 0.2610 0.2616 0.2627 0.2s 0.2649 0.2661 0.2666 0.2672 0.2684 0.269.5 0.2707 0.2719 0.2731 0.2743 0.2755 0.2768 0.2780 0.2792 0.2805 0.2818 0.2831 0.2844 0.2857 0.2870 0.2883 0.2897 0.2910 0.2924 0.2938 0.2952 0.2966 0.2980 0.2995 0.3009

n

n

k

k

1.67

1.50 1.50 1.51 1.51 1.52 1.53 1.54 1.55 1.57 1.58 1.60 1.61 1.63

3.19

3.39 3.42 3.45 3.48 3.52 3.55 3.59 3.62 3.66 3.70 3.74 3.78 3.83

1.69

1.75

1.81

1.88

3.56

3.68 1.65 1.68 1.70 1.73 1.76

1.95

2.07 2.12 2.19 2.25 2.33 2.41 2.51 2.60 2.71 2.83 2.96 3.10 3.25 3.41 3.59 3.76 3.92 4.04 4.15 4.23 4.32 4.40 4.45 4.47 4.49

3.36

3.80

3.88 3.93 3.98 4.03 4.08

3.95 1.79 1.83 1.87 1.92 1.97 2.03 2.10 2.18 2.26 2.35 2.45 2.56 2.68 2.81 2.96 3.13 3.30 3.48 3.64 3.80 3.93 4.03 4.13 4.20 4.25 4.29 4.33

4.10 4.16 4.23 4.29 4.3.5 4.41 4.47 4.52 4.58 4.62 4.66 4.70 4.72 ,1.72 4.70 4.64 4.54

4.44 4.34 4.23

4.14 4.03 3.89 3.78 3.69

4.14 4.20 4.26 4.33 4.39 4.46 4..52 4.59 4.66 4.72 4.78 4.84 4.90 4.94 4.98 5.00 5.00 4.97 4.91 4.83 4.73 4.62 4.51 4.40 4.30 4.21 4.11

Silicon-Germanium Alloys (SixGel_x) T A B L E I(c)

631 (Continued)

SixGel_x, x = 0 . 8 0

eV 4.10 4.08 4.06 4.04 4.02 4.00 3.98 3.96 3.95 3.94 3.92 3.90 3.88 3.86 3.85 3.84 3.82 3.80 3.78 3.76 3.75 3.74 3.72 3.70 3.68 3.66 3.65 3.64 3.62 3.60 3.58 3.56 3.55 3.54 3.52 3.50 3.48 3.46 3.44 3.42 3.40 3.38 3.36 3.34 3.32 3.30 3.28

cm -1 33,060 32',900 32,740 32,580 32,420 32,260 32,100 31,930 31,850 31,770 31,610 31,450 31,290 31,130 31,050 30,970 30,810 30,640 30,480 30,320 30,2-10 30,160 30,000 29,&10 29,680 29,520 29,430 29,350 29,190 29,030 28,870 28,710 28,630 28,550 28,390 28,220 28,060 27,900 27,740 27,580 27,420 27,260 27,100 26,930 26,770 26,610 26.450

~m 0.3024 0.3039 0.3054 0.3069 0.3084 0.3100 0.3115 0.3131 0.3139 0.3147 0.3163 0.3179 0.3195 0.3212 0.3220 0.3229 0.3246 0.3263 0.3280 0.3'297 0.3.306 0.3315 0.3333 0.3351 0.3.369 0.3388 0.3397 0.3406 0.3425 0.34.14 0.3463 0.3483 0.3493 0.3502 0.3522 0.35.t2 0.3563 0.3583 0.3604 0.3625 0.3647 0.3668 0.3690 0.3712 0.373,1 0.3757 0.3780

n

n

k

k

4.51 4.52 4.52 4.53 4.53 4.53

4.36 4.38 4.40 4.41 4.43 4.44 4.45 4.46

3.60 3.52 3.45 3.38 3.32 3.26

4.03 3.95 3.88 3.81 3.75 3.69 3.64 3.59

4.54

4.54

3.13 4.47 4.48 4.50 4.51 4.52

4.54

4.55

2.95 4.53 4.54 4.56 4.57 4.59

4.57

4.60

2.76

3.21 3.19 3.16 3.14 3.12

2.71 4.69 4.71 4.73 4.75 4.77

4.71

4.77 4.80 4.84 4.88 4.92 4.98 5.03 5.10 5.17 5.24 5.32 5.40

2.88

3.36 3.32 3.30 3.27 3.24

2.81 4.60 4.62 4.64 4.65 4.67

4.63

4.66

3.03

3.55 3.50 3.46 3.42 3.39

2.67

3.10 3.08 3.06 3.05 3.03

2.64 4.81 4.82 4.86 4.90 4.94 4.98 5.03 5.09 5.15 5.21 5.29 5.37 5.,16 5.55

2.61 2.61 2.60 2.59 2.58 2.57 2.55 2.53 2.50 2.,16 2.42 2.36

3.05 3.01 3.00 2.99 2.98 2.97 2.96 2.95 2.93 2.91 2.89 2.86 2.82 2.76

632

J. HumliEek et al. TABLE I(c)

(Continued)

SixGel-x, x = 0.80 eV 3.26 3.24 3.22 3.20 3.18 3.16 3.14 3.12 3.10 3.08 3.06 3.05 3.04 3.02 3.00 2.98 2.96 2.95 2.94 2.92 2.90 2.88 2.86 2.85 2.84 2.82 2.80 2.78 2.76 2.75 2.74 2.72 2.70 2.68 2.66 2.65 2.64 2.62 2.60 2.58 2.56 2.55 2.54 2.52 2.50 2.48 2.46

cm -1 26,290 26,130 25,970 25,810 25,640 25,48n 25,320 25,160 25,000 24,840 24,680 24,600 24,510 24,350 24,190 24,O30 23,870 23,790 23,710 23,550 23,390 23,220 23,060 22,980 22,900 22,740 22,580 22,420 22,260 22,180 22,090 21,930 21,770 21,610 21,450 21,370 21,290 21,130 20,970 20,800 20,640 20,560 20,480 20,320 20,160 20,000 19,840

tLm 0.3803 0.3827 0.3850 0.3875 0.3899 0.3924 0.3949 0.3974 0.4000 0.4025 0.4052 0.4065 0.4078 0.4105 0.4133 0.4161 0.4189 0.4203 0.4217 0.4246 0.,1275 0.4.305 0.4.335 0.4350 0.4366 0.4397 0.4428 0.4460 0.4492 0.4509 0.4525 0.4558 0.4592 0.4626 0.4661 0.4679 0.4696 0.4732 0.4769 0.4806 0.4843 0.4862 0.4881 0.4920 0.4959 0.4999 0.5040

n

n

k

k

5.48 5.55 5.62 5.68 5.73 5.76 5.79 5.80 5.79

5.64 5.73 5.82 5.89 5.96 6.01 6.06 6.10 6.11 6.11 6.08

2.28 2.19 2.10 1.99 1.87 1.75 1.62 1.48 1.34

2.70 2.61 2.52 2.41 2.30 2.17 2.03 1.88 1.73 1.58 1.44

1.03

5.66

5.49

6.04 5.99 5.92 5.86 5.78

0.65

5.32

5.15

5.71 5.64 5.57 5.50 5.44

0.55

0.79 0.73 0.66 0.60 0.56

0.47

5.00

4.88

0.81

1.30 1.17 1.06 0.95 0.87

5.38 5.32 5.26 5.21 5.16

0.43

0.51 0.48 0.44 0.42 0.39

4.76

4.65

5.11 5.06 5.02 4.97 4.93

4.56

4.50

4.89 4.85 4.82 4.78 4.75

4.44

4.39

4.71 4.68 4.65 4.62 4.59

0.37 0.34 0.32 0.30

Silicon-Germanium Alloys (SixGel_x) TABLE l(c)

633

(Continued)

SixGel-x, x = 0.80 eV 2.45 2.44 2.42 2.40 2.38 2.36 2.34 2.32 2.30 2.28 2.26 2.24 2.22 2.20 2.18 2.16 2.14 2.12 2.10 2.08 2.06 2.04 2.02 2.00 1.98 1.96 1.94 1.92 1.90 1.88 1.86 1.84 1.82 1.80 1.78 1.76 1.74 1.72 1.70 1.68 1.66 1.60 1.50 1.49 1.48 1.47 1.46

cm -1 19,760 19,680 19,510 19,350 19,190 19,030 18,870 18,710 18,550 18,380 18,220 18,060 17,900 17,740 17,580 17,420 17,260 17,090 16,930 16,770 16,610 16,450 16,290 16,130 15,970 15,800 1.5,640 15,480 15,320 15,160 15,000 14,840 14,670 14,510 14,350 14,190 14,0.30 13,870 13,710 13,550 13,380 12,900 12,090 12,010 11,930 11,850 11,770

ll.m 0.5061 0.5081 0.5123 0.5166 0.5209 0.5254 0.5299 0.5344 0.5391 0.5438 0.5486 0.5535 0.5585 0.5636 0.5687 0.5740 0.5794 0.5848 0.5904 0.5961 0.6019 0.6078 0.6138 0.6199 0.6262 0.6326 0.6391 0.6458 0.6526 0.6595 0.6666 0.6738 0.6812 0.6888 0.6.965 0.7045 0.7126 0.7208 0.7293 0.7380 O. 7469 0.7749 0.8266 0.8321 0.8377 0.8,134 0.8492

n

n

k

k

4.35

4.30

4.23

4.16

4.11

4.05

4.00

3.96

3.91

4.57 4.54 4.51 4.49 4.46 4.44 4.41 4.39 4.37 4.35 4.33 4.31 4.29 4.27 4.25 4.23 4.21 4.19 4.18 4.16 4.14 ,t.13 4.11 4.10 4.08 4.07 4.05 4.04 4.03 4.01 4.00 3.99 3.98 3.97 3.95 3.94 3.93 3.92 3.91 3.90

3.87 3.84 6.43E-03 [4] 6.19E-03 .5.96E-03 5.73E-03

634

J. Humli~.ek et al. TABLE I(c) (Continued) SixGel-x, x = 0.80

eV 1.45 1.44 1.43 1.42 1.41 1.40 1.39 1.38 1.37 1.36 1.35 1.34 1.33 1.32 1.31 1.30 1.29 1.28 1.27 1.26 1.25 1.24 1.23 1.22 1.21 1.20 1.19 1.18 1.17 1.16 1.15 1.14 1.13 1.12 1.11 1.10 1.09 1.08 1.07 1.06 1.05 1.04 1103 1.02 1.01 1.00 0.99

cm- 1 11,690 1.1,61o 11,530 11,45o 11,37o 11,29o 11,21o 11,13o 11,050 1o,96o lO,88o lO,8OO 1o,72o lO,64o 10,560 lO,48o 10,400 10,320 10,24o 10,16o 1o,98o lO,OOO 9,921 9,840 9,759 9,679 9,598 9,517 9,437 9,356 9,275 9,195 9,114 9,033 8,953 8,872 8,791 8,711 8,630 8,549 8,469 8,388 8,307 8,227 8,146 8,065 7,985

p.m o.8551 o.861o 0.8670 o.8731 0.8793 0.8856 0.8920 0.8984 0.9050 o.9117 o.9184 0.9253 0.9322 0.9393 0.9465 0.9537 o.9611 0.9686 0.9763 o.98,10 0.9919 0.9999 1.008 1 .o16 1.o24 1.033 1 .o41 1.o5o 1.059 1.068 1.078 1.087 1.097 1.1o7 1.116 1.127 1.137 1.148 1.158 1.169 1.18o 1.192 1.2(~3 1.215 1.227 1.239 1.252

n

3.80

n

k

k 5.5oE-03 5.28E-O3 5.O6E-03 4.84E-03 4.63E-O3 4.42E-o3 4.21E-O3 4.Ol E-03

3.81E-o3 3.61E-o3 3.42E-o3 3.23E-03 3.05 E-03 2.87E-03 2.70E-03

3.76

2.53E-03 2.36E-O3 2.20E-03 2.04E-03 1.89E-03 1.74E-03 1.60E-03 1.47E-O3 1.33E-03 1.21E-03

3.73

3.69

3.65

1.09E..03 9.73E-04 8.64E-04 7.61E-04 6.64 E-04 5.74E-04 4.90E-04 4.13E-04 3.42E-04 2.79E-04 2.22E-04 1.74E-04 1.32E-04 9.89E-05 7.34E-05 5.63E-05 4.55E-05 3.61E-05 2.76E-o5 2.O2E-O5 1.39E-O5 8.76E-O6

Silicon-Germanium Alloys (SixGel_x)

635

TABLE I(c) ( C o n t i n u e d ) SixGel-x, x = 0.80

eV

cm- 1

,um

n

0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.1487 0.1463 0.1438 0.1413 0.1388 0.1363 0.1339 0.1314 0.1289 0.1264 O. 1239 0.1215 0.1190 0.1165 O. 1141 0.1116 0.1091 O. 1066 0.1041 0.1017 0.1000 0.09919 0.09671 0.09423 0.09175 0.08927 0.08679 0.08555 0.08..131 0.08183 0.07935 0.07687 0.07439 0.07191 0.07067 0.06943 0.06695 0.06447 0.06199

7,259 6,452 5,646 4,839 4,033 3,226 2,420 1,613 1,200 1,180 1,160 1,140 1,120 1,100 1,080 1,060 1,040 1,020 1,000 980 960 940 920 900 880 860 840 820 807 800 780 760 740 720 700 690 680 660 640 620 600 580 570 560 540 520 500

1.377 1.549 1.771 2.066 2.479 3.099 4.132 6.199 8.333 8.,174 8.620 8.772 8.928 9.091 9.259 9.434 9.615 9.804 10.00 10.20 10.41 10.63 10.86 11.11 11.36 11.62 11.90 12.19 12.39 12.49 12.82 13.15 13.51 13.88 14.28 14.49 14.70 15.15 15.62 16.12 16.66 17.24 17.54 17.85 18.51 19.23 20.00

3.63 3.60 3.58 3.56 3.55 3.53 3.52 3.52

n

k

k

1 . o 5 E - o 5 [111 1.48E-O5 1.92E-O5 2.53E-O5 3.18E-O5 3.85E-O5 3.77E-o5 3.36E-o5 2.78E-o5 2.51E-o5 3.22E-O5 6.47E-O5 9.31E-o5 1.05E-o4 1.27E-o4 1.53E-o4 1.91E-o4 2.o6E-O4 1.84E-o4 1.52E-o4

3.51 1.41E-O4 1.45E-o4 1.59E-o4 1.8oE-o4 2.15E-O4 2.37E-o4 2.37E-o4 2.28E-O4 2.21E-o4 2.63E-o4 3.63E-o4 4.46E-04 5.ooE-o4 5.13E-O4 5.17E-o4 5.16E-O4 5.30E-04 7.53E-o4

636

J. Humli(:ek et al.

TABLE l(c) (Continued) SixGel-x, x = 0.80 eV 0.06075 0.05951 0.05827 0.05703 0.05579 0.05455 0.05331 0.05207 0.05145 0.05083 0.05021 0.04959 0.04897 0.04835 0.04711 0.04587 0.04463 0.04339 0.04215 0.03968 0.03720 0.03472 0.03224 0.02976 0.02728 0.02604 0.02480 0.02356 0.02232

cm -1 490 480 470 460 450 440 430 420 415 410 405 400 395 390 380 370 360 350 340 320 300 280 260 240 220 210 200 190 180

#m 20.40 20.83 21.27 21.74 22.22 22.72 23.25 23.81 24.09 24.39 24.69 25.00 25.31 25.64 26.31 27.02 27.78 28.57 29.41 31.24 33.33 35.70 38.45 41.66 45.44 47.61 50.00 52.62 55.54

n

n

k

k 9.45E-04 1.01E-03 9.21E-04 6.81E-04 6.06E-04 6.33E-04 5.97E-04 5.61E-04 5.55E-04 6.93E-04 9.63E-04 1.06E-03 8.23E-04 6.58E-04 4.42E-04 3.16E-04 2.91E-04 3. I 1 E-04 2.85E-04 1.98E-04 1.88E-04 1.64 E-04 1.24E-04 7.93E-05 2.84E-05 2.68E-04 4.28E-04 1.24E-03 1.49E-03

a T h e reference fi'om whid~ the values were e x t r a c t e d is given in brackets n e x t to the first value. T h e nominal alloy compositions are: x = 0 . 8 0 3 [6], x=0.782 [8], x = 0 . 8 0 0 o b t a i n e d by i n t e r p o l a t i n g the d a t a of B r a u n s t e i n et al. [4], x = 0 . 8 0 0 [11].

Lead Tin Telluride (PbSnTe) H. PILLER Louisiana State University Baton Rouge, Louisiana

There are many alloys of Pb~_xSnxTe. We can tabulate data only for a very few on the Pb-rich side. Also, it is not possible to find the same x value over a wide spectral range. The reflectivity of the lead-salt narrow-gap alloy PbSnTe was studied by Korn and Braunstein [1, 2]. Measurements of near-normal reflectivity were made at room temperature in the photonenergy range of 0.5-13 eV. Czochralski-grown p-type bulk samples were used in the measurements. Samples were cut, ground, and hand-polished to produce optical-quality surfaces. After an electrochemical etch, samples were immediately placed in a vacuum (at least 10 -5 tort), and spectral scans were begun. Consistent results were obtained upon repreparation of sample surfaces. The maximum estimated absolute reflectance error throughout the spectral range was at most 2%. The optical constants n and k were obtained through a Kramers-Kronig computation. Pbl_xSnxTe can be grown throughout the range 0~~2>~4~tm do not correspond to those of Heitmann and Ritter [3] and Kim and Vedam [17]. Namely, the values of n found by Mouchart et al. at 2 = 4 ~tm and the wavelengths close to that are relatively high compared with the refractive-index values published for the wavelengths of the visible in Refs. [3] and [17]. Moreover, dispersion of n presented by Mouchart et al. is rather pronounced. (An extrapolation of the data of Mouchart et al. to the visible and near-UV region gives values of n substantially higher than those presented in Refs. [3] and [17]). These discrepancies may be explained by the fact that the films analyzed by Mouchart et al. [14] were probably formed with a mixture of thorium fluoride and thorium oxide. For example, ThO2 films are characterized by considerably higher values of the refractive index than the ThF4 films in the visible and near-UV region: the values of n characterizing the ThO2 films lie in the interval 1.75-1.9 from the visible to the near UV region [21]. Other, more reliable data of ThF4 films are not available in the mid-IR region at present. Values of n and k of ThF4 films for the far-IR region have not been presented yet, since spectrophotometric or Raman measurements have not been performed in this region (that is, optical-phonon frequencies of ThF4 are not known).

1054

I. Ohlidal and K. Navr&til

In conclusion, it should be noted that thorium and its compounds are radioactive. In general, no risk of external radiation damage is present. However, incorporation of thorium compounds into the human organism does present danger. Recommended maximum permissible limits for thorium compounds in the body are outlined in Ref. [3]. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

J. Ebert, H. Panhorst, H. Kiister and H. Welling, Appl. Opt. 18, 818 (1979). H. K. Pulker, Appl. Opt. 18, 1969 (1979). W. Heitmann and E. Ritter, Appl. Opt. 7, 307 (1968). C. S. Barrett, "Structure of Metals," McGraw-Hill, New York, (1952). D. E. Aspens, Thin Solid Films 89, 249 (1982). K. L. Chopra, S. K. Sharma, and V. N. Yadava, Thin Solid Films 20, 209 (1974). M. Harris, H. A. Macleod, S. Ogura, E. Pelletier, and B. Vidal, Thin Solid Films 57, 173 (1979). I. Ohlidal and K. Navr~itil, Thin Solid Films 74, 51 (1980). D. A. G. Bruggeman, Ann. Phys. (Leipzig) 24, 636 (1935). D. E. Aspnes, J. B. Theeten, and F. Hottier, Phys. Rev. B 20, 3292 (1979). A. Heisen, Optik 18, 59 (1961). H. K. Pulker, Thin Solid Films 34, 343 (1976). H. Ahrens, H. Welling, and W. Scheel, Appl. Phys. 1, 69 (1973). J. Mouchart, G. Lagier, and B. Pointu, Appl. Opt. 24, 1808 (1985). J. C. Manifacier, J. Gasiot, and J. P. Fillard, J. Phys. E 9, 1002 (1976). M. E. Pedinoff, M. Braunstein, and O. M. Stafsudd, Appl. Opt. 16, 1849 (1977). S. Y. Kim and K. Vedam, Thin Solid Films 166, 325 (1988). P. J. Martin, H. A. Macleod, R. P. Netherfield, C. G. Pacey, and W. G. Sainty, Appl. Opt. 22, 178 (1983). G. A. AI-Jumaily, J. J. McNally, J. R. McNeil, and W. C. Herrmann, Jr., J. Vac. Sci. Technol. A 3,651 (1985). G. A. AI-Jumaily, L. A. Yazlovitsky, T. A. Monney, and A. Smajikiewicz, Appl. Opt. 26, 3752 (1987). P. H. Lissberger, Rep. Prog. Phys. 39, 197 (1970).

Thorium Fluoride (ThF4)

101

1055

,

'

I'

'''f

,

'

,'

i'

'''l

100

n,k

10-1

10-2

10-3 10-1

,

,

I

. . . .

I

,

,

,

[

,

, ,,I

100 101 WAVELENGTH [Nm]

Fig. 1. Log-log plot of n (solid line) and k (dashed line) versus wavelength in micrometers for thorium fluoride.

1056

I. Ohl[dal and K. Navr&til

.,-4 +.~ O O

v 9.,,-

N

O 9

CO ~ O Lf~ "~IP" t ' ~ O J - - - ,--- O 9

9

9

9

9

9

9

9

9

O

9

9

C ~ (~'~ Cr~ CO C ~ a ~ i,...- i.,,- D.-- ~ 9

9

9

9

9

9

9

9

9

9

D.... D--. 9

9

I..,

o~

p.

E o~

r--v

~r.., r - - ' l

L_j 9-.- O ~ a ~ I,-- ~,J~ ~.O u'~ u'~ " ~ 9

9

9

9

9

9

9

',ql~ ',~- '~1~ r

4

9

9

9

9

("~ ~"~ c"~ ~"~ (""~ 0,,11 0,,I 0 J N 9

9

9

9

9

9

9

9

9

9

( ' ~ ('~1 0,J 9

9

9

o,.u O

N

C'~ "~1~' L t ' ~ 0 9

O

9

O

9

O

9

O

It"-- a:) (:~'~ O

9

O

9

O

9

O

9

O

9

O

,,-, N 9

O

9

O

~"~ ,,~1t-I.f'~ ~:) I",- a:) C ~ , O 9

O

9

O

9

O

9

O

9

O

9

O

9

O

9

O

9

P

N

9

O

O

~"~ ~'~ ',,~1"U"~

9

O

9

O

9

9

O

O

n

p,

"7

IA"~ 1"-- ~.O O 9,~1" .,ml- ~ O O

~ C'~ , - - CO e ' ~ If'~ I r ~ O , - - r , - D,-- N ',ml- O ~ - - ,~0- e,'~ O J O ~ C,'~ ,,el" I ~ I'--- O

, - - ,~1- O e,'~.O O

O'~ O LC~ ' ~ O,J O~1 e"~ OO O J O'~ D..- O J

N

N

to .,~1~ -,~!,- ,,~I,- .,,~11,.t~',~ t~'~ (',,~ (','~ ~,",~ C,~ Cr~ ~'~ N

N

N

N

N

N

N

N

N

N

N

Thorium Fluoride (ThF4)

1057

i)

.r-I +.~ 0 0

9--9--

0

0%00

,-.

9..

(M,-L._.JL._J 0 t'-~O

9-.

V---- q . .

lur% ~1'- ( ~ , o a ~ - 0

9---- 9--- 9--. 9---

9---- V - o

GO (:I:) r ' - ~ 0

V--. 9---

",e"

9--- 9---- 9---- 9---

I''3

C~J

oa

9---

t._.J ~D

I

!

0

0

9..-

(~

i.----i

N

0a N

0a N

0,1 , - - , . - ,-.- 0 0 N N N N N N

0a

0a ,U ~ ur~ ur% i~r% ur~ u ~ ur% 9

E,-

9

9

9

9

9

9

,F-. 9-- ,e.-. 9 - - ~e--. 9 . - ,~..

%0 r'-- cO c~

TE~

o

o

o

o

o

o

~ (~

I " " ('% CO 0 I'~" ( ~ 0 0

,-. --- N

o

o

o

o

' , ~ ',~1' ~.0 a : ) o

Oa "~1"~0 0 ",~" e O 0 0 0 ~ % 0 ~ 0 r.-- t-.- I ' . - CO Cr~ 0

o

0

o

o

o

O 0 CO , - - 0 ' ~ N I'-.- ,.-- .t~-0 ~ ( ~ - - - ~-.- I / ~ "~1"~:)

0

0

0

0

( : ~ ILr~ N ~ O

0

o

0

(~i,--- 0,--

~0 00~J

~.- ~ -

0

I~

~,-- , -

"qt" ( ~

0 Od

I"- 0

, - - ~.0 ~ -

( M I~'% (E) I/'% ~-- 0

",el" f " ~ ~4::) 0

9--.

9-,,

9--

0 0 o

r ' . - (7~ q..- ~ 0 0 (X) 0 0 (~D ~ ( ~ r - - I'..- r.-. C ~ t J ~ r..- .-t- I'--.. f,'~ o d ~ o % c o t'.-- ~ u~ ~ ( ~ (%l , . - o (:E) ~ o

0 9- - 0 O a C~J N

0 0 C ~ Cr% O ~ ( : 0 CO I " - r ' . - ~ : ) Oa Oa

Lt'% CO ( ~ 0 0 ( ~ ( ~ (X) ( ~ ( X ) 9

9

9

9

9

, - - t".- ',el" ~.0 q-.- "~l" ( : 0 ~ 0 f~O E ) ( : ~ I ~ " t "-- f " % ~ O 9

9

9

9

9

9

9

9

O J O,J O,J O J C'%J O J O J O J O,J C'~ O J O J O J

9

9

Oa ,-.

~---

9

9

9,-,

9

q-.-

9

9--

9

~-- q-- 9 - - ,,-- ~ -

9---

9

v-,.

9

9

9

4

9

e - - v-.. , - - ~,- 0

0

Water (H20) MARVIN R. QUERRY and DAVID M. WIELICZKA Department of Physics University of Missouri-Kansas City Kansas City, Missouri and

DAVID J. SEGELSTEIN Bell Communications Research Red Bank, New Jersey One could easily prepare a list of several hundred primary references to the scientific literature on spectral measurements of the optical properties of water and compilations of water's refractive index and absorption coefficient. In this critique, however, we only highlight such activities during the past century. Readers interested in more detail should consult bibliographies in the references cited here. To those who contributed to the scientific literature on the optical properties of water and are not explicitly acknowledged here, we apologize. We conclude the critique with two of our previously unpublished compilations of n and k for water. In 1892, Rubens [1] presented refractometer measurements of the refractive index n(2) of water to a long-wavelength limit of 1.256 ~tm. These measurements of n(2) were extended to a new long-wavelength limit of 2.327 ~tm by Seegert's [2] doctoral work in 1908. In 1908 and 1909, Rubens and Ladenberg [3] published their measurements of the reflectance and absorption spectra, and determined what was later discovered to be incorrect dispersion n(2) of water out to a wavelength of 18 ~tm. Others who measured the infrared reflectance spectrum of water in 1906, 1915, 1918, and 1931, respectively, were Pfund [4], Gehrts [5], Brieger [6], and Weingeroff [7]. General reviews of the infrared absorption spectrum of water were published in 1917, 1929, 1930, and 1934 by Fowle [8], Lecompte [9], Schaefer and Matossi [10], and Dawson and Hulburt [11], respectively. Compilations of the optical properties of water were published in 1929 by Becquerel and Rossignol [12], and in 1940 by Dorsey [13]. One can gain a historical and technical perspective from Dorsey's thorough compilation of water's optical properties. He presented data, gleaned from about 320 references, that spanned the spectral range from low frequency (dielectric constants) to the X-ray region (absorption coefficients). For example, Dorsey's Table 137 provides values for n(2) in the HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS II

1059

Copyright 9 1991 by Academic Press. All rights of reproduction in any form reserved. ISBN 0-12-544422-2

1060

Marvin R. Querry et al.

wavelength range 0.154 n m - l l mm. For the visible and ultraviolet regions, Dorsey reviewed several dispersion relations for the refractive index of water at temperatures in the range of 0-80 ~ Of particular interest to Dorsey were the data of Martens [14], Duclaux and Jeantet [15], Ch6neveau [16], and Tilton and Taylor [17]. Among those who also published works prior to 1941 on the optical properties of water were Paschen [18], 1894; Coblentz [19], 1905, 1906, and 1911; Goldhammer [20], 1913; Collins [21], 1922 and 1939; Plyler [22], 1924; Dreisch [23], 1924; Hulburt [24], 1928; Cartwright [25], 1935; Williams et al. [26], 1936; and Fox and Martin [27], 1940. Centeno's [28] tabulation in 1941 of n and k for water, and the compilation previously published by Dorsey in 1940, were benchmarks for the future of such activity and were the standard of such compilations for nearly three decades. Centeno gleaned data from 59 articles, including all those previously noted here, to determine n and k for water in the 0.115118 gm and 1-18 ~tm wavelength regions, respectively. Centeno's values of n(2) in the ultraviolet, visible, and near infrared were from Refs. [13], [16], and [17]. Centeno corrected the infrared absorption spectrum of a 90% water and 10% glycerin mixture previously measured in 1909 by Rubens and Ladenburg [3], to obtain the absorption and extinction coefficients a(2) = 4:rk(2)/2 and k(2) of water. Possessing corrected values for k(2) and the normal-incidence infrared reflectance spectrum R(2) [3] of water, he determined n(2) by use of

e~

[ ( n - 1)2 + k el [(n + i)2 + k2l 9

The scientific literature on the optical properties of water fell silent through the 1940s. During the 1950s the following articles on the absorption and reflectance of water appeared in the literature: studying absorption were Curcio and Petty [29], 1951; Blout and Lenormant [30], 1953; Plyler and Aquista [31], 1954; Adams and Katz [32], Ceccaldi et al. [33], and Potts and Wright [34], 1956; Hyman et al. [35], 1957; Bocharov and Krutikov [36], and Waggener [37], 1958; and Goulden [38], 1959; Kislovskii [39], 1959, studied reflectance. Kislovskii [39], who concluded his paper with a dateline of Christmas Day, 1958, measured the reflectance spectrum and determined the first fairly reliable values of n and k of water in the 2.1-3.4 ~tm wavelength range. He used semiempirical conceptual models to compute n in the 182.9 nm-1000 Km wavelength region. He also used the conceptual models to calculate k in the 2.0~tm-1000 Km wavelength region. Kislovskii's determinations of k, the measurements of a(2) by Curcio and Petty [29] in the 0.7-2.5 ~tm wavelength region, and similar measurements of a(2) by

Water (H20)

1061

Plyler and Aquista [31] in the 2-42 ~tm wavelength region, provided the standard values of k for the next six to nine years. A systematic error, however, was later discovered in the work of Plyler and Aquista. The error was corrected in part by measurement and publication of "Absolute Absorption Coefficients of Liquid Water at 2.95 ~tm, 4.7 ~tm, and 6.1 ~tm" by Plyler and Griff [40] in 1965. The advent of the space age in 1957, the development of space technology in the 1960s and 1970s, and the consequent new opportunities for remote sensing of planetary atmospheres and surfaces created renewed interest in the optical properties of natural mineral materials. On Earth, water is a ubiquitous substance, and it was thus essential that some members of the scientific community again devote their interest to the optical properties of water. For the past three decades, an accurate compilation of N = n + ik of liquid water and ice over the broadest possible range of frequency v or wavelength 2 has thus been a topic of considerable interest, particularly to those engaged in the endeavor and to those who used and continue to use the data in a host of varying theoretical constructs. During the 1960s the following articles on the optical properties of water appeared in the literature: McDonald [41], 1960; Ackermann [42], 1961; Stanevich and Yaroslavskii [43], 1961; Bayly et al. [44], 1963; Kondratyev et al. [45], 1964; Plyler and Griff [40], 1965; Thompson [46], 1965; Bagdade and Tinkham [47], 1965; Draegert et al. [48], 1966; and Kondratyev [49], 1969, all studied absorption. Studying reflectance were McAlister [50], 1964; Pontier and Dechambenoy [51], 1965-1966; Painter et al. [52], 19681969; Querry et al. [53], 1969; and Zolotarev et al. [54], 1969. Pontier and Dechambenoy [51] measured reflectance spectra of water for angles of incidence of 0 degrees (1965), and 50 and 60 degrees (1966), in the 1-38 ~tm and 1-28 ~tm regions, respectively. They also measured the absorption spectrum a(2) in the 1-40 ~tm wavelength region (1966). The reflectance and absorption spectra, and the appropriate Fresnel equations, were used to determine n and k throughout the 1-40 ~tm wavelength region. In 1968, Irvine and Pollack [55] published a more extensive tabulation of n and k for water and ice in the 0.2-200 ~tm wavelength region. Although they seemed to have overlooked the work by Pontier and Dechambenoy [51], Irvine and Pollack considered data from references [21, 23, 25, 27-48] to prepare their tabulation of the optical constants of water. In 1969 and 1970, Zolotarev et al. [54, 56] presented a tabulation of N(2) for the 1-1 x 106 ~tm wavelength region. In the 1970 paper [56], they used four methods to determine n and k: measurements of absorption, reflectance, internal reflectance spectra, and Kramers-Kronig (K-K) analysis. Sources of spectral values of k used in the K-K analysis to determine n were the following:

1062

Wavelength Region 0.2~ 41 > 4.1 > 4.1 3.41 • 10 -3 1 x 10 -3 6.82 x 10 -5 1.1 x 10 -5 2.56 X 10 -6 2.13 x 10 -6 1.93 x 10 -6 1.67 • 10 -6 1.57 x 10 -6 1.33 x 10 -6 1.14 x 10 -6 1.11 x 10 -6 1.08 x 10 -6 9.2 x 10 -7 6.6 • 10 -7

[34, 35] [36] [28] [23] [20-22] [1, 33] [33] [32] [33] [26, 29] [1, 33] [33] [25, 33] [30] [33] [27] [33] [24, 31, 33]

t Values for electric field polarization perpendicular to c-axis.

frequency shift. Assuming the lineshape to be Lorentzian, one can d e r i v e from Eq. (6) Ao9

=

2K

(12)

where A~o is the full width at half maximum of the line. Equation (12) can be expressed in terms of the absorption coefficient ao = 47r~Ao or the skin depth 6 - 1/ao, yielding the following practical relation: Am = 2___KK=

rl

A0

.

(13)

2rrr/6

First, we emphasize that the relation in Eq. (12) is exact only when the lineshape is Lorentzian, a case applicable for low-absorbing materials such as Si. For highly absorbing systems such as Ge, the expression in Eq. (7) describes more accurately the measured lineshape, and Eq. (12) is just an approximation. We show in Fig. 3 the lineshape of the Brillouin lines of Si and Ge as measured by Sandercock [1]. Here we contrast the symmetric line observed for Si with the asymmetric line of Ge, a point already made by Dresselhaus and Pine [ 13].

5. Determination of the Optical Constants by Brillouin Scattering

t

133

si (~oo)

L

R

L

I

-i

-4eoo

,

I

o

,,,

I

I

I

2

3

Ge(i,i)

L

T

R

T

L

t

Ge000)

L

|

i

|

i

-8 - 6 - 4

i

i

L

i

-2

1..

[

O

i

t

2

|

|

4

|

t

6

i

9

I

m

8

F'F~QUENCY SHIFTS(an-') Fig. 3. Bri]]ouin peaks of Si and Ge obtained by Sandercock [1].

Second, we note that the ratio, Ao~/O, can be obtained directly from the measured line. Sood and Cardona [16] measured the change in ratio Aw/12 in Si as a function of concentration of a n-type dopant, which changes its absorption coefficient. We show in Fig. 4 their results. The determination of K or ~ from the measured line requires previous knowledge of r/or K from another technique. If both ~ and K are unknown, one needs the phonon velocity, v~, which can be determined from ultrasonic measurements, for example. This approach was followed by Sandercock [1 ] in his analysis of his

134

Hacbne Boukari and Nicholas Lagakos 30

lo s o

i~" /Pure Si

~" lr;

"''-

1

.~

,1 . . . . . .

i

3

I

I

~

6

I

!

lo

Ne ( x 1021cm-3 ) Fig. 4. Optical absorption coefficient of Si is shown as a function of concentration of dopant. The data were determined from the broadening of the measured Brillouin peaks. (From Sood and Cardona [161 with permission from the publisher.)

Si and Ge measurements. In this way, the Brillouin technique works as a complementary technique. Third, Table I indicates further that the ratio in Eq. (12) becomes larger in highly absorbing materials such as metals. It may then be difficult to observe the peak, as its width can be larger than its frequency shift. Indeed, as K is increased, the linewidth increases, extending the wings of the Brillouin peak to interfere with the next order of the Fabry-Perot lines. Another way, if the skin depth is of the order of the wavelength of the beam in the scattering medium (6 = Ao/rl), Eq. 13 yields Ato/g2 ~ 1/6, showing a big broadening effect already. Since for metals, Aoj/g] _ 2, this is probably why bulk acoustic phonons have not been reported for metals. Practically speaking, materials with 0.01 < A~o/O < 1 show probably a measurable linewidth dependence on absorption.

VII

BRILLOUIN SCATTERING IN THIN SLABS AND OPTICAL FIBERS

In the second part of this chapter, Brillouin scattering is considered in thin samples, such as thin slabs and single-mode optical fibers, where the dimension of the sample becomes of the same order as the light wavelength. Contrary to the bulk-sample case where both light and phonon energy and momentum are well defined, in thin samples, the light scattered over the thickness of the film develops an envelope of frequencies (and wavevectors), which, therefore, encompass more phonon energies and wavevectors. First, the formulas for calculating the refractive index in slabs are given for several convenient scattering geometries. Then, some examples of Brillouin spectra from the literature are considered, and the refractive index of the

5. Determination of the Optical Constants by Brillouin Scattering

135

samples is calculated using the given formulas. Finally, phonon propagation in thin slabs and single-mode fibers is discussed.

SPECIAL CASE OF TRANSPARENT THIN SLABS

VIII

The refractive index ~ of a sample can be determined from the Brillouin shift .Q of an acoustic excitation or phonon from one of the two following relationships: g 2 - 47rvsr/sin0~ ~o 2-'

q -

47rr/sin 0~ ~o -2-'

(14)

where vs is the velocity of sound of the phonon generated by the light scattering, )to is the wavelength of the incident light in vacuum, and 0s is the scattering angle in the sample. The dispersion curve for acoustic phonons is given by ~ = o~ = q v s, where oJ is often used, and q is the phonon wavevector. The Brillouin scattering in thin samples can be done by using side illumination [37-40] or edge illumination [40].

Side Illumination (90 ~ External Scattering)

Most Brillouin spectra in thin samples have been measured by using the side illumination, which is shown schematically in Fig. 5. In this figure, light is incident from the side of the sample (which has 1.458 refractive index, as that of silica), and the scattered light is observed at 90 ~. The internal scattered angle is not 90 ~ As can be seen from Fig. 5 (where the notation of Wang e t al. [37] has been used), two different phonons are generated (created or destroyed depending on whether it is Stokes or anti-Stokes) with the refracted and reflected beams to produce the scattered light inside the sample. Note that for a typical material with r / - 1.5, at a reflection most of the light is refracted out of the slab; only ---4% is reflected for modest angles of incidence. As shown in Fig. 5, in the first case, the phonon has q~ wavevector, and the scattering angle is 0~; while in the second case, the phonon has q2 wavevector, and the scattering angle is 02. The two scattering angles can be determined from the following equations: 01 -- /3 -~- "y,

02 -- 7 r - 13 + T,

(15)

where/3 and 7 are related to the refractive index 7): ~sin/3 = sina,

~/sin7 = cosa,

(16)

A

136

Hac6ne Boukari and Nicholas Lagakos

~J

E~

.E~J

r~

5. Determination of the Optical Constants by Brillouin Scattering

137

where a is the incident angle (not to be confused with the absorption coefficient, ao) and/3 is the refraction angle (not to be confused with the phase coefficient,/3o). In the second Eq. (16), cosa = sina', since a 90 ~ scattering is considered. Here, a' is the angle formed by the external scattered light and the normal to the sample surface. Note that in Fig. 5 the q2 phonon is involved with near-backscattering internally. Using the above equations, it can be shown that the phonon velocity Vs [Eq. (14)] can be expressed in terms of r/and the incident angle a as follows [37]: ff~l/~i Vs'l -- [2,0 2 +

(17)

sin2c~ - 2(,0 4 - ,02 _~_ 1sin22a)l/2]l/2' ~2~i

Vs'2 = [2,0 2 + s i n 2 a

+ 2(,04 -- ,02 + 41_sin22o~)l/211/2 '

(18)

where ~'~1 and g~2 are the Brillouin shifts corresponding to the phonons with q l and q2 wavevectors, respectively. For many transparent materials, one usually knows ,0 in advance, so the phonon velocity is determined. For a cubic crystal oriented so that the principal axes are along the slab edges, these phonons will, in general, have different velocities. For an isotropic sample (v~,~ = v~,2), from the above Eqs. (17) and (18), the following equation is obtained ff~l 2

2,02 _jr_ sin2a - 2(,04 __ ,02 _~_ 1sin22a ) 1/2

022

2'02 _q_ sin2a + 2('04 __ ,02 _~_ 88 2 2a) 1/29

(19)

Fused silica can serve as an example of an isotropic sample. From this equation the refractive index r/can be found from the Brillouin shifts g]l and ~ 2 for any incident angle a. 45~ ~ Scattering When the incident angle a is 45 ~ Eq. (19) has a very simple form. This geometry, in which the sample bisects the incident and the scattered light (45~176 has been used extensively in the literature to obtain the refractive index of thin samples and films [37-40]. In this case, /3 = y and a = 45 ~ (see Fig. 5). The q2 phonon is involved in internal backscattering. Then, from Eqs. (14), (15), and (16) one obtains:

~l~i

V~,l = X / 2 '

v~,2 =

~2/~i

2r/

(20)

(21)

138

Hac~ne Boukari and Nicholas Lagakos

Therefore, for an isotropic sample the refractive index "0 can be obtained from the following equation: 1

~-~2

'0 - ~/2 g2~"

B

(22)

Edge Illumination (90 ~ External Scattering)

This geometry, in which light is incident at the edge of the sample, has been discussed extensively by Takagi and Gammon [40], and it is shown schematically in Fig. 6. Our notation continues to be that found by Wang et al. [37]. One of the positive characteristics of this geometry is that the intensity of the phonon peaks is much stronger than that with the side illumination [40]. This is probably due to the larger volume per reflection implied by Fig. 6 compared to Fig. 5. Similarly to the side-illumination case, as can be seen from Fig. 6, two different phonons are involved in the scattering of the refracted and the reflected beams inside the sample. In the first case, the phonon has q l wavevector, and the scattering angle (inside the sample) is 0"1, while in the second case the phonon has q2 wavevector, and the scattering angle is 0*2. The two scattering angles can be determined by the following equations: 0* 1 : ~ -

/3-

~,

0* 2 = 71" - /3 -'1- )t,

(23)

where/3 and 3, are related to the refractive index r/through Eq. (16). Note that in Fig. 6, the q2 phonon is involved with internal near-backscattering. Using Eqs. (14), (16), and (23) one obtains: 1 V's,1 = [2'02_ sin2a + 2(I-/4- "02 + lsin22a)l/211/2' ~"]*2}ki V*s'2 = [2~ 2 + s i n 2 a + 2('04 -- "02 _~ 1sin220~)1/2]1/2"

(24)

(25)

Note that Eqs. (18) and (25) are the same. From Eqs. (24) and (25) for an isotropic sample, we obtain the following equation: 0"21

2'02

~"]'22

2'02 + sin2a + 2('04

-

s i n 2 a

+ 2('04

--

'02 -[- 88

-- "02 _.!_ 88

1/2

(26)

From this equation, the refractive index r/can be found from the Brillouin shifts g2*~ and g2*e for any incident angle ct.

5. Determination of the Optical Constants by Brillouin Scattering

139

I I I I

--....

4-.,J

'$

t~

/ 1"~

.o

\ /

!

_=

140

Hac~ne Boukari and Nicholas Lagakos

SCATTERED LIGHT

INCIDENT LIGHT

SAMPLE

I \

/ \

?K, Ki

Fig. 7. Side illumination with 180~ external backscattering.

450-45 ~ Scattering In this geometry we h a v e / 3 - 7 and a - 45 ~ The q2 phonon is involved exactly in backscattering internally. Then, from Eqs. (14), (16), and (23) one obtains: 1

Vs, 1 =

~/-~(2rl2 - 1)1/2,

(27)

~~*2A/ " 2~

(28)

Vs,2

--

~

Note that Eqs. (21) and (28) are the same. Therefore, the refractive index of an isotropic sample can be found from the following equation: 1 - ~/211 - ~-~'21/~'~*22]1/2"

C

Special Geometries

1

External Backscattering

(29)

Two examples of external backscattering (rather than 90 ~ extemal scattering) are shown in Figs. 7 and 8, where backscattering is done from the side and the edge of the thin slab. For extemal backscattering for both side and edge illumination, the same phonon with wavevector q2 is observed as described by Eqs. (18) and (25) or for the 450-45 ~ position, by Eqs. (21) and (28). Note that since only one q2 phonon is observed (and q2 is along k~), Eqs. (17) and (24) are not pertinent. These results hold for an isotropic material or for a cubic material with the principal crystal axes along the edges of the thin slab. For an isotropic sample it can be shown from selection-rule analysis that for the backscattering geometry (180~ only the longitudinal phonon will be present, while no transverse phonon can be seen [41].

5. Determination of the Optical Constants by Brillouin Scattering

141

K s _ . , - - - " " ~ i N CI DENT

I L,OHT

-

Fig. 8. Edge illumination with 180~ external backscattering. Classical 90 ~ Scattering

S

2

For the traditional 90 ~ scattering from a bulk sample shown in Fig. 9, = 90 ~ in Eq. (14) and, thus" ~-]Ai v~ = x/2~"

(30)

Note that the phonon propagates at 45 ~ to the plate edge. Contrary to the backscattering case, in the 90 ~ scattering both the longitudinal phonon and the transverse phonons (only one transverse phonon for an isotropic sample) will be present.

Examples

D

Here, some Brillouin spectra from thin plates and films are used from the literature to calculate their refractive index. Isotropic Samples Glass Plate The Brillouin spectra of a thin glass plate (150 /xm thick) for the 450-45 ~ geometry for the side and edge illumination, is shown in

INCIDENT LIGHT

SAMPLE

SCATTERED LIGHT

Ki K,

Fig. 9. Classical 90~ scattering.

1

142

Hac6ne Boukari and Nicholas Lagakos

I

L

i

(a)

t

I

,-It-

(b)

9

m

I0 GHI

I

L

!

F

o

_I u8

I

Fig. 10. Typical Brillouin spectra of thin glass plate (150/_~m thick). FSR = 77; a = 45 ~ (a) Side illumination; (b) Edge illumination. (After Takagi and Gammon [40].)

Fig. 10 [40]. The spectra were obtained using the VV scattering geometry, where only the longitudinal phonons can be seen for an isotropic sample. In Fig. 10(a) the side illumination was used to obtain the ratio of the Brillouin shifts of the two longitudinal phonons L 1 and L2 (with q l and q2 wavevectors in Figs. 5 and 6). Then, using this ratio from Eq. (22), the refractive index of the isotropic thin glass plate is calculated to be 1.52. In Fig. 10(b) the edge illumination was used to obtain the ratio of the Brillouin shifts of the two longitudinal phonons L*I and L*2. Using this ratio, from Eq. (29) the refractive index of the thin glass plate is calculated to be 1.51, which is close to the 1.52 obtained from the side illumination. While this type of glass was not described, the fused-silica refractive index is 1.458, which is close to that obtained above.

Polymer Film: Mylar-PET The Brillouin spectra obtained from a polymer film, Mylar-PET, which is a commercial film used as a transparency sheet for overhead projectors can be seen elsewhere [40]. In this case, the side illumination and the 45o-45 ~ scattering geometry were used to obtain

5. Determination of the Optical Constants by Brillouin Scattering

143

the Brillouin spectra shown in Fig. 11. From the ratio of the Brillouin shifts of the two longitudinal phonons L1 and L2, the refractive index of the isotropic Mylar-PET film is found from Eq. (22) to be 1.28. Most plastics have refractive indexes in the range 1.42-1.65, so this value appears too low [42, 43]. DNA Films The refractive index and the sound speed of DNA films has been studied using backscattering and 450-45 ~ geometry [39]. It was found that the refractive index decreases as the relative humidity of the DNA films increases (for dry DNA, r / = 1.50).

Uniaxially Stretched Films For nonisotropic films, the formulas given above for determining the refractive index are not expected to hold. PET Film Brillouin scattering has been used in polyethylene terephalate (PET) films to study the sound velocity and the elastic constants as a function of the film stretching ratio [38]. The refractive index of the unoriented PET film is 1.588 at 4880 A, which is within the range of plastics [42, 43]. It was found that, even though the refractive index of the film is not sensitive to stretching, the elastic constants strongly depend on the stretching ratio of the uniaxially stretched film [38]. The refractive index of the uniaxially stretched PET was found from the Brillouin spectra (side excitation; 45 ~ 45 ~ geometry) [38] using Eq. (22) to be 1.06, a value too low for plastics [42, 43]. This discrepancy is due probably to the fact that Eq. (22) holds only for isotropic samples. Polyvinylidene Fluoride (PVFz) Films Brillouin scattering was also used to study the sound velocity and the elastic constants of uniaxially stretched PVF2 films [37]. Similar to the PET case, while the refractive index is not

L

I

!

1

,J Fig. 11. Brillouin spectra of polymer film Mylar-PET with side illumination. F S R 40 GHz. (After Takagi and Gammon [40].)

=

2

144

Hac~ne Boukari and Nicholas Lagakos

sensitive to stretching ( r / = 1.43), the sound velocity and the elastic constants are sensitive to the stretching ratio. From the Brillouin spectra (side illumination; 45o-45 ~ geometry) seen elsewhere [37], it was found (using Eq. (22)) that the refractive index of the PVF2 in the form I is 1.05, while in the form II it is 1.02, values too low for plastics. As in the previous case, this discrepancy is due probably to the fact that the film is not isotropic.

E

Side Illumination at Normal Incidence

Bfillouin scattering has been used to study phonon propagation in thin films. Following Sandercock [44], for the case of backscattefing (180 ~ perpendicular to the film surface, we have the following distinct regimes. a. With a real refractive index r/, for a semi-infinite thick film, the Bfillouin spectrum is typical with the normal Bfillouin shift 12 = 2rlkovs, as illustrated in Fig. 12(a), where some instrumental and phononlifetime broadening of the peaks has been assumed.

)

(b)

,_Z IIlILIIILL

ILIlll LI.L!It, (0)

~

A

A

1

s2o Frequency shift Q,

Fig. 12. (a) A normal Brillouin spectrum, (b) the expected spectrum from a thin supported film, (c) the possible frequency shifts in the spectrum from a thin unsupported film, and (d) the observable spectrum from a thin unsupported film. (After Sandercock [44].)

5. Determination of the Optical Constants by Brillouin Scattering

145

b. With a real rl, there is a special case of a film supported on a substrate with the special properties that the substrate has the same acoustic impedance (so phonons can propagate across the boundary freely), but the substrate has very different elasto-optic properties, so that scattering occurring in the film is distinguishable from scattering in the substrate. Such a film/substrate sample probably does not exist, so the spectrum in Fig. 12(b) is not observable. Since film and substrate have the same acoustic properties, all values of q perpendicular to the surface are allowed. However, the Brillouin peaks are broadened due to the finite summation of the scattered electric fields as the incident beam traverses the film. The line shape is given by a sin2x/x 2, as shown schematically in Fig. 12(b). Here, x = (g2o - g2)d/2vs, where d is the film thickness, g]o is the Brillouin shift corresponding to the phonon frequency, and ~ is the variable frequency. c. If further, the film is unsupported, the boundary conditions require that the component of the phonon wavevector perpendicular to the surface be an integral multiple of rr/d. In this case, the Brillouin spectrum would have peaks at the frequency shifts ~ = qvs = pTrvJd (p is an integer), as shown in Fig. 8(c). d. Moreover, these phonon lines have a relative amplitude determined by the sinZx/x 2 envelope (Fig. 8(b)). Thus, the resulting Brillouin spectrum will be similar to that shown in Fig. 12(d), where again, some instrumental and lifetime broadening has been assumed. The exact form of the peaks of Fig. 12(d) depends on the thickness d, but in general they show triplet or quadruplet structure within the main peak of the sin2x/x 2 envelope. With reference to the dispersion curve shown schematically in Fig. 13, the quantized acoustic-phonon modes would be along the q-axis, while the measured phonon frequencies would occur along the w-axis. Furthermore, the observed modes would be attenuated severely. e. With a complex index of refraction n, a near-Lorentzian broadening of the Brillouin peaks occurs, as has been discussed in Section IVB. Therefore, the Brillouin components in Fig. 12(d) should be broadened. From the above discussion the main conclusion is that while normal Brillouin scattering (from a large volume) determines precisely both the energy (from the spectrum) and the momentum (from the scattering geometry) of the phonon involved, the backscattering from a thin film normal to the surface measures phonons within a range of momenta and correspondingly, having a range of energies. The predicted triplet in the Brillouin spectrum of a thin unsupported film (Fig. 12(d)) has been observed experimentally in MnzPS 4 films [44]. Figure 14 shows the observed Brillouin spectrum from two specimens of MnzPS 4 of thickness 1.1 and 2.0/xm. In both these spectra the triplets are clear; as the thickness of the film increases, however, the secondary peaks of the triplet get much weaker (Fig. 14(b)).

146

Hac~ne Boukari and Nicholas Lagakos

,i

IIIIII

IIIIII

q (=p~/d ; p:integer) Fig. 13. Frequency vs. wavevector for acoustic phonons propagating perpendicular to the surface of a thin unsupported film. While the incident light energy and wavevector are well defined, the light backscattered over the thickness of the film contains an envelope of frequencies which contain several of the phonons, thus giving the multiple structure.

IX

BRILLOUIN SCATTERING IN OPTICAL FIBERS

Brillouin scattering has also been used to study phonon propagation in optical fibers. Most of the work has been done with single-mode fibers in the core of which only one optical mode (photon) propagates. The core is surrounded by the cladding (clad) which has a slightly lower refractive index than that of the core, so that light can propagate in the core in a totalinternal-reflection mode. In the simplest case of the backscattering geometry, which we consider here and in which only the longitudinal phonon is present, both the phonon and the photon are propagating almost along the fiber axis. Similar to the thin-film case, phonon propagation in single-mode fibers is expected to be strongly affected by the boundary conditions at the core-clad interface, since light is confined to propagate in the fiber core, which has a diameter (few microns) of the same order as the propagating light wavelength. Indeed, it has been found that [45]: the normal acoustic modes are linear combinations of transverse and longitudinal phonons in both the core and the clad, and the relative field amplitudes are dependent on the boundary conditions at both surfaces. Second, the density of phonon states is not uniform in q and to space. Finally, the wavevector is not conserved in this acousto-optic interaction for a direction normal to the surface, i.e., the radial direction, and scattering is allowed over a continuum of phonons.

5. Determination of the Optical Constants by Brillouin Scattering

147

(a)

-I.I

-I0

-09

09

llo

(b)

-I.I

9

-1,0

-0

19

//

, ~

09

1.0

,,,

i

I.I

FREQUENCY SHIFT (cm-') Fig. 14. Observed spectra from (a) and (b) two specimens of Mn2PS 4 of thickness 1.1 and 2.0/~m, respectively. After Sandercock [44].

Even with the added complexities of a cylinder compared with a flat plate, some comparison can be made with the results for plates given in Eqs. (14-28).

Examples A Brillouin scattering in optical fibers has been done using 180 ~ and 90 ~ geometry, which are shown schematically in Fig. 15.

148

Hac~ne Boukari and Nicholas Lagakos

MICROSCOPE OBJECTIVE

oo

9

ft 3

,.

.,J" ... p,

FIBER CORE

MICROSCOPE OBJECTIVE

FIBER CORE

,, .........

i

i 1

i OUT Fig. 15. Schematic of Bfi]louin scattering in an optical fiber (a) ]80 ~ scattering; (b) 90 ~ scattering. After Thomas et al. [45].

1

180 ~ S c a t t e r i n g

Using the Brillouin backscattering shown in Fig. 15, in which only the longitudinal phonon can be seen, phonon propagation has been studied in single-mode fibers (3/.~m core diameter), which is shown in Fig. 16 [45]. In this figure we can see the longitudinal phonons propagating in the core only and in the core-clad space. In Thomas et al. [45] the refractive indexes of the core and the clad are given to be 1.462 and 1.457, respectively. From the spectrum of Fig. 16 and the refractive indexes, the sound velocities were calculated using Eq. (28) (backscattering phonon): V c - 6.026 • 105

cm/sec,

Vc~- 5.160 • 105

cm/sec.

(31)

Using Eq. (28), which has been derived for thin slabs, is an approximation for the cylindrical fibers, and it probably holds only for the optical rays passing through the fiber axis. These rays propagate in planes containing the

5. Determination of the Optical Constants by Brillouin Scattering

149

CORE LONG. LASE R T000

CLAOBING

+16

tONG.

9

~

,I ,.~ I

9

.

~

"...

,'F

9

9 ..

,.

o.

9 ; :.

...

..,,~ -tJ~" 9 .,.

9 .

.

I

9. ,

t t ;~

. 9

J

0

A

A

30

35

FREQUENCY SHIFT (GHZ) Fig. 16. Experimental Brillouin spectrum from a single optical fiber (180 ~ scattering geometry). (After Thomas et al. [45].)

fiber axis. For the skew rays which do not pass through the fiber axis, the above equations would not hold; however, the skew modes attenuate fast in a fiber, and in the steady state of propagation only the rays passing through the center are present. Even though the exact composition of the fiber was not given, as it can be seen below, the velocities listed in Eq. (31) correspond to standard fibers. In this example, the refractive indexes were given, and the sound velocities were found. However, the opposite is also true: Given the sound velocity, the refractive index can be found using the formula of Eq. (25) or (28).

90 ~ Scattering

Single-Mode Fiber Using the 90 ~ scattering shown in Fig. 15, the Brillouin spectrum was obtained in the fiber which was also used in Fig. 16, which is shown schematically in Fig. 17. This spectrum contains both the longitudinal phonons of the core and the clad and the transverse phonon of the core. For the Brillouin shift of the longitudinal core phonon (23.9 GHz), using Eq. (30), it is found that rl = 1.463, a value very close to 1.462 found from the backscattering experiment of Fig. 16. Single-Mode Fiber Preforms The elastic and elasto-optic (Pockels) coefficients of single-mode fibers have been studied doing Brillouin scattering

2

150

Hac~ne

Boukari and Nicholas Lagakos

CORE LONG.

~.

".

9

. g

LASER

o

9

9

"

'

""

#

.

CORE

CLADDING

.'

.."

o

..

9

'

9" 9

'

TRANS. :

LONG..

..

~

.

9

9

1000

-;-

9

.

.'

*

. ,.

9

9 -

o

"

'~

.,

.'.,,:

"

.;..,,.."

.

:""----o'--"* II

"'..--"'"" IIII

9

.,

.

'

9

I

I

0

30 FREQUENCY (GHZ)

Fig. 17. Experimental Brillouin scattering from an optical fiber (classical 90~ scattering geometry). (After Thomas et al. [45].) (90 ~ on the glass preforms from which the fibers were drawn [46]. The preforms were cylinders 5 cm long and 7 mm in diameter with optically polished end faces. Two different fiber preforms were studied, one with numerical aperture N A = 0.1 composed of a fused-silica core with traces of GeO2, and a clad of 5% B20 3 -at- 95% SiO 2 and another with N A = 0.15 composed of a 3% GeO2 + 97% SiO2 core, and a 15% B20 3 + 85% SiO 2 clad. The longitudinal (L) and the transverse (T) phonons were studied from the (V,V) and the (V,H) Brillouin spectra, respectively. The sound velocities of the phonons were calculated from Eq. (30), where the refractive index had been obtained using the prism method. From the two sound speeds, for an isotropic sample the elastic moduli, i.e., Young's modulus Y and Poisson's ratio or, can be calculated using the following formulas: (3VL2 -- 4V2 ) llL 2 -2v~ Y = Pv 2 _ 2 ' O" = (32) VC2 Vr 2(VL2 - V2) ' where p is the glass density. The Pockels coefficients were also found by comparing the Brillouin spectra of the preforms to that of a fused-silica preform, whose Pockels coefficients are known, using the following formula for the L phonon [6]: I-

Kr/gP122 PVs

2

(33) '

where K is a geometrical and temperature factor, and P12 is the Pockels coefficient. Keeping the scattering geometry of the fused silica and the preforms identical, the K factor was eliminated by dividing the intensities of the spectra of the fused silica by those of the preforms [46]. This demonstrates the power of the Brillouin-scattering spectra from which, not only all the elastic moduli can be determined at once, but also the (very difficult to

5. Determination of the Optical Constants by Brillouin Scattering

151

measure) Pockels coefficients can by obtained. It should be also noticed from Eq. (33) that the intensity of the Brillouin spectra has the very strong dependence on the 8th power of the refractive index a~. The results of these experiments on the two fiber preforms for the coefficients of the longitudinal phonon for the core and the clad are listed below: Fiber N A = O. 1: r L - 1.4580,

~z = 1.4546,

vc,L = 5.944 • 105 cm/sec, 5.605 • 105 cm/sec.

(34)

v~,L = 5.806 • vct,L = 5.164 •

(35)

VcI,L -

Fiber N A = 0.15: t i c - 1.4624,

a%z- 1.4547,

105 cm/sec, 105 cm/sec.

A close comparison of the sound velocities in Eq. (31) to those in Eqs. (34) and (35) identifies the composition of the fiber used in the 180 ~ scattering to be close to that of the 0.15 NA. Also the refractive indexes of these two glasses are very close (r/c. = 1.462 vs. 1.4624, r/cz = 1.457 vs. 1.4547).

Edge Illumination (90 ~ External Scattering) Pyrex Glass The edge illumination (90 ~ external scattering) has been used [40] to obtain the Brillouin spectrum of the longitudinal phonon of a 600 p~m diameter glass rod (Pyrex), which is shown in Fig. 18(a). The incidence angle was --~45~ for the 450-45 ~ geometry [47]. If the slab model holds for the cylinder, which seems to be the case in the examples above, two phonons should be observed in the cylinder. If I21"2/I22 *e = 0.844 is obtained from Fig. 18(a), r / = 1.8 is calculated from Eq. (29), while for Pyrex -- 1.471 [48]. The correct value for the refractive index of Pyrex can be found from Eq. (26) with a = 20 ~ or 69 ~ for the angle of incidence. Only with these incidence angles for a 90 ~ external scattering, can the peaks shown in Fig. 18(a) correspond to LI* and Le* phonons discussed in Section VIII.B, as has been assumed by Takagi and Gammon [40]. Optical Fibers The Brillouin spectra of a multimode and a singlemode fiber are also given and are shown in Figs. 18(b) and 18(c), respectively [40]. For the multimode fiber, ~1"2/~2 .2 = 0.855 (Fig. 18(b)), which gives -q = 1.8 for the 45~ ~ geometry; this is too large for an optical fiber, which is probably a high-silica glass. Conversely, Eq. (26) gives a = 17 ~ or 72 ~ (with ~ = 1.458 for silica), which is not in agreement with experiment. For the single-mode fiber, ~1"2/~2 . 2 = 0.849 (Fig. 18(c)), which gives - 1.8 for the 450-45 ~ geometry; again, this is too large for an optical fi-

3

152

Hac~ne Boukari and Nicholas Lagakos

.....

II

II

I

I

I

(o)

'

i

-II-

LI .-II,. "2 (b) ,

I

iL .

.

.

.

.

,

t !

I

I

I '

(c)

1 i

i i

Fig. 18. Brillouin spectra with edge illumination at FSR = 77 GHz; (a) glass rod (600 ~m), (b) multimode optical fiber, and (c) single-mode optical fiber. After Takagi and Gammon [40].

ber. Conversely, Eq. (26) gives c~ = 19 ~ or 71 ~ which is not in agreement with experiment. However, for the case of the fibers another possibility is that the two longitudinal phonons correspond to the core and the clad phonons, similar to the ones observed in Figs. 16 and 17, and not to the L~* and L2* phonons, as were assigned by Tagaki and Gammon [40].

5. Determination of the Optical Constants by Brillouin Scattering

153

REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

J. R. Sandercock, Phys. Rev. Lett. 28, 237 (1972). B. Chu, "Laser Light Scattering." Academic Press, New York, 1974. B. Berne and R. Pecora, "Dynamic Light Scattering." Wiley, New York, 1976. G. B. Benedek and K. Fritsch, Phys. Rev. 149, 647 (1966). A. Griffin, Rev. Mod. Phys. 40, 167 (1968). M. Born and K. Huang, "Dynamic Theory of Crystal Lattices." Oxford Univ. Press (Clarendon) Oxford, 1954. A. S. Borovik-Romanov and N. W. Kreines, Phys. Rep. 81, 351 (1982). N. L. Rowell and G. I. Stegeman, Phys. Rev. B 18, 2598 (1978). J. M. Vaughan, "The Fabry-Perot Interferometer." Hilger, London, 1989. G. Hernandez, "Fabry-Perot Interferometers." Cambridge Univ. Press, Cambridge, 1986. N. W. Ashcroft and N. D. Mermin, "Solid State Physics." Sanders College, Philadelphia, 1976. R. Loudon, J. Phys. C: Solid State Phys. 11, 403 1978. G. Dresselhaus and A. S. Pine, Solid State Commun. 16, 1001 (1975). M. Born and E. Wolf, "Principles of Optics," Chap. 7, p. 328. Pergamon, London, 1959. C. S. Zha, R. J. Hemsley, H. K. Mao, T. S. Duffy, and C. Meade, Phys. Rev. B 50, 13105 (1994). A. K. Sood and M. Cardona, Solid State Commun. 49, 299 (1984). E. D. Palik, H. Boukari, and R. W. Gammon, Appl. Opt. 34, 58 (1995). H. Boukari, E. D. Palik, and R. W. Gammon, Appl. Opt. 34, 69 (1995). E. D. Palik, H. Boukari, and R. W. Gammon, Appl. Opt. 35, 48 (1996). A. S. Pine, "Light Scattering in Solids: Topics in Applied Physics," (M. Cardona, ed.), Vol. 28, Chap. 6. Springer-Verlag, Berlin, 1975. A. S. Pine, Phys. Rev. B 5, 2997 (1972). A. S. Pine, Phys. Rev. B 5, 3003 (1972). J. Zuk, H. Kiefte, and M. J. Clouter, J. Appl. Phys. 72, 3504 (1992). R. Bhadra, M. Grimsditch, I. K. Schuller, and F. Nizzoli, Phys. Rev. B 39, 12456 (1989). R. Jorna, D. Visser, V. Bortolani, and F. Nizzoli, J. Appl. Phys. 65, 718 (1989). B. Hillebrands, P. Baumgart, R. Mock, G. Gtintherodt, and P. S. Bechthold, J. Appl. Phys. 58, 3167 (1985). J. A. Bell, W. R. Bennett, R. Zanoni, G. I. Stegeman, C. M. Falco, and F. Nizzoli, Phys. Rev. B 35, 4127 (1987). G. Carlotti, D. Fioretto, L. Palmieri, G. Socino, L. Verdini, and E. Verona, IEEE ultrasonic symposium, p. 1201, (1989). M. Grimsditch, R. Bhadra, and I. K. Schuller, Phys. Rev. Lett. 58, 1216 (1987). S. Subramanian, R. Sooryankumar, G. A. Prinz, B. T. Jonker, and Y. U. Idzerda, Phys. Rev. B 49, 17319 (1994). E Nizzoli, R. Bhadra, O. E de Lima, M. B. Brodsky, and M. Grimsditch, Phys. Rev. B 37, 1007 (1988). M. Elena, M. Bonelli, C. E. Bottani, G. Ghislotti, A. Miotello, P. Mutti, and P. M. Ossi, Thin Solid Films 236, 209 (1993). J. R. Sandercock, Solid State Commun. 26, 547 (1978). J. M. Karanikas, R. Sooryakumar, and J. M. Phillips, J. Appl. Phys. 65, 3407 (1989). J. M. Karanikas, R. Sooryakumar, and J. M. Phillips, Phys. Rev. B 39, 1388 (1989). L. E. McNeil, M. Grimsditch, and R. H. French, J. Am. Ceram. Soc. 76, 1132 (1993). C. H. Wang, D. B. Cavanaugh, and Y. Hagashigaki, J. Polym. Sci., (Polym. Phys. Ed.) 19, 941-950 (1981). D. B. Cavanaugh and C. H. Wang, J. Appl. Phys. 52, 5998 (1981). M. B. Hakin, S. M. Lindsay, and J. Powell, Biopolymers 23, 1185 (1984). Y. Takagi and R. W. Gammon, J. Appl. Phys. 61, 2030 (1987).

154

Hac~ne Boukari and Nicholas Lagakos

41. J. Pelous and R. Vacher, Solid State Commun. 16, 279 (1975); R. Vacher and J. Pelous, Phys. Rev. B 14, 823 (1976). 42. J. D. Lytle, Polymeric optics, in "Handbook of Optics, Devices, Measurements and Properties," (M. Bass, ed.), 2nd Ed., Vol. 2, Chap. 34. McGraw-Hill, New York, 1995. 43. B. H. Billings, Optics, in "American Institute of Physics Handbook," (D. E. Gray, ed.), 3rd Ed., Section 6, pp. 6-108. McGraw-Hill, New York, 1982. 44. J. R. Sandercock, Phys. Rev. Lett. 29, 1735 (1972). 45. P. J. Thomas, N. 1. Rowell, H. M. van Driel, and G. I. Stegeman, Phys. Rev. B 19, 4986 (1979). 46. N. Lagakos, J. A. Bucaro, and R. Hughes, Appl. Opt. 19, 3668 (1980). 47. Y. Takagi, private communication (1996). 48. J. Schroeder, J. Non-Cryst. Solids 40, 549 (1980).

Chapter Doped n-Type Silicon (n-Si) MARK AUSLENDER and SHLOMO HAVA Ben-Gurion University of the Negev Department of Electrical and Computer Engineering Beer-Sheva, Israel

I. Introduction II. Contributions to Dielectric Function: Optical Properties versus Optical Constants II1. Semiclassical Theory of Free-Carrier Contribution to the Dielectric Function: The Drude Approximation IV. Free-Carrier Contribution to the Dielectric Function in Quantum Absorption Regime: Generalized Drude Approach V. Experimental Data Overview VI. Generalized Drude Approximation for n-Si A. Electron-Phonom Scattering B. Electron-Impurity Scattering VII. Tabulation of Optical Constants of Doped n-Si References

155 156 158 159 162 166 167 168 171 185

INTRODUCTION

The importance of knowing optical constants was excellently outlined in the introduction to HOC I [ 1]. In the case of silicon, one can also point out technologically important applications of this knowledge. A partial list includes the determination of epilayer thickness [2] and the losses in all-silicon integrated optical devices [3], silicon-wafer thermometry [4], IR emissivity, and absorbance enhancement in silicon gratings [5, 6] To be self-contained, this critique includes an overview of dielectricfunction partition and the relation of optical constants to measured optical properties (Section II), semiclassical, and quantum absorption theory (Sections III and IV). In order to avoid multiple repetition when citing textbook materials, we refer in advance to well-known books on optics [7], semiconductor optics [8], and physics [9], and quantum mechanics [10]; in Section IV we also introduce the generalized Drude approximation. Section V is an 155 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS III

Copyright 9 1998 by Academic Press. All rights of reproduction in any form reserved ISBN 0-12-544423-0/$25.00.

I

156

Mark Auslender and Shlomo Hava

overview of experimental data and the Drude formula with empirically adjusted parameters. In Section VI we present a first-principle theory of freecarrier contribution to optical constants in doped n-Si based on the generalized Drude approximation. We outline the calculations and emphasize the difficulties that the theory is faced with when being used to describe available experimental data.

CONTRIBUTIONS TO DIELECTRIC FUNCTION: OPTICAL PROPERTIES VERSUS OPTICAL CONSTANTS

Electromagnetics of homogeneous material is completely defined by the dielectric function (DF): e = el + ice, a function of radiation frequency to or vacuum wavelength )t = 27rc/to. The basic quantity of optics, however, is complex refractive index ~ - n + ik, where the refractive index, n, and extinction coefficient, k, are related to e~ and 42 via equations n 2-

2nk

k2 = =

El,

E 2.

(1) (2)

The lattice and free-carriers, which arise from doping, give additive contributions to the DF: E =

E1 +

Ec.

(3)

Here, e~ = (n 1 + ikl)2 is the lattice contribution, and ec is that due to the carriers .47ro'(to) e~ = 1 ~ ,

(4)

to

where or(to) is conductivity at optical frequencies. Though doping also affects elvia band-gap reduction [ 11], in the IR this effect is negligible as compared to ec. The tables of nl and kl for intrinsic silicon were presented by Edwards [12]; such tables for Si: O2 were also reported [13]. In the near and mid IR, n~ decreases from 3.4490 at 5000 cm-1 to 3.4201 at 400 cm-1, the largest values of kl -'~ 10 -3 are achieved in a narrow band near 606 cm-~. It seems that this variation is negligible on the scale of e~ for doping in excess of 1016 cm-3; in most papers, n 1 is taken to be constant, and kl is neglected. Experimentally, n and k are determined using measurements of various optical properties of quality samples with simple geometry. For smooth

6. Doped n-Type Silicon (n-Si)

157

opaque samples, the only quantity of interest is the complex amplitude reflectance r - -

X/~-I

= X/R exp(iqb),

(5)

1

where R is the Fresnel reflectance of the sample's surface R -

]r] 2 = (n - 1)2 + k 2 (n + 1) 2 + k 2'

(6)

and @ is the argument of the complex reflectance 2k tan ( 4 ) ) - n2 + k 2 -

1"

(7)

The spectrum of R is measured using reflectometry techniques; with R measured, qb is determined using Kramers-Kronig analysis (KKA) [14]. With both spectra available, Eqs. (6, 7) are solved simultaneously to yield n and k. For a transparent slab, the apparent reflectance Ra and transmittance T~ are measured. In many practical cases, interference oscillations (fringes) in the spectra of Ra and Ta are averaged out. This leads to simple equations:

Ra-R

[

1 + I 1 - r212

T ~ - ] 1 - p]a

exp(-2otd) 1 - R2 exp(-2ad)

exp(-ad) 1 - R 2 exp(-2o~d)'

1 '

(8)

(9)

in which d is the thickness of the slab, and a =

4~'rk

(10)

?t

is the EM power attenuation coefficient. It is referred to as the absorption coefficient in the case n > k (E 1 > e2). At n >> k, r 2 = R in Eqs. (8, 9), and spectral absorbance A = 1 - Ra - Ta takes the following form when used in silicon-wafer thermometry [4]:

A =

(1 - R)[ 1 - e x p ( - ad)] 1 - R e x p ( - c~d)

.

(11)

158

Mark Auslender and Shlomo Hava

III SEMICLASSICAL THEORY OF FREE-CARRIER CONTRIBUTION TO THE DIELECTRIC FUNCTION: THE DRUDE APPROXIMATION

An approach to calculating ec, which combines an account of band structure and scattering with the merit of being simple, is based on quasi-classical Boltzmann transport equation. This is restricted by the requirement that the radiation quantum is much smaller than the mean electron energy: hto

t

= m~

Tac

, 3m_L

.

3m•

(17)

-

The quantity mc is called optical or conductivity effective mass. With the use of cyclotron-resonance data, the average in Eq. (15) is calculated trivially to give mc = 0.26mo. If, for some reasons, ~'Ocand %c are close to each other, the two limits Eq. (13) and Eq. (14) may be interpolated by the simple formula 2

2

n 10gp - ~l - o9(oo + i~/)'

1 3' - -'1"

(18)

which is the celebrated Drude (also Drude-Lorentz or Drude-Zener) equation, and ~"is the relaxation time [8, 9, 16]. For isotropic ~', the closeness of ~'d~ and ~'a~ is certainly realized when carriers are degenerate. In n-Si with the valley density-of-state effective mass md _ ml{/3_m12/3 ~ 0.33mo, no degeneracy occurs unless N --> 102~ cm -3. In order to validate the Drude equation at all concentrations, the relation ~'dc/%c = 1.13, valid for combined DA-phonon and point-defect scattering in non-degenerate statistics, is used. For ionized-impurity scattering another reason must be sought, because ~'dc/~'ac >> 1 even in the isotropic band. At heavy doping, the allowance of Eq. (18) implies the existence of a mechanism which 'binds' one-particle currents into collective variable, total current, which relaxes as a whole. This mechanism could be electron-electron (e-e) interaction, which smears an individual electron momentum but conserves the total one. Then the response of the electron plasma on an external EM field will be described in terms of total momentum, provided that the electron momentum exchange in e-e collisions becomes faster than the electron momentum relaxation due other mechanisms, as N increases over some value. However, this picture is only partially adequate because the e-e momentum-relaxation rate is of the same order of magnitude as that due to ionized impurities, though detailed estimates are not widely known especially for the anisotropic band.

FREE-CARRIER CONTRIBUTION TO THE DIELECTRIC FUNCTION IN QUANTUM ABSORPTION REGIME: GENERALIZED DRUDE APPROACH

The Boltzmann equation becomes invalid at ho) > (E), since in this case radiation influences the scattering process. In the quantum regime, the act of

IV

160

Mark Auslender and Shlomo Hava

scattering is a transition between initial state with momentum k and energy E(k) and final state with momentum k' and energy E ( k ' ) with radiation quanta absorbed or emitted: E(k') = E(k) ___ hto. In the quantum regime, ec was calculated in the terms of transition probabilities using time-dependent perturbation theory. The first calculations of this type were reported 40 years ago by Fan et al. in connection with the first IR-absorption measurements on Ge [17] and Si [18]. For DA-phonon scattering, the absorption coefficient (Eq.(10)) obeys a power law" n a ~ to-1.5 ~ hl.5 at hto > (E). At (E)/h > to > 1/~', quantum theory gives the same result as semiclassical theory or the Drude approximation at tot > 1 (Eqs. (14, 18) with Eqs. (2, 10)) n a ~ - - ( n2 top )2 ~ h2. CT

(19)

to

DO-phonon scattering in n-Ge was also considered by Fan et al. [ 17]; phononassisted absorption in a many-valley conduction band was studied by Meyer [19] and Rosenberg and Lax [20]. The method was recapitulated by Dumke [21 ] in 1960. Ionized impurity-assisted absorption in the Born approximation was studied by Meyer [ 19] and Rosenberg and Lax [20], and with Coulomb wave functions by Visvanathan [22] and Donovan [23]. It was found that in the quantum regime, n a ~ to-3.5 ~ h3.5. The transition to Eq. (19) may not occur even after (E) exceeds hto. The reason is, though the electron wave packet as a whole starts to move at thermal equilibrium, its breadth 8r is still determined by the oscillations in radiation field: 6r = ~ / h / 2 m * to, m* being an effective mass. So at 8r < rs (screening length) the electrons 'feel' an unscreened impurity for which n a ~ to-3 ~ h3 in the classical frequency region [22, 23], and the limit of Eq. (19) is reached only at 8r > r s. The calculations using transition probabilities require throwbacks to first principles, and are recast each time a new mechanism is considered. From the late 1960s to 1970s, studies devoted to the derivation of generalized Boltzmann equations, which could recover both quantum and classical regimes, were carried out. Having no room to review this subject here, we refer to the contribution of Jensen [24]. Note that at o~" >> 1, the quantum Boltzmann equation and the transition-probability methods give close results. Kubo quantum-admittance formalism [25] with second quantization allowed researchers to automate the calculations at high frequencies in a reasonably correct, unified form. The Kubo formula for dynamical conductivity [25] reads o-(to) - fo~ exp(itot) ~o~1/~7"(jx( t _ ihs)Jx(O))~dtds.

(20)

Here, J~ is x - ( o r any) component of electric-current operator, A(t) = e x p ( i t ~ T h ) A e x p ( -

it~q{Th),

(21)

6. Doped n-Type Silicon (n-Si)

161

where ~%~is a Hamiltonian, and (...)~ stands for thermodynamic average with the temperature T. Using integration by parts with respect to time t, in Eq. (20) the free-carrier contribution (Eq. (4)) can be identically transformed to 2 Ec --

~'-~P

002

I

1 Jr-

.,~oo~

i~o

1 '

2 _ ~'-~p 4rr

1/kT

s

{Jx(ihs)Jx(O))rds

0

(22)

with /-/(09) -- 4rrOp 2 f~ exp(iwt) f 1/kT{ix (t + ihs)Jx(O))rdtds, 0

(23)

0

where the upper dot denotes the time derivative: i

A ( t ) - ~[~,, A(t)]. Eq. (23) is convenient for the calculation at high frequencies. Indeed, H(to) is at least of second order with respect to the interactions, since the electriccurrent operator commutes with the kinetic energy part of ~.. Thus, at o~r >> 1 the interactions can be neglected when considering the correlation function in Eq. (23). Then the lowest-order estimate of Eq. (23)//~2)(w) = E(~o) is obtained in a closed form as a sum of the contributions of different scattering mechanisms. It is expedient to note that in this approximation, the e-e interaction does not contribute to F(to) for the isotropic band, where Jx is proportional to the total momentum. With H(oJ) replaced by F(~o), Eq. (22) becomes a fragment of the highfrequency expansion of

ec -

Op2 w[o9+ iF(w)]

~

(24)

The extention of Eq. (24) to all to's is called the generalized Drude approximation (GDA). The structure of this type may formally be obtained from exact Eq.(22) using the Moil projection ansatz [26]. Of course, the prescription of the structure by no means validates GDA. The validity of GDA in the whole frequency range was extensively discussed by Sernelius [27]. It was applied to simple metals [28], and to an isotropic model of doped n-Si [29, 30]. GDA appears to be a good approximation if the e-e collisions dominate, as outlined in Section III. Since g2p = n ] 6 o p in the leading approximation, the total DF (Eq. (10)) in GDA differs from the Drude formula (Eq. (19)) only in the presence of the relaxation-memory function /-'(o9) instead of constant relaxation rate % If the valleys are uncoupled, GDA must be applied separately to each valley. The calculations for the valleys of (100), and {010), (001) types give, in

162

Mark Auslender and Shlomo Hava

general, two different contributions Fll(o)), and F• tal DF in this case is 4 rrNe 2

respectively. The to-

8 rrNe 2

e - e1 -3mll~O[w + i/-'ll(~

- 3m_t_w[w +/F_t_(w)]"

(25)

At the high-frequency limit only the average relaxation function mc [/-'H ( t ~

/'(o))- T

mll

2/-'j_(o)) ]

m•

(26)

enters Eq. (25), thus comprising the first term of exact perturbation expansion as required. In the presence of a scattering, that couples the valleys, Eq. (25) must be changed (Eq. (3.8) [27]).

V

EXPERIMENTAL DATA OVERVIEW

There exists a body of literature on the optical properties of variously doped n-Si crystals in the IR. The equilibrium solubility of atoms of As, P, and Sb in Si does not exceed 5 x 102~ -3 [31]. The use of advanced ionimplantation technology increased the active dopant density far above this (up to 5 X 1021 cm -3) and allowed plasma effects to be observed in the near IR [32-34]. Basically, these studies used reflection and/or transmission measurements. For bulk-doped samples at N < 1019cm -3, it is technologically possible to prepare slabs suitable for transmission measurements. With the use of Eqs. (6-10), optical constants are retrieved from measured values of R a and Ta. As is seen from Eq. (14), extrapolating measured el to zero from the region n >> k gives the plasma frequency, and hence the optical effective mass me. Using this method, Spitzer and Fan [18] found mc = 0.27mo for N = 3.6 X 10 TM cm -3, in excellent agreement with the value cited in Section II. It was pointed out by Fistul [31] that for heavier doping, the method has low reliability, as was seen from the result me = 0.44mo of Cardona et al. for N = 6.5 X 1019 and 1.1 x 1020 cm -3 [35]. At doping in excess of 1019 cm -3, reflectometry was mostly used [33, 36-39]. A comprehensive study of reflection was made by Howarth and Gilbert [36], who measured 16 n-Si samples doped in the range 7.4 x 1018cm -3 -1.67 x 102~ -3. For ionimplanted n-Si layers, combined measurements of reflection and transmission were carried out by Streltsov and Titov [40], Miyao et al. [32], and Aspnes et al. [34]. Two approaches to processing reflectivity data of bulk-doped samples were used. The first one, which relied on the Drude model, was proposed by

6. Doped n-Type Silicon (n-Si)

163

Kukharski and Subashiev [38], and advocated by Lambert [41]. In this approach, O~p and 3, are found from the reflectivity m i n i m u m frequency, r n (wavelength Am~,), and the value of reflectivity minimum, Rmin. With known O~p, one can calculate m c if N is measured and vice versa; this method is an improvement of widely used approximations: (n - 1)2 >>/(2 or n --~ 1, k --~ 0 [17, 18, 35-37]. The method of Kukharski and Subashiev [38] was used also for doped n-Si layers in a homogeneous approximation [40] and modified by Slaoui and Siffert [33] for concentration profiles. The second approach is KKA giving n and k independently of the physical model (see Section II). However, as a rule, the retrieved n and k were not displayed, and again, the Drude model was used. Spitzer et al. [37] assumed m c - 0.28m o and studied the influence of heat treatment on y in a sample with N - 7.5 • 1019 cm -3. Barta [39] expressed m c and 7 via el and e2, which were known after KKA (see Eqs. (1, 2)). The parameters were calculated at the zero of e~, which gave" m~ - 0.3 lm o and m~ - 0.35m o for two samples with N - 9 • 1019 cm -3 and 1.6 • 102~ cm -3, respectively, and about the same value of the relaxation parameter 3, --~ 1.09 • 1014 s -1 for both samples. A variety of spectra of the absorption coefficient c~ was observed in n-Si. Dubrovsky in his thesis cited by Fistul [31] in the range of N --~ (0.1 1) • 102~ -3 and reported the dependence c~ ~ A35, which is characteristic for ionized-impurity scattering (see Section IV). Balkanski et al. [42] measured c~ ~ A2 at 4 ~ m < A < 6 ~ m in bulk-doped samples with N - 6 • 1018, 1019 cm-3. On the contrary, in ion-implanted layers with N ~ (0.5 2) • 102~ -3, the dependence c~ ~ A 15 was observed at 2p~m < A < 10~m [40]. Later this was also observed in laser-annealed poly-Si layers with a much larger concentration [34]. Based on the fact that this dependence is obtained for DA-phonon scattering (see Section VI), Streltsov and Titov [40] asserted that the phonon mechanism dominates the scattering processes in n-Si doped in excess of 1019 cm -3. On the one hand, the researchers were disposed to retrieve some physical information rather than tabulate optical constants. On the other hand, such restricted information can hardly be used in practice. For a fast and accurate approximation of optical constants in a wide range of N and o~, Humlf~ek and Wojtechovsky [43] proposed a KKA-adjusted Drude model for a best fit of reflectivity data. Using the data for the samples of Howarth and Gilbert [36] and three of their own [43], they found fitting values of % and ~/for each, and derived linear relations: 3, = 113 + 0.208 ~ov

(Si: Sb),

COp (Si: As), 3, = 194 + 0.174 LOp (Si: P), 3, = 153 + 0.265

(27)

where ~Op and 31 are in units cm-1, and N -> 5 • 1018 cm -3. The authors sought a dependence between m~, as calculated using the definition of w v,

164

Mark Auslender and Shlomo Hava

and N. Due to a large scatter, the result (Eq. (15) [43]) seemed so unphysical that they recommended substituting the value m c -- 0.26m o for interpolation of optical constants using Eq. (27). Thus, at N --- 1020 c m - 3 it can be put in the Drude equation (Eq. (18)): nl2tOp2_ 3.452 X 10-13 N cm -2,

(28)

where N is in units c m - 3 . Eq. (28) can be validated by fittting O0p of Huml/rek and Wojtechovsky [43] linearly to N. The result is shown in Fig. 1. We display only the fit for Si" As, which yields m~ - 0.27mo. It fits the data reasonably well up to --~ 8 X 1019 c m - 3 . The data for Si" Sb also have a small scatter and are fitted with m c - 0.28m o. The data for Si" P have larger scatter especially at the end concentrations. The linear fit using the data for N - 1020 cm -3 gives mc 0.29mo. The point N - 1.67 • 1020 cm -3 (not shown in Fig. 1) drops drastically below this fit. Including this point in the fit increases the value of me to 0.31mo; the calculation using only this point gives mc - 0.34mo, which are about the same as obtained by Barta [39]. Since m c must be independent of impurity type, we conclude that its apparent increase at N -< 1020 c m - 3 is a result of experimental uncertainty. This agrees with band-structure calculations of van Driel [15], which predicted different behavior of mc: nearto-constant m~ - (0.26 - 0.27) mo at N > 1, Eq.(30) gives the asymptotic form of the absorption coefficient (see Section IV): rto~(ep) ~

16Ne2(Tr'h~~1/2 kT

9cT/,nctO 2

'

~"= "r~exp(-r/)F1/z(r/)'

(33)

where To is DA-phonon mobility relaxation time for light doping. In Streltsov and Titov [40] the absorption spectra retrieved from the transmission data for ion-implanted Si: P layers with N = (0.5 - 2) • 1020 c m - 3 w e r e shown to fit Eq. (33). Using qualitative arguments, the authors stated that ionizedimpurity scattering should 'switch-off' in the spectral interval under study, so observed absorption is due to acoustic phonons; the same conclusion was also drawn by Aspnes et al. [34]. At N = 1020 c m - 3 the fit by Streltsov and Titov [40] needed ~"about 15 times smaller than ~'o, and the formula in Eq. (33) gives a factor --~ 3. The rest, however, is not necessary due to the increase of electron-phonon interaction as was claimed by Streltsov and Titov [40]. Actually, any short-range interaction leads to the same spectral dependence as given by Eq. (33), but with another To. It might be the short-range part of electron-donor interaction (see next subsection). In this case, the asymptotic form in Eq. (33) is almost independent of T, whereas in the case of phonons, it is linear with respect to the temperature. Thus, the origin of Al5-absorption spectrum can not be identified until its temperature dependence is studied experimentally. More seriously, the concept of Aspnes et al. [34] and Streltsov and Titov [40] implies impurity scattering to play no role also at DC. Indeed, the fit values of 7" attributed by Streltsov and Titov [40] solely to electron-phonon interaction correspond to the mobilities in the range (61-88) cm 2 V - i s -1, typical for the measured mobility at the studied concentrations.

B

Electron-Impurity Scattering A key problem in the theoretical description of electron-donor scattering is to determine the electron-donor interaction potential. Usually it is taken in the form of screened Coulomb potential Vi(r) - - e e x p ( - Ir - RD I/rs), E l l r - RDI

(34)

6. Doped n-Type Silicon (n-Si)

169

where e is the electron charge, RD is a donor position, and r S is a screening length. The lowest-order calculation of Eq. (23) with respect to this potential can be performed easily no matter what model for rs is chosen. In the Debye-Thomas-Fermi model and isotropic valleys, the calculation was carried out in our paper [29]. In a theory that seems more consistent, the screening is not introduced but is derived in perturbative electron-gas calculations, electron-impurity, and e-e interactions being treated simultaneously. For many-valley bands such a calculation of Eq. (23) was performed by Sernelius [27] using a diagrammatic method. His result for the impurity contribution can be transformed to: /-'(vei) ( 0 . ) ) -

d 3q 6e2 Im J((27r)3q2av(q)Vi(q;og)V~(q,O)[xo(q,w)- Xo(q,O)],

md09

(35)

where all(q) - cos20, a• - 1/2 sin20, Xo(q,og) is the wavevectorfrequency-dependent electron polarizability in one parabolic valley (Lindhard function) with effective mass md; Vi(q, w) _

- 4"n'e

elq 2

mllall md

(36)

m j_a •

+ 8,rre2 ~Xo(qk, co)

md

k= 1

+ ~

is the space-time Fourier transform of the effective interaction potential, where: q~ = q, q2 = q K1/2 [c~ + K-l(cos2q ~ + K-lsinZq~)] 1/2, q3 - q K1/2 [cos20 + K-l(sin2q~ + K-lcos2q))] 1/2, K - mfl/m • 0 and q~ being the polar and azimuthal angles, respectively. In the isotropic valleys Eq. (35) reduces to our result [29] by putting o) = 0, q = 0 in the polarizability term of Eq. (36), that is also obtained using transition-probability method. Sernelius [27] obtained an expression for an e-e contribution that describes 'friction' damping of relative momentum of differently oriented valleys. It changes the structure of Eq. (25) at general o~, but at high frequencies contributes only an additive term to Eq. (26). Both screening models take into consideration only the long-range part of electron-donor interaction potential. For uncompensated n-Si at T = 300K, because of full donor ionization, they predict independence of the scattering rate from impurity type. At the same time, there exists experimental evidence of such dependence for the mobility [31] and for IR reflectance [34] (see Section III). The dependence is probably due to a short-range potential, for which at least two sources can be pointed out. The first is the difference between the atomic potentials of Si and the donor atom, which is responsible for the deviation of dopant ionization energy from the Bohr energy, and for the difference in those for different dopants. These substitution potentials for the three dopants in Si were estimated by Vifia and Cardona [11 ]. The second is a local deformation in the vicinity of the impurity that affects the valleys through the deformation potential the same way as the lattice

170

Mark Auslender and Shlomo Hava

vibrations do. We have not found a reliable estimation of this effect in the literature. The near-IR absorption spectra [34, 40] offer an additional evidence that a short-range potential emerges with doping (see Subsection VIA). The calculation of DC mobility is a crucial test for the model, because without agreeement between theory and experiment at DC, the disagreement will appear also in the IR. As is well known, the mobility calculated in the Born approximation with respect to the Debye-Thomas-Fermi potential, and isotropic-valley approximation (Brooks-Herring mobility) disagrees strongly with experimental data [45]. The calculations in GDA with (q, ~o)-screening model (Eqs. (25, 35, 36)) include triple integration of complicated function with respect to q, 0, q~. This is a formidable task for desktop-computer simulations (a Cray-type supercomputer was used by Serrelius [27]). In our case K 1/2 ~ 2.27, the mean values of q2.3 appear to be close to q, so we could put q2,3 = q in Eq. (35), but retain angular dependence through the functions a,(q). After this, the angle integration is carried out analytically. The relaxation functions are expressed via the integrals with respect to the reduced-energy variable x as in Eq. (30), but with much more complicated integrands. Despite this complexity, they still allow a fast desktop-computer calculation. The calculations become much faster in the Debye-ThomasFermi limit. Neglecting e-e scattering, the DC relaxation time is found using GDA extrapolated to ~o = 0: 1- -

mc 3mllFil(0)

+

2mc

.

(37)

3m•177

At maximum e-e scattering, 1-changes from this value to F-1(0), where/-'(to) is given by Eq. (26) [27]. We calculated 1-for N - 1017, 10 TM, 1019, and 10 20 cm -3 and obtained: 1-= 1.31 X 10 -13, (7.22-7.07) X 10 -14, (3.903.77) X 10 -14, and (2.58-2.51) x 10 -14 S, respectively. The second value in the brackets shows the effect of e-e scattering, which thus proves to be small. Corresponding 'experimental' values of ~" calculated using Eqs. (27-29) are: 1.08 • 10 -13, 4.22 X 10 -14, (1.83-1.69) X 10 -14, and (1.09-1.02) • 10 -14 s, respectively. In the last two data, the first value corresponds to Si: P and the second to Si: As; close values are extracted from the Irvin resistivity curves [46]. We see systematic discrepancy between the theoretical and experimental values of 1" by a factor which increases with the increase of N from --~ 1.2 to --~ 2.3 -2.5. The Debye-Thomas-Fermi model gives a larger factor (3.5-3.7 at N - 1020 cm-3). To the best of our knowledge this discrepancy has not yet been resolved, and is still discussed in the literature [47]. Thus, the disagreement between optical constants calculated by GDA using the screening models and those calculated using the empirically adjusted Drude approach [43], arises even if the static values of the relaxation functions are substituted into Eq. (25). Including full frequency de-

6. Doped n-Type Silicon (n-Si)

171

pendence does not make the situation better. Moreover, as seen from Fig. 4, the relaxation functions appear to strongly decrease with decreasing A, so their values at short wavelengths are too small to account for observed absorption. In addition, at heavy doping (N > 3 • 1018 c m - 3 ) the relaxation functions acquire peaks near the plasma wavelength, which are more pronounced for F• (see Figs. 4, 5). Mathematically, these peaks arise from the poles of V~(q,to) in the complex (q,to)-plane. Physically, they display the response of plasma waves excited by scattered electrons in the screening electron cloud, a feature that cannot appear in the Debye-Thomas-Fermi model. These peaks produce weak humps on the reflectivity curves near hm~., which have never been observed experimentally. In our opinion, this is one more reason to recast the donor potential issue in n-Si.

TABULATION OF OPTICAL CONSTANTS OF DOPED n-Si

We tabulated n and k for practical use at nine concentrations in the range from 1016 c m - 3 to 10 20 cm -3, with roughly a factor-of-three increase for each step, using the fit of Huml/6ek and Wojtechovsky [43] given by 260 '

I

'

I

,

I

'

I

i

I

""

I

'

I

,

I

220

180

140

100

60

20

,

0

5

10

I

15

WAVELENGTH

i

20

I

25

(lim)

Fig. 4. Theoretical spectra of (100) relaxation function/'11 calculated using Eqs. (35, 36) for three electron concentrations: (A) N = l019 cm-3; (o) N = 3 X l0 J9 cm-3; (D) N = 1020 cm- 3

VII

172

Mark Auslender and Shlomo Hava 720 640 560 480 400 320 -4 240 160

80 0

5

10

15

20

25

WAVELENGTH (gm) Fig. 5. Theoretical spectra of (010) relaxation function F• calculated using Eqs. (35, 36) for three electron concentrations: (A) N = 1019 c m - 3 ; (0) N -- 3 X 1019 cm-3; (n) N = 102o cm- 3.

Eqs. (27, 28) and Drude Eq. (18). The results are placed in Tables I to IX in the columns headed by abbreviation 'HW'. For heavy doping we display different values for Si: P and Si: As. The fit for Si: Sb (first line of Eq. (27)) was not used because we found that three data points available for this dopant fit a linear function of N better than that of X/N. Each table covers a spectral interval from 2/xm either to 100/xm at light and moderate dopings, or to 25~m at heavy doping. The tables can be easily extended to wider intervals using Eqs. (27, 28, 18). For sake of comparison, in each table we added the columns with theoretical values calculated by us using Eqs. (35, 36) with Eq. (25) as outlined above; they are abbreviated by 'GDA'. It is seen that at light and moderate doping the disagreement between theory and the empirical fit is not so strong. It is more or less concordant with the disagreement at DC discussed above. In some cases, where the fit overestimates % theory may be in a better agreement with direct experimental data (see Fig. 6). At heavy doping (Tables VII-IX), the disagreement becomes more drastic especially near the peaks of the relaxation functions. Fig. 7 shows some plots of n and k.

n-Si 10 2 0 v

z uJ

101

i.I.. t.l_ i.U

9 0 Z

0 ~0

_o 10o

0

I-13.. rr O 03 133
..rr < Z ....., C3 rr

!

E

\

! 'k

l

L

X %

l x 1 0 -3 ,,,,,--- no, P .-.-

O l x 1 0 "4

l /

ko, P

-.....

no, A

--,,

ko.A

l x 1 0 -s

l x 1 0 . 6 , i~ lx10 1

,.. , . . . . . . . . . . . . . . lx10 ~

,

.......

lx101

I,. . . . . . . . . . lx102

lx103

WAVELENGTH (,urn) Fig. 1. Log-log plot of ordinary no and ko versus wavelength in micrometers for e-GaSe. The curves no, A and ko, A are calculated from the sum of Eqs. (1), (2), (4), and e]~ = 1.25. The curves no,P and ko,P are calculated from Eq. (8).

480

David F. Edwards lx101

_.r lx10 o

|

l x 1 0 -1

tl I!

c c

>.n-" ,< Z ..=..., E3 nO < n"

ii

I

tl;i

I!

lx10-2. t

\ ~

I

lx10-3 Q

\

I

kX

UJ

\ \

/ l x 1 0 -4.

l x 1 0 -s

----.

ke, P

....

ne, A

....

k e,A

i,

lX10 1

,

i

, , ............... lX10 ~

\

t

ne, P

lx10"6 ,

%

/

I

,

...............

lX101

lX102

lX103

WAVELENGTH (l.trn) Fig. 2. Log-log plot of extraordinary rte and k e v e r s u s wavelength in micrometers for e-GaSe. The curves n~,A and ke,A are calculated from the sum of Eq. (7) and e ~ = 4.9. The curves n~,P and ke,P are calculated from Eq. (8).

Gallium Selenide (GaSe)

481

9,...,


~

"] -:i

.-]

=

.-.]

__

"~. :J

I..U o

1017 cm -3) [15]. Within the 0.2-1.3 eV energy range, optical constants do not vary much. The refractive index for this range was found equal to 3.3 +__ 0.1 [7-9]. The fundamental absorption edge for bulk Zn3P2 was investigated from 1.3 to 1.6 eV elsewhere [8, 9, 16-19]. Data presented in these papers are in good agreement, except for results published by Sobolev and Syrbu [19]. For energy higher than 1.6 eV, only transmission data for thin films are available [8-12]. The high-quality thin films of Zn3P 2 with thicknesses below 0.5 ~m were obtained by thermal vacuum evaporation onto the glass substrate. A standard optical system was used in the optical measurements. The values of extinction coefficient given in Table II for energies smaller than 1.6 eV are derived directly from the absorption coefficient determined by means of transmission measurements [5, 15]. High-quality Zn3P 2 single

Zinc Phosphide (Zn3P2)

611

crystals of two types, with hole concentrations higher than 1017 cm -3 and lower than 1016 cm -3, were used in the measurements. The transmission was measured with high-resolution spectrometers, according to the wavenumber range: with a GDM 1000 from 0.77 to 1.3 p~m, with a Carry II from 1.2 to 2.5 ~m, and with a Beckman spectrophotometer from 2.5 to 10 txm. In the visible and ultraviolet regions the values of n and k are provided by Kramers-Kronig analysis of the reflectivity spectra for single crystals of Zn3P2. The published reflectivity spectra were measured within different energy ranges, namely by Sobolev and Syrbu [19] within the energy range of 1-12 eV, by Misiewicz and Jezierski [20] within the 4-11 eV energy range, by Fagen [8] from 1.4 to 5 eV, by Jezierski et al. [21] from 2.2 to 5.8 eV, and by Misiewicz et al. [7] from 1.2 to 5.2 eV. The values of reflectivity presented by Sobolev and Syrbu [19] are much smaller than those reported by Misiewicz and Jezierski [20], especially for energies higher than 3.5 eV. This difference is related to the quality of the samples. Therefore, the Kramers-Kronig analysis was performed only for the spectrum with higher reflectivity values [20]. The n and k values are given in Table II for energies greater than 5 eV. For the lower energies K-K analysis was based on the reflectivity spectrum chosen from the spectra published by Jezierski et al. [21]. The chosen spectrum was characterized by the highest reflectivity value (R = 0.524) at the second maximum. This value indicates a good-quality sample with a well-prepared surface. The surface preparation involved polishing with aluminum-oxide powder of decreasing grain size (1.0, 0.3, and 0.05 /xm) and etching in 1% bromine-methanol solution [21]. It is worthwhile to notice that the highest value obtained by Fagen (R = 0.5 at the second peak) [8] is very close to that just mentioned. The values from K-K analysis for the spectrum chosen from those measured by Jezierski et al. [21] are tabulated for the energy range 1.93-5.7 eV. For energies lower than 2.2 eV, the reflectivity was calculated by K-K analysis [22] from a knowledge of the absorption coefficient in this range. For energies lower than 1.93 eV, the n values given in Table II were characteristic of the K-K analysis of the reflectivity spectra measured down to 1.2 eV, for example, the spectrum given by Misiewicz et al. [7]. These values of n are decreasing to the value 3.32 with decreasing energy. The spectra for n and k for unpolarized light over a wide spectral range are presented in Fig. 1. The measured optical spectra of Zn3P 2 showed anisotropy, which was observed for the fundamental absorption edge [16, 18] and in the reflectivity spectra within the visible and ultraviolet energy range [23]. The measurements were performed at room temperature by using x-ray oriented Zn3P 2 single crystals obtained from a gas-transport growth method. A doublegrating Carl-Zeiss GDM-1000 monochromator equipped with linear polarizers from Carl-Zeiss and a Hamamatsu photomultiplier were utilized. Within the 0.9-1.0 eV energy range, a HgCdTe-based detector was used. The de-

612

J. Misiewicz and K. Jezierski

t e r m i n e d values of n e, n o and ke, ko are also i n c l u d e d in Table II and presented in Figs. 2 and 3. T h e small birefringence in Zn3P 2 was m e a s u r e d within the 0 . 5 - 1 . 4 4 e V e n e r g y range [24, 25]. T h e b i r e f r i n g e n c e value increased f r o m 0.0156 at 0.5 eV, t h r o u g h 0.0165 at 1.1 e V and 0.0176 at 1.30 eV, up to 0.0200 at 1.44 eV. REFERENCES

1. E. K. Arushanov, Prog. Cryst. Growth Charact. 25, 131 (1992). 2. J. M. Pawlikowski, Rev. Solid State Sci. 2, 581 (1988); J. Misiewicz, L. Bryja, K. Jezierski, J. Szatkowski, N. Mirowska, Z. Gumienny, and E. Placzek-Popko, Microelectron. J. 25, 23 (1994). 3. J. Misiewicz, J. M. Wr6bel, and B. E Clayman, Solid State Commun. 66, 747 (1988). 4. J. Misiewicz, A. Lemiec, K. Jezierski, J. M. Wr6bel, and B. E Clayman, Infrared Phys. Technol. 35, 775 (1994). 5. J. Misiewicz, J. Phys. Condens. Matter 1, 9283 (1989). 6. G. Pangilinan, R. Sooryakumar, and J. Misiewicz, Phys. Rev. B 44, 2582 (1991). 7. J. Misiewicz, J. Wr6bel, and B. Sujak-Cyrul, Opt. Appl. 10, 75 (1980). 8. E. A. Fagen, J. App.l. Phys. 50, 6505 (1979). 9. L. Zdanowicz, W. Zdanowicz, D. Petelenz, and K. Kloc, Acta Phys. Pol. A57, 159 (1980). 10. K. R. Murali, B. S. V. Gopalam, and J. Sabhanadri, Thin Solid Films 136, 275 (1986). 11. J. L. Deiss, B. Elidrissi, M. Robino, and R. Weil, Appl. Phys. Lett. 49, 969 (1986). 12. A. Weber, E Sutter, and H. von Kanel, J. Appl. Phys. 75, 7448 (1994). 13. A. Subrahmanyam, K. R. Murali, B. S. V. Gopalam, and J. Sobhanadri, Phys. Status Solidi A 88, 681 (1985). 14. R. S. Radautsan, N. N. Syrbu, J. J. Nebola, and K. J. Volodina, Sov. Phys. Solid State 19, 1290 (1977). 15. J. Misiewicz, J. Phys. Chem. Solids 50, 1013 (1988). 16. V. Munoz, D. Decroix, A. Chery, and J. M. Besson, J. Appl. Phys. 60, 3282 (1986). 17. J. M. Pawlikowski, J. Misiewicz, and N. Mirowska, J. Phys. Chem. Solids 40, 1027 (1979). 18. J. Misiewicz, J. Phys. Condens. Matter 2, 2053 (1990). 19. V. V. Sobolev and N. N. Syrbu, Phys. Status Solidi B 64, 423 (1974). 20. J. Misiewicz and K. Jezierski, Solid State Commun. 70, 465 (1989). 21. K. Jezierski, J. Misiewicz, and E Kr61icki, Phys. Status Solidi A 112, K135 (1989). 22. K. Jezierski, J. Phys. C: Solid State Phys. 17, 475 (1984). 23. J. Misiewicz, J. Wr6bel, and K. Jezierski, J. Phys. C: Solid State Phys. 17, 3091 (1984). 24. J. Misiewicz and J. A. Gaj, Phys Status Solidi B 105, K23 (1981). 25. J. Misiewicz, N. Mirowska, and Z. Gumienny, Phys. Status Solidi A 83, K51 (1984).

Zinc Phosphide (Zn3P2) III

I I

I

I

illll

I I

)

I

613

I ........

I ........

I........

I ........

I ........

-

+ w o

0 + w D(~

"- ,~ -> 0.219/xm: n 2 = 1.4500 +

0.2000 ~2 + ~2 _ 0.015129

0.3651 ~2 ~2 _ 0.022650

+

0.3224 ,~2

(2)

~2 _ 0.033124 '

,~ is also in micrometers. This formula gives wavelengths of ultraviolet absorption peaks that agree closely with those obtained by direct measurements [7, 8] but gives no information concerning the infrared absorption peak. Extrapolated values of n for the long wavelengths, that is, for the IR wavelengths, as given by the Radhakrishnan formula have large uncertainties in contrast to those given by the Li formula expressed with Eq. (1). This means that the Radhakrish-

Rubidium Bromide (RbBr)

847

nan dispersion formula is not an adequate one for use in a wide wavelength range, that is, in both the mid-IR and the far-IR regions in particular. Thus, the Radhakrishnan formula only correlates the dispersion in both the visible and the near UV with absorption bands in the UV region. Sprockhoff [3] measured the values of n of RbBr using the method of minimum deviation as well. He determined the refractive-index value of the RbBr single crystal at the following wavelengths: 0.486, 0.589 and 0.656/xm (see Table I). The errors of the refractive-index data calculated using Eq. (1) are dependent on wavelength. (The greatest and/or smallest errors in n are equal to ___ 0.02 in the regions 0.21-0.22/xm and 40.00-60.00 ~m and/or _ 0.002 in the region 0.40-1.50 /xm.) In the Sprockhoff and Kublitzky data the errors of the values of n can be estimated as _ 0.001. Pai et al. [4] determined the spectral dependences of n and k characterizing RbBr bulk samples (i.e., single crystals) by means of amplitude and phase reflection measurements made by dispersive Fourier transform spectroscopy in the region 222.2 ~m -> A -> 50/zm. The errors in the optical constants n and k evaluated in this way are also dependent on wavelength. (The greatest and/or smallest errors in both n and k are equal to + 0.10 and/or _+ 0.01 at the wavelength of about 116/zm and/or in the interval 93/xm _> )t -> 50/xm [4].) Spectral dependences of n and k in the vicinity of the maximum value of k at the wavelength of the fundamental infrared absorption peak &z are related to the TO phonon mode. The fine structures in the curves of n and k in the far-IR region may be caused by multiple-phonon absorption processes. In the visible there is very good agreement between the refractive-index data calculated using the Li formula [2] and measured by Kublitzky [5] on one side and those presented by Sprockhoff [3] on the other side (see Table I). Similarly, one can see a relatively good agreement between the values of n calculated by means of the Li formula and those determined by Pai et al. [4] in the region 54.00 /zm _> )t _> 50.00 /zm (see Table I). Both the facts mentioned give support for the correctness and reliability of the opticalconstant data presented for the RbBr single crystal in this paper. From Eq. (1) and the spectral dependence of k determined by Pai et al. [4], one can see that there is a good agreement between the value of/~I used in the Li formula [2] and that implied by the data of Pai et al. It should be noted that in the parts of the IR region in which RbBr is a weakly absorbing material (k -< 0.1), more precise values of k can be determined by transmittance measurements of sufficiently thin RbBr slabs than by means of the experimental techniques used by Pai et al. [4]. Thus far, these measurements of the transmittance of the RbBr slabs have not been performed in the IR region. Unfortunately, measurements enabling us to evaluate the values of the extinction coefficient k of the RbBr single crystals or RbBr thin films in the UV region have not been carried out, either. The optical-constant data presented here are the room-temperature

848

Ivan Ohlidal and Miloslav Ohlidal

data, as pointed out earlier. However, strictly speaking, there are small differences at temperatures corresponding to the data summarized. Namely, the refractive-index data obtained by means of the Li dispersion formula [2], the values of n determined by Sprockhoff [3] and the values of n and k published by Pai et al. [4] correspond to 293 K, 298 K, and 300 K, respectively. Both the Radhakrishnan formula [6] and the Kublitzky data correspond to 308 K. This means that at first Li reduced the Kublitzky refractiveindex data from 308 to 293 K to obtain his formula in the form presented. He performed these corrections by using empirical parameters to construct a formula for dn/dT, enabling him to carry out the mentioned corrections of the Kublitzky values of n [2] (dn/dT denotes the derivative of the refractive index with respect to temperature T). In conclusion, it is possible to state that the optical-constant data summarized here represent the best values of the refractive index n characterizing the RbBr single crystal in the region 222.2/xm --- h -> 0.21/.~m and the best values of the extinction coefficient k of the same material in the region 222.2 tzm -> h -> 50/xm. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8.

D. E. McCarthy, Appl. Opt. 7, 1243 (1968). H. H. Li, J. Phys. Chem. Ref Data 5, 329 (1976). M. Sprockhoff, Neues Jahrb. Mineral Geol. Palaeontol., Beilageband 18, 117 (1904). K. E Pai, T. J. Parker, N. E. Tornberg, R. P. Lowndes, and W. G. Chambers, Infrared Phys. 18, 199 (1978). A. Kublitzky, Ann. Phys. 20, 793 (1934). T. Radhakrishnan, Proc. Indian Acad. Sci. Sect. A 27, 165 (1948). R. Hilsch and R. W. Pohl, Z. Phys. 59, 812 (1930). E. G. Schneider and H. M. O'Bryan, Phys. Rev. 51, 293 (1937).

Rubidium

Bromide

(RbBr)

10

849

'

'

'

'

''"1

'

'

'

'

''"1

'

'

'

'

''"1

I

L

I I I I I I I I I I I I

n,k

I

|

|

0.1

I I I I I I tI i I I I I I I I I I

0.01

,

0,1

, ,,,,,,I

,

1

, ,,,,,,I

,

.......

10

WAVELENGTH

'

,

100

[ p,m ]

Fig. 1. Log-log plot of n (solid line) and k (dashed line) versus wavelength in micrometers for RbBr.

850

Ivan Ohlidal and Miloslav Ohl[dal TABLE I Values of n and k for R u b i d i u m B r o m i d e as Obtained from Various References a

eV

cm - - 1

5.904 5.848 5.794 5.740 5.687 5.636 5.585 5.535 5.486 5.438 5.391 5.344 5.299 5.254 5.209 5.166 5.123 5.081 5.040 4.999 4.959 4.920 4.881 4.843 4.806 4.769 4.732 4.696 4.661 4.626 4.592 4.558 4.525 4.492 4.460 4.428 4.397 4.366 4.335 4.305 4.275 4.246 4.217 4.189 4.161

47,620 47,170 46,730 46,300 45,870 45,450 45,050 44,640 44,250 43,860 43,480 43,100 42,740 42,370 42,020 41,670 41,320 40,980 40,650 40,320 40,000 39,680 39,370 39,060 38,760 38,460 38,170 37,880 37,590 37,310 37,040 36,760 36,500 36,230 35,970 35,710 35,460 35,210 34,970 34,720 34,480 34,250 34,010 33,780 33,560

/zm 0.210 0.212 0.214 0.216 0.218 0.220 0.222 0.224 0.226 0.228 0.230 0.232 0.234 0.236 0.238 0.240 0.242 0.244 0.246 0.248 0.250 0.252 0.254 0.256 0.258 0.260 0.262 0.264 0.266 0.268 0.270 0.272 0.274 0.276 0.278 0.280 0.282 0.284 0.286 0.288 0.290 0.292 0.294 0.296 0.298

n 1.962 [2] 1.936 1.913 1.893 1.875 1.858 1.844 1.830 1.818 1.807 1.796 1.787 1.777 1.769 1.761 1.754 1.747 1.740 1.734 1.728 1.722 1.717 1.712 1.707 1.703 1.698 1.694 1.690 1.686 1.683 1.679 1.676 1.673 1.669 1.666 1.664 1.661 1.658 1.655 1.653 1.651 1.648 1.646 1.644 1.642

(continued) a

References given in brackets.

Rubidium Bromide (RbBr)

851 TABLE I

(Continued)

Rubidium Bromide

eV 4.133 4.065 4.000 3.936 3.875 3.815 3.757 3.701 3.647 3.594 3.542 3.493 3.444 3.397 3.351 3.306 3.263 3.220 3.179 3.139 3.100 3.024 2.952 2.883 2.818 2.755 2.695 2.638 2.583 2.551 2.530 2.480 2.431 2.384 2.339 2.296 2.254 2.214 2.175 2.138 2.105 2.101 2.066 2.000 1.937 1.890

cm-

1

33,330 32,790 32,260 31,750 31,250 30,770 30,300 29,850 29,410 28,990 28,570 28,170 27,780 27,400 27,030 26,670 26,320 25,970 25,640 25,320 25,000 24,390 23,810 23,260 22,730 22,220 21,740 21,280 20,830 20,580 20,410 20,000 19,610 19,230 18,870 18,520 18,180 17,860 17,540 17,240 16,980 16,950 16,670 16,130 15,630 15,240

/xm 0.300 0.305 0.310 0.315 0.320 0.325 0.330 0.335 0.340 0.345 0.350 0.355 0.360 0.365 0.370 0.375 0.380 0.385 0.390 0.395 0.400 0.410 0.420 0.430 0.440 0.450 0.460 0.470 0.480 0.486 0.490 0.500 0.510 0.520 0.530 0.540 0.550 0.560 0.570 0.580 0.589 0.590 0.600 0.620 0.640 0.656

1.640 1.635 1.630 1.626 1.622 1.618 1.615 1.612 1.609 1.606 1.603 1.601 1.598 1.596 1.594 1.592 1.590 1.588 1.587 1.585 1.583 1.580 1.578 1.575 1.573 1.571 1.569 1.567 1.566 1.565 [31 1.564 [21 1.563 1.561 1.560 1.559 1.558 1.557 1.556 1.555 1.554 1.553 [31 1.553 [2] 1.552 1.551 [2] 1.550 1.548 [3]

(continued)

852

Ivan Ohlidal and Miloslav Ohlfdal TABLE I (Continued) Rubidium Bromide

eV 1.879 1.823 1.771 1.722 1.675 1.631 1.590 1.550 1.512 1.476 1.442 1.409 1.378 1.348 1.319 1.292 1.265 1.240 1.181 1.127 1.078 1.033 0.9919 0.9537 0.9184 0.8856 0.8551 0.8266 0.7749 0.7293 0.6888 0.6526 0.6199 O.5636 O.5166 0.4769 0.4428 0.4133 0.3542 0.3100 O.2755 0.2480 0.2066 0.1771 0.1550 0.1378

--1 cm

15,150 14,710 14,290 13,890 13,510 13,160 12,820 12,500 12,200 11,900 11,630 11,360 11,110 10,870 10,640 10,420 10,200 10,000 9,524 9,091 8,696 8,333 8,000 7,692 7,407 7,143 6,897 6,667 6,250 5,882 5,556 5,263 5,000 4,545 4,167 3,846 3,571 3,333 2,857 2,500 2,222 2,000 1,667 1,429 1,250 1,111 ,,,

/xm 0.660 0.680 0.700 0.720 0.740 0.760 0.780 0.800 0.820 0.840 0.860 0.880 0.900 0.920 0.940 0.960 0.980 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.60 1.70 1.80 1.90 2.00 2.20 2.40 2.60 2.80 3.00 3.50 4.00 4.50 5.00 6.00 7.00 8.00 9.00

1.548 [2] 1.547 1.546 1.546 1.545 1.544 1.543 1.543 1.542 1.542 1.541 1.541 1.540 1.540 1.539 1.539 1.539 1.538 1.538 1.537 1.537 1.536 1.536 1.535 1.535 1.535 1.535 1.534 1.534 1.534 1.533 1.533 1.533 1.532 1.532 1.532 1.532 1.532 1.531 1.531 1.530 1.530 1.529 1.528 1.527 1.526

Rubidium Bromide (RbBr)

853 TABLE I

(Continued)

Rubidium Bromide eV 0.1240 0.1127 0.1033 0.09537 0.08856 0.08266 0.07999 0.07749 0.07514 0.07293 0.07085 0.06888 0.06702 0.06526 0.06358 0.06199 0.06048 0.05904 0.05767 0.05636 0.05510 0.05391 0.05276 0.05166 0.05061 0.04959 0.04862 0.04769 0.04679 0.04592 0.04509 0.04428 0.04350 0.04275 0.04203 0.04133 0.04065 0.04000 0.03936 0.03875 0.03815 0.03757 0.03701 0.03647 0.03594 0.03542

cm-

1

,000 909.1 833.3 769.2 714.3 666.7 645.2 625.0 606.1 588.2 571.4 555.6 540.5 526.3 512.8 500.0 487.8 476.2 465.1 454.5 444.4 434.8 425.5 416.7 408.2 400.0 392.2 384.6 377.4 370.4 363.6 357.1 350.9 344.8 339.0 333.3 327.9 322.6 317.5 312.5 307.7 303.0 298.5 294.1 289.9 285.7

/xm lO.O 11.0 12.0 13.0 14.0 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0 25.5 26.0 26.5 27.0 27.5 28.0 28.5 29.0 29.5 30.0 30.5 31.0 31.5 32.0 32.5 33.0 33.5 34.0 34.5 35.0

1.525 1.524 1.522 1.521 1.519 1.517 1.516 1.515 1.514 1.513 1.511 1.510 1.509 1.508 1.506 1.505 1.504 1.502 1.501 1.499 1.498 1.496 1.495 1.493 1.491 1.489 1.488 1.486 1.484 1.482 1.480 1.478 1.476 1.474 1.471 1.469 1.467 1.465 1.462 1.460 1.457 1.455 1.452 1.449 1.446 1.444

(continued)

854

Ivan Ohlfdal and Miloslav Ohlidal TABLE I

(Continued)

Rubidium Bromide -1

/.~m

0.03493 0.03444 0.03397 0.03351 0.03306 0.03263 0.03220 0.03179 0.03139 0.03100 0.03061 0.03024 0.02988 0.02952 0.02917 0.02883 0.02850 0.02818 0.02786 0.02755 0.02725 0.02695 0.02666 0.02638 0.02610 0.02583 0.02556 0.02530 0.02505 0.02480

281.7 277.8 274.0 270.3 266.7 263.2 259.7 256.4 253.2 250.0 246.9 243.9 241.0 238.1 235.3 232.6 229.9 227.3 224.7 222.2 219.8 217.4 215.1 212.8 210.5 208.3 206.2 204.1 202.0 20O

35.5 36.0 36.5 37.0 37.5 38.0 38.5 39.0 39.5 40.0 40.5 41.0 41.5 42.0 42.5 43.0 43.5 44.0 44.5 45.O 45.5 46.0 46.5 47.0 47.5 48.0 48.5 49.0 49.5 50.00

0.02418

195

51.28

0.02356

190

52.63

0.02294

185

54.05

0.02232 0.02170 0.02108 0.02046 0.02009 O.01984 0.01959 0.01934 0.01909

180 175 170 165 162 160 158 156 154

55.56 57.14 58.82 60.61 61.73 62.50 63.29 64.10 64.94

eV

cm

1.441 1.438 1.435 1.432 1.429 1.425 1.422 1.419 1.415 1.412 1.408 1.405 1.401 1.397 1.393 1.389 1.385 1.381 1.376 1.372 1.368 1.363 1.358 1.354 1.349 1.344 1.338 1.333 1.328 1.322 1.31 [4] 1.308 [2] 1.29 [4] 1.291 [2] 1.27 [4] 1.272 [2] 1.25 [4] 1.22 1.18 1.16 1.13 1.11 1.09 1.07 1.04 1.00

0.01 [4] 0.01 [4] 0.02 [4] 0.02 [4] 0.03 0.04 0.04 0.04 0.05 0.05 0.06 0.07 0.09

Rubidium Bromide (RbBr)

855 TABLE I

(Continued)

Rubidium Bromide eV 0.01885 0.01860 0.01835 0.01810 0.01785 0.01761 0.01736 0.01711 0.01686 0.01661 0.01637 0.01612 0.01587 0.01562 0.01537 0.01513 0.01488 0.01463 0.01438 0.01413 0.01389 0.01364 0.01339 0.01314 0.01289 0.01265 0.01240 0.01215 0.01190 0.01165 0.01141 0.01116 0.01091 0.01066 0.01041 0.01017 0.00992 0.00967 0.00942 0.00917 0.00893 0.00868 0.00843 0.00806 0.00744 0.00682 0.00620 0.00558

cm

-1

152 150 148 146 144 142 140 138 136 134 132 130 128 126 124 122 120 118 116 114 112 110 108 106 104 102 100 98.0 96.0 94.0 92.0 90.0 88.0 86.0 84.0 82.0 80.0 78.0 76.0 74.0 72.0 70.0 68.0 65.0 60.0 55.0 50.0 45.0

/xm 65.79 66.67 67.57 68.49 69.44 70.42 71.43 72.46 73.53 74.63 75.76 76.92 78.13 79.37 80.65 81.97 83.33 84.75 86.21 87.72 89.29 90.91 92.59 94.34 96.15 98.04 100.0 102.0 104.2 106.4 108.7 111.1 113.6 116.3 119.0 122.0 125.0 128.2 131.6 135.1 138.9 142.9 147.1 153.8 166.7 181.8 200.0 222.2

0.99 0.99 0.97 0.96 0.96 0.95 0.91 0.88 0.80 0.72 0.75 0.84 O.87 0.84 0.81 0.78 0.69 0.55 0.50 0.49 0.50 0.50 0.45 0.42 0.44 0.47 0.49 0.54 0.56 0.59 0.73 1.25 2.66 4.26 4.27 4.08 3.62 3.12 2.87 2.78 2.78 2.76 2.73 2.65 2.53 2.47 2.41 2.39

0.10 0.11 0.13 0.14 0.17 0.18 0.18 0.20 0.21 0.27 0.36 0.38 0.37 0.44 0.53 0.52 0.56 0.67 0.76 0.86 0.99 1.14 1.30 1.46 1.61 1.76 1.99 2.33 2.68 3.05 3.46 4.12 4.56 3.31 1.95 1.17 0.52 0.29 0.25 0.23 0.25 0.27 0.27 0.25 0.25 0. I9 0.18 0.13

Rubidium Iodide (Rbl) IVAN OHLiDAL and DANIEL FRANTA Department of Solid State Physics Faculty of Sciences Masaryk University Brno Brno, Czech Republic

Single crystals of rubidium iodide exhibit the cubic structure of NaC1 (0 5h-Fm3m space group). Rubidium iodide crystals are widely used for manufacturing various optical components employed especially in the infrared region. This is due to the fact that RbI crystals exhibit a broad range of transparency from the near UV (0.24/xm) to the far IR (64/xm). RbI is the most hygroscopic material of the rubidium halides. Considerable care has to be devoted to preparing surfaces of the optical components fabricated from RbI crystals, since these surfaces play an important role in their transparency properties. The gradual decrease in transmittance at shorter wavelengths is due to surface scattering that is caused by roughness of the surfaces of the components prepared by usual procedures of polishing, and not by absorption or scattering within the material itself [1]. This is probably caused by the fact that usual polishing procedures yield roughness of surfaces of the RbI components, exhibiting relatively large values of the heights of the surface irregularities. There are few data of the optical constants available for RbI, similar to the other rubidium halides. The room-temperature values of the refractive index n and the extinction coefficient k tabulated here (see Table I and Fig. 1) were obtained from the following papers: the 0.25/.zm (4.959 eV) to 0.18 /.zm (6.888 eV) results are from the paper of Baldini and Rigaldi [2]; the 0.26 /xm (4.769 eV) to 0.23 ~m (5.391 eV) values of n and k are obtained on the basis of measurements presented in the article of Fr6hlich et al. [3]; the 64/xm (0.01937 eV) to 0.24 tzm (5.166 eV) refractive-index data were calculated by means of the dispersion formula presented in the paper of Li [4]; the 0.656 tzm (1.890 eV) to 0.486/xm (2.551 eV) values of n are from the paper of Sprockhoff [5]; and the 222.2 txm (45 cm -~) to 52.63 ~m (190 cm -~) values of n and k were published by Pai et al. [6]. 857 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS III

Copyright 9 1998 by Academic Press. All rights of reproduction in any form reserved ISBN 0-12-544423-0/$25.00.

858

Ivan Ohlfdal and Daniel Franta

The values of n and k presented by Baldini and Rigaldi [2] were obtained by interpreting spectral dependences of the reflectances and transmittance of a RbI thin film with the thickness of 0.042/.~m (the error of the thickness was __+0.003/zm). The RbI film was prepared by sublimation onto a fusedsilica plate in the vacuum chamber employed simultaneously for the optical measurements (temperature of the fused-silica substrate during the deposition of the RbI film is not specified in the paper of Baldini and Rigaldi [2], but one can assume that it was about room temperature). Thus, the spectral dependences of both the reflectances R and R' of the film for incidence of light from the ambient side and the substrate side, respectively, could be measured together with the transmittance T of the film corresponding to incidence of light from the ambient side in the spectral region of interest (R 4: R' and T = T', if the film under investigation is absorbing, where T' denotes the transmittance of the film corresponding to incidence of light from the substrate side). The authors measured the values of these three different optical quantities, that is, the values of R, R' and T, at selected wavelengths at near-normal incidence of light. They then determined the value of the thickness and the values of n and k by solving the three equations expressing R, R', and T by means of a numerical procedure at the wavelengths of interest without any auxiliary measurements [2]. The spectral dependences of the optical constants of the RbI film determined in this way correspond to absorption caused by excitons in this region. Uncertainty of the opticalconstant data was reported to be about 1%. It should be noted that the wavelengths of the ultraviolet absorption peaks derived from the work by Baldini and Rigaldi [2] are inconsistent with those observed for bulk material of RbI [7, 8]. In the region 0.260/zm -> A -> 0.230 ~m, we calculated the values of the refractive index n and extinction coefficient k by means of the values of parameters characterizing resonances on the lower exciton-polariton branch in RbI single crystals evaluated with three-photon difference-frequency generation (TP-DFG) by Fr6hlich et al. [3]. Namely, as a modem method of nonlinear optical spectroscopy, the TP-DFG method opens up new possibilities to study elementary excitations in solids and, moreover, in the corresponding spectral region this method allows the determination of the spectral dependences of the optical constants of thick bulk samples of the single crystals in a reliable way. Therefore, the values of positions and widths of the resonance peaks (i.e., the values of the parameters characterizing resonances) determined for the RbI single crystal by the TP-DFG method at room temperature in the paper of Fr6hlich et al. [3] were used to calculate the spectral dependences of n and k of RbI for the region of the exponential tail of the excitonic absorption edge, that is, for the region mentioned earlier. Equations enabling us to carry out these calculations are introduced in the paper mentioned [3]. Unfortunately, uncertainties in the values of the parameters determined with the TP-DFG method (and needed for calculating n and k)

Rubidium Iodide (Rbl)

859

are not introduced in the paper of Fr6hlich et al., and so it is not possible to estimate an accuracy of the optical constants calculated. Thus, it is impossible to compare the values of n and k determined by Baldini and Rigaldi [2] with those calculated on the basis of the TP-DFG method in an exact way. In spite of this fact, it is evident that there are nonnegligible differences between the Baldini and Rigaldi data and the data calculated. This statement is especially true for the region 0.238 ~m - A >- 0.230 ~m (see Table I). One can expect that the values of n and k calculated on the basis of the TP-DFG measurements are more reliable than those determined by Baldini and Rigaldi. The values of the optical constants obtained by Baldini and Rigaldi are probably influenced by defects frequently occurring in thin solid films. The RbI thin film studied by Baldini and Rigaldi probably exhibited polycrystalline structure (structure of this film was not examined), and thus the defects such as columnar structure, impurities, and roughness of the boundaries could considerably affect the correctness of the data presented. This conclusion is also supported by the fact that the wavelengths of the ultraviolet absorption peaks corresponding to the Baldini and Rigaldi data are inconsistent with those observed for the RbI single crystals in direct measurements [7, 8]. In the region 64.00/xm - A - 0.24/xm, that is, in the region of transparency of RbI, the refractive-index data obtained on the basis of the Li resuits [4] are introduced. This author employed the important refractiveindex data published by Kublitzky [9] to obtain the following empirical dispersion formula for the refractive index of RbI: n 2

= 1.60563 +

0.00947A 2 + (0.120)2

~t2 __

0.00136A 2 /~2 __ (0.156)

+ 2

0.13707A 2 + (0.223)2

/~2 __

flf~|ft72)~v.,_,.v.._,,~ 2 + (0.134)2

/~2 __

0.41864A 2 + /~2 __ (0.179) 2

0.41771A 2 + /~2 __ (0.187) 2

(1)

2.36091A 2 (132.45) 2,

/~2 __

A denotes the wavelengths in micrometers. Thus, this formula of the Sellmeier type was utilized for calculating the values of n within the spectral region of transparency of Rb! [4]. These refractive-index values correspond to the RbI single crystal, since Kublitzky [9] measured the values of n characterizing RbI prisms by the method of minimum deviation at selected wavelengths in the spectral region 0.578 ~m -> A -> 0.254 p~m. Sprockhoff [5] also measured the values of n of RbI by means of the method of minimum deviation. He determined the refractive-index values of the single crystal for the following three wavelengths: 0.486, 0.589, and 0.656 ~m (see Table I).

860

Ivan Ohlfdal and Daniel Franta

The uncertainties of the refractive-index values determined by means of Eq. (1) are dependent on wavelength. (The smallest and/or greatest uncertainties in n are equal to +0.002 in the region 1.5/xm -> A -> 0.4/zm and/or +0.02 in regions 0.25 ~m -> A -> 0.24 txm and 64 tzm -> A -> 50 ~m). The errors of n in Sprockhoff's measurements can be estimated as +0.001. One can see that the values of n determined using Eq. (1) agree very well with those calculated on the basis of the TP-DFG measurements. Similarly, there is a very good agreement between values of n determined by means of Eq. (1) and those measured by Sprockhoff. These facts gives significant support for the correctness and reliability of the n values of bulk RbI presented here for both the visible and the region 0.23-0.38 ~m. It should be noted that in the region 0.546/xm >- h -> 0.25/xm by treating the Kublitzky data, Radhakrishnan [10] obtained the empirical dispersion formula of the Sellmeier type expressing the spectral dependence of n of the RbI single crystal in the following form: n 2=

1.6017 +

0.01749A 2

+

h 2 -- 0.015625

0.83466A 2 A2 - 0.033489

+

0.13917A 2

(2)

A2 - 0.049729'

A is also in micrometers. This formula yields the wavelengths of three ultraviolet absorption peaks, which agree with those studied for bulk material by direct measurements [7, 8] but gives no information concerning the infrared absorption peak. Thus, the Radhakrishnan formula [10] can not be used to calculate the values of n in a wide spectral region. Namely, the extrapolated refractiveindex values for the IR wavelengths using this formula have large uncertainties compared to those obtained using the Li formula [4]. This is caused by the fact that Radhakrishnan only correlated his formula with the wavelengths of the absorption peaks in the UV region found in the independent way, while Li performed the correlation of his formula not only with these ultraviolet wavelengths, but also with the wavelength of the fundamental infrared absorption peak AI found by direct measurements. This fact enables us to use the Li formula in the IR region in which RbI is a nonabsorbing material. It should be noted that the values of the adjustable parameters of both the empirical dispersion formulas mentioned (e.g., the value of 0.00947 in the Li formula) were found by fitting the Kublitzky data [9] by means of the least-squares method. Pai et al. [6] determined the spectral dependences of n and k characterizing RbI bulk samples (i.e., the single crystals) using amplitude and phase reflection measurements made by dispersive Fourier transform spectroscopy in the region 222.2 ~m -> A -> 52.63 ~m. The errors of n and k determined in this way are also dependent on wavelength. (The greatest and/or smallest

Rubidium Iodide (Rbl)

861

errors in both n and k are equal to ___0.13 and/or +__0.01 at the wavelength of about 132 /~m and/or in the interval 106 /~m >- )~ >- 54 /~m [6].) Spectral dependences of n and k in the vicinity of the maximum value of k at the wavelength of the fundamental infrared absorption peak AI are related to onephoton absorption, corresponding to the TO phonon mode. The fine structures of the curves expressing the spectral dependences of n and k in the region 60-90/.~m may be caused by multiple-phonon processes of absorption. In the region 64.52/xm _> ,~ >_ 52.63 ~m, very good agreement exists between the refractive-index data presented by Pai et al. [6] and those calculated using the Li formula [4] (see Table I). This agreement again gives support for the correctness and reliability of the n and k values of bulk RbI introduced for the infrared region here. However, it is necessary to point out that more precise values of k will be able to be determined using transmittance measurements of sufficiently thin RbI slabs in the intervals of the infrared region in which RbI is weakly absorbing (k 0.600 /xm questionable. The measurements of Sasson and Arakawa [33] are used for the near IR, although this latter work dealt with the low-temperature melt, and it is unclear how similar the k-values are to those for the roomtemperature solid; they likely provide an upper bound. It is worth noting that the absorption edge seen in these works is responsible for the characteristic yellow color of sulfur. Because of preferential absorption of shorter wavelengths, the light emerging from sulfur at room temperature appears yellow. As the temperature rises, thermal broadening of the band centered at about 4 eV carries the absorption edge further into the visible, and the hue darkens. Conversely, the absorption edge recedes into the UV with decreasing temperature, rendering sulfur crystals colorless below about - 5 0 ~ C. Further into the IR, the n-values obtained by Fuller et al. [ 10] for incident irradiance normal to the (111) crystal plane and with polarization perpendicular to the c-axis are used. The refractive indices were obtained from Kramers-Kronig analysis, and data were also collected for the orthogonal polarization and for a cryptocrystalline melt freeze. By far the most prominent features in the reflectance spectra of Fuller et al. are the bands centered near 190 and 240 cm-~, respectively. Because of extrapolations beyond experimental range that are required by K-K analysis, it is difficult to be sure of the uncertainty in k below 250 cm-~, but based on several different extrapolations, k(190) was not found to change by more than about _0.01 and k(240) varied by no more than about +0.003. A strong dependence on polarization of the band at 240 cm -a was observed, and the E1 band at

Orthorhombic Sulfur o~-S

905

190 cm-1 appears only as a shoulder on what is a very strong B 2 absorption in the case of _L or ab polarization. The band arising from the v5 fundamental at 470 cm-1 was easily distinguished in all spectra. Values of k(470) were found to range between 0.016 (kil) and 0.019 (k• Rather than lying somewhere between k H and k• the value of kc was found to be 0.023. This anomalously high value is attributed to both particle-size effects and experimental uncertainty. The most important combination band appearing in the transmittance and diffuse reflectance spectra of c~-S is the V~o + v3 combination, which is centered at about 845 cm-1. As calculated from transmission spectra of various authors, peak values of k(845) 0.001 to 0.004. All other absorption bands are quite weak. Their influence on R and n is on the order of at most 0.3%, and any k values obtained via the KramersKronig relations from such faint structures are quite unreliable. Above 675 cm-1 numerical values for k were obtained by electronically scanning graphs presented in Barrow's paper. Unpolarized light incident on the (110) plane was used because the transmission of the band at 845 c m - ] was too low at the desired polarization to allow the absorption index to be extracted. After correcting for reflection losses with the use of measurements made by Fuller et al., values for k were taken from the spectra of Srb and Va~ko between 675 and 500 cm-1. Below 500 cm-1, the k values provided by Fuller et al. [10] are used, but supplemental values are also provided from the Srb and Va~ko's work between 450 and 410 cm-1. Data for optical constants in the far IR are not available, though work is planned for this region. In view of mm-far-IR instrumentation planned by NASA, there are additional features, especially absorption at around 74 cm-~, that warrant investigation. To provide some idea of the optical constants associated with more arbitrary polarizations or crystallographic orientations, those of the melt freeze are included in the tabulated values. Attention is called to the differences between the two sets of data at energies below about 250 cm-1. Crystal structure [42] and electric-vector polarization are discussed in the caption of Fig. 1. ACKNOWLEDGMENTS

We are grateful to Dr. W. R. Hunter (SFA, Inc., 1401 McCormick Dr., Largo, MD 20774) for calculation of n and k in the far-UV-x-ray region, and we also wish to thank Philip Gabriel for his assistance in electronically scanning many of the graphs from which the tabulated data were obtained. REFERENCES

1. B. Meyer, Elemental sulfur. Chem. Rev. 76, 367-388 (1976). 2. R. Stuedel, Homocyclic sulfur molecules. Top. Curr. Chem. 102, 147-176 (1981).

906

Kirk A. Fuller, Harry D. Downing, and Marvin R. Querry

3. E. D. Eastman and W. C. McGavock, Heat capacity and entropy of rhombic and monoclinic sulfur. J. Am. Chem. Soc. 59, 145-151 (1937). 4. E. D. West, Heat capacity of sulfur from 25 to 450K, the heats and temperatures of transition and fusion. J. Am. Chem. Soc. 81, 29-37 (1959). 5. B. Meyer, Solid allotropes of sulfur. Chem. Rev. 64, 429-451 (1964). 6. A. T. Young, Venus cloud microphysics. Icarus 56, 568-577 (1983). 7. J. I. Moses and D. B. Nash, Phase transformations and the spectral reflectance of solid sulfur: Can metastable sulfur allotropes exist on Io? Icarus 89, 277-304 (1991). 8. A. L. Sprague, D. M. Hunten, and K. Lodders, Sulfur at Mercury, elemental at the poles and sulfides in the regolith. Icarus 118, 211-215 (1995). 9. A. R. Adams and W. E. Spear, Charge transport in orthorhombic sulphur crystals. J. Phys. Chem. Solids 25, 1113-1118 (1964). 10. K. A. Fuller, H. D. Downing, and M. R. Querry, Infrared optical properties of orthorhombic sulfur. Appl. Opt. 30, 4081-4093 (1991). 11. K.A. Fuller, Elemental sulfur and its optical properties in the infrared. M. S. Thesis, Stephen E Austin State University, Nacogdoches, Texas (1984). 12. A. T. Young, No sulfur flows on Io. Icarus 58, 197-226 (1984). 13. B. Meyer, "Sulfur, Energy and Environment." Elsevier, New York, 1979. 14. W. W. Coblentz, Optical notes. Phys. Rev. 19, 89-97 (1904). 15. R. B. Barnes, Measurements in the long wavelength infrared from 20 /xm to 135 ~m. Phys. Rev. 39, 562-575 (1932). 16. H. J. Bernstein and J. Powling, The vibrational spectra and structure of inorganic molecules. II. Sulfur $8, sulfur chloride $2C12, phosphorous P4. J- Chem. Phys. 18, 10181023 (1950). 17. G. M. Barrow, The infrared spectra of oriented rhombic sulfur crystals with polarized radiation. J. Chem. Phys. 21, 219-222 (1953). 18. V. D. Neff and T. H. Walnut, Effect of temperature on the intensity and structure of bands in the infrared spectrum of rhombic sulfur. J. Chem. Phys. 35, 1723-1729 (1961). 19. I. Srb and A. Va~ko, Remark on the infrared spectrum of rhombic sulfur. J. Chem. Phys. 37, 1892-1893 (1962). 20. C. E. MacNeill, Infrared transmittance of rhombic sulfur. J. Opt. Soc. Am. 53, 398-399 (1963). 21. D. W. Scott, J. P. McCullough, and F. H. Kruse, Vibrational assignment and force constants of $8 from a normal coordinate treatment. J. Mol. Spectrosc. 13, 313-320 (1964). 22. G. Gautier and M. Debeau, Spectres vibration d'un monocristal de soufre orthorhombique. Spectrochim. Acta Part A 30, 1193-1198 (1974). 23. A. Anderson and P. G. Boczar, Far infrared spectrum of orthorhombic sulfur. Chem. Phys. Lett. 43, 506-511 (1976). 24. R. Sasson, R. Wright, E. T. Arakawa, B. N. Khare, and C. Sagan, Optical properties of solid and liquid sulfur at visible and infrared wavelengths. Icarus 64, 368-374 (1985). 25. B. Kurrelmeyer, The photoelectric conductivity of sulphur. Phys. Rev. 30, 893-910 (1927). 26. W. E. Spear and A. R. Adams, Photogeneration of charge carriers and related optical properties in orthorhombic sulphur. J. Chem. Phys. Solids 27, 281-289 (1966). 27. B. E. Cook and W. E. Spear, The optical properties of orthorhombic sulphur in the vacuum ultraviolet. J. Phys. Chem. Solids 30, 1125-1134 (1969). 28. R. L. Emerald, R. E. Drews, and R. Zallen, Polarization-dependent optical properties of orthorhombic sulfur in the ultraviolet. Phys. Rev. B 14, 808-813 (1976). 29. B. Meyer, M. Gouterman, D. Jensen, T. Oommen, and T. Stroyer-Hansen, The spectrum of sulfur and its allotropes. Adv. Chem. Ser. 110, 53-72 (1972). 30. N. M. Nelson, D. C. Pieri, S. M. Bologa, D. B. Nash, and C. Sagan, The reflection spectrum of liquid sulfur: Implications for Io. Icarus 56, 409-413 (1983). 31. J. Gradie and J. Veverka, Photometric properties of sulfur. Icarus 58, 227-245 (1984).

Orthorhombic Sulfur o~-S

907

32. D. B. Nash, Mid-infrared reflectance spectra (2.3-22/xm) of sulfur, gold, KBr, MgO, and Halon. Appl. Opt. 25, 2427-2433 (1986). 33. R. Sasson and E. T. Arakawa, Temperature dependence of index of refraction, reflection, and extinction coefficient of liquid sulfur in the 0.4-2.0/xm wavelength range. Appl. Opt. 25, 2675-2680 (1986). 34. M. Wong, The infrared optical constants of molten sulfur at 150 ~ and 200~ M. S. Thesis Stephen E Austin State University, Nacogdoches, Texas, (1984). 35. A. M. Kellas, The determination of the molecular complexity of liquid sulphur. J. Chem. Soc. 113, 903-922 (1918). 36. L. Brillouin, "Wave Propagation and Group Velocity," Academic Press, New York, 1960. 37. J. D. Jackson, "Classical Electrodynamics," 2nd Ed. Wiley, New York, 1975. 38. L. W. Pinkley, P. P. Sethna, and D. Williams, Optical constants of water in the infrared: Influence of temperature. J. Opt. Soc. Am. 67, 494-499 (1977). 39. L. W. Pinkley and D. Williams, The infrared optical constants of sulfuric acid at 250K. J. Opt. Soc. Am. 66, 122-124 (1976). 40. C. W. Robertson, H. D. Downing, B. Curnutte, and D. Williams, Optical constants of solid ammonia in the infrared. J. Opt. Soc. Am. 65, 432-435 (1975). 41. B. L. Henke, E. M. Gullikson, and J. C. Davis, X-ray interactions: Photoabsorption, scattering, transmission, and reflection at E = 50-30000 eV, Z = 1-92. At. Data Nucl. Data Tables 54, 181 (1993). 42. A. S. Cooper, W. L. Bond, and S. C. Abrahams, The lattice and molecular constants in orthorhombic sulfur. Acta Crystallogr. 14, 1008 (1961).

908

Kirk A. Fuller, Harry D. Downing, and Marvin R. Querry

10 0

:?'.....: o

10-1 .. ~ ,.,

10-2

-"

..! ~

,

,,.,.

o

10 .3

." -

. .

.

i

.

!ii.'

V

.

_

10 .4

, ,

..

~

!

~

fi

(m)

10-5

o,

1

10-8 (tI~)

10-7 10-4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-3

10-2

10-1

WAVELENGTH

10 0

101

10 2

(gm)

Fig. 1. Log-log plot of n (

) and k (. . . . ) versus wavelength in micrometers for a-S. The symmetry of the orthorhombic crystals is Fddd with lattice constants of Cooper et al. [42] a'b'c=

10.46" 12.87" 24.49.

The c-axis of the orthorhombic crystal lies along the length of the crystal. The ab-plane contains the middle horizontal parallelogram shown in the figure, with the a-axis lying along the diagonal that extends out of the plane of the figure. The primary tabulated data (columns 4 and 5) involve incident radiation with electric vector contained in the ab-plane. The direction of incidence for the primary data is along the c-axis for the UV-VIS region and perpendicular to it in the IR.

Orthorhombic Sulfur oL-S

909 TABLE I

Values for n and k for O r t h o r h o m b i c Sulfur O b t a i n e d from Various R e f e r e n c e s a

eV .300E .292E .284E .277E .270E .263E .256E .249E .242E .236E .230E .224E .218E .212E .206E .201E 196E 191E 186E 181E 176E 171E 167E 162E 158E 154E

+ + + + + + + + + + + + + + + + + + + + + + + + + +

150E +

146E 142E 138E 135E 131E 128E 124E 121E l18E

+ + + + + + + + +

115E +

cm- ~ 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05

l12E + 05 109E + 05 106E + 05 103E + 05 100E + 05 .978E + 04 .952E + 04 .927E + 04

.242E .236E .229E .223E .217E .212E .206E .201E .195E .190E 9185E

+ + + + + + + + + + +

180E +

176E 171E 167E 162E 158E 154E

+ + + + + +

150E +

146E 142E 138E 135E 13IE 128E 124E 121E l18E l15E

+ + + + + + + + + +

112E +

109E 106E .103E .100E .976E .951E .926E .901E .878E .855E .832E .810E .789E .768E .748E

+ + + + + + + + + + + + + + +

09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 09 08 08 08 08 08 08 08 08 08 08 08

/xm .413E-04 .425E-04 .436E-04 .448E-04 .460E-04 .472E-04 .485E-04 .498E-04 .512E-04 .526E-04 .540E-04 .554E-04 .569E-04 .585E-04 .601E-04 .617E-04 .633E-04 .651E-04 .668E-04 .686E-04 .705E-04 .774E-04 .743E-04 .764E-04 .784E-04 .805E-04 .827E-04 .850E-04 .878E-04 .896E-04 .920E-04 .945E-04 .971E-04 .997E-04 102E-03 105E-03 108E-03

n• 1.000 [41 ] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

111E-03

1.000

114E-03 117E-03 120E-03 123E-03 127E-03 130E-03 134E-03

1.000 1.000 1.000 1.000 1.000 1.000 1.000

k•

nsupp

ksupp

.123E-08 [41] 9137E-08 .153E-08 .171E-08 9190E-08

.212E-08 .236E-08 .263E-08 .293E-08 .327E-08 .364E-08 .406E-08 .452E-08 .503E-08 .560E-08 .624E-08 .695E-08 .773E-08 .860E-08 .957E-08 107E-07 118E-07

132E-07 147E-07 163E-07 181E-07 .201E-07 .224E-07 .248E-07 .276E-07 .307E-07 .341E-07 .378E-07 .420E-07 .467E-07 .518E-07 .575E-07 .639E-07 .709E-07 .787E-07 .873E-07 .967E-07 .107E-06 9119E-06

9132E-06

(continued) a References given in brackets.

910

Kirk A. Fuller, Harry D. Downing, and Marvin R. Querry TABLE I

(Continued)

Orthorhombic Sulfur eV .903E + .879E + .856E + .833E + .811E + .790E + .769E + .749E + .729E + .710E + .691E + .673E + .655E + .638E + .621E + .605E + .589E + .573E + .558E + .544E + .529E + .515E + .502E + .489E + .476E + .463E + .451E + .439E + .428E + .416E + .405E + .395E + .384E + .374E + .364E + .355E + .345E + .336E + .327E + .319E + .310E + .302E + .294E + .287E + .279E + .272E +

cm

04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04

.728E .709E .690E .672E .654E .637E .620E .604E .588E .573E .558E .543E .529E .515E .501E .488E .475E .463E .450E .438E .427E .416E .405E .394E .384E .374E .364E .354E .345E .336E .327E .318E .310E .302E .294E .286E .279E .271E .264E .257E .250E .244E .237E .231E .225E .219E

-1

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

ftm 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 08

137E-03 141E-03 145E-03 149E-03 153E-03 157E-03 161E-03 166E-03 170E-03 175E-03 179E-03 .184E-03 .189E-03 .194E-03 .200E-03 .205E-03 .211E-03 .216E-03 .222E-03 .228E-03 .234E-03 .241E-03 .247E-03 .254E-03 .261E-03 .268E-03 .275E-03 .282E-03 .290E-03 .298E-03 .306E-03 .314E-03 .323E-03 .331E-03 .340E-03 .350E-03 .359E-03 .369E-03 .379E-03 .389E-03 .399E-03 .410E-03 .421E-03 .433E-03 .444E-03 .456E-03

n•

k•

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

9146E-06 9162E-06 9180E-06 9199E-06 .221E-06 .245E-06 .271E-06 .301E-06 .333E-06 .369E-06 .409E-06 .453E-06 .502E-06 .556E-06 .616E-06 .682E-06 .755E-06 .835E-06 .924E-06 9102E-05 .113E-05 .125E-05 .138E-05 .153E-05 9169E-05 9187E-05 .207E-05 .228E-05 .252E-05 .278E-05 .307E-05 .339E-05 .374E-05 .413E-05 .455E-05 .502E-05 .553E-05 .609E-05 .671E-05 .738E-05 .812E-05 .893E-05 .982E-05 9108E-04 9119E-04 9130E-04

nsupp

ksupp

Orthorhombic Sulfur oL-S

911 TABLE I

(Continued)

O r t h o r h o m b i c Sulfur eV

cm-~

/xm

n•

k•

08 08 08 08 08 08 08 08 08 08 08

.469E-03 .481E-03 .495E-03 .508E-03 .522E-03 .536E-03 .550E-03 .565E-03 .580E-03 .596E-03 .612E-03

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

.143E-04 . 157E-04 .172E-04 .186E-05 .207E-05 .230E-05 .255E-05 .282E-05 .313E-05 .346E-05 .383E-05

159E + 08 155E + 08 151E + 08

1.000 1.000 1.000

.423E-05 .467E-05 .516E-05

08 08 08 08 08 08 08 08 08 08 08 08 08 08 08 07 07 07 07 07 07 07 07 07

.629E-03 .646E-03 .663E-03 .681E-03 .700E-03 .719E-03 .738E-03 .758E-03 .778E-03 .799E-03 .821E-03 .843E-03 .866E-03 .890E-03 .914E-03 .938E-03 .964E-03 .990E-03 102E-02 104E-02 107E-02 110E-02 113E-02 116E-02 119E-02 123E-02 126E-02

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

.570E-05 .630E-05 .695E-05 .766E-05 .844E-05 .930E-05 .102E-04 .113E-04 .125E-04 .137E-04 . 152E-04 .167E-04 .185E-04 .204E-04 .225E-04 .248E-04 .274E-04 .303E-04 .334E-04 .369E-04 .407E-04 .449E-04 .496E-04 .547E-04

959E + 03

.774E + 07

129E-02

1.000

.603E-04

934E + 03 .910E + 03

.753E + 07 .734E + 07

133E-02

1.000

.665E-04

136E-02

0.999

.733E-04

.886E .862E .840E .817E .796E

.714E + 07 .695E + 07 .677E + 07

140E-02 144E-02 148E-02

.659E + 07 .642E + 07

152E-02 156E-02

0.999 0.999 0.999 0.999 0.999

.808E-04 .891E-04 .981E-04 .108E-03 .119E-03

.264E .258E .251E .244E .238E .231E .225E .219E .214E

+ + + + + + + + +

04 04 04 04 04 04 04 04 04

.208E .203E 197E 192E 187E 182E 177E 173E 168E 164E 159E 155E 151E 147E 143E 139E 136E 132E 129E 9125E 9122E .l19E .l16E .l13E .ll0E 107E 104E 101E 985E

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +

04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 03

+ + + + +

03 03 03 03 03

.213E .208E .202E 197E 192E 187E 182E 177E 172E 168E 163E

147E 143E 139E 136E 132E 128E 125E 122E l l9E ll5E I12E 109E 107E 104E 101E .984E .958E .933E .908E .884E .861E .838E .816E .795E

+ + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + +

nsupp

ksupp

(continued)

912

Kirk A. Fuller, Harry D. Downing, and Marvin R. Querry TABLE I

(Continued)

O r t h o r h o m b i c Sulfur ,,

eV .775E .755E .735E .715E .696E .678E .660E .643E .626E .609E .593E .578E .563E .548E .533E .519E .506E .492E .479E .467E .454E .442E .431E .419E .408E .398E .387E .377E .367E .357E .348E .339E .330E .321E .313E .305E .297E .289E .281E .274E .267E .259E .253E .246E .240E .233E

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

cm

03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

.625E .609E .592E .577E .562E .547E .532E .518E .505E .492E .479E .466E .454E .442E .430E .419E .408E .397E .387E .376E .366E .357E .347E .338E .329E .321E .312E .304E .296E .288E .281E .273E .266E .259E .252E .246E .239E .233E .227E .221E .215E .209E .204E .198E .193E .188E

-1

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

/xm 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07

9160E-02 9164E-02 .169E-02

.173E-02 .178E-02 .183E-02 .188E-02

.193E-02 .198E-02 .203E-02 .209E-02 .215E-02 .220E-02 .226E-02 .232E-02 .239E-02 .245E-02 .252E-02 .259E-02 .266E-02 .273E-02 .280E-02 .288E-02 .296E-02 .304E-02 .312E-02 .320E-02 .329E-02 .338E-02 .347E-02 .356E-02 .366E-02 .376E-02 .386E-02 .397E-02 .407E-02 .418E-02 .430E-02 .441E-02 .453E-02 .465E-02 .478E-02 .491E-02 .504E-02 .518E-02 .532E-02

n•

k•

0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.998 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.996 0.996 0.996 0.996 0.996 0.996

.131E-03 .144E-03 .159E-03 .175E-03 .192E-03 .210E-03 .231E-03 .255E-03 .279E-03 .306E-03 .335E-03 .367E-03 .402E-03 .440E-03 .480E-03 .525E-03 .573E-03 .625E-03 .683E-03 .746E-03 .814E-03 .887E-03 .965E-03 .105E-02 .114E-02 .124E-02 .135E-02 .146E-02 .157E-02 .170E-02 .183E-02 .198E-02 .214E-02 .231E-02 .249E-02 .269E-02 .291E-02 .314E-02 .339E-02 .361E-02 .381E-02 .401E-02 .422E-02 .444E-02 .467E-02 .491E-02

nsupp

ksupp

Orthorhombic Sulfur o~-S

913 TABLE I

(Continued)

Orthorhombic

eV .227E .221E .215E .210E .204E 199E 193E 188E 183E 179E 174E 169E 165E 161E 156E 152E 148E 144E

+ + + + + + + + + + + + + + + + + +

140E +

137E 133E 130E 126E 123E

+ + + + +

120E +

cm 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

117E + 03 113E + 03 110E + 03

108E + 03 105E + 03 102E + 03 993E + 02 .966E + 02 .941E + 02 .916E + 02 .892E + 02 .869E + 02 .846E + 02 .823E + 02 .802E + 02 .781E + 02 .760E + 02 .740E + 02 .721E + 02 .702E + 02 .683E + 02

183E 178E 174E 169E 165E

-1

+ + + + +

160E +

156E 152E 148E 144E

+ + + +

140E +

137E 133E 129E 126E 123E

+ + + + +

119E +

l16E l13E l l0E 108E 105E 102E .991E .965E .940E .915E .890E .867E .844E .822E .801E .779E .759E .739E .719E .701E .682E .664E .647E .630E .613E .597E .582E .566E .551E

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +

/~m 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 07 06 06 06 06 06 06 06 06 06 06 06 06 06 06 06 06 06 06 06 06 06 06 06

.546E-02 .561E-02 .576E-02 .592E-02 .608E-02 .624E-02 .641E-02 .658E-02 .676E-02 .694E-02 .713E-02 .733E-02 .752E-02 .773E-02 .793E-02 .815E-02 .837E-02 .860E-02 .883E-02 .907E-02 .931E-02 .957E-02 .982E-02 .101E-01 .104E-01 9106E-01 9109E-01

.112E-01 .l15E-01 .l18E-01 .122E-01 .125E-01 .128E-01 .132E-01 .135E-01 .139E-01 .143E-01 9147E-01 .151E-01 .155E-01 .159E-01 .163E-01 9168E-01

.172E-01 .177E-01 9182E-01

n_L

0.996 0.996 0.996 0.996 0.996 0.997 0.997 0.997 0.998 0.998 0.999 1.001 1.004 1.006 1.001 1.000

0.999 0.998 0.997 0.996 0.996 0.995 0.994 0.994 0.993 0.992 0.991 0.991 0.990 0.989 0.988 0.988 0.987 0.986 0.985 0.984 0.983 0.982 0.981 0.980 0.979 0.977 0.976 0.975 0.973 0.972

Sulfur

k•

nsupp

ksupp

9517E-02 .544E-02 .573E-02 .606E-02 .646E-02 .688E-02 .732E-02 .778E-02 .825E-02 .873E-02 .924E-02 .978E-02 .103E-01 .928E-03 .988E-03 .105E-02 112E-02 120E-02 128E-02 138E-02 147E-02 157E-02 168E-02 179E-02 191E-02

.204E-02 .218E-02 .231E-02 .246E-02 .264E-02 .283E-02 .303E-02 .321E-02 .340E-02 .361E-02 .382E-02 .403E-02 .426E-02 .449E-02 .474E-02 .497E-02 .519E-02 .546E-02 .575E-02 .597E-02 .618E-02

(continued)

914

Kirk A. Fuller, Harry D. Downing, and Marvin R. Querry TABLE I

(Continued)

O r t h o r h o m b i c Sulfur

eV .665E + 02 .648E + 02 .631E + 02 .614E + 02 .598E + 02 .582E + 02 .567E + 02 .557E + 02 .537E + 02 .523E + 02 .l18E + 02 l17E + 02 l16E + 02 115E + 02 l14E + 0 2 l13E + 02 112E + 02 l l l E + 02 110E + 02 109E + 02 108E + 02 107E + 02 106E + 02 105E + 02 104E + 02 103E + 02 102E + 02 101E + 02 100E + 02 990E + 01 .980E + 01 .970E + 01 .960E + 01 .950E + 01 .940E + 01 .930E + 01 .920E + 01 .910E + 01 .900E + 01 .890E + 01 .880E + 01 .870E + 01 .860E + 01 .850E + 01 .840E + 01 .830E + 01

cm

.536E .523E .509E .495E .482E .469E .457E .449E .433E .422E .952E .944E .936E .928E .920E .911E .903E .895E .887E .879E .871E .863E .855E .847E .839E .831E .823E .815E .806E .798E .790E .782E .774E .766E .758E .750E .742E .734E .726E .718E .710E .702E .694E .686E .678E .669E

-1

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

06 06 06 06 06 06 06 06 06 06 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05

~m

n•

k•

9186E-01 .191E-01 .197E-01 .202E-01 .207E-01 .213E-01 .219E-01 .225E-01 .231E-01 .237E-01 .105E + 00 .106E + 00 .107E + 00 .108E + 00 .109E + 00 . l l 0 E + 00 . l l l E + 00 .112E + 00 .l13E + 00 .114E + 00 .l15E + 00 .l16E + 00 .l17E + 00 .II8E + 00 .l19E + 00 .120E + 00 .122E + 00 .123E + 00 .124E + 00 .125E + 00 .126E + 00 .128E + 00 .129E + 00 .131E + 00 132E + 00 133E + 00 135E + 00 136E + 00 138E + 00 139E + 00 141E + 00 142E + 00 144E + 00 .146E + 00 .148E + 00 .149E + 00

0.970 0.968 0.967 0.965 0.963 0.960 0.958 0.955 0.953 0.950 1.340 [28] 1.344 1.352 1.358 1.359 1.358 1.352 1.348 1.338 1.332 1.326 1.318 1.308 1.296 1.281 1.272 1.272 1.301 1.358 1.432 19 1.599 1.705 1.802 1.877 1.947 2.009 2.040 2.047 2.045 2.049 2.061 2.079 2.089 2.078 2.044

.637E-02 .653E-02 .665E-02 .673E-02 .688E-02 .699E-02 .693E-02 .693E-02 .693E-02 .686E-02 .795E + 00 [28] .793E + 00 .789E + 00 .783E + 00 .778E + 00 .775E + 00 .771E + 00 .770E + 00 .775E + 00 .782E + 00 .798E + 00 .815E + 00 .834E + 00 .853E + 00 .880E + 00 .930E + 00 101E + 01 109E + 01 l16E + 01 121E + 01 124E + 01 125E + 01 125E + 01 .122E + 01 .l17E + 01 .l12E + 01 .106E + 01 .986E + 00 .914E + 00 .851E + 00 .807E + 00 .778E + 00 .742E + 00 .692E + 00 .635E + 00 .577E + 00

nsupp

ksupp

Orthorhombic Sulfur o~-S

915 TABLE I

(Continued)

Orthorhombic

eV

cm-

n_L

/~m

.820E .810E .800E .790E .780E .770E .760E .750E .740E .730E .720E .710E .700E .690E .680E .670E .660E

+ + + + + + + + + + + + + + + + +

01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

.661E .653E .645E .637E .629E .621E .613E .605E .597E .589E .581E .573E .565E .556E .548E .540E .532E

+ + + + + + + + + + + + + + + + +

05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05

.650E .640E .630E .620E .610E .600E .590E .580E .570E .560E .550E .540E .530E .520E .510E .500E .490E .480E .470E .460E .450E .440E .430E .420E

+ + + + + + + + + + + + + + + + + + + + + + + +

01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

.524E .516E .508E .500E .492E .484E .476E .468E .460E .452E .444E .436E .428E .419E .411E .403E .395E .387E .379E .371E .363E .355E .347E .339E

+ + + + + + + + + + + + + + + + + + + + + + + +

05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05

.410E + 01

.331E + 05

.400E + 01

.323E + 05

.390E + 01 .380E + 01 .370E + 01

.315E + 05 .306E + 05 .298E + 05

Sulfur

nsupp

k•

2.005 1.962 1.907 1.848 1.790 1.737 1.692 1.659 1.648 1.672 1.723 1.782

9182E + .185E + .188E +

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

.191E .194E .197E .200E .203E .207E .210E .214E .218E .221E .225E .230E .234E .238E .243E .248E .253E .258E .264E .269E .275E .282E .288E .295E

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

1.923 1.951 1.984 2.018 2.043 2.063 2.085 2.097 2.103 2.102 2.101 2.096 2.087 2.082 2.077 2.076 2.074 2.062 2.042 2.015 1.985 1.990 2.065 2.150

.302E + 00 .310E + 00

2.217 2.275

.552E + 00 .539E + 00

.318E + 00

2.322

.498E + 00

.326E + 00 .335E + 00

2.353 2.380

.435E + 00 .370E + 00

151E 153E 155E 157E 159E 161E 163E 165E 168E 170E 172E 175E 177E

+ + + + + + + + + + + + +

91 8 0 E +

+ + + + + + + + + + + + + + + + + + + + + + + +

1.833 1.868 1.883 1.891 1.903

.526E .484E .450E .423E .405E .399E .423E .473E .534E .593E .653E .702E .727E .725E

+ + + + + + + + + + + + + +

00 00 00 00 00 00 00 00 00 00 00 00 00 00

.709E .684E .670E .672E .678E .680E .667E .655E .643E .623E .592E .566E .541E .515E .487E .461E .438E .421E .410E .403E .393E .378E .363E .356E .370E .435E .516E

+ + + + + + + + + + + + + + + + + + + + + + + + + + +

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

ksupp

(continued)

916

Kirk A. Fuller, Harry D. Downing, and Marvin R. Querry

(Continued)

TABLE I

Orthorhombic Sulfur ,,

eV .360E .350E .344E .340E .330E .320E .315E .292E .277E .263E .248E .243E .238E .234E .230E .225E .221E .218E .214E .210E .207E .191E .177E .165E 155E 146E 138E 130E 124E l18E l13E 108E 103E 992E .954E .918E .886E .855E .827E .800E .775E .751E .729E .709E .689E .670E

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ,,,

cm 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 00 00 00 00 00 00 00 00 00 00 00 00 00

.290E .282E .277E .274E .266E .258E .254E .236E .223E .212E .200E 9197E 9192E .189E 9185E 9182E .179E 175E 172E 169E 167E 154E 143E 133E 125E l18E lllE 105E 100E .952E .909E .870E .833E .800E .769E .741E .714E .690E .667E .645E .625E .606E .588E .571E .556E .540E

-1 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

/~m 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 05 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04

.344E .354E .360E .365E .376E .387E .394E .425E .448E .471E .500E .510E .520E .530E .540E .550E .560E .570E .580E .590E .600E .650E .700E .750E .800E .850E .900E .950E .100E .105E .ll0E .l15E .120E .125E .130E .135E .140E .145E .150E .155E 160E 165E 170E 175E 180E 185E

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

n j_ 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

2.395 2.376 2.360 2.339 2.293 2.247 2.230 2.160 2.130 2.090 2.030 [24] 2.020 2.020 2.010 2.010 2.000 2.000 1.990 1.990 1.980 1.990 1.980 1.970 1.970 1.960 1.955 1.950 1.950 1.950 1.947 1.945 1.942 1.940 1.940 1.940 1.940 1.940 1.940 1.940 1.940 1.940 1.940 1.940 1.940 1.940 1.940

k• .313E + 00 .249E + 00 .287E-01

.313E-02 [26] .338E-03 .356E-04 .375E-05 .174E-05 [25] 9122E-05 .931E-06 .844E-06 .752E-06 .711E-06 .668E-06 .624E-06 .600E-06 .587E-06 .350E-06 [33] .330E-06 .330E-06 .320E-06 .312E-06 .305E-06 .298E-06 .290E-06 .294E-06 .298E-06 .301E-06 .305E-06 .309E-06 .312E-06 .316E-06 .320E-06 .321E-06 .322E-06 .323E-06 .323E-06 .324E-06 .325E-06 .326E-06 .327E-06 .328E-06

nsupp

ksupp

Orthorhombic Sulfur o~-S

917 TABLE I

(Continued)

O r t h o r h o m b i c Sulfur eV .653E .636E .620E .558E .496E .434E .372E .310E .248E .186E .185E .184E .182E .181E .180E .178E .177E 176E 175E 174E 173E 172E 172E 171E 170E 170E 169E 169E 168E 167E 167E 166E 166E 165E 164E 164E 163E 162E 162E 161E 161E 160E 159E 159E 158E 157E

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

cm

-1

.526E .513E .500E .450E .400E .350E .300E .250E .200E 150E 149E 148E 147E 146E 145E 144E 143E 142E 141E 140E 140E 139E 138E 138E 138E 137E 136E 136E 136E 135E 134E .134E .134E .133E .132E .132E .132E .131E .130E .130E .130E .129E .128E .128E 9128E 9127E

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

fLm 04 O4 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04

.190E .195E .200E .222E .250E .286E .333E .400E .500E .667E .671E .676E .680E .685E .690E .694E .699E .704E .709E .714E .717E .719E .722E .725E .727E .730E .733E .735E .738E .741E .743E .746E .749E .752E .755E .758E .761E .763E .766E .769E .772E .775E .778E .781E .784E .787E

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

n• 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

1.940 1.940 1.926 [10] 1.926 1.926 1.926 1.926 1.926 1.926 1.926 1.925 1.925 1.925 1.925 1.925 1.925 1.925 1.925 1.924 1.924 1.924 1.924 1.924 1.924 1.924 1.924 1.924 1.924 1.924 1.924 1.923 1.923 1.923 1.923 1.922 1.921 1.921 1.920 1.921 1.921 1.921 1.922 1.923 1.923 1.924 1.926

k•

nsupp

ksupp

.328E-06 .329E-06

.288E-04 [ 17] .303E-04 .315E-04 .329E-04 .356E-04 .374E-04 .396E-04 .429E-04 .465E-04 .554E-04 .660E-04 .790E-04 .926E-04 .108E-03 .127E-03 .148E-03 .166E-03 .192E-03 .211E-03 .217E-03 .218E-03 .216E-03 .213E-03 .206E-03 .184E-03 .158E-03 9132E-03

2.003 [ 1O] 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.003 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002

(continued)

918

Kirk A. Fuller, Harry D. Downing, and Marvin R. Querry TABLE I

(Continued)

Orthorhombic Sulfur

eV

cm

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

.126E .126E .126E .125E .124E .124E .124E .123E .122E .122E .122E .121E .120E .120E .120E .119E .l18E .l18E .l18E .l17E .l16E .l16E .l16E .l15E .l14E

140E + 00

l13E l12E l12E l12E lllE ll0E ll0E ll0E .109E .108E .108E .108E .107E .106E .106E .106E .105E .104E .104E

157E 156E 156E 155E 154E 154E 153E 153E 152E 151E 151E 150E 149E 149E 148E 147E 147E 146E 146E 145E 144E 144E 143E 143E 142E 141E 141E 140E 139E 138E 138E 137E 136E 136E .135E .134E .134E .133E .133E .132E .131E .131E .130E .130E .129E

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

--1

+ + + + + + + + + + + + + + + + + + + + + + + + +

114E + 114E +

+ + + + + + + + + + + + + + + + + + +

/xm 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04

.791E .794E .797E .800E .803E .806E .810E .813E .816E .820E .823E .826E .830E .833E .837E .840E .844E .848E .851E .855E .858E .862E .866E .870E .873E .877E .881E .885E .889E .893E .897E .901E .905E .909E .913E .917E .922E .926E .930E .935E .939E .943E .948E .952E .957E .961E

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01

n•

k•

1.926 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927

9101E-03 .771E-04 .624E-04 .457E-04 .329E-04 .258E-04 9187E-04 9154E-04 9132E-04 9110E-04 .979E-05 .815E-05 .734E-05 .610E-05 .584E-05 .506E-05 .508E-05 .429E-05 .433E-05 .435E-05 .357E-05 .277E-05 .343E-05 .279E-05 .317E-05 .302E-05 .367E-05 .319E-05 .431E-05 .449E-05 .456E-05 .482E-05 .464E-05 .547E-05 .690E-05 .812E-05 .872E-05 9102E-04 91 0 9 E - 0 4

.133E-04 9152E-04 9175E-04 .207E-04 .214E-04 .233E-04 .234E-04

nsupp 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002 2.002

ksupp

Orthorhombic Sulfur o~-S

919 TABLE I

(Continued)

Orthorhombic

eV 9128E .128E 9127E 9126E .126E .125E .125E .124E .123E .123E .122E .122E .121E .120E .120E .l19E .l18E l18E l17E

+ + + + + + + + + + + + + + + + + + +

cm- ~ 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

116E + 00

l16E l15E l15E l14E l13E l13E

+ + + + + +

00 00 00 00 00 00

112E + 00 l12E + 00

l l l E + 00 ll0E

+ 00

110E + 00

109E 108E 108E 107E

+ + + +

00 00 00 00 1 0 7 E + 00

106E + 00 105E + 00 1 0 5 E + 00

104E + 00 104E + 00 103E + 00 1 0 2 E + 00 .102E + 00 .101E + 00 .100E + 00

.104E + 04 .103E + 04 .102E + 04 9I 0 2 E + 04 .102E + 04 .101E + 04 .100E + 04 .100E + 04 .995E + 03 .990E + 03 .985E + 03 .980E + 03 .975E + 03 .970E + 03 .965E + 03 .960E + 03 .955E + 03 .950E + 03 .945E + 03 .940E + 03 .935E + 03 .930E + 03 .925E + 03 .920E + 03 .915E + 03 .910E + 03 .905E + 03 .900E + 03 .895E + 03 .890E + 03 .885E + 03 .880E + 03 .875E + 03 .870E + 03 .865E + 03 .860E + 03 .855E + 03 .850E + 03 .845E + 03 .840E + 03 .835E + 03 .830E + 03 .825E + 03 .820E + 03 .815E + 03 .810E + 03

/.zm .966E .971E .976E .980E .985E .990E .995E .100E .101E .101E .101E

+ + + + + + + + + + +

91 0 2 E +

.103E + .103E + .104E + 91 0 4 E +

n• 01 01 01 01 01 01 01 02 02 02 02 02 02 02 02 02

91 0 5 E + 0 2 91 0 5 E + 0 2 91 0 6 E + 0 2 91 0 6 E + 0 2 91 0 7 E + 0 2 91 0 8 E + 02

.108E .109E .109E .ll0E .lllE .lllE .l12E .l12E l13E

+ + + + + + + + +

02 02 02 02 02 02 02 02 02 1 1 4 E + 02

114E + 02 115E + 02 l16E + 02 116E + 02 1 1 7 E + 02

l18E + 02 l18E + 02 l19E + 02 1 2 0 E + 02 121E + 02 121E + 02 122E + 02 123E + 02 124E + 02

1.927 1.927 1,.927 1.927 1.927 1.927 1.927 1.927 1.927 1.927 1.926 1.926 1.926 1.926 1.926 1.926 1.926 1.926 1.926 1.926 1.926 1.925 1.926 1.926 1.926 1.926 1.926 1.926 1.926 1.926 1.926 1.926 1.926 1.926 1.925 1.924 1.924 1.924 1.924 1.925 1.926 1.926 1.926 1.926 1.926 1.926

Sulfur

k• .223E-04 .187E-04 .156E-04 .135E-04 .107E-04 .118E-04 .124E-04 .130E-04 .260E-04 .304E-04 .331E-04 .289E-04 .349E-04 .395E-04 .433E-04 .579E-04 .912E-04 .161E-03 .296E-03 .430E-03 .434E-03 .315E-03 .198E-03 .150E-03 .157E-03 .262E-03 .312E-03 .280E-03 .301E-03 .420E-03 .607E-03 .848E-03 .102E-02 .119E-02 .131E-02 .167E-02 .237E-02 .334E-02 .419E-02 .307E-02 .220E-02 .149E-02 .109E-02 .810E-03 .647E-03 .486E-03

nsupp

ksupp

2.002 2.002 2.002 2.002 2.001 2.001 2.001 2.001 2.001 2.001 2.001 2.001 2.001 2.001 2.000 2.000 2.000 1.999 1.998 1.998 2.000 2.001 2.002 2.003 2.003 2.002 2.001 2.001 2.002 2.003 2.001 1.999 1.997 1.996 1.995 1.994 1.995 1.995 1.997 1.998 2.000 2.003 2.004 2.005 2.005 2.005

(continued)

920

Kirk A. Fuller, Harry D. Downing, and Marvin R. Querry TABLE I

(Continued)

O r t h o r h o m b i c Sulfur

eV .998E-01 .992E-01 .986E-01 .979E-01 .973E-01 .967E-01 9961E-01

.955E-01 .949E-01 .942E-01 .936E-01 .930E-01 .924E-01 .918E-01 .911E-01 .905E-01 .899E-01 .893E-01 .887E-01 .880E-01 .874E-01 .868E-01 .862E-01 .856E-01 .849E-01 .843E-01 .837E-01 .831E-01 .825E-01 .818E-01 .812E-01 .806E-01 .800E-01 .794E-01 .787E-01 .781E-01 .775E-01 .769E-01 .763E-01 .756E-01 .750E-01 .744E-01 .738E-01 .732E-01 .725E-01 .719E-01

cm

.805E .800E .795E .790E .785E .780E .775E .770E .765E .760E .755E .750E .745E .740E .735E .730E .725E .720E .715E .710E .705E .700E .695E .690E .685E .680E .675E .670E .665E .660E .655E .650E .645E .640E .635E .630E .625E .620E .615E .610E .605E .600E .595E .590E .585E .580E

-1

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

/xm

03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

124E 125E 126E 127E 127E 128E 129E

+ + + + + + +

130E +

131E 132E 132E 133E 134E 135E 136E 137E 138E 139E

+ + + + + + + + + +

140E +

141E 142E .143E .144E .145E .146E .147E .148E .149E

+ + + + + + + + +

.150E +

.151E .153E .154E .155E .156E .158E .159E

+ + + + + + +

160E +

161E 163E 164E 165E 167E 168E

+ + + + + +

170E +

171E + 172E +

n t_ 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02

19 1.926 19 1.926 1.926 1.925 1.925 1.925 1.925 1.925 1.924 1.924 1.924 1.924 1.923 1.923 1.923 1.923 1.923 1.922 1.921 1.920 1.919 1.919 1.919 1.919 1.918 1.918 1.919

1.920 1.922 1.924 1.927 1.930 1.931 1.933 1.933 1.932 1.931 1.930 1.930 1.930 1.930 1.930 1.929 1.929

kL .297E-03 .200E-03 .141E-03 .103E-03 .716E-04

.442E-04 .296E-04 .257E-04 .262E-04 .269E-04 .344E-04 .422E-04 .525E-04 .695E-04 .964E-04 .134E-03 .201E-03 .425E-03 .593E-03 .612E-03 .502E-03 .447E-03 .306E-03 [ 19] .364E-03 .478E-03 .569E-03 .671E-03 .872E-03 .103E-02 .111E-02 .994E-03 .666E-03 .440E-03 .368E-03 .373E-03 .443E-03 .560E-03 .609E-03 .524E-03 .409E-03 .378E-03 .401E-03 .494E-03 .705E-03 .801E-03 .628E-03

nsupp 2.005 2.005 2.005 2.005 2.004 2.004 2.004 2.004 2.004 2.004 2.004 2.004 2.004 2.004 2.004 2.004 2.004 2.004 2.004 2.004 2.004 2.004 2.003 2.003 2.003 2.003 2.002 2.001 2.000 1.999 1.998 1.998 2.00 l 2.004 2.009 2.013 2.013 2.013 2.016 2.019 2.019 2.019 2.016 2.014 2.012 2.010

ksupp

Orthorhombic Sulfur oL-S

921 TABLE I

(Continued)

Orthorhombic

eV .713E-01 .707E-01 .701E-01 .694E-01 .688E-01 .682E-01 .676E-01 .670E-01 .663E-01 .657E-01 .651E-01 .645E-01 .639E-01 .632E-01 .626E-01 .620E-01 .614E-01 .608E-01 .601E-01 .595E-01 .589E-01 .583E-01 .576E-01 .570E-01 .564E-01 .558E-01 .552E-01 .545E-01 .539E-01 .533E-01 .527E-01 .521E-01 .514E-01 .508E-01 .502E-01 .496E-01 .490E-01 .483E-01 .477E-01 .471E-01 .465E-01 .459E-01 .452E-01 .446E-01 .440E-01 .434E-01

cm

-1

.575E .570E .565E .560E .555E .550E .545E .540E .535E .530E .525E .520E .515E .510E .505E .500E .495E .490E .485E .480E .475E .470E .465E .460E .455E .450E .445E .440E .435E .430E .425E .420E .415E .410E .405E .400E .395E .390E .385E .380E .375E .370E .365E .360E .355E .350E

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

~m 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

174E + 02 1 7 5 E + 02 177E + 02 179E + 02 1 8 0 E + 02 182E + 02 184E + 02 185E + 02 187E + 02 189E + 02 1 9 0 E + 02 192E + 02 .194E + 02 .196E + 02 9198E + 02 .200E + 02 .202E + 02 .204E + 02 .206E + 02 .208E + 02 .210E + 02 .213E + 02 .215E + 02 .217E + 02 .220E + 02 .222E + 02 .225E + 02 .227E + 02 .230E + 02 .233E + 02 .235E + 02 .238E + 02 .241E + 02 .244E + 02 .247E + 02 .250E + 02 .253E + 02 .256E + 02 .260E + 02 .263E + 02 .267E + 02 .270E + 02 .274E + 02 .278E + 02 .282E + 02 .286E + 02

n_L 1.928 1.927 1.926 1.925 1.924 1.924 1.923 1.923 1.923 1.923 1.923 1.923 1.923 1.923 1.923 1.922

Sulfur

k j_ .485E-03 .443E-03 .476E-03 .650E-03 .936E-03 91 1 3 E - 0 2

.108E-02 .986E-03 .907E-03 .880E-03 .966E-03 9107E-02 91 1 3 E - 0 2

.120E-02 9125E-02 .900E-02 [ 10]

1.921

91 0 2 E - 0 1

1.920 1.921 1.921 1.924 1.927 1.931 1.934 1.935 1.936 1.935 1.934 1.931 1.929 1.928 1.927 1.926 1.925 1.925 1.925 1.925 1.925 1.924 1.924 1.924 1.924 1.924 1.924 1.923 1.923

.l14E-01 .137E-01 .159E-01 .172E-01 91 8 6 E - 0 1

.176E-01 .166E-01 .137E-01 .108E-01 .835E-02 .590E-02 .505E-02 .420E-02 .470E-02 .520E-02 .585E-02 .650E-02 .725E-02 .800E-02 .835E-02 .870E-02 .915E-02 .960E-02 .995E-02 .103E-01 .108E-01 .l13E-01 .l17E-01 .121E-01

nsupp 2.009 2.007 2.009 2.010 2.008 2.006 2.007 2.009 2.007 2.004 2.003 2.003 2.003 2.002 2.002 2.002 2.0O 1 2.001 1.995 1.990 1.995 2.000 2.016 2.033 2.027 2.021 2.016 2.011 2.011 2.011 2.010 2.009 2.009 2.008 2.008 2.008 2.008 2.008 2.007 2.007 2.007 2.007 2.007 2.006 2.005 2.004

ksupp

.130E-02 [19] .133E-02 91 3 4 E - 0 2

.460E-02 [ 10] .920E-02 91 6 0 E - 0 1

.229E-01 .194E-01 91 6 0 E - 0 1

.800E-02 .181E-02 [19] .205E-02 .253E-02 .278E-02 .263E-02 .242E-02 .207E-02 .173E-02 .169E-02

(continued)

922

Kirk A. Fuller, Harry D. Downing, and Marvin R. Querry TABLE

I

(Continued)

Orthorhombic

eV .428E-01 .421E-01 .415E-01 9 .403E-01 .397E-01 .391E-01 .384E-01 .378E-01 .372E-01 .366E-01 .360E-01 .353E-01 .347E-01 .341E-01 .335E-01 .329E-01 .322E-01 .316E-01 .310E-01 .304E-01 .298E-01 .291E-01 .285E-01 .279E-01 .273E-01 .267E-01 .260E-01 .254E-01 .248E-01 .242E-01 .236E-01 .229E-01 .223E-01

cm

-1

.345E .340E .335E .330E .325E .320E .315E .310E .305E .300E .295E .290E .285E .280E .275E .270E .265E .260E .255E .250E .245E .240E .235E .230E .225E .220E .215E .210E .205E .200E 9195E .190E 9185E

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

91 8 0 E

+

/xm 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03 03

.290E .294E .299E .303E .308E .312E .318E .323E .328E .333E .339E .345E .351E .357E .364E .370E .377E .385E .392E .400E .408E .417E .425E .435E .444E .455E .465E .476E .488E .500E .513E .526E .540E .556E

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02

Sulfur

n•

k•

19 19 19 19 1.921 1.921 1.921 1.920 1.919 1.919 1.918 1.917 1.916 1.915 1.912 1.909 1.906 1.902 1.911 1.919 1.939 1.958 1.975 1.993 1.997 2.001 1.998 1.995 1.999 2.003 2.016 2.028 2.037 2.046

.126E-01 9132E-01 9136E-01 .139E-01 91 4 6 E - 0 1

.152E-01 .159E-01 .167E-01 .178E-01 91 8 8 E - 0 1

.201E-01 .214E-01 .232E-01 .250E-01 .286E-01 .321E-01 .422E-01 .524E-01 .657E-01 .789E-01 .844E-01 .900E-01 .810E-01 .720E-01 .587E-01 .455E-01 .388E-01 .322E-01 .318E-01 .315E-01 .233E-01 91 5 0 E - 0 1

.750E-02

nsupp

2.003 2.002 2.000 1.999 1.998 1.998 1.996 1.994 1.993 1.992 1.990 1.988 1.986 1.984 1.980 1.976 1.971 1.966 1.955 1.944 1.969 1.993 2.013 2.033 2.020 2.006 1.991 1.976 1.964 1.951 1.968 1.985 2.080 2.176

ksupp

.600E-03 [ 10] .120E-02 .162E-01 .313E-01 .458E-01 .603E-01 .451E-01 .299E-01 .158E-01 9160E-02 9145E-02 9130E-02 .281E-01 .550E-01 .843E-01 .l14E + 00 0.912E-01 .686E-01

Cubic Thallium(I) Halides WILLIAM J, TROPF Applied Physics Laboratory The Johns Hopkins University Laurel, Maryland

Most of the univalent thallium [thallium(I) or thallous] halides have the cubic cesium-chloride structure (space group Pm3m or O~). Exceptions are thallium fluoride and the low-temperature (below 170 ~ C) form of thallium iodide, which are both orthorhombic. Cubic forms of the thallium(I) halides are important optical materials because of their high refractive index, farinfrared transparency, relative insolubility in water, and moderate strength. Thallium halides are miscible in all concentrations; eutectic mixtures are typically used to improve composition homogeneity [1]. The cubic eutectic mixtures of thallium-halide crystals with the designations KRS-5 and KRS-6 ~ are especially important. The usual composition of KRS-5 is 54.3 molepercent TII and of KRS-6 is 70.2 mole percent T1C1, with the balance being thallium bromide for both. Single crystals of pure and mixed thallous halides are grown by the Stockbarger technique. Table I summarizes the physical properties of thallous-halide crystals.

THALLIUM CHLORIDE

Thallium chloride has a direct band gap of 3.4 eV [1]. Unlike most materials, the band gap increases with temperature, by about 0.02 eV from 0 to 200 K. Optical constants in the region of fundamental electronic absorption from the band gap to about 20 eV were determined by Hinson and Stevenson [2] and Frandon and LaHaye [3] from Kramers-Kr6nig analyses of reflection and energy-loss spectra, respectively. Absorption immediately above the band gap has been determined from transmission measurements by Martienssen [4] and Tutihasi [5]. These results are given in Table II. i KRS is an abbreviation for the German term Kristalle aus dem SchmeltzfluB, or crystal from the fused melt. 923 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS III

Copyright 9 1998 by Academic Press. All rights of reproduction in any form reserved ISBN 0-12-544423-0/$25.00.

I

924

William J. Tropf

Strong absorption begins with exciton formation at the band gap followed by interband transitions between the valence band of chlorine and the conduction band of thallium. A thallium-valence-electron plasma oscillation occurs around 7.2 eV and a chlorine 4p electron plasma resonance occurs near 12.4 eV. Smakula [6] reports low-level absorption at the ultraviolet and infrared edges of transparency. Refractive index in the visible transparent region has been measured by Barth [7] (method unknown, probably index matching), Schrrter [8] (minimum-deviation method), and Takahei and Kobayashi [9] (prism refraction method). Barth's data are lower that those from other sources by 0.015 to 0.020. The later two references give dispersion equations only. McCarthy [10] gives interferometrically measured refractive indices in the infrared at 45 ~ C, which have been corrected to room temperature (20 ~ C) by adding 0.006. The dispersion equation of Schrrter [8] is n2(~) -- 1 0.07858 = 0.47856 + -- 0.00881 )t 2 n2(~) + 2 /~2 _ 0.08277

(1)

for the 0.435 to 0.660/zm range, and the dispersion equation of Takahei and Kobayashi [9] is n2(A) = (2"146) 2 +

1.780 0.0778 )2 + (M0.2651 - 1 (M0.3532) 2 -

1

(2)

for the 0.390 to 0.635 ~m range. Wavelengths are in micrometers in both cases. Table II has the data of Barth and McCarthy and selected calculated values from the dispersion equations. The temperature change of the refractive index due to electronic absorption, dno~/dT, of thallium chloride is determined by Takahei and Kobayashi [9] to be - 1 5 4 • 10-6/K over the interval 40 to 300 K and as - 1 9 4 • 10-6/K by Shanker and Sundaraj [11] (using n~ = 2.146). This latter dn/dT value was used to correct the McCarthy data (see earlier discussion) to room temperature. Low-level infrared absorption of thallium chloride, up to the region of fundamental lattice vibration absorption, was measured by Hidaka et al. [12]. They model the absorption coefficient in the 500 to 1100 cm-1 region with the equation

(3)

/3(v) = / 3 o e x p ( - V/Vo)

with/30 = 6 cm-1 Vo = 100 cm-~ and v is in wave numbers Infrared reflection measurements have been made by Mitsuishi et al. [13], Lowndes [14], and Claudel et al. [15]. Thallium chloride, like the other cubic thallous halides, has only a single (degenerate) Raman-inactive, infrared~

9

Cubic Thallium(I) Halides

925

active optical mode (F1,). Lowndes uses a model of the complex refractive index (n) of the form V2LO- 112- iTLOV n 2 ( v ) - e ( v ) - e~v2 ~ _ v2 - iTTOV,

(4)

where the transverse and longitudinal frequencies of thallium chloride are VTo = 62 cm-1 and VLO = 173 cm -1, with widths of ~/TO = 7.7 cm -1 and 7Lo = 20 cm-1, and e~ = 4.60. Claudel et al. [15] used a classical oscillator model of the form n Z ( v ) - e ( v ) - e~ + ~"

4 7r~woj 2

"7 4oj

p2

(5)

~ i ~j ~

with two far-infrared modes to represent their data. Their model parameters are Pl = 2.20, P2 = 0.05, Vvol = 63 cm -1, VTO2 = 120 cm -1, ~/1 = 0.08, 3'2 --0.6, and e~ = 4.4. Both models, which generally agree within 10%, are used to calculate the optical constants given in Table II. The room-temperature, low-frequency (1-100 kHz) dielectric constant was measured as e o - - 3 2 . 7 ___ 0.2 (or n = 5.72 +_ 0.02) by Lowndes [16], eo = 32.4 + 0.1 (or n = 5.70 _+ 0.01) by Lowndes and Martin [17], and eo = 32.3 _+ 0.2 (or n = 5.68 _+ 0.02) by Samara [18] using capacitance techniques. The low-frequency dielectric constant has a large, negative temperature dependence: deo/dT = - 0 . 0 1 2 K -1 [16-18] and deo/dT= - 0 . 0 1 2 1 6 K -1 [11] (or dn/dT = -0.0011 K - l ) . Table II gives all the optical-constant data for thallium chloride and Fig. 1 shows the data graphically.

THALLIUM BROMIDE

Thallium bromide has a direct band gap of 3.0 eV [1]. Like thallium chloride, the band gap increases with temperature, by about 0.02 eV from 0 to 200 K. Optical constants in the region of fundamental electronic absorption from the band gap to about 20 eV were determined by Hinson and Stevenson [2] and Frandon and LaHaye [3] from a Kramers-Kronig analysis of reflection and energy-loss spectra, respectively. These results are given in Table III. Strong absorption begins with exciton formation at the band gap followed by interband transitions between the valence band of bromine and the conduction band of thallium. A thallium-valence-electron plasma oscillation occurs around 6.7 eV and bromine 4p electron plasma resonance occurs near 11.8 eV.

II

926

William J. Tropf

Smakula [6] reports low-level absorption at the ultraviolet and infrared edges of transparency. Refractive index in the visible transparent region has been measured by Barth [7], Schr6ter [8], and Takahei and Kobayashi [9]. Barth's data is lower that the other sources by 0.020. The latter two references give dispersion equations only. McCarthy [10] gives refractive indices in the infrared at 45 ~ C which have been corrected to room temperature (20 ~ C) by adding 0.006. The dispersion equation of SchrtJter [8] is n2(A)- 1

n2(,h) + 2

= 0.48484 +

0.10279 A2 - 0.090000

-- 0.0047896A 2

(6)

for the 0.435 to 0.660 txm range, and the dispersion equation of Takahei and Kobayashi [9] is n2(A) - (2.273) 2 +

2.582 (M0.2736) 2 -

1

+

0.0789 (M0.3986) 2 -

1

(7)

for the 0.390 to 0.635 /xm range. Wavelengths are in micrometers in both cases. Table III has the data of Barth and McCarthy and selected calculated values from the dispersion equations. The temperature change of the refractive index due to electronic absorption, d n J d T , of thallium bromide is determined by Takahei and Kobayashi [9] to be - 1 8 0 • 10-6/K over the interval 40 to 300 K and as - 2 2 9 • 10-6/K by Shanker and Sundaraj [11] (using n~ = 2.27). This latter dn/dT value was used to correct the McCarthy data (see above) to room temperature. Infrared reflection measurements have been made by Mitsuishi et al. [ 13], Lowndes [14], and Claudel et al. [15]. Lowndes uses the model of the complex refractive index (n) given by Eq. (4) with the transverse and longitudinal frequencies of thallium bromine are VTO = 47 cm -~ and VLo -- 116 cm -~, with widths are YTO- 4.0 cm -~ and ~/co = 17.5 cm -~, and E~ = 5.15. Claudel et al. [15] uses the classical oscillator model of Eq. (5) with two far-infrared modes to represent their data. Their model parameters are p ~ = 1.98, p 2 = 0 . 0 2 , VTO~--47.5 cm-~, VTO2=137.5 cm-~, ~/~ = 0.10, 3/2 = 0.5, and e~ = 5.4. Both models give similar results, with agreement typically 20%. The Lowndes model is used to calculate the optical constants given in Table III. Pai et al. [ 19] measured the room-temperature complex dielectric constant of thallium bromide between 22 and 190 cm -~ using dispersive Fourier transform spectroscopy. These data are also given in Table III. The room-temperature, low-frequency (1-100 kHz) dielectric constant of thallium bromide was measured as eo = 30.6 ___ 0.2 (or n = 5.53 _+ 0.01) by Lowndes [16] and Lowndes and Martin [17], and E o - 30.0 ___0.2 (or

Cubic Thallium(I) Halides

927

n = 5.48 _+ 0.02) by Samara [18] using capacitance techniques. The lowfrequency dielectric constant has a large, negative temperature dependence: d e o / d T = - 0 . 0 1 3 K -1 [16, 17], dEo/dT = - 0 . 0 1 1 K -1 [18], and d ~ o / d T = -0.01295 K - l [11] (or d n / d T ~ -0.0012 K-l). Table III gives all the optical constant data for thallium bromide and Fig. 2 shows the data graphically.

THALLIUM IODIDE

Little is known about the optical properties of thallium iodide. Thallium iodide is orthorhombic at room temperature and 1 atmosphere (see Table I) and does not revert to the cubic (CsC1) structure until temperature is raised to 170 ~ C or pressure to 5000 atm. The orthorhombic form (space group 17 has four molecules per unit cell. The thallium and iodide Cmcm or D2h) atoms both occupy 4(c) sites (C2v symmetry). The band gap of thallium iodide is 2.7 eV (orthorhombic structure) or about 2.1-2.2 eV for the cubic structure (extrapolated from pressure-dependent absorption measurements [20]). Optical constants in the region of fundamental electronic absorption from the band gap to about 22 eV were determined by Frandon and LaHaye [3] from a Kramers-Kronig analysis of an energy-loss spectrum of a thin film of thallium iodide measured in transmission. The authors state that the measured crystals (T1C1, T1Br, and TII) all have cubic symmetry. Most likely the thallium iodide film is orthorhombic (optical constants are consistent with the band gap) and polycrystalline. These results are given in Table IV. Barth [7] measured the refractive index (method unknown) of metastable cubic thallium iodide at room temperature by quickly cooling and carefully handling the crystals. (Note that comparison of Barth' s measurements of the refractive index of T1C1 and T1Br, made at the same time, are lower than those of other researchers; see above). His values are given in Table IV. Unpolarized infrared reflection measurements of orthorhombic TII were made by Claudel et al. [15], probably with a polycrystalline sample. They used the classical-oscillator model of Eq. (5) with two far-infrared modes to represent their data. The model parameters are 0 1 - 1.12, 0 2 - 0.05, VTO~ -- 51 cm -~, Vvo2 -- 9 0 c m -~, ~1 - - 0.02, ")/2 - - 0 . 5 , and e~ - 6.1. These results indicate that the optical modes of different crystallographic directions are very similar. This model was used to provide the far-infrared optical constants in Table IV. Samara [18] measured the room-temperature, low-frequency (10 kHz) dielectric constant of polycrystalline orthorhombic thallium iodide as eo = 21.2 + 0.2 (or n - 4.60 + 0.02) and extrapolated the dielectric constant of cubic thallium iodide as e o - 29.6 _+ 0.5 (or n - 5.44 _+ 0.05) from its

III

928

William J. Tropf

pressure variation using a capacitance technique. Von Hippel [21] gives eo = 21.8 (1 kHz-10 MHz) for (orthorhombic) TII at 25 ~ C and eo = 37.3 (10 MHz) for (cubic) TII at 193 ~ C. Note that the orthorhombic values agree well with the value of 20.9 obtained from the model of Claudel et al. [15]. Table IV gives all the optical-constant data for thallium iodide, and Fig. 3 shows the data graphically.

IV

MIXED THALLOUS HALIDES KRS-5 AND KRS-6

The mixed crystals KRS-5 and KRS-6 are less studied than T1C1 and T1Br, but more optically important. Optical properties of these materials were originally studied by Barth [7], and later they were developed in Germany during World War II (see Hettner and Leisegang [22]). Further study of these mixtures improved optical uniformity by accurately finding the eutectic compositions [23]. The actual composition of some measured materials given later may differ from the eutectic mixture and have different optical constants. Takahei and Kobayashi [9] measured low-temperature reflectance of pure and mixed crystals of T1C1-T1Br and derived dispersion equations for several mixtures at different temperatures. Their results give the band gap of KRS-6 as 3.25 eV. Absorption-edge measurements of Smakula [6] and roomtemperature reflectance measurements of Tropf [24] indicate a band gap of ~2.45-2.50 eV for KRS-5. Tropf [24] measured the absorption coefficient of KRS-5 at the visible edge of transparency between 20 and 209 ~ C and fit his data to an Urbachrule model of the form /3(v, T) = ~oexp

(-o'shc(~'g-

v)/kBT),

(8)

where [3(v,T) is the absorption coefficient as a function of wave number (v) and temperature (T). Fixed quantities are Planck's constant (h), Boltzmann's constant (kB), and the speed of light (c). Material-dependent parameters are the slope parameter, trs = 0.9418, a constant over the measured temperature range;/3o = 1.8063 cm- ~" and Vg = 17,474 cmThe only measurement of room-temperature visible absorption of KRS-6 are the absorption-edge transmission and laser-calorimetry measurements of Moil and Izawa [25]. Their measurements are included in Table VI and were fitted to the sum of two exponential absorption edges with absorption coefficient,/3(v), each having the form *

/3(v) =/3o exp(v/Vo),

(9)

where/3o = 1 • 10 -34 c m - 1 and Vo = 301.3 cm-~ for the fundamental absorption-edge component and /3o = 5.62 • 10 - 6 c m - 1 and Vo = 2258.3

Cubic Thallium(I) Halides

929

cm-1 for the weak-absorption-tail component. These two components combine to model the absorption edge from 15,000 to 25,000 cm-1, with fundamental absorption dominating above 23,000 cm -1. (Extrapolated values to 26,600 cm-~ are given in Table VI.) The KRS-5 Urbach-edge model of Tropf [24], Eq. (8), gives roomtemperature (295 K) constants /30 = 2.5 • 10 -35 cm -1 and Vo = 217.7 cm-~, when recast into the form of Eq. (9). These values were used to calculate the imaginary part of the refractive index of KRS-5 at the electronic absorption edge between 17,000 and 19,000 cm -1 (2.10 to 2.35 eV). The refractive index of KRS-5 was measured using the minimiumdeviation method by Hettner and Leisegang [22] for a composition 56 mol % TII, by Tilton et al. [26] for compositions of 58 and 58.3 mol % TII, and at three near-ambient temperatures by Rodney and Malitson [27] for a composition of 54.3 mol % thallium iodide. Barth [7] also provided data for several T1Br-TII mixtures including 50 and 60 mol % thallium iodide. Rodney and Malitson's measurements at 25 ~ C are given in Table V. Tilton et al. provide a dispersion equation and Rodney and Malitson give a five-term Sellmeier equation for their data. Tropf [24] developed this equally accurate three-term Sellmeier dispersion equation based on the Rodney and Malitson 25 ~ C measurements (0.57 to 39.4/xm): n2()t)- 1 =

3.7442390A 2 h2 _ (0.2079603) 2

+

0.9189162A 2 h 2 - (0.3765643) 2 +

(10)

12.5444602A 2 ~ 2 __

(165.6525518)2"

Wavelength (h) is in micrometers. He also gives three-term Sellmeier models for their data at 19 and 31 ~ C, as well as a less-accurate, temperaturedependent Sellmeier model. The refractive index of KRS-6 was measured using the minimum-deviation method by Hettner and Leisegang [22] for a highly noneutectic composition, 60 mol % T1C1 (which will have a higher index) and by Takahei and Kobayashi [9] for a composition with 68 mol % T1C1. The dispersion equation of Takahei and Kobayashi is n2(h)

=

(2.186) 2 +

1.877 0.0296 + (M0.2828) 2 - 1 (M0.3748) 2 - 1

(ll)

for the 0.390 to 0.635/xm range; wavelengths are in micrometers. The data of Hettner and Leisegang and calculations from the visible-wavelength dispersion equation of Takahei and Kobayashi are given in Table IV. Barth [7] used a Lorentz-Lorenz molar-refractivity formulation to estimate the visible refractive index of mixed T1Br-TII crystals of various com-

930

William J. Tropf

positions. Craig [28] later refined Barth's calculation and obtained a good fit to his data. Rodney and Malitson [27] give d n / d T for KRS-5 as - 2 5 4 • 10-6/K at 0.577 /xm, decreasing to - 2 3 3 X 10-6/K at 11.025 ~m, and to - 1 5 4 • 10-6/K at 39.38 ~m. Tropf [24] measured an average d n / d T for KRS-5 as - 2 3 2 • 10-6/K over the 700-1000 cm -~ (10-14.3 ~m) spectral range and the temperature interval of 22 to 209 ~ C, consistent with the Rodney and Malitson results. The temperature change of the refractive index due to electronic absorption, dn~/dT, of KRS-6 (68 mol % T1C1) was determined by Takahei and Kobayashi [9] to be - 1 6 0 x 10-6/K over the interval 40 to 300 K. Afanas'ev and Nosov [29] give d n / d T = - 2 5 4 • 10-6/K for KRS-5 and d n / d T = - 2 3 4 • 10-6/K for KRS-6 at 0.633/zm. Absorption of KRS-5 and KRS-6 in the transparent region has been extensively studied, primarily for infrared fibers. Available data includes the laser-calorimetry measurements of Harrington et al. [30] on crystalline KRS-5 at HF and DF laser wavelengths; transmission measurements on long polycrystalline KRS-5 fibers at 10.6 ~m by Harrington and Standlee [31]; laser calorimetry at 1, 5, and 10/xm by Belousov et al. [32] on crystalline KRS-5 and KRS-6; carbon-monoxide laser calorimetry measurements of Nosov et al. [33] on crystalline KRS-5 and KRS-6; and differentialspectrophotometry measurements of Hidaka et aL [12] on crystalline KRS-5. This last set of measurements covers a broad spectral range (500-1150 cm-~) and is included in Table V. Typical measured infrared absorption coefficients in the transparent range of these materials are on the order of 10-3 to 10-S/cm. Absorption at the infrared edge of transparency was measured by Smakula [6] and by Dianov et al. [34] for both KRS-5 and KRS-6, and for KRS-5 alone by Hidaka et al. [12] and by Tropf [24]. The infrared absorption coefficient,/3(v), can also be modeled as a exponential edge using Eq. (3). For KRS-5 at room temperature, Hidaka et aL give / 3 0 - 5400 and vo - 2 5 cm-~; Dianov et al. give ]30 = 20700 and vo = 25 cm-~; and Tropf gives /30 = 7700 and vo = 27.1 c m - ~. For KRS-6, Dianov et al. give ]30 = 10810 and Vo = 43 cm-~. Values of the imaginary refractive index at the infrared absorption edge in Tables IId and IIe come from Tropf [24] and Dianov et al. [34] for KRS-5 and KRS-6, respectively. (Note: Dianov et al. also gives model parameters for the visible absorption edges of KRS-5 and KRS-6. These values appear to be in error.) There is little data on optical constants of KRS-5 and KRS-6 in the region of fundamental optical absorption. The reflectance measurements of Mitsuishi et al. [13] were fit to a the dispersion model of Eq. (4), and the following parameters provided a good fit: for KRS-5, VTO = 43 cm -~, VLO = 103 c m - ~, "YTO= 7 c m - ~, ~/CO-- 25 c m - ~, and e~ = 5.66; for KRS-6, VTO 58 c m - ~, PLO = 149 cm-~, "YTO -- 8 c m - ~, ~LO = 35 cm-~, and E~ = 4.85. These values were used to calculate far-infrared optical constants given in Tables V and VI. =

Cubic Thallium(I) Halides

931

The room-temperature, low-frequency (measurement method unknown, accuracy of 2% or better) dielectric constants of KRS-5 (58 mol % TII) and KRS-6 (60 mol % T1C12) are given by von Hippel [21] as E o - 32.5 (or n - 5.70) for frequencies from 10 kHz to 10 MHz and e o - 31.8 (or n - 5.64) at 100 kHz, respectively. Von Hippel also gives deo/dT = -0.0091 K-~ for KRS-5 at 1 MHz and d E o / d T - -0.0075 K-~ for KRS-6 at 10 MHz. Tables V and VI give all the optical constant data for KRS-5 and KRS-6, respectively. Figures 4 and 5 shows the same data graphically. REFERENCES

1. C. W. Jurgensen and H. G. Drickamer, High-pressure studies of the absorption edges of three thallous halides. Phys. Rev. B 30, 7202-7205 (1984). 2. D. C. Hinson and J. R. Stevenson, Optical constants of T1C1 and T1Br with a comparison of the Kramers-Krrnig and "two-angle" methods of data analysis. Phys. Rev. 159, 711716 (1967). 3. J. Frandon and B. LaHaye, Electron loss characteristics of T1C1, T1Br, TII and calculated optical constants from 3 to 25 eV. J. Phys (Paris) 33, 229-235 (1972). 4. W. Martienssen, The optical absorption edge in ionic crystals. J. Phys. Chem. Solids 8, 294-296 (1959). 5. S. Tutihasi, Ultraviolet absorption in thallous halides. J. Phys. Chem. Solids 12, 344-348 (1960). 6. A. Smakula, Synthetic crystals and polarizing materials. Opt. Acta 9, 205-222 (1962). 7. T. E W. Barth, Optical properties of mixed crystals. Am. J. Sci. 219, 135-146 (1930); also see T. Barth, Some new immersion melts of high refraction. Am. Mineral. 14, 358361 (1929). 8. H. Schrrter, On the refractive indices of some heavy-metal halides in the visible and calculation of interpolation formulas for dispersion. Z. Phys. 67, 24-36 (1931). 9. K. Takahei and K. Kobayashi, Reflectivity spectra of T1C1-T1Br mixed crystals, J. Phys. Soc. Jpn. 43, 891-898 (1977). 10. D. E. McCarthy, Refractive index measurements of T1Br and T1C1 in the IR. Appl. Opt. 4, 878-879 (1965). 11. J. Shanker and R. Sundaraj, Dielectric and anharmonic behavior of silver halides, thallous halides, alkaline earth fluorides, and lead fluoride crystals. Phys. Status Solidi B 115, 67-74 (1968). 12. T. Hidaka, T. Morikawa, and J. Shimada, Spectroscopic small loss measurements on infrared transparent materials. Appl. Opt. 19, 3763-3766 (1980). 13. A. Mitsuishi, Y. Yamada, and H. Yoshinaga, Reflection measurements on reststrahlen crystals in the far-infrared region. J. Opt. Soc. Am. 52, 14-16 (1962). 14. R. P. Lowndes, Anharmonicity in the silver and thallium halides: Far-infrared dielectric response. Phys. Rev. B 6, 1490-1498 (1972). 15. J. Claudel, A. Handi, P. Strimer, and P. Vergnat, Absorption spectra and reflection of the thallium halides in the far infrared at low temperatures. J. Phys. Chem. Solids 29, 15391544 (1968). 16. R. P. Lowndes, Anharmonicity in the silver and thallium halides: Low-frequency dielectric response. Phys. Rev. B 6, 4667-4674 (1972). 17. R. P. Lowndes and D. H. Martin, Dielectric constants of ionic crystals and their variations with temperature and pressure. Proc. R. Soc. (London) A 316, 351-375 (1970). 2 Composition of 60% T1Br, 40% T1C1 stated; this is undoubtedly an error. Molar percentage assumed corresponding to the composition of this era.

932

William J. Tropf

18. G. A. Samara, Temperature and pressure dependence of the dielectric constants of the thallous halides. Phys. Rev. 165, 959-969 (1968). 19. K. E Pai, T. J. Parker, N. E. Tornberg, and R. P. Lowndes, Determination of the complex refractive indices of solids in the far-infrared by dispersive fourier transform spectroscopy--II. Pseudo-displacement ferroelectrics. Infrared Phys. 18, 327-336 (1978). 20. J. C. Zahner and H. G. Drickamer, The effect of pressure on the absorption edge in heavymetal halides. J. Phys. Chem. Solids 11, 92-96 (1959). 21. A. R. von Hippel, ed., "Dielectric Materials and Applications." Wiley, New York, 1954. 22. G. Hettner and G. Leisegang, Dispersion of mixed crystals T1Br-TII (KRS 5) and T1C1T1Br (KRS 6) in the infrared. Optik 3, 305-314 (1948). 23. A. Smakula, J. Kalnajs, and V. Sils, Inhomogeneity of thallium halide mixed crystals and its elimination. J. Opt. Soc. Am. 43, 698-701 (1953). 24. A. Z. Tropf, M. E. Thomas, and W. J. Tropf, Optical properties of KRS-5. To be published in Proc. SPIE 3060 (1997). 25. H. Mori and T. Izawa, A new loss mechanism in ultralow loss optical fiber materials. J. AppL Phys. 51, 2270-2271 (1980). 26. L. W. Tilton, K. Plyler, and R. E. Stephens, Refractive indices of thallium bromide-iodide crystals for visible and infrared radiant energy. J. Res. Natl. Bur. Stand. (U.S.) 43, 81-86 (1949). 27. W. S. Rodney, and I. H. Malitson, Refraction and dispersion of thallium bromide iodide. J. Opt. Soc. Am. 46, 956-961 (1956). 28. H. Craig, A simple model for refractive indices in mixed crystals and glasses. Geochim. Comochim. Acta 56, 3001-3006 (1992). 29. I. I. Afanas'ev and V. B. Nosov, Temperature increments of the refractive indices of certain single crystals. Sov. J. Opt. Technol. 46, 281-283 (1979). 30. J. A. Harrington, D. A. Gregory, and W. E Otto, Jr., Infrared absorption in chemical laser window materials. Appl. Opt. 15, 1953-1959 (1976). 31. J.A. Harrington and A. G. Standlee, Attenuation at 10.6 mm in loaded and unloaded polycrystalline KRS-5 fibers. Appl. Opt. 22, 3073-3078 (1983). 32. A. P. Belousov, E. M. Dianov, I. S. Lisitskii, T. M. Nesterova, V. G. Plotnichenko, and V. K. Sysoev, Single crystals of thallium halides with optical losses below 10 dB/km. Soy. J. Quantum Electron. 12, 496-497 (1982). 33. V. B. Nosov, G. T. Petrovskii, M. V. Serzhantova, and A. V. Shatilov, Calorimetric measurements of the volume and surface absorption of IR materials in the 5-6 mm spectral region. Sov. J. Opt. Technol. 56, 238-240 (1989). 34. E. M. Dianov, N. S. Lisitskii, V. G. Plomichenko, V. B. Sulimov, and V. K. Sysoev, Evaluation of minimum possible optical losses in thallium halides. Opt. Spectrosc. 56, 281-283 (1984); also see E. M. Dianov, N. S. Lisitsky, V. G. Plotnichenko, V. K. Sysoev, V. B. Sulimov, and L. N. Butvina, Evaluation of optical loss minima in thallium halide crystals. Fiber Integr. Opt. 5, 125-133 (1985). (Note error in Table 1 of the latter reference.)

Cubic Thallium(I) Halides . . . . . .

933

I

. . . . . . . .

I

a) 101

oo o~ 10 ~

,

,

,

, , , I

,

.

.

.

.

.

.

.

10 -]

. . . . . .

10 ~

I

I

10 o

. . . . . . . .

,

,

i

,

i

l l l l

|

i

,

,

,

, , , I

101 W A V E L E N G T H (gm)

,

,

. . . . . .

10 2

I

10 3

!

b)

10 -2

10

/

.4

/

10 -6

10-8

. . . .

|ll

10 -1

I

|

t

,

|

, | 1 1

10 ~

,

,

. . . . . .

I

,

,

,

10 ] W A V E L E N G T H (gm)

,

J iiii

i

10 2

,

,

|

,

, , , I

10 3

Fig. 1. (a) Log-log plot of the index of refraction n versus wavelength in micrometers for thallium(I) chloride. (b) Log-log plot of the extinction coefficient k versus wavelength in micrometers for thallium(I) chloride. Lines are data sets of many points; scattered data are indicated by circles.

i

934

William J. Tropf . . . . . .

.

I

.

.

.

.

.

.

.

I

.

.

.

.

.

.

.

.

I

a) 101 ----.-_.__

0000

10 ~

i

i

i

~,,I

i

I

i

i

~

. . . . . .

I

i

, , . I

10 "1

10 ~

.

.

.

.

.

.

.

.

i

i

i

i

,

,.I

.

.

.

.

,

1 01 W A V E L E N G T H (lain)

,

,,1

10 2

.......

io3

'

I

b)

10 ~

10 -2 0

/

10 -4

10 "6

10 "8

,

~

, , , , I

10 1

*

L

,

,

,

, , , I

10 ~

,

,

,

,

, , , , I

,

,

J

l0]

~

, J A , I

10 2

,

t

. . . . . .

I

,

10 3

W A V E L E N G T H (pm) Fig. 2. (a) Log-log plot of the index of refraction n versus wavelength in micrometers for thallium(I) bromide. (b) Log-log plot of the extinction coefficient k versus wavelength in micrometers for thallium(I) bromide. Lines are data sets of many points; scattered data are indicated by circles.

Cubic Thallium(I) Halides . . . . . .

101

935

I

_ a)

10 ~

10 -I

i

i

i

i

,I

I I

10 0

10 -1

. . . . . .

101

i

.

.

.

.

.

.

.

.

I

.

.

I

.

.

.

.

.

.

.

101 W A V E L E N G T H (lam)

.

.

.

.

.

I

'

i

!

*

i

10 2

.

.

I

I

III

10 3

.

.

.

.

.

.

.

.

!

b)

/

10 ~

10 -1

10 -2

10 -3 10 "1

10 ~

101 W A V E L E N G T H (~tm)

10 2

10 3

Fig. 3. (a) Log-log plot of the index of refraction n versus wavelength in micrometers for thallium(I) iodide. (b) Log-log plot of the extinction coefficient k versus wavelength in micrometers for thallium(I) iodide. Lines are data sets of many points; scattered data are indicated by circles.

936

William J. Tropf . . . . . .

I

a) 10 ]

10 ~

j

. . . . .

J

.i

i

,

i

10 .2

. . . .

I

i

i

i

i

j

10 ~

|,,I

.

.

.

.

.

.

101

.

iI

i

i.,

. . . . .

I

10 2

10 3

10 2

10 3

W A V E L E N G T H (pm)

97

9

m , , . ]

.

.

.

.

.

'~'1

.

.

.

.

.

.

.

.

I

b) 10 ~

10 2

/

10 -4

10 -6

10 -8

10 q

10 ~

101 W A V E L E N G T H (~tm)

Fig. 4. (a) Log-log plot of the index of refraction n versus wavelength in micrometers for KRS-5. (b) Log-log plot of the extinction coefficient k versus wavelength in micrometers for KRS-5.

Cubic Thallium(I) Halides . . . . . .

937

I

a) 101

10 ~

......

. . . . . .

10' "1

10~ ~

. . . . . . . .

1" 01 . . . . . . . W A V E L E N G T H (~tm)

02

. . . . . . .

10 3

i . . . . . . . . . .

!

b) 10 ~

10 -2

/

10 -4

1 0 "6

lO-S

......

,

10 "1

I

I

I

|

|

, I l l

10 ~

,

I

. . . . . .

I

.

.

.

.

.

.

.

101 W A V E L E N G T H (~tm)

.

I

10 2

.

.

.

.

.

.

.

.

I

10 3

Fig. 5. (a) Log-log plot of the index of refraction n versus wavelength in micrometers for KRS-6. (b) Log-log plot of the extinction coefficient k versus wavelength in micrometers for KRS-6.

938

William J. Tropf TABLE 1 Properties of Pure and Mixed Thallium(I) Halides

Material

Crystal Structure and Space Group

T1F

Orthorhombic: Fmmm

T1C1 T1C1-T1Br T1Br T1Br-TII TII TII

Cubic: Pm3m Cubic: Pm3m Cubic: Pm3m Cubic: Pm3m Cubic: Pm3m Orthorhombic: Cmcm

Lattice Constants (/~) a = c a-a -a-a -a = a = c

5.495; b = 6.080; -- 5.180 3.84247 3.88946 3.98588 4.11826 4.2099 a 4.582; b = 12.92; = 5.251

Density (g/cm 3)

Melting Point (~

8.573

326

7.020 7.142 7.455 7.365 7.373 7.079

430.2 422.9 460.5 413.9 441.8 -.- 170 b

a extrapolated to room temperature and pressure. b Phase change from orthorhombic to cubic.

TABLE II Values of n and k for Thallium Chloride from Various References eV 23.800 23.600 23.400 23.200 23.000 22.800 22.600 22.400 22.200 22.000 21.800 21.600 21.400 21.200 21.000 20.800 20.600 20.400 20.200 20.000 19.800 19.600 19.400 19.200 19.000 18.800 18.600

cm- 1 191958.4 190345.3 188732.2 187119.1 185506.0 183892.9 182279.9 180666.8 179053.7 177440.6 175827.5 174214.4 172601.3 170988.2 169375.1 167762.0 166148.9 164535.8 162922.7 161309.6 159696.5 158083.4 156470.3 154857.2 153244.1 151631.0 150017.9

/xm 0.05209 0.05254 0.05299 0.05344 0.05391 0.05438 0.05486 0.05535 0.05585 0.05636 0.05687 0.05740 0.05794 0.05848 0.05904 0.05961 0.06019 0.06078 0.06138 0.06199 0.06262 0.06326 0.06391 0.06458 0.06526 0.06595 0.06666

n

k

0.821 [3] 0.821 0.822 0.820 0.817 0.814 0.810 0.804 0.798 0.791 0.787 0.805 0.825 0.831 0.830 0.820 0.805 0.779 0.752 0.736 0.739 0.750 0.762 0.767 0.747 0.729 0.716

0.162 [3] 0.162 0.164 0.165 0.167 0.169 0.173 0.180 0.189 0.203 0.221 0.234 0.234 0.222 0.206 0.194 0.183 0.180 0.193 0.219 0.244 0.262 0.274 0.264 0.257 0.266 0.286

n

Cubic Thallium(I) Halides

939 TABLE II (Continued) Thallium Chloride

eV 18.400 18.200 18.000 17.900 17.800 17.700 17.600 17.500 17.400 17.300 17.200 17.100 17.000 16.900 16.800 16.700 16.600 16.500 16.400 16.300 16.200 16.100 16.000 15.900 15.800 15.700 15.600 15.500 15.400 15.300 15.200 15.100 15.000 14.900 14.800 14.700 14.600 14.500 14.400 14.300 14.200 14.100 14.000 13.900 13.800 13.700 13.600

cm -1 148404.8 146791.7 145178.6 144372.1 143565.5 142759.0 141952.5 141145.9 140339.4 139532.8 138726.3 137919.7 137113.2 136306.6 135500.1 134693.5 133887.0 133080.4 132273.9 131467.3 130660.8 129854.2 129047.7 128241.1 127434.6 126628.0 125821.5 125014.9 124208.4 123401.8 122595.3 121788.8 120982.2 120175.7 119369.1 118562.6 117756.0 116949.5 116142.9 115336.4 114529.8 113723.3 112916.7 I12110.2 111303.6 110497.1 109690.5

/zm

0.06738 0.06812 0.06888 0.06927 0.06965 0.07005 0.07045 0.07085 0.07126 0.07167 0.07208 0.07251 0.07293 0.07336 0.07380 0.07424 0.07469 0.07514 0.07560 0.07606 0.07653 0.07701 0.07749 0.07798 0.07847 0.07897 0.07948 0.07999 0.08051 0.08104 0.08157 0.08211 0.08266 0.08321 0.08377 0.08434 0.08492 0.08551 0.08610 0.08670 0.08731 0.08793 0.08856 0.08920 0.08984 0.09050 0.09117

n

k

0.702 0.693 0.685 0.683 0.681 0.681 0.688 0.703 0.714 0.722 0.718 0.700 0.676 0.644 0.617 0.592 0.559 0.490 0.471 0.486 0.627 0.803 0.908 1.079 1.152 1.199 1.172 1.109 1.021 0.963 0.894 0.853 0.829 0.824 0.844 0.980 1.105 1.181 1.180 1.151 1.134 1.108 1.058 1.012 0.982 0.951 0.925

0.309 0.332 0.356 0.370 0.383 0.399 0.411 0.422 0.426 0.420 0.411 0.408 0.417 0.437 0.465 0.485 0.517 0.596 0.665 0.742 0.846 0.910 0.900 0.795 0.755 0.695 0.555 0.456 0.431 0.422 0.443 0.465 0.507 0.560 0.630 0.743 0.706 0.660 0.557 0.431 0.370 0.332 0.320 0.318 0.315 0.312 0.308

0.596 [2] 0.579 0.561 0.550 0.547 0.538 0.516 0.484 0.449 0.429 0.434 0.458 0.505 0.633 0.778 0.898 0.953 0.974 0.967 0.946 0.924 0.904 0.884 0.864 0.846 0.830 0.816 0.803 0.804 0.818 0.906 0.984 1.045 1.086 1.104 1.111 1.111 1.108 1.102 1.095 1.082 1.066 1.048

0.17312] 0.171 0.176 0.189 0.197 0.203 0.208 0.212 0.229 0.272 0.329 0.422 0.518 0.643 0.710 0.622 0.544 0.501 0.474 0.457 0.445 0.445 O.449 0.457 0.468 0.480 0.505 0.545 0.637 0.749 0.657 0.580 0.528 0.489 0.454 0.421 0.394 0.371 0.353 0.335 0.321 0.312 0.303

(continued)

940

William J. Tropf TABLE II

(Continued)

Thallium Chloride

eV 13.500 13.400 13.300 13.200 13.100 13.000 12.900 12.800 12.700 12.600 12.500 12.400 12.300 12.200 12.100 12.000 11.900 11.800 11.700 11.600 11.500 11.400 11.300 11.200 11.100 11.000 10.900 10.800 10.700 10. 600 10.500 10.400 10.300 10.200 10.100 10.000 9.900 9.800 9.700 9.600 9.500 9.400 9.300 9.200 9.100 9.OOO 8.900

cm

--I

108884.0 108077.4 107270.9 106464.3 105657.8 104851.2 104044.7 103238.1 102431.6 101625.1 100818.5 100012.0 99205.4 98398.9 97592.3 96785.8 95979.2 95172.7 94366.1 93559.6 92753.0 91946.5 91139.9 90333.4 89526.8 88720.3 87913.7 87107.2 86300.6 85494.1 84687.5 83881.0 83074.4 82267.9 81461.3 80654.8 79848.3 79041.7 78235.2 77428.6 76622.1 75815.5 75009.0 74202.4 73395.9 72589.3 71782.8

ftm 0.09184 0.09253 0.09322 0.09393 0.09465 0.09537 0.09611 0.09686 0.09763 0.09840 0.09919 0.09999 0.1008 O. 1016 0.1025 0.1033 0.1042 0.1051 0.1060 0.1069 0.1078 0.1088 0.1097 0.1107 O.1117 0.1127 0.1137 0.1148 O.1159 0.1170 0.1181 0.1192 0.1204 0.1216 O.1228 O.1240 0.1252 0.1265 0.1278 0.1292 0.1305 0.1319 0.1333 0.1348 0.1362 0.1378 0.1393

n

k

0.903 0.881 0.857 0.834 0.810 0.782 0.753 0.710 0.671 0.628 0.593 0.573 0.576 0.592 0.621 0.666 0.712 0.771 0.837 0.912 0.980 1.025 1.049 1.075 1.085 1.087 1.086 1.033 0.987 0.949 0.912 0.878 0.852 0.829 0.817 0.803 0.793 0.795 0.798 0.800 0.802 0.807 0.813 0.825 0.882 0.985 0.987

0.307 0.310 0.314 0.319 0.326 0.320 0.311 0.318 0.346 0.380 0.426 0.489 0.576 0.657 0.721 0.767 0.807 0.843 0.852 0.839 0.815 0.798 0.769 0.738 0.699 0.654 0.604 0.584 0.566 0.552 0.551 0.559 0.572 0.589 0.610 0.641 0.675 0.701 0.728 0.757 0.785 0.813 0.838 0.867 0.863 0.910 0.862

1.029 1.006 0.981 0.950 0.915 0.874 0.820 0.762 0.719 0.690 0.671 0.665 0.667 0.681 0.697 0.734 0.771 0.807 0.844 0.878 0.908 0.937 0.964 0.991 1.024 1.056 1.086 1.114 1.115 1.107 1.082 1.041 0.993 0.952 0.917 0.885 0.858 0.838 0.828 0.859 0.885 0.899 0.905 0.911 0.920 0.938 0.948

0.296 0.290 0.287 0.285 0.290 0.299 0.310 0.323 0.348 0.379 0.429 0.479 0.525 0.564 0.600 0.634 0.667 0.697 0.719 0.732 0.737 0.732 0.716 0.692 0.652 0.615 0.578 0.544 0.511 0.483 0.462 0.458 0.462 0.469 0.481 0.497 0.516 0.547 0.599 0.652 0.675 0.682 0.691 0.714 0.780 0.785 0.774

Cubic Thallium(I) Halides

941 TABLE II

(Continued)

Thallium Chloride eV 8.800 8.700 8.600 8.500 8.400 8.300 8.200 8.100 8.000 7.900 7.800 7.700 7.600 7.500 7.400 7.300 7.200 7.100 7.000 6.900 6.800 6.700 6.600 6.500 6.400 6.300 6.200 6.100 6.000 5.900 5.800 5.700 5.600 5.500 5.400 5.300 5.200 5.100 5.000 4.900 4.800 4.700 4.600 4.500 4.400 4.300 4.200

cm -~

/xm

70976.2 70169.7 69363.1 68556.6 67750.0 66943.5 66136.9 65330.4 64523.8 63717.3 62910.7 62104.2 61297.6 60491.1 59684.6 58878.0 58071.5 57264.9 56458.4 55651.8 54845.3 54038.7 53232.2 52425.6 51619.1 50812.5 50006.0 49199.4 48392.9 47586.3 46779.8 45973.2 45166.7 44360.1 43553.6 42747.0 41940.5 41133.9 40327.4 39520.9 38714.3 37907.8 37101.2 36294.7 35488.1 34681.6 33875.0

O. 1409 0.1425 0.1442 0.1459 0.1476 0.1494 O. 1512 0.1531 0.1550 O. 1569 O. 1590 0.1610 0.1631 O. 1653 0.1675 0.1698 O. 1722 0.1746 0.1771 0.1797 0.1823 0.1851 0.1879 0.1907 O. 1937 0.1968 0.2000 0.2033 0.2066 0.2101 0.2138 0.2175 0.2214 0.2254 0.2296 0.2339 0.2384 0.2431 0.2480 0.2530 0.2583 0.2638 0.2695 0.2755 0.2818 0.2883 0.2952

0.942 0.910 0.876 0.850 0.832 0.819 0.810 0.808 0.799 0.792 0.790 0.785 0.785 0.799 0.813 0.831 0.850 0.866 0.877 0.897 0.917 0.944 1.142 1.482 1.623 1.587 1.475 1.377 1.296 1.270 1.320 1.369 1.460 1.557 1.648 1.727 1.801

0.780 0.759 0.783 0.815 0.843 0.871 0.899 0.926 0.964 1.004 1.049 1.085 1.124 1.172 1.217 1.268 1.317 1.365 1.422 1.492 1.572 1.654 1.779 1.941 1.802 1.618 1.403 1.418 1.497 1.594 1.684 1.734 1.810 1.875 1.927 1.952 1.966

0.634 [5] 0.630 0.621 0.602

0.953 0.941 0.919 0.896 0.868 0.842 0.818 O.798 0.793 0.800 0.808 0.820 0.837 0.857 0.878 0.908 0.955 1.014 1.085 1.148 1.205 1.254 1.293 1.324 1.339 1.395 1.410 1.431 1.454 1.580 1.735 1.798 1.834 1.868 1.906 1.957 2.020 2.100 2.207 2.325 2.408 2.428 2.438 2.449 2.458 2.453 2.450

0.742 0.719 0.708 0.711 0.724 0.755 0.792 0.833 0.876 0.923 0.977 1.034 1.081 1.124 1.184 1.248 1.303 1.347 1.382 1.402 1.400 1.395 1.392 1.396 1.407 1.411 1.415 1.441 1.499 1.560 1.594 1.548 1.499 1.472 1.449 1.439 1.441 1.447 1.444 1.402 1.262 1.178 1.112 1.053 0.997 0.941 0.884

(continued)

942

William J. Tropf TABLE II (Continued) Thallium Chloride

eV

cm

-1

/.zm

4.100 4.000 3.900 3.800 3.700 3.600 3.500

33068.5 32261.9 31455.4 30648.8 29842.3 29035.7 28229.2

0.3024 0.3100 0.3179 0.3263 0.3351 0.3444 0.3542

0.572 0.534 0.497 0.462 0.434 0.436 0.531

3.480 3.460 3.440 3.420 3.400 3.380 3.360 3.340 3.320 3.300 3.280 3.263

28067.9 27906.6 27745.3 27583.9 27422.6 27261.3 27100.0 26938.7 26777.4 26616.1 26454.8 26315.8

0.3563 0.3583 0.3604 0.3625 0.3647 0.3668 0.3690 0.3712 0.3734 0.3757 0.3780 0.3800

0.533 0.424 0.212 0.0890 0.0412 0.0187 0.0083 0.0039 0.0019 0.0009 0.0004

eV

cm

--1

25809.5 25641.0 25003.0 24390.2 24196.4 22935.8 18315.0 17301.0 16977.9 15384.6 13333.3

0.3875 0.3900 0.4000 0.4100 0.4133 0.436 0.546 0.578 0.589 0.650 O.750

eV

cm -1

/xm

1150.0 1075.0 1050.0 1025.0 1000.0 975.0 950.0 925.0 900.0 875.0 850.0

2.656

2.550

0.835 0.789 0.747 0.711 0.683 0.683 0.703 0.495 [4] 0.433 0.324 0.210 0.111 0.0590 0.0248 0.00961 0.00401 0.00153 5.73E-4 2.91E-4 3.74E-6 [6]

/.zm

3.200 3.179 3.100 3.024 3.O00 2.844 2.271 2.145 2.105 1.907 1.653

0.1426 0.1333 0.1302 0.1271 0.1240 0.1209 0.1178 0.1147 0.1116 0.1085 0.1054

2.451 2.454 2.461 2.468 2.467 2.474 2.633

8.696 9.302 9.524 9.756 10.00 10.26 10.53 10.81 11.11 11.43 11.76

2.49

2.4112 [81 2.2856 2.2681 2.2629 2.2402 2.2155

2.199 [101

2.5474 [9] 2.5050 2.4714 2.4618 2.4079 2.2826 2.2640 2.2586 2.2345 2.2094

3.51E-8 [12] 7.41E-8 1.13E-7 1.58E-7 2.41E-7 3.45E-7 2.62E-7 1.32E-7 2.25E-7 4.56E-7 4.68E-7

2.46 2.42 2.400 [7] 2.270 2.253 2.247 2.223 2.198

3.03E-5 [41 8.25E-7 [61 1.91E-7

Cubic Thallium(I) Halides

943 TABLE II

(Continued)

Thallium Chloride eV 0.1023 0.0994 0.0992 0.0961 0.0930 0.0899 0.0868 0.0837 0.0806 0.0775 0.0744 0.0713 0.0682 0.0676 0.06509 0.06199 0.05889 0.05579 0.05269 0.04959 0.04649 0.04339 0.04030 0.03720 0.03410 0.03100 0.02790 0.02480 0.02356 0.02232 0.02108 0.02046 0.01984 0.01922 0.01860 0.01798 0.01736 0.01674 0.01612 0.01550 0.01488 0.01426 0.01364 0.01302 0.01240 0.01178 0.01116

cm-J 825.0 802.0 800.0 775.0 750.0 725.0 700.0 675.0 650.0 625.0 600.0 575.0 550.0 545.0 525.0 500.0 475.0 450.0 425.0 400.0 375.0 350.0 325.0 300.0 275.0 250.0 225.0 200.0 190.0 180.0 170.0 165.0 160.0 155.0 150.0 145.0 140.0 135.0 130.0 125.0 120.0 115.0 110.0 105.0 100.0 95.0 90.0

/~m 12.12 12.47 12.50 12.90 13.33 13.79 14.29 14.81 15.38 16.00 16.67 17.39 18.18 18.35 19.05 20.00 21.05 22.22 23.53 25.00 26.67 28.57 30.77 33.33 36.36 40.00 44.44 50.00 52.63 55.56 58.82 60.61 62.50 64.52 66.67 68.97 71.43 74.07 76.92 80.00 83.33 86.96 90.91 95.24 100.0 105.3 111.1

4.23E-7 2.197 4.05E-7 6.34E-7 7.57E-7 1.22E-6 1.86E-6 1.84E-6 1.81E-6 1.91E-6 2.31E-6 2.86E-6 3.93E-6 2.188 6.23E-6 2.029 [14] 2.015 2.000 1.981 1.959 1.931 1.896 1.852 1.794 1.715 1.604 1.437 1.157 0.984 0.751 0.498 0.414 0.362 0.330 0.310 0.299 0.293 0.291 0.295 0.302 0.313 0.331 0.354 0.387 0.432 0.495 0.588

0.128 [141 0.196 0.255 0.372 0.635 0.817 1.003 1.187 1.369 1.553 1.741 1.936 2.140 2.358 2.593 2.850 3.136 3.458 3.829 4.267 4.798

1.979 [15] 1.966 1.950 1.930 1.907 1.878 1.843 1.797 1.736 1.654 1.537 1.358 1.049 0.848 0.563 0.334 0.295 0.278 0.273 0.276 0.283 0.293 0.305 0.317 0.327 0.332 0.334 0.331 0.329 0.333 0.350 0.390

5.71E-5 [6] 1.11E-4 2.16E-4 4.24E-4

0.113 [15] 0.171 0.321 0.687 0.882 1.066 1.240 1.410 1.579 1.748 1.921 2.099 2.286 2.489 2.714 2.973 3.275 3.636 4.073 4.615

(continued)

944

William J. Tropf TABLE II

(Continued)

Thallium Chloride

eV 0.01054 0.00992 0.00930 0.00868 0.00806 0.00744 0.00682 0.00620 0.00558 0.00496 0.00434 0.00372 0.00310 0.00248 0.00186 0.00124 0.00062 0.0

cm- ~ 85.0 80.0 75.0 70.0 65.0 60.0 55.0 50.0 45.0 40.0 35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0

/zm 117.6 125.0 133.3 142.9 153.8 166.7 181.8 200.0 222.2 250.0 285.7 333.3 400.0 500.0 666.7 1000.0 2000.0

n

k

0.731 0.974 1.443 2.573 6.400 13.222 11.316 9.433 8.303 7.575 7.077 6.722 6.464 6.277 6.143 6.053 6.001 5.985 5.72 [16]

5.466 6.349 7.599 9.521 12.135 7.565 2.509 1.150 0.649 0.410 0.277 0.194 0.138 0.098 0.067 0.042 0.020 0.000

n

k

0.469 0.625 0.968 1.955 7.200 14.443 10.692 8.841 7.807 7.151 6.703 6.383 6.150 5.980 5.859 5.778 5.731 5.716 5.68 [18]

5.312 6.259 7.666 10.087 14.464 5.057 1.481 0.716 0.428 0.284 0.200 0.146 0.107 0.078 0.055 0.035 0.017 0.000

TABLE III Values of n and k for Thallium Bromide from Various References

eV 23.800 23.600 23.400 23.200 23.000 22.800 22.600 22.400 22.200 22.000 21.800 21.600 21.400 21.200 21.000 20.900 20.800 20.700 20.600 20.500 20.400 20.300

cm -~ 191958.4 190345.3 188732.2 187119.1 185506.0 183892.9 182279.9 180666.8 179053.7 177440.6 175827.5 174214.4 172601.3 170988.2 169375.1 168568.5 167762.0 166955.4 166148.9 165342.3 164535.8 163729.2

/xm 0.05209 0.05254 0.05299 0.05344 0.05391 0.05438 0.05486 0.05535 0.05585 0.05636 0.05687 0.05740 0.05794 0.05848 0.05904 0.05932 0.05961 0.05990 0.06019 0.06048 0.06078 0.06108

n

k

n

0.804 [3] 0.807 0.811 0.821 0.830 0.836 0.837 0.828 0.821 0.813 0.803 0.791 0.782 0.787 0.800 0.805 0.809 0.812 0.813 0.813 0.810 0.807

0.188 [3] 0.197 0.205 0.212 0.214 0.209 0.198 0.191 0.188 0.188 0.191 0.201 0.217 0.238 0.248 0.247 0.245 0.243 0.241 0.238 0.235 0.232

1.13512] 1.131 1.127 1.123 1.120 1.115 1.111 1.106

0.369 [2] 0.368 0.369 0.369 0.371 0.373 0.375 0.377 gb

Cubic Thallium(I) Halides

945 TABLE III (Continued) Thallium Bromide

eV 20.200 20.100 20.000 19.900 19.800 19.700 19.600 19.500 19.400 19.300 19.200 19.100 19.000 18.900 18.800 18.700 18.600 18.500 18.400 18.300 18.200 18.100 18.000 17.900 17.800 17.700 17.600 17.500 17.400 17.300 17.200 17.100 17.000 16.900 16.800 16.700 16.600 16.500 16.400 16.300 16.200 16.100 16.000 15.900 15.800 15.700 15.600

cm -~ 162922.7 162116.2 161309.6 160503.1 159696.5 158890.0 158083.4 157276.9 156470.3 155663.8 154857.2 154050.7 153244.1 152437.6 151631.0 150824.5 150017.9 149211.4 148404.8 147598.3 146791.7 145985.2 145178.6 144372.1 143565.5 142759.0 141952.5 141145.9 140339.4 139532.8 138726.3 137919.7 137113.2 136306.6 135500.1 134693.5 133887.0 133080.4 132273.9 131467.3 130660.8 129854.2 129047.7 128241.1 127434.6 126628.0 125821.5

/a,m

0.06138 0.06168 0.06199 0.06230 0.06262 0.06294 0.06326 0.06358 0.06391 0.06424 0.06458 0.06491 0.06526 0.06560 0.06595 0.06630 0.06666 0.06702 0.06738 0.06775 0.06812 0.06850 0.06888 0.06927 0.06965 0.07005 0.07045 0.07085 0.07126 0.07167 0.07208 0.07251 0.07293 0.07336 0.07380 0.07424 0.07469 0.07514 0.07560 0.07606 0.07653 0.07701 0.07749 0.07798 0.07847 0.07897 0.07948

n

k

0.803 0.798 0.792 0.784 0.775 0.765 0.752 0.736 0.718 0.706 0.702 0.725 0.758 0.773 0.782 0.781 0.775 0.768 0.761 0.751 0.738 0.723 0.706 0.680 0.678 0.686 0.696 0.701 0.704 0.704 0.701 0.694 0.688 0.680 0.671 0.661 0.647 0.650 0.635 0.651 0.693 0.795 0.925 0.997 1.016 1.024 1.018

0.230 0.228 0.227 0.225 0.225 0.225 0.231 0.242 0.256 0.274 0.300 0.323 0.333 0.329 0.325 0.316 0.306 0.297 0.294 0.293 0.295 0.301 0.317 0.343 0.364 0.382 0.397 0.398 0.400 0.403 0.410 0.420 0.433 0.446 0.462 0.482 0.509 0.524 0.586 0.629 0.671 0.723 0.748 0.713 0.653 0.593 0.537

1.102 1.097 1.093 1.088 1.082 1.077 1.072 1.067 1.063 1.058 1.052 1.046 1.040 1.032 1.025 1.016 1.008 0.999 0.991 0.983 0.976 0.973 0.970 0.970 0.973 0.982 0.983 0.972 0.939 0.903 0.866 0.843 0.841 0.850 0.916 1.054 1.211 1.392 1.543 1.622 1.591 1.510 1.431 1.378 1.342 1.313 1.303

0.378 0.380 0.382 0.385 0.388 0.390 0.394 0.397 0.400 0.404 0.410 0.415 0.421 0.426 O.433 0.440 0.446 0.453 0.463 0.473 0.482 0.496 0.511 0.525 0.538 0.551 0.545 0.543 0.549 0.566 0.597 0.674 0.772 0.860 0.958 1.033 0.996 0.914 0.787 0.694 0.554 0.478 0.462 0.469 0.481 0.494 0.510

(continued)

946

William J. Tropf TABLE III

(Continued)

Thallium Bromide

eV 15.500 15.400 15.300 15.200 15.100 15.000 14.900 14.800 14.700 14.600 14.500 14.400 14.300 14.200 14.100 14.000 13.900 13.800 13.700 13.600 13.500 13.400 13.300 13.200 13.100 13.000 12.900 12.800 12.700 12.600 12.500 12.400 12.300 12.200 12.100 12.000 11.900 11.800 11.700 11.600 11.500 11.400 11.300 11.200 11.100 11.000 10.900

cm

--1

125014.9 124208.4 123401.8 122595.3 121788.8 120982.2 120175.7 119369.1 118562.6 117756.0 116949.5 116142.9 115336.4 114529.8 113723.3 112916.7 112110.2 111303.6 110497.1 109690.5 108884.0 108077.4 107270.9 106464.3 105657.8 104851.2 104044.7 103238.1 102431.6 101625.1 100818.5 100012.0 99205.4 98398.9 97592.3 96785.8 95979.2 95172.7 94366.1 93559.6 92753.0 91946.5 91139.9 90333.4 89526.8 88720.3 87913.7

/xm 0.07999 0.08051 0.08104 0.08157 0.08211 0.08266 0.08321 0.08377 0.08434 0.08492 0.08551 0.08610 0.08670 0.08731 0.08793 0.08856 0.08920 0.08984 0.09050 0.09117 0.09184 0.09253 0.09322 0.09393 0.09465 0.09537 0.09611 0.09686 0.09763 0.09840 0.09919 0.09999 0.1008 0.1016 0.1025 0.1033 0.1042 0.1051 0.1060 0.1069 0.1078 0.1088 0.1097 0.1107 0.1117 0.1127 0.1137

n

k

O.992 0.968 0.938 0.878 0.813 0.784 0.795 0.815 0.863 0.935 0.992 1.064 1.148 1.152 1.129 1.075 1.017 0.960 0.912 0.876 0.861 0.864 0.872 0.887 0.920 0.966 0.984 0.981 0.976 0.940 0.902 0.861 0.838 0.815 0.797 0.780 0.761 0.744 0.729 0.719 0.719 0.724 0.732 0.762 0.811 0.862 0.912

0.497 0.465 0.446 0.458 0.486 0.533 0.579 0.628 0.667 0.699 0.683 0.645 0.576 0.524 0.460 0.412 0.395 0.384 0.398 0.425 0.451 0.484 0.519 0.549 0.564 0.566 0.527 0.496 0.463 0.443 0.433 0.432 0.443 0.456 0.481 0.511 0.544 0.580 0.616 0.655 0.697 0.742 0.799 0.844 0.897 0.928 0.938

1.293 1.291 1.290 1.296 1.308 1.329 1.368 1.420 1.477 1.549 1.566 1.529 1.488 1.464 1.444 1.428 1.420 1.412 1.407 1.403 1.395 1.387 1.378 1.368 1.356 1.337 1.316 1.293 1.265 1.233 1.198 1.161 1.120 1.072 1.026 0.992 0.985 0.987 0.997 1.003 1.009 1.015 1.024 1.035 1.051 1.082 1.146

0.531 0.555 0.581 0.611 0.651 0.705 0.756 0.790 0.755 0.649 0.491 0.418 0.395 0.388 0.390 0.391 0.390 0.389 0.388 0.387 0.383 0.377 0.367 0.355 0.343 0.334 0.324 0.318 0.315 0.313 0.310 0.323 0.360 0.401 0.450 0.494 0.525 0.550 0.573 0.595 0.618 0.642 0.669 0.697 0.726 0.757 0.800

Cubic Thallium(I) Halides

947 T A B L E III

(Continued)

Thallium Bromide

eV 10.800 10.700 10.600 10.500 10.400 10.300 10.200 10.100 10.000 9.900 9.800 9.700 9.600 9.500 9.400 9.300 9.200 9.100 9.000 8.900 8.800 8.700 8.600 8.500 8.400 8.300 8.200 8.100 8.000 7.900 7.800 7.700 7.600 7.500 7.400 7.300 7.200 7.100 7.000 6.900 6.800 6.700 6.600 6.500 6.400 6.300 6.200

cm-

/.~m

87107.2 86300.6 85494.1 84687.5 83881.0 83074.4 82267.9 81461.3 80654.8 79848.3 79041.7 78235.2 77428.6 76622.1 75815.5 75009.0 74202.4 73395.9 72589.3 71782.8 70976.2 70169.7 69363.1 68556.6 67750.0 66943.5 66136.9 65330.4 64523.8 63717.3 62910.7 62104.2 61297.6 60491.1 59684.6 58878.0 58071.5 57264.9 56458.4 55651.8 54845.3 54038.7 53232.2 52425.6 51619.1 50812.5 50006.0

0.1148 O. 1159 O. 1170 O. 1181 0.1192 0.1204 0.1216 0.1228 0.1240 0.1252 0.1265 0.1278 0.1292 0.1305 0.1319 0.1333 0.1348 0.1362 O. 1378 O. 1393 0.1409 0.1425 0.1442 O. 1459 0.1476 0.1494 O. 1512 0.1531 O. 1550 O. 1569 O. 1590 0.1610 0.1631 0.1653 O. 1675 0.1698 0.1722 0.1746 0.1771 0.1797 O. 1823 O. 1851 0.1879 0.1907 O. 1937 O. 1968 0.2000

n

k

0.960 1.015 1.044 1.070 1.089 1.101 1.115 1.124 1.127 1.125 1.121 1.112 1.096 1.078 1.061 1.037 1.008 0.985 0.983 0.981 0.978 0.976 0.978 0.982 0.986 0.990 0.995 0.999 1.004 1.006 1.008 1.012 1.014 1.017 1.022 1.028 1.027 1.032 1.049 1.042 1.056 1.279 1.443 1.476 1.396 1.313 1.188

0.935 0.909 0.892 0.870 0.848 0.827 0.805 0.784 0.766 0.747 0.728 0.713 0.702 0.693 0.688 0.691 0.709 0.726 0.747 0.768 0.789 0.808 0.823 0.840 O.857 0.875 0.891 0.904 0.917 0.931 0.947 0.964 0.983 1.005 1.026 1.045 1.074 1.108 1.152 1.223 1.262 1.404 1.446 1.294 1.078 1.001 1.033

1.217 1.285 1.342 1.395 1.455 1.499 1.538 1.553 1.559 1.547 1.522 1.489 1.459 1.432 1.407 1.381 1.360 1.339 1.319 1.297 1.273 1.249 1.231 1.214 1 198 1 188 1 177 1 161 1 141 1 124 1 116 1 108 1 111 1.127 1.145 1.162 1.186 1.219 1.257 1.321 1.429 1.607 1.736 1.844 1.905 1.946 1.972

0.837 0.861 0.881 0.892 0.892 0.867 0.827 0.784 0.737 0.687 0.642 0.606 0.579 0.563 0.550 0.540 0.537 0.536 0.541 0.546 0.553 0.573 0.593 0.616 0.653 0.692 0.730 0.768 0.801 0.837 0.861 0.876 0.913 0.966 1.023 1.083 1.141 1.203 1.268 1.337 1.419 1.592 1.643 1.611 1.531 1.452 1.387

(continued)

948

William J. Tropf TABLE III

(Continued)

Thallium Bromide eV 6.100 6.000 5.900 5.800 5.700 5.600 5.500 5.400 5.30O 5.200 5.100 5.000 4.900 4.800 4.700 4.600 4.500 4.400 4.300 4.200 4.100 4.O00 3.900 3.800 3.700 3.600 3.500 3.400 3.300 3.200 3.100 3.000 2.900 2.818 2.800 2.755 2.700 2.600 eV 2.844 2.271 2.145 2.105 1.907 1.653

cm

--1

49199.4 48392.9 47586.3 46779.8 45973.2 45166.7 44360.1 43553.6 42747.0 41940.5 41133.9 40327.4 39520.9 38714.3 37907.8 37101.2 36294.7 35488.1 34681.6 33875.0 33068.5 32261.9 31455.4 30648.8 29842.3 29035.7 28229.2 27422.6 26616.1 25809.5 25003.0 24196.4 23389.9 22727.3 22583.3 22222.2 21776.8 20970.2 cm

--1

22935.8 18315.0 17301.0 16977.9 15384.6 13333.3

/xm 0.2033 0.2066 0.2101 0.2138 0.2175 0.2214 0.2254 0.2296 0.2339 0.2384 0.2431 0.2480 0.2530 0.2583 0.2638 0.2695 0.2755 0.2818 0.2883 0.2952 0.3024 0.3100 0.3179 0.3263 0.3351 0.3444 0.3542 0.3647 0.3757 0.3875 0.4000 0.4133 0.4275 0.4400 0.4428 0.4500 0.4592 0.4769

n

k

n

1.129 1.133 1.159 1.190 1.240 1.288 1.329 1.374 1.421 1.464 1.506 1.548 1.590 1.645 1.698 1.747 1.805 1.874 1.957

1.136 1.246 1.332 1.399 1.443 1.479 1.502 1.530 1.554 1.576 1.597 1.620 1.643 1.680 1.718 1.755 1.801 1.850 1.900

1.989 2.003 2.019 2.039 2.061 2.139 2.155 2.152 2.281 2.478 2.601 2.651 2.689 2.726 2.776 2.860 2.904 2.936 2.952 2.955 2.960 2.970 2.913 2.858 2.843 2.825 2.795 2.760 2.767 2.790 2.857 2.819 2.709 6.8E-6 [6] 2.637 8.2E-7 2.580 2.539

2.7111 [8] 2.4744 2.4448 2.4364 2.4001 2.3634

2.6917 [9] 2.4743 2.4462 2.4380 2.4021 2.3652

2.652 [7] 2.452 2.424 2.418 2.384 2.350

/.~m 0.436 0.546 0.578 0.589 0.650 0.750

k 1.370 1.354 1.339 1.369 1.401 1.383 1.343 1.364 1.495 1.550 1.428 1.313 1.238 1.191 1.141 1.056 0.991 0.918 0.836 0.742 O.649 0.576 0.482 0.441 0.417 0.387 0.339 0.291 0.284 0.285 0.289 0.180

Cubic Thallium(I) Halides

949 TABLE III

(Continued)

Thallium B r o m i d e eV O. 1242 0.0889 0.0627 0.0508 0.0372 0.0341 0.0310 0.0279 0.0248 0.0236 0.0229 0.0223 0.0217 0.0211 0.0205 0.0198 0.0192 0.0186 0.0183 0.0181 0.0179 0.0176 0.0174 0.0171 0.0169 0.0166 0.0164 0.0161 0.0159 0.0156 0.0154 0.0151 0.0149 0.0146 0.0144 0.0141 0.0139 0.0136 0.0134 0.0131 0.0129 0.0126 0.0124 0.0122 0.0119 0.0117 0.0114

cm - 1 1002.0 717.0 506.0 410.0 300.0 275.0 250.0 225.0 200.0 190.0 185.0 180.0 175.0 170.0 165.0 160.0 155.0 150.0 148.0 146.0 144.0 142.0 140.0 138.0 136.0 134.0 132.0 130.0 128.0 126.0 124.0 122.0 120.0 118.0 116.0 114.0 112.0 110.0 108.0 106.0 104.0 102.0 100.0 98.0 96.0 94.0 92.0

/.Lm 9.98 13.95 19.76 24.39 33.33 36.36 40.00 44.44 50.00 52.63 54.05 55.56 57.14 58.82 60.61 62.50 64.52 66.67 67.57 68.49 69.44 70.42 71.43 72.46 73.53 74.63 75.76 76.92 78.13 79.37 80.65 81.97 83.33 84.75 86.21 87.72 89.29 90.91 92.59 94.34 96.15 98.04 lO0.O0 102.04 104.17 106.38 108.70

n

k

2.343 [10] 2.326 2.326 2.326

1.82 [19] 1.78 1.74 1.70 1.64 1.58 1.53 1.46 1.38 1.35 1.36 1.35 1.33 1.30 1.28 1.25 1.23 1.23 1.21 1.20 1.14 1.05 0.99 0.91 0.82 0.75 0.69 0.65 0.62 0.59 0.59 0.60 0.60 0.58 0.58 0.59 0.60 0.62

0.02 [19] 0.02 0.03 0.03 0.04 0.04 0.04 0.04 0.04 0.03 0.04 0.04 0.04 0.05 0.05 0.04 0.03 0.02 0.02 0.02 0.10 O. 18 0.22 0.28 0.35 0.44 0.55 0.68 0.82 0.93 1.03 1.13 1.24 1.36 1.46 1.55 1.67 1.80

1.907 [15] 1.861 1.835 1.805 1.772 1.736 1.695 1.648 1.595 1.533 1.506 1.477 1.446 1.413 1.378 1.340 1.300 1.256 1.209 1.158 1.102 1.043 0.978 0.910 0.838 0.765 0.695 0.631 0.577 0.531 0.495 0.466 0.443 0.425 0.411 0.400 0.392 0.386 0.383

0.0001 [6] 0.0002 0.00039 0.00077 0.00155

0.519 [15] 0.587 0.667 0.760 0.862 0.971 1.083 1.196 1.311 1.428 1.545 1.665 1.787 1.912 2.041

(continued)

950

William J. Tropf TABLE III

(Continued)

Thallium Bromide

eV 0.0112 0.0109 0.0107 0.0104 0.0102 0.0099 0.0097 0.0094 0.0092 0.0089 0.0087 0.0O84 0.0082 0.0079 0.0077 0.0074 0.0072 0.0069 0.0067 0.0064 0.0062 0.OO6O 0.0057 0.0055 0.0052 0.0050 0.0047 0.0045 0.0042 0.0040 0.0037 0.0035 0.0032 0.0030 0.0027 0.00248 0.00186 0.00124 0.00062 0.0

cm

--1

90.0 88.0 86.0 84.0 82.0 80.0 78.0 76.0 74.0 72.0 70.0 68.0 66.0 64.0 62.0 60.0 58.0 56.0 54.0 52.0 50.0 48.0 46.0 44.0 42.0 40.0 38.0 36.0 34.0 32.0 30.0 28.0 26.0 24.0 22.0 20.0 15.0 10.0 5.0 0.0

/~m

111.11 113.64 116.28 119.05 121.95 125.00 128.21 131.58 135.14 138.89 142.86 147.06 151.52 156.25 161.29 166.67 172.41 178.57 185.19 192.31 200.00 208.33 217.39 227.27 238.10 250.00 263.16 277.78 294.12 312.50 333.33 357.14 384.62 416.67 454.55 500.00 666.67 1000.0 2000.0

n

0.62 0.63 0.65 0.69 0.79 0.97 1.07 1.08 1.08 1.25 1.21 1.26 1.73 2.05 2.26 2.47 2.81 3.16 4.21 5.17 5.37 6.73 7.19 7.19 6.96 6.77 6.50 6.34 6.19 6.08 5.98 5.90 5.85 5.82 5.76

5.53 [16] 5.53 [15]

k

1.92 2.02 2.11 2.19 2.33 2.46 2.59 2.72 2.96 3.10 3.14 3.73 4.11 4.34 4.63 4.85 5.18 6.13 7.13 8.44 8.68 6.55 5.10 3.64 2.51 1.99 1.66 1.53 1.34 1.16 1.08 0.95 0.87 0.77 0.60

0.381 0.381 0.384 0.388 0.394 0.403 0.414 0.428 0.446 0.469 0.497 0.533 0.579 0.639 0.719 0.830 0.990 1.235 1.647 2.429 4.233 9.167 14.658 13.099 11.137 9.808 8.891 8.226 7.722 7.329 7.015 6.758 6.546 6.369 6.220 6.095 5.860 5.711 5.628 5.601 5.48 [18]

2.174 2.313 2.457 2.610 2.770 2.941 3.124 3.321 3.535 3.768 4.026 4.313 4.637 5.009 5.441 5.956 6.585 7.379 8.427 9.879 11.932 13.595 8.314 3.386 1.673 0.974 0.627 0.431 0.309 0.229 0.174 0.134 0.105 0.083 0.066 0.053 0.030 0.017 0.007 0.0

Cubic Thallium(I) Halides

951 TABLE IV

Values of n and k Obtained for Polycrystalline Orthorhombic (and Cubic as Noted) Thallium Iodide eV

cm- ~

21.500 21.200 21.000 20.800 20.600 20.400 20.200 20.000 19.800 19.600 19.400 19.200 19.000 18.800 18.600 18.400 18.200 18.000 17.800 17.600 17.400 17.200 17.000 16.800 16.600 16.500 16.400 16.300 16.200 16.100 16.000 15.900 15.800 15.700 15.600 15.500 15.400 15.300 15.200 15.100 15.000 14.900 14.800 14.700 14.600

173407.8 170988.2 169375.1 167762.0 166148.9 164535.8 162922.7 161309.6 159696.5 158083.4 156470.3 154857.2 153244.1 151631.0 150017.9 148404.8 146791.7 145178.6 143565.5 141952.5 140339.4 138726.3 137113.2 135500.1 133887.0 133080.4 132273.9 131467.3 130660.8 129854.2 129047.7 128241.1 127434.6 126628.0 125821.5 125014.9 124208.4 123401.8 122595.3 121788.8 120982.2 120175.7 119369.1 118562.6 117756.0

/xm 0.05767 0.05848 0.05904 0.05961 0.06019 0.06078 0.06138 0.06199 0.06262 0.06326 0.06391 0.06458 0.06526 0.06595 0.06666 0.06738 0.06812 0.06888 0.06965 0.07045 0.07126 0.07208 0.07293 0.07380 0.07469 0.07514 0.07560 0.07606 0.07653 0.07701 0.07749 0.07798 0.07847 0.07897 0.07948 0.07999 0.08051 0.08104 0.08157 0.08211 0.08266 0.08321 0.08377 0.08434 0.08492

n 0.775 [3] 0.768 0.763 0.759 0.755 0.751 0.745 0.738 0.730 0.722 0.713 0.704 0.691 0.676 0.662 0.678 0.683 0.681 0.675 0.669 0.662 0.645 0.613 0.616 0.742 0.788 0.809 0.823 0.836 0.845 0.852 0.855 0.852 0.842 0.830 0.818 0.806 0.774 0.730 0.704 0.710 0.730 0.769 0.823 0.880

0.110 [3] 0.115 0.120 0.125 0.130 0.135 0.141 0.147 0.153 0.161 0.169 0.180 0.192 0.213 0.235 0.253 0.261 0.268 0.278 0.291 0.310 0.335 0.381 0.456 0.530 0.529 0.517 0.496 0.474 0.453 0.431 0.411 0.395 0.382 0.368 0.352 0.335 0.333 0.364 0.408 0.453 0.490 0.524 0.539 0.539

(continued)

952

William J. Tropf TABLE IV

(Continued)

Orthorhombic (and Cubic as Noted) Thallium Iodide

eV

cm - 1

14.500 14.400 14.300 14.200 14.100 14.000 13.900 13.800 13.700 13.600 13.500 13.400 13.300 13.200 13.100 13.000 12.900 12.800 12.700 12.600 12.500 12.400 12.300 12.200 12.100 12.000 11.900 11.800 11.700 11.600 11.500 11.400 11.300 11.200 11.100 11.000 10.900 10.800 10.700 10.600 10.500 10.400 10.300 10.200 10.100 10.000 9.900

116949.5 116142.9 115336.4 114529.8 113723.3 112916.7 112110.2 111303.6 110497.1 109690.5 108884.0 108077.4 107270.9 106464.3 105657.8 104851.2 104044.7 103238.1 102431.6 101625.1 100818.5 100012.0 99205.4 98398.9 97592.3 96785.8 95979.2 95172.7 94366.1 93559.6 92753.0 91946.5 91139.9 90333.4 89526.8 88720.3 87913.7 87107.2 86300.6 85494.1 84687.5 83881.0 83074.4 82267.9 81461.3 80654.8 79848.3

/zm 0.08551 0.08610 0.08670 0.08731 0.08793 0.08856 0.08920 0.08984 0.09050 0.09117 0.09184 0.09253 0.09322 0.09393 0.09465 0.09537 0.09611 0.09686 0.09763 0.09840 0.09919 0.09999 0.1008 0.1016 0.1025 0.1033 0.1042 0.1051 0.1060 0.1069 0.1078 0.1088 0.1097 0.1107 0.1117 0.1127 0.1137 0.1148 0.1159 0.1170 0.1181 0.1192 0.1204 0.1216 0.1228 0.1240 0.1252

n 0.927 0.964 0.988 0.988 0.963 0.928 0.895 0.862 0.831 0.802 0.769 0.741 0.724 0.706 0.686 0.673 0.666 0.661 0.657 0.656 0.657 0.658 0.658 0.658 0.658 0.659 0.660 0.662 0.665 0.668 0.671 0.673 0.676 0.679 0.681 0.684 0.688 0.692 0.696 0.698 0.698 0.698 0.700 0.706 0.712 0.718 0.732

0.524 0.493 0.437 0.365 0.314 0.287 0.267 0.252 0.249 0.258 0.270 0.290 0.308 0.329 0.353 0.376 0.395 0.414 0.432 0.449 0.465 0.479 0.493 0.508 0.522 0.536 0.549 0.563 0.576 0.589 0.603 0.617 0.632 0.647 0.663 0.678 0.690 0.702 0.714 0.728 0.746 0.764 0.780 0.799 0.818 0.837 0.861

Cubic Thallium(I) Halides

953 TABLE IV

(Continued)

Orthorhombic (and Cubic as Noted) Thallium Iodide eV 9.800 9.700 9.600 9.500 9.400 9.300 9.200 9.100 9.000 8.900 8.800 8.700 8.600 8.500 8.400 8.300 8.200 8.1 O0 8.000 7.900 7.800 7.700 7.600 7.500 7.400 7.300 7.200 7.1 O0 7.000 6.900 6.800 6.700 6.600 6.500 6.400 6.300 6.200 6.1 O0 6.000 5.900 5.800 5.700 5.600 5.500 5.400 5.300 5.200

cm

-1

79041.7 78235.2 77428.6 76622.1 75815.5 75009.0 74202.4 73395.9 72589.3 71782.8 70976.2 70169.7 69363.1 68556.6 67750.0 66943.5 66136.9 65330.4 64523.8 63717.3 62910.7 62104.2 61297.6 60491.1 59684.6 58878.0 58071.5 57264.9 56458.4 55651.8 54845.3 54038.7 53232.2 52425.6 51619.1 50812.5 50006.0 49199.4 48392.9 47586.3 46779.8 45973.2 45166.7 44360.1 43553.6 42747.0 41940.5

/~m O. 1265 0.1278 O. 1292 O. 1305 O. 1319 O. 1333 0.1348 O. 1362 O. 1378 O. 1393 O.1409 O.1425 O. 1442 O. 1459 0.1476 O.1494 O. 1512 O. 1531 O. 1550 0.1569 O. 1590 O. 1610 O. 1631 O. 1653 O. 1675 O. 1698 O. 1722 O. 1746 O. 1771 O. 1797 O. 1823 O. 1851 O. 1879 O. 1907 O. 1937 O. 1968 0.2000 0.2033 0.2066 0.2101 0.2138 0.2175 0.2214 0.2254 0.2296 0.2339 0.2384

n 0.746 0.760 0.780 O.799 0.823 0.847 0.870 0.892 0.913 0.938 0.957 0.976 0.994 1.012 1.031 1.048 1.064 1.077 1.088 1.093 1.096 1.095 1.090 1.081 1.066 1.050 1.027 1.016 1.011 1.010 1.012 1.014 1.016 1.020 1.026 1.034 1.042 1.051 1.060 1.074 1.088 1.102 1.116 1.141 1.167 1.199 1.233

0.884 0.908 0.924 0.940 0.955 0.965 0.975 0.983 0.992 0.996 0.996 0.996 0.995 0.994 0.992 0.987 0.982 0.976 0.970 0.961 0.952 0.942 0.931 0.922 0.919 0.918 0.926 0.949 0.977 1.010 1.041 1.071 1.1 O0 1.129 1.158 1.189 1.219 1.246 1.273 1.307 1.341 1.375 1.409 1.450 1.488 1.527 1.567

(continued)

954

William J. Tropf TABLE IV

(Continued)

Orthorhombic (and Cubic as Noted) Thallium Iodide eV 5.1 O0 5.000 4.900 4.800 4.700 4.600 4.500 4.400 4.300 4.200 4.100 2.271 2.105 1.907 1.90 1.80 1.70 1.60 1.50 1.40 1.30 1.20 1.10 1.00 0.90 0.80 0.70 0.60 0.50 0.0248 0.0242 0.0236 0.0229 0.0223 0.0217 0.0211 0.0205 0.0198 0.0192 0.0186 0.0180 0.0174 0.0167 0.0161 0.0155 0.0149 0.0143

cm --1 41133.9 40327.4 39520.9 38714.3 37907.8 37101.2 36294.7 35488.1 34681.6 33875.0 33068.5 18315.0 16977.9 15384.6 15324.4 14517.9 13711.3 12904.8 12098.2 11291.7 10485.1 9678.6 8872.0 8065.5 7258.9 6452.4 5645.8 4839.3 4032.7 200.0 195.0 190.0 185.0 180.0 175.0 170.0 165.0 160.0 155.0 150.0 145.0 140.0 135.0 130.0 125.0 120.0 115.0

t Cubic thallium iodide

ftm 0.2431 0.2480 0.2530 0.2583 0.2638 0.2695 0.2755 0.2818 0.2883 0.2952 0.3024 0.546 0.589 0.650 0.653 0.689 0.729 0.775 0.827 0.886 0.954 1.033 1.127 1.240 1.378 1.550 1.771 2.066 2.480 50.00 51.28 52.63 54.05 55.56 57.14 58.82 60.61 62.50 64.52 66.67 68.97 71.43 74.07 76.92 80.00 83.33 86.96

n 1.269 1.311 1.363 1.412 1.457 1.512 1.572 1.633 1.713 1.796 1.882 2.85t [7] 2.78t 2.72t 2.790 [31 2.740 2.700 2.660 2.620 2.590 2.570 2.540 2.520 2.500 2.490 2.470 2.450 2.440 2.430 2.230 [15] 2.216 2.200 2.183 2.164 2.143 2.120 2.093 2.064 2.031 1.993 1.950 1.900 1.842 1.774 1.693 1.597 1.482

k 1.609 1.654 1.704 1.751 1.795 1.847 1.891 1.927 1.982 2.040 2.078

O.lO4115] o.135 o.18o

Cubic Thallium(I) Halides

955 TABLE IV

(Continued)

Orthorhombic (and Cubic as Noted) Thallium Iodide

eV

cm -1

0.0136 0.0130 0.0124 0.0118 0.0112 0.0105 0.0099 0.0093 0.0087 0.0081 0.0074 0.0068 0.0062 0.0056 0.0050 0.0043 0.0037 0.0031 0.0024 0.0018 0.0012 0.00062 0.0

110.0 105.0 100.0 95.0 90.0 85.0 80.0 75.0 70.0 65.0 60.0 55.0 50.0 45.0 40.0 35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0

/xm 90.91 95.24 100.0 105.3 111.1 117.6 125.0 133.3 142.9 153.8 166.7 181.8 200.0 222.2 250.0 285.7 333.3 400.0 500.0 666.7 1000.0 2000.0

n 1.344 1.182 1.005 0.834 0.679 0.526 0.385 0.278 0.214 0.196 0.251 0.656 17.685 8.373 6.584 5.779 5.320 5.029 4.836 4.706 4.623 4.576 4.561 4.60[ 18] 5.44t [18]

0.248 0.353 0.512 0.734 1.008 1.342 1.770 2.314 3.008 3.950 5.444 8.853 4.132 0.316 0.128 0.074 0.048 0.034 0.024 0.016 0.010 0.005 0.0

TABLE V Values of n and k for KRS-5 from Various References

eV

cm -1

2.3557 2.2937 2.2317 2.1697 2.1489 2.1413 2.1077 1.9257 1.7949 1.4550 1.3862 1.2228 1.1697 1.0985

19000.0 18500.0 18000.0 17500.0 17332.2 17270.7 17000.0 15531.6 14477.0 11735.4 11180.7 9862.1 9434.0 8860.1

/zm 0.5263 0.5405 0.5556 0.5714 0.57696 0.57902 0.5882 0.64385 0.69075 0.85212 0.8944 1.01398 1.0600 1.12866

n

2.62758 [27] 2.62505

k 8.38E-3 [241 8.66E-4 8.95E-5 9.26E-6 4.33E-6 3.27E-6 9.59E-7

2.56454 2.53470 2.47448 2.46470 2.44416 8.44E-11 [31] 2.43090

(continued)

956

William J. Tropf TABLE V

(Continued)

KRS-5

eV 0.90679 0.88878 0.81061 0.73225 0.72540 0.68384 0.62933 0.53322 0.36266 0.29118 0.22380 0.18527 0.14878 0.14258 0.13638 0.13018 0.12750 0.12399 0.11779 0.11697 0.11236 0.11159 0.10539 0.09919 0.09299 0.08679 0.08676 0.08277 0.08059 0.08009 0.07439 0.07126 0.06827 0.06819 0.06199 0.06027 0.05690 0.05579 0.05448 0.05269 0.05205 0.04959 0.04928 0.04774 0.04656 0.04339 0.04159

cm

--1

7313.7 7168.5 6538.O 5906.0 5850.7 5515.5 5075.9 4300.7 2925.0 2348.5 1805.1 1494.3 1200.0 1150.0 1100.0 1050.0 1028.4 1000.0 950.0 943.4 906.2 900.0 850.0 800.0 750.0 700.0 699.8 667.6 650.0 646.0 600.0 574.7 550.7 550.0 500.0 486.1 458.9 450.0 439.4 425.0 419.8 400.0 397.5 385.1 375.5 350.0 335.5

/xm 1.3673 1.395 1.52952 1.6932 1.7092 1.81308 1.9701 2.3252 3.4188 4.258 5.54 6.692 8.333 8.696 9.091 9.524 9.724 10.000 10.526 10.60 11.035 11.111 11.765 12.500 13.333 14.286 14.29 14.98 15.385 15.48 16.667 17.40 18.16 18.182 20.000 20.57 21.79 22.222 22.76 23.529 23.82 25.00 25.16 25.97 26.63 28.571 29.81

2.41385 2.41242 2.40658 2.40143 2.40105 2.39851 2.39553 2.39076 2.38386 2.38142 3.53E-10 2.37680 5.64E-7 [12] 7.00E-7 8.97E-7 9.25E-7 2.37132 8.24E-7 6.35E-7 4.22E-9 [31 ] 2.36857 5.03E-7 [ 12] 3.67E-7 2.58E-7 2.18E-7 1.48E-7 2.36023 2.35824 1.58E-7 2.35667 6.13E-7 2.35022 2.34755 5.05E-7 4.95E-7 2.33816 2.33290 6.13E-7 2.32851 2.88E-7 2.32338 1.64E-7 2.31696 2.31260 2.30902 1.01E-6 2.28988

Cubic Thallium(I) Halides

957 TABLE V (Continued) KRS-5

eV 0.04030 0.03911 0.03757 0.03720 0.03596 0.03410 0.03301 0.03148 0.03100 0.02976 0.02852 0.02728 0.02604 0.02480 0.02356 0.02232 0.02108 0.01984 0.01860 0.01736 0.01612 0.01587 0.01562 0.01537 0.01513 0.01488 0.01463 0.01438 0.01413 0.01389 0.01364 0.01339 0.01314 0.01289 0.01265 0.01240 0.01215 0.01190 0.01165 0.01141 0.01116 0.01091 0.01066 0.01041 0.01017 0.00992 0.00967

cm- 1 325.0 315.5 303.0 300.0 290.0 275.0 266.2 253.9 250.0 240.0 230.0 220.0 210.0 200.0 190.0 180.0 170.0 160.0 150.0 140.0 130.0 128.0 126.0 124.0 122.0 120.0 118.0 116.0 114.0 112.0 110.0 108.0 106.0 104.0 102.0 100.0 98.0 96.0 94.0 92.0 90.0 88.0 86.0 84.0 82.0 80.0 78.0

~m 30.769 31.70 33.00 33.333 34.48 36.364 37.56 39.38 40.00 41.67 43.48 45.45 47.62 50.00 52.63 55.56 58.82 62.50 66.67 71.43 76.92 78.13 79.37 80.65 81.97 83.33 84.75 86.21 87.72 89.29 90.91 92.59 94.34 96.15 98.04 100.0 102.0 104.2 106.4 108.7 111.1 113.6 116.3 119.0 122.0 125.0 128.2

4.39E-6 2.27730 2.26821 8.49E-6 2.25740 2.23243 2.21621 2.21 [ 13] 2.19 2.17 2.15 2.13 2.10 2.06 2.02 1.97 1.91 1.83 1.73 1.60 1.56 1.53 1.49 1.45 1.41 1.36 1.31 1.26 1.21 1.15 1.09 1.03 0.98 0.92 0.87 0.82 0.78 0.75 0.72 0.70 0.69 0.68 0.67 0.67 0.67 0.68

8.75E-5 [24] 1.25E-4 2.06E-4 2.42E-4 3.64E-4 5.49E-4 8.30E-4 1.26E-3 1.91E-3 2.91E-3 4.44E-3 6.80E-3

0.63 [13] 0.69 0.75 0.83 0.92 1.01 1.12 1.23 1.35 1.47 1.59 1.73 1.86 2.00 2.15 2.30 2.46

(continued)

958

William J. Tropf TABLE V

(Continued)

KRS-5

eV 0.00942 0.00917 0.00893 0.00868 0.00843 0.00818 0.00794 0.00769 0.00744 0.00719 0.00694 0.00670 0.00645 0.00620 0.00595 0.00570 0.00546 0.00521 0.00496 0.00471 0.00446 0.00422 0.00397 0.00372 0.00248 0.00124 0.0

cm-~ 76.0 74.0 72.0 70.0 68.0 66.0 64.0 62.0 60.0 58.0 56.0 54.0 52.0 50.0 48.0 46.0 44.0 42.0 40.0 38.0 36.0 34.0 32.0 30.0 20.0 10.0 0.0

~m 131.6 135.1 138.9 142.9 147.1 151.5 156.3 161.3 166.7 172.4 178.6 185.2 192.3 200.0 208.3 217.4 227.3 238.1 250.0 263.2 277.8 294.1 312.5 333.3 500.0 1000.0

0.69 0.70 0.72 0.75 0.78 0.83 0.88 0.96 1.05 1.18 1.35 1.59 1.95 2.52 3.48 5.22 8.11 10.70 10.92 10.08 9.21 8.50 7.95 7.51 6.30 5.83 5.70 5.70 [21]

2.64 2.82 3.01 3.22 3.45 3.70 3.97 4.28 4.63 5.03 5.49 6.04 6.71 7.52 8.48 9.42 9.42 7.09 4.20 2.46 1.54 1.03 0.72 0.53 0.15 0.05 0.0

TABLE VI Values of n and k for KRS-6 from Various References

eV

cm --1

3.3000 3.2000 3.1000 3.0508 3.0240 2.9771 2.9559 2.8955 2.8437 2.5407 2.4098

26616.1 25809.5 25003.0 24606.0 24390.2 24012.0 23840.9 23353.3 22935.8 20491.8 19436.3

/xm 0.3757 0.3875 0.4000 0.4064 0.4100 0.4165 0.4194 0.4282 0.4360 0.4880 0.5145

n

2.5754 [9] 2.5516 2.5417 2.5155 2.4951 2.4021 2.3716

0.0697 [25] 0.00493 0.000349 9.46E-5 4.66E-5 1.35E-5 7.71E-6 1.56E-6 1.90E-7 1.26E-7

Cubic Thallium(I) Halides

959 TABLE VI

(Continued)

KRS-6 /xm

eV

cm -~

2.2708 2.1821 2.1451 2.1039

18315.0 17599.4 17301.0 16969.3

0.5460 0.5682 0.5780 0.5893

2.0664 1.9160 1.9075 1.7712

16666.7 15453.6 15384.6 14285.7

0.6000 0.6471 0.650 0.700

1.6531 1.5498 1.3776 1.2399 1.1697 1.1271 1.0332 0.9537 0.8856 0.8266 0.7749 0.7293 0.6888 0.6526 0.6199 0.5636 0.5166 0.4769 0.4428 0.4133 0.3542 0.3100 0.2755 0.2480 0.2238 0.2066 0.1771 0.1550 0.1378 0.1240 0.1170 0.1127 0.10332 0.09537 0.08856 0.08266 0.07749

13333.3 12500.0 lllll.1 10000.0 9434.0 9090.9 8333.3 7692.3 7142.9 6666.7 6250.0 5882.4 5555.6 5263.2 5000.0 4545.5 4166.7 3846.2 3571.4 3333.3 2857.1 2500.0 2222.2 2000.0 1805.1 1666.7 1428.6 1250.0 llll.l 1000.0 943.4 909.1 833.3 769.2 714.3 666.7 625.0

0.750 0.80 0.90 1.00 1.06 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.20 2.40 2.60 2.80 3.00 3.50 4.00 4.50 5.00 5.54 6.00 7.00 8.00 9.00 10.00 10.60 11.00 12.00 13.00 ! 4.00 15.00 16.00

2.3437 2.3280 2.3218 2.3154 2.3367 2.3294 2.2884 2.2873 2.2982 2.2708 2.2582 2.2660 2.2510 2.2404

6.16E-8

[22] [9]

2.71E-08

[22] [9] [22]

6.77E-10 [32] 2.2321 2.2255 2.2212 2.2176 2.2148 2.2124 2.2103 2.2086 2.2071 2.2059 2.2039 2.2024 2.2011 2.2001 2.1990 2.1972 2.1956 2.1942 2.1928 6.17E-10 2.1900 2.1870 2.1839 2.1805 2.1767 1.18E-09 2.1723 2.1674 2.1620 2.1563 2.1504 2.1442

(continued)

960

William J. Tropf TABLE VI

(Continued)

KRS-6

eV 0.07293 0.06888 0.06526 0.06199 0.05904 0.05636 0.05391 0.05166 0.04959 0.04649 0.04339 0.04030 0.03720 0.03410 0.03100 0.02976 0.02852 0.02728 0.02604 0.02480 0.02356 0.02232 0.02108 0.01984 0.01959 O.O1934 0.01909 O.01885 0.01860 0.01835 0.01810 0.01785 0.01761 0.01736 0.01711 0.01686 0.01661 0.01637 0.01612 0.01587 0.01562 0.01537 0.01513 0.01488 0.01463 0.01438 0.01413

-1

~m

588.2 555.6 526.3 500.0 476.2 454.5 434.8 416.7 400.0 375.0 350.0 325.0 300.0 275.0 250.0 240.0 230.0 220.0 210.0 200.0 190.0 180.0 170.0 160.0 158.0 156.0 154.0 152.0 150.0 148.0 146.0 144.0 142.0 140.0 138.0 136.0 134.0 132.0 130.0 128.0 126.0 124.0 122.0 120.0 118.0 116.0 114.0

17.00 18.00 19.00 20.00 21.00 22.00 23.00 24.00 25.00 26.67 28.57 30.77 33.33 36.36 40.00 41.67 43.48 45.45 47.62 50.00 52.63 55.56 58.82 62.50 63.29 64.10 64.94 65.79 66.67 67.57 68.49 69.44 70.42 71.43 72.46 73.53 74.63 75.76 76.92 78.13 79.37 80.65 81.97 83.33 84.75 86.21 87.72

c m

2.1377 2.1309 2.1236 2.1154 2.1067 2.0976 2.0869 2.0752 2.07 [13] 2.05 2.03 2.00 1.96 1.91 1.83 1.80 1.76 1.71 1.65 1.58 1.49 1.39 1.25 1.09 1.06 1.02 0.99 0.95 0.92 0.89 0.85 0.82 0.79 0.77 0.74 0.72 0.70 0.69 0.67 0.66 0.65 0.64 0.63 0.63 0.62 0.62 0.62

1.96E-4 [34] 3.74E-4 7.17E-4 1.38E-3 2.68E-3 5.22E-3

0.34 [131 0.40 0.49 0.62 0.65 0.69 0.73 0.77 0.82 0.87 0.93 0.99 1.05 1.12 1.19 1.26 1.33 1.41 1.48 1.56 1.64 1.73 1.81 1.90 1.99 2.09 2.18 ,

,

,

Cubic Thallium(I) Halides

961 TABLE VI

(Continued)

KRS-6 eV 0.01389 0.01364 0.01339 0.01314 0.01289 0.01265 0.01240 0.01215 0.01190 0.01165 0.01141 0.01116 0.01091 0.01066 0.01041 0.01017 0.00992 0.00967 0.00942 0.00917 0.00893 0.00868 0.00843 0.00818 0.00794 0.00769 0.00744 0.00719 0.00694 0.00670 0.0O645 0.00620 0.00595 0.00570 0.00546 0.00521 0.00496 0.00372 0.00248 0.00124 0.0

cm

-1

ll2.0 110.0 108.0 106.0 104.0 102.0 100.0 98.0 96.0 94.0 92.0 90.0 88.0 86.0 84.0 82.0 80.0 78.0 76.0 74.0 72.0 7O.O 68.0 66.0 64.0 62.0 60.0 58.0 56.0 54.0 52.0 50.0 48.0 46.0 44.0 42.0 40.0 30.0 20.0 10.0 0.0

/.zm 89.29 90.91 92.59 94.34 96.15 98.04 100.0 102.0 104.2 106.4 108.7 111.1 113.6 116.3 119.0 122.0 125.0 128.2 131.6 135.1 138.9 142.9 147.1 151.5 156.3 161.3 166.7 172.4 178.6 185.2 192.3 200.0 208.3 217.4 227.3 238.1 250.0 333.3 500.0 1000.0

0.62 0.62 0.63 0.63 0.64 0.65 0.66 0.68 0.69 0.71 0.74 0.76 0.80 0.84 0.89 0.94 1.02 1.10 1.22 1.36 1.56 1.82 2.20 2.76 3.65 5.15 7.59 10.55 12.00 11.64 10.75 9.91 9.20 8.63 8.17 7.78 7.47 6.46 5.97 5.73 5.66 5.63 [21]

2.28 2.39 2.49 2.61 2.73 2.85 2.98 3.12 3.26 3.42 3.58 3.76 3.95 4.16 4.39 4.64 4.92 5.23 5.58 5.97 6.44 6.98 7.62 8.38 9.27 10.18 10.61 9.33 6.42 3.96 2.50 1.66 1.16 0.85 0.63 0.49 0.38 0.13 0.05 0.02 0.0

Yttrium Aluminum Garnet

(Y3AI5012)

WILLIAM J. TROPF Applied Physics Laboratory The Johns Hopkins University Laurel, Maryland

Yttrium aluminum garnet (YAG or yttrogarnet), is a synthetic, nonmagnetic material with the chemical formula Y3A15012 or Y3A12(A104)3. It is a host medium for rare-earth-doped, solid-state lasers, a low-loss material for acoustic devices, and a gemstone. YAG crystals have body-centered cubic crystalline structure with space group Ia3d (or OhlO). The unit cell has an edge dimension of 12.008 A [1] and contains eight formula units, giving a theoretical density of 4.554 g/cm 3. The yttrium atoms occupy the 24(c) sites (222 or D2 point symmetry), aluminum(l) atoms occupy the 16(a) sites (3 o r C3i[$6] point symmetry), aluminum(2) atoms occupy the 24(d) sites (4 or $4 point symmetry), and the oxygen atoms occupy the 96(h) sites (1 or C~ point symmetry). YAG undergoes a phase change at 2193 K. Optical constants above the electronic band gap have been extensively studied by Tomiki et al. [2-5] using Kramers-Kronig analyses of reflectance spectra. Professor Tomiki provided approximately 750 values of the complex dielectric constant at 297 K [4], from the band edge at 6.4 eV to 116 eV, in digital form with three decimal places. Selected values of the complex refractive index (n and k) determined from these data are given in Table I. Yttrium aluminum garnet has a direct band gap at 6.5 eV. The lowestenergy feature in the electronic absorption is excitonic absorption (see later discussion). Interband transitions from the upper valence band of oxygen (2p 6) to the conduction bands of yttria (4d + 5s states) dominate the structure from 7 to 15 eV. Optical constants between 15 and 26 eV are mainly due to interband transitions from the oxygen valence bands to conduction bands of yttrium (5d + 6s levels) and aluminum (3s + 3p levels). The valence electrons form a plasma oscillation centered at 23.4 eV. Inner-core yttria electron transitions occur in the 26 to 35 eV region. At energies above 29.3 eV, absorption is mainly due to transitions from oxygen (2s 2) bands to the aluminum (3p) conduction band. Plasma oscillations from m

963 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS III

Copyright 9 1998 by Academic Press. All rights of reproduction in any form reserved ISBN 0-12-544423-0/$25.00.

964

William J. Tropf

the full bands of yttria (4p 6) and oxygen (2S 2) peak near 36.2 eV. At higher energies, yttria and aluminum electron transitions dominate, especially the prominent aluminum L2, 3 transitions near 77 eV. Tomiki et al. [3, 5] determined the temperature dependence of exciton absorption between 10 and 297 K from transmission measurements (6.5-7.0 eV) and fitted their data to an Urbach-rule formulation, /3(E, T) =/30 e x p ( - trs(T)(E o - E ) / k B T ) ,

(la)

where/3(E,T) is the absorption coefficient as a function of energy (E) and temperature (T), and kB is Boltzmann's constant. The slope parameter O-s(T) is given by cr~(T) = trio (2kT/Eso) tanh(E~o/2kT).

(lb)

Constants of Eq. (1) are / 3 o = 4 . 0 9 3 • cm-1, E o = 7 . 0 4 7 eV, OsO = 0.558, and Eso = 36.5 meV [5]. Values of the imaginary part of the refractive index between 6.4 and 6.6 eV given in Table I are calculated from this equation. Slack et al. [6] measured YAG absorption at 300 K in the vicinity of the Urbach absorption edge. Imaginary index values between 4.5 and 6.6 eV in Table I are taken from his measurements. Values below 6.4 eV are dominated by extrinsic absorption but are similar to those reported by Tomiki et al. [2, 3, 5]. Refractive-index measurements in the transparent region are surprisingly sparse. The most comprehensive data set, covering 0.4 to 4.0 /xm, is the minimum-deviation prism measurement of Bond [7]. Other measurements (primarily from interferometric techniques) are those of Shchavelev et al. [8], McCollum et al. [9], and Wilson [10], which are restricted to the visible. Data given in Table I are from Bond [7], Shchavelev et al. [8], and McCollum et al. [9], Several sources give Sellmeier dispersion relationships for YAG in the form Ai ~2

(2)

1

where the constants A i and/~i are given in Table II. These dispersion formulas are either based on the data of Bond (see [2, 11]) or on visible measurements [12, 13]. Several sources give the temperature dependence of the refractive index [14-16]. All are restricted to the visible or near infrared. Table III lists values of d n / d T with references. Absorption from the infrared edge of transparency through the region of fundamental lattice vibrations was measured by Slack et al. [6] and DeShazer et al. [17]. Imaginary-refractive-index values from 840 to 2350 cm -~ derived from these measurements are included in Table I.

Yttrium Aluminum Garnet (Y3AI5012)

965

The fundamental infrared lattice vibrations are studied by Hurrell et al. [18], Thirumavalavan et al. [19], Gledhill et al. [20], and Hofmeister and Campbell [21]. Hurrell et al. give the optical irreducible representation as F = 5A1,(-) + 3alg(R) + 5Azu(- ) -k- 5Azg(- ) + I O E , ( - ) (3) + 8Eg(R) + 14F~g(-) + 17F~,(/R) + 14Fzg(R) + 16F2,(-) and make the following assignments to the Raman-active modes: A lg vibrations at 373, 561, and 783 cm-~; Eg vibrations at 162, 310, 340, 403, 531, 537, 714, and 758 cm-~; and 11 of 14 F2g modes located at 144, 218, 243, 259, 296, 408, 436, 544, 690, 719, and 857 cm -1. Table IV summarizes measured locations of the infrared-active modes from several sources. Optical constants in Table I for the region of fundamental lattice vibrations are derived from the models of Gledhill et al. [20] and Hofmeister and Campbell [21]. Gledhill et al. measured room-temperature reflectivity and fitted the data to a dispersion model of the form

~LOj.- p2--i'YLOy F/2(U)- E(I))- E~ j~. 2 , 9 VTOj -

(4)

vZ--iyTOjV

where VTO and VLO are the transverse and longitudinal infrared-active optical mode frequencies, TTO and YLO are the respective mode widths, and e~ the high-frequency (electronic) contribution to the dielectric constant. Hofmeister and Campbell also made reflectivity measurements, preformed a Kramers-Kr6nig analysis on their data, and fit the resulting optical constants to a Lorentzian (classical) oscillator model of the form n 2 ( v ) - e ( v ) - e~ + j~.

vz-iv:v '

(5)

where VTo is transverse infrared-active optical mode frequency, 7 is the mode width, f is the oscillator strength, and e~ the high-frequency (electronic) contribution to the dielectric constant. These models were used to calculate the real index between 200 and 2350 cm-~ and the imaginary index between 200 and 835 c m - ~ (or to 1000 c m - ~ for [21]). Real refractive index and absorption coefficient below 100 cm-~ for Nddoped YAG were measured by Zhan and Coleman [22] using channeledspectrum (interference) technique, who also provide a Sellmeier-type dispersion relationship for this regime, n 2 - 9.42 + 0.983 ~ ,,o2_,,2 ,

(6)

where v is in wave numbers, and Vo is the lowest-frequency infrared-active vibration (set to 123 cm-~). In Table I, real-refractive-index values are cal-

966

William J. Tropf

culated f r o m this equation, and the i m a g i n a r y values are derived f r o m the m e a s u r e d absorption coefficient. The l o w - f r e q u e n c y (1 M H z ) dielectric constant was d e t e r m i n e d by Shannon et al. [23] as ~ = 10.60 _+ 0.05 (giving n = 3.26 _+ 0.01) using a parallelplate capacitance technique. This value is included in Table I, as well as limiting values f r o m the infrared m o d e l s of Gledhill et al. [20] and H o f m e i s ter and C a m p b e l l [21], and the refractive-index m o d e l o f Z h a n and C o l e m a n [22] b a s e d on Eq. (6). Figure 1 shows a c o m p o s i t e of the c o m p l e x refractive index o f yttrium a l u m i n u m garnet. ACKNOWLEDGMENT

The author thanks Professor T. Tomiki, University of Ryukyus (retired), for graciously providing digital data from his measurements. REFERENCES

1. H. S. Yoder and M. L. Keith, Complete substitution of aluminum for silicon: The systems 3MnO x A1203.3SiO2-3YeO3.5A1203. Am. Mineral. 36, 519-533 (1951). 2. T. Tomiki, J. Tamashiro, M. Hiraoka, N. Hirata, and T. Futemma, A determination of the spectra of Y3A15012 (YAG)--Reflectivity and intrinsic tail absorptivity in VUV region. J. Phys. Soc. Jpn. 57, 4429-4433 (1988). 3. T. Tomiki, E Fukudome, M. Kaminao, M. Fujisawa, Y. Tanahara, and T. Futemma, Optical spectra of Y3A150~e (YAG) single crystals in the vacuum ultraviolet region. J. Phys. Soc. Jpn. 58, 1801-1810 (1989). 4. T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, H. Fukutani, H. Kato, J. Tamashiro, T. Miyahara, and A. Yonesu, Optical spectra of Y3A15012(YAG) single crystals in the vacuum ultraviolet region. II. J. Phys. Soc. Jpn. 62, 1388-1400 (1993). 5. T. Tomiki, Y. Isa, Y. Kadekawa, Y. Ganaha, N. Toyokawa, T. Miyazato, M. Miyazato, T. Kohatsu, H. Shimabukuro, and J. Tamashiro, Optical absorption of single crystals of Y3A15O12 and Y3A15012:Nd3+ in the UV fundamental absorption edge region. J. Phys. Soc. Jpn. 65, 1106-1113 (1996). 6. G.A. Slack, D. W. Oliver, R. M Chrenko, and S. Roberts, Optical absorption of Y3A15012 from 10- to 55,000 cm -~ wave numbers. Phys. Rev. 177, 1308-1314 (1969). 7. W. L. Bond, Measurement of the refractive index of several crystals. J. AppL Phys. 36, 1674-1677 (1965). 8. O. S. Shchavelev, V. A. Babkina, and Z. S. Mal'tseva, Thermo-optic properties, expansion coefficient and refractive index of yttrium aluminum garnet. Sov. J. Opt. Technol. 40, 623-624 (1973). 9. B. C. McCollum, W. R. Bekebrede, M. Kestigian, and A. B. Smith, Refractive index measurements on magnetic garnet films. Appl. Phys. Lett. 23, 702-703 (1973). 10. K. E. Wilson, Thermo-optics of nonlinear crystals and laser materials. Ph.D. Dissertation, University of Southern California, Los Angeles (January 1980). 11. W. J. Tropf, M. E. Thomas, and T. J. Harris, Properties of crystals and glasses, in "Optical Society of America Handbook of Optics" (M. Bass, ed.), 2nd Ed., Chap. 33, McGrawHill, New York, 1994. 12. S. H. Wemple and W. J. Tabor, Refractive index behavior of garnets. J. Appl. Phys. 44, 1395-1396 (1973).

Yttrium Aluminum Garnet (Y3AI5012)

967

13. E. V. Zharikov, Yu. S. Privis, R A. Studenikin, V. A. Chikov, V. D. Shigorin, and I. A. Shcherbakov, Temperature-wise measurements of refractive indices of rare-earth garnets. Sov. Phys. Crystallogr. 34, 712-714 (1989). 14. J. D. Foster and L. M. Osterink, Index of refraction and expansion thermal coefficients of Nd:YAG. Appl. Opt. 7, 2428-2429 (1968). 15. D. D. Young, K. C. Jungling, T. L. Williamson, and E. R. Nichols, Holographic interferometry measurement of the thermal refractive index coefficient and thermal expansion coefficient of Nd:YAG and Nd:YALO. IEEE J. Quantum Electon. QE-8, 720-721 (1972). 16. R R. Stoddart, R E. Ngoepe, P. M. Mjwara, J. D. Comins, and G. A. Saunders, Hightemperature elastic constants of yttrium aluminum garnet. J. Appl. Phys. 73, 7298-7301 (1993). 17. L. G. DeShazer, S. C. Rand, and B. A. Wechsler, Laser crystals, in "CRC Handbook of Laser Science and Technology," (M. J. Weber, ed.), Vol. 5. pp. 281-338, CRC Press, Boca Raton, Florida, 1988. 18. J. R Hurrell, S. R S. Porto, I. E Chang, S. S. Mitra, and R. E Bauman, Optical phonons of yttrium aluminum garnet. Phys. Rev. 173, 851-856 (1968). 19. M. Thirumavalavan, J. Kumar, E D. Gnanam, and E Ramasamy, Vibrational spectra of Y3A15012 crystals grown from Ba- and Pb-based Flux systems. Infrared Phys. 26, 101103 (1986). 20. G. A. Gledhill, P. M. Nikolid, A. Hamilton, S. Stojilkovid, V. Blagojevid, P. Mihajlovid, and S. Djurid, FIR optical properties of single crystal Y3A15Oj2 (YAG). Phys. Status Solidi B 163, K123-K128 (1991). 21. A. M. Hofmeister and K. R. Campbell, Infrared Spectroscopy of yttrium aluminum, yttrium gallium, and yttrium iron garnets. J. Appl. Phys. 72, 638-646 (1992). 22. Y. Zhan and P. D. Coleman, Far-IR properties of Y3A150~2, LiYF 4, Cs2NaDyC16, and Rb2NaYF 6. Appl. Opt. 23, 548-551 (1984). 23. R. D. Shannon, M. A. Subramanian, T. H. Allik, H. Kimura, M. R. Kokta, M. H. Randles, and G. R. Rossman, Dielectric constants of yttrium and rare-earth garnets, the polarizability of gallium oxide, and the oxide additive rule. J. Appl. Phys. 67, 3798-3802 (1990).

968

William J. Tropf

101

.

.

.

.

.

.

.

.

I

a)

10 ~

10 -~ 10 .2

,

,

i

,,,l

, , , | I

.

,

,

. . . .

,I

10 ~

10 -1

,

i

,

|

, , , . l

10 3

10 2

10 ~

W A V E L E N G T H (~tm)

.

.

.

.

.

.

.

.

I

.

.

.

.

.

.

.

.

!

10 0 _ b )

10 -2

10 .4

10 .6 .

10 .2

.

.

.

.

.

,,I

10 l

i

i

i

|

1 , , ,

I

,

,

l

i

i

l

i l l

10 ~ 10 t 1' W A V E L E N G T H (~tm)

i

.

.

.

.

.

.

.

I

10 2

,

,

,

. . . . .

10 3

Fig. 1. (a) Log-log plot of the index of refraction n versus wavelength in micrometers for yttrium aluminum garnet. (b) Log-log plot of the extinction coefficient k versus wavelength in micrometers for yttrium aluminum garnet. Lines are data sets ~of many points; scattered data are indicated by circles.

Yttrium Aluminum Garnet (Y3AI5012)

969 TABLE I

Values of n and k for Yttrium Aluminum Garnet (YAG) from Various References eV 115.982 114.758 113.539 112.315 111.098 109.877 108.654 107.439 106.416 105.403 104.593 103.571 102.552 101.535 100.523 99.506 98.998 98.487 97.981 97.473 96.962 96.456 95.941 95.439 94.928 94.422 93.914 93.404 92.894 92.388 91.875 91.367 90.858 90.348 89.844 89.333 88.827 88.315 87.808 87.301 86.794 86.286 85.773 85.266 84.759 84.252 83.740

cm - ! 935451 925578 915747 905874 896059 886211 876347 866547 858296 850126 843593 835350 827131 818929 810766 802564 798466 794345 790264 786167 782045 777964 773810 769761 765640 761559 757461 753348 749235 745154 741016 736919 732813 728700 724635 720514 716432 712303 708214 704124 700035 695938 691800 687711 683622 679533 675403

/xm 0.01069 0.01080 0.01092 0.01104 0.01116 0.01128 0.01141 0.01154 0.01165 0.01176 0.01185 0.01197 0.01209 0.01221 0.01233 0.01246 0.01252 0.01259 0.01265 0.01272 0.01279 0.01285 0.01292 0.01299 0.01306 0.01313 0.01320 0.01327 0.01335 0.01342 0.01349 0.01357 0.01365 0.01372 0.01380 0.01388 0.01396 0.01404 0.01412 0.01420 0.01428 0.01437 0.01446 0.01454 0.01463 0.01472 0.01481

n 0.978 [4] 0.977 0.976 0.975 0.974 0.973 0.972 0.971 0.969 0.967 0.966 0.965 0.963 0.961 0.959 0.958 0.959 0.959 0.960 0.961 0.962 0.962 0.962 0.963 0.964 0.965 0.966 0.966 0.965 0.965 0.964 0.962 0.961 0.959 0.959 0.958 0.958 0.957 0.956 0.956 0.954 0.950 0.948 0.951 0.954 0.955 0.956

k

n

0.028 [4] 0.028 0.028 0.029 0.029 0.029 0.029 0.030 0.030 0.030 0.032 0.033 0.033 0.035 0.038 0.041 0.043 0.044 0.046 0.046 0.047 0.047 0.047 0.048 0.049 0.049 0.048 0.047 0.046 0.045 0.044 0.044 0.044 0.045 0.046 0.046 0.047 0.048 0.049 0.048 0.048 0.048 0.054 0.059 0.058 0.058 0.056

(continued)

970

William J. Tropf TABLE I (Continued) Yttrium Aluminum Garnet (YAG)

eV 83.234 82.723 82.218 81.709 81.201 80.688 80.182 79.672 79.163 78.646 78.160 77.641 77.153 76.633 76.116 75.605 75.097 74.586 74.078 73.573 73.061 72.553 71.946 71.334 70.723 70.115 69.502 68.892 68.281 67.751 67.150 66.849 66.249 65.649 65.050 64.451 63.851 63.251 62.650 62.048 61.449 60.849 60.251 59.648 59.049 58.45O 57.850

cm -1 671322 667201 663128 659022 654925 650787 646706 642593 638488 634318 630398 626212 622276 618082 613912 609791 605693 601572 597475 593402 589272 585175 580279 575343 570415 565511 560567 555647 550719 546444 541597 539169 534330 529491 524659 519828 514989 510150 505302 500447 495616 490776 485953 481090 476259 471427 466588

/.~m 0.01490 0.01499 0.01508 0.01517 0.01527 0.01537 0.01546 0.01556 0.01566 0.01576 0.01586 0.01597 0.01607 0.01618 0.01629 0.01640 0.01651 0.01662 0.01674 0.01685 0.01697 0.01709 0.01723 0.01738 0.01753 0.01768 0.01784 0.01800 0.01816 0.01830 0.01846 0.01855 0.01872 0.01889 0.01906 0.01924 0.01942 0.01960 0.01979 0.01998 0.02018 0.02038 0.02058 0.02079 0.02100 0.02121 0.02143

n 0.954 0.952 0.950 0.948 0.947 0.949 0.951 0.951 0.951 0.949 0.945 0.947 0.953 0.957 0.957 0.955 0.955 0.954 0.953 0.953 0.952 0.951 0.950 0.949 0.948 0.948 0.946 0.944 0.943 0.942 0.941 0.940 0.939 0.938 0.936 0.935 0.934 0.933 0.932 0.929 0.928 0.926 0.926 0.921 0.919 0.917 0.916

k 0.055 0.055 0.056 0.058 0.061 0.064 0.065 0.064 0.064 0.063 0.068 0.074 0.076 0.072 0.069 0.068 0.068 0.068 0.068 0.067 0.068 0.068 0.068 0.068 0.069 0.069 0.069 0.070 0.071 0.072 0.072 0.073 0.073 0.075 0.076 0.076 0.078 0.078 0.078 0.080 0.081 0.083 0.082 0.083 0.086 0.088 0.088

Yttrium Aluminum Garnet (Y3AI5012)

971

TABLE I (Continued) Yttrium Aluminum Garnet (YAG) eV 57.249 56.650 56.049 55.450 54.851 54.251 53.650 53.051 52.449 51.851 51.250 50.650 50.051 49.450 48.850 48.251 47.650 47.050 46.450 45.849 45.250 44.650 44.051 43.450 42.850 42.249 41.650 41.051 40.449 39.850 39.249 38.650 38.050 37.450 36.850 36.250 35.650 35.050 34.450 33.850 33.250 32.650 32.050 31.450 30.850 30.250 29.650

cm -~ 461741 456909 452062 447231 442400 437560 432713 427882 423026 418203 413356 408517 403685 398838 393999 389167 384320 379481 374642 369794 364963 360124 355292 350445 345606 340758 335927 331096 326241 321409 316562 311731 306892 302052 297213 292374 287534 282695 277856 273016 268177 263338 258499 253659 248820 243981 239141

~m 0.02166 0.02189 0.02212 0.02236 0.02260 0.02285 0.02311 0.02337 0.02364 0.02391 0.02419 0.02448 0.02477 0.02507 0.02538 0.02570 0.02602 0.02635 0.02669 0.02704 0.02740 0.02777 0.02815 0.02854 0.02893 0.02935 0.02977 0.03020 0.03065 0.03111 0.03159 0.03208 0.03258 0.03311 0.03365 0.03420 0.03478 0.03537 0.03599 0.03663 0.03729 0.03797 0.03868 0.03942 0.04019 0.04099 0.04182

n 0.913 0.911 0.909 0.906 0.904 0.901 0.898 0.895 0.893 0.889 0.886 0.882 0.879 0.875 0.871 0.867 0.863 0.859 0.854 0.849 0.844 0.838 0.834 0.829 0.823 0.818 0.808 0.793 0.790 0.773 0.768 0.772 0.772 0.770 0.765 0.765 0.775 0.808 0.854 0.900 0.942 0.970 0.992 1.000 0.991 0.978 0.973

k 0.090 0.092 0.094 0.096 0.098 0.100 0.101 0.104 0.106 0.109 0.112 0.116 0.118 0.122 0.126 0.129 0.133 0.138 0.143 0.148 0.153 0.160 0.167 0.173 0.181 0.188 0.190 0.207 0.228 0.230 0.264 0.285 0.303 0.323 0.349 0.383 0.424 0.468 0.491 0.498 0.487 0.471 0.451 0.421 0.401 0.401 0.405

(continued)

972

William J. Tropf TABLE I (Continued) Yttrium Aluminum Garnet (YAG)

eV 28.960 28.302 27.673 27.072 26.496 25.944 25.414 24.906 24.418 23.948 23.496 23.061 22.642 22.238 21.847 21.471 21.107 20.755 20.415 20.085 19.767 19.611 19.458 19.307 19.158 19.012 18.868 18.726 18.587 18.449 18.313 18.180 18.048 17.918 17.790 17.664 17.540 17.417 17.296 17.177 17.059 16.943 16.828 16.715 16.604 16.494 16.385

cm- ~ 233576 228269 223196 218349 213703 209251 204976 200879 196943 193152 189507 185998 182619 179360 176207 173174 170238 167399 164657 161995 159430 158172 156938 155720 154518 153341 152179 151034 149913 148800 147703 146630 145566 144517 143485 142469 141469 140476 139501 138541 137589 136653 135726 134814 133919 133032 132153

/xm 0.04281 0.04381 0.04480 0.04580 0.04679 0.04779 0.04879 0.04978 0.05078 0.05177 0.05277 0.05376 0.05476 0.05575 0.05675 0.05775 0.05874 0.05974 0.06073 0.06173 0.06272 0.06322 0.06372 0.06422 0.06472 0.06521 0.06571 0.06621 0.06671 0.06720 0.06770 0.06820 0.06870 0.06920 0.06969 0.07019 0.07069 0.07119 0.07168 0.07218 0.07268 0.07318 0.07368 0.07418 0.07467 0.07517 0.07567

n 0.977 0.982 0.981 0.964 0.938 0.913 0.889 0.872 0.870 0.877 0.869 0.852 0.838 0.826 0.820 0.825 0.857 0.894 0.915 0.940 0.972 0.988 1.005 1.025 1.045 1.064 1.082 1.099 1.114 1.126 1.142 1.161 1.181 1.205 1.230 1.254 1.277 1.295 1.313 1.331 1.350 1.372 1.399 1.427 1.455 1.481 1.499

k 0.407 0.408 0.393 0.381 0.383 0.394 0.414 0.447 0.483 0.501 0.511 0.534 0.566 0.604 0.651 0.712 0.765 0.795 0.819 0.854 0.882 0.895 0.909 0.922 0.930 0.937 0.943 0.948 0.952 0.960 0.971 0.980 0.990 0.997 0.999 1.000 0.995 0.992 0.989 0.987 0.986 0.986 0.984 0.974 0.959 0.938 0.907

Yttrium Aluminum Garnet (Y3AI5012)

973

TABLE I

(Continued)

Yttrium Aluminum Garnet (YAG)

eV

cm - 1

16.279 16.173 16.068 15.965 15.864 15.763 15.664 15.566 15.470 15.374 15.280 15.187 15.095 15.004 14.914 14.825 14.737 14.651 14.565 14.480 14.396 14.314 14.232 14.151 14.071 13.992 13.914 13.837 13.760 13.685 13.610 13.536 13.463 13.390 13.319 13.248 13.178 13.109 13.040 12.972 12.905 12.838 12.772 12.707 12.643 12.579 12.516

131298 130443 129596 128765 127951 127136 126338 125547 124773 123999 123241 122490 121748 121014 120289 119571 118861 118167 117474 116788 116111 115449 114788 114135 113489 112852 112223 111602 110981 110376 109771 109174 108586 107997 107424 106851 106287 105730 105174 104625 104085 103545 103012 102488 101972 101456 100948

/~m

0.07616 0.07666 0.07716 0.07766 0.07816 0.07866 0.07915 0.07965 0.08015 0.08065 0.08114 0.08164 0.08214 0.08263 0.08313 0.08363 0.08413 0.08463 0.08513 0.08563 0.08612 0.08662 0.08712 0.08762 0.08811 0.08861 0.08911 0.08960 0.09011 0.09060 0.09110 0.09160 0.09209 0.09260 0.09309 0.09359 0.09408 0.09458 0.09508 0.09558 0.09608 0.09658 0.09708 0.09757 0.09807 0.09857 0.09906

n

1.511 1.518 1.518 1.510 1.498 1.487 1.474 1.461 1.446 1.429 1.408 1.384 1.364 1.351 1.342 1.337 1.338 1.344 1.353 1.364 1.375 1.386 1.397 1.407 1.416 1.426 1.431 1.435 1.435 1.433 1.431 1.428 1.426 1.424 1.423 1.423 1.424 1.428 1.433 1.438 1.447 1.459 1.469 1.480 1.492 1.505 1.518

k

0.882 0.851 0.821 0.793 0.775 0.759 0.744 0.734 0.723 0.716 0.713 0.715 0.729 0.745 0.763 0.783 0.803 0.821 0.837 0.847 0.855 0.861 0.866 0.867 0.868 0.867 0.863 0.861 0.857 0.857 0.858 0.862 0.868 0.876 0.885 0.894 0.906 0.918 0.929 0.941 0.955 0.965 0.973 0.982 0.990 0.997 1.001

(continued)

974

William J. Tropf TABLE I (Continued) Yttrium Aluminum Garnet (YAG)

eV 12.453 12.391 12.330 12.269 12.209 12.149 12.090 12.032 11.974 11.917 11.860 11.804 11.748 11.719 11.609 11.502 11.462 11.356 11.252 11.150 11.050 10.952 10.856 10.761 10.668 10.576 10.486 10.398 10.311 10.225 10.141 10.058 9.977 9.897 9.818 9.740 9.664 9.589 9.515 9.442 9.370 9.299 9.229 9.161 9.093 9.027 8.961

cm

-1

100439 99939.4 99447.4 98955.4 98471.4 97987.5 97511.7 97043.9 96576.1 96116.3 95656.6 95204.9 94753.3 94519.4 93632.2 92769.2 92446.5 91591.6 90752.8 89930.1 89123.6 88333.1 87558.9 86792.6 86042.5 85300.5 84574.6 83864.9 83163.2 82469.5 81792.0 81122.6 80469.3 79824.1 79186.9 78557.8 77944.8 77339.9 76743.0 76154.3 75573.5 75000.9 74436.3 73887.9 73339.4 72807.1 72274.8

/.zm 0.0996 0.1001 0.1006 0.1011 0.1016 0.1021 0.1026 0.1030 0.1035 0.1040 0.1045 0.1050 0.1055 0.1058 0.1068 0.1078 0.1082 0.1092 0.1102 0.1112 0.1122 0.1132 0.1142 0.1152 O. 1162 O. 1172 0.1182 0.1192 0.1202 O.1213 O.1223 0.1233 0.1243 0.1253 0.1263 0.1273 0.1283 0.1293 0.1303 0.1313 0.1323 0.1333 0.1343 0.1353 0.1364 0.1373 0.1384

n 1.531 1.543 1.554 1.565 1.576 1.585 1.592 1.600 1.611 1.616 1.621 1.627 1.632 1.636 1.648 1.658 1.668 1.702 1.733 1.763 1.800 1.835 1.871 1.903 1.929 1.954 1.983 2.004 2.023 2.044 2.073 2.094 2.113 2.126 2.145 2.160 2.169 2.189 2.204 2.213 2.240 2.257 2.277 2.294 2.307 2.336 2.352

k 1.005 1.007 1.010 1.011 1.013 1.012 1.013 1.016 1.019 1.014 1.021 1.022 1.028 1.031 1.038 1.058 1.070 1.085 1.093 1.102 1.109 1.105 1.103 1.090 1.080 1.071 1.061 1.044 1.035 1.027 1.019 0.998 0.984 0.967 0.959 0.941 0.929 0.926 0.910 0.901 0.901 0.882 0.873 0.856 0.845 0.838 0.815

Yttrium Aluminum Garnet (Y3AIsO12)

975

TABLE I

(Continued)

Yttrium Aluminum Garnet (YAG) eV

cm -~

8.896 8.833 8.770 8.708 8.647 8.586 8.527 8.468 8.410 8.391 8.334 8.279 8.224 8.169 8.116 8.063 8.010 7.959 7.908 7.858 7.808 7.759 7.710 7.663 7.615 7.569 7.523 7.477 7.432 7.388 7.344 7.301 7.258 7.215 7.173 7.132 7.091 7.051 7.011 6.971 6.932 6.893 6.855 6.817 6.780 6.743 6.600

71750.5 71242.4 70734.3 70234.2 69742.2 69250.2 68774.3 68298.5 67830.7 67677.4 67217.7 66774.1 66330.5 65886.9 65459.4 65032.0 64604.5 64193.2 63781.8 63378.5 62975.3 62580.1 62184.9 61805.8 61418.6 61047.6 60676.6 60305.6 59942.6 59587.8 59232.9 58886.1 58539.3 58192.4 57853.7 57523.0 57192.3 56869.7 56547.1 56224.5 55909.9 55595.4 55288.9 54982.4 54684.0 54385.5 53232.2

/xm 0.1394 0.1404 0.1414 0.1424 0.1434 0.1444 0.1454 0.1464 0.1474 0.1478 0.1488 0.1498 0.1508 0.1518 0.1528 0.1538 0.1548 0.1558 0.1568 0.1578 0.1588 0.1598 0.1608 0.1618 0.1628 0.1638 0.1648 0.1658 0.1668 0.1678 0.1688 0.1698 0.1708 0.1718 0.1728 0.1738 0.1748 0.1758 0.1768 0.1779 0.1789 0.1799 0.1809 0.1819 0.1829 0.1839 0.1879

n 2.371 2.399 2.411 2.412 2.440 2.446 2.486 2.522 2.578 2.578 2.583 2.581 2.574 2.577 2.585 2.593 2.598 2.565 2.533 2.497 2.467 2.444 2.433 2.421 2.415 2.412 2.409 2.402 2.396 2.386 2.373 2.356 2.342 2.326 2.313 2.298 2.285 2.275 2.261 2.252 2.241 2.228 2.217 2.208 2.195 2.183

k 0.803 0.785 0.754 0.739 0.734 0.715 0.717 0.687 0.640 0.613 0.567 0.522 0.487 0.460 0.428 0.387 0.331 0.275 0.242 0.219 0.209 0.206 0.202 0.195 0.190 0.180 0.165 0.149 0.133 0.115 0.0974 0.0838 0.0726 0.0630 0.0549 0.0479 0.0431 0.0369 0.0318 0.0271 0.0205 0.0159 0.0129 0.00806 0.00406 0.00204 1.44E-4 [5]

n

k

0.0020 [6] 1.51E-4

(continued)

976

William J. Tropf TABLE I

(Continued)

Yttrium Aluminum Garnet (YAG) eV 6.550 6.500 6.450 6.400 6.250 6.000 5.750 5.500 5.250 5.000 4.750 4.500 4.250 4.000 3.750 3.500 3.250 3.0996 3.0614 2.8568 2.8437 2.7552 2.5511 2.4797 2.11 2.105 2.0664 1.89 1.7712 1.5498 1.3776 1.2399 1.1271 1.0332 0.8856 0.7749 0.6888 0.6199 0.5636 0.5166 0.4769 0.4428 0.4133 0.3875 0.3647 0.3444 0.3263

cm

--1

52828.9 52425.6 52022.3 51619.1 50409.3 48392.9 46376.5 44360.1 42343.8 40327.4 38311.0 36294.7 34278.3 32261.9 30245.6 28229.2 26212.8 25000 24691.4 23041.5 22935.8 22222.2 20576.1 20000 17018.4 16977.9 16666.7 15243.9 14285.7 12500 11111.1 10000 9090.9 8333.3 7142.9 6250.0 5555.6 5000.0 4545.5 4166.7 3846.2 3571.4 3333.3 3125.0 2941.2 2777.8 2631.6

/.~m 0.1893 0.1907 0.1922 0.1937 0.1984 0.2066 0.2156 0.2254 0.2362 0.2480 0.2610 0.2755 0.2917 0.3100 0.3306 0.3542 0.3815 0.4000 0.4050 0.4340 0.4360 0.4500 0.4860 0.5000 0.5876 0.5890 0.6000 0.6560 0.70 0.80 0.90 1.00 1.10 1.20 1.40 1.60 1..80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80

n

2.120

k 5.71E-5 2.26E-5 8.96E-6 3.55E-6

6.25E-5 2.78E-5 1.37E-5 6.01E-6 3.03E-6 2.18E-6 1.84E-6 1.44E-6 9.06E-7 6.85E-7 5.67E-7 4.23E-7

2.088 2.059 2.033 2.009 1.987 1.968 1.950 1.933 1.918 1.905 1.892 1.881 1.871 1.8650 [7] 1.85977 [8] 1.85282 1.85246 1.8532 1.84367 1.8450 1.83260 1.83247 1.8347 1.82785 1.8285 1.8245 1.8222 1.8197 1.8170 1.8152 1.8121 1.8093 1.8065 1.8035 1.8004 1.7970 1.7935 1.7896 1.7855 1.7810 1.7764 1.7713 1.7659

Yttrium Aluminum Garnet (Y3AI5012)

977

TABLE I

(Continued)

Yttrium Aluminum Garnet (YAG) eV 0.3100 0.2914 0.2901 0.2889 0.2876 0.2864 0.2852 0.2839 0.2827 0.2814 0.2802 0.2790 0.2777 0.2765 0.2752 0.2740 0.2728 0.2715 0.2703 0.2690 0.2678 0.2666 0.2653 0.2641 0.2628 0.2616 0.2604 0.2591 0.2579 0.2566 0.2554 0.2542 0.2529 0.2517 0.2505 0.2492 0.2480 0.2467 0.2455 0.2443 0.2430 0.2418 0.2405 0.2393 0.2381 0.2368 0.2356

cm -1 2500.0 2350.0 2340.0 2330.0 2320.0 2310.0 2300.0 2290.0 2280.0 2270.0 2260.0 2250.0 2240.0 2230.0 2220.0 2210.0 2200.0 2190.0 2180.0 2170.0 2160.0 2150.0 2140.0 2130.0 2120.0 2110.0 2100.0 2090.0 2080.0 2070.0 2060.0 2050.0 2040.0 2030.0 2020.0 2010.0 2000.0 1990.0 1980.0 1970.0 1960.0 1950.0 1940.0 1930.0 1920.0 1910.0 1900.0

/.~m

n

k

4.00 4.2553 4.2735 4.2918 4.3103 4.3290 4.3478 4.3668 4.3860 4.4053 4.4248 4.4444 4.4643 4.4843 4.5045 4.5249 4.5455 4.5662 4.5872 4.6083 4.6296 4.6512 4.6729 4.6948 4.7170 4.7393 4.7619 4.7847 4.8077 4.8309 4.8544 4.8780 4.9020 4.9261 4.9505 4.9751 5.0000 5.0251 5.0505 5.0761 5.1020 5.1282 5.1546 5.1813 5.2083 5.2356 5.2632

1.7602 1.76 [20] 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.72 1.72 1.72 1.72 1.72

4.51E-6 [6] 4.97E-6 5.63E-6 6.30E-6 6.99E-6 7.76E-6 8.63E-6 9.55E-6 1.05E-5 1.13E-5 1.23E-5 1.33E-5 1.45 E-5 1.58E-5 1.70E-5 1.83E-5 1.95E-5 2.09E-5 2.23E-5 2.35E-5 2.46E-5 2.56E-5 2.65E-5 2.64E-5 2.59E-5 2.50E-5 2.38E-5 2.28E-5 2.19E-5 2.10E-5 2.01E-5 1.89E-5 1.89E-5 2.02E-5 2.24E-5 2.45E-5 2.64E-5 2.84E-5 3.08E-5 3.30E-5 3.51E-5 3.71E-5 3.91E-5 4.11E-5 4.30E-5 4.49E-5

1.77 [21 ] 1.77 1.77 1.77 1.77 1.77 1.77 1.77 1.77 1.77 1.77 1.77 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.76 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.74 1.74

(continued)

978

William J. Tropf TABLE I

(Continued)

Yttrium Aluminum Garnet (YAG) eV 0.2343 0.2331 0.2319 0.2306 0.2294 0.2281 0.2269 0.2257 0.2244 0.2232 0.2219 0.2207 0.2195 0.2182 0.2170 0.2157 0.2145 0.2133 0.2120 0.2108 O.2095 0.2083 0.2071 0.2058 0.2046 0.2033 0.2021 0.2009 0.1996 0.1984 0.1971 0.1959 0.1947 0.1934 0.1922 0.1909 0.1897 0.1885 0.1872 0.1860 0.1847 0.1835 0.1823 0.1810 0.1798 0.1785 0.1773

cm

-1

1890.0 1880.0 1870.0 1860.0 1850.0 1840.0 1830.0 1820.0 1810.0 1800.0 1790.0 1780.0 1770.0 1760.0 1750.0 1740.0 1730.0 1720.0 1710.0 1700.0 1690.0 1680.0 1670.0 1660.0 1650.0 1640.0 1630.0 1620.0 1610.0 1600.0 1590.0 1580.0 1570.0 1560.0 1550.0 1540.0 1530.0 1520.0 1510.0 1500.0 1490.0 1480.0 1470.0 1460.0 1450.0 1440.0 1430.0

/xm 5.2910 5.3191 5.3476 5.3763 5.4054 5.4348 5.4645 5.4945 5.5249 5.5556 5.5866 5.6180 5.6497 5.6818 5.7143 5.7471 5.7803 5.8140 5.8480 5.8824 5.9172 5.9524 5.9880 6.0241 6.0606 6.0976 6.1350 6.1728 6.2112 6.2500 6.2893 6.3291 6.3694 6.4103 6.4516 6.4935 6.5359 6.5789 6.6225 6.6667 6.7114 6.7568 6.8027 6.8493 6.8966 6.9444 6.9930

n 1.72 1.72 1.72 1.72 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.71 1.70 1.70 1.70 1.70 1.70 1.70 1.70 1.69 1.69 1.69 1.69 1.69 1.68 1.68 1.68 1.68 1.68 1.68 1.67 1.67 1.67 1.67 1.66 1.66 1.66 1.66 1.65 1.65 1.65 1.65 1.64 1.64 1.64 1.64 1.63

k 4.74E-5 5.02E-5 5.29E-5 5.62E-5 5.97E-5 6.30E-5 6.63E-5 6.99E-5 7.38E-5 7.77E-5 8.31E-5 8.85E-5 9.53E-5 1.02E-4 1.10E-4 1.19E-4 1.29E-4 1.40E-4 1.53E-4 1.70E-4 1.79E-4 1.88E-4 2.16E-4 2.55E-4 3.34E-4 4.65E-4 6.05E-4 7.27E-4 8.56E-4 9.51E-4 9.84E-4 1.01E-3 1.18E-3 1.64E-3 2.09E-3 2.54E-3 2.75E-3 2.82E-3 2.87E-3 3.15E-3 3.33E-3 3.49E-3 3.66E-3 3.88E-3 3.97E-3 3.94E-3 4.02E-3

1.74 1.73 1.73 1.73 1.73 1.73 1.73 1.73 1.72 1.72 1.72 1.72 1.72 1.72 1.72 1.71 1.71 1.71 1.71 1.71 1.71 1.70 1.70 1.70 1.70 1.70 1.70 1.69 1.69 1.69 1.69 1.68 1.68 1.68 1.68 1.68 1.67 1.67 1.67 1.66 1.66 1.66 1.66 1.65 1.65 1.65 1.64

Yttrium Aluminum Garnet (Y3AI5012)

979

TABLE I

(Continued)

Yttrium Aluminum Garnet (YAG) eV 0.1761 0.1748 0.1736 0.1723 0.1711 0.1699 0.1686 0.1674 0.1661 0.1649 0.1637 0.1624 0.1612 0.1599 0.1587 0.1575 0.1562 0.1550 0.1537 0.1525 0.1513 0.1500 0.1488 0.1475 0.1463 0.1451 0.1438 0.1426 0.1413 0.1401 0.1389 0.1376 0.1364 0.1351 0.1339 0.1327 0.1314 0.1302 0.1289 0.1277 0.1265 0.1252 0.1240 0.1234 0.1227 0.1221 0.1215

cm-1 1420.0 1410.0 1400.0 1390.0 1380.0 1370.0 1360.0 1350.0 1340.0 1330.0 1320.0 1310.0 1300.0 1290.0 1280.0 1270.0 1260.0 1250.0 1240.0 1230.0 1220.0 1210.0 1200.0 1190.0 1180.0 1170.0 1160.0 1150.0 1140.0 1130.0 1120.0 1110.0 1100.0 1090.0 1080.0 1070.0 1060.0 1050.0 1040.0 1030.0 1020.0 1010.0 1000.0 995.0 990.0 985.0 980.0

/xm 7.0423 7.0922 7.1429 7.1942 7.2464 7.2993 7.3529 7.4074 7.4627 7.5188 7.5758 7.6336 7.6923 7.7519 7.8125 7.8740 7.9365 8.0000 8.0645 8.1301 8.1967 8.2645 8.3333 8.4034 8.4746 8.5470 8.6207 8.6957 8.7719 8.8496 8.9286 9.0090 9.0909 9.1743 9.2593 9.3458 9.4340 9.5238 9.6154 9.7087 9.8039 9.9010 0.000 0.050 0.101 0.152 0.204

n 1.63 1.63 1.62 1.62 1.61 1.61 1.61 1.60 1.60 1.60 1.59 1.59 1.58 1.58 1.57 1.57 1.56 1.56 1.55 1.54 1.54 1.53 1.53 1.52 1.51 1.50 1.50 1.49 1.48 1.47 1.46 1.45 1.44 1.43 1.42 1.41 1.39 1.38 1.36 1.35 1.33 1.31 1.30 1.29 1.28 1.26 1.25

k 3.84E-3 3.83E-3 3.89E-3 3.94E-3 3.88E-3 3.50E-3 2.47E-3 2.18E-3 2.07E- 3 2.10E-3 2.18E-3 2.28E-3 2.72E-3 2.96E-3 3.21E-3 3.58E-3 4.07E-3 4.43E-3 4.49E-3 4.41E-3 4.48E-3 4.74E-3 4.99E-3 5.25E-3 5.51E-3 5.80E-3 6.17E-3 6.51E-3 6.90E-3 7.31E-3 7.69E-3 8.04E-3 8.22E-3 8.02E- 3 7.98E-3 8.18E-3 8.88E-3 9.72E-3 0.0105 0.0116 0.0120 0.0125 0.0142 0.0148 0.0157 0.0161 0.0166

1.64 1.64 1.63 1.63 1.63 1.62 1.62 1.61 1.61 1.60 1.60 1.60 1.59 1.59 1.58 1.57 1.57 1.56 1.56 1.55 1.54 1.54 1.53 1.52 1.52 1.51 1.50 1.49 1.48 1.47 1.46 1.45 1.44 1.43 1.41 1.40 1.39 1.37 1.36 1.34 1.32 1.30 1.28 1.27 1.26 1.24 1.23

0.0122 [20] 0.0127 0.0133 0.0139 0.0146

(continued)

980

William J. Tropf TABLE I

(Continued)

Yttrium Aluminum Garnet (YAG) eV 0.1209 0.1203 0.1196 0.1190 0.1184 0.1178 0.1172 0.1165 0.1159 0.1153 0.1147 0.1141 0.1134 0.1128 0.1122 0.1116 0.1110 0.1103 0.1097 0.1091 0.1085 0.1079 0.1072 O.1066 O. 1060 0.1054 0.1048 0.1041 0.1035 O.1029 0.1023 0.1017 0.1010 O.1004 0.0998 0.0992 0.0986 0.0979 0.0973 0.0967 0.0961 0.0955 0.0948 0.0942 0.0936 0.0930 0.0924

cm

--1

975.0 970.0 965.0 960.0 955.0 950.0 945.0 940.0 935.0 930.0 925.0 920.0 915.0 910.0 9O5.0 900.0 895.0 890.0 885.0 880.0 875.0 870.0 865.0 860.0 855.0 850.0 845.0 840.0 835.0 830.0 825.0 820.0 815.0 810.0 805.0 800.0 795.0 790.0 785.0 780.0 775.0 770.0 765.0 760.0 755.0 750.0 745.0

/xm 10.256 10.309 10.363 10.417 10.471 10.526 10.582 10.638 10.695 10.753 10.811 10.870 10.929 10.989 11.050 11.111 11.173 11.236 11.299 11.364 11.429 11.494 11.561 11.628 11.696 11.765 11.834 11.905 11.976 12.048 12.121 12.195 12.270 12.346 12.422 12.500 12.579 12.658 12.739 12.821 12.903 12.987 13.072 13.158 13.245 13.333 13.423

n 1.24 1.23 1.22 1.20 1.19 1.17 1.16 1.14 1.13 1.11 1.09 1.07 1.05 1.03 1.00 0.97 0.95 0.92 0.88 0.85 0.80 0.76 0.71 0.65 0.58 0.50 0.41 0.32 0.27 0.24 0.23 0.23 0.25 0.27 0.32 0.39 0.51 0.74 1.14 1.47 1.34 1.02 0.72 0.54 0.48 0.49 0.55

k 0.0171 0.0171 0.0172 0.0174 0.0182 0.0187 0.0196 0.0206 0.0216 0.0225 0.0242 0.0259 0.0285 0.0384 0.0427 0.0462 0.0492 0.0528 0.0571 0.0637 0.0938 0.116 0.158 0.216 0.250 0.298 0.353 0.413 0.419 [19] 0.546 0.668 0.788 0.909 1.04 1.17 1.32 1.49 1.65 1.67 1.30 0.864 0.704 0.775 0.984 1.22 1.44 1.66

1.22 1.20 1.19 1.18 1.16 1.14 1.12 1.11 1.09 1.07 1.04 1.02 0.99 0.97 0.94 0.90 0.87 0.83 0.79 0.74 0.68 0.62 0.55 0.46 0.35 0.25 O.20 0.17 0.16 0.16 0.16 0.18 0.20 0.23 0.28 0.37 0.56 0.93 1.50 1.62 1.31 0.94 0.64 0.47 0.41 0.40 0.44

0.0153 0.0160 0.0169 0.0178 0.0188 0.0198 0.0210 0.0223 0.0238 0.0254 0.0272 0.0293 0.0316 0.0343 0.0374 0.O410 0.0453 0.0505 0.0568 0.0648 0.0752 0.0894 0.110 0.143 0.204 0.313 0.446 0.574 0.694 0.810 0.926 1.05 1.17 1.31 1.47 1.66 1.87 2.05 1.85 1.24 0.879 0.808 0.945 1.18 1.43 1.67 1.92

Yttrium Aluminum Garnet (Y3AI5012)

981

TABLE I

(Continued)

Yttrium Aluminum Garnet (YAG) eV 0.0917 0.0911 0.0905 0.0899 0.0893 0.0886 0.0880 0.0874 0.0868 0.0862 0.0855 0.0849 0.0843 0.0837 0.0831 0.0825 0.0818 0.0812 0.0806 0.0800 0.0794 0.0787 0.0781 0.0775 0.0769 0.0763 0.0756 0.0750 0.0744 0.0738 0.0732 0.0725 0.0719 0.0713 0.0707 0.0701 0.0694 0.0688 0.0682 0.0676 0.0670 0.0663 0.0657 0.0651 0.0645 0.0639 0.0632

cm- ~ 740.0 735.0 730.0 725.0 720.0 715.0 710.0 705.0 700.0 695.0 690.0 685.0 680.0 675.0 670.0 665.0 660.0 655.0 650.0 645.0 640.0 635.0 630.0 625.0 620.0 615.0 610.0 605.0 600.0 595.0 590.0 585.0 580.0 575.0 570.0 565.0 560.0 555.0 550.0 545.0 540.0 535.0 530.0 525.0 520.0 515.0 510.0

/xm 13.514 13.605 13.699 13.793 13.889 13.986 14.085 14.184 14.286 14.388 14.493 14.599 14.706 14.815 14.925 15.038 15.152 15.267 15.385 15.504 15.625 15.748 15.873 16.000 16.129 16.260 16.393 16.529 16.667 16.807 16.949 17.094 17.241 17.391 17.544 17.699 17.857 18.018 18.182 18.349 18.519 18.692 18.868 19.048 19.231 19.417 19.608

n 0.67 0.88 1.24 1.73 1.91 1.52 1.10 1.09 1.44 2.04 2.63 2.92 2.92 2.80 2.65 2.50 2.37 2.25 2.13 2.03 1.93 1.84 1.75 1.66 1.57 1.48 1.38 1.27 1.15 1.02 0.87 0.72 0.65 0.79 1.66 1.87 1.40 1.06 0.80 0.62 0.54 0.51 0.54 0.63 0.84 1.03 0.91

k 1.88 2.09 2.24 2.11 1.57 1.24 1.49 1.98 2.39 2.53 2.26 1.75 1.29 0.973 0.762 0.619 0.519 0.448 0.396 0.358 0.329 0.309 0.295 0.286 0.284 0.287 0.297 0.316 0.350 0.409 0.515 0.712 1.030 1.485 1.744 0.739 0.471 0.486 0.613 0.816 1.04 1.26 1.47 1.68 1.81 1.69 1.56

0.53 0.71 1.03 1.60 2.22 2.25 1.77 1.31 1.35 2.60 4.93 4.31 3.65 3.22 2.92 2.70 2.52 2.36 2.23 2.11 2.01 1.90 1.80 1.71 1.61 1.51 1.40 1.28 1.15 1.00 0.81 0.56 0.39 0.39 0.62 1.53 1.46 0.93 0.51 0.32 0.26 0.24 0.26 0.30 0.40 0.62 0.92

2.19 2.48 2.78 2.94 2.58 1.94 1.71 2.11 3.03 4.17 2.70 1.00 0.520 0.337 0.247 0.195 0.162 O.140 0.125 0.114 0.106 0.101 0.0983 0.0974 0.0986 0.102 0.108 O.119 0.136 O. 165 0.224 0.374 0.701 1.09 1.55 1.59 0.678 0.501 0.672 0.984 1.25 1.49 1.71 1.94 2.17 2.38 2.31

(continued)

982

William J. Tropf TABLE I

(Continued)

Yttrium Aluminum Garnet (YAG) eV 0.0626 0.0620 0.0614 0.0608 0.0601 0.0595 0.0589 0.0583 0.0577 0.0570 0.0564 0.0558 0.0552 0.0546 0.0539 0.0533 0.0527 0.0521 0.0515 0.0508 0.0502 0.0496 0.0490 0.0484 0.0477 0.0471 0.0465 0.0459 0.0453 0.0446 0.0440 0.0434 0.0428 0.0422 0.0419 0.0417 0.0414 0.0412 0.0409 0.0407 0.0404 0.0402 0.0399 0.0397 0.0394 0.0392 0.0389

cm

-1

505.0 500.0 495.0 490.0 485.0 480.0 475.0 470.0 465.0 460.0 455.0 450.0 445.0 440.0 435.0 430.0 425.0 420.0 415.0 410.0 405.0 400.0 395.0 390.0 385.0 380.0 375.0 370.0 365.0 360.0 355.0 350.0 345.0 340.0 338.0 336.0 334.0 332.0 330.0 328.0 326.0 324.0 322.0 320.0 318.0 316.0 314.0

~m 19.802 20.000 20.202 20.408 20.619 20.833 21.053 21.277 21.505 21.739 21.978 22.222 22.472 22.727 22.989 23.256 23.529 23.810 24.096 24.390 24.691 25.000 25.316 25.641 25.974 26.316 26.667 27.027 27.397 27.778 28.169 28.571 28.986 29.412 29.586 29.762 29.940 30.120 30.303 30.488 30.675 30.864 31.056 31.250 31.447 31.646 31.847

n 0.74 0.66 0.65 0.68 0.75 0.87 1.07 1.08 1.15 1.46 2.01 2.90 3.59 2.87 2.36 3.02 3.79 3.97 3.68 3.11 2.22 1.47 2.84 3.47 3.90 2.64 2.51 4.98 4.43 3.87 3.42 2.97 2.39 1.59 1.43 1.55 2.05 3.16 4.69 5.40 5.27 4.96 4.66 4.41 4.19 4.01 3.84

k 1.66 1.85 2.05 2.26 2.47 2.67 2.81 2.88 3.23 3.60 3.89 3.87 2.93 2.11 2.77 3.24 2.83 2.05 1.40 1.01 0.988 2.24 3.34 2.27 1.61 1.44 3.34 2.45 1.14 0.770 0.626 0.588 0.698 1.39 2.04 2.81 3.64 4.27 3.96 2.74 1.80 1.27 0.974 0.796 0.681 0.604 0.551

0.81 0.53 0.39 0.36 0.38 0.43 0.55 0.76 1.20 2.32 5.27 6.56 5.22 3.89 3.81 5.84 5.19 4.42 3.81 3.21 2.43 1.60 4.41 3.82 4.12 2.37 6.40 4.90 4.14 3.71 3.36 3.01 2.55 1.75 1.33 1.23 1.56 2.72 5.44 6.16 5.49 4.95 4.57 4.30 4.09 3.91 3.76

2.07 2.19 2.49 2.8I 3.16 3.55 4.02 4.63 5.45 6.55 6.72 3.52 1.93 1.91 3.36 2.51 1.00 0.588 0.454 0.450 0.670 2.12 3.55 1.85 0.841 1.32 3.40 0.393 0.198 0.153 0.147 O. 172 0.260 0.661 1.24 2.17 3.26 4.52 4.39 2.01 0.932 0.535 0.355 0.260 0.204 0.170 0.148

Yttrium Aluminum Garnet (Y3AI5012)

983

TABLE I

(Continued)

Yttrium Aluminum Garnet (YAG) eV 0.0387 0.0384 0.0382 0.0379 0.0377 0.0374 0.0372 0.0369 0.0367 0.0365 0.0362 0.0360 0.0357 0.0355 0.0352 0.0350 0.0347 0.0345 0.0342 0.0340 0.0337 0.0335 0.0332 0.0330 0.0327 0.0325 0.0322 0.0320 0.0317 0.0315 0.0312 0.0310 0.0307 0.0305 0.0303 0.0300 0.0298 0.0295 0.0293 0.0290 0.0288 0.0285 0.0283 0.0280 0.0278 0.0275 0.0273

cm -~ 312.0 310.0 308.0 306.0 304.0 302.0 300.0 298.0 296.0 294.0 292.0 290.0 288.0 286.0 284.0 282.0 280.0 278.0 276.0 274.0 272.0 270.0 268.0 266.0 264.0 262.0 260.0 258.0 256.0 254.0 252.0 250.0 248.0 246.0 244.0 242.0 240.0 238.0 236.0 234.0 232.0 230.0 228.0 226.0 224.0 222.0 220.0

/.zm 32.051 32.258 32.468 32.680 32.895 33.113 33.333 33.557 33.784 34.014 34.247 34.483 34.722 34.965 35.211 35.461 35.714 35.971 36.232 36.496 36.765 37.037 37.313 37.594 37.879 38.168 38.462 38.760 39.063 39.370 39.683 40.000 40.323 40.650 40.984 41.322 41.667 42.017 42.373 42.735 43.103 43.478 43.860 44.248 44.643 45.045 45.455

n 3.68 3.53 3.38 3.21 3.03 2.81 2.54 2.20 1.85 1.84 2.59 4.67 5.80 5.40 4.96 4.64 4.41 4.23 4.09 3.98 3.88 3.79 3.72 3.65 3.59 3.53 3.47 3.42 3.37 3.32 3.27 3.22 3.17 3.12 3.07 3.01 2.94 2.87 2.79 2.70 2.58 2.44 2.23 1.95 1.65 1.74 3.29

k 0.516 0.494 0.485 0.491 0.516 0.574 0.692 0.948 1.52 2.49 3.68 4.04 2.28 1.17 0.738 0.539 0.433 0.369 0.327 0.298 0.277 0.262 0.250 0.240 0.233 0.227 0.222 0.219 0.217 0.215 0.215 0.216 0.218 0.221 0.226 0.233 0.242 0.256 0.275 0.302 0.342 0.409 0.527 0.774 1.36 2.48 3.91

3.62 3.49 3.36 3.22 3.06 2.88 2.64 2.29 1.72 1.25 1.90 5.61 5.95 5.07 4.59 4.30 4.10 3.96 3.84 3.75 3.67 3.60 3.53 3.48 3.42 3.38 3.33 3.28 3.24 3.20 3.15 3.11 3.06 3.02 2.97 2.91 2.85 2.78 2.70 2.60 2.48 2.31 2.06 1.64 1.14 1.46 3.96

0.134 0.126 0.125 0.129 0.141 0.167 0.219 0.338 0.697 1.894 3.754 4.466 1.202 0.463 0.252 0.164 0.120 0.0941 0.0776 0.0665 0.0587 0.0529 0.0485 0.0452 0.0426 0.0407 0.0392 0.0382 0.0375 0.0372 0.0372 0.0377 0.0386 0.0401 0.0423 0.0454 0.0499 0.0563 0.0657 0.0801 0.103 0.145 0.231 0.464 1.30 2.80 4.10

(continued)

984

William J. Tropf TABLE I

(Continued)

Yttrium Aluminum Garnet (YAG) eV 0.0270 0.0268 0.0265 0.0263 0.0260 0.0258 0.0255 0.0253 0.0250 0.0248 0.0245 0.0243 0.0241 0.0238 0.0236 0.0233 0.0231 0.0228 0.0226 0.0223 0.0221 0.0218 0.0216 0.0213 0.0211 0.0208 0.0206 0.0203 0.0201 0.0198 0.0196 0.0193 0.0191 0.0188 0.0186 0.0183 0.0181 0.0179 0.0176 0.0174 0.0171 0.0169 0.0166 0.0164 0.0161 0.0159 0.0156

cm

-1

218.0 216.0 214.0 212.0 210.0 208.0 206.0 204.0 202.0 200.0 198.0 196.0 194.0 192.0 190.0 188.0 186.0 184.0 182.0 180.0 178.0 176.0 174.0 172.0 170.0 168.0 166.0 164.0 162.0 160.0 158.0 156.0 154.0 152.0 150.0 148.0 146.0 144.0 142.0 140.0 138.0 136.0 134.0 132.0 130.0 128.0 126.0

/a,m 45.872 46.296 46.729 47.170 47.619 48.077 48.544 49.020 49.505 50.000 50.505 51.020 51.546 52.083 52.632 53.191 53.763 54.348 54.945 55.556 56.180 56.818 57.471 58.140 58.824 59.524 60.241 60.976 61.728 62.500 63.291 64.103 64.935 65.789 66.667 67.568 68.493 69.444 70.423 71.429 72.464 73.529 74.627 75.758 76.923 78.125 79.365

n 5.65 4.81 4.24 3.91 3.69 3.53 3.40 3.29 3.19 3.10 3.02 2.93 2.84 2.74 2.63 2.51 2.35 2.17 1.95 1.88 2.61 2.78 1.95 1.14 0.92 1.13 1.97 5.40 7.93 6.56 5.71 5.21 4.87 4.64 4.46 4.31 4.20 4.10 4.02 3.95 3.89 3.83 3.78 3.72 3.67 3.60 3.50

k 1.92 0.559 0.267 0.176 0.142 0.129 0.125 0.126 0.131 O. 138 O. 148 O. 160 O.177 0.199 0.230 0.274 0.344 0.466 0.713 1.24 1.68 0.743 0.588 1.20 2.33 3.56 5.20 6.80 2.64 0.887 0.423 0.245 0.159 0.111 0.0822 0.0636 0.0509 0.0421 0.0357 0.0310 0.0275 0.0249 0.0230 0.0220 0.0222 0.0250 0.0376

5.29 4.50 4.05 3.78 3.60 3.46 3.35 3.26 3.18 3.10 3.02 2.95 2.87 2.79 2.69 2.59 2.45 2.27 2.01 1.58 2.33 2.36 1.62 0.74 0.57 0.78 1.87 8.97 7.04 5.74 5.11 4.73 4.47 4.29 4.15 4.04 3.95 3.87 3.80 3.75 3.69 3.65 3.60 3.56 3.51 3.46 3.39

1.35 0.469 0.239 0.149 0.106 0.0817 0.0672 0.0581 0.0523 0.0489 0.0472 0.0470 0.0483 0.0513 0.0568 0.0660 0.0822 O. 114 0.198 0.576 1.49 0.491 0.444 1.14 2.38 3.82 6.22 5.77 0.998 0.387 0.211 0.135 0.0953 0.0718 0.0567 0.0463 0.0389 0.0335 0.0294 0.0262 0.0239 0.0221 0.0210 0.0206 0.0214 0.0247 0.0358

Yttrium Aluminum Garnet (Y3AI5012)

985

TABLE I

(Continued)

Yttrium Aluminum Garnet (YAG) eV

cm

-1

/xm

0.0154 0.0151 0.0149 0.0146 0.0144 0.0141 0.0139 0.0136 0.0134 0.0131 0.0129 0.0126 0.0124

124.0 122.0 1,20.0 118.0 116.0 114.0 112.0 110.0 108.0 106.0 104.0 102.0 100.0

80.645 81.967 83.333 84.746 86.207 87.719 89.286 90.909 92.593 94.340 96.154 98.039 100.00

0.0118 0.0112 0.0105 0.0099 0.0093 0.0087 0.0081 0.0074 0.0068 0.0062 0.0056 0.0050 0.0043 0.0037 0.0031 0.0

95.0 90.0 85.0 80.0 75.0 70.0 65.0 60.0 55.0 50.0 45.0 40.0 35.0 30.0 25.0 0.0

105.26 111.11 117.65 125.00 133.33 142.86 153.85 166.67 181.82 200.00 222.22 250.00 285.71 333.33 400.0

n

k

3.27 4.23 3.92 3.77 3.70 3.65 3.62 3.59 3.57 3.55 3.53 3.51 3.50 3.510 [22] 3.443 3.396 3.362 3.335 3.314 3.298 3.284 3.273 3.263 3.255 3.249 3.243 3.239 3.235 3.232 3.26 [201 3.225 [22]

0.129 1.01 0.0933 0.0412 0.0282 0.0227 0.0197 0.0178 0.0165 0.0155 0.0147 0.0141 0.0135 0.0167 [22] 0.0140 0.0121 0.0108 0.0099 0.00922 0.00838 0.00688 0.00542 0.00437 0.00375 0.00323 0.00252 0.00179 0.00125 0.00060 0.0

3.24 3.18 3.82 3.64 3.56 3.52 3.49 3.46 3.44 3.42 3.41 3.39 3.38

0.0901 0.994 0.123 0.0373 0.0214 0.0157 0.0129 0.0112 0.0100 0.00916 0.00849 0.00794 0.00748

3.18121] 3.26 [23]

0.0

TABLE II Sellmeier Dispersion Equation Constants for Yttrium Aluminum Garnet Term [(Eq. (2)]

Tomiki [2]

Tropf [11]

)t 1 Az A1 A2

0.11227

0.1095 17.825 2.293 3.705

0.1117

0.1110

2.2883

2.2736

0.4-4

0.4-0.9

0.4-0.6

Range (/xm)

2.2831

0.4-1

Wemple [121

Zharikov [13]

986

William J. Tropf TABLE III Thermo-optical Coefficients of Yttrium Aluminum Garnet

Wavelength (txm) 0.4579 0.4765 0.4880 0.4965 0.5017 0.508 0.5146 0.546 0.6328

0.644 1.06 1.0642

dn/dT (10- 6/K)

Temperature (~

Reference

11.9 12.0 11.6 11.4 11.4 10.2 10.6 12.2 11.5 10.4 9.86 7.3 a 9.4 10.0 9.1

30-45 30-45 30-45 30-45 30-45 20-120 30-45 27 23-372 30-45 75-110 27-58 20-120 23-372 30-45

[10] [ 10] [10] [ 10] [ 10] [8] [ 10] [16] [ 13] [ 10] [15] [14] [8] [13] [ 10] . . . .

a Neodynium-doped YAG.

TABLE IV Transverse (and Longitudinal) Optical Phonon Frequencies in Yttrium Aluminum Garnet

Mode

Gledhill [20]

Hoffmeister [21 ]

Flu

122.25(122.65) 163.4(172.3) 177.5(180.2) 219.0(224) 289.7 (296.0 ) 330.2(340) 372.5(378) 387.3(388) 394.5(402.5) 427.8(438.3) 445.5(471.5) 472.3( 510.8) 516.5(549.0) 569.2(585.0) 692.0(712.0) 723(765) 782(840.5)

121.7(122.2) 164.2(173.0) 177.9(179.8) 219.6(224.7) 288.8(295.8) 328.5(338.0) 374.3(380.4) 388.3(391.2) 395.1 (402.4) 430.6(438.6) 451 (547.8'?) -509.7(506.3?) 565( 582.6) 690/3(708.4) 720.8(766.5) 784.9(857.7)

F1. F~,, El. F l,, FI, F1, FI. Fl. F1. F~. F 1. FI. F 1,

Flu F1.

Flu

Hurrell [ 18] 119(125) 165(181) 217.8(227) 289(298) 329(341) 370(379) 392(403) 429(437) 466(471) 482(505) 521 (551) 569.5(591.8 ) 690(707) 737(769) 811 (921)?

Zircon (ZrSi04) EDWARD D, PALIK Institute for Physical Science and Technology and R. KHANNA Department of Chemistry and Biochemistry University of Maryland College Park, Maryland

The interest in zircon is primarily as a gemstone. Many varieties and colors are found all over the world. The colors are due to the impurities, which give line absorption in the visible. We will not discuss these extrinsic effects in any detail. Many of the properties of zircon are summarized by Deer et al. [1]. ZrSiO4 is a tetragonal crystal with space group symmetry I4/amd (D4h). The clear crystal is termed high zircon, but containing impurities, it may have various colors. A somewhat amorphous phase made up of some SiO2 and ZrO2 is called low zircon and bears a green color [2]. The crystal form should have four E, (EIc) and three A2u (E lie) vibration modes in the infrared spectral region. The IR from 200 to 1100 cm - - 1 has been studied by Dawson et al. [3]. Gem-quality crystals from Ceylon were oriented, cut, and mechanically polished. An RIIC Michelson interferometer FS-720 was used at low frequencies, while single-beam and double-beam grating spectrometers (both built in the King's College physics department) were used at higher frequencies. The samples were either at 300 K or 100 K. Wire-grid polarizers were made of gold lines on AgC1 or polyethylene sheets. The reflectivity spectrum was Kramers-Kronig (K-K) analyzed, but no mention was made of including the high-frequency reflectivity above the band gap. No Lorentz-oscillator analysis of reflectivity was made (which is often useful for comparison). The absorption spectrum was also measured for samples of thickness 0.3 to 3 mm with the c-axis in the plane of the samples. The data were presented in small graphs of e' (real dielectric function) and g' (imaginary dielectric function) for polarization II (e) and _L(o) to the c-axis. These graphs were expanded by photocopying and read, and the 300 K data are listed in Table I. There is a significant 987 HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS III

Copyright 9 1998 by Academic Press. All rights of reproduction in any form reserved ISBN 0-12-544423-0/$25.00.

988

Edward D. Palik and R. Khanna

uncertainty in n and k, especially where these parameters are changing rapidly. Although we include n to three figures and k to two figures, the reading uncertainty is surely +20%. There is also an uncertainty in the wave number of probably _+3 cm-~. Four Eu modes are observed. Three A2u modes are seen with a hint of a fourth one. The A2u modes (Ell c) have been measured by Gervais et al. [4]. The sample was a natural crystal, mechanically polished and then annealed at 1300 K. Both a K-K analysis and an oscillator model have been used to fit the reflectivity. The oscillator model gave the complex dielectric function as

AE/(.0. 2 E =

e' -

i g'=

e~

+

. (co2 _ w2 + iFjco)

The parameters at 295 K for four oscillators are given in Table II. Note that group theory predicts only three A2u modes, and a fourth mode has been added at high frequency for the model fit. The dc dielectric constant is calculated to be e'o(e) = 10.71. We have calculated n(e) and k(e) to extend and fill in the gaps of the data of Dawson et al. [3] and also give them in Table I. The magnitudes of n and k are similar for the two sets of data, but do differ by +20% in places if account is taken of the mode frequencies being slightly different. This gives slightly shifted spectra. Dawson et aL [3] measured the transmittance of thin plates at 100 and 300 K with polarized radiation. They displayed absorption spectra with arbitrary but linear ordinate. They also gave the peak absorption coefficient = 4rrk/:t at three points in the spectrum. (We assume these are lowtemperature data; if this is the wrong choice, there are up to 20% differences.) These are (_1_: 887 cm -~, 85000 cm-1; 1900 cm -~, 70 cm-~; 2750 cm -a, ---0 cm-a); (11:994 cm -1, 93000 cm-1; 1900 cm -~, 100 cm-1; 2750 cm-~, 1.4 cm-1). Assuming we can make the comparison of absorption coefficient with arbitrary absorption, we have calculated k in the two- and threephonon absorption region near 1900 and 2800 cm-~, respectively. We did not use a grid fine enough to show all the details of the observed spectral lines. The results are listed in Table I. We could find no near-IR, visible data for n(o) and n(e) obtained by prism minimum-deviation methods. Many books on gems give the indices of refraction at the Na D line (0.5890 /zm) but usually do not quote the source. Clear high zircon has n(o) = 1.925 and n ( e ) = 1.984 in one gem book [5] and is given a range of values n(o) = 1.924-1.934 and n(e) = 1.970-1.977 in another book [1]. However, because of the ubiquitous presence of impurities, which change the color of the gem, these constants can vary drastically; therefore, we should focus on clear high zircon. Vance [6] has measured the visible and near-IR transmittance of various

Zircon (ZrSi04)

989

gem stones of low zircon at 15 and 290 K. The spectra contain many weak, sharp lines attributed to impurity atoms of U and Th. It is believed that high zircon containing U and Th degenerates into low zircon over the years because of the radioactive decay of the impurity atoms. Only the lowtemperature spectra are given in graphs, and they are labeled as o-polarized. The atomic impurities often appear only in o- or rr polarization. We assume this refers to and perpendicular and parallel to the c-axis of the crystal. The c-axis was presumably located in the plane of these gem samples. One ospectrum shows the onset of band-edge absorption. Even though the spectrum was obtained at 15 K, we have smoothed out the impurity structure and show k(o) in Table II, since we can find no detailed study of the absorption edge at room temperature. This is not quite the correct thing to do, since low zircon is an "amorphous" mixture of S i O 2 and ZrO2 (but showing a c-axis). But the band gap of crystalline S i O 2 is ~ 9 eV, while that of crystalline ZrO2 (zirconia) is --~ 3 eV. We would expect a mixture to have a band gap somewhere in between these limits. Deer et al. [1] gives several references to optical absorption studies, but none of these were quantitative regarding k. We could find no data above the band gap into the UV that could yield the optical constants and, therefore, cannot obtain an accurate value of the band gap. The data listed in Table I are plotted in Fig. 1. The ordinary n(o) and k(o), primarily from Dawson et al. [3], are shown in Fig. la; the extraordinary n(e) and k(e), primarily from Dawson et al. [3] are shown in Fig. lb; the extraordinary n(e) and k(e) from Gervais et al. [4] are shown in Fig. l c for comparison. ACKNOWLEDGMENT

We thank James E. Shigley, Director of Research, Gemological Institute of America, 1630 Stewart St., Santa Monica, CA 90404-4088 for kindly providing us with a copy of the zircon section of Deer et al. [1]. REFERENCES

1. W. A. Deer, R. A. Howie, and J. Zussman, "Rock-Forming Minerals," Vol. 1A, 2nd Ed., Orthosilicates, p. 431. Longmans, London, 1982. 2. R. Webster, "Gems," p. 131. Newnes-Butterworth, London, 1975. 3. E Dawson, M. M. Hargreave, and G. R. Wilkinson, The vibrational spectrum of zircon (ZrSiO4). J. Phys. C: Solid State Phys. 4, 240 (1971). 4. E Gervais, B. Piriou, and E Cabannes, Anharmonicity in silicate crystals: temperature dependence of Au type vibrational modes in ZrSiO 4 and LiA1Si206. J. Phys. Chem. Solids 34, 1785 (1973). 5. R. T. Liddicoat, Jr., "Handbook of Gem Identification," p. 272. Gemological Institute of America, Santa Monica, CA, 1969. 6. E. R. Vance, The anomolous optical absorption spectrum of low zircon. Minerol. Mag. 39, 709 (1974).

990

Edward D. Palik and R. Khanna |

101

'

'

'

'

'

I

(a) o

10 o

87~o

AoA

10-1

o A o

Ao

9

o v

o

o ~,,

10-2

o r.

10.3

10

,r

-4

%

10 -s

|

a

|

|

a

=

|

I

I

I

I

I

"

'

1

10

1

WAVELENGTH (gm) '

101

'

'

'

'

I

'

'

'

'

'

I

(b) o

100

~

o

10-1

"oo

~

~

t

o o

v( D

t-

10 .

o

2

10-3 k 10 -4

10 -5

10-6

. . . . . .

J

1

,

,

. . . . . .

I

10

WAVELENGTH (ILtm) Fig. 1. (a) Log-log plot of n(o) ( 0 0 0 ) and k(o) (AAA) versus wavelength in micrometers for zircon, primarily from Dawson et al. [3]. (b) Log-log plot of n(e) (OOO) and k(e) (&kA) versus wavelength in micrometers for zircon, primarily from Dawson et al. [3].

Zircon

(ZrSiO4)

'

101

991

'

'

'

'

'

I

"

|

|

=

a

i

|

a

|

"

"

a= =

-(c) o

100

10-1

v( 1 )

t-

1 0.

2

10-3

1 0 -4

'1

=

!

10-5

10. 6

. . . . . .

I 1

,

,

,

,

,

, , ,I

I

I

10

W A V E L E N G T H (ILtm) Fig. 1, eont'd. (c) Log-log plot of n(e) (0(3(3) and k(e) (AA&) versus wavelength in micrometers for zircon, primarily from Gervais et al. [4].

I

992

Edward D. Palik and R. Khanna

~J

~a

~J

.=.

O

V'b I,th

O

~a

I C'~I C'~ C'a C",I C'~ C',I C'~I Cq

C'~I C'q ,--.~

Zircon (ZrSi04)

993

994

Edward D. Palik and R. Khanna

O

O

I

Zircon

(ZrSi04)

995

c~

_~ = = = = = = = = = = __. _~ ~ ~ d ~2 ~ _~ ~2 _~ _~ ~2 _~ _~ _~ ~2 ~ _~ _~ __: __:

0

0

Lt'~ 0

~r~ 0

~

0

0

~

0

0

0

0

0

0

0

0

0

ddddddddddddddooddddddddddddoddd

0

0

~

0

0

0

0

0

0

0

0

~::~ 0

996

Edward D. Palik and R. Khanna

~ 1 7 6 1 7 6 1 7 6 1~7~6~1~7~6 1 7 6

r--- (:~

9

O

O

::L

I ~j

v'~

~O

P.-- v ~

,~

O~

t-,.-

~

~

~::) ~::~ t ~..

Zircon

(ZrSi04)

997

o = =o= o = =o=o = =o=o = =o=o = =o o o o o ~ ~ ~ o o = = ~ = = = o = ~o o o - - ~ -

~

~ ~ ~ ~

0

o

~

~

oo oo oo

. ~

0

~

~

0

~

~

~

~

0

~

~

0

~

~

~ ,

0

d

o ~

0

~ ~

0

o

~

~

~

~ ~

0

d

o ~

0

0

d

d

~ ~

0

d

0

-

0

d

-

~

0

d

~

~ ~

0

~

0

d

d

d

~ o

0

~ ~

0

d

0

~

~

0

d

~

0

0

d

d

~

0

d

~ "

0

0

d

" ~ ~

0

d

998

Edward D. Palik and R. Khanna

9

~J

O

O

I

~1C',l

~

0

~"q C',l Cq C'q C~l ('~1Cq

0

0

0

~

0

~

O,I C',l s

0

~

0

Zircon

(ZrSi04)

999 TABLE II Oscillator Parameters for A2u Vibrational Modes at 295 K; ~ = 3.8

rj 5.75 0.36 0.78 0.022

(cm - j )

(cm -1)

339 605.7 977 1020

9.5 9.5 12.2 65

Handbook of Optical Constants of Solids, Volumes t It and III SUBJECT INDEX AND CONTRIBUTOR INDEX

Handbook of Optical Constants of Solids, Volumes I, II, and III SUBJECT INDEX AND CONTRIBUTOR INDEX Indexed by

EDWARD J. PRUCHA

FOREWORD TO THE SET IVAN P. KAMINOW

@ Academic Press San Diego London Boston New York Sydney Tokyo Toronto

This book is printed on acid-free paper.

S

Copyright © 1998 by Academic Press All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. ACADEMIC PRESS 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA 1300 Boylston Street, Chestnut Hill, MA 02167, USA http://www.apnet.com Academic Press Limited 24-28 Oval Road, London NWI 7DX, UK http://www.hbuk.co.uk/ap/

Library of Congress Cataloging-in-Publication Data

Library of Congress CIP data pending.

ISBN 0-12-544424-9

Printed in the United States of America 97 98 99 00 01 IC 9 8 7 6

5

4

3

2

1

Contents Foreword Ivan P. Kaminow

vii

Contributor Index

SUbject Index

3

v

Foreword IVAN P. KAMINOW INTRODUCTION This set of five volumes (three volumes edited by Edward D. Palik, and accompanying index, and a volume by Gorachand Ghosh) is a unique resource for any science and technology library. It provides materials researchers and optical device designers with reference facts in a context not available elsewhere. The singular functionality of the set derives from the unique format for the three core volumes that comprise the Handbook of Optical Constants of Solids (HOC). The Handbook satisfies several essential needs: First, it affords the most comprehensive database of the refractive index and extinction (or loss) coefficient of technically important and scientifically interesting materials. This data has been critically selected and evaluated by authorities on each material. Second, the dielectric constant database is supplemented by tutorial chapters covering the basics of dielectric theory and reviews of experimental techniques for each wavelength region and material characteristic. As an additional resource, two of the tutorial chapters summarize the relevant characteristics of each of the materials in the database. The data in the core volumes have been collected and analyzed over a period of 12 years with the most recent completed in 1997. The volumes systematically define the dielectric properties of 143 of the most engaging materials, including metals, semiconductors, and insulators. Together, the three Palik books contain nearly 3000 pages, with about two-thirds devoted to the dielectric-constant data. The tutorial chapters in the remaining onethird of the pages contain a wealth of information, including some dielectric data. Hence, the separate volume, Subject Index and Contributor Index, which is included as part of the set, substantially enhances the utility of the Handbook and in essence, joins all the Palik volumes into one unit. This is of great importance to the users of the set. A final volume rounds out the set. The Handbook of Thermo-Optic Coefficients of Optical Materials with Applications collects refractive-index measurements and their temperature dependence for a large number of crystals and glasses. Mathematical models represent these data, and in tum, are used in the design of nonlinear optical devices. As I outline in more detail below, the community of materials researchers, spectroscopists, and optical device designers has reason to welcome this vii

viii

Foreword

five-volume reference tool. The selected materials include all those of technical importance for the design of classical and photonic components. Every technical library will want this unique resource on its shelves.

HANDBOOK OF OPTICAL CONSTANTS OF SOLIDS The optical constants referred to in the title above are the components of the complex dielectric constant E

= E 1 + iE2 = (n + ik)2.

To be specific, the refractive index (n) and the extinction (or loss) coefficient (k) are tabulated against wavelength. The defined optical range comprises wavelengths from millimeters (10 - 3 m) to angstroms (10 -10m ), a potential seven-order span, although few materials have been measured over the full range. The solids include elemental metals and semiconductors, compound semiconductors, oxides and other insulators, even two liquids, mercury and water. For the most part, the solids are single crystals or polycrystalline thin films. Since thin films are sensitive to deposition conditions, the Handbook values are best taken only as a guide. Ideally, the films actually employed in an application should be measured directly. Glasses, which can vary widely in composition, are not included. An exception is the technologically important and chemically simple silica (Si0 2 ) glass. Since the resonant modes or band structure of the medium define the spectral variation of dielectric constant, the dielectric constant is fundamental to other optical properties as well. In addition, the dielectric behavior is key to any device design. Hence, the constants in this Handbook are of interest to both condensed-matter scientists and optical-device designers in academics and industry. The wide spectral range covered by the databases for each material is essential for designing nonlinear optical interactions. Electro-optic modulation, second-harmonic generation, and four-wave mixing comprise as many as four widely-spaced discrete wavelengths. Edward D. Palik edited the series, Handbook of Optical Constants of Solids. He spent most of his career as a materials researcher and spectroscopist at the Naval Research Laboratory in Washington, D.C. Since his retirement in 1988, he has been associated with the Institute for Physical Science and Technology of the University of Maryland at College Park. The format and goals set for HOC differ from traditional handbooks that tabulate measurements from the literature in narrow ranges of wavelength. Compilers of such tables tend to take the most recent data as most reliable, with the tacit assumption that the ultimate user will eventually check the original sources for reliability. It certainly is a good idea for the user to confirm the original

Foreword

ix

source, especially if accuracy is essential to his purpose. However, many users are not sufficiently expert in the measurement and materials fields to make a valid judgment, and those who are qualified may not have the time or resources to gather and evaluate data from the widespread literature. The plan of HOC is to select experts in the experimental techniques peculiar to each spectral range and class of materials, and have them review those methods. Other experts summarize the relevant theory of dielectrics. Finally, a group of materials experts each survey, or critique, the literature for a particular material, selecting the best data in the widest possible spectral range for the tables.

HOC I The format of the first volume (832 pages) in the set, HOC I, follows this plan with 11 tutorial chapters on basic theory and measurement methods, and 28 critiques covering 37 specific materials. Examples of measurement methods include vacuum ultraviolet reflectivity, ellipsometry, thin-film techniques, and techniques for the millimeter range. The critiques evaluate, combine, and present measurements from diverse sources. They cover 13 metals, including aluminum, platinum, gold, silver, copper, and tungsten; 14 semiconductors, including silicon (both crystalline and amorphous, used in solar cells and imaging tubes), germanium, gallium arsenide and indium phosphide (employed in lasers and photodiodes), and cadmium telluride; and 12 insulators, including diamond, lithium fluoride, lithium niobate (used in electro-optic modulators and harmonic generators), silicon dioxide (both crystalline and glass, used in optical fiber and planar waveguides), silicon nitride, and sodium chloride. The measurements are generally at a nominal "room temperature." The "critiquers" are careful to account for their choices of data and to explain the limitations in precision and accuracy. The tabulated values of nand k may have been measured directly or may have been derived from Kramers-Kronig relations or from Sellmeier models or other representations of the wavelength dispersion in the dielectric material. The measurements are given in both graphical and tabular forms, along with the mathematical representations. Each format has its own uses: The graphs show at a glance where the resonances, or band edges, and loss peaks occur, and where the transmission windows can be found. The tables allow precise numbers to be read off or interpolated. And the mathematical models allow the materials to be represented in computer simulations. The positions and shapes of resonance peaks may also be compared with spectroscopic measurements, such as infrared absorption, and Raman and Brillouin scattering.

x

Foreword

HOC II The second volume (1120 pages), HOC II, continues the original format. The 14 tutorial chapters elucidate "convention confusions" and basics in dielectrics, as well as review the attenuated-total-reflection method, measurements in the ultraviolet and x-ray regions, and calculations of nand k in semiconductors. A tabulation of handy parameters characterizing the materials in HOC I and HOC II is also included among the tutorials. These parameters comprise crystal structure, symmetry class, irreducible representations for acoustic and optic modes plus their resonance frequencies, plasma frequency for metals, band-gap energy for semiconductors, and dc dielectric constant. The 41 critiques cover 13 metals, including the alkalis, chromium, iron, niobium, tantalum, beryllium, graphite, and mercury; 14 semiconductors, including aluminum gallium arsenide (used in lasers and photodiodes), silicon-germanium alloys, lead tin telluride (used in infrared imaging), selenium, tellurium, and ~-silicon carbide; and 19 insulators, including aluminum oxide (a popular coating in optical devices), barium titanate, beryllium oxide (an electrical insulator and good thermal conductor), magnesium fluoride, potassium dihydrogen phosphate plus isomorphs (used as electro-optic switches and harmonic generators in high-power applications), strontium titanate, polyethylene (a common infrared window material and the only polymer in the Handbook), and water. The water tabulation takes the prize for widest spectral range, spanning nine orders from 10- 8 m (100A) to 10m (30MHz).

HOC III A new volume (1024 pages), HOC III, includes materials missing from the earlier volumes and updates measurements included in those volumes. The tutorial chapters also fill some gaps, including discussions of far-infrared measurements by Fourier transform spectroscopy, photothermal techniques to determine the extinction coefficient, the application of Brillouin scattering to determine optical constants in otherwise difficult circumstances, and theory and measurements of doped n-silicon. A chapter on handy optical parameters for the materials in HOC I, II, and III supplements the corresponding tabulation in HOC II. The 39 critiques cover 19 metals, including titanium, indium, tin, and cesium; 8 rare earths, including erbium; 23 semiconductors (loosely defined), including updates for undoped silicon and silicon-germanium alloys, numerous nitrides, bismuth germanium oxide (a popular nonlinear crystal), chalcopyrites like cadmium germanium arsenide and silver gallium selenide, silver halides (used in photographic film), zinc

Foreword

xi

phosphide, and zinc arsenide; and 17 insulators, including an extensive update of aluminum oxide, barium fluoride, strontium fluoride, calcite (a natural birefringent crystal used for waveplates and polarizers), cesium halides, lead fluoride, lithium tantalate (an efficient electro-optic and acousto-optic crystal), potassium niobate (a nonlinear optics crystal), sulfur, thallium halides, yttrium aluminum garnet (the host for Nd:YAG lasers), and zircon (a gemstone).

HANDBOOK OF THERMO-OPTIC COEFFICIENTS OF OPTICAL MATERIALS WITH APPLICATIONS The Handbook of Thermo-Optic Coefficients by Gorachand Ghosh of the Electrotechnical Laboratory of MITI in Tskuba, Japan is an excellent companion to the HOC. The Ghosh tables are a bonus for those in need of additional and/or corroborating (real) refractive index numbers. He has determined Sellmeier coefficients for some 70 technologically interesting crystals, some that are found in HOC, and others found in commercial glasses by fitting to published experimental results. He tabulates the results over a wide range of wavelengths, comparing theory and experiment. In addition, he presents tables of the thermo-optic coefficient (dn/dT) for these materials. Finally, he presents mathematical expressions for n as a function of wavelength and temperature. With these representations, he is able to define design criteria for a number of temperature-dependent nonlinear components. They include phase-matched parametric mixing devices, a thermo-optic switch, and an optical-fiber temperature sensor.

CONCLUSION

The three volumes of HOC distill measurements of the optical constants of some 143 materials, including 43 elements. By self-selection, most of the materials of industrial or academic importance are included. These materials are employed as gemstones; as classical optics components, such as windows, lenses, waveplates, and polarizers; and in photonic devices such as lasers, modulators, and nonlinear media. If a material has useful or exceptional optical properties, and if it can be produced in a robust and pure form, its optical constants will surely be thoroughly measured. Most of them are scrutinized in HOC. Those materials not yet included in HOC only await qualified critiquers willing to devote the effort to analyze them for a chapter to be included in a possible HOC IV. For the convenience of libraries and researchers, all three volumes of HOC

xii

Foreword

are offered as a set. In addition, two valuable supporting volumes are included in the set: the Handbook of Optical Constants of Solids, Volume I, II, and III: Subject Index and Contributor Index and the Handbook of ThermoOptic Coefficients of Optical Materials with Applications. These five books will be an invaluable addition to any researcher's library. April, 1997

IVAN

P.

KAMINOW

Handbook of Optical Constants of Solids/ Volumes t It and III SUBJECT INDEX AND CONTRIBUTOR INDEX

Contributor Index Roman numerals indicate volume number.

A Addamiano, A., 1:597 Alterovitz, Samuel A., 11:705, 837 Amirtharaj, P. M., 11:655 Apell, P., 11:97 Arakawa, E. T., 11:421, 461; 111:341 Artus, Luis, 111:573 Ashok, 1., 11:789, 957 Aspnes, D. E., 1:89 Auslender, Mark, 111:155

B Barth, 1., 11:213 Bauer, G., 1:517, 535 Bezuidenhout, D. F., 11:815 Biaggio, Ivan, 111:821 Birch, 1. R., 11:957 Birken, H. G., 11:279 Blessing, C., 11:279 Bloomer, 1., 11:151 Borghesi, A., 1:445, 525; 11:449, 469 Boukari, Hacene, III: 121 Briggs, Matthew E., 111:99

c Callcott, T. A., 11:421 Cardona, M., 11:213 Chang, Yun-Ching, 11:421 Choyke, W. 1., 1:587 Cotter, T. M., 11:899

D DamHio, Alvaro 1., 111:761 Destro, Marcia A. F., 111:761 Doll, Gary L., 111:425 Downing, Harry D., 111:899

E Edwards, David F., 1:547, 665; 11:501, 597, 805, 1005; llI:473, 489, 531,753 Eldridge, John E., 1:775; 11:853; 111:731, 743 Fink, 1., 11:293 Fonseca, Vicente, 111:777 Forouhi, A. R., 11:151 Franta, Daniel, III: 857 French, R. H., 111:373 Fuller, Kirk A., 111:899

G Geist, Jon, 111:519 Gervais, Fran~ois, 11:761, 1035; 111:777 Glembocki, O. J., 1:503; 11:489, 513 Guizzetti, G., 1:445, 525; 11:449, 513

H Hadni, Armand, 111:553 Hava, Shlomo, III: 155 Holm, R. T., 1:479, 491; 11:21 Huffman, Donald R., 11:921 Humllcek, 1., 11:607; 111:537 Hunderi, 0., 11:97 Hunter, W. R., 1:69, 275, 675; 11:341; 111:233

2

Contributor Index

I Inagaki, T., 11:341, 461; 111:341 Inokuti, Mitio, 1:369

J Jensen, B., 1:169; 11:124 Jezierski, K., 111:595, 609 Johnson, R. L., 11:213

K Khanna, R., 111:871,987 Krenn, H., 1:517, 535 Kunz, C., 11:279

L Lagakos, Nicholas, III: 121 Loughin, S., 111:373 Lukes, F., 11:279 Lynch, David W., 1:189, 275; 11:341; 111:233

M Mandelis, Andreas, 111:59 Maslin, K. A., 111:13 Mirjalili, G., III: 13 Misiewicz, 1., 111:595, 609 Mitra, Shashanka S., 1:213 Moore, W. J., 111:717

N Navratil, K., 11:1021, 1049

o Ohlidal, Ivan, II: 1021, 1049; 111:845, 857 Ohlidal, Miloslav, ITI:845

p Palik, Edward D., 1:3, 409, 429, 479, 587, 597, 675, 695, 703, 775; 11:3, 313, 489, 691, 709, 989; 111:3, 187,403, 507, 871, 987 Parker, T. 1., 111:13 Pelletier, E., 11:57 Pfluger, J., 11:293 Philipp, H. R., 1:665, 719, 749, 765, 771 Piaggi, A., 11:469, 477 Piller, H., 1:503, 571; 11:559, 637, 725 Potter, Roy F., I: 11, 465

Q Querry, Marvin R., II: 1059; 111:899

R Ribarsky, M. W., 1:795 Ribbing, Carl G., 11:875; 111:351 Rife,1. C., 111:445, 459, 637 Roessler, David M., 11:921 Roos, Arne, 11:875; 111:351

s Savvides, N., 11:837 Schmidt, E., 11:607 Scott, Marion L., 11:203 Segelstein, David J., II: 1059 Shamir, Joseph, 1:113 Shiles, E., 1:369 Simonis, George 1., I: 155 Smith, D. Y., 1:35, 369 Smith, F. W., 11:837 Sprokel, G. 1., 11:75 Swalen, 1. D., 11:75

T Takarabe, Kenichi, 11:489, 513 Tatchyn, Roman, 11:247 Tazawa, Masato, 111:553 Temple, P. A., 1:135 Teng, Ye Yung, 11:789 Thomas, Michael E., IT:177, 777,883, 899, 1079; IIT:653, 683, 807, 883 Treacy, D. 1., 1:623, 641 Tropf, William 1., 11:777, 883, 899, 1079; 111:653, 683, 701, 923, 963

v Varaprasad, P. L. H., 11:789

W Ward, L., 11:435, 579, 737; 111:287 White, Richard H., 11:501,597, 805, 1005 Wieliczka, David M., 11:1059 Wong, Chuen, 11:789 Woollam, John A., 11:705, 837

Subject Index Page locator key: Roman numerals indicate volume number. Bold page numbers indicate a table. Italic page numbers indicate a figure or caption.

A a-As 2 S3 • See Arsenic Sulfide, amorphous a-As 2 Se 3 • See Arsenic Selenide, amorphous Abeles method and reflection phase changes, 11:43 and thin films, 11:58 Absolute reflection spectra Lead Fluoride and, 111:762 Absorbing materials scattering in, 111:127-128 Absorptance defined, 11:51 of lamelliform, described, 1:22 laser calorimetry and, 1:135-153, 142 in layered slab, 1:25-26 measurement of, I: 140 optical-properties measurement and, 1:18-19 Ruthenium and, III:254 temperature dependence and, 1:382 wedged-film laser calorimetry and, 1:139 Absorptance measurements Aluminum, 1:204, 206 wedged-film laser calorimetry and, 1:141-143 Absorption Aluminum and, 1:376-378, 381

Aluminum Antimonide and, II:501 Aluminum Nitride and, 111:376 Aluminum Oxide and, II:761 Aluminum Oxynitride and, 11:778 Barium Titanate and, 11:790 Bismuth Germanium Oxide and, 111:404 Bismuth Silicon Oxide and, III:404 Cadmium Germanium Arsenide and, 111:446 Cadmium Selenide and, 11:559 Cadmium Sulphide and, 11:580, 582 Cesium Bromide and, 111:719 Cesium Chloride and, 111:732 Cesium Iodide and, 11:853-854 Copper Gallium Sulfide and, 111:461-462 defined, 11:51 f sum rules and, 1:61-64 index of refraction and, II: 181-185 of infrared radiation by crystal, 1:254 interferometry and, 1:129 Iron Pyrite and, 111:509-510 Lithium Tantalate and, 111:778 Magnesium Aluminum Spinel and, 11:885 Magnesium Fluoride and, 11:900 Mercury Cadmium Telluride and, 11:662 3

4

SUbject Index

Absorption (Continued) multiphonon, 1:232-254 Polyethylene and, 11:959 quantum-mechanical theory of, 11:152 reflectance and, 11:204 reststrahlen region and, 1:219 Rubidium Bromide and, 111:846 Rubidium Iodide and, 111:857 Sodium Fluoride and, II: 1021, 1023 Sodium Nitrate and, 111:872 sum rules and, 1:38 Tellurium and, 11:711 Thallium Bromide and, 111:926 Thallium Chloride and, 111:923 Thallous Halides and, 111:930 Tin Telluride and, 11:725 transparency and, II: 185-199 values of n and k and, 1:69-70 Water and, 11:1060, 1061 Yttrium Aluminum Garnet and, 111:963 Yttrium Oxide and, 11:1080 Zinc Selenide and, 11:737 Zinc Telluride and, 11:737 See also Burstein-Moss effect Absorption bands Calcium Fluoride and, 11:823 insulator transparencies and, 1:47 Orthorhombic Sulfur and, 111:904-905 Potassium Bromide and, 11:990 Absorption coefficients, III: 122 Aluminum and, 1:378 Aluminum Arsenide and, 11:490 Aluminum Gallium Arsenide and,

11:514 Aluminum Nitride and, 111:374,376 Aluminum Oxide and, 11:762; 111:654,

655, 656, 657 amorphous Silicon and, II: 168 Barium Fluoride and, 111:684, 685, 686 Barium Titanate and, 11:791 Beryllium and, 11:422 Bismuth Germanium Oxide and,

111:404, 406 Bismuth Silicon Oxide and, 111:404 Cadmium Germanium Arsenide and,

111:446-447 Cadmium Selenide and, 11:560 Cadmium Sulphide and, 11:580, 584

Calcium Carbonate and, 111:701 Calcium Fluoride and, II: 817, 820, 821 Cerium and, 111:292 Cesium Chloride and, 111:733 Cesium Fluoride and, III:? 44 Cesium Iodide and, 11:856 Chromium and, 1:135 Cobalt and, 11:437 Copper Gallium Sulfide and,

111:460-461 determination of, 1:163, 229-230 Drude theory and, I: 175 electron-hole interaction and, 1:202 equation for, II: 186 and extinction coefficient, II: 154 in far-infrared region, II: 188 frequency dependence of, 1:244 Gallium Antimonide and, 11:598 Gallium Telluride and, 111:490-491 index-of-refraction measurements and,

1:156 Indium and, 111:262 Iron Pyrite and, 111:508 Kramers-Kronig analysis and,

1:229 lattice disorder and, II: 119 Lead Fluoride and, 111:763, 764 Lead Tin Telluride and, 11:637 Lithium Tantalate and, 111:778-779 Magnesium Fluoride and, 11:900, 902 Magnesium Oxide and, 11:920 -922,

926-928, 934, 936 Manganese and, 111:249 Mercury Cadmium Telluride and,

11:656, 659, 660 for multilayer samples, 11:110-111 for multiphonon events, 1:235 one-electron model and, I: 192 optical conductivity and, I: 173 Orthorhombic Sulfur and, 111:900-901,

904 in polar crystalline insulators, II:178 polar crystals and, 1:214 Polyethylene and, 11:960, 961-962,

965 Potassium Bromide and, 11:990 Potassium Iodide and, III: 807-809 Potassium Niobate and, 111:822

5

Subject Index

rare-earth metals and, 111:290-291 Samarium and, 111:293 Silicon, doped n-type and, Ill: 157 Silicon and, 1:209 Silicon-Germanium alloys and, 11:608;

111:538 Silver Chloride and, 111:554 Silver Gallium Selenide and, 111:575 Silver Iodide and, 111:555-556 spectroscopic measurement of,

111:59-94 Strontium Fluoride and, 111:883, 886 Tellurium and, 11:711, 712 Thallium Chloride and, 111:924 Thallous Halides and, 111:928, 930 Thorium Fluoride and, II: 1051 Tin and, 111:269 Tin Telluride and, 11:726 Titanium and, 111:240 in transparent regime, 1:230-231 Urbach rule and, explained, 11:152 Vanadium and, 11:478 Water and, II: 1059 wavelength dependence and, II: 127 wedged-film laser calorimetry and,

1:140 Yttrium Aluminum Garnet and, 111:964,

965-966 Yttrium Oxide and, 11:1083 Zinc Arsenide and, 111:595 Zinc Germanium Phosphide and,

111:638 Zinc Phosphide and, 111:610-611 Zinc Selenide and, 11:738, 741-742 Zinc Telluride and, 11:744-746 Zircon and, 111:988 Absorption constants introduced, I: 8 Absorption curves Magnesium Fluoride and, 11:900 Absorption edges Cadmium Sulphide and, 11:581, 584 Dysprosium and, 111:298 Lead Fluoride and, 111:762 rare-earth metals and, 111:290 Samarium and, 111:293 Terbium and, 111:297 Thallous Halides and, lll:928

Thulium and, 111:301 Zinc Selenide and, 11:742 Absorption measurements of crystal quartz, I: 167 for metallic Aluminum, 1:369 optical-properties measurement and,

1:375 Potassium Niobate and, 111:824 Silver Gallium Selenide and, 111:574 Silver Gallium Sulfide and, 111:576 Ytterbium and, 111:302 Absorption mechanisms optical-properties measurement and,

1:17-18 Absorption peaks Aluminum Oxide and, 111:654 Cadmium Sulphide and, 11:580, 586 Cesium and, 111:341 Cesium Chloride and, 111:732 Erbium and, 111:300 Gadolinium and, 111:294, 296 rare-earth metals and, 111:290 Rubidium Iodide and, 111:861 Terbium and, 111:297 Ytterbium and, 111:302 Absorption processes above fundamental band gap, 1:4 sum rules for, 1:36, 45 Absorption spectra Cadmium Selenide and, 11:560 for metallic Aluminum, 1:388 of superlattices, II: 102-105 superlattices and, II: 107 Zircon and, 111:987, 988 Absorptive loss measurement of, 1:135-153 Absorptivity defined, 11:51 Palladium and, 11:470 Accuracy of measurements index of refraction and, I: 155-156 Accuracy versus precision introduced, 1:5-6 a-C:H. See Carbon films (Diamond-like) Acoustic damping Bismuth Germanium Oxide and,

111:403 Bismuth Silicon Oxide and, 111:403

6

Subject Index

Acoustic devices low-loss material for, 111:963 Acoustic modes Iron Pyrite and, 111:507 Potassium Niobate and, III: 824 Acoustic-phonon energy Barium Fluoride and, 111:684 Potassium Iodide and, 111:808 Strontium Fluoride and, 111:884 Acousto-optical methods and measurement of absorption in thin layers, 11:70 Actinides, 111:289 ADP. See Ammonium Dihydrogen Phosphate Adsorption process Calcium Fluoride and, 11:815 one-phonon, 1:214 Ag. See Silver AgBr. See Silver Bromide AgCI. See Silver Chloride a-Ge:H. See Germanium, amorphous hydrogenated AgGaS 2 • See Silver Gallium Sulfide AgGaSe2 • See Silver Gallium Selenide AgI. See Silver Iodide Ahrenkel algorithm Kramers-Kronig analysis and, I: 109 AIOTER method. See Angles of incidence, total-extemal-reflectance Airy formula, function Fizeau method and, I: 117 Airy function interferometry and, 111:128, 131 AI. See Aluminum; Metallic Aluminum AlAs. See Aluminum Arsenide AIGaAs. See Aluminum Gallium Arsenide Alkali halides absorption and, 1:251 absorption versus photon energy for,

1:243 excitons in, 1:208 impurities in, 1:257 index of refraction and, II: 180 optical-phonon spectra of, 1:777 reflection measurements of, 111:20- 31 vibrational modes in, 1:255-257

Alkali metals, 111:303

critique, 11:342-344 Cesium and, 111:341 Lithium and, 111:342 Allotropes, 111:899-900 See also Orthorhombic Sulfur AI 2Mg04 • See Aluminum Magnesium Oxide AIN. See Aluminum Nitride A1 2 0 3 . See Aluminum Oxide ALON. See Aluminum Oxynitride a-So See Orthorhombic Sulfur AISb. See Aluminum Antimonide Alumina, 111:653 See also Aluminum Oxide Aluminum, metallic. See Metallic Aluminum Aluminum (AI)

critique, 1:369-383 absorption and, 1:63 chemical etching of, I: 104 complex index of refraction for, 1:390 dispersion analysis of, 1:60 evaporated layer measurement of, 1:80,

80-81, 83-84

finite energy sum rule and, 1:50 interband absorption and, 1:204, 206 interdiffusion with Gold, 1:86 optical constants of, 1:5 optical parameters determination of,

11:204, 208-210 optical parameters for, 11:318 oxygen adsorption by, 1:83-85 reflectance in UHV and, 11:206-207, 207 thin films, reflectivity of, 11:81 unbacked film measurement of,

1:82-85 values of nand k for, 1:74, 388,

395-401; 11:211 See also Metallic Aluminum Aluminum Antimonide (AISb)

critique, 11:501-505 absorption process and, 1:238, 239 dispersion analysis of, 11:502-504 optical parameters for, 1:221-222;

11:320 transparency frequency of, 1:231

7

Subject Index

two-phonon combination bands in,

critique, 11:777-780

1:240 values of n and k for, 11:507, 508-511

hemispherical emissivity for, 11:787 lattice-vibration parameters of, 11:782 optical parameters for (Alz30z7Ns),

Aluminum Arsenide (AlAs)

critique, 11:489-492 values of nand k for, 11:493, 494-499 Aluminum Gallium Arsenide (AIGaAs) superlattices and, II: 102, 118 Aluminum Gallium Arsenide (AIGaAs) alloy optical parameters for, 11:323 Aluminum Gallium Arsenide (AlxGal_~s)

critique, 11:513-515 index of refraction and, 11:138-147,

143, 144, 148 values of n and k for, 11:517, 518-558 Aluminum Magnesium Oxide (AlzMg0 4 ) optical parameters for, 11:320, 324;

III:198 See also Magnesium Aluminum Spinel Aluminum Nitride (AIN)

critique, 111:373-377 values of n and k for, 111:380, 381-401 Aluminum Oxide (AIO z). See Sapphire Aluminum Oxide (Alz0 3)

critique, 11:761-763; 111:653-657 absorption coefficient of, 1:253 absorption coefficients of, 111:681 far-infrared absorption and, 1:251, 252 index of refraction and, II:187 LO phonon frequencies for, 111:682 multiphonon absorption and, 11:193 optical parameters for, 11:322, 329; 111:204 TO phonon frequencies for, 111:682 photothermal deflection and,

111:107-109 in polycrystalline samples, 1:374-375 Sellmeier dispersion equations for,

III:681 static dielectric constant of, II:186 superlattices and, II: 118 temperature dependence and, II:189 thermo-optical coefficients of, 111:681 values of nand k for, 11:765, 766-775;

111:660-661, 662-680 Aluminum Oxynitride (ALON) Spinel

11:321; III:198 values of n and k for, II: 781, 783-787 Alumma 995, 1:160 AlxGal_~s. See Aluminum Gallium Arsenide Ammonium Dihydrogen Phosphate (ADP) (NH4 HzP04 )

critique, II: 1005-1008 Hertzberger-type dispersion formula coefficients for, 11:1018-1019 optical parameters for, 11:322, 328;

111:213 values of n and k for, 11:1010-1011,

1014-1015, 1017 Amorphous dielectrics Kramers-Kronig analysis and,

11:166-169, 301 optical-constants calculation and, II: 163 quantum-mechanical absorption theory and, 11:153 Amorphous materials absorption coefficients and, 1:251 Cadmium Germanium Arsenide and,

111:445 effective-medium theory and, 1:105 infrared absorption and, 1:259-262 interferometry and, 1:132 Orthorhombic Sulfur and, 111:900, 903 photothermal deflection and,

III: 109-110, 112-113 Silicon Dioxide, 111:987, 989 Thorium Fluoride and, II: 1049 Zirconium Oxide and, III:987., 989 See also Individual elements Amorphous semiconductors Kramers-Kronig analysis and,

11:166-169 optical-constants calculation and, II: 163 quantum-mechanical absorption theory and, 11:153 Urbach tail in, 1:202 Amorphous superlattices versus crystalline superlattices, II: 110 optical properties of, II: 109-112

8

Subject Index

Ampere's law expanded, and optical constants, I: 17 Maxwell's extension of, I: 11-12 optical-properties measurement and,

1:17 Analysis chambers reflectometers and, 11:205 Analytic functions sum rules and, 1:42, 51-52 Analyzers ellipsometry and, 1:95 Anderson localization in random potential, II: 120 Angle evaporation SiO x experiments with, 11:92 Angles of incidence Aluminum Gallium Arsenide and,

11:515 Aluminum Nitride and, 111:375 attenuated total reflection and, 11:76 Bismuth Germanium Oxide and,

111:406 Bismuth Silicon Oxide and, 111:406 Cadmium Selenide and, 11:560 Calcium Fluoride and, II: 817 Cesium and, 111:342 Cobalt and, 11:438 correction term determination and,

1:101 Cubic Silicon Carbide and, 11:705 defeating laser coherence effects and,

1:137-138 and electric-field vectors, 11:46 and evanescent waves, 11:79 formulas for, 1:30-33 Fresnel coefficients and, 1:24 Gallium Telluride and, 111:490 Graphite and, 11:450-451 interferometry and, I: 116 isoreflectance curves and, 1:74 in lamelliform, 1:18, 20, 22, 24, 30-33 Lead Fluoride and, 111:763 Liquid Mercury and, 11:462 Magnesium Oxide and, 11:921 measurement of, 11:62 optical-properties measurement and,

1:18 overlayers and, 1:99

Palladium and, 11:472 Polyethylene and, 11:958, 966 Potassium and, 11:366 Potassium Bromide and, 11:992 and prism experiments, 11:81 reflectance and, 1:71, 75, 76, 85 Ruthenium and, 111:254 Selenium and, 11:692 in Silicon Dioxide superlattice, II: 117 Silicon-Oxide reflectance and, 1:375 Sodium Nitrate and, III: 871, 873 in superlattice systems, 11:99, 119, 121 total-external-reflectance (AIOTER) method, 11:206 values of nand k and, 1:75, 76, 79;

11:203-212 Vanadium and, 11:478 Water and, 11:1061 Zinc Selenide and, 11:741 See also Critical angle of incidence; Brewster's angle; Pseudo-Brewster angle See also Individual critiques Angles of refraction discussed, 11:203-204 Angles of rotation Bismuth Germanium Oxide and,

111:403 Bismuth Silicon Oxide and, 111:403 Anions and cations Silver Gallium Selenide and, 111:573 Silver Gallium Sulfide and, 111:573 Anisotropic dyes attenuated-total-reflection experiments with, 11:89 Anisotropic films attenuated-total-reflection experiments and, 11:88 Anisotropy Aluminum Nitride and, 111:375 Aluminum Oxide and, 11:761; 111:653,

655 Antimony and, 111:274 Beta-Gallium Oxide and, 111:754 Boron Nitride and, 111:425 calculation of, II: 70 Chromium and, 11:374 index determination and, 11:65-67

Subject Index Indium and, 111:261 Iron and, 11:385 Lithium Tantalate and, 111:777 loss of, I: 193 Magnesium and, 111:235 Manganese and, 111:249 metals and, 111:234-235 Polyethylene and, 11:958 Potassium Niobate and, 111:822 rare-earth elements and, 111:287 Rhenium and, 111:278 Ruthenium and, 111:253 in superlattice systems, 11:99, 105, 107 Tin and, 111:268 Titanium and, 111:240 Ytterbium and, 111:302 Zinc Arsenide and, 111:596 Zinc Phosphide and, 111:610 Anisotropy measurement of Tantalum Pentoxide, 11:70 of Titanium Dioxide, 11:70 of Zinc Sulfide, 11:70 Annihilation operators absorption and, 1:234 Anomalous skin effect discussed, 1:276, 277-279 Antibonding states photon energies and, II: 153, 156 Antiferromagnetism Chromium and, 11:374 Dysprosium and, 111:299 Erbium and, 111:300-301 rare-earth elements and, III:288 Antimony (Sb) critique, 111:274-275 values of n and k for, 111:275, 276-278 Antireflection coatings birefringence and, I: 165 film on a substrate samples of, 1:5 Antiresonances Barium Titanate and, 11:793 APL model absorption coefficient and, II:189 Arachidic acid attenuated-total-reflection experiments and, 11:88 ARL model EMX-SM electron-probe microanalyzer

9 Aluminum Gallium Arsenide and, 11:515 Ar-0 2 plasma saline oxidation and, 11:90 Arsenic Selenide, amorphous (a-As 2Se3) optical parameters for, 11:323, 330 Arsenic Selenide, crystalline (As 2Se3) optical properties of, II: 10 values of nand k for, 1:626, 627-632 Arsenic Selenide (As 2 Se3) critique, 1:623-625 interferometry and, 1:132 optical parameters for, 11:323 values of nand k for, 1:626, 627-639 Arsenic Sulfide, amorphous (a-As 2 S3) optical parameters for, 11:323 Arsenic Sulfide, crystalline (As 2 S3) values of n and k for, 1:645, 647-656 Arsenic Sulfide (As 2 S3) critique, 1:641-644 dispersion analysis of, 1:643 optical parameters for, 11:322; 111:219 spectroscopic measurements of, 111:82-84,83-84 values of n and k for, 1:645-646, 647-663 a-See See Selenium, amorphous Ashcroft-Sturm model of interband spectrum, 1:370, 381, 383 a-Si. See Silicon, amorphous a-Si:H. See Silicon, hydrogenated a-SiH4 • See Silicon, hydrogenated amorphous a-Si 3N4 . See Silicon Nitride, amorphous a-SiO. See Silicon Oxide, amorphous a-Si0 2 . See Silicon Dioxide, amorphous As 2 S 3. See Arsenic Sulfide; Arsenic Sulfide, crystalline; Vitreous Arsenic Sulfide As 2 Se3. See Arsenic Selenide; Arsenic Selenide, crystalline; Vitreous Arsenic Selenide Athermal materials Barium Fluoride and, 111:684 a-ThF4' See Thorium Fluoride, amorphous a-Ti0 2 • See Titanium Dioxide, amorphous Atomic absorption edges Aluminum Arsenide and, 11:489

10

SUbject Index

Atomic absorption edges (Continued) Aluminum Gallium Arsenide and, 11:514 Chromium and, 11:375 Hafnium Nitride and, 111:357 Potassium and, 11:366 Zirconium Nitride and, 111:357 Atomic arrangement Barium Fluoride and, lll:683 Strontium Fluoride and, III: 883 Atomic coordinates Aluminum Oxide, 111:653 Atomic energy levels and f sum rules, 1:63 Atomic motions determination of, 1:218 Atomic-scale measurements electric-dipole excitations and, I: 16-18 macroscopic optical constants and, 1:16-18 Atomic-scattering coefficients Bromine and, 111:719 Cesium Bromide and, 111:717, 719 Cesium Chloride and, 111:733 Cesium Fluoride and, lll:744 Polyethylene and, 11:958 Atomic-scattering factors. See Henke UV and X-ray optical constants Atomic systems resonance excitations in, II: 155 Atomic-vibration region Lithium Tanatalate and, Ill:779 Atoms optical absorption spectrum of, I: 190 ATR. See Attenuated total reflection (ATR) Attenuated total reflection (ATR) absorption and, 1:32-33 defined, 11:75-76 formulas for, 1:32-33 and Gold films, 11:90 on liquid crystal cells, 11:92 optical analysis of, 11:81-84 Palladium and, 11:472 physical optics of, 11:76 Sodium and, 11:356 for thick Silicon Oxide films on Gold,

11:91

Attenuation in dielectric response function, 1:221 index of refraction and, I: 158 interferometry and, I: 163 Au. See Gold Auger-electron-spectroscopy Palladium and, 11:469 Autocollimation technique Silicon and, 1:547 Avogadro's constant Beta-Gallium Oxide and, Ill:754

B B. See Boron Backscattering, III: 126, 128 external, 111:140-141, 141 BaF2 • See Barium Fluoride Balanced-bridge techniques versus Mach-Zehnder interferometer, 1:163 Ballantyne's permittivity Barium Titanate and, 11:792-793 Balzer BA-500 chamber Lead Fluoride and, Ill:765 Band-edge-shift calorimetry absorption-coefficient measurement and, 1:231 Lithium Tantalate and, 111:778 Band-gap absorption Aluminum Nitride and, 111:374-375 Barium Fluoride and, 111:683-684 critical points and, I: 196 Cuprous Oxide and, II: 875 Drude theory and, I: 173-174 impurities and, 1:257 introduced, 1:4 Iron Pyrite and, 111:509 Potassium Iodide and, 111:807 Thallous Halides and, 111:928 Band-gap (edge) absorption Aluminum Antimonide and, 11:503 Aluminum Arsenide and, 11:490 Aluminum Gallium Arsenide and, 11:513 Aluminum Nitride and, 111:374-375 Aluminum Oxynitride and, 11:778 Cadmium Germanium Arsenide and, 111:446-447

Subject Index Cadmium Selenide and, ll:559 Cadmium Telluride and, 1:410 Calcium Fluoride and, II:816 Copper Gallium Sulfide and, 111:460-462 Cubic Silicon Carbide and, ll:706 direct interband transitions and, ll:128 Gallium Antimonide and, ll:599 Gallium Arsenide and, 1:431 Gallium Phosphide and, 1:447 Germanium and, 1:467 Indium Antimonide and, 1:493 Iron Pyrite and, lll:510 Lead Selenide and, 1:518 Lead Telluride and, 1:537 Lead Tin Telluride and, 11:637 Lithium Fluoride and, 1:676 Lithium Niobate and, 1:695 Magnesium Aluminum spinel and, 11:884 Magnesium Oxide and, 11:920 Mercury Cadmium Telluride and, 11:655, 659-663, 665 Orthorhombic Sulfur and, 111:904 Potassium Bromide and, 11:991-992 Potassium Chloride and, 1:704 Potassium Niobate and, 111:821-822 quantum-density-matrix formulation and, II: 132, 138 Silicon and, 1:551 Silicon Carbide and, 1:587 Silicon Dioxide (type u, Crystalline) and, 1:721 Silicon-Germanium alloys and, 11:607, 609; Ill:538 Silicon layer thickness and, 11:111 Sodium Chloride and, 1:778 Strontium Fluoride and, 111:883-884 superlattices and, 11:98, 110-111 Tellurium and, 11:709, 711-712 Thallium Bromide and, 111:924, 925 Thallium Chloride and, 111:923 Titanium Dioxide and, II: 169 Zinc Germanium Phosphide and, 111:637, 638-639 Zinc Sulfide and, 1:598 Zircon and, 111:988-989

11 Band-gap energy Barium Fluoride and, 111:684 versus photon energy, II: 127 Potassium Iodide and, 111:807-808 quantum-density-matrix formulation and, 11:147 Silver Gallium Sulfide and, 111:576 Urbach tail and, II: 179 Band gaps Bismuth Germanium Oxide and, 111:404 Bismuth Silicon Oxide and, lll:404 Calcium Fluoride and, 11:816 Copper Gallium Sulfide and, 111:459 Magnesium Fluoride and, 11:900 Polyethylene and, 11:958 Thallium Iodide and, lll:927 Yttrium Oxide and, 11:1079 Zinc Germanium Phosphide and, 111:639-640 Bands of attenuation Orthorhombic Sulfur and, lll:901 Band structures electron-hole interaction and, 1:199, 201 interband transitions and, 1:372 Lithium Tantalate ano, 111:777 methods of studying, ll:725 one-electron model and, I: 190 in polar crystalline insulators, 11:178 Tellurium and, 11:709 Zinc Germanium Phosphide and, 111:637 Band-to-band transitions finite-energy sum rules and, 1:53 Barium (Ba), Ill:302 Barium Fluoride (BaF2 ) critique, 111:683-686 multiphonon absorption and, ll:195 oscillator-model parameters and, 111:699 Sellmeier-model parameters for, 111:698 thermo-optical coefficients of, 111:699 transparency frequency of, 1:231 values of n and k for, lll:688, 689-698 Barium Titanate (BaTi0 3 ) critique, 11:789-793 damping constants for, ll:796

12 Barium Titanate (Continued) optical parameters for, 11:322, 327; 111:215 values of mode frequencies for, 11:796 values of n and k for, 11:795, 797-803 Barriers parallel light and, II: 107 perpendicular light and, II: 107 BaTi03 • See Barium Titanate Be. See Beryllium Beam splitters interferometry and, 1:123, 127, 129-130, 130, 164 Mylar, 1:156 polarizing interferometers and, I: 157 See also Interferometers; Kosters prisms Beckman spectrometer Zinc Phosphide and, 111:610 Beckman spectrophotometer Zinc Phosphide and, 111:610 Beckman UV 5270 interferometer Lead Fluoride and, 111:766 Beer-Lambert law attenuation and, 1:221 BeF2 • See Beryllium Fluoride Bell-jar evaporation technique Lead Fluoride and, 111:765 BeO. See Beryllium Oxide Berreman effect in metal-insulator superlattice, II: 118 thin-film study and, 11:115-116 Berreman modes Gallium Arsenide and, II: 118 Berreman thickness described, II: 116 Beryllium (Be) critique, 11:421-425 Henke model and, 11:432-433 interband absorption and, 1:204 optical parameters for, 11:321 summary of experimental information for, 11:427 values of n and k for, 11:427, 428-431 Beryllium Fluoride (BeF2 ) absorption coefficient of, 1:245, 247 Beryllium Oxide (BeO), 1:160 critique, 11:805-807

Subject Index

dispersion parameters for, 11:808 optical parameters for, 11:321 in samples, 11:422 values of n and k for, 11:808, 809-814 See also Ceramic Beryllium Oxide Beta-Gallium Oxide (f3-Ga2 0 3 ) critique, 111:753-754 values of n and k for, 111:756, 757-760 f3-magnesia. See Magnesium Oxide f3-Sic. See Cubic Silicon Carbide BGO. See Bismuth Germanium Oxide Biaxial crystals Gallium Telluride and, 111:490 Bi 12Ge02o ' See Bismuth Germanium Oxide Bimetallic junction interface plasmon frequency and, II: 114 Biological material dielectric properties of, I: 165 Birefringence Aluminum Nitride and, 111:374-375 Aluminum Oxide and, 111:655-656 Barium Titanate and, 11:791 Beta-Gallium Oxide and, Ill:753 Cadmium Germanium Arsenide and, Ill:445 Cadmium Selenide and, 11:561 Calcium Carbonate and, 111:701 Copper Gallium Sulfide and, 111:459 Magnesium Fluoride and, 11:899 near-millimeter wavelength and, 1:164-165, 167 Orthorhombic Sulfur and, Ill:899, 901, 903 Polyethylene and, 11:967 Selenium and, 11:691 Silver Gallium Sulfide and, 111:577 Sodium Nitrate and, III: 871 Thorium Fluoride and, II: 1050 Zinc Germanium Phosphide and, 111:637-638 Zinc Phosphide and, Ill:611 Bi 12Si0 2o ' See Bismuth Silicon Oxide Bismuth Germanium Oxide (Bi 12GeO zo ) critique, 111:403-408 optical parameters for, 111:200 values of nand k for, 111:410, 416-423

13

SUbject Index Bismuth Silicon Oxide (Bi 12Si0 2o)

critique, 111:403-408 values of n and k for, 111:409,

411-416 BK7 Schott glass substrate Lead Fluoride and, 111:764 Blackbody sources overmoded nonresident cavities and,

1:165 Black Oxide. See Cupric Oxide Bloch functions electron-hole interactions and, I: 199 free-carrier absorption and, I: 173 one-electron model and, I: 190, 193 quantum-density-matrix formulation and, 11:129-130 BN. See Boron Nitride Bohr radius in bulk material, 11:97 Boltzmann constant absorption coefficients and, II: 187 Aluminum Oxide and, 111:654 Barium Fluoride and, 111:684 Potassium Iodide and, III: 808 Silicon-Germanium alloys and, 11:608 Strontium Fluoride and, 111:884 Thallous Halides and, 111:928 Yttrium Aluminum Garnet and, 111:964 Boltzmann equation discussed, 1:278 in free-carrier absorption experiment,

1:173 for free-electron gas, II: 121 frequency-independent electron scattering and, II: 127 photon energies and, I: 171 scattering and, 111:159-160 See also Drude theory Bonding-dependent redistribution Zinc Germanium Phosphide and,

111:638 Bonding states photon energies and, II: 153, 156 Polyethylene and, 11:957 Born approximation, III: 160 optical-properties determination and,

11:297 Born-Mayer functions, 1:244, 245

Boron (B) absorption spectrum from, 1:258 Boron Nitride (BN)

critique, 111:425-429 values of n and k for, 111:430 values of nand k for (cubic),

111:431-437 values of n and k for (hexagonal),

111:438-443 Bose-Einstein expression Aluminum Nitride and, 111:374 Boules Lithium Tantalate and, 111:778 Boundary conditions Brillouin scattering and, III: 145 laser calorimetry and, I: 144 Bound electrons electron-hole interaction and, 1:200 quantum-mechanical treatment of, I: 17 rare-earth metals and, 111:289 semiclassical quantum-mechanical treatment of, I: 17 Bragg equation Brillouin scattering and, III: 126 layer thickness and, II: 101 Raman scattering and, 11:108-109 in wave guides, II: 86 Bravais cell ionic mass and, 1:216 Bremsstrahlung radiation Magnesium Oxide and, 11:920 Brewster's angle ellipsometry and, I: 101 and phase changes, 11:46, 47, 48, 50 reflectance and, 1:71, 76 Tellurium and, 11:712 Bridgeman technique Mercury Cadmium Telluride and,

11:659 Sodium Nitrate and, 111:873 Zinc Selenide and, 11:741 Brillouin scattering optical-constants determination and, III: 121-152 Brillouin zone, 1:371-372 Boron Nitride and, 111:426 Cadmium Selenide and, 11:563 Cadmium Sulphide and, 11:581

14 Brillouin zone (Continued) Cesium Iodide and, 11:856 Cobalt and, 11:439 Copper and, 1:205 index of refraction and, II: 106 interband absorption and, I: 191, 207 lead salts and, 1:535 Lead Telluride and, 1:535 Lead Tin Telluride and, 11:637 liquid Aluminum and, 1:376-377 Molybdenum and, 1:205 one-electron model and, I: 190 optical-property measurement and, 11:152 phonons and, 1:238, 253 point-by-point analyses, discussed, 11:152 pressure considerations and, 1:382-383 Raman scattering and, 11:108-109 rare-earth metals and, 111:290 semiconductors and, 111:32 Silicon and, IIT:531-532 Silver Gallium Selenide and, 111:573 Silver Gallium Sulfide and, IIT:573 superlattices and, II: 102 Tin Telluride and, 11:725 Zinc Selenide and, 11:739 Zinc Telluride and, 11:745 Bromellite. See Beryllium Oxide Bromine (Br) atomic scattering coefficients and, IIT:719 Bruggeman effective-medium approximation ellipsometry and, 1:97, 106-107 Bruggeman formula Thorium Fluoride and, II: 1050 BSO. See Bismuth Silicon Oxide Bulk absorption coefficient materials measurement of, 1:135, 139-141, 143, 153 Bulk modulus transparency and, 1:248-249 Burstein-Moss effect Indium Antimonide and, 1:492 Indium Arsenide and, 1:479 Lead Sulfide and, 1:527

Subject Index

Mercury Cadmium Telluride and, 11:663 quantum-density-matrix formulation and, 11:148 Silicon and, 1:551

c C. See Carbon; Cubic Carbon; Graphite CaC0 3 . See Calcium Carbonate Cadmium Arachidate (Cd-arachidate) index calculations of, 11:88 optical constants of, 11:89 Cadmium (Cd) interband absorption and, 1:204 Cadmium Germanium Arsenide (CdGeAs 2 ) critique, 111:445-448 values of n and k for, 111:450, 451-458 Cadmium Selenide (CdSe) critique, 11:559-563 index of refraction and, IT: 148 optical parameters for, 11:321 spectroscopic measurements of, 111:82 values of nand k for, 11:564, 565-578 Cadmium Selenide Sulphide (CdSexS 1_x) index of refraction and, II: 148 Cadmium Sulphide, cubic (CdS), 11:585 Cadmium Sulphide, hexagonal (CdS), 11:579-582 values of n and k for, 11:591-594 Cadmium Sulphide (CdS) critique, 11:579-586 corrected information for, 111:6, 7 index of refraction and, II: 148 optical parameters for, 11:321 spectroscopic measurements of, III:81, 81-82 values of nand k for, 11:589, 590-597 Cadmium Telluride spectroscopic measurements of, 111:73 Cadmium Telluride (CdTe) critique, 1:409-413; 111:33-34 dispersion analysis of, 1:410-412 free-carrier absorption and, I: 171 index of refraction and, II: 148 optical parameters for, 11:320 reststrahlen region and, 1:249, 250 Urbach rule and, 1:202

Subject Index

values of n and k for, 1:414, 415-427; 11:656, 657; 111:34, 35 Cadmium Zinc Sulfide (Cd~nl_xTe) index of refraction and, II: 148 CaF2 • See Calcium Fluoride Cahan-Spanier design photometric ellipsometers and, 1:93-94 Calcite, 111:657 See also Calcium Carbonate Calcite crystal polarization introduced, 11:27 Calcium (Ca) interband absorption and, 1:207 Calcium Carbonate (CaC0 3 ), 111:701 critique, 111:701-702 LO phonon frequencies for, 111:715 TO phonon frequencies for, 111:715 Sellmeier dispersion equation constants for, 111:714 Sodium Nitrate and, 111:871 thermo-optical constants of, 111:714 values of nand k for, III: 704 -705, 706-713 Calcium Fluoride (CaF2 ) critique, 11:815-825 absorption coefficient of, 1:245, 247 dispersion equation constants for, 11:829 multiphonon absorption and, 11:196 optical parameters for, 11:319 substrate absorptance and, 1:142, 143 transparency frequency of, 1:231 values of n and k for, 11:828, 830-835 Calcspar. See Calcium Carbonate Calorimetry, I: 135 absorption-coefficient measurement and, 1:231, 232 Cobalt and, 11:440 Gallium Arsenide and, 1:430 Iridium and, 1:296 Lead Fluoride and, 111:764 Molybdenum and, 1:303 Nickel and, 1:314 Palladium and, 11:471 Potassium Chloride and, 1:704 Rhodium and, 1:342 Tungsten and, 1:357 Vanadium and, 11:477-478 See also Laser calorimetry

15 Capacitance bridges Barium Fluoride and, 111:686 Potassium Iodide and, III: 810 Strontium Fluoride and, III:886 Capacitance techniques Thallium Bromide and, 111:927 Thallium Chloride and, 111:924 Yttrium Aluminum Garnet and, 111:965 Carbon (C) optical parameters for, 11:320 superlattices and, II: 108 See also Cubic Carbon (Diamond) Carbon Films (Arc-evaporated) critique, 11:837-840 values of n and k for, 11:841-843, 852 Carbon films (Diamond-like) critique, 11:837-840 versus Diamond films, 11:837 dispersion relation parameters for, 11:843 values of n and k for, 11:841-843, 844-851 Carbon Films (Diamond-like), amorphous (a-C:H) optical parameters for, 11:323 Carbon films (Diamond-like), amorphous (a-C:H) photothermal deflection and, III: 109 Carbon Tetrachloride (CCI4) correction term determination of, I: 101 Carcinotron system Barium Titanate and, 11:792 Carl-Zeiss GDM-1000 monochromator Zinc Arsenide and, 111:596 Zinc Phosphide and, 111:610 Carrier concentrations Aluminum Gallium Arsenide and, 11:515 Drude theory and, I: 175 index of refraction and, II: 148 infrared dispersion and, 1:264 Lead Tin Telluride and, 11:638-639 optical-properties determination and, 11:177-178 quantum-density-matrix formulation and, 11:134, 146 Tin Telluride and, 11:726

16

Subject Index

Cartesian coordinate system and polarization, 11:28 for wave equations, 11:23 Casting techniques Polyethylene and, 11:966 Cauchy dispersion equations electron-energy-Ioss spectroscopy and,

11:299 Lead Fluoride and, 111:765 and wavelength measurement, 11:65 Cauchy's principle-value integral index of refraction and, II: 161 Cauchy theorem integration finite-energy sum rules and, 1:45 Causality principle, 1:228 described, 11:152, 160 Cavity degradation index of refraction and, I: 163 Cavity-resonance technique Selenium and, 11:691 Cavity technique Tantalum and, 11:409 CCI4 . See Carbon Tetrachloride Cd. See Cadmium Cd-arachidate. See Cadmium Arachidate CdGeAs 2 . See Cadmium Germanium Arsenide CdS. See Cadmium Sulphide CdSe. See Cadmium Selenide CdTe. See Cadmium Telluride Ceo See Cerium Central-cell corrections electron-hole interactions and, I: 199 Central-limit theorem multiphonon absorption and, II: 190,

196 Ceramic beryllia. See Beryllium Oxide Ceramic Beryllium Oxide values of nand k for, 11:811 See also Beryllium Oxide Ceramic materials Aluminum Nitride and, 111:373 Hafnium Nitride and, 111:351 photothermal deflection and, III: 112 Zirconium Nitride and, 111:351 Cerenkov radiation phase velocity of light and, 11:298

Cerium (Ce), 111:287

critique, 111:292-293 values of n and k for, 111:306, 314-315 Cesium Bromide (CsBr) critique, III: 717-719 transparency frequency of, 1:231 values of nand k for, 111:720, 721-730 Cesium Chloride (CsCI)

critique, 111:731-733 interband absorption and, 1:208 Thallium Iodide and, 111:927 values of n and k for, 111:734, 735-741 Cesium (Cs) critique, III: 341-344 interband absorption and, 1:208 values of n and k for, 111:345, 346-350 Cesium Fluoride (CsF)

critique, 111:743-744 values of nand k for, 111:745, 746-751 Cesium Iodide (CsI)

critique, 11:853-857 optical parameters for, 11:320 transparency frequency of, 1:231 values of nand k for, 11:858, 859-874 Cgs units formula development and, 1:215 optical-properties measurement and,

1:13 Chalcogens Gallium Selenide and, 111:473 Gallium Telluride and, 111:489 Chalcopyrite structures Cadmium Germanium Arsenide and,

111:445 Copper Gallium Sulfide and, 111:459 Silver Gallium Selenide and, 111:573 Silver Gallium Sulfide and, 111:573 Zinc Germanium Phosphide and,

111:637 Chamber technique Lead Fluoride and, 111:765-766 Potassium Bromide and, 11:991 Channel-spectrum analysis Beta-Gallium Oxide and, 111:754 Cadmium Sulphide and, 11:585 Cubic Carbon and, 1:666 Lead Fluoride and, 111:765 Polyethylene and, 11:964

Subject Index

Silicon and, 1:547-549 Silicon Dioxide (crystalline) and, 1:720 Titanium Dioxide and, 1:795 Characteristic matrix described, 11:82 parallel-sided slabs and, I: 15 Chemical etching on semiconductor surfaces, I: 104 Chemical polishing solutions CP-4 (HN0 3 :HF:CH3 COOH 5:3:3), 1:104 See also Polishing techniques Chemical transport method Beta-Gallium Oxide and, lll:753 Chemical-vapor-deposition technique (CVD) Cadmium Telluride and, 1:412 Zinc Selenide and, 11:737 Zinc Telluride and, 11:737 Chemical-vapor-transport technique Iron Pyrite and, 111:510 C 13 H 3F. See Methyl Fluoride (CH 2 )n. See Polyethylene (C 2 H4 )n. See Polyethylene Chromatic abberation Calcium Fluoride and, II:818 Chromium (Cr) critique, 11:374-376 absorption measurement of, and calorimetry, I: 135 optical parameters for, 11:319 values of nand k for, 11:377, 378-385 Circuit theory and temporal phenomena, 11:24 Circular polarization, 11:34 discussed, 11:30-37 momentum of, 11:35 spatial representation of, 11:33 terminology used to classify, 11:35-37 Clamped dielectric permittivity values Potassium Niobate and, III:824 Clausius-Mossotti equation, III: 114 Beta-Gallium Oxide and, 111:754 Clebsch-Gordon coefficients critical points and, 1:233 Closed resonant cavity near-millimeter wavelength measurement and, 1:161

17 Co. See Cobalt Coated materials wedged-film laser calorimetry and, 1:140-141 Coating-materials characterizations equations for, 1:138-139 Cobalt (Co) critique, 11:435-440 values of nand k for, 11:422, 443-448 Coherent combinations thin-film absorbtance and, 1:147 Collective excitations optical properties and, II: 112-118 Collimated monochromatic light measurement of polarization states of, 11:62 Color centers Calcium Fluoride and, 11:816 Color standard Nitrides and, 111:351 Columnar structures Thorium Fluoride and, II: 1049-1050 Common-path interferometers described, I: 120 Compensators ellipsometry and, 1:93-95 Complex amplitude reflectance interferometry and, I: 115 Complex dielectric constants Aluminum Oxide and, 111:653-654 Antimony and, 111:274 Beryllium Oxide and, 11:806 Cadmium Selenide and, 11:560 Calcium Fluoride and, 11:820 in compound semiconductors, 11:125-126 Drude model for, I: 169 ellipsometry and, 1:91 Fresnel coefficients and, 1:24 Gallium Selenide and, 111:474-476 Indium and, 111:261 Magnesium and, 111:235 metals and, 111:233 quantum-density-matrix formulation of, 11:128-138 Rhenium and, 111:278 Ruthenium and, 111:253

18

SUbject Index

Complex dielectric constants (Continued) scattering rates and, I: 176 of semiconductor model, I: 169-170 Silicon and, 111:531 sum rules and, 1:36-37 superconvergence sum rules and, 1:38 Thallium Bromide and, 111:926 Tin and, 111:268 Titanium and, 111:240 Yttrium Aluminum Garnet and, 111:963 Complex dielectric response function Polyethylene and, 11:958 Complex field coefficients electromagnetic waves and, I: 89 Complex functions optical-properties measurement and,

1:13 Complex index of refraction, III: 122 Aluminum Nitride and, 111:376 Aluminum Oxide and, 111:654 Cadmium Selenide and, 11:560 Calcium Carbonate and, 111:702 Cesium Chloride and, 111:733 Cesium Fluoride and, 111:744 derivation of, 11:160 Drude-Zener theory and, I: 169-170 electronic transitions and, II: 177 Gallium Selenide and, 111:473-476 Gallium Telluride and, 111:490 Hafnium Nitride and, 111:353 Lead Fluoride and, 111:764 measurement of, 1:155-167 metals and, 111:233 one-electron model and, I: 192 overlayers and, I: 100 reflectance and, 11:204 Rhenium and, 111:278 Silicon and, 111:533 Strontium Titanate and, II: 1035 sum rules and, 1:37 Thallium Bromide and, 111:926 Thallium Chloride and, 111:925 Yttrium Oxide and, 11:1083-1084 Zirconium Nitride and, 111:353 See also Index of refraction, temperature dependence Complex numbers discussed, 11:25

Complex reflectance ratio electromagnetic waves and, 1:90 ellipsometry and, 1:91 Complex refractive index dielectric constants and, II: 125-126,

127 See also Index of refraction Computer modeling infrared reflectivity and, 11:762 laser calorimetry and, 1:136-137 Lead Fluoride and, 111:765 Mercury Cadmium Telluride and,

11:660 reflectometers and, 11:470 steepest-descent method and, 1:221 Computer software accuracy of, 11:26 and complex numbers, 11:26 equation fit and, II: 164, 166, 170 Henke UV and X-ray optical constants,

111:8 optical constants, III: 12 optical-properties measurement and,

11:62-63 Optimatr, 111:6-7 rare-earth metals and, 111:288 and thin films, 11:58 values of nand k and, 11:207-208 Condensation Thorium Fluoride and, II: 1049 Condensed materials electronic states of, I: 11 vibrational states of, I: 11 Condensed matter optical-properties measurement and,

1:17 Condensed-state physics constitutive equations and, 1:24 intrinsic microscopic parameters of,

1:24 Conduction absorption f sum rules and, 1:62 Conduction bands, III: 160 Cadmium Sulphide and, 11:579-580,

582 Cesium Iodide and, 11:856 Iron Pyrite and, 111:509 Magnesium Fluoride and, 11:900

19

Subject Index

Mercury Cadmium Telluride and,

11:662 photon energies and, II: 153, 156 quantum-density-matrix formulation and, 11:130, 132-134, 137 Thallium Bromide and, 111:925 Thallium Chloride and, 111:924 Yttrium Aluminum Garnet and, 111:963 Yttrium Oxide and, 11:1080 Zinc Germanium Phosphide and,

111:637 Zinc Selenide and, 11:738 See also Valence bands Conduction electrons sum rules and, 1:50-51, 57 Conductivity Aluminum Oxynitride and, 11:779 Cadmium Selenide and, 11:561 Cesium and, 111:342 discussed, 11:23 Erbium and, 111:300-301 Gadolinium and, 111:294-295 Kubo formula for, III: 160 Lead Fluoride and, 111:761 rare-earth metals and, 111:290 Terbium and, 111:297 Conductivity effective mass described, 111:159 Conductors Beta-Gallium Oxide and, 111:753 Lead Fluoride and, 111:762 sum rules and, 1:41 Confinement effect superlattices and, II: 104 Consistency importance of, 11:52 Continued-fraction approach superlattice optical properties and,

II: 112 Contrast Brillouin scattering and, III: 129 Conventions discussed, 11:51-52 Convergence sum rules and, 1:42 See also Superconvergence Conversion factors Potassium Niobate and, 111:824

Coordinate systems choice of, II: 38 and incident waves, 11:46 and reflected waves, 11:46 Copper Bromide (CuBr) transparency frequency of, 1:231 Copper Chloride (CuCI) transparency frequency of, 1:231 Copper (Cu)

critique, 1:280 band structure of, 1:206 interband transitions in, 1:204, 205 optical parameters for, 11:318 optical-properties calculations of, 1:205 photothermal deflection and, III: 109 temperature dependence of reflectance,

1:382 values of n and k for, 1:282, 283-286;

11:173, 174, 174 Copper Gallium Sulfide (CuGaS 2 )

critique, 111:459-462 values of nand k for, 111:463, 464-471 Copper Oxides. See Cupric Oxide; Cuprous Oxide Core-electron excitation interband absorption and, 1:201 Core-level absorption process, 1:4 metallic Aluminum and, 1:370 Coming 7059 amorphous Silicon and, 1:573 Coming 7940 (fused silica), 1:159, 160 Coming 9606 (glass ceramic), 1:160 Coming 7971 (titanium silicate), 1:159, 160 Correction terms ellipsometry and, I: 101 local-field effects and, 1:203 Potassium Niobate and, 111:823 Corundum, 111:653 See also Aluminum Oxide Coulomb guage, I: 172 quantum-density-matrix formulation and, 11:129 Coulombic field Lithium Tantalate and, 111:779 Coulomb interactions sum rules and, 1:50

20

Subject Index

Coulomb potential Silicon, doped n-type and, III: 168 Coulomb states interband absorption and, 1:200, 201 Coulomb wave functions, III: 160 Coupling rare-earth elements and, 111:288 Covalent bonding Hafnium Nitride and, 111:351 Zinc Germanium Phosphide and, 111:637 Zirconium Nitride and, 111:351 Cr. See Chromium Critical angle defined, 11:204 Critical-angle measurements Polyethylene and, 11:958 Potassium and, 11:366 values of nand k for, 1:79-81 Critical angle of incidence Aluminum and, 1:80 formulas for, 1:30-33 in lamelliform, 1:30-33 See also Angles of incidence Critical points identification of, 1:207 introduced, 1:194, 196 in phonon branches, 1:233, 253 in two-dimensional crystals, 1:200 Critical-point states photon energies and, II: 153, 156 Critical-point transition energy for Silicon, II: 169 Cryptocrystalline materials Orthorhombic Sulfur and, 111:900, 903 Crystal lattices Bloch functions in, II: 129-130 Chromium and, 11:374 Crystalline dielectrics Kramers-Kronig analysis and,

11:169-171 optical-constants calculation and, II: 163 quantum-mechanical absorption theory and, 11:153 Crystalline materials Aluminum Oxide and, 111:653-654 Beta-Gallium Oxide and, 111:753 Cadmium Germanium Arsenide and,

111:445

Cesium Bromide and, 111:719 Copper Gallium Sulfide and, 111:459 frequency-distribution function and,

1:215 Gallium Phosphide and, 1:448 Gallium Selenide and, 111:473 index of refraction of, I: 102, 167;

11:164 Lead Fluoride and, 111:761 Polyethylene and, 11:957, 960 Silicon-Germanium, 11:607 Zircon and, 111:989 See also Individual elements Crystalline quartz near-millimeter wavelength and, 1:165-167, 166 Crystalline semiconductors Kramers-Kronig analysis and,

11:169-171 optical-constants calculation and, II: 163 quantum-mechanical absorption theory and, 11:153 Crystalline semiconductor superlattices optical properties of, II: 109-112 Crystalline solids Brillouin zone and, 1:191 Crystalline superlattices versus amorphous superlattices, II: 110 Crystallinity Aluminum and, 1:376-377 Boron Nitride and, 111:425 deposited films and, 1:8 Crystals described, 1:215 impurities and, 1:257-258 with substitutional impurities,

1:255-259 Crystals, optical interactions in one-phonon processes and, 1:215 Cs. See Cesium CsBr. See Cesium Bromide CsCl. See Cesium Chloride CsF. See Cesium Fluoride CsI. See Cesium Iodide Cu. See Copper Cubic Carbon (Diamond) critique, 1:665-668 critical points and, I: 196, 208, 233

SUbject Index dispersion analysis of, 1:666-667 index of refraction of, 1:382-383, 393 transparency frequency of, 1:231 values of nand k for, 1:669, 670-673;

11:169 Cubic crystals Barium Fluoride and, 111:683 Bismuth Germanium Oxide and,

111:403 Bismuth Silicon Oxide and, 111:403 Boron Nitride and, 111:425, 426-428 Cesium Fluoride and, 111:743 Cobalt and, 11:435, 439 interband absorption and, I: 192 local-field effects and, 1:203 Potassium Iodide and, III: 807 Rubidium Bromide and, 111:845 Rubidium Iodide and, 111:857 Silver Bromide and, 111:554 Silver Chloride and, 111:553 Silver Iodide and, 111:556 Strontium Fluoride and, 111:883 Taylor series and, II: 136 Thallium Iodide and, 111:927 unpolarized radiation and, II: 130-132 Yttrium Oxide and, 11:1079 Zinc Telluride and, 11:745 Cubic lattices Calcium Fluoride and, 11:815 Cubic materials Cerium and, 111:288 Cesium and, 111:341 rare-earth elements and, 111:287 Tin and, 111:268 Ytterbium and, 111:302 Cubic perovsite structure Lithium Tantalate and, 111:777 Cubic semiconductors Iron Pyrite and, 111:507 Cubic Silicon Carbide (f3-SiC)

critique, 11:705-706 index of refraction and, 11:707 optical parameters for, 11:320 Cubic Silicon Dioxide (Si0 2) optical-constants calculation of,

11:169 Cubic Thallium(l) Halides

critique, 111:923-931

21 Cubic Thallium Iodide (TlI) values of n and k for, 111:951-955 See also Thallium Iodide Cubic Zinc Selenide (ZnSe) optical parameters for, 11:320 Cubic Zinc Sulfide (ZnS)

critique, 1:598-600 values of n and k for, 1:605, 606-616 See also Hexagonal Zinc Sulfide; Zinc Sulfide CuBr. See Copper Bromide CuC!. See Copper Chloride CuGaS 2 • See Copper Gallium Sulfide CuO. See Cupric Oxide Cu20. See Cuprous Oxide Cupric Oxide (CuO)

critique, 11:875-878 optical parameters for, 11:323, 331 spectroscopic measurements of,

111:82 values of nand k for, 11:879, 882 Cuprous Oxide (Cu2 0)

critique, 11:875-878 optical parameters for, 11:321 values of n and k for, 11:879, 880-881 Curie temperature Ammonium Dihydrogen Phosphate and, 11:1007 Barium Titanate and, 11:789, 793 Erbium and, 111:300 Lithium Tantalate and, 111:777 Potassium Dihydrogen Phosphate and,

11:1007 rare-earth elements and, 111:288 Strontium Titanate and, II: 1035 Terbium and, 111:297 Thulium and, 111:301 CVD. See Chemical vapor deposition Cyclo-octasulfur, 111:899 See also Orthorhombic Sulfur Cyclotron resonance, 11:36, 36 Cylindrical lens attenuated-total-reflection experiments with, 11:93 Czochralski technique Aluminum Oxide and, 111:653, 655-656 Bismuth Germanium Oxide and,

111:404, 406

22

SUbject Index

Czochralski technique (Continued) Bismuth Silicon Oxide and, 111:404, 406 Gallium Phosphide and, 1:445, 448 Lead Tin Telluride and, 11:637 Lithium Niobate and, 1:696 Lithium Tantalate and, 111:777-778 Magnesium Aluminum spinel and,

11:884

D Damped harmonic oscillators Gallium Selenide and, 111:474, 476 Silicon and, 111:532 Damped oscillators absorption coefficients and, 1:244-245 dielectric response function and, 1:223 reflection spectra of, 1:218, 229 reststrahlen spectrum of, 1:221, 223 and surface-plasmon waves, 11:79 Damped phonons, 1:222 Damping Aluminum Oxide and, 11:763 Beryllium Oxide and, 11:805 Cadmium Germanium Arsenide and,

111:448 Cesium Iodide and, 11:854 Lithium Tantalate and, 111:779 metals and, 111:233 Potassium Niobate and, 111:824 Silicon, doped n-type and, III: 169 Damping coefficients Lithium Tantalate and, 111:779 Damping constants acoustic phonons and, 111:122 Aluminum Gallium Arsenide and,

11:515 Barium Titanate and, 11:792 Boron Nitride and, 111:426 Calcium Fluoride and, 11:820 Sodium Nitrate and, 111:872-873 Tellurium and, 11:710 Damping factors dielectric response function and, 1:220 infrared spectra and, 1:224 oscillator models and, 1:245 pseudoharmonic approximations and,

1:226 in superlattices, II: 113, 118

Damping in time, 11:24 Damping parameters Iron Pyrite and, 111:508 Zinc Phosphide and, 111:610 Dc conductivity index of refraction and, II: 180 Silver Iodide and, 111:556 Dc conductivity rule, 1:40, 41, 61 Dc conductivity sum rule tests for, 1:61 Dc mobility as function of carrier concentration,

1:171,173,185 DeBell method laser calorimetry and, I: 143, 145 Debye frequency for polyatomic crystals, 1:249 Debye relaxation effects frequency level and, I: 156 Debye temperature of Beryllium, 11:421 Debye-Thomas-Fermi model Silicon, doped n-type and, 111:170-171 Decimal place reliability introduced, 1:5 Defect absorptions incident radiation intensity and, II: 178 Magnesium Aluminum spinel and,

11:887 Smakula's equation and, 1:47-49 spectral region and, II: 178 temperature considerations and, II: 178 Yttrium Oxide and, 11:1081, 1084 Defect analysis of Cuprous Oxide, II: 875 Magnesium Aluminum Spinel and,

11:884 Potassium Dihydrogen Phosphate and,

11:1006 Sodium Fluoride and, II: 1022 See also Error analysis Defects Potassium Iodide and, 111:809 Rubidium Iodide and, 111:859 Strontium Fluoride and, 111:886 Zinc Arsenide and, 111:595 Zinc Germanium Phosphide and,

111:637

23

Subject Index

Defects and disorders infrared absorption by, 1:254-262 Defects and impurities transparency and, 1:213-214 Degenerate n-type material quantum-density-matrix formulation and, 11:135 Degenerate p-type material quantum-density-matrix formulation and, 11:133-134 Degenerate semiconductors exclusion principle and, II: 133 Degrees of freedom of crystals, 1:215 Gallium Telluride and, 111:491 Silicon-Germanium alloys and, 11:607 de Haas-van Alphen measurements,

1:371 Density considerations, III: 122 Aluminum Gallium Arsenide and,

11:513 Barium Fluoride and, 111:683 Lead Fluoride and, 111:763 Yttrium Aluminum Garnet and, 111:963 Density of states as normal distribution function, 1:245 superlattices and, II: 108 Zirconium Nitride and, 111:352 See also Joint density of states Deoxyribonucleic acid. See DNA Deposition rates Lead Fluoride and, 111:765 wedged-film laser calorimetry and,

1:141 DESYaccelerator. See Deutsches Electronen-Synchrotron Deuterated crystals. See Ammonium Dihydrogen Phosphate; Potassium Dihydrogen Phosphate Deutsches Electronen-Synchrotron, 11:761 Potassium Bromide and, 11:992 Vanadium and, 11:478 DFTS. See Dispersive Fourier-transform spectroscopy; Fourier-transform spectroscopy Diamond. See Cubic Carbon Diamond films versus Carbon films, 11:837

Diamond-like Carbon films. See Carbon films (Diamond-like) Diamond structures Tin and, 111:268 Diatomic crystals infrared radiation of, 1:216 optic branches of, 1:215 reststrahlen band in, 1:219 Diatomic lattices impurities and, 1:255 optical-branch vibrations of, 1:215 Dichroism Polyethylene and, 11:967 Dielectric constants Aluminum and, 1:392, 402-406 Aluminum Antimonide and,

11:501-502 Aluminum Arsenide and, 11:490 Aluminum Oxide and, 11:762-763;

111:657 Aluminum Oxynitride (ALON) Spinel and, 11:782 Barium Titanate and, 11:792 Beta-Gallium Oxide and, 111:754 Boron Nitride and, 111:426-429 Cadmium Germanium Arsenide and,

111:448 Cadmium Selenide and, 11:559, 562 Cadmium Sulphide and, 11:584 Calcium Carbonate and, Ill:702 Calcium Fluoride and, II: 820 Cesium Chloride and, 111:732 Cesium Fluoride and, 111:743, 744 complex numbers and, 11:26 Copper Gallium Sulfide and, 111:462 described, 11:294 Drude damping and, II: 113, 118 in Drude model, 1:369-370 Drude theory and, I: 172, 174 effective-medium theory and, 1:105 effect of nonlocal, 11:120-122 electron-energy-Ioss spectroscopy and,

11:294 electron-hole interactions and, I: 199 ellipsometry and, 1:97 Erbium and, III:300 finite-energy sum rules and, 1:45 free-carrier effects and, I: 18-19

24

Subject Index

Dielectric constants (Continued) Gallium Antimonide and, 11:597, 600 Gallium Selenide and, 111:474 Gallium Telluride and, 111:490 Hafnium Nitride and, 111:352 imaginary component of, I: 163; II: 113 imaginary component of, in superlattices, 11:100 Iron Pyrite and, 111:510 Kramers-Kronig analysis and, 1:203,

380, 402-406 Kramers-Kronig integral and, I: 192 Lead Fluoride and, 111:762-763 Lead Tin Telluride and, 11:638-639 Lithium Tantalate and, 111:779 Magnesium Aluminum Spinel and,

11:886 Magnesium Oxide and, 11:922 Manganese and, 111:249 Mercury Cadmium Telluride and,

11:655,656 for metal-insulator system, 11:114-115 metals and, 111:233, 235 for noble metals, 1:205 optical-constants calculations and,

1:187-188 optical-properties measurement and,

1:17-18 optical responses of superlattices and,

11:98, 105 Palladium and, 11:472 photon energies and, 1:216 Polyethylene and, 11:958 Potassium Bromide and, 11:991 Potassium Iodide and, 111:809 Potassium Niobate and, 111:824-825,

831 radiation absorption and, 1:218 resonance frequency and, 1:235 Rhenium and, 111:278 room temperature and, 1:380-381 Ruthenium and, 111:254 Selenium and, 11:691 semiconductors and, III:32 Silicon, doped n-type and, III: 156-157 Silicon and, 111:531 Silver Iodide and, 111:555 Sodium Nitrate and, 111:873

Strontium Fluoride and, 111:886 Strontium Titanate and, II: 1035, 1036 sum rules and, 1:44-45 superlattice systems and, 11:98-99,

119 Tellurium and, 11:709 Thallium Bromide and, 111:926 Thallium Chloride and, 111:924, 925 Thallium Iodide and, 111:927 Thallous Halides and, 111:930 Water and, 11:1059, 1063 wavelength measurement and, 1:160,

162 Yttrium Aluminum Garnet and, 111:963,

965 Yttrium Oxide and, 11:1082 Zinc Germanium Phosphide and,

111:640 Zinc Phosphide and, 111:610 Zinc Selenide and, 11:742 Zinc Telluride and, 11:744 Zircon and, 111:987 Zirconium Nitride and, 111:352 See also Individual critiques; Complex dielectric constants; Static dielectric constants Dielectric constants, dc Cesium and, 111:343 Graphite and, 11:449, 452 imaginary versus real component of,

11:125-127 Indium Arsenide and, 1:480 Indium Phosphide and, 1:506 Lithium Fluoride and, 1:677 Lithium Niobate and, 1:697 Potassium Bromide and, 11:989 Silicon Carbide and, 1:589 Sodium Chloride and, 1:775 Tellurium and, 11:711 Zircon and, 111:988 Dielectric ellipsoid Sodium Nitrate and, III: 872 Dielectric films attenuated total reflection and, 11:90 Dielectric-function models Aluminum Oxide and, 11:763 Dielectric interface laser calorimetry and, I: 143

25

Subject Index

Dielectric interference filters Magnesium Fluoride and, 11:900 Dielectric loss functions Lead Fluoride and, 111:762 Dielectric materials optical anisotropy induced by, 11:69 Dielectric optical waveguides discussed, II:84 Dielectric parameters Magnesium Oxide and, 11:919 Dielectric polarization in crystals, 1:216 Dielectric response function damping factor and, 1:220 dispersion theory and, 1:226 infrared dispersion and, 1:264 ion interaction and, 1:217 Lithium Tantalate and, 111:778 Dielectrics optical-property calculation of, 11:151,

153 Dielectrics, amorphous. See Amorphous dielectrics Dielectrics, crystalline. See Crystalline dielectrics Dielectric slab, parallel sided lamelliform and, I: 19 Dielectric strength Aluminum Oxide and, 111:653 Dielectric susceptibility optical-properties measurement and,

1:12 in rationalized system of units, I: 12 Dielectric theory optical-properties determination and,

11:297 Differential spectrophotometry absorption-coefficient measurement and, 1:231 Diffraction measurement accuracy and, I: 157 Diffraction analysis Magnesium Aluminum spinel and,

11:883 Magnesium Oxide and, 11:920 measurement accuracy and,

1:155-156 Yttrium Oxide and, 11:1079

Diffuse-reflection technique Lead Fluoride and, 111:761 Dipole matrix element extinction coefficient and, II: 154, 156 optical-constants calculation and, 11:164 Dipole matrix elements Bloch states and, I: 193 density of states and, 1:261 interpolation scheme for, I: 194, 205 superlattices and, 11:108, 120 Dipole measurements temperature dependence and, II: 183,

187, 198 Dirac theory electron density of states and, I: 175 Direct band gaps Aluminum Oxide and, 111:654 Copper Gallium Sulfide and, 111:459 Cuprous Oxide, 11:875 Thallium Bromide and, 111:924 Thallium Chloride and, 111:923 Yttrium Aluminum Garnet and, 111:963 Zinc Germanium Phosphide and,

111:637 Disordered lattices described, 1:254 Dispersion analysis Aluminum Oxide and, 111:655 Ammonium Dihydrogen Phosphate and, 11:1006-1007 Brillouin scattering and, III: 125 Cadmium Germanium Arsenide and,

111:448 Calcium Carbonate and, Ill:702 Calcium Fluoride and, 11:818, 820 Cesium Bromide and, 111:717 Cesium Chloride and, 111:732 Cesium Fluoride and, Ill:743 Cesium Iodide and, 11:855 Copper Gallium Sulfide and, 111:461 Gallium Selenide and, 111:487 Lead Fluoride and, 111:762 Magnesium Fluoride and, 11:902-903 Magnesium Oxide and, 11:929-930,

937 metals and, 111:234 Orthorhombic Sulfur and, 111:902 Polyethylene and, 11:962-963

26

Subject Index

Dispersion analysis (Continued) Potassium Bromide and, 11:990 Potassium Dihydrogen Phosphate and,

11:1006-1007 Potassium Niobate and, III: 823 Rubidium Bromide and, 111:845-847 Rubidium Iodide and, 111:857, 859-860 Silver Gallium Selenide and, 111:575 Silver Gallium Sulfide and, 111:577 Sodium Fluoride and, II: 1022 Strontium Fluoride and, 111:883-884 Thallium Bromide and, 111:926 Thallium Chloride and, 111:924 Thallous Halides and, 111:929 Thorium Fluoride and, II: 1049, 1051 Yttrium Aluminum Garnet and,

111:964-965 Yttrium Oxide and, 11:1080, 1083 Dispersion curves Calcium Fluoride and, 11:817, 820-821 Zinc Selenide and, 11:740 Dispersion equations Thallium Chloride and, 111:924 Dispersion formulas Aluminum Gallium Arsenide and,

11:515 Cadmium Selenide and, 11:563 Gallium Antimonide and, 11:599 Zinc Selenide and, 11:741 Dispersion relations high-frequency limits and, 1:40 Kramers-Kronig analysis and,

1:227-229 sum rules and, 1:55-60 surface polaritons and, II: 113-114 Dispersion theory Aluminum Antimonide and, 11:502-504 dielectric response function and, 1:226 optical-constants calculation and,

11:164 optical-constants measurement and,

1:377-378 Silicon Dioxide (crystalline) and, 1:720 Silver Gallium Selenide and, 111:575 sum rules and, 1:55-60 Dispersive Fourier-transform spectroscopy (DFTS) far-infrared region and, 111:13-55

See also Fourier-transform spectroscopy Dispersive processes Fourier transform spectroscopy and,

1:157-161 sum rules for, 1:36 Distillation Liquid Mercury and, 11:462-463 Vanadium and, 11:478 DLC. See Carbon films (Diamond-like) DNA (deoxyribonucleic acid) index of refraction of, III: 143 Doping Aluminum Oxide and, 111:653 Bismuth Germanium Oxide and,

111:404 Bismuth Silicon Oxide and, 111:404 Cadmium Germanium Arsenide and,

111:447 Copper Gallium Sulfide and, 111:459 Strontium Titanate and, II: 1035 Yttrium Oxide and, 11:1079, 1081,

1082 Zinc Germanium Phosphide and,

111:639 Doppler shift line Brillouin scattering and, III: 124 DPNA dye, 11:88 Drude absorption Cesium and, III:341 Drude approximation Silicon, doped n-type and, 111:158-159 Drude damping superlattices and, II: 113, 118 Drude dielectric functions metal-metal superlattices and, II: 114 Drude equations rare-earth metals and, 111:289 Drude metal superlattices and, 11:99, 101 Drude models, 11:26 Aluminum and, 1:376-377, 380, 389, 390, 394; 11:211-212 Cobalt and, 11:435 for complex dielectric constants,

1:169-170,177,185 damping-coefficient conductivity in,

1:370

27

Subject Index

dielectric response function and, 1:264 Dysprosium and, 111:299 finite-energy sum rules and, 1:49, 50 free-carrier density of semiconductors and, 11:126 Gadolinium and, 111:296 Hafnium Nitride and, 111:352-354,

Dyes attenuated-total-reflection experiments and, 11:88-89 Dynasil. See Fused silica Dysprosium (Dy), 111:287

critique, 111:298-300 values of nand k for, 111:310, 327-332

356 Indium Antimonide and, 1:492 intraband spectrum and, 1:369-370 Lead Selenide and, 1:517 Lead Sulfide and, 1:526 Lead Telluride and, 1:536 Liquid Mercury and, 11:461-464 Lithium and, 11:345, 347, 350 Palladium and, 11:471 Potassium and, 11:365-366,369 Rhenium and, 111:278 Rhodium and, 1:342 Ruthenium and, lll:254 Silicon, doped n-type and, III: 162-163 Silicon-Germanium alloys and, 11:610 Sodium and, 11:355, 359 sum rules and, 1:40 superlattices and, II: 119 Tantalum and, 11:408 Titanium and, 111:240 Titanium Nitride and, 11:309 Tungsten and, 1:358 Vanadium and, 11:478 Vanadium Nitride and, 11:309 Zirconium Nitride and, 111:352-354,

356 Drude theory electron scattering and, I: 381 index of refraction and, 11:143 lattice dielectric constants and,

11:148 quantum-density-matrix formulation and, 11:132 quantum result for k and, II: 127-128 values of n and k and, I: 171 See also Individual critiques Drude values Cesium and, 111:343 Drude-Zener theory, 111:159 quantum extension of, I: 169-188 Dy. See Dysprosium

E Edge-defined, film-fed growth techniques (EFG) Aluminum Oxide, 111:653, 656 EELS. See Electron-energy-Ioss spectroscopy Effective-mass model electron-hole interactions and, I: 199 Effective-medium theory bulk and thin-film effects and,

1:104-108 Sodium and, 11:355 for superlattices, 11:98-100, 110,

118-119 Thorium Fluoride and, II: 1050 EFG. See Edge-defined, film-fed growth techniques Eigenmodes surface polaritons ana, II: 113 Eigenstates one-electron model and, I: 190, 193 in perturbing electromagnetic fields,

1:173 quantum-density-matrix formulation and, 11:129 Eigenvalues multiphonon absorption and, II: 190 Elastic-restoring forces atomic motions and, 1:218 Elastic-solid theory, 11:37 Electrical conductivity Cobalt and, 11:439 described, 1:12 introduced, 1:12 Electrical fields impact on optical constants of, I: 16-17 Electrical measurements Zinc Phosphide and, 111:610 Electric-dipole approximation interband transitions and, 1:191-192

28

Subject Index

Electric-dipole excitations model, I: 16-18 Electric-dipole matrix element superposition and, 1:201 Electric-dipole selection rule infrared absorption and, 1:254 optical phonons and, 1:232 Electric fields deformation of liquid crystal in, 11:91 distribution, and temperature, 1:137-138 local-field effects and, 1:203 and reflected waves, 11:42 in single-layer films, 1:138,147 Electric permittivity described, I: 12 Electric polarization Beta-Gallium Oxide and, 111:754 Electrolyte electroreflectance Zinc Selenide and, 11:739 Electrolyte immersion ellipsometry and, I: 104 Electromagnetic interactions quantum-mechanical treatment of,

11:152 Electromagnetic radiation interaction of, with matter, I: 11-16 lattice vibrations and, 1:214-215 optical-properties measurement and,

1:18 propagation through matter and, I: 17 source of, and Iamelliform, I: 18 values of n and k and, II: 166 Electromagnetic theory optical-properties determination and,

1:89-92 Electromagnetic wave equation plane-wave solutions to, 11:22 Electromagnetic waves field vectors of, 11:26-27 one-electron model and, 1:190-191 parallel-sided slabs and, I: 14 as polaritons, 1:217 Polyethylene and, 11:961 Electron analysis Thorium Fluoride and, II: 1049 Electron-band structure Mercury Cadmium Telluride and,

11:655 Silicon-Germanium alloys and, 11:607

Electron density Drude theory and, I: 174-175 f sum rules and, 1:61 Electron-energy-Ioss spectroscopy (EELS) Aluminum Nitride and, 111:375 Aluminum Oxide and, 111:654 Copper Oxides and, 11:877 Graphite and, 11:450-451 Hafnium Nitride and, 111:355 Magnesium Oxide and, 11:925-926 metals and, 111:234 optical-constants determination by,

11:293-310 Potassium Bromide and, 11:991 Vanadium and, 11:477-478 Zirconium Nitride and, 111:355 Electron-hole interactions critical-points metamorphosis and, 1:202 interband absorption and, 1:198-203 Electronic absorption Thallium Bromide and, 111:926 Thallium Chloride and, 111:924 Thallous Halides and, lll:930 Yttrium Aluminum Gamet and,

Ill:963 Electronic band transitions in polar crystalline insulators, 11:178 Electronic energy-band structure Gallium Selenide and, 111:474 Electronic excitations Lithium Tantalate and, 111:778 Electronic structure Aluminum Nitride and, 111:373 Electronic transitions absorption in solids and, 1:213 index of refraction and, II: 180 optical-properties determination and,

11:177 Yttrium Oxide and, 11:1080 Zinc Germanium Phosphide and,

111:638 Electronic well-well coupling superlattices and, II: 102, 105 Electron-impurity scattering Silicon, doped n-type and, III: 168-171 Electron-phonon scattering Silicon, doped n-type and, 111:167-168

Subject Index Electron-plasma oscillation Thallium Bromide and, 111:925 Thallium Chloride and, 111:924 Electron-plasma resonance Thallium Bromide and, 111:925 Thallium Chloride and, 111:924 Electron relaxation frequencies rare-earth metals and, 111:290 Electron relaxation time Drude theory and, I: 174 Electrons local-fie1d effects and, 1:203 oscillators for, I: 17-18 quantum-mechanical treatment of, 11:152 rare-earth metals and, 111:289 scattering mechanism and, II: 128-129, 138 Electrons and holes and circular polarization, 11:36 Electron scattering Drude theory and, 1:381; 11:126 supedattices and, II: 101 Electron transitions Cadmium Sulphide and, 11:580, 584 Cobalt and, 11:439 Graphite and, 11:452 Electro-optics Aluminum Gallium Arsenide and, 11:513 index of refraction and, II: 148 Lithium Tantalate and, 111:777 Mercury Cadmium Telluride and, 11:666 Electrostatic interactions dielectric polarization and, 1:216-217 Electrostatic regime superlattices and, II: 114 Ellipsometers discussed, 1:93; 11:213-245 types of, 1:93-95, 99, 102 Ellipsometric interferometry, I: 124 Ellipsometric measurements Aluminum and, 1:377-379, 395-401 Copper and, 1:280 dispersion analysis and, 1:59 pseudodielectric function and, 1:96 Silicon and, 111:519 Silicon-Germanium alloys and, 111:538

29 Ellipsometric methods and index measurement, 11:67, 69 Ellipsometry Aluminum Antimonide and, 11:501, 504 Aluminum Arsenide and, 11:489 Aluminum Gallium Arsenide and, 11:514 Aluminum Nitride and, 111:376-377 versus attenuated total reflection, II:75 benefits of, described, 11:559 Beryllium and, 11:423-424 Cadmium Selenide and, 11:559, 561 Cadmium Sulphide and, 11:583 Carbon films and, 11:839 Cerium and, 111:292 Cesium and, 111:342 Chromium and, 11:375 Cobalt and, 11:438, 440 Copper and, 1:280 described, 1:18 discussed, 1:276 Dysprosium and, 111:299 and electromagnetic wave propagation, 11:51 Erbium and, III: 300 Gadolinium and, 111:294, 296 Gallium Antimonide and, 11:597 Gallium Arsenide and, 1:431 Gallium Phosphide and, 1:448 Gallium Selenide and, 111:473-474 Germanium and, 1:467 Gold and, I: 106, 287 Graphite and, 11:452 Hafnium Nitride and, 111:355, 357 Indium Antimonide and, 1:491 Indium Arsenide and, 1:479 Indium Phosphide and, 1:503 Lead Fluoride and, 111:765 Liquid Mercury and, 11:461, 462-464 Lithium and, 11:345 Manganese and, 111:249 Mercury Cadmium Telluride and, 11:655,657 Mercury Telluride and, 11:657 metals and, 111:234 optical-properties determination and, 1:89-110 Potassium and, 11:364-365

30 Ellipsometry (Continued) reflectance and sum rules and, 1:55 Ruthenium and, 111:254 Silicon and, 1:100, 550; 111:521, 531 Silicon-Germanium alloys and, 11:609-610 Silicon Nitride (noncrystalline) and, 1:772 Sodium and, 11:354-356 Tantalum and, 11:408 temperature dependence and, 1:381 Terbium and, 111:297-298 Thorium Fluoride and, II: 1050, 1052-1053 Tin and, 111:269 Tin Telluride and, 11:726 Titanium Dioxide and, 11:168 Vanadium and, 11:477 Ytterbium and, III: 303 Yttrium Oxide and, II: 1081 Zirconium Nitride and, 111:355, 357 Ellipsometry principle phase changes and, II:48 Elliptical polarizer described, 1:93 EMA. See Bruggeman effective-medium approximation Emission Calcium Carbonate and, 111:702 Emissivity Yttrium Oxide and, 11:1083-1084 Emittance Magnesium Aluminum spinel and, 11:887 optical-properties measurement and, 1:22 EM radiation. See Electromagnetic radiation Energy bands Cadmium Sulphide and, 11:581 described, 1:195 interband absorption and, 1:205 Lead Fluoride and, 111:763 quantum-mechanical absorption theory and, 11:152 Energy-band structure Aluminum Antimonide and, 11:503 Energy conservation, III: 125

Subject Index

Energy density wedged-film laser calorimetry and, 1:140-145, 146 Energy dependence Aluminum Nitride and, 111:373 Energy dissipation lattice waves and, 1:220 Energy gaps Aluminum Antimonide and, 11:501 Gallium Antimonide and, 11:598 Gallium Selenide and, 111:477 Gallium Telluride and, 111:490 Lead Tin Telluride and, 11:638 Zinc Selenide and, 11:738-739, 741 Zinc Telluride and, 11:738 Energy-loss measurements Aluminum and, 1:378 dispersion analysis and, 1:60 Graphite and, 11:451 Polyethylene and, 11:958 sum rules and, 1:37 Energy-loss spectra Thallium Bromide and, III:925 Thallium Chloride and, 111:923 Thallium Iodide and, 111:927 Entrance surface films calorimetry and, 1:147-149 Envelope technique Lead Fluoride and, 111:765 Epitaxial films, 1:5 Aluminum Arsenide and, 11:491 Cubic Cadmium Sulphide and, 11:586 Cubic Silicon Carbide and, 11:705 Lead Tin Telluride and, 11:637 Mercury Cadmium Telluride and, 11:655 Tin Telluride and, II:725 Epitaxial structures Silicon-Germanium alloys and, III:537 Equilibrium density matrix in compound semiconductors, II: 128 in perturbing electromagnetic fields, 1:172 Equilibrium deviation in compound semiconductors, II: 128 Er. See Erbium

31

SUbject Index Erbium (Er), 111:287

critique, 111:300-301 values of nand k for, 111:311, 332-334 Error analysis Magnesium Fluoride and, 11:902 Sodium Fluoride and, II: 1022-1023 See also Defect analysis Error determination Cadmium Selenide and, 11:561 Cubic Silicon Carbide and, 11:706 Graphite and, 11:451 Liquid Mercury and, 11:462 Mercury Cadmium Telluride and,

ll:657, 661 values of n and k measurement and,

11:206 Zinc Selenide and, 11:741 Error locations f sum rules and, 1:62 Errors Beta-Gallium Oxide and, 111:754 Cesium Bromide and, lll:719 Cesium Chloride and, 111:731 Iron Pyrite and, lll:510 Potassium Niobate and, 111:822, 823 Rubidium Bromide and, 111:847 See also Uncertainty Europium (Eu), 111:287, 302 Eutectic compositions Thallous Halides and, 111:928 Evanescent waves decay of, 11:75 generation of, 11:76 SiOx experiments with, 11:92 and surface-plasmon waves, 11:79 and waveguides, 11:84 Evaporated films Lead Fluoride' and, 111:763 polarimetric studies of, 1:374 and prism experiments, 11:81-82 Sodium and, 11:354 Evaporated metal films optical-properties determination of, 1:96 sputtered Aluminum and, 1:373 Evaporated thin films Cadmium Sulphide and, 11:580 Evaporation methods Iron Pyrite and, 111:511

Evaporation-rate dependence Vanadium and, 11:478 Evaporation rates Lead Fluoride and, 111:764-766 Exchange coupling rare-earth elements and, 111:288 Exchange interactions sum rules and, 1:50-51 Excitation energy Cadmium Selenide and, 11:560 Excitation levels Cadmium Sulphide and, 11:581 Excitations Rubidium Iodide and, 111:858 Exciton absorption process Calcium Fluoride and, 11:816 Cesium Bromide and, 111:719 electron-hole interaction and, 1:200,

202-203 Gallium Antimonide and, 11:598 introduced, 1:4 one-electron model and, I: 196, 198 Exciton effect Potassium Chloride and, 1:704 Exciton energy Gallium Selenide and, 111:477 Exciton formation Thallium Bromide and, 111:925 Thallium Chloride and, 111:924 Excitonic absorption Aluminum Oxide and, 111:654 Exciton peaks Magnesium Fluoride and, 11:900 Magnesium Oxide and, 11:921-923 Yttrium Oxide and, 11:1079-1080 Exciton-phonon coupling model,

1:203 Excitons Cadmium Selenide and, 11:562 Cadmium Sulphide and, 11:585 Liquid Mercury and, 11:462 Magnesium Oxide and, 11:921-922 semiconductor superlattices and,

11:102-108 Silver Gallium Sulfide and, 111:576 Sodium Fluoride and, II: 1021 Zinc Selenide and, 11:742-743 Zinc Telluride and, 11:747

32 Exciton surface polaritons described, 11:89 Exclusion principle n-type materials and, 11:132-133 Exit surface films calorimetry and, 1:147-149 Exponential absorption edges, 1:203 Extinction coefficients, III: 122 Aluminum Antimonide and, 11:501-503 Aluminum Nitride and, 111:376-377 Barium Titanate and, 11:790-791 Beryllium Oxide and, 11:805 Calcium Carbonate and, III: 704-705 Calcium Fluoride and, 11:817, 820 Carbon films and, 11:838 Cesium Bromide and, 111:717-719 Cesium Chloride and, 111:731 Cesium Fluoride and, 111:743 Cubic Silicon Carbide and, 11:706 defined, II: 152 derivation of, 11:154-160 in dielectric response function, 1:220 of dielectrics, determination of, 11:69 Drude model and, I: 170 Gallium Antimonide and, 11:590 Gallium Selenide and, 111:474-475 Gallium Telluride and, 111:490 and index of refraction, II: 127 infrared reflectivity and, 1:226 introduced, 1:7 Lead Fluoride and, 111:764, 766 Lithium Tantalate and, 111:777 measurement of, 1:81-85 for metallic Aluminum, 1:388 Nickel and, 1:313 photothermal deflection of, 111:99-116 Polyethylene and, 11:961-962 Potassium Iodide and, 111:810 Rubidium Bromide and, III:845, 847-848 Rubidium Iodide and, 111:857-858 Selenium and, 11:691-692 Silicon, doped n-type and, III: 156 Silicon and, 111:520-522, 525, 528, 533 Silicon-Germanium alloys and, 11:608-609,611

Subject Index

Silver Chloride and, 111:554 Silver Gallium Selenide and, 111:575 Silver Gallium Sulfide and, 111:576 Sodium Fluoride and, II: 1023 sum rules and, 1:44 of thin film layers, 11:57 Thorium Fluoride and, II: 1050-1051 Water and, 11:1060 wedged-film laser calorimetry and, 1:140 Yttrium Oxide and, 11:1081 Zinc Arsenide and, 111:595-596 Zinc Phosphide and, 111:609-610 Zinc Selenide and, 11:737 Zinc Telluride and, 11:737 See also Individual critiques Extinction values Silicon-Germanium alloys and, 111:538 Extreme ultraviolet region (XUV) Magnesium Fluoride and, 11:900 optical-constants determination and, 11:203 Yttrium Oxide and, 11:1079 Extrinsic absorption Yttrium Aluminum Garnet and, 111:964 Extrinsic scattering in polar crystalline insulators, II:178 versus Rayleigh scattering, II: 178

F Fabrication techniques wavelength measurement and, 1:155 Fabry-Perot cavities wavelength measurement and, 1:161-162, 162, 166 Fabry-Perot interferences of Gallium Arsenide, II: 118 Fabry-Perot interferometers Brillouin scattering and, III: 123-124,

128-129, 130 Fano line shapes described, 1:201 Faraday's law of induction electromagnetic-wave propagation and, 1:11 optical-properties measurement and, 1:11

33

Subject Index Far-infrared region (FIR) Beryllium and, 11:422, 424 Bismuth Germanium Oxide and, 111:406 Bismuth Silicon Oxide and, 111:406 Cesium Chloride and, 111:731 Cesium Fluoride and, 111:743 Cesium Iodide and, II: 85 3 dispersive Fourier-transform spectroscopy and, III: 13-55 Gallium Telluride and, 111:491 Iron Pyrite and, 111:507 Lead Tin Telluride and, 11:638 Mercury Cadmium Telluride and,

11:663-666 Orthorhombic Sulfur and, 111:901,

904-905 Polyethylene and, 11:960-961,

966-968 Potassium Bromide and, 11:989 Potassium Iodide and, III: 808 Silicon-Germanium alloys and, 111:537 Silver Chloride and, 111:554 Silver Iodide and, 111:556 Sodium Nitrate and, 111:873 Strontium Titanate and, II: 1035 Tellurium and, 11:710 Thallium Iodide and, 111:927 Thallous Halides and, 111:930 Thorium Fluoride and, II: 1053 Zinc Phosphide and, 111:609 Far-ultraviolet region (FUV) Orthorhombic Sulfur and, 111:903 Fe. See Iron FE axis. See Ferroelectric properties FECO method optical measurement and, I: 118-119,

126, 131 Fermi-Dirac distribution function interband absorption and, I: 191, 196,

198 Fermi-energy measurement conduction-band edge and, 11:133-135 discussed, 1:278 Drude theory and, I: 171 Fermi integrals Silicon, doped n-type and, III: 167 Fermi level Cesium and, 111:343

Cobalt and, 11:436 conduction-band edge and, 11:127 extinction coefficients and, II: 157 Gadolinium and, 111:295 Hafnium Nitride and, 111:352 rare-earth elements and, 111:287,

290 in refractory compounds, 11:303 Zirconium Nitride and, 111:352 Fermi surface for Copper, 1:205 Kane theory and, 1:176 Fermi velocity superlattices and, II: 120 Fermion commutator absorption rules and, 1:38 Ferroelectric properties discussed, II: 1035 Lithium Tantalate and, 111:777 Potassium Niobate and, 111:821 Ferromagnetism Dysprosium and, 111:299 Gadolinium and, 111:295 rare-earth elements and, 111:288 FeS2. See Iron Pyrite FeSi2. See Iron Disilicide Field-independent scalar quantities optical constants and, I: 12 Film deposition Lead Fluoride and, 111:765 Film morphology optical-properties measurement and,

1:375-376 Film on substrate Carbon films and, 11:839 Lithium and, 11:346 reflectance experiment on, 1:23-24 samples, 1:5 small absorption constants determination of, 1:4 Film thickness interferometers and, I: 119 Palladium and, 11:469 for Silver films deposited on a prism,

11:84 of surface films, 1:148-152 Vanadium and, 11:478 See also Thickness considerations

34

Subject Index

Finesse Brillouin scattering and, III: 129 reflective, III: 122 Finite-energy dispersion relations sum rules and, 1:54 Finite-energy rules f sum rules and, 1:61 Finite-energy sum rules, 1:45-51 FIR. See Far-infrared region Fizeau method optical measurement and, 1:117-118,

126, 131 Sodium Fluoride and, II: 1022 Flame-fusion technique Aluminum Oxide, 111:653 Magnesium Aluminum Spinel and,

11:884 Yttrium Oxide and, 11:1079 See also Vemeuil method Flame photometry Silver Iodide and, 111:555 Floating-zone technique Gallium Phosphide and, 1:445 Fluorescence Magnesium Fluoride and, 11:901 Fluorides index of refraction and, II: 180 Fluorite structures selection rules for, 1:234 Flux-growth technique Barium Titanate and, 11:789 Potassium Niobate and, 111:821 Fool's gold. See Iron Pyrite Forbidden energy band gap Calcium Fluoride and, II: 816 Forbidden-frequency bands. See Stopgaps Forbidden transitions Iron Pyrite and, 111:508-509 Formulas for absorbing layers, 1:25 for Fresnel coefficients, 1:24-25 for layered slabs, 1:25 for nonabsorbing layers with symmetry,

1:26 Formvar (polyvinyl formal resin) Silver Chloride and, 111:554

Fourier components of electron distribution functions, I: 173 in free-carrier absorption experiment,

1:173 in metallic Aluminum, 1:371 Fourier integrals dispersion analysis and, 1:56 Fourier series dispersion analysis and, 1:56 Fourier spectrophotometry Silver Iodide and, 111:556 Fourier spectrum ellipsometry and, 1:94 Fourier-transform infrared spectrophotometer Magnesium Oxide and, 11:938 Fourier-transform interferometry Barium Titanate and, 11:792 Zinc Phosphide and, 111:609-610 Fourier-transform spectroscopy (FrS) Bismuth Germanium Oxide and,

lII:407 Bismuth Silicon Oxide and, 111:407 Cadmium Telluride and, 1:411 Cesium Bromide and, lII:717 Cesium Iodide and, 11:854 dispersive measurements and,

1:157-161, 167 Gallium Telluride and, 111:491 Germanium and, 1:467 index-of-refraction measurements and,

1:156-161 Indium Arsenide and, 1:480 Indium Phosphide and, 1:506 Iron Pyrite and, 111:508 Lead Selenide and, 1:517 Lithium Tantalate and, 111:779 nondispersive measurements and,

1:157-158, 167 Polyethylene and, 11:961-963 Potassium Bromide and, 11:990 Potassium Chloride and, 1:703 Potassium Iodide and, 111:809-810 Rubidium Bromide and, IIl:847 Rubidium Iodide and, lII:860 Silicon, doped n-type and, III: 169 Silicon and, 1:550

SUbject Index Sodium Fluoride and, II: 1022-1023 Strontium Titanate and, II: 1036 sum rules and, 1:41 See also Dispersive Fourier-transform spectroscopy Frank-Oseen theory attenuated-total-reflection curves and, 11:92 Free-carrier absorption Aluminum Antimonide and, 11:501 Aluminum Oxynitride and, 11:779 below fundamental band gap, 1:4 Calcium Fluoride and, II:819 Gallium Antimonide and, 11:598 Graphite and, 11:452 Iron Pyrite and, 111:509, 510 Mercury Cadmium Telluride and, 11:663 in narrow-gap semiconductors, 1:263 quantum theory of, I: 172-174 of radiation, II: 126 in semiconducting compounds, 1:171-172 Strontium Titanate and, II: 1035 sum rules and, 1:40 Tellurium and, 11:709 two-phonon process and, 1:239 in wide-gap semiconductors, 1:45 Zinc Germanium Phosphide and, 111:640 See also Individual critiques Free-carrier concentrations in samples, 1:179 Selenium and, 11:691 of semiconductors, 111:32 Tellurium and, 11:710 Free-carrier contributions quantum absorption and, 111:159-162 values of n and k for, 1:4 Free-carrier density Gallium Arsenide and, I: 184 Indium Arsenide and, 1:186 Indium Phosphide and, 1:181 in polar semiconductors, I: 169-172 Free-carrier effects Gallium Antimonide and, ll:600 Yttrium Oxide and, 11:1082

35 Free-carrier impurities Cadmium Sulphide and, 11:581 Free-carrier plasma edge Aluminum Gallium Arsenide and, 11:515 Free-carrier response superlattices and, II: 118 Free carriers classical-oscillator case for electron and, 1:17 near-millimeter index-of-refraction measurement and, I: 165 optical-properties determination and, 11:177 optical-properties measurement and, 1:17-18 quantum-density-matrix formulation and, 11:148 semiconductor crystals and, 111:33 Free-electron absorption rare-earth metals and, 111:289 Free-electron gas interband absorption and, I: 190 Free electrons Cesium and, III: 341 Hafnium Nitride and, 111:352 optical-constants calculation and, II: 153 rare-earth metals and, lll:289 Ytterbium and, ITI:303 Zirconium Nitride and, 111:352 Free-electron theory Cobalt and, 11:439 Free-space impedance optical-properties measurement and, 1:13 Free-space permeability described, I: 12 Free-space permittivity described, I: 12 optical-properties measurement and, 1:13 Free-space resonant cavities near-millimeter wavelength measurement and, I: 161-163 Free-space wavelengths optical-properties measurement and, 1:13

36 Free-space wave velocity optical-properties measurement and, I: 13 Frenkel exciton model electron-hole interactions and, I: 199 Frequency considerations Drude theory and, 1:175-176 temperature dependence and, 1:248 Frequency dependence absorption coefficients and, IT: 187 Cesium Iodide and, 11:854 index of refraction and, II: 179, 182 Lead Fluoride and, 111:762 wire-grating beam splitters and, I: 163 Frequency-dependent conductivity optical-properties measurement and, I: 13 Frequency-dependent damping constants Sodium Nitrate and, 111:872 Frequency-dependent electron scattering rate Drude theory and, I: 185 Frequency-dependent relaxation rate Boltzmann equation and, II: 127 free-carrier absorption and, I: 173 Frequency-distribution function crystalline samples and, 1:215 Frequency ranges Fourier transform spectroscopy and, 1:156, 167 Fresnel amplitude reflection coefficients discussed, 11:37-50 Fresnel coefficients, I: 18; 11:39 in attenuated-total-reflection structures, 11:82-83 basic formulas for, 1:24-25 optical-properties measurement and, 1:18, 379 parallel-sided slabs and, I: 16 and prism evaporation, 11:81 reflectivity and, I: 12 transmittivity and, I: 12 in waveguides, II: 86 Fresnel convention and phase changes, 11:48-50 Fresnel equations discussed, 1:278 dispersion analysis and, 1:59 dispersion relations and, 1:56 Graphite and, 11:450-451

Subject Index

Magnesium Oxide and, IT:921, 931, 935 Palladium and, 11:470 Polyethylene and, 11:961 radiation reflection and, 1:219 Sodium and, IT:356 Vanadium and, 11:477 Water and, 11:1061 Fresnel reflection complex index of refraction and, 1:70 Germanium and, 1:467 sign conventions used in, 11:22 Fresnel relations metals and, 111:234 reflectance experiment and, 1:23 Fresnel terms in matrix multiplication of layers model, 1:16 Fringe patterns Fizeau method and, I: 118 Frustrated total reflectance absorption and, 1:30-32 formulas for, 1:30-32 I-sum rule, 1:36-38, 44, 46 and dielectric function, 1:381 for dipole matrix elements, 1:51 loss-function determination and, II: 300 tests for, 1:61-64 See also Sum rules FrS. See Fourier-transform spectroscopy Fundamental absorption band Calcium Fluoride and, 11:816 Fundamental absorption edge Cadmium Selenide and, 11:561 Calcium Fluoride and, 11:816 electronic transitions and, 1:214 Gallium Selenide and, 111:474 Gallium Telluride and, 111:489 Lead Fluoride and, 111:762-763 optical-constants calculation and, II: 163 quantum effects near, II: 127 Thallous Halides and, 111:928-929 Zinc Arsenide and, IIT:595-596 Zinc Phosphide and, IIT:610 Fundamental band gaps absorption processes of, 1:4, 172 Aluminum Gallium Arsenide and, 11:514

37

Subject Index

interband absorption and, I: 197 Iron Pyrite and, 111:508 Mercury Cadmium Telluride and, 11:663 Potassium Bromide and, 11:991 Selenium and, 11:692 Silver Gallium Selenide and, 111:574 Silver Gallium Sulfide and, 111:574,

576 Sodium Nitrate and, 111:872 Thallium Bromide and, 111:925 Tin Telluride and, 11:725 See also Individual critiques Fundamental electronic absorption Thallium Bromide and, 111:924 Thallium Chloride and, 111:923 Fundamental electronic transitions Zinc Germanium Phosphide and,

111:638 Fundamental energy gap Zinc Telluride and, 11:744 Fundamental lattice vibrations Aluminum Oxide and, 111:656-657 Aluminum Oxynitride and, 11:778 Barium Fluoride and, 111:685 Gallium Selenide and, 111:475 Magnesium Fluoride and, 11:901-902 Potassium Iodide and, III: 809 Yttrium Aluminum Garnet and,

111:964-965 Yttrium Oxide and, 11:1081-1082 See also One-phonon region Fused quartz interferometry and, I: 131 Fused-quartz heat method Iron Pyrite and, 111:511 Fused silica Brillouin scattering and, III: 142 correction term determination of, I: 101 Dynasil, 11:58 index of refraction for, 1:159 interdiffusion probability and, I: 86 photothermal deflection and, III: 107-109 Rubidium Iodide and, 111:858 sample preparation and, II: 1053 Fusion-casting method Magnesium Aluminum Spinel and,

11:884

G Ga. See Gallium GaAIAs. See Gallium Aluminum Arsenide GaAs. See Gallium Arsenide Gadolinium (Gd), 111:287

critique, 111:294-297 values of n and k for, 111:308, 318-323 Gaertner L 117 ellipsometer Lead Fluoride and, 111:765 Gaertner spectrometer Calcium Fluoride and, II: 818 Gallium Aluminum Arsenide (GaAIAs) absorption spectrum of, II:103, 105,

108 band structure of, 11:102 spectroscopic measurements of,

111:84-85, 85 superlattices and, 11:97, 101-102, 105,

108 Gallium Antimonide (GaSb)

critique, 11:597-601; 111:44-47 optical-constants calculation of, II: 169 optical parameters for, 11:320 optical-properties determination of,

1:101-102 values of nand k for, 11:602, 603-606;

111:45-46, 46 Gallium Arsenide (GaAs) critique, 1:429-432; 111:40-41 absorption coefficient and, 1:249 absorption spectrum of, 11:103, 105,

108 band structure of, 11:102 critical-point phonons and, 1:239, 240 dispersion analysis of, 1:60, 430 ellipsometric measurement of, I: 101 free-carrier absorption and, I: 171 free-carrier concentrations and, I: 182,

183 index of refraction and, 11:148 infrared absorption spectrum of, 1:238 infrared reflection spectrum of, 1:266 multiphonon absorption in, 1:6 optical-constants calculations of, 1:177;

11:169 optical constants of, I: 184

38

Subject Index

Gallium Arsenide (Continued) optical parameters for, 11:320 oxide effect on, 1:98 photothermal deflection and,

III: 109-110 quantum effects in, II: 127 reflectance measurements on, I: 109 reflectivity curves of, 1:185 reststrahlen region and, 1:187-188 in samples, 11:489-490 selection rules for, 1:233-234 spectroscopic measurements of,

111:74-75 superlattices and, 11:97, 101-102, 105 three-phonon processes in, 1:242 transparency frequency of, 1:231 two-phonon processes in, 1:241 values of nand k for, 1:4, 176, 184,

433, 434-443; 111:42-43, 44 wavevector dependence and, 1:267 Gallium (Ga) interband absorption and, 1:204 Gallium Oxide, beta. See Beta-Gallium Oxide Gallium Phosphide (GaP)

critique, 1:445-449; Ill:38-40 dielectric parameters for, 1:450 optical-constants calculation of,

11:169 optical parameters for, 11:320 photothermal deflection and, III: 110 spectroscopic measurements of,

111:75-76 transparency frequency of, 1:231 values of nand k for, 1:450, 451-464;

111:39-40, 41 Gallium Selenide (GaSe)

critique, 111:473-477 index of refraction and, 111:487 optical parameters for, 111:201, 205 parameters used for, 111:487 spectroscopic measurements of, 111:79, 80 values of nand k for, 111:479-480,

481-487 Gallium Telluride (GaTe)

critique, 111:489-491 optical parameters for, 111:220

values of n and k for, 111:493-496,

497-505 /3-Ga2 0 3 . See Beta-Gallium Oxide GaP. See Gallium Phosphide GaSb. See Gallium Antimonide Gas-cell-microphone photoacoustics,

111:60-64 GaSe. See Gallium Selenide Gas-transport method Zinc Arsenide and, 111:596 GaTe. See Gallium Telluride Gaussian form multiphonon absorption and, II: 190,

198 Gaussian function Brillouin scattering and, III: 131 Gaussian units electron-hole interaction and, 1:202 one-electron model and, I: 190, 192 optical-properties measurement and,

1:13 Gd. See Gadolinium GDA. See Generalized Drude approximation Ge. See Germanium Generalized Drude approximation (GDA) described, III: 161 See also Drude theory Ge0 2 • See Germanium Dioxide Germanium, amorphous hydrogenated (a-GE:H) photothermal deflection and, III: 110 Germanium Dioxide (Ge0 2) reststrahlen spectrum of, 1:223 Germanium (Ge)

critique, 1:465-469 atomic motions determination of, 1:218 Brillouin scattering and, 111:133 chemical etching of, I: 104 dispersion analysis of, 1:466, 468 infrared absorption and, 1:254, 260 multiphonon interactions and, 1:236 one-electron calculation and, 1:207 optical-constants calculation of, 11:169 optical parameters for, 11:320 selection rules for, 1:234 transparency frequency of, 1:231 values of n and k for, 1:470, 471-478

39

Subject Index

Germanium Diselenide (GeSe2 ) spectroscopic measurements of, 111:94 Germanium Selenide (GeSe) photothermal deflection and, III: 110 Germanium-Silicon (Ge-Si) alloy optical parameters for, 11:323 Germanium Sulphide (GeS) phonons and, 1:203 GeS. See Germanium Sulphide GeSe 2 • See Gallium Diselenide Glass absorption coefficients and, 1:251 effective-medium theory and, I: 105 interdiffusion probability and, 1:86 reflectance and, 11:204 See also Silicon Dioxide Glass prisms Liquid Mercury and, 11:463 Glow-discharge decomposition optical-constants calculation and, II: 168 Glow-discharge lamps Zinc Arsenide and, 111:595 Golay cell detector interferometry and, I: 164 Gold (Au)

critique, 1:286-287 effective-medium theory and, 1:106,

107 layer thickness, attenuated total reflection and, 11:88 optical parameters for, 11:318 permittivity of, 11:81 values of nand k for, 1:289, 290-295;

111:7-8 Gold (Au) film Cadmium Arachidate on, 11:88 and liquid crystals, 11:90 photothermal deflection and, III: 111 Gradation forms of, 11:66 Gradient-furnace technique Magnesium Aluminum spinel and,

11:884 Grain boundaries effective-medium theory and, 1:105-106 film morphology and, 1:376 Graphite (C)

critique, 11:449-453

Boron Nitride and, 111:425 optical parameters for, 11:321, 326;

111:201 values of n and k for, 11:454, 455-460 Graphs and tables introduced, 1:6-7 Grating interferometers Silver Iodide and, 111:556 Grating spectrometers Aluminum Oxide and, 11:762 Bismuth Germanium Oxide and,

111:404, 406 Bismuth Silicon Oxide and, 111:404,

406 Potassium Bromide and, 11:991-992 Grating-spectrometer techniques beam splitters and, 1:157, 158, 159,

163 Cadmium Telluride and, 1:412 versus Fourier transform spectroscopy,

1:156 Potassium Bromide and, 11:989-990 Green' s-function technique dynamical behavior of impurities and,

1:257 Group theory Barium Fluoride and, 111:686 Bismuth Germanium Oxide and,

111:406 Bismuth Silicon Oxide and, lll:406 dipole matrix elements and, I: 193 infrared spectra of crystal and, 1:233 Magnesium Fluoride and, 11:901 Potassium Iodide and, III: 809 Yttrium Oxide and, II: 1081 Growth parameters Calcium Fluoride and, 11:815 Guided-modes technique introduced, 11:76 Lead Fluoride a~d, 111:765 Guided-wave techniques for transparent layers, 11:69 Gyratron sources optical-constants determination of, 1:4

H H. See Hydrogen de Haas-van Alphen measurements, 1:371

40

Subject Index

Hafnium Dioxide (Hf0 2 ) protective coatings of, 1:375 Hafnium Nitride (HfN) critique, 111:351-357 values of n and k for, 111:361-362,

367-369 Hagen-Rubens law Lead Selenide and, 1:517 Halides absorption and, lll:719 Rubidium Bromide and, III: 845 Rubidium Iodide and, 111:857 Thallium(I), 111:923-931 Hamamatsu photomultiplier Zinc Arsenide and, 111:596 Zinc Phosphide and, 111:610 Hamiltonians, 1:51, 172 in compound semiconductors, II: 128 electron-hole interactions and, I: 199 in Kubo formula, III: 161 one-electron model and, 1:190-191, 193 in perturbing electromagnetic fields, 1:172 polar scattering and, I: 174 Harmonic approximations electric-dipole selection rules and, 1:232 reststrahlen band in, 1:219 Harmonic-oscillator models dispersion analysis and, 1:58 optical-property measurement and, 11:152, 164 Zinc Germanium Phosphide and, Ill:639-640 Harmonics Cadmium Germanium Arsenide and, 111:445,447 Copper Gallium Sulfide and, 111:459 Zinc Germanium Phosphide and, 111:638 Hartree-Fock-Slater (HFS) model f sum rules and, 1:63 HDPE (High-density Polyethylene). See Polyethylene Heat-exchanger method (HEM) Aluminum Oxide, 111:653, 656

Heat treatment Lead Fluoride and, 111:762 Helium-Neon (HeNe) laser ellipsometric measurement and, 1:93 Helix rotation of, 11:35 temporal behavior of, 11:33-35 HEM. See Heat-exchanger method Henke UV and X-ray optical constants access via computer and, 111:8 for Aluminum Arsenide, 11:489 for Aluminum Gallium Arsenide, 11:514 for Antimony, 111:275 for Beryllium, 11:422, 425 for Cadmium Germanium Arsenide, 111:446 for Cadmium Sulphide, 11:579 for Cerium, 111:292 for Cesium, 111:343 for Cesium Bromide, 111:717 for Cesium Chloride, 111:731, 733 for Cesium Iodide, 11:853, 857 for Chromium, 11:375 for Cobalt, 11:438 for Copper Gallium Sulfide, 111:460 for Dysprosium, 111:298, 300 for Erbium, 111:300, 301 for Cesium Fluoride, 111:744 for Gadolinium, 111:294, 296 for Hafnium Nitride, 111:356, 357 for Indium, 111:262 for Iridium, 1:296 for Iron, 11:386 for Lithium Fluoride, 1:675 for Manganese, 111:249 for Molybdenum, 1:303 for Nickel, 1:313 for Niobium, 11:397 for Orthorhombic Sulfur, 111:903 for Osmium, 1:324 for Platinum, 1:335 for rare-earth metals, 111:290 for Rhenium, 111:278 for Rhodium, 1:342 for Ruthenium, 111:254 for Samarium, 111:293

Subject Index

for Selenium, 11:692 for Silicon, 1:552 for Silicon-Germanium alloys, Ill:537 for Silver, 1:350 for Terbium, 111:297 for Thulium, 111:301 for Titanium, 111:241 for Tungsten, 1:357 for Ytterbium, 111:302 for Zinc Germanium Phosphide, Ill:638 for Zinc Selenide, 11:738 for Zinc Sulfide, 1:552 for Zinc Telluride, 11:743 for Zirconium Nitride, 111:356-357 Herzberger-type equations Aluminum Antimonide and, 11:504 Beryllium Oxide and, 11:806 Bismuth Germanium Oxide and, lll:406 Cadmium Telluride and, 1:412 Cubic Carbon and, 1:666 Gallium Antimonide and, ll:590 Potassium Dihydrogen Phosphate and, 11:1006 Silicon and, 1:548 Heterogeneous systems optical properties measurement of, 1:105 Heterostructure Aluminum Arsenide and, 11:491 Gallium Arsenide and, ll:491 Hexagonal Cadmium Sulphide (CdS), 11:579-582 values of nand k for, 11:591-594 Hexagonal Lithium (Li), 11:345 Hexagonal materials Aluminum Oxide and, 111:653 Beryllium and, 11:421 Boron Nitride and, 111:425, 428-429 Cadmium Selenide and, 11:559 Cadmium Sulphide and, 11:579 Cobalt and, 11:435 Gadolinium and, 111:295-296 Gallium Telluride and, 111:489 Iridium, 1:296 Magnesium and, 111:235 rare-earth elements and, 111:287-288 Rhenium and, 111:278

41 Silver Iodide and, 111:555 Yttrium Oxide and, 11:1079 Hexagonal Zinc Sulfide (ZnS) critique, 1:600-601 optical parameters for, 11:321 values of n and k for, 1:605, 617-619 See also Cubic Zinc Sulfide; Zinc Sulfide HtN. See Hafnium Nitride Hf0 2 • See Hafnium Dioxide Hg. See Liquid Mercury; Mercury HgTe. See Mercury Telluride Hg1_xCdxTe. See Mercury Cadmium Telluride High vacuum evaporated films in, 1:373, 379 Hilbert transformation explained, II: 152-153 index of refraction and, II: 160 Potassium Iodide and, III: 807 sum rules and, 1:43 Hilger UV grating monochromator Bismuth Germanium Oxide and, 111:404 Hilger-Watts E580 Silver Chloride and, Ill:554 HIP treatment. See Hot isostatic pressing method H 20. See Water Homogeneity Boron Nitride and, Ill:425 and index determination, 11:65-67 interferometry and, I: 116 See also Inhomogeneity Homopolar crystals atomic-motions determination of, 1:218 defects and, 1:254 multiphonon absorption and, 1:232 one-phonon absorption in, 1:236 Hot-forge process Calcium Fluoride and, 11:823 Hot isostatic pressing method (HIP) Calcium Fluoride and, 11:823 Hot-pressed technique Cadmium Telluride and, 1:412 Magnesium Aluminum Spinel and, 11:884 Zinc Sulfide and, 1:601-602

42

Subject Index

Hot-wall technique substrates and, 1:535 Humphreys-Owen reflectance methods, 1:71-72, 75 Hydrogen-cyanide laser measurements Polyethylene and, 11:964-965 Hydrogen (H) unbound continuum states of, I: 199 water-vapor reduction and, 1:375 Hyperbolic excitons defined, 1:201

I lAD. See lon-assisted deposition Iceland spar. See Calcium Carbonate Image displacement method Barium Titanate and, 11:790 Immersion method Cesium Fluoride and, ITI:743 Impurities Aluminum Nitride and, 111:373 Iron Pyrite and, 111:511 Potassium Iodide and, ITI:809 Silver Iodide and, IIT:555 spectral-behavior results and, I: 17 Strontium Fluoride and, 111:886 Zinc Germanium Phosphide and, 111:637-640 Zircon and, 111:988-989 See also Defects and disorders Impurity absorption below fundamental band gap, 1:4 incident radiation intensity and, II: 178 spectral region and, II: 178 temperature considerations and, II: 178 Impurity atoms lattice potential energy function and, 1:255 Impurity dominated plateau polar crystals and, 1:214 Impurity molecules Potassium Bromide and, 11:990 Impurity scattering Drude theory and, 1:175,177 In. See Indium

InAs. See Indium Arsenide Incidence reflectance and sum rules and, 1:55 Incidence of light Rubidium Iodide and, 111:858 Incident beam polarization Orthorhombic Sulfur and, 111:902 Incident dielectric functions Fresnel coefficients and, 1:24 Incident photon intensity extinction coefficient and, II: 154 Incident radiant flux Orthorhombic Sulfur and, IIT:900 Incident radiation polarization state and, 1:80 Incident waves rare-earth metals and, ITI:289 Incipient ferroelectric property discussed, 11:1035 Index determination of Titanium Oxide (Ti0 2 ), IT:65 Index of extinction temperature dependence and, II: 179 Index of reflection Orthorhombic Sulfur and, IIT:902 Index of refraction, ITI: 122 Aluminum and, 1:378-379, 388, 395-401; 11:209 Aluminum Antimonide and, 11:501, 502-504 Aluminum Gallium Arsenide and, 11:138-147, 143, 144 Aluminum Nitride and, 111:373, 376-377 Aluminum Oxide and, 11:761; 111:654-657 Aluminum Oxynitride and, 11:778-780 Ammonium Dihydrogen Phosphate and, II: 1006 Barium Fluoride and, 111:684-686 Barium Titanate and, 11:791 Beryllium Oxide and, 11:805-806 Beta-Gallium Oxide and, 111:753-754 Bismuth Germanium Oxide and, 111:403 Bismuth Silicon Oxide and, 111:403

43

SUbject Index Cadmium Germanium Arsenide and,

111:447 Cadmium Selenide and, 11:560-562 Cadmium Sulphide and, 11:585 Calcium Carbonate and, 111:702, 704-705 Calcium Fluoride and, 11:817-821, 823 Carbon films and, 11:838 Cesium Bromide and, 111:717-718 Cesium Chloride and, 111:731-732 Cesium Fluoride and, 111:743 Cesium Iodide and, 11:855 compound semiconductors and,

11:125-148 Copper Gallium Sulfide and, 111:461 Copper Oxides and, II: 877 crystal quartz and, I: 165 Cubic Silicon Carbide, 11:707 Cubic Silicon Carbide and, 11:705 defect absorption and, 1:48 defined, II: 151 derivation of, 11:160-163 Diamond and, 1:382-383 in dielectric response function, 1:220 DNA (deoxyribonucleic acid) and,

111:143 fused silica and, III: 142 Gallium Antimonide and, 11:590, 598,

600 Gallium Arsenide and, 11:138, 141-147 Gallium Selenide and, 111:473-477, 487 Gallium Telluride and, 111:490 Hafnium Nitride and, 111:353 and index of the dielectric, II: 80 Indium and, 111:262 Indium Gallium Arsenic Phosphide and, 11:138-147, 139, 140, 142, 143 Indium Phosphide and, 11:138-147, 139, 140 infrared reflectivity and, 1:226 interferometers and, I: 119 interferometry and, I: 114 Iron Pyrite and, 111:508 Kramers-Kronig analysis and, 11:153,

300

Lead Fluoride and, 111:761-762,

764-766 Lithium Tantalate and, 111:777-778 Magnesium Aluminum Spinel and, 11:884-886, 889 Magnesium Fluoride and, 11:899-903 Magnesium Oxide and, 11:925,

929-931, 930, 933 Manganese and, 111:249 measurement of, 1:155-167; III: 125-128, 140 measurement of, introduced, 1:5 Mercury Cadmium Telluride, 11:138-147, 146 Mercury Cadmium Telluride and,

11:656,660 metals and, 111:233 Nickel and, 1:313 one-electron model and, I: 192 optical-property measurement and,

11:151 Orthorhombic Sulfur and, 111:900,

902-903 overlayers and, 1:99-100, 102 versus photon energy, II:144 versus photon energy in GaAs samples, 1:183 Polyethylene and, 11:957-959, 961,

963, 965-966, 968 Potassium Dihydrogen Phosphate and, 11:1006 Potassium Iodide and, 111:808, 810 Potassium Niobate and, III: 821, 823-824, 830 refractory compounds and, 11:303 Rhenium and, 111:278 Rhodium films and, 11:60-64 Rubidium Bromide and, 111:845-846,

848 Rubidium Iodide and, 111:857-860 Scandium Oxide films and, 11:60-64 Sellmeier dispersion equations, explained, II: 152 sensitivity and accuracy of, 11:64-67 Silicon, doped n-type and, III: 156 Silicon and, 1:99-100, 547, 569; 111:519, 522-524, 528, 531

44 Index of refraction (Continued) Silicon-Germanium alloys and, 11:608, 610-611; 111:538 Silver Bromide and, 111:555 Silver Chloride and, 111:553 Silver Gallium Selenide and, 111:574 Silver Gallium Sulfide and, 111:577 Silver Iodide and, 111:555 Sodium Fluoride and, II: 1022-1024 Sodium Nitrate and, 111:872 Strontium Fluoride and, 111:884, 886 Strontium Titanate and, II: 1035-1036 sum rules and, 1:36, 44-45 superlattice absorption and, II: 106 Tellurium and, 11:709, 711 Thallium Bromide and, 111:926 Thallium Chloride and, IIT:924 Thallium Iodide and, 111:927 Thallous Halides and, 111:929, 930 of thin dielectric film, II:84 of thin film layers, IT:57 of thin layers, discussed, 11:58, 71 Thorium Fluoride and, II: 1049-1052 Tin and, 111:269 Tin Telluride and, 11:726 Water and, II: 1059-1060 Yttrium Aluminum Gamet and, 111:964-966 Yttrium Oxide and, 11:1079-1080, 1082 Zinc Germanium Phosphide and, 111:639 Zinc Phosphide and, 111:609-610 Zinc Selenide and, 11:737, 739 Zinc Telluride and, 11:737, 745 Zircon and, 111:988 Zirconium Nitride and, 111:353 See also Individual critiques; Complex index of refraction See also Values of n Index of refraction, temperature dependence Aluminum and, 1:381 Aluminum Oxide and, 11:187 Aluminum Oxynitride and, 11:187 Germanium and, 1:466 Indium Antimonide and, 1:493

Subject Index Lead Sulfide and, 1:526 Magnesium Aluminum Spinel and, 11:187 Magnesium Oxide and, 11:187 Niobium and, 11:396 Potassium and, 11:364 Potassium Chloride and, 1:705 Silicon and, 1:548, 569 Silicon Dioxide and, IT:187 Sodium and, 11:345 Sodium Chloride and, II:187 Tantalum and, 11:409 Yttrium Oxide and, 11:187 Zinc Sulfide and, II:187 See also Individual critiques; Complex index of refraction Indium Antimonide (InSb) critique, 1:491-493; 111:53 index of refraction and, II: 148 optical-constants calculation of, 11:169 optical parameters for, 11:320 Rollin detector for, 1:158, 159 spectra in sample of, 1:103 spectroscopic measurements of, 111:76,

77-78 transparency frequency of, 1:231 values of n and k for, 1:9, 494, 495-502; 111:53, 54-55 Indium Arsenide (InAs) critique, 1:479-480; 111:50-52 dispersion analysis of, 1:479-480 free-carrier absorption and, I: 171 free-carrier concentration in samples of, 1:185 optical-constants calculations of, 1:177; 11:169 optical constants of, I:186 optical parameters for, 11:320 reflectance measurements on, I: 109 reststrahlen region and, 1:187-188 values of nand k for, 1:4, 481, 482-489; 111:50, 51-52 Indium Gallium Arsenide Phosphide

(In1_xGaxAsyP l-y) index of refraction and, 11:138-148, 139

45

Subject Index

Indium (In)

critique, 111:261-262 doping with, 11:638 interband absorption and, 1:204 values of n and k for, 111:263, 264-268 Indium Phosphide (InP)

critique, 1:503-506; 111:47-50 dispersion analysis of, 1:504 free-carrier absorption and, I: 171 free-carrier concentration of, I: 179 index of refraction and, 11:138-148 optical-constants calculation of, 1:177; II: 169 optical constants of, I: 181 optical parameters for, 11:320 photothermal deflection and, III: 111 quantum effects in, II: 127 relaxation rate for impurity scattering,

1:178 relaxation rate for polar scattering,

1:179 reststrahlen region and, I: 187-188 spectroscopic measurements of,

111:78-79 values of n and k for, 1:4, 507,

508-516; 111:47, 48-49 wavevector dependence and, 1:267 Indium Selenide (InSe) spectroscopic measurements of, 111:94 Inertial sum rules, 1:41, 43 tests for, 1:60-61 See also Sum rules Inference method Cadmium Selenide and, 11:561 Infrared absorption defects and, 1:263 by defects and disorders, 1:254-262 optical-constants calculation and, II: 153 Infrared absorption peaks Rubidium Bromide and, 111:846 Rubidium Iodide and, III: 861 Infrared band edge Calcium Fluoride and, 11:819 Infrared dispersions by plasmons, 1:263-267 Infrared lattice absorption Silicon-Germanium alloys and, 11:610

Infrared measurements index of refraction and, I: 155-156, 167 interband absorption and, 1:371 photolithography and, I: 162-163 Infrared propagation of surface-plasmon waves, 11:79 Infrared radiation dispersion of, 1:216, 263-267 Infrared reflection measurements Lithium Tantalate and, 111:778 Infrared region (IR) Aluminum Antimonide and, 11:501 Aluminum Arsenide and, 11:492 Aluminum Nitride and, 111:376 Aluminum Oxide and, 11:761 Antimony and, 111:274 Cadmium Germanium Arsenide and,

111:448 Cadmium Selenide and, 11:560-561 Cadmium Sulphide and, 11:579, 584-585 Calcium Carbonate and, 111:702 Carbon films and, 11:839 Chromium and, 11:375 Cobalt and, 11:435 Copper Gallium Sulfide and,

111:461-462 Cubic Silicon Carbide and, 11:706 Cupric Oxide and, 11:878 Cuprous Oxide and, 11:875 electromagnetic radiation propagation and, 1:14 electronic absorption edge and, 1:214 Indium and, 111:261 Iron Pyrite and, 111:507 Lead Fluoride and, 111:761 Lithium Tantalate and, 111:779 Magnesium Aluminum Spinel and,

11:884 Magnesium Fluoride and, 11:899 Magnesium Oxide and, 11:919,

934-939 Manganese and, 111:249 Niobium and, 11:397 optical properties of solids in,

1:214-215 Orthorhombic Sulfur and, 111:900-901 phonon absorption and, 1:234

46

Subject Index

Infrared region (Continued) Potassium and, 11:366 Potassium Iodide and, 111:808-809 Rubidium Bromide and, 111:846-847 Rubidium Iodide and, 111:857 semiconductor absorption and, 1:22 Silver Bromide and, 111:555 Silver Gallium Selenide and, 111:575 Sodium and, 11:356 Strontium Fluoride and, III: 883 Strontium Titanate and, II: 1035 Thallium Bromide and, 111:926 Thallium Chloride and, 111:924 Thallium Iodide and, 111:927 Thallous Halides and, 111:930 Thorium Fluoride and, II: 1049, 1051 Zinc Arsenide and, 111:595 Zinc Germanium Phosphide and,

111:639-640 Infrared windows in polar crystalline insulators, 11:178 Inhomogeneity effective-medium theory and, 1:104-105 See also Homogeneity Inhomogeneous layers and index measurement, 11:67 InP. See Indium Phosphide In phase defined, 11:46-48 InSb. See Indium Antimonide InSe. See Indium Selenide In situ evaporation technique Cesium Chloride and, 111:733 Cesium Fluoride and, 111:744 Potassium Bromide and, 11:992 In situ reflectance measurement Cesium Iodide and, 11:856 Palladium and, 11:470 Tin Telluride and, 11:727 vacuum deposition and, 11:205 Instrumentation use calorimetry and, 1:135-138 collinear-photothermal-deflectionapparatus and, III: 100 error determination and, 11:206 interferometers and, 111:101 optical-properties determination and,

1:92-95

photothermal-deflection apparatus and,

111:105-106, 108 synchrotron radiation and, 1:277 See also Calorimetry; Ellipsometry; Laser calorimetry; Monochromators; Reflectometers; Spectrometers Insulating films Berreman effects and, 11:115-116 Insulating systems sum rules and, 1:42-43 Insulator films on substrates, 1:5 Insulators Beta-Gallium Oxide and, 111:753 dispersion analysis and, 1:59 infrared dispersion and, 1:263 large band gap, models for, I: 199 lattice vibrations and, II: 178 Lead Fluoride and, 111:761 loss-function determination and, 11:300 phonons and, II: 178 sum rules and, 1:39 temperature-dependence of optical phonons in, II: 179 Insulators, optical properties of superlattices and, 11:99 Intensity described, 1:90 Intensity ratio measurement reflectivity, I: 18 Interband absorption Cesium and, 111:342 introduced, 1:4, 6 Manganese and, 111:249 metallic Aluminum and, 1:369-370,

380-381, 389 metals and, 111:233 one-electron model and, I: 190-198 optical properties of solids and,

1:189-209 rare-earth metals and, 111:289 Rhenium and, 111:278 Ruthenium and, 111:254 temperature dependence and, 1:381-382 Titanium and, 111:240 Interband electronic transitions, I: 18 Interband hole absorption Tellurium and, 11:711

47

Subject Index

Interband peaks Lithium and, 11:346 Potassium and, 11:365 Interband region Aluminum and, 1:376-377 Aluminum Antimonide and, 11:504 amorphous materials and, II: 169 Ashcroft-Sturm model of, 1:370, 381,

383 Cobalt and, 11:439 discussed, II: 151, 153 free-electron contributions in, II: 158 parallel-band effect and, 1:371-372 Silicon-Germanium alloys and, 11:609 Sodium and, 11:355 Interband transitions absorption and, 1:197 Aluminum Nitride and, 111:375 Aluminum Oxide and, 111:654 Beryllium and, 11:422 of bound electrons, II: 154 Cadmium Selenide and, 11:560 Cesium and, 111:342, 343 Cesium Iodide and, II: 856 Chromium and, 11:375 Cobalt and, 11:436, 439 in Copper, 1:205 electron excitation and, I: 189 Erbium and, 111:300 Graphite and, 11:451, 453 Hafnium Nitride and, 111:352 Iron and, 11:386 Iron Pyrite and, 111:511 joint densities of states and, 11:304 Lithium and, 11:347 Magnesium Fluoride and, 11:900 Mercury Cadmium Telluride and,

11:662,664 Niobium and, 11:397 Potassium Bromide and, 11:989 pressure considerations and, 1:382-383 rare-earth metals and, 111:290 Silicon and, 111:531 superlattices and, II: 107-108 Terbium and, 111:297 Thallium Bromide and, 111:925 Thallium Chloride and, 111:924 Titanium and, 111:240

Yttrium Aluminum Garnet and, 111:963 Yttrium Oxide and, II: 1080 Zirconium Nitride and, 111:352 Interferograms, I: 159 Sodium Chloride and, 1:776 Interferometers Brillouin scattering and, 111:123-124,

128-129 Calcium Fluoride and, 11:818 development of, 1:373 Gallium Phosphide and, 1:448 index-of-refraction measurements and, 1:156-161, 157 Lead Fluoride and, 111:763, 766 parameter definitions on, 1:128 Polyethylene and, 11:966 Silver Iodide and, 111:556 thermal effects and, I:125 thin-film parameters determination and,

1:113-133 Zinc Phosphide and, 111:610 Interferometry Barium Titanate and, 11:792 Cadmium Selenide and, 11:562 Cesium Bromide and, 111:718 Cesium Iodide and, II: 854 and index determination, 11:63 Lead Fluoride and, 111:764 Liquid Mercury and, 11:464 Lithium Fluoride and, 1:675 Lithium Tantalate and, 111:778 Magnesium Oxide and, 11:929 Palladium and, 11:470 Silicon Dioxide (glass) and, 1:750 Sodium Fluoride and, II: 1022 Thallium Chloride and, 111:924 Thorium Fluoride and, II: 1051 Yttrium Aluminum Garnet and, 111:964 Zinc Selenide and, 11:741 Zircon and, 111:987 Internal-reflection spectroscopy (IRS) and prism geometry, 11:76 International System of Units (SI units) discussed, 11:22 Intraband absorption of Metallic Aluminum, 1:394 Intraband transitions Calcium Fluoride and, 11:816

48

Intraband transitions (Continued) Graphite and, 11:453 plasma frequency for, 1:370 Zinc Arsenide and, III:595 Intrinsic material parameters index of refraction and, 11:181 optical-properties measurement and, 1:16-18 Inversions quantum-density-matrix formulation and, 11:135-136 and sum rules, 1:43 Inversion wavelength Cadmium Sulphide and, 11:582 Inl_xGaxAsyPl_y See Indium Gallium Arsenide Phosphide lon-assisted deposition (lAD) Thorium Fluoride and, II: 1053 See also Sample preparation Ionic conductivity Lead Fluoride and, 111:761 Ionic crystals reststrahlen band for, 1:219 Ionic motions in a solid frequency spectrum of, 1:215 Ionic solids absorption coefficients and, 1:249 absorption spectrum in, 1:243 electromagnetic radiation and, 1:218 photon-phonon coupling in, 1:216 Ionization Yttrium Oxide and, II: 1080 Ionized impurity scattering quantum density matrix and, I: 171 Ir. See Iridium IR determinations values of n and k for, 1:8, 377-379 Iridium (Ir) critique, 1:296 optical parameters for, 11:318 values of nand k for, 1:297, 298-303 Iron Disilicide (FeSi2 ) photothermal deflection and, 111:111-112 Iron (Fe) critique, 11:385-386 optical parameters for, 11:319 values of nand k for, 11:388, 389-396

Subject Index

Iron Pyrite (FeS 2) critique, 111:507-511 optical parameters for, III:199 oscillator models and, 111:512 values of nand k for, 111:513, 514-517 IRS. See Internal-reflection spectroscopy Irtran 1, 11:899 See also Magnesium Fluoride Irtran 4, 11:743 See also Zinc Selenide Irtran 5, II:931, 935 See also Magnesium Oxide Isoreflectance curves oblique incidence and, 1:73-75 parallel components and, 1:85-86 values of nand k and, 1:73, 78 Isotropic free carriers optical-properties measurement and, I: 18 Isotropic intensity transmission Bismuth Germanium Oxide and, 111:403 Bismuth Silicon Oxide and, 111:403 Isotropic materials Antimony and, 111:274 Gadolinium and, 111:295 Indium and, 111:261 local-field effects and, 1:203 Magnesium and, 111:235 Manganese and, 111:249 optical constants of, explained, 1:69-70 Ruthenium and, Ill:253 Tin and, 111:268 Titanium and, 111:240

J Jahoda's proof dispersion relations and, 1:56 Joint density of states absorption coefficients and, 1:235 Hafnium Nitride and, 111:352 interband absorption and, 1:209; 11:309 interband transitions and, 11:304, 309 optical-properties measurement and, 1:17 rare-earth metals and, 111:290 in transparent regime, 1:248 Zirconium Nitride and, 111:352 See also Density of states

49

Subject Index

Jones 2 X 2 matrix attenuated-total-reflection structures and, 11:82-83

K K. See Potassium Kane model Lead Tin Telluride and, 11:638 Mercury Cadmium Telluride and, 11:662 Kane theory band structure of, I: 171, 176 electron density of states and, I: 175 free-carrier absorption and, I: 173 in free-carrier absorption experiment,

1:173, 176 quantum-density-matrix formulation and, 11:130 KBr. See Potassium Bromide KCl. See Potassium Chloride KDP. See Potassium Dihydrogen Phosphate K-edge Potassium Bromide and, 11:992 K-edge spectra dielectric function and, 1:380 KH2P04 • See Potassium Dihydrogen Phosphate KI. See Potassium Iodide Kinetic energy and wave equations, 11:24 Kirchhoff's law emittance and, 1:22 See also Absorptance optical-properties measurement and,

1:22 KK analysis. See Kramers-Kr6nig analysis KNb0 3 • See Potassium Niobate Known incidence method Potassium Dihydrogen Phosphate and,

11:1005 KO-l, 11:899 See also Magnesium Fluoride K6sters prisms interferometers and, I: 120, 122,

123-126 Kramers-Kr6nig analysis of Aluminum, 1:378

of Aluminum Oxide, 11:761-762;

111:654 of Arsenic Selenide, 1:623 of Barium Fluoride, 111:683 of Barium Titanate, 11:789, 792 of Beryllium, 11:424-425, 426 of Beryllium Oxide, 11:807 of Bismuth Germanium Oxide, 111:404,

408 of Bismuth Silicon Oxide, 111:407-408 of Boron Nitride, 111:427-429 of Cadmium Germanium Arsenide,

111:446 of Cadmium Selenide, 11:559, 561-562 of Cadmium Sulphide, 11:580-581 of Cadmium Telluride, 1:410 of Calcium Carbonate, 111:701-702 of Calcium Fluoride, 11:821 calculations and, II: 128 of Carbon films, 11:839 of Cesium, 111:342-343 of Cesium Chloride, lll:731 of Cesium Iodide, 11:854-856 of Chromium, 11:374-375 of Cobalt, 11:437-439 of Copper Gallium Sulfide, 111:460-461 of Copper Oxides, II: 877 correction technique for, 11:925 dielectric functions and, 1:203 discussed, 1:276 dispersion analysis and, 1:59 dispersion relations and, 1:227-229 dispersion-relations sum rules and,

1:36,52 of Gallium Arsenide, 1:431 of Gallium Phosphide, 1:445 of Gallium Selenide, 111:474, 476 of Gallium Telluride, 111:490 of Germanium, 1:467 of Gold, 1:286 of Graphite, 11:450-452 of Hafnium Nitride, 111:353-354, 356 of hydrogenated amorphous Silicon,

11:167 of Indium Antimonide, 1:491 of Indium Phosphide, 1:503 introduced, 1:5-6 inversions and sum rules, 1:43

50 Kramers-Kronig analysis (Continued) of Iridium, 1:296 of Iron, 11:386 of Iron Pyrite, 111:507-508, 510 of Lead Fluoride, 111:763 of Lead Selenide, 1:517 of Lead Sulfide, 1:527 of Lead Telluride, 1:537 of Lead Tin Telluride, 11:637 of Liquid Mercury, 11:461, 464 of Lithium, II:346-347 of Lithium Fluoride, 1:675 of Lithium Niobate, 1:695 of Lithium Tantalate, 111:778-780 of Magnesium, 111:235 of Magnesium Oxide, 11:921-925, 927, 931, 935, 937-938 of Mercury Cadmium Telluride, II:658, 664 of metals, 111:234 of Molybdenum, 1:205, 304 of Nickel, 1:313 one-electron model and, I: 192 one-phonon temperature dependence and, 11:182 optical constants and, 11:152-153 optical-constants measurement and, 1:378-379 optical data and, 1:56 optical-properties determination and, 11:299-300 of Orthorhombic Sulfur, 111:901, 904 of Osmium, 1:324 of Palladium, II:470-471 of Platinum, 1:334 of Polyethylene, 11:958 of Potassium Bromide, II:990-991 of Potassium Chloride, 1:704 of Potassium Iodide, III: 807 of Potassium Niobate, III: 822 for reflection, 1:8, 108-110; 111:221 of Rhodium, 1:342 of Ruthenium, 111:254 of Selenium, II:692, 693 of Silicon, 1:551; 111:522, 531 of Silicon, doped n-type, III: 157, 163 of Silicon (type a), 1:571 of Silicon Carbide, 1:587

Subject Index

of Silicon Dioxide (type a), 1:750 of Silicon Dioxide (type-x), 1:720 of Silicon-Germanium alloys, 11:609; 111:538 of Silicon Monoxide, 1:776 of Silicon Nitride, 1:771 of Silver, 1:350 of Silver Bromide, 111:555 of Silver Chloride, 111:553-554 of Silver Gallium Selenide, 111:574 of Silver Gallium Sulfide, 111:576 of Silver Iodide, 111:555-556 of Sodium Chloride, 1:778 of Sodium Nitrate, III: 871, 872 of Strontium Fluoride, III: 883 of Strontium Titanate, II: 1036 superconvergence sum rules and, 1:38-39 of Tantalum, 11:409 of Tellurium, 11:712 of Thallium Bromide, 111:925 of Thallium CWoride, 111:923 of Thallium Iodide, 111:927 of Tin Telluride, 11:725, 726 of Titanium, 111:240 of Titanium Carbide, 11:309 of Titanium Dioxide, 1:797 of Titanium Nitride, 11:309 of Tungsten, 1:358 of Vanadium, 11:477, 478 of Vanadium Carbide, 11:309 of Vanadium Nitride, 11:309 of Water, II: 1061-1063 of Ytterbium, 111:302 of Yttrium Aluminum Garnet, 111:963, 965 of Zinc Arsenide, 111:595-596 of Zinc Germanium Phosphide, 111:638 of Zinc Phosphide, 111:610 of Zinc Selenide, 11:738-740 of Zinc Sulfide, 1:598 of Zinc Telluride, 11:744, 746 of Zircon, 111:987, 988 of Zirconium Nitride, 111:353-356 Kramers-Kronig dispersion relations index of refraction and, II:160-161 Titanium Dioxide and, II: 169

51

Subject Index Kramers-Kronig transform procedure (transformation) sum-rule tests and, 1:60-61 Kretschmann prism attenuated-total-reflection experiments and, 11:92-93 Kronecker products of space-group irreducible representations, 1:233 Kroneker delta quantum-density-matrix formulation and, 11:132 KRS-5. See Thallous Halide (KRS-5) KRS-6. See Thallous Halide (KRS-6) K-shell absorption f sum rules and, 1:62 Kubo formula for conductivity, III: 160

L Lambert absorption coefficients Water and, II: 1063 Lambert's law absorptance and, 1:139-140 extinction-coefficient measurement and, 1:81 Lamellae in experimental determination of optical parameters, I: 19-22 Lamellar systems formulas for, 1:24-33 Lamelliform, 1:14 described, 1:18-19 formulas and, I: 14-16 optical-properties measurement and, 1:19-24 radiation absorption and, 1:22 Langmuir-Blodgett films optics of, 11:76 Langmuir-Blodgett technique and molecular films, 11:87-88 and prism experiments, 11:82 Langmurr-Blodgettwedge attenuated-total-reflection experiments with, 11:93 Lanthanides, 111:287, 289-290 Lanthanum Oxide Yttrium Oxide and, 11:1079

LA phonons Gallium Arsenide and, 11:108-109 Laser analysis Gallium Antimonide and, 11:598 Lead Fluoride and, 111:763-764, 766 low-loss material for, 111:963 Polyethylene and, 11:964-966 Potassium Dihydrogen Phosphate and, 11:1005 Yttrium Oxide and, 11:1079 Zinc Germanium Phosphide and, 111:637 Laser applications Calcium Fluoride and, 11:815, 822-825 Cesium Iodide and, II: 853 Copper and, 11:876 Magnesium Fluoride and, 11:899 Laser beams and Xenon lamps Liquid Mercury and, 11:465 Laser calorimetry absorption-coefficient measurement and, 1:231 Aluminum Oxide and, 111:655-656 Arsenic Selenide and, 1:625 Arsenic Sulfide and, 1:643 Barium Fluoride and, 111:685 Calcium Fluoride and, 11:823-824 Cesium Iodide and, 11:856 electric-field considerations in, 1:143-147 Magnesium Aluminum spinel and, 11:886 Magnesium Oxide and, 11:928, 933 Sodium Chloride and, 1:778 Strontium Fluoride and, III: 884 -885 Thallous Halides and, 111:928, 930 thin-film absorptance measurements and, I: 135-153 typical, 1:136 wedged film and, I: 139-143 Laser coherence effects defeating, I: 137-138 in substrates, 1:137 Laser interferometers Calcium Fluoride and, II:818 film on substrate values of n and k, 1:4 response linearization and, I: 126 thin-film parameters determination and, I: 114, 116

52 Laser-window materials introduced, 1:9 Lattice band parameters Gallium Antimonide and, 11:600 Lattice constants Aluminum Gallium Arsenide and, 11:513 Barium Fluoride and, 111:683 Calcium Fluoride and, II:815 degenerate semiconductors and, 11:132-133, 138 Drude theory and, II: 148 index of refraction and, II: 143 Indium Gallium Arsenide Phosphide and, 11:143 Lead Fluoride and, 111:762 Lead Tin Telluride and, 11:637 Mercury Cadmium Telluride and, 11:138-147 Potassium Iodide and, III:807 quantum-density-matrix formulation and, 11:133, 136 Strontium Fluoride and, 111:883 Lattice dielectric constants Drude theory and, 1:174, 187; 11:126 reststrahlen contribution to, I: 187 Lattice dimensions Calcium Carbonate and, 111:701 Lattice disorder effect of, 11:118-120 Lattice modes Gallium Telluride and, 111:491 Lattice oscillators Mercury Cadmium Telluride and, 11:665 Lattice parameters Aluminum Nitride and, 111:373 Beta-Gallium Oxide and, 111:753 Potassium Niobate and, 111:821 Lattice-parameter variations Potassium Dihydrogen Phosphate and, 11:1006 Lattice properties Lithium Tantalate and, 111:777 Lattice reflection bands Gallium Antimonide and, 11:598 Lattice resonance semiconductor crystals and, 111:33

Subject Index

Lattice symmetry, I: 17 defects and, 1:254 selection rules for, I: 17 Lattice tail polar crystals and, 1:214 Lattice vibration absorption Thallium Chloride and, 111:924 Lattice vibration modes Bismuth Germanium Oxide and, 111:406 Bismuth Silicon Oxide and, 111:406 Lattice vibration region Iron Pyrite and, 111:507 Lattice vibrations absorption in solids and, 1:213 Aluminum Arsenide and, 11:491 Aluminum Oxide and, 111:656-657 Aluminum Oxynitride and, 11:778-779 Cadmium Sulphide and, 11:579 Calcium Fluoride and, 11:820 Cubic Silicon Carbide and, 11:706 electromagnetic radiation and, 1:214-215 Gallium Selenide and, 111:475 impurities and, 1:255 Magnesium Aluminum Spinel and, 11:886 Magnesium Fluoride and, 11:901 optical-properties determination and, 11:177 as phonons, 1:215 in polar crystalline insulators, 11:178 pole-fit model and, 11:185 Potassium Iodide and, 111:808-809 Silicon-Germanium alloys and, 111:538 Strontium Fluoride and, Ill:885 Tellurium and, 11:710 Yttrium Aluminum Garnet and, Ill:964-965 Yttrium Oxide and, 11:1080, 1082 Zinc Telluride and, 11:747 See also Multiphonon absorption processes; Superlattices Lattice vibrations, oscillator models Cadmium Telluride and, 1:411 Gallium Arsenide and, 1:429 Gallium Phosphide and, 1:446 index of refraction and, II: 179, 182

53

Subject Index Indium Antimonide and, 1:492-493 Indium Phosphide and, 1:504 Lead Telluride and, 1:536 Lithium Fluoride and, 1:677 Silicon Carbide and, 1:588 temperature dependence and, II: 190 Titanium Dioxide and, 1:796 Yttrium Oxide and, 11:184 Zinc Sulfide and, 1:600 Lattice waves as polaritons, 1:217 Law of index dispersion by a limited equation, 11:65 Law of motion, 1:41 sum rules and, 1:36, 40 See also Motion equation Layer characteristics of thin films, 11:57-58 Layered crystals Gallium Telluride and, 111:489 Layered structures, 1:18-19 absorptance of, 1:18-21 collective modes in, 11:112 optical-properties measurement and,

1:18-19 reflectance of, 1:18-21 transmittance of, 1:18-21 Layer-labeling system laser calorimetry and, 1:143, 144 Layer thickness calorimetry and, 1:148-152 of evanescent wavelengths, 11:82 superlattices and, 11:98, 109, 119 values of nand k and, 1:78, 81, 83-85 See also Thickness considerations Layer uniformity reflectance measurement and, I: 86 -87 Layer width factor material engineering and, 11:97 LB. See Langmuir-Blodgett technique LDPE (Low-density Polyethylene). See Polyethylene Lead Fluoride (PbF2 )

critique, 111:761-766 values of nand k for, 111:767, 768-776 Lead Lanthanum Zirconate Titanate Ceramics (PLZT) photothermal deflection and, III: 112

Lead (Pb) interband absorption and, 1:204 Lead Selenide (PbSe)

critique, 1:517-518 index of refraction and, II: 148 optical parameters for, 11:318 values of nand k for, 1:519, 520-523 Lead Sulfide (PbS)

critique, 1:525-528 index of refraction and, II: 148 optical parameters for, 11:318 values of n and k for, 1:529, 530-533 Lead Telluride (PbTe)

critique, 1:535-538 index of refraction and, II: 148 infrared reflection spectrum of, 1:265 optical parameters for, 11:318 values of nand k for, 1:540, 541-546 Lead Tin Telluride (PbSnTe)

critique, 11:637-639 optical parameters for, 11:323 values of n and k for, 11:640, 641-653 Least-squares fitting Aluminum Antimonide and, 11:502 Ammonium Dihydrogen Phosphate and, 11:1006 Beryllium Oxide and, 11:806 Beta-Gallium Oxide and, 111:754 crystalline material and, II: 164, 165,

166, 169, 171 Gallium Selenide and, 111:477 Graphite and, 11:450-451 Magnesium Oxide and, 11:921,931 Palladium and, 11:470 Potassium Dihydrogen Phosphate and,

11:1006 Silver Chloride and, 111:553 LEC. See Liquid encapsulation Czochralski technique L-edge structure dielectric function and, 1:380 Potassium Bromide and, 11:992 Leiss monochromator Lead Fluoride and, 111:762 Tellurium and, 11:712 Leurgans method laser calorimetry and, 1:143, 145 Li. See Lithium

54 LiF. See Lithium Fluoride Light Rubidium Iodide and, 111:858 Light, free-space velocity of described, I: 12 Light polarizations Potassium Niobate and, 111:822-823 Light scattering Copper Oxides and, 11:876 elastic, II: 112 inelastic, II: 112 Polyethylene and, 11:957 Sodium Fluoride and, II: 1022 Light-source emissions Magnesium Oxide and, 11:923 Light sources calorimetry and, I: 138, 140 coherence of, I: 116 reflectance and, 11:204 spectroscopy and, 1:158 stability of, I: 116 wedged-film laser calorimetry and, I: 141 Light waves linear momentum of, 11:35 LiNb0 3 • See Lithium Niobate Lindhard function Silicon, doped n-type and, III: 169 Linear polarization, 11:29 discussed, 11:28-30 phase shifting and, I: 160 reflection of, 11:47 Zinc Arsenide and, 111:596 Zinc Phosphide and, 111:610 Linear portion of plots extrapolation of, II: 163 Linear response theory sum rules and, 1:40 Line-tuned HF cw gas laser calorimetry and, I: 143 Lin-Okubo weighting function, 1:44 Liquid encapsulation Czochralski technique (LEC) Gallium Phosphide and, 1:445 See also Czochralski technique Liquid Mercury (Hg)

critique, 11:461-465 optical parameters for, 11:323 values of n and k for, 11:466, 467-468

SUbject Index Liquid Sulfur (S), 111:900 See also Orthorhombic Sulfur LiTa0 3 • See Lithium Tantalate Lithium cells Gallium Selenide and, 111:473 Lithium Fluoride (LiF)

critique, 1:675-678 absorption versus photon energy for,

1:243 dispersion analysis of, 1:676 optical constants of, II: 12-19 optical parameters for, 11:318 protective coatings of, 1:375 and surface-plasmon oscillation (SPO) resonance curves, 11:85 transparency frequency of, 1:231 values of n and k for, 1:679, 680-693 Lithium (Li)

critique, 11:345-348 absorption spectrum from, 1:258 Drude-model parameters, 11:350 index of refraction and, II: 180 optical parameters for, 11:319 values of n and k for, 11:349, 351-354 See also Hexagonal Lithium Lithium Niobate (LiNb0 3 ) critique, 1:695-697 versus Lithium Tantalate, 111:777 optical parameters for, 11:322, 328;

111:205 values of nand k for, 1:698, 699-702 Lithium Tantalate (LiTa0 3)

critique, 111:777-780 values of nand k for, 111:781, 783-805 Littrow mirror Silver Chloride and, 111:554 Littrow-type mount monochromator Potassium Dihydrogen Phosphate and,

11:1005 Lloyd's mirror experiment and phase changes, 11:48 Local-field effects ellipsometry and, I: 105 Localization in superlattices, II: 120 Longitudinal optical damping (LO) Aluminum Gallium Arsenide and,

11:515

55

Subject Index

Lithium Tantalate and, 111:779 Longitudinal optical frequencies (La) Aluminum Oxide and, 111:656 Magnesium Fluoride and, 11:901-902 Thallium Bromide and, 111:926 Thallium Chloride and, 111:925 Yttrium Oxide and, 11:1082 Longitudinal optical modes (La) Aluminum Gallium Arsenide and, 11:515 Aluminum Oxide and, 11:762-763 Boron Nitride and, 111:425-426 Cadmium Selenide and, 11:562 Gallium Selenide and, 111:476 Yttrium Aluminum Garnet and, 111:965 Longitudinal optical phonon energy (La) Aluminum Gallium Arsenide and,

11:513 Longitudinal optical phonon frequency (La) Yttrium Aluminum Garnet and, 111:986 Zinc Selenide and, 11:741 Longitudinal optical phonons (La) Brillouin scattering and, III: 141-142 Gallium Arsenide and, 11:108-109 Zinc Phosphide and, 111:610 Longitudinal optical resonance (LO) normal incidence reflectance and, II: 115 Longitudinal optical vibrational modes (La) Potassium Niobate and, III: 824 Longitudinal polar lattice mode (LO) Sodium Nitrate and, III: 873 Lorentzian energy dependence extinction coefficient and, 11:155 Lorentzians Brillouin scattering and, III: 127 electron-hole interaction and, 1:201 Lorentz-Lorenz equation attenuated-total-reflection experiments with, 11:89 ellipsometry and, I: 106 Orthorhombic Sulfur and, 111:902-904 Thallous Halides and, 111:929 values of nand k and, II: 179-180 Lorentz-oscillator models, 11:26 Aluminum Arsenide and, 11:491 Arsenic Selenide and, 1:623

Cadmium Telluride and, 1:410-411 Cesium Chloride and, 111:731 Cesium Iodide and, II: 854 formula for lattice vibrations, introduced, 1:7 Indium Antimonide and, 1:493 Iron Pyrite and, 111:507-508, 510,

512 optical constants and, I: 187-188 Potassium Bromide and, 11:990 Selenium and, 11:692 Silver Bromide and, 111:555, 560 Silver Chloride and, 111:554, 560 Silver Iodide and, 111:556, 560 Sodium Nitrate and, 111:873 superconvergence sum rules and, 1:38 Tellurium and, 11:710-711 values of n and k and, II: 179-180 Yttrium Aluminum Garnet and,

111:965 Zinc Selenide and, 11:738, 741 Zinc Telluride and, 11:738, 746 See also Oscillator models Lorentz-type equations Cadmium Sulphide and, 11:584-585 Loss function described, 11:294 determination of, 11:298-299 Titanium Carbide and, 11:304 Titanium Nitride and, 11:304 Vanadium Carbide and, 11:304 Vanadium Nitride and, 11:304 Loss tangent determination of, I: 163 Lossy material and complex numbers, 11:25 L-shell absorption fsum rules and, 1:62-63 LST. See Lyddane-Sachs-Teller relation Lucite (methyl methacrylate resin) Silver Chloride and, 111:554 Luminescence Aluminum Nitride and, 111:373, 375 Calcium Fluoride and, II: 816 Magnesium Oxide and, 11:926-927 Luminescence spectrum superlattices and, II: 108 Luminiferous ether concept, 11:37

56

Subject Index

Lyddane-Sachs-Teller relation (LST) Boron Nitride and, 111:427 Copper Oxides and, 11:877 described, 1:218 one-phonon temperature dependence and, 11:183 phonon symmetry and, 1:222 reststrahlen spectrum and, 1:221 Silicon Carbide and, 1:589

M Mach-Zehnder interferometer near-millimeter wavelength measurement and, 1:163-164, 164 Macroscopic dielectric functions superlattices and, II: 118 Macroscopic measurements of optical constants, 1:16-18 Macroscopic optical properties of matter describing of, I: 12 Macroscopic response in superlattice, II: 101 Magnesium Aluminum Spinel (MgAI 20 4 )

critique, 11:883-887 hemispherical emissivity for, 11:897 index of refraction and, II: 187 index of refraction of, 11:889 infrared lattice vibration frequencies for, 11:891, 892 values of dn/dT for, 11:897 values of n and k for, 11:890, 893-896 Magnesium Fluoride (MgF2)

critique, 11:899-903 infrared lattice vibration parameters for,

11:907 optical parameters for, 11:322 protective coatings of, 1:375 values of dne/dT for, 11:918 values of dno/dT for, 11:918 values of nand k for, 1:85, 87; 11:906,

908-917 Magnesium (Mg)

critique, 111:235 evaporated layer measurement of, I: 80 values of n and k for, 1:74; 111:236,

237-239 Magnesium Oxide (MgO)

critique, 11:919-939

extinction coefficients of, 1:228 far-infrared absorption and, 1:251 index of refraction and, II: 187 infrared reflectivity of, 1:224, 225, 226,

227 optical parameters for, 11:319 transparency frequency of, 1:231 values of n and k for, 11:942, 943-955 Magnetic fields impact on optical constants of, I: 16-17 interband absorption and, 1:190-191 Magnetic permeability described, 1:12 introduced, 1:12 Malus' plane of polarization defined, 11:29 Mandel' shtam-Brillouin scattering Cesium Iodide and, II: 856 Manganese (Mn)

critique, 111:249-250 values of nand k for, 111:250, 251-253 Many-body effects described, 1:190 Maser material Beta-Gallium Oxide and, 111:753 Material engineering superlattices and, 11:97 Material equations described, 1:12 Matrix multiplication model reflectance experiment and, 1:23-24 Matrix multiplication of layers model,

1:15-16 Matter dynamics interaction with light and, I: 36 Maxwell-Garnett theory ellipsometry and, I: 105-107 Palladium and, 11:471 Maxwell's equations attenuated-total-reflection structures and, 11:82 Brillouin scattering and, III: 124 effective-medium theory and, 1:105 electromagnetic wave propagation and,

1:11 introduced, 1:7 ion interaction and, 1:217 material equations and, I: 13

57

Subject Index

metals and, 111:234 optical constants and, I: 12 optical properties and, 11:62 optical-property measurement and, 1:13 parallel-sided slabs and, 1:14 plane-wave solution in, I: 170 for propagation through material, 1:24 propagation velocities and, I: 12 and surface-plasmon waves, 11:77 MBE technology superlattices and, 11:97 McDonald function Silicon, doped n-type and, III: 167 Measurement accuracy requirements for, 1:155-156 Measurement redundancy errors and, 1:72 Measurement techniques of Scandium Oxide (Sc02), II:59 Mechanical properties of Potassium Iodide, III: 807 Mechanical strength Aluminum Oxide and, lII:653 Mechanical stylus and index determination, 11:63 Melt-growth method Barium Titanate and, 11:789 Melting point of Aluminum Oxide, 111:653 of Barium Fluoride, 111:683 Cesium and, 111:341 Hafnium Nitride and, 111:351 of Indium, 111:261 of Lead Fluoride, 111:761, 762 of Potassium Iodide, III: 807 of Sodium Nitrate, III: 872 of Strontium Fluoride, 111:883 of Thallium Bromide, 111:938 of Thallium Chloride, 111:938 of Thallium Fluoride, 111:938 of Thallium Iodide, 111:938 Zirconium Nitride and, 111:351 Mercury Cadmium Telluride (HgCdTe) alloy optical parameters for, 11:323 Mercury Cadmium Telluride (Hg1_xCdxTe)

critique, 11:655-666 values of n and k for, 11:668, 669-689

Mercury Cadmium Telluride (HgxCd1_xTe) index of refraction and, 11:138-148,

146 Mercury (Hg) optical parameters for, 11:323 Mercury Telluride (HgTe) optical parameters for, 11:320 values of n and k for, 11:657 Metal gratings anomalies in the diffraction spectra of,

11:76 Metallic Aluminum (AI)

critique, 1:369-383 absolute values and, 1:392 dielectric function for, 1:402-406 Drude parameters for intraband absorption of, 1:394 extinction coefficient for, 1:388 f sum rules and, 1:61-62, 62 optical functions of, 1:60 optical-properties measurement of,

1:369 reflectance of, 1:389, 391 temperature dependence of, 1:382, 388 values of n and k for, 1:395-401 See also Aluminum Metals carrier concentration in, II: 178 critical points in, I: 196 free-carrier effects and, II: 177-178 Kramers-Kronig analysis and,

11:171-174 loss-function determination and, 11:300 optical-constants calculation and, II: 163 optical constants of, II: 153 optical-property calculation of, 11:151 photon absorption and, 11:153 Metals, optical properties of discussed, 1:40 interband absorption and, I: 192 superlattices and, 11:99 Metastable phases Beta-Gallium Oxide and, 111:753 Methyl Fluoride (C 13H 3F) laser index of refraction and, I: 162 Mg. See Magnesium MgAI 2 0 4 • See Magnesium Aluminum Spinel

58 MgF2 • See Magnesium Fluoride MgO. See Magnesium Oxide Michelson interferometers, I: 126 Cadmium Sulphide and, 11:585 Cesium Bromide and, 111:718 Cesium Iodide and, II: 854 Lithium Tantalate and, lll:778 with Mylar beam splitters, I: 155, 157 Polyethylene and, 11:966 Titanium Dioxide and, 1:795 Zircon and, 111:987 Microwave closed resonant cavity near-millimeter wavelength measurement and, I: 161 Microwave dielectric determination overmoded nonresident cavities and, 1:165 Microwave dielectrometer Barium Titanate and, 11:793 Microwave frequencies Yttrium Oxide and, 11:1083 Microwave region Barium Titanate and, ll:793 static dielectric constant and, II: 185 Microwaves measurement accuracy and, 1:155-156 Microwave techniques Potassium Iodide and, III: 81 0 Millimeter-wave region Lithium Tantalate and, 111:777 Minimum angle-of-deviation method Tellurium and, 11:711 Minimum-deviation measurements Aluminum Arsenide and, 11:490 Aluminum Oxide and, 111:655 Barium Fluoride and, 111:684 Barium Titanate and, 11:790 Bismuth Germanium Oxide and, 111:404 Bismuth Silicon Oxide and, 111:404 Cadmium Germanium Arsenide and, 111:447 Cadmium Sulphide and, 11:581 Calcium Fluoride and, 11:823 Cesium Bromide and, 111:718 Cesium Chloride and, 111:732 Cesium Iodide and, 11:855 Copper Gallium Sulfide and, Ill:461

Subject Index

Gallium Antimonide and, 11:598 Lead Fluoride and, 111:762 Magnesium Oxide and, 11:929 Potassium Dihydrogen Phosphate and, 11:1006 Silicon and, 111:523 Silver Chloride and, 111:554 Silver Gallium Selenide and, 111:575 Silver Gallium Sulfide and, 111:576, 577 Strontium Titanate and, II: 1036 Tellurium and, 11:711 Thallium Chloride and, 111:924 Thallous Halides and, 111:929 Yttrium Aluminum Garnet and, 111:964 Zinc Germanium Phosphide and, 111:639 Zinc Selenide and, 11:740 Zinc Telluride and, 11:745 Mirrors Cesium and, 111:341 Cesium Bromide and, 111:718 development of, 1:373 Fourier transform spectroscopy and, 1:157-158 interferometry and, I: 126 Iron Pyrite and, 111:507 Lead Tin Telluride and, 11:638 Lithium Fluoride and, 1:676 Magnesium Fluoride and, 11:900 Palladium and, 11:470 Potassium Dihydrogen Phosphate and, 11:1005 reflection measurements and, Ill:15 Silver Chloride and, 111:554 Sodium Chloride and, 1:776 Thorium Fluoride and, II: 1049, 1051 See also Reflectance; Reflection MKSA (meter, kilogram, second, ampere) system of units described, 11:22 Mks units conversions optical-properties measurement and, 1:13 Mn. See Manganese Mo. See Molybdenum Mobility Silicon, doped n-type and, 111:165, 167

Subject Index

Mode-stirring devices overmoded nonresident cavities and, 1:165 Molecular films attenuated-total-reflection experiments with, II:87-89 Mole ratio Lithium Niobate and, 1:695 Molten Sulfur, 111:901-902 See also Orthorhombic Sulfur Molybdenum (Mo) critique, 1:303-304 interband absorption and, 1:205, 207 optical parameters for, 11:319 values of nand k, II:87 values of n and k for, 1:305, 306-313 Momentum conservation, III: 125 Momentum transfer free-carrier absorption and, I: 172 interband transitions and, I: 191 optical-properties determination and, 11:296, 298 Monochromatic-beam transmittance, 1:230 Monochromatic measurements Polyethylene and, 11:966 Monochromatic radiation Zinc Selenide and, 11:739 Monochromatic shearing interferometers discussed, I: 122 Monochromators Arsenic Sulfide and, 1:644 Barium Titanate and, 11:790, 792 Bismuth Germanium Oxide and, 111:404 Calcium Fluoride and, II:815 extreme ulraviolet measurements and, 11:206 Gallium Arsenide and, 1:431 Iron Pyrite and, 111:509-510 Lead Fluoride and, 111:762 Magnesium Oxide and, 11:936 Polyethylene and, 11:958 Potassium Bromide and, 11:992 Potassium Chloride and, 1:703 Potassium Dihydrogen Phosphate and, 11:1005 Selenium and, 11:692 Silver Chloride and, 111:553

59 Strontium Titanate and, II: 1036 Tellurium and, 11:712 Titanium Dioxide and, 1:796 use of, 1:94 Wadsworth mount for, 11:992 Zinc Arsenide and, III:596 Zinc Phosphide and, 111:610 Monoclinic structures Gallium Telluride and, 111:489, 492 Mori projection ansatz, III: 161 Morphology optical-properties measurement and, 1:375-376 Morse interatomic potential temperature dependence and, II: 190 Morse potentials, 1:244-245 Motion equation dielectric constants and, II: 128 infrared dispersion and, 1:264 lattice waves and, 1:220 quantum density matrix of, I: 171-173 Motion of atoms optical-branch vibrations and, 1:215 MQW material. See Multiple-quantum-well material MSFTI. See Multiple-slit Fourier-transform interferometry Multibounce effects of substrates, I: 147 Multilayer systems optical response of, 11:98-101 Multiphonon absorption processes, 1:232-254 Aluminum Oxide and, 111:656-657 Aluminum Oxynitride and, 11:778-779 Arsenic Selenide and, 1:624 Arsenic Sulfide and, 1:642 Barium Fluoride and, 111:685 below fundamental band gap, 1:4 Bismuth Germanium Oxide and, 111:406 Bismuth Silicon Oxide and, 111:406 Boron Nitride and, III:428 Cadmium Sulphide and, 11:584 Cadmium Telluride and, 1:412 Calcium Fluoride and, 11:819-820 Cesium Iodide and, II:856

60 Multiphonon absorption processes (Continued) Cubic Carbon (Diamond) and, 1:667 Gallium Arsenide and, 1:430 Gallium Phosphide and, 1:446 Germanium and, 1:465 Indium Arsenide and, 1:480 Indium Phosphide and, 1:505 Lithium Fluoride and, 1:676 Magnesium Aluminum Spinel and, 11:885-886 Magnesium Fluoride and, 11:901-902 Magnesium Oxide and, 11:935 in polar crystalline insulators, 11:178 pole-fit model and, II: 185 Potassium Bromide and, 11:989 Potassium Chloride and, 1:703 Potassium Iodide and, 111:808, 809 Rubidium Bromide and, 111:847 Selenium and, 11:692 Silicon and, 1:550 Silicon-Germanium alloys and, IIT:538 Sodium Chloride and, 1:777 Sodium Fluoride and, II: 1023 Strontium Fluoride and, 111:885 Tellurium and, 11:711 transparency and, 1:213-214 Yttrium Oxide and, 11:1082 Zinc Selenide and, IT:742 Zinc Sulfide and, 1:601 See also Phonon absorption Multiphonon difference bands pole-fit model and, II: 185 Multiphonon electronic transitions transparency and, 1:213-214 Multiphonon model quantum-mechanical Morse potential, 11:190-199 Multiphonon sum-band model temperature dependence and, II: 198 Multiphonon sum bands pole-fit model and, II: 185 Multiphonon transitions Zinc Phosphide and, 111:610 Multiple-beam interferometers Lead Fluoride and, 111:764

Subject Index

Multiple-quantum-well material (MQW) superlattices and, IT: 102 Multiple-slit Fourier-transform interferometry (MSFfI), 11:247-274 See also Fourier-transform interferometry Murmann's formulae Silver Gallium Selenide and, 111:575

N N 2 • See Nitrogen gas Na. See Sodium NaBr. See Sodium Bromide NaCl. See Sodium Chloride NaP. See Sodium Fluoride NaN0 3 . See Sodium Nitrate Nb. See Niobium NbC. See Niobium Carbide NbN. See Niobium Nitride Ne. See Neon Near-infrared region (NIR) Aluminum Oxynitride and, 11:778-779 Barium Fluoride and, 111:684 Barium Titanate and, 11:791 Calcium Fluoride and, 11:816 Cuprous Oxide and, 11:876 Lead Fluoride and, 111:762 Magnesium Aluminum Spinel and, 11:884 Magnesium Oxide and, IT:926-929 Mercury Cadmium Telluride and, 11:656-659 Orthorhombic Sulfur and, 111:901 Polyethylene and, 11:959-960 Potassium Iodide and, III: 808 Selenium and, 11:692 Silicon and, 111:519 Silicon-Germanium alloys and, 111:537 Sodium Nitrate and, III: 872 Tellurium and, 11:711 Water and, 11:1060 Yttrium Aluminum Garnet and, 111:964 Zinc Selenide and, 11:739 Zircon and, 111:988-989 Near-millimeter wave analysis Polyethylene and, 11:968

61

Subject Index Near-millimeter wave properties (NMMW) introduced, 1:155 Near-normal-incidence reflectivity Bismuth Germanium Oxide and, 111:404 Iron Pyrite and, 111:509 Lead Fluoride and, 111:762 Orthorhombic Sulfur and, 111:900, 902 Near-ultraviolet region (NVV) Aluminum Oxide and, 111:655 Cesium Bromide and, 111:719 Magnesium Oxide and, 11:926-929 Thorium Fluoride and, II: 1053 Zinc Selenide and, 11:737 Zinc Telluride and, 11:737 Nebraska conventions described, 11:22 Neel temperature Erbium and, 111:300 rare-earth metals and, 111:288 Samarium and, 111:293 Nematic liquid crystals surface tilt angle of, 11:90-93 Neon (Ne) excitons and, I: 199 Neutron-diffraction studies Aluminum Oxynitride and, 11:777 NH4 H 2P04 . See Ammonium Dihydrogen Phosphate Ni. See Nickel Nickel (Ni)

critique, 1:313-314 attenuated-total-reflection experiments with, 11:87 optical parameters for, 11:318 values of n and k for, 1:316, 317-323 NINA Lead Fluoride and, 111:763 Niobium Carbide (NbC) dielectric properties of, 11:303 Niobium (Nb)

critique, 11:396-397 optical parameters for, 11:319 values of nand k for, 11:399;

11:400-408 Niobium Nitride (NbN) dielectric properties of, 11:303

Nitrogen gas (N2) overlayer thickness and, I: 102 NMMW properties. See Near-millimeter wave properties Noise interference and ellipsometry, I: 123 Noise reduction Potassium Iodide and, III: 810 Nondispersive measurements Fourier transform spectroscopy and,

1:157-158 Nonlaser coherence interferometry and, I: 116 Nonlaser interferometers described, I: 117 Nonlinear crystals Potassium Dihydrogen Phosphate and,

11:1005 Nonlinear regression technique Cadmium Sulphide and, 11:581 Non-polarized radiation reflection and, 1:70 Normal distribution functions density of states and, 1:245 Normal incidence Sodium Nitrate and, 111:872 Normal-incidence illumination calorimetry and, I: 149 Normal-incidence reflection Beryllium and, 11:424 Cadmium Sulphide and, 11:583 Cesium and, 111:343 Cubic Cadmium Sulphide and, 11:586 Graphite and, 11:449, 451 Liquid Mercury and, 11:465 Magnesium Oxide and, 11:922 optical-properties determination and,

1:93 optical-properties studies and, 11:110,

120 Palladium and, 11:470, 471 refractive-index measurement and,

1:70-71 Water and, II: 1060 NOTS absolute reflectometer Lead Fluoride and, 111:764 n-Si. See Silicon, doped n-type

62

Subject Index

n-type crystals Cadmium Germanium Arsenide and, 111:447 Copper Gallium Sulfide and, 111:459 Nuclear resonance water in Silicon films and, 1:138 Null ellipsometers discussed, 1:93-94, 102 overlayers and, 1:99 Null method Liquid Mercury and, 11:463

o Oblique-incidence reflection refractive-index measurement and,

1:71-75 use of, 11:69 Ohm's law, 1:12 optical constants and, I: 17 optical-properties measurement and,

1:17 One-electron effects described, I: 190 One-electron model hyperbolic exciton and, 1:201 interband absorption and, I: 190-198 One-phonon absorption processes defects and, 1:254 structurally disorders solids and, 1:214 One-phonon red wing difference bands pole-fit model and, 11:185, 187-188 One-phonon region Barium Fluoride and, 111:685 Cadmium Germanium Arsenide and,

Ill:448 Potassium Iodide and, lll:809 Strontium Fluoride and, 111:885 See also Fundamental lattice vibrations One-phonon transition Silver Chloride and, 111:554 Opacity Barium Fluoride and, 111:685 Cesium Bromide and, 111:719 introduced, I:8 Strontium Fluoride and, 111:885 See also Transparency Optical characteristics superlattices and, 11:98

Optical conductivity versus dc conductivity, I: 17 free-carrier absorption and, I: 173 Metallic Aluminum and, 1:394 perturbation theory and, I: 175 Optical-constants determination access via computer and, 111:12 angles of incidence and, I: 72 Brillouin scattering and, III: 121-152 causality principles and, 1:228 experimental techniques, introduced,

1:4 explained, 11:151-153 far- IR technique of, 1:4 interferometry and, 11:247-274 macroscopic equations and, 1:24 measurement of, discussed, 1:276 photoemmision and, 11:279-292 sample preparation, introduced, 1:4 submillimeter-wave region technique of, 1:4 in superlattice systems, 11:99 See also Values of n and k Optical data analysis of sum rules, 1:55-64 Optical-density measurements Cesium Bromide and, 111:719 Optical energy band gap discussed, 11:152-153, 166, 169 extinction coefficients and, II: 158 Optical fibers Brillouin scattering in, 111:146-152, 148-150, 152 Optical functions dispersion analysis and, 1:56-60 Optical modes of crystals, 1:216 Optical phonons versus acoustical phonons, 1:253 occupation probability, Drude theory and, 1:174 reflection spectrum and, 1:227-229 Optical profilometer Palladium and, 11:469 Optical propagation stopgaps and, 11:101 Optical properties calculation, categories for, II: 152

63

Subject Index

determination of, overlayer effects and,

1:96-104 importance of, 11:293 measurement of, and calorimetry, I: 135 measurement of, formulas for interpreting, 1:24-33 measurement of, introduced, I: 11-16 measurement of, oxide formation and,

1:375 of metals, discussed, 111:233 one-electron model and, 1:190-198 of rare-earth metals, 111:288-297 of solids, limitations of, 1:36 See also Individual critiques Optical-properties determination physical processes of, II: 177 of superlattices, II: 122 Optical sum rules physical interpretation of, 1:36-45 Optical thickness film index, 1:150-152 Optical transition modes direct, II: 163 forbidden, II: 163 indirect, II: 163 Optics nomenclature explained, 1:8 Optovac Strontium Fluoride and, 111:884 Organic thin films optics of, 11:76 Orthorhombic crystals Polyethylene and, 11:959 Orthorhombic structures Lead Fluoride and, 111:761 Potassium Niobate and, III: 821 Orthorhombic Sulfur (a-S)

critique, 111:899-905 values of nand k for, 111:908, 909-922 Orthorhombic Thallium Iodide (TlI)

critique, 111:927-928 values of nand k for, 111:951-955 Os. See Osmium Oscillator dispersion analysis Beryllium Oxide and, 11:805 Calcium Carbonate and, 111:702 Oscillator models absorption coefficients and, 1:244-245 Aluminum Antimonide and, 11:503

Aluminum Arsenide and, 11:490 Arsenic Sulfide and, 1:641, 643 Barium Fluoride and, 111:685-686 Barium Titanate and, 11:792 Bismuth Germanium Oxide and, 111:407 Bismuth Silicon Oxide and, 111:407 Boron Nitride and, 111:428-429 Cadmium Germanium Arsenide and,

111:448 Cadmium Selenide and, 11:561, 563 Cadmium Telluride and, 1:410-411 Cesium Fluoride and, 111:744 Copper Gallium Sulfide and,

111:461-462 Copper Oxides and, II: 877 Cubic Silicon Carbide and, 11:706 Gadolinium and, 111:296 Gallium Antimonide and, 11:598 Gallium Arsenide and, 1:429-430 Gallium Phosphide and, 1:446 Gallium Selenide and, 111:476 Graphite and, 11:452-453 Indium Phosphide and, 1:505-506 Iron Pyrite and, 111:507, 512 Lead Selenide and, 1:517 Lead Telluride and, 1:536 Lead Tin Telluride and, 11:638 Lithium Fluoride and, 1:676-677 Lithium Niobate and, 1:696 Magnesium Oxide and, 11:935, 937 Potassium Iodide and, III: 809 Potassium Niobate and, 111:823-824 Selenium and, 11:692 semiconductors and, 111:32, 34-55 Silicon and, 111:532-533 Silicon Carbide and, 1:588 Sodium Chloride and, 1:777 Sodium Nitrate and, 111:872-873 Strontium Fluoride and, 111:885-886 superlattices and, 11:99 Tellurium and, 11:710 Thallium Bromide and, 111:925-926 Thallium Chloride and, 111:924-925 Thallium Iodide and, 111:927 Titanium Dioxide and, 1:796 Yttrium Aluminum Gamet and, 111:965 Zinc Germanium Phosphide and,

111:639

64

SUbject Index

Oscillator models (Continued) Zinc Phosphide and, 111:609 Zircon and, 111:988 See also Lorentz-oscillator models Oscillators optical-properties measurement and,

1:17-18,229 See also Damped oscillators; Lattice vibrations, oscillator models Oscillator strength Aluminum Oxide and, 11:763 crystal potential energy and, 1:245 f sum rules and, 1:61-63 Lithium Tantalate and, 111:779 optical definition of (equation), 1:37 sum rules and, 1:37 superlattices and, II: 108 temperature and, 1:381 Osmium (Os) critique, 1:324 optical parameters for, 11:321 values of nand k for, 1:325, 326-333 Out of phase defined, 11:46-48 Overflow method Liquid Mercury and, 11:462 Overlayer effects in optical-properties determination, 1:96-104 Overmoded nonresonant cavities near-millimeter wavelength and, 1:165 Oxidation Aluminum, 1:373-375 Aluminum Nitride and, 111:373 ellipsometry and, 1:97-99, 101 Hafnium Nitride and, 111:355 metals and, 111:234 optical constants of, 1:85-87 in polarized nonnormal-incidence reflection techniques, 1:4 rare-earth metals and, 111:290 Titanium and, 111:240 Zirconium Nitride and, 111:355

p Packing density Lead Fluoride and, 111:764 Thorium Fluoride and, II: 1053

Palladium (Pd)

critique, 11:469-472 attenuated-total-reflection experiments with, 11:87 optical parameters for, 11:318 values of nand k for, 11:473, 474-476 Parabolic dispersion relation quantum-density-matrix formulation and, II: 134-135 Parallel alignment Cadmium Sulphide and, 11:580 Graphite and, 11:449 in liquid crystals, ll:90-92 Polyethylene and, 11:961 Parallel-band effect interband spectrum and, 1:371-372 Parallel beams defeating laser-coherence effects and,

1:137 Parallel measurements Lithium Tantalate and, 111:778 reflection and, 1:70 Parallelopipeds isoreflectance curves and, 1:74 Parallel polarization Lithium Tantalate and, 111:779 Maxwell's equations and, 1:14 optical-properties measurement and,

1:14 Parallel polarized light Boron Nitride and, 111:425 Potassium Niobate and, 111:824 Parallel-sided slabs formulas for, 1:25-26, 28 lamelliform and, 1:19-21, 25-26 Parametric oscillation Zinc Germanium Phosphide and,

111:637 Particle-physics conventions, 11:36-37 Partition function absorption coefficients and, II: 187 Pauli principle f sum rules and, 1:63 sum rules and, 1:51 Pb. See Lead PbF2 • See Lead Fluoride PbS. See Lead Sulfide PbSe. See Lead Selenide

65

Subject Index

PbSnTe. See Lead Tin Telluride PbTe. See Lead Telluride Pd. See Palladium Periclase. See Magnesium Oxide Perkin-Elmer grating spectrometer Tellurium and, 11:710-711 Perkin-Elmer infrared spectrometer Silver Chloride and, 111:554 Perkin-Elmer Model 301 spectrophotometer Iron Pyrite and, 111:507 Perkin-Elmer 112 monochromator Iron Pyrite and, 111:509 Perkin-Elmer PE 350 Spectrophotometer Silver Iodide and, 111:556 Perkin-Elmer spectrometer Lead Fluoride and, 111:763, 766 Permeability of free space, 11:23 Permittivity Barium Fluoride and, 111:685 Barium Titanate and, 11:792-793 and evanescent waves, 11:79 of free space, 11:23 Potassium Iodide and, III: 807, 809 Potassium Niobate and, 111:824 Strontium Fluoride and, 111:885 in wave guides, II: 85 Perovskite structures Potassium Niobate and, III: 821 Perpendicular polarization Boron Nitride and, 111:425-426 Maxwell's equations and, I: 14 optical-properties measurement and,

1:14 See also TM modes Perturbation, oscillatory external sum rules and, 1:53 Perturbation energy dielectric constants and, II: 128-130 Perturbation technique Arsenic Sulfide and, 1:643 Perturbation theory, III: 160 in free-carrier absorption experiment, 1:172-173 interband absorption and, I: 197 time-dependent, I: 171; II: 154-155 PES. See Photoacoustic spectroscopy PET. See Polyethylene terephalate films

Phase changes Cobalt and, 11:438 and Fresnel coefficient, 11:41-50 and p polarization, 11:45 for steel, 11:44 Phase coefficient, 111:122 Phase conjugation interferometry and, I: 116-117 Phase differences dispersion relations and, 1:229 Phase incoherence insufficient resolution of, 1:21 optical-properties measurement and,

1:21-24 Phase modulation signal-to-noise ratio and, I: 157, 158,

159 Phase shifts dispersion analysis and, 1:59 interferometry and, 1:115-116,

123-124, 126, 131, 163 Polyethylene and, 11:961 in wave guides, 11:86 Phase transformations Lead Fluoride and, 111:762 Yttrium Aluminum Garnet and,

111:963 Phase transitions Cerium and, 111:292 Gadolinium and, 111:295 Gallium Selenide and, Ill:477 Lithium Tantalate and, 111:779 metals and, 111:234 Potassium Niobate and, 111:821 Silver Iodide and, 111:556 Sodium Nitrate and, 111:871 Phonon absorption Aluminum and, 1:381 interband absorption and, I: 197-198,

201 introduced, 1:6 in polar insulators, 1:45 scattering and, I: 196 Silicon and, 1:209 Phonon-assisted electronic transitions transparency and, 1:213-214 Phonon energy Silicon-Germanium alloys and, 11:608

66

SUbject Index

Phonon frequencies Aluminum Gallium Arsenide and,

11:513 Cadmium Selenide and, 11:561 Tellurium and, 11:710 Zinc Phosphide and, 111:610 Phonon frequency Magnesium Aluminum Spinel and,

11:885 Phonon processes interaction with infrared radiation and,

1:215, 233, 255 Phonon resonances in infrared spectrum, 1:232 Phonons absorption probability and, 1:234 Aluminum Gallium Arsenide and,

11:515 Cadmium Sulphide and, 11:579 defined, 1:215 Erbium and, 111:301 Mercury Cadmium Telluride and,

11:665-666 scattering and, II: 128 semiconductor superlattices and,

11:108-109 Terbium and, 111:298 Phonon scattering Drude theory and, I: 171 Phonon self-energy functions Lithium Tantalate and, 111:779 Phonon sidebands electron-hole interaction and,

1:202-203 Photoabsorption Aluminum Oxide and, 111:654 Photoacoustic measurements Calcium Fluoride and, 11:823 Magnesium Oxide and, II:928 in semiconductors, III:59-94 Silicon and, 1:549 Photoacoustic spectroscopy (PES),

III:60-67 Photoconductivity measurements Orthorhombic Sulfur and, 111:901 Zinc Phosphide and, 111:609 Photodiodes ellipsometers and, 1:94

Photoelastic modulators optical-properties determination and,

1:94-95 Photoemission technique, 11:279-292 Photolithography crossed-wire meshes and, 1:162 Photoluminescence Calcium Fluoride and, II:816 Photometers, I: 126 Magnesium Oxide and, 11:927 Photometric measurements development of, 1:92-95 Mercury Cadmium Telluride and,

11:655 metals and, 111:234 overlayers and, I: 103 Vanadium and, 11:478 Photomultipliers ellipsometers and, 1:94 interferometry and, III: 129 Magnesium Oxide and, 11:922 Zinc Arsenide and, 111:596 Zinc Phosphide and, 111:610 Photon absorption extinction coefficients and, II: 157 quantum-mechanical absorption theory and, II: 152, 154 temperature dependence and, II: 198 Photon density matrix Drude theory and, I: 173 Photon energies Drude theory and, I: 171 fundamental band gap and, 1:213-267 Hafnium Nitride and, 111:357 interband absorption and, I: 196 metals and, 111:234 rare-earth metals and, 111:289-291 Zirconium Nitride and, 111:357 Photon energy bound-electron transitions and, 11:152 Carbon films and, II:83 8 in compound semiconductors, II: 127 versus index of refraction, II:144 interband region and, II: 152 perturbation theory and, II: 154 Photon-energy region Cadmium Selenide and, 11:560 Gallium Antimonide and, 11:598

67

Subject Index

Lead Fluoride and, 111:763 Potassium Niobate and, 111:822 Silicon-Germanium alloys and, 11:608;

111:537 Zinc Arsenide and, 111:595 Zinc Phosphide and, 111:609 Photonic band structure concept of, II: 101 Photon interactions quantum-mechanical treatment of,

11:152-153 Photon-phonon interaction plasma effects and, 1:215 Photons scattering mechanism and, 11:128-129,

131 Photopyroelectric spectroscopy (PPES),

111:67-69 Photothermal measurements of absorption in thin layers, 11:70 extinction coefficients measurements and, 111:99-116 in semiconductors, 111:59-94 Photovoltaic devices Gallium Selenide and, 111:473 Physics, solid-state Anderson localization and, II: 120 Physics of condensed state introduced, I: 12 Piezoelectric photoacoustics, 111:64-67 Piezoelectric transducers, 1:122 interferometry and, 1:126 Piezo-modulation experiments parallel-band transition and, 1:372 Planar carbon chains, 11:957 Planck's constant absorption coefficients and, II: 187 optical-properties determination and,

11:296 Thallous Halides and, 111:928 Plane of incidence (POI), 1:14 electromagnetic waves and, 1:90 optical-properties measurement and, I: 14 reflection and, 1:70 Sodium Nitrate and, 111:872 Plane of polarization defined, 11:29 Faraday-cell modulation of, 1:93

Plane-parallel samples defeating laser-coherence effects and,

1:137 interferometry and, 1:160, 162 Plane-polarized waves described, 11:29 Plane-wave solutions for wave equations, 11:23 Plasma absorption process, 1:4 ellipsometry and, 1:107, 378 finite-energy sum rules and, 1:45-46,

49, 54 Plasma frequency Aluminum and, 1:370 Aluminum Gallium Arsenide and,

11:515 Beryllium and, 11:423-424 Cadmium Sulphide and, 11:581 Carbon films and, 11:837-838 Cobalt and, 11:439 Drude theory and, 11:126-127 Erbium and, 111:301 Gadolinium and, 111:296 Graphite and, 11:452 Indium Antimonide and, 1:492 Lead Selenide and, 1:517 Lead Tin Telluride and, 11:638 Magnesium Oxide and, 11:925 rare-earth metals and, 111:290 Samarium and, 111:294 semiconductors and, 111:32 Silicon, doped n-type and, 111:163,

164 Silicon and, 1:552 Silicon Carbide and, 1:588 superlattice reflective properties and,

11:113 Tellurium and, 11:710 Tin Telluride and, 11:725-726 Ytterbium and, 111:303 Zinc Sulfide and, 1:599 Plasma oscillations Aluminum Oxide and, 111:654 Yttrium Aluminum Garnet and,

111:963 Plasma resonances Hafnium Nitride and, 111:353-354 rare-earth metals and, 111:289

68

Subject Index

Plasma resonances (Continued) Thulium and, 111:302 Zirconium Nitride and, 111:353-354 Plasmon energy Zinc Telluride and, 11:744 Plasmon resonance attenuation of, 11:93 Plasmons described, 11:294 Dysprosium and, 111:299-300 exciting of, 1:189 infrared dispersion by, 1:263-267 rare-earth metals and, 111:290 superlattices and, 11:98 Terbium and, 111:298 Ytterbium and, 111:303 Plasmon surface polaritons described, 11:89 Plasmon waves on smooth surfaces excitation methods for, 11:79 Platinum (Pt)

critique, 1:333-334 attenuated-total-reflection experiments with, 11:87 optical parameters for, 11:318 superlattices and, II: 118 values of n and k for, 1:335, 336-341 PLZT. See Lead Lanthanum Zirconate Titanate Ceramics PMMAdye attenuated-total-reflection experiments with, 11:88 Pockels cells optical-properties determination and,

1:94,95 Pockels coefficient, III: 122 POI. See Plane of incidence Poisson's ratio, 111:122 Polar crystals absorption gap in, 1:214 anharmonic radiation and, 1:232 infrared dispersion by, 1:215-226 optical properties of, 11:178 Strontium Titanate and, II: 1036 Polarimetric technique Cobalt and, 11:439 Orthorhombic Sulfur and, 111:900

Polarimetry Aluminum and, 1:374 Erbium and, 111:300 Gadolinium and, 111:295-296 optical-properties determination and,

1:90-91 Orthorhombic Sulfur and, 111:901 Terbium and, 111:297 Polar insulators finite-energy sum rules and, 1:53 Polaritons described, 1:217 superlattices and, 11:98 Polarizabilities on atomic scale introduced, I: 12 Polarization Barium Titanate and, 11:789 Beta-Gallium Oxide and, 111:754 birefringence and, I: 164-165 Boron Nitride and, 111:425, 428 Cadmium Germanium Arsenide and,

111:446 Cadmium Selenide and, 11:562 Cadmium Sulphide and, 11:580 Calcium Carbonate and, 111:701 Copper Gallium Sulfide and, 111:460 discussed, 11:26-37 ellipsometry and, 1:90 Gadolinium and, 111:294-295 Gallium Telluride and, 111:490 Graphite and, 11:450 index of refraction and, II: 180 interferometry and, I: 115-116, 124 interferometers and, 1:157, 158 Lithium Tantalate and, 111:777 Mach-Zehnder interferometers and,

1:163 Magnesium Fluoride and, 11:899-900 Orthorhombic Sulfur and, 111:900,

902-905 Potassium Niobate and, 111:821, 822 reflectance and, 1:70 Sodium Nitrate and, III: 871 Tellurium and, 11:711 Zinc Germanium Phosphide and,

111:640 Zinc Phosphide and, 111:609 Zircon and, 111:987, 988-989

Subject Index

See also Parallel polarization; Perpendicular polarization Polarization by reflection discovery of, 11:27 Polarization ellipse attributes of, 1:90 reflectometry and, 1:90 Polarization solutions models for, I: 14-16 Polarization states Dysprosium and, 111:299 of electromagnetic waves, 11:27 reflectance and, 1:72 reflectometry and, 1:93 Polarization techniques Terbium and, 111:297 Polarization terminology discussed, 11:27 Polarized light Bismuth Germanium Oxide and, 111:403 Bismuth Silicon Oxide and, 111:403 Cadmium Selenide and, 11:559, 563 Gallium Selenide and, 111:474 Hafnium Nitride and, 111:355 Selenium and, 11:692 Zirconium Nitride and, 111:355 See also Unpolarized light Polarized light measurements Cadmium Selenide and, 11:559, 563 Polyethylene and, 11:959 Potassium Bromide and, 11:992 Selenium and, 11:692 Polarized non-normal-incidence reflection techniques in ultraviolet-visible spectral region, 1:4 Polarized radiation measurements Copper Gallium Sulfide and, 111:460 Polarized synchrotron light measurements Strontium Titanate and, II: 1036 Polarized synchrotron radiation Cadmium Selenide and, 11:562 Polarizers described, 1:93-95 Polar-optical-mode scattering Drude theory and, I: 176 Polar phonons superlattice excitons and, II: 104

69 Polar scattering Drude theory and, I: 171-172, 174 free-carrier absorption and, I: 176-179 Polar semiconductors Drude-Zener theory in, 1:169-188 infrared dielectric function for, 11:24 optical properties of, 11:26 Polar transitions Lithium Tantalate and, 111:779-780 Pole-fit model index of refraction and, II: 181-182, 185 Polish-abolishment opinion, 1:8 Polishing techniques, I: 104 Aluminum Nitride and, 111:373 Aluminum Oxide and, 11:762 Barium Titanate and, 11:790 Bismuth Germanium Oxide and, 111:406 Bismuth Silicon Oxide and, 111:406 Boron Nitride and, 111:426 Cadmium Selenide and, 11:559-560 Calcium Fluoride and, 11:817, 824-825 Cesium Chloride and, 111:731 Cesium Iodide and, II: 854 Cobalt and, 11:436, 439-440 Graphite and, II:451-452 Indium and, 111:261 Iron Pyrite and, 111:510 Lead Fluoride and, 111:762-763 Manganese and, III:249 Mercury Cadmium Telluride and, 11:657-658 Orthorhombic Sulfur and, 111:900 Palladium and, 11:469, 471 Potassium Bromide and, 11:990 Potassium Niobate and, 111:822 Rubidium Bromide and, 111:845 Rubidium Iodide and, 111:857 Sodium Nitrate and, III: 871, 873 Strontium Titanate and, II: 1035 Tellurium and, 11:710 Vanadium and, 11:478 Zinc Phosphide and, 111:610 See also Sample preparation Polycrystalline materials Aluminum, metallic, 1:370 Aluminum, polished, 1:372-373

70 Polycrystalline materials (Continued) Aluminum, thin-film, 1:373-376 Aluminum Oxynitride and, 11:778 Antimony and, 111:274 Beryllium and, 11:422 Beryllium Oxide and, 11:807 Boron Nitride and, 111:425 Cadmium Selenide and, 11:563 Cadmium Sulphide and, 11:582-583 Cadmium Telluride and, 1:409 Cerium and, 111:292 Cobalt and, 11:435, 438-439 Cuprous Oxide and, 11:875 effective-medium theory and, I: 105 Erbium and, 111:300 Indium and, lll:261 Indium Antimonide and, 1:491 interband absorption and, I: 192 Iridium and, 1:296 Iron Pyrite and, 111:508, 511 Magnesium Fluoride and, 11:899 metals and, 111:235 Orthorhombic Sulfur and, 111:901, 904 reactive sputtering and, 11:303 Rhenium and, 111:278 Rubidium Iodide and, 111:859 Ruthenium and, 111:253 scattering and, II: 179 Silicon-Germanium alloys and, 11:609 Silver Iodide and, 111:555 Sodium Nitrate and, III: 872 Spinel and, 11:885 Thallium Iodide, 111:927-928 Thorium Fluoride and, II: 1049 Tin and, 111:268-269 Tin Telluride and, 11:726 Titanium and, 111:240 Vanadium and, 11:478 Yttrium Oxide and, 11:1079 Zinc Arsenide and, 111:595 Zinc Germanium Phosphide and, 111:638 Polyethylene (CH 2 )n optical parameters for, 11:323, 331; Ill:216 Polyethylene (C 2H 4 )n critique, 11:957-968 low-frequency vibration mode dispersion of, 11:971

Subject Index

values of n and k for, 11:970, 972-987 Polyethylene terephalate films (PET) Brillouin scattering and, 111:143, 143 Polyfluorocarbon film from tetrafluoroethylene in RF plasma, 11:90 Polymers absorption coefficients and, 1:251 binding states in, 11:301 Polyethylene and, 11:960, 962 Polystyrene dispersion analysis of, 1:60 Polyvinylidene Fluoride films (PVF2 ) Brillouin scattering and, 111:143-144 Pores and voids Thorium Fluoride and, II: 1049 Potassium Bromide (KBr) critique, 11:989-992; 111:25-27 absorption versus photon energy for, 1:243 impurities and absorption in, 1:256 optical parameters for, 11:319 phonon processes for, 1:245, 246 transparency frequency of, 1:231 values of nand k for, 11:993, 994-1004; 111:25, 26 Potassium Chloride (KCI) critique, 1:703-705; 111:23-25 absorption versus photon energy for, 1:243 dispersion analysis of, 1:704 ellipsometry and, I: 100 impurities and absorption in, 1:256, 257 laser-window-material measurement of, 1:9 optical parameters for, 11:318 transparency frequency of, 1:231 values of nand k for, I: 707, 707-718; 111:22, 24 Potassium Dihydrogen Phosphate (KH2P04 ) (KDP) critique, II: 1005-1008 Hertzberger-type dispersion formula coefficients for, II: 1018-1019 index of refraction for, 11:1020 lattice-vibration modes of, 11:1020 optical parameters for, 11:322, 327; 111:211

71

Subject Index values of n and k for, 11:1010-1011,

1012-1013, 1016 Potassium Iodide (KI)

critique, 111:27, 807-810 impurities and absorption in, 1:256 oscillator model parameters for, 111:820 Sellmeier model parameters for, 111:820 thermo-optical coefficients of, 111:820 values of nand k for, 111:27, 28, 812,

813-819 Potassium (K)

critique, 11:364-366 Drude-model parameters for, 11:369 optical parameters for, 11:319 values of nand k for, 11:368, 370-374 Potassium Niobate (KNb0 3 )

critique, 111:821-825 dielectric constants and, 111:831 index of refraction and, 111:830 optical parameters for, 111:217 values of n and k for, 111:827-829,

832-843 Power attenuation coefficient Silicon, doped n-type and, III: 157 Poynting vector optical-properties measurement and,

1:18 PPES. See Photopyroelectric spectroscopy p- polarization Cesium and, 111:343 ellipsometry and, 1:90-91, 97 and Fresnel coefficients, 11:40-41 and prism evaporation, II: 81 Sodium Nitrate and, 111:871 superlattices and, 11:100, 113, 119, 121 and surface-plasmon waves, 11:78 transmission measurements on thin films, 1:223 Practical cgs units conversions optical-properties measurement and,

1:13 Press-forging method Magnesium Aluminum spinel and,

11:884 Pressure coefficients Gallium Selenide and, 111:477 Pressure considerations Aluminum Antimonide and, 11:505

Gallium Antimonide and, 11:600 Tin Telluride and, 11:727 Pressure dependence Ammonium Dihydrogen Phosphate and, II: 1008 Copper Oxides and, II: 875 Lead Fluoride and, 111:762, 764-765 Magnesium Aluminum Spinel and,

11:884 Magnesium Fluoride and, 11:902 optical-constants measurements and,

1:382-383, 393 Potassium Dihydrogen Phosphate and,

11:1008 Silver Gallium Selenide and, 111:574 Silver Gallium Sulfide and, 111:576 Yttrium Oxide and, 11:1081 Zinc Germanium Phosphide and,

111:639 Prism, minimum deviation Aluminum Oxide and, 111:655 Arsenic Selenide and, 1:625 Arsenic Sulfide and, 1:643 Barium Fluoride and, 111:684 Bismuth Germanium Oxide and,

111:404 Bismuth Silicon Oxide and, 111:404 Brillouin scattering and, III: 131 Cadmium Telluride and, 1:410 Calcium Carbonate and, 111:702 Cesium Bromide and, 111:718 Cesium Chloride and, 111:732 Gallium Arsenide and, 1:430 Gallium Phosphide and, 1:446-447 Germanium and, 1:466 Indium Phosphide and, 1:504 Lead Fluoride and, 111:762 Lithium Fluoride and, 1:676 Lithium Niobate and, 1:695-696 Magnesium Oxide and, 11:929 Potassium Bromide and, 11:990 Potassium Chloride and, 1:704 Potassium Dihydrogen Phosphate and,

11:1005 Rubidium Bromide and, 111:846 Rubidium Iodide and, 111:859 Rutile and, 1:796-797 Silicon and, 1:547

72

Subject Index

Prism, minimum deviation (Contin.ued) Silicon Dioxide (crystalline) and, 1:719 Silicon Dioxide (glass) and, 1:749 Silicon Dioxide (type a), 1:750 Silicon Dioxide (type x), 1:720 Silver Chloride and, 111:554 Silver Gallium Selenide and, 111:575 Silver Gallium Sulfide and,

111:576-577 Sodium Fluoride and, II: 1022 Strontium Fluoride and, 111:884 Strontium Titanate and, II: 1036 Titanium Dioxide and, 1:796 Yttrium Aluminum Garnet and, 111:964 Zinc Selenide and, 11:741 Zircon and, 111:988 Prism coupling Liquid Mercury and, 11:462-464 metalldielectric (air) interface and,

11:80 Palladium and, 11:472 Prism experiments attenuated total reflection and, 11:81-84 Prism geometry and internal-reflection spectroscopy,

11:76 Prism monochromators Strontium Titanate and, II: 1036 Prism refraction analysis Thallium Chloride and, 111:924 Prisms attenuated-total-reflection experiments with, 11:93 Barium Titanate and, 11:792 Bismuth Germanium Oxide and,

111:404 Cadmium Selenide and, 11:561 Cadmium Sulphide and, 11:581-582 Cesium Iodide and, 11:855 evanescent waves and, 11:79 Gadolinium and, 111:295 Gallium Antimonide and, 11:598 index measurement and, 11:69 interferometry and, I: 163 Lead Fluoride and, 111:762, 765 liquid crystals and, 11:90 Lithium Tantalate and, 111:778 photometric ellipsometry and, 1:94

Polyethylene and, 11:966 Potassium Niobate and, 111:823 Selenium and, 11:692 Silicon and, 1:547 Silicon Dioxide (type a), 1:750 Silver Iodide and, lll:556 Tellurium and, 11:711 Zinc Selenide and, 11:740 Zinc Telluride and, 11:745 See also Kosters prisms; Wollaston prisms Prism spectrometers Aluminum Oxide and, 11:762 Lithium Tantalate and, 111:778-779 Propagating transverse waves as polaritons, 1:217 Propagation attenuation measurements of, 11:69 Propagation distances and surface-plasmon waves, 11:79 Propagation vectors superlattices and,· II: 107 Propagation velocities, I: 12 Pseudo-Brewster angle Germanium and, 1:467 values of nand k and, 1:71-72, 77 See also Brewster's angle Pseudodielectric function ellipsometric measurement and, 1:96 Pseudo-dielectric functions Aluminum Antimonide and, 11:501 discussed, 11:597 Gallium Antimonide and, 11:597 Pseudopotential band calculation model, 1:204,206

Pseudo-unit-cell model mixed-crystal behavior and, 1:259 Pte See Platinum p-terphenyl, 11:957 p-type crystals Cadmium Germanium Arsenide and,

111:447 Copper Gallium Sulfide and, 111:459 Purity Boron Nitride and, 111:425 PVF2 • See Polyvinylidene Fluoride Pyrex glass Brillouin scattering and, III: 151

73

Subject Index Pyrolysis Silicon Nitride (noncrystalline) and,

1:772 Pyrolytic Graphite. See Graphite (C) PZT. See Piezoelectric

Q Quantum band theory Silicon and, 111:520 Quantum density matrix formulas for, II: 128 Quantum density-matrix equation of motion Drude theory and, I: 176 equilibrium radiation field and, I: 172 Quantum infrared detectors Mercury Cadmium Telluride and,

11:655 Quantum-mechanical description of free-carrier absorption, I: 171 Quantum-mechanical Morse-potential multiphonon model, II: 190-199 Quantum-mechanical treatments optical-property measurement and, II: 152 Quantum-mechanical wells in Gallium Arsenide layers,

11:101-102 Quantum mechanics development of, 1:37 and electromagnetic theory, 11:23-24 Quantum size effects amorphous superlattices and, II: 110,

112, 122 in compound semiconductors, II: 127 Quantum theory of free-carrier absorption, I: 171-174 Quantum transitions Gadolinium and, 111:296 rare-earth elements and, 111:287 Quantum-well confinement Silicon layer thickness and, II: 111 Quantum-well structures Aluminum Gallium Arsenide and,

11:513 index of refraction and, II: 148 Quartz monitor Lead Fluoride and, 111:765

Quartz prisms Liquid Mercury and, 11:463-464 See also Prism, minimum deviation; Prisms Quasi-classical Boltzmann equation, I: 171 See also Drude theory Quasistatic theory ellipsometry and, I: 105 Quench-anneal process Mercury Cadmium Telluride and,

11:662

R Radar Water and, II: 1063 Radiation absorption of, 1:218 reflectance and, I: 193 reflection of, 1:218-219 Radiation analysis Polyethylene and, 11:967 Potassium Bromide and, 11:991 Radiation effects Magnesium Fluoride and, 11:901 Radiation intensity absorption and, II: 178 Radiation transmission lamelliform and, I: 19-21 Radioactive materials Thorium Fluoride and, II: 1054 Radio-wave propagation attenuated total reflection and, 11:76 RAE. See Rotating-analyzer ellipsometers Raman activity Aluminum Oxide and, lll:657 Barium Fluoride and, 111:686 Boron Nitride and, 111:425 Strontium Fluoride and, lll:885 Yttrium Aluminum Garnet and, 111:965 Raman frequencies Calcium Carbonate and, 111:702 Raman region Iron Pyrite and, 111:507 Raman-scattering measurements Cadmium Selenide and, 11:561 Calcium Fluoride and, ll:821-822 Cuprous Oxide and, II: 875 discussed, 11:76

74

Subject Index

Raman-scattering measurements (Continued)

Indium Arsenide and, 1:480 Indium Phosphide and, 1:505 integrated optics and, 11:76 Lead Fluoride and, 111:762 Magnesium Fluoride and, 11:901 plasmon-phonon interaction and, 1:267 Potassium Dihydrogen Phosphate and, 11:1008 superlattices and, II: 108-109 Thorium Fluoride and, II: 1053 Yttrium Oxide and, II: 1081 Zinc Germanium Phosphide and, 111:639 Zinc Phosphide and, 111:610 Raman spectroscopy Aluminum Nitride and, III:376 attenuated total reflection and, II: 84 Copper Gallium Sulfide and, 111:461 Raman spectrum Aluminum Arsenide and, 11:491-492 Barium Titanate and, 11:792 Selenium and, 11:691 Terbium and, 111:298 Raman techniques Erbium and, III:301 Random-element isodisplacement model (REI)

mixed-crystal behavior and, 1:259 Rare-earth elements, 111:287 Rationalized systems optical-properties measurement and, 1:12 Rayleigh scattering density fluctuations and, III: 124 importance of, in long-path optical fibers, II: 178 importance of, in UV transparent materials, II: 178 in polar crystalline insulators, 11:178 Rb. See Rubidium RbBr. See Rubidium Bromide RbCl. See Rubidium Chloride RbI. See Rubidium Iodide RCE. See Rotating-compensator ellipsometers Re. See Rhenium

ReAs 2 Se3 • See Rhenium Arsenic Selenide Recursion relation laser calorimetry and, 1:145-146 Red Oxide. See Cuprous Oxide Reduction coefficients critical points and, 1:233 Red-wing shifting Aluminum Oxide and, 111:657 Brillouin scattering and, III: 125 pole-fit model and, 11:185, 187-188 Strontium Fluoride and, 111:886 Yttrium Oxide and, 11:1082 Reflectance Aluminum and, 1:373, 379, 393, 395-401; 11:206-207 Aluminum Antimonide and, 11:501 Aluminum Oxynitride and, 11:778 Beryllium and, 11:422-423 Beryllium Oxide and, 11:807 Bismuth Germanium Oxide and, 111:406 Bismuth Silicon Oxide and, 111:405 Boron Nitride and, 111:427 Cadmium Germanium Arsenide and, 111:448 Cadmium Sulphide and, II:580-582, 584 Calcium Fluoride and, 11:817 Chromium and, 11:375 Cobalt and, 11:438-439 Copper Gallium Sulfide and, III:462 Copper Oxides and, 11:877 Cubic Cadmium Sulphide and, 11:586 Cubic Silicon Carbide and, 11:706 defined, 11:51 ellipsometry and, I: 101 evaporated films and, 1:373 extreme ultraviolet region and, 11:203-204 of film on substrate, 1:23 Gallium Antimonide and, 11:597-598 Gallium Telluride and, 111:490 interferometry and, I: 115 Iron and, 11:386 in layered slab, 1:26-30 layered structures of, 1:18-22, 26-30 Lead Fluoride and, 111:766 Liquid Mercury and, 11:461, 464

Subject Index Magnesium Oxide and, 11:919, 921-923, 931 measurement of, 11:62-64 measurement of, and dispersion analysis, 1:58 optical-properties measurement and, 1:18-19 Orthorhombic Sulfur and, lll:900 overlayers and, I: 103 Palladium and, 11:469, 472 Potassium and, 11:365-366 Potassium Iodide and, III:807 rare-earth metals and, 111:290 Rubidium Iodide and, 111:858 Silicon, doped n-type and, III: 157 Silicon-Germanium alloys and, 11:609-610 Sodium Fluoride and, II: 1021 Sodium Nitrate and, 111:872 Strontium Fluoride and, 111:883 temperature dependence and, 1:382; 11:183 Thallous Halides and, 111:928 Thorium Fluoride and, II: 1050-1051 values of n and k and, 1:69-70, 79, 87 Vanadium and, 11:477 Water and, 11:1060-1061 Zinc Selenide and, 11:738, 740 Zinc Telluride and, 11:744 Reflectance curves Cadmium Sulphide and, 11:581 roughness and, 1:376, 389 two-media measurements and, 1:86 Reflectance formulas nonabsorbance and, 1:26 for single layer on slab, 1:26-30, 28 Reflectance measurements Aluminum Oxide and, Ill:654 Barium Fluoride and, 111:683, 685 Bismuth Silicon Oxide and, 111:405 Boron Nitride and, Ill:428 Cesium and, 111:342-343 Copper Gallium Sulfide and, 111:460 dispersion analysis and, 1:59 Dysprosium and, 111:298 Gadolinium and, 111:294 Gallium Selenide and, 111:475-476

75 Gallium Telluride and, 111:489 Hafnium Nitride and, 111:353-355 Lead Fluoride and, 111:764 Magnesium Oxide and, 11:936 Ruthenium and, 111:254 on Silicon, I: 109 Silicon and, 111:523 Silicon-Germanium alloys and, 111:537 Strontium Fluoride and, 111:885 Terbium and, Ill:297 Thallous Halides and, 111:930 Zinc Germanium Phosphide and, 111:639-640 Zirconium Nitride and, 111:353-355 Reflectance methods discussed, 1:276 optical-properties determination and, 1:92 refractive-index measurement and, 1:70-71 single-medium, 1:85-87 two-medium, 1:85-87 Reflectance spectra Cadmium Selenide and, 11:562 Hafnium Nitride and, ID:355 Orthorhombic Sulfur and, lll:901, 903-905 Potassium Iodide and, III:809 Strontium Fluoride and, lll:886 Zinc Selenide and, 11:739 Zirconium Nitride and, 111:355 Reflection absorption coefficients and, 1:229 Aluminum Gallium Arsenide and, 11:514 Calcium Carbonate and, 111:701-702 Calcium Fluoride and, II:820 Cesium Bromide and, Ill:718 Cesium Iodide and, 11:856 defined, 11:51 free-carrier density of semiconductors and, 11:126 introduced, 1:8 redistribution of electric fields and, 1:139 Thallium Bromide and, 111:925 Thallium Chloride and, 111:923

76 Reflection coatings film on substrate samples of, 1:5 Reflection coefficients and electric fields, 11:38 laser calorimetry and, I: 144 parameters of, 11:50 Polyethylene and, 11:961 and p-polarization, ll:40-41 and s-polarization, 11:40 superlattices and, II: 113 See also Fresnel coefficients, 11:38 Reflection electron scattering overlayers and, I: 100 Reflection measurements Cadmium Germanium Arsenide and, Ill:446 f sum rules and, 1:62 Lithium Tantalate and, 111:777-778 multi-oscillator fit and, 1:222 optical-constants determination and, 11:184 Rubidium Bromide and, 111:847 Silicon, doped n-type and, III: 162 Silver Iodide and, 111:555 Thallium Bromide and, Ill:926 Thallium Iodide and, 111:927 Yttria and, 11:185 Reflection peaks Iron Pyrite and, 111:510 Reflection phase measurement of, I: 124 Reflection phase change. See Phase changes Reflection spectroscopy plasmon-phonon interaction and, 1:267 sum rules and, 1:51-55 Reflection spectrum Lead Fluoride and, 111:763 Reflection techniques electromagnetic theory and, 1:89-92 model dependency and, 1:96 Reflectivity Aluminum Arsenide and, 11:491-492 Aluminum Gallium Arsenide and, 11:514, 515 Aluminum Oxide and, 11:762 attenuated-total-reflection experiments and, 11:88

Subject Index Barium Titanate and, 11:789, 791, 793 Bismuth Germanium Oxide and, 111:403 Cesium Chloride and, Ill:731, 733 Cesium Iodide and, 11:854-855 defined, 11:51 of evanescent waves, 11:75-76 Gallium Arsenide and, 11:491 Graphite and, 11:450, 452 of ionic diatomic crystals, 1:219 Iron Pyrite and, lll:511 Lead Tin Telluride and, 11:637, 638 Lithium Tantalate and, 111:779 Mercury Cadmium Telluride and, 11:656,658 of Metallic Aluminum, 1:395-401 optical constants and, 1:228-229 optical-constants determination and, 11:302 Potassium Niobate and, III: 822, 824 and prism experiments, 11:81 Selenium and, 11:691 Silicon, doped n-type and, III: 166 Silver Gallium Selenide and, 111:574 Silver Iodide and, lll:556 Sodium Nitrate and, III: 872 superlattices and, 11:99-100, 100, 119, 119 and surface-plasmon resonance, 11:94 Tellurium and, 11:710-711 Tin Telluride and, 11:725 values of nand k and, I: 176, 185 Yttrium Aluminum Garnet and, lll:965 Reflectivity curves in Gallium Arsenide samples, 1:185 Reflectivity measurements Boron Nitride and, 111:426 complex dielectric constants and, I: 169 Drude theory and, I: 171 Iron Pyrite and, 111:507-509 Lead Fluoride and, 111:763 Potassium Bromide and, 11:991 Silver Chloride and, 111:553 Silver Gallium Sulfide and, 111:576 Sodium Nitrate and, 111:871 Strontium Titanate and, II: 1035 Zinc Phosphide and, 111:609-610 Reflectivity spectra Zircon and, 111:987

77

Subject Index

Reflectometers Cadmium Selenide and, 11:562 Cadmium Sulphide and, 11:580 Cesium Iodide and, 11:856 Lead Fluoride and, lll:764 Magnesium Oxide and, 11:922 Mercury Cadmium Telluride and,

11:658 optical-properties determination and,

1:92 Osmium and, 1:324 Palladium and, 11:470 Silicon Carbide and, 1:587 Silver Chloride and, 111:553 in situ measurements and, 11:205 Tungsten and, 1:357 Reflectometry versus attenuated total reflection, 11:75 versus ellipsometry, 1:91-92 optical-properties determination and,

1:90-91 Reflectors metal mesh, 1:162 Refraction in dielectric response function, 1:221 interferometers and, I: 120 Refractive index. See Index of refraction Refractory compounds optical properties of, 11:295, 303-310 REI. See Random-element isodisplacement model Relaxation frequencies Gadolinium and, 111:296 Hafnium Nitride and, 111:353-354 Zirconium Nitride and, 111:353-354 Relaxation modes Sodium Nitrate and, 111:873 Relaxation rates Boltzmann equation and, I: 171,

173-174; 11:127 Drude theory and, I: 171, 177; III: 161 frequency dependent, I:182 Hafnium Nitride and, 111:356-357 for impurity scattering, I:178, 180 for polar scattering, 1:179-180 Potassium and, 11:366 Silicon, doped n-type and, 111:166-167 Zirconium Nitride and, 111:356-357

Residual gas bulk inclusion of, 1:375 Resins Silver Chloride and, 111:554 Resolution Calcium Fluoride and, II: 818 Resonance Boron Nitride and, 111:426 Carbon films and, 11:838 Cesium Iodide and, 11:853 Copper Oxides and, II: 877 dielectric response function and,

1:222 Iron Pyrite and, 111:508 Magnesium Fluoride and, 11:900-901 Polyethylene and, 11:960 Thallium Bromide and, 111:925 Thallium Chloride and, 111:924 Resonance absorption peaks Gadolinium and, 111:296 Resonance frequency Beryllium Oxide and, 11:805 dielectric constants and, 1:235 optical properties of crystals and, 1:216,

226 Selenium and, 11:691 Resonance location introduced, 1:8 Resonance minimum shifts attenuated total reflection and, II: 84 Resonance structure Potassium Niobate and, 111:824 Resonant absorption of infrared radiation, 1:257 Resonant cavity interferometry index-of-refraction measurement and,

1:163 Resonant conditions extinction coefficients and, II: 155 Resonant frequencies near-millimeter-wavelength measurement and, I: 161, 163 Resonant modes frequency spectrum and, 1:255 Response linearization interferometry and, I: 126 Reststrahlen phenomenon described, 1:219

78

Subject Index

Reststrahlen region absorption and, 1:243 alkali halides and, 1:259 Aluminum Arsenide and, 11:491 Aluminum Oxide and, 111:657 Arsenic Selenide and, 1:624 Beryllium Oxide and, 11:807 Beta-Gallium Oxide and, 111:754 Bismuth Germanium Oxide and,

lII:406 Bismuth Silicon Oxide and, 111:406 Cadmium Telluride and, 1:410 Calcium Fluoride and, II: 819-820 Cesium Bromide and, 111:718 Cesium Chloride and,

111:731-732 Cesium Iodide and, 11:853-854 Cubic Silicon Carbide and, 11:706 Gallium Arsenide and, 1:429, 431;

11:118 Gallium Selenide and, 111:475-476 Gallium Telluride and, 111:491 Indium Antimonide and, 1:493 Indium Arsenide and, 1:480 Indium Phosphide and, 1:505 infrared absorption and, 1:249-254 Magnesium Oxide and, 11:935-938 Mercury Cadmium Telluride and,

11:663-666 optical-constants calculations and,

1:187-188, 221 Potassium Bromide and, 11:990 Potassium Chloride and, 1:703 Potassium Dihydrogen Phosphate and,

11:1008 Selenium and, 11:692 semiconductors and, 111:32 Silicon Carbide and, 1:588 Silver Gallium Selenide and, IIl:575 Sodium Chloride and, 1:776 Sodium Nitrate and, 111:872 spectral regions and, 1:214 superlattices and, II: 118 Tellurium and, 11:710 Zinc Sulfide and, 1:600 RF plasma tetrafluoroethylene in, 11:90 Rh. See Rhodium

Rhenium Arsenic Selenide (ReAs 2 Se3 ) optical properties of, II: 10 See also Arsenic Selenide, crystalline Rhenium (Re)

critique, 111:278 values of n and k for, 111:279, 280-286 Rhodium (Rh)

critique, 1:342 experiments with, 11:58-60 index measurements of, 11:61 optical parameters for, 11:318 thickness measurements of, 11:61 values of nand k for, 1:343, 344-349 Rhombohedral structures Antimony and, 111:274 Rigaku's Bragg-Brentano x-ray diffractometer Lead Fluoride and, 111:765 RMS deviation Silver Gallium Selenide and, 111:575 Silver Gallium Sulfide and, IIl:577 RMS errors Aluminum Antimonide and,

11:504-505 Beryllium Oxide and, 11:806 Copper Oxides and, 11:876 Gallium Antimonide and, 11:598, 600 Potassium Dihydrogen Phosphate and,

11:1006 Rock salt optic branches of, 1:215 Rotating-polarizer method Liquid Mercury and, 11:462 Rotating-analyzer ellipsometers (RAE) described, 1:94-95 Rotating-compensator ellipsometers (RCE) described, 1:95 Rotating-polarizer ellipsometers (RPE) described, 1:94-95 Liquid Mercury and, 11:462 Roughness Aluminum and, 1:373, 376 Cadmium Selenide and, 11:560 Cesium and, III: 341 Cuprous Oxide as, 11:876 effective-medium theory and, 1:105 ellipsometry and, 1:91-92

79

Subject Index

Gallium Antimonide and, 11:597 interferometry and, I: 131-132 Lithium and, 11:347 metals and, 111:234 models of, 1:97, 98 optical-properties measurement and, 1:8, 375-376; 11:205 Palladium and, 11:469-470 Rubidium Bromide and, 111:845 Rubidium Iodide and, lll:857 Silicon and, I: 104 Sodium and, 11:354 Sodium Fluoride and, II: 1021 Thorium Fluoride and, II: 1050 Vanadium and, 11:478 RPE. See Rotating-polarizer ellipsometers Ru. See Ruthenium Rubidium Bromide (RbBr) critique, 111:845-848 discussed, 111:29 values of n and k for, 111:849, 850-855 Rubidium Chloride (RbCI) critique, 111:29 values of n and k for, 111:29, 30-31 Rubidium Iodide (RbI) critique, 111:857-861 discussed, 111:29 values of n and k for, 111:862, 863-870 Rubidium (Rb) halides, absorption and, 111:719 Ruby, 111:653, 656 See also Aluminum Oxide Ruthenium (Ru) critique, 111:253-254 values of n and k for, 111:255, 256-261 Rutile (Ti0 2) critique, 1:795-797 See also Titanium Dioxide

s Samarium (Sm), 111:287 critique, 111:293-294 values of nand k for, Ill:307, 315-317 Sample preparation Beta-Gallium Oxide and, 111:753 Bismuth Germanium Oxide and, 111:406 Bismuth Silicon Oxide and.. 111:406

Carbon films and, 11:838 Cesium Chloride and, 111:731 Cesium Fluoride and, Ill:743 Cesium Iodide and, II:854 Cubic Thallium(l) Halides and, 111:923 Iron Pyrite and, 111:508 Lead Fluoride and, 111:761-762 Magnesium Fluoride and, 11:901 Magnesium Oxide and, 11:936 optical-properties determination and, 11:295 Orthorhombic Sulfur and, lll:901 Polyethylene and, 11:958, 966 Potassium Bromide and, 11:990 Potassium Niobate and, 111:821-822 Rubidium Bromide and, 111:845 Rubidium Iodide and, 111:857-858 Silicon-Germanium alloys and, 111:538 Sodium Fluoride and, II: 1021 Sodium Nitrate and, Ill:871, 873 Strontium Titanate and, II: 1035 of surface, importance of, 1:8 Thorium Fluoride and, II: 1051 Yttrium Oxide and, 11:1079 Zinc Phosphide and, 111:610 Zircon and, 111:987-988 See also Polishing techniques; Individual critiques Sapphire, 11:761; 111:653 Beryllium and, 11:423 Cadmium Sulphide and, 11:583 ellipsometry and, 11:355 Liquid Mercury and, 11:465 multiphonon absorption and, 11:192, 196, 198 sample preparation and, II: 1053 as substrate, lll:766 Zinc Telluride and, 11:745 See also Aluminum Oxide Sb. See Antimony Scandium Oxide (Sc0 2) experiments with, 11:58-60 index measurements of, 11:59 Scattering acoustical-mode phonons and, II: 128 Aluminum Oxide and, lll:655 Cadmium Germanium Arsenide and, 111:446

80

Subject Index

Scattering (Continued) Cadmium Selenide and, 11:561 Calcium Fluoride and, 11:823 Cesium Iodide and, 11:856-857 Copper Gallium Sulfide and, 111:460 dependence for ionized impurity, I: 171 electron-energy-Ioss spectroscopy and,

11:294, 295, 298-299 electrons and, I: 196 Graphite and, 11:450-451 Liquid Mercury and, 11:461 Magnesium Oxide and, 11:920-921 Mercury Cadmium Telluride and,

11:665 metals and, III:233 optical-constants determination and, III: 121-152 optical-mode phonons and, 11:128 Orthorhombic Sulfur and, 111:901-902 Palladium and, 11:470 phonons and, I: 197-198 Polyethylene and, 11:957-958 Rubidium Iodide and, 111:857 Selenium and, 11:692 Silicon, doped n-type and, III: 163,

167-171 Sodium and, 11:355, 357 Sodium Fluoride and, II: 1022 Thorium Fluoride and, II: 1051 Vanadium and, 11:478 Zinc Germanium Phosphide and,

111:638 See also Raman-scattering measurements Scattering angle, III: 122 Scattering mechanism free-carrier absorption and, I: 172 inelastic, I: 174 wavelength dependence and, II: 127 Scattering rates Drude theory and, I: 175 optical-constants calculations and,

1:177, 177, 179 Scattering theory Cesium Chloride and, 111:733 Cesium Fluoride and, 111:744 Schering bridge circuits Lead Fluoride and, 111:763

Schrodinger equations, 11:24 absorption rules and, I: 38 multiphonon absorption and, II: 190 one-electron model and, I: 190 sum rules and, 1:41 time dependent, 1:41 Sc0 2 • See Scandium Oxide Screening effect ellipsometry and, I:106 See See Selenium Selection rules defects and, 1:254 electron-hole interaction and, I: 199 one-electron model and, I: 193 optical phonons and, 1:232 Selenium, amorphous (a-Se) photothermal deflection and, III: 110 Selenium (Se)

critique, 11:691-693 optical parameters for, 11:322, 329;

111:207 oscillator parameters for, 11:696 spectroscopic measurements of, III: 84 values of nand k for, 11:695, 697-703 Self-shadowing crystallites Aluminum and, 1:376 Sellmeier-type dispersion equations Aluminum Antimonide and, 11:504 Aluminum Oxide and, 111:655 Ammonium Dihydrogen Phosphate and, 11:1006-1007 Arsenic Sulfide and, 1:643 Barium Fluoride and, 111:684 Barium Titanate and, 11:790 Beryllium Oxide and, 11:806 Bismuth Germanium Oxide and,

111:404 Bismuth Silicon Oxide and,

111:404-405 Boron Nitride and, 111:427 Cadmium Germanium Arsenide and,

111:447 Cadmium Sulphide and, 11:581, 583 Cadmium Telluride and, 1:410, 412 Calcium Carbonate and, 111:702 Calcium Fluoride and, 11:818, 823 Cesium Bromide and, 111:718 Cesium Chloride and, 111:732

81

Subject Index

Cesium Fluoride and, 111:743 Cesium Iodide and, 11:855 Copper Gallium Sulfide and, 111:461 Cubic Carbon and, 1:666-667 discussed, II: 152 Gallium Antimonide and, 11:599 Gallium Arsenide and, 1:430 Germanium and, 1:466, 468 Indium Arsenide and, 1:479-480 Indium Phosphide and, 1:504 Lead Fluoride and, 111:762 Lithium Fluoride and, 1:676 Magnesium Aluminum Spinel and,

11:884 Magnesium Fluoride and, 11:900, 902 Mercury Cadmium Telluride and,

11:660 optical-property measurement and,

11:152 Potassium Bromide and, 11:990 Potassium Chloride and, 1:704 Potassium Dihydrogen Phosphate and,

11:1006-1007 Potassium Iodide and, 111:808-809 Potassium Niobate and, 111:823-824 Rubidium Bromide and, 111:846 Rubidium Iodide and, III: 859-860 Silicon, doped n-type and, III: 166 Silicon and, 1:548 Silicon Carbide and, 1:588 Silicon Dioxide (glass) and, 1:750 Silver Gallium Selenide and, 111:575 Silver Gallium Sulfide and, 111:577 Sodium Chloride and, 1:777 Sodium Nitrate and, 111:872 Strontium Fluoride and, 111:884, 886, 896 Thallous Halides and, 111:929 Yttrium Aluminum Garnet and,

111:964-965 Yttrium Oxide and, 11:1080, 1083 Zinc Germanium Phosphide and,

111:639 Zinc Selenide and, 11:738, 740 Zinc Sulfide and, 1:599 Zinc Telluride and, 11:738, 745 Semiconductor diode laser double-heterostructure, II: 127

Semiconductors Cadmium Germanium Arsenide and,

111:445 complex dielectric constants of, I: 169 Copper Gallium Sulfide and, 111:459 covalently bonded, 1:207 Cuprous Oxide, 11:875-876 direct band-gap, 11:691 direct band gap energy of, I: 172 free-carrier absorption in, I: 172 free-carrier concentration in, I: 8 free-carrier effects and, II: 177-178 Gallium Antimonide and, 11:597 Gallium Selenide and, 111:473 heavily doped, I: 198 highly doped, 1:202-203 hyperbolic exciton in, 1:201 infrared dispersion and, 1:263 infrared properties study of, 1:22 interband absorption and, I: 196 lattice vibrations and, II: 178 loss-function determination and, 11:300 narrow transitions regions on, I: 104 optical properties of, I: 18 optical-property calculation of, II: 151 oxide-thickness measurement on, 1:94 phonons and, 11:178 quantum free-carrier-absorption theory of, 1:4, 172-174 reflection measurements of, 111:32-55 small band gap, 1:198 Tin Telluride and, 11:725 Zinc Arsenide and, 111:595 Zinc Germanium Phosphide and,

111:637 Zinc Selenide and, 11:737 Zinc Telluride and, 11:737 Semiconductors, amorphous. See Amorphous semiconductors Semiconductors, compound complex dielectric constant and, II: 125 Semiconductors, crystalline. See Crystalline semiconductors Semiconductor slab lamelliform and, 1:19-21 Semiconductor slab, parallel sided lamelliform and, I: 19

82 Semiconductor superlattices excitons and, II: 102-108 introduced, II: 101-102 optical properties of, 11:100-112, 122 phonons and, IT:I08-109 single-particle excitations and, 11:102-108 Semiconfocal cavity configuration index-of-refraction measurement and, 1:161, 162, 166, 167 Seya-Namioka monochromator Iron Pyrite and, 111:510 sf exchange coupling rare-earth elements and, 111:288 Shearing interferometers optical measurement and, I: 119-121, 121, 126 Si. See Silicon SiC. See Silicon Carbide Signal-to-noise ratio spectroscopy and, 1:156, 159, 161 Silane (SiH4 ) preparation of, IT: 168 Silicon, amorphous (a-Si) critique, 1:571-573 index of refraction for, I: 102 optical parameters for, II: 167, 323 photothermal deflection and, III: 112 spectroscopic measurements of, 111:85-93,86-92 values of n and k for, 1:574, 575-586; 11:167 Silicon, doped n-type (n-Si) critique, 111:155-172 values of n and k for, 111:173, 174-185 Silicon, hydrogenated amorphous (SiH4 ) Kramers-Kronig analysis and, II: 167 Silicon, hydrogenated (a-Si:H) photothermal deflection and, III: 112 Silicon Carbide (SiC) critique, 1:587-589 dispersion analysis of, 1:588 infrared absorption and, 1:261-262 optical parameters for, 11:321 values of n and k for, 1:590, 590-595; 11:170, 172, 172 See also Cubic Silicon Carbide

Subject Index

Silicon Dioxide, amorphous (a-Si0 2) optical parameters for, 11:323, 330 Silicon Dioxide, crystalline (Si0 2) dispersion analysis of, 1:720 values of n and k for, II: 170, 171 Silicon Dioxide (glass) (Si02 ) critique, 1:749-752 Brillouin scattering and, III: 131 dispersion analysis of, 1:750 photothermal deflection and, 111:107-109 values of nand k for, 1:753, 753-763 Silicon Dioxide (Si02 ) absorption dependence of, 11:117 attenuated-total-reflection experiments and, 11:92 Berreman effect and, IT: 116 ellipsometry and, I: 100, 102 film, saline oxidation and, 11:90 index of refraction and, II:187 optical parameters for, 11:322; ITl:207 optical properties of, II:12-19 reflectance of, II:117 Silver (Ag) film on, 11:93 Silicon Dioxide (Si0 2) (type-

CdTe

~

tJ

~

V)

'=C'i CU ~

KRS-5

0

t'i

LBO

~"--

~

tr) f"'OI4

~ CsBr

--=- ADP Silica

0.1

1.0

10.0

Wavelength f)lID] Figure 2.2: Refractive index versus wavelength for some important optical materials.

14

CHAPTER 2. REFRACTIVE INDEX

dispersion relation. However, the fitting accuracy should be the dominant consideration in choosing a dispersion formula having fewer optical parameters. As we have analyzed the refractive indices of the various optical materials, the two-pole Sellmeier dispersion formula, Eq. (2.16) is the most accurate of the dispersion formulas, and can fit index data to an accuracy consistent with the measurements and the accuracy needed for most optoelectronic applications. The normal approach for calculating the Sellmeier coefficients is to first find the initial values of the parameters and then to add corrections by an iterative process so as to minimize the deviation between the measured and the computed values. We are able to evaluate the Sellmeier coefficients by fitting the measured data obtained from the quoted values (and sometimes from the other form of dispersion relations) to an accuracy better or consistent with, the experimental accuracy. The Sellmeier coefficients A, B, C, and D are evaluated by first taking a reasonably estimated value for E, the lattice absorption frequency. The choice of the value for E is not critical since the materials stop transmitting long before the onset of lattice absorption frequency. The average electronic absorption band gaps (Eag ) are calculated from the Sellmeier coefficients C by the following equation E ag

_ 1.2398 -

ve ·

(2.17)

The estimated band gaps are also shown in Tables 2.1-2.3. These values are within the measured values of the electronic absorption gaps. The values of these evaluated band gaps are required to estimate the excitonic band gaps, which will be described in the next chapter to characterize the thermo-optic coefficients of the optical materials. The conversion of wavelength and energy of any electromagnetic wave is required in many opto-electronic applications. This equation is given by A = 1.2398,

E

(2.18)

where A is the wavelength in 11m and E is the energy in eV.

2.5 Sellmeier Coefficients The room temperature values of refractive indices are used to evaluate the Sellmeier coefficients A, B, C, D, and E for mostly all optical materials.

2.5. SELLMEIER COEFFICIENTS

15

Sometimes, other forms of Sellmeier equations are used to evaluate the Sellmeier coefficients in the present form. The computed Sellmeier coefficients are shown in Tables 2.1, 2.2, and 2.3 at room temperature for crystals (linear and nonlinear), semiconductors, glasses (fiber glass and optical glasses), and tellurite-based high index glasses, respectively. As we have noticed, the band edge data of refractive indices differ in the third decimal place for nonlinear crystals and in the fourth decimal place for higher index glasses. The refractive index changes in a third decimal place for a change of 1°C ambient temperature at the band-edge region of some optical materials. The five evaluated Sellmeier coefficients are used to calculate the refractive index for any wavelength lying within the normal transmission region of the optical materials. In Chapter 3, thermo-optic coefficients of several optical materials have been analyzed. The values of thermo-optic coefficients for several wavelengths are available without having the room temperature refractive index values for some materials. Therefore, the corresponding room temperature values of the refractive index are calculated from the Sellmeier coefficients for analyzing thermo-optic coefficients.

25.0

25.0

20.0 23.9 25.0

Ag3 AsS 3

AgBr Agel AgGaS 2

33.0

24.8

33.0

33.0

33.0

AD*P [NH4O2P04] Ag3 AsS 3

[ND4 0 2 As04] ADP [NH4 H2P04] ADP

AD*A

[NH4~As04]

ADA

e

0

e

0

e

0

e

0

e

0

e

0

e

0

e

0

Temp. PolarMaterials (OC) ization (CrystaV Semiconductor) 1.77351582 1.79381654 1.98983870 1.67385409 1.10453338 1.18225569 1.31484546 1.29423990 1.21960960 1.04697126 12.6344855 11.9169211 8.97904251 8.98318525 8.63304669 4.64845739 10.03 13.10

0.95173751 0.78810870 0.81273393 0.84365085 1.10385100 0.82412401 0.84995622 0.74447222 0.89209958 0.94636494 3.52626894 2.71084707 5.28411478 3.81009590 1.21022462 1.69645766 2.1686 1.527

1.49117717 1.48811668 1.60831944 1.42043624 1.18918067 1.33838706 1.45233409 1.41882936 1.38701774 1.20368487 3.96061147 3.63610700 2.19843460 2.53564322 3.48460089 2.31173639 3.6280 4.0172

)

C(XI0

B

A

2

D 1.22764415 0.44890395 0.79297224 0.27903481 1.30107754 0.69630836 1.30189736 0.51068171 1.01112621 0.41558974 1.54935351 0.59005174 1.64807314 0.95414797 0.35646114 4.28025210 2.1753 2.1699

Sellmeier Coefficients

Table 2.1: Sellmeier Coefficients for Some Crystals and Semiconductors; [n A is in ,um] Absorp- Refs. tion band n and gap (eV) Eqn.

[9, *] 9.31 9.26 [9, *] 0.4-1.1 8.79 9.58 [9, *] 0.4-1.1 11.8 11.4 0.21-1.2 [16, *] 10.8 10.9 0.4-1.1 [9, *] 11.2 12.1 0.6-7.5 3.49 [20, *] 3.59 0.63-4.6 4.14 [21, *] 0.59-4.6 4.14 0.49-0.67 4.22 [22, *] 0.57-20.6 5.75 [23, *] 0.49-12 3.92 [24, 19] 3.43 0.4-1.1

Range (Jlm)

+B/(I-C/A 2 ) + D/(I-E/A 2 )

36 36 36 36 49 49 36 36 49 49 900 900 900 900 16 5000 950 950

E

2=A

,

~

>


~

0

e x y

25.0

Cl-ID0 3

e

0

25.0

HgS

0

25.0 20.0

e

0

Ge HgGa2 S4

DLAP 25.0 (Deuterated LArginine Phosphate) GaAs 25.0 25.0 GaN 20.0 GaP GaP 25.0 GaSb 25.0 GaSe 23.0

y z x y z

A B

Coefficients

D

2.37884165 0.34472179 2.51806693 0.63845298 1.44978445 0.23019106 2.12775193 0.48252292 2.28197845 0.38814926 1.9347 16.663 4.1 6.5536 9.7650944 4.4062455 9.44041466 0.8387802 160.70651 0.6824531 7.79448739 4.48612725 2.64903269 3.55416460 0.21307 44.105 9.56864600 0.28313910 9.21463300 0.28124737 1.1378 13.28 12.84 1.5604 5.48091354 0.42016876 5.02832424 0.33429415

C (X 10 2)

Sellmeier

1.96937278 0.45091898 2.05112076 0.39200228 1.41859292 0.81635259 1.71820457 0.71259363 1.69216712 0.75583480 7.4969 3.5 1.75 3.6 5.7610532 3.3346794 3.09558131 5.99865272 0.7546437 13.095068 2.40622747 5.04000694 5.04419996 0.73385456 6.72880 9.28156 3.44829864 2.47671447 3.39565649 2.33212283 2.7896 4.1506 4.3736 4.0101 2.55425206 0.71628581 2.46046031 1.21369374

Temp. PolarMaterials (CrystaV COC) ization Semiconductor)

Table 2.1: (Continued)

"36 36 36 36 36 1382 319 1500 300 100 3000 3000 3870 225 225 705 705 25 25

E

Absorp- Refs. tion band n and gap (eV) Eqn.

8.04 7.81 [60, *] 0.25-1.35 10.3 8.5 8.21 1.4--11 [61 ] 3.04 [62] 4.84

~

g

tv tv

33.0

25.0

33.0

KD*P (KD2 P04) KD*P

KDPRb

KfA (KTiOAs04)

20.0

20.0 KI 24.0 KLN (K3Li2NbsOlS) 20.0 KM (COOK-CHOH-CH 2 COOK.l.5H 2O) 22.0 KNO (KNb03)

24.8

KDP

y

x

z

y

x

z

y

x

e

0

e

0

e

0

e

0

e

0

Temp. PolarMaterials (OC) ization (CrystaV Semiconductor)

Table 2.1: (Continued)

B

1.46289336 .0.79550715 1.42190536 0.71047582 1.59068702 0.65000325 1.43418215 0.69205653 0.98363343 1.26014242 1.58774984 0.54292865 0.81960177 1.43173192 1.00383613 1.13065110 1.66617420 0.98625098 3.708 1.0 3.349 1.0 0.8299 1.542 0.7473 1.470 0.8519 1.339 2.54088371 2.29332835 2.62583723 2.36053534 2.55261698 1.86721183 2.3039444 0.8472855 0.7878492 2.3880145

A 1.27782716 1.21984109 1.49871678 1.23182859 0.81911791 1.48828489 0.80682241 0.81234817 3.81692057 4.601 3.564 2.013561 1.803649 1.428025 5.62190899 6.42922649 5.40028525 4.8326702 5.5830324

D

7.30699101 8.29653561 5.64155101 3.0688893 3.3986856

----------------

1.10560172 0.27051559 1.11286794 0.41594387 0.68954938 0.39228493 1.37946279 0.32442632 1.86575318

Coefficients

C (X 10 2)

Sellmeier

300 300 300 250250

----------------

36 36 64 64 36 36 49 49 7000

E

Absorp- Refs. tion band n and gap (eV) Eqn.

[16, *] 11.0 11.2 [9, *] 0.4-1.1 10.1 11.2 [76, *] 0.40-0.69 13.7 10.2 [9, *] 0.4-1.1 13.8 13.8 0.25-50 6.35 [56, *] [77] 0.4-5 5.78 6.57 [78] 0.24-1.4 8.74 9.23 10.4 0.4-3.4 5.23 [79, 80] 4.89 5.34 [81, *] 0.35-4 5.64 5.25 0.2-1.4

Range (J.1m)

N

~

N

~

~

~ tij

ii

~

(J

~

~

~

~

~

~

25.0

20.0 23.0

LFM (LiCOOH.H 2O)

LiF UO (LilO3 )

20.0

20.0

LAP

e

0

z

y

x

z

y

x

z

y

e x

(L-Arginine Phosphate) LBO (LiB 30 5 )

0

z

y

x

z

20.0

20.0 20.0

LaP3

(KTiOP04 )

KTa0 3 laP

Temp. PolarMaterials (OC) ization (CrystaV Semiconductor)

Table 2.1: (Continued)

B 1.0920291 3.590987 1.0011186 0.9874869 1.0539416 1.537600 1.514900 0.66350234 0.84010856 0.78967786 1.0109932 1.0388217 1.1365228 0.40475012 0.50339981 0.59550781 0.91190354 1.3420521 1.2525669

A

2.3548285 1.000014 2.0009584 2.0411662 2.2541202 1.0 1.0 1.57994331 1.59917136 1.66879020 1.4426279 1.5014015 1.4489240 1.43733506 1.65545000 1.66828005 1.01369880 2.0730090 1.6658390

D

3.8746096 5.8457976 3.724897 0.000004 1.7740435 4.0294037 1.8828437 4.3432843 2.2516434 5.5303637 0.776161 ---0.770884 ---1.77268882 0.37256988 1.89167472 0.71647713 2.24949793 0.54785154 1.1210197 1.2363218 1.2157100 1.7567133 1.5830069 1.1676746 1.69196273 0.01695506 2.33456179 0.43120589 2.52926288 0.51969132 0.55023818 7.14251615 1.5915063 3.5155686 2.8115791 0.6580167

C (X 10 2)

Sellmeier Coefficients

36 36 36 91 91 91 36 36 36 1100 190 190

-------

250 20 150 150 150

E

[56, *] [88, 89]

[87, *]

[86,33]

[60, *]

[85]

[82, *] [83, 84]

Absorp- Refs. tion band n and gap (eV) Eqn.

5.13 6.42 6.18 5.95 5.27 0.35-0.70 14.1 14.1 0.25-1.4 9.31 9.02 8.27 0.29-1.7 11.7 11.2 11.5 0.23-1.3 9.5 8.1 7.8 0.1-10 16.7 0.5-5 6.61 7.4 0.4-1.06 0.2-10.6

(J1m)

Range

~ >
~

g

~

tv

20.0

x

(CloHIIN306)

MDB

23.0

y z

MAP

e x

e

0

e

0

e

0

e

0

e

0

e

0

0

25.0

25.0

25.0

(congruent! y 24.5 melt) LNO:MgO 25.0 (7%MgO:LiNb03) 25.0 LID (LiTa03) LiYF4 25.0

UO (LilO3) LNO (LiNb03) (pure)

Temp. PolarMaterials (OC) ization (CrystaV Semiconductor)

Table 2.1: (Continued)

2.03132 1.83086 2.30868466 2.17136846 2.27819911 2.25826059 2.38952126 2.31555508 2.00386576 2.40205217 1.98794841 2.15687934 1.38757 1.31021 2.16757084 2.30650608 2.73830634 2.16007371

A

C(XI0 3.50823 3.13810 4.53622670 4.12238100 4.51938833 4.25064669 4.70532927 4.39945088 4.20429003 4.47232281 3.48752303 3.69273851 0.931 0.876 16.8505230 17.9514184 15.9882765 6.37947255

B 1.37623 1.08807 2.59943258 2.40239261 2.64261396 2.31529513 2.51367167 2.26509259 2.85969043 2.14162463 2.51909534 2.36800178 0.70757 0.84903 0.10542484 0.22784362 0.61735732 0.72471419

2 )

D, 1.06745 0.554582 7.85093573 6.35196621 7.77870589 6.24472815 7.88014969 6.36660092 6.47078590 6.28571221 6.74867738 6.71815687 0.18849 0.53697 0.99126222 1.12305937 3.18448050 0.18117133

Sellmeier Coefficients

169 158.76 300 300 300 300 300 300 250 250 300 300 51 135 64 64 64 36

E

0.43-1.2

0.5-2

0.23-2.6

0.45-4

0.4-4

0.40-3.1

0.42-4

0.42-4

0.5-5

Range (J.lm)

--

[90] 6.62 7.0 [91, *] 5.82 6.11 5.83 [92, *] 6.01 5.72 193, *] 5.91 [94, *] 6.05 5.86 [31, *] 6.64 6.45 12.9 [95] 13.2 [96, *] 3.0 2.9 3.1 4.91 [97, *]

Absorp- Refs. tion band n and gap (eV) Eqn.

N

N

Ul

~

~

n tii

~

~

(]

~ ~

~

~

~

20.0

25.0 25.0

19.0

20.0 20.0 20.0

25.0

20.0

(Meta-dinitrobenzene) mNA

MgAl 2 0 4 MgBaF4

MgF2

MgO NaBr NaCI

NaCOOH

NaF

x y z

e

0

x y z

y z x y z

Temp. PolarMaterials (OC) ization (CrystaV Semiconductor)

Table 2.1: (Continued)

B

2.02036204 0.68508036 1.95937253 0.20285306 2.46752274 0.18691228 2.66462581 0.16286509 2.80751576 0.15347567 1.93704783 0.95795641 1.11497468 0.96202532 2.13246088 -0.0000172 1.32842222 0.81777778 1.30618931 0.58007346 1.29108974 0.62728316 1.53180538 1.42502805 1.59778665 0.99998568 1.47469012 0.85594247 1.44354608 0.88670071 1.25793545 0.64465461 1.25686750 0.84428339 1.24795357 1.07632137 0.32785 1.41572

A )

D

5.91484437 0.22989248 6.76086193 0.32565318 1.19780255 15.9965171 1.27902389 17.1893347 1.76534933 17.4861506 1.83214061 3.76543269 0.79 ---9.60889563 -0.0003404 0.90 ---0.77355208 2.24075551 0.75322929 2.41840547 1.52465664 6.98416419 2.81766115 3.74557330 2.03004560 3.75408845 1.98488774 3.76346189 1.20293934 0.06407803 1.44481248 0.02913995 1.72301349 0.07580585 1.3689 3.18248

2

Coefficients

C(XI0

Sellmeier

550 550 650 5500 4000 4000 64 64 64 1646

----

100

----

36 36 64 64 64 300

E

Absorp- Refs. tion band n and gap (eV) Eqn.

0.15-17

0.36-5.6 0.21-34 0.2-20 0.2-20 0.24-2.3

0.2-7.0

13.1 14.1 [101, *] 14.3 [102, *] 10.0 [56, *] 7.39 [45, *] 8.7 [56, *] 8.8 [103, *] 11.3 10.3 9.45 [56] 10.6

5.1 4.77 [98, *] 0.42-2 3.1 3.0 2.96 [99, *] 0.35-5.5 9.16 [100, *] 0.185-12 14.0

Range (J1m)

tv

>

~

@

00

N

25.0

25.0

25.0

Ti02

Ti02

Tl 3 AsSe 3 [TAS] TlBr Tl[Br: CI] KRS-6 Tl[Br: I] KRS-5 TICI Y3 AI S0 12 (YAG) Y2 0 3

25.0 25.0 19.0 25.0 25.0 20.0

20.0

Te02

Te

25.0 25.0 25.0

SrTi03

e

0

e

0

e

0

e

0

e

0

e

A

2.29721810 2.86912089 2.42187038 18.5346 29.5222 2.71688346 2.88752546 2.84860910 3.17419736 3.23385217 4.51423711 6.03737957 8.40171923 3.83190858 0.45587304 2.84207150 3.74680347 1.80034564 0.99859062

Temp. PolarMaterials (OC) ization (CrystaV Semiconductor)

Table 2.1: (Continued)

1.20510636 2.34233183 2.76968545 4.3289 9.3068 2.0362633 2.49103113 3.06181302 4.01135254 2.72670030 2.77391366 5.19399305 1.62283661 1.44039164 4.36355477 2.82918542 1.02028243 1.49343976 2.57931374

B 4.03047127 7.06002312 6.28759701 398.10 257.66 5.95328147 6.06531999 8.00134067 8.36350921 8.39522371 9.79744383 34.6307307 82.3603918 11.266023 4.45018360 10.1422791 9.73387282 1.77099120 1.92329098

D 2.52723226 10.9753287 7.53814810 3.7800 9.2350 0.11344622 0.12676676 -0.03198089 -0.09750392 3.38150919 5.31788562 2.90084653 2.05812267 1.30884878 7.90340049 14.3405647 1.98890646 3.50903401 3.72479973

Coefficients

C (X 10 2)

Sellmeier

300 300 300 11813 13521 16 16 25 25 100 100 2500 2500 16 9500 31000 16 300 500

E

0.54--0.65 0.58-24 0.58-39 0.43-0.66 0.4--4 0.2-12

2-12

0.45-2.4

0.43-1.53

0.4--1

0.43-3.8 0.45-3.8 4-14

Range (j1m)

6.18 4.67 4.95 0.62 0.77 5.08 5.03 4.38 4.29 4.28 3.96 2.1 1.4 3.69 5.88 3.89 3.97 9.32 8.94

[22, [120, [121, [22, [31, [122,

*] *] *] *] *] *]

[119, *]

[31, *]

[45, *]

[118, *]

[45, *] [31, *] [19]

Absorp- Refs. tion band n and gap (eV) Eqn.

N

N \0

~

~

rs~~

~

(]

~

~tij

~

~

~

25.0

21.6 25.0 25.0

20.3 25.0

25.0 25.0

25.0

Zno

~-ZnS

ZnSe ZnSiAs2

ZnTe Znwo4

2'J02 : 12%Y2 0 3

*-

My fit

Zns a-Zns

20.0

ZnGeP2

y z

x

e

0

e

0

e

0

e

0

Temp. PolarMaterials (OC) ization (CrystaV Semiconductor)

Table 2.1: (Continued)

4.47330 4.63318 2.85349562 2.80512169 2.23696722 2.57477308 3.41601054 3.46184666 3.34955729 4.6066 4.9091 4.29515331 2.72421908 2.40450786 2.65345900 2.05122399

A 5.26576 5.34215 0.84269535 0.94272630 2.86316280 2.53722297 1.74084980 1.72082603 2.57689896 5.6912 5.5565 2.98859685 1.73800676 2.11263371 2.43455523 2.41401832

B

D

1.49085 13.381 1.45785 14.255 9.46396061 2.17891980 9.03655516 2.09919541 5.18497733 2.86977560 5.70100072 0.41111677 7.16458591 -0.0023589 7.19323780 0.11687750 9.38826749 2.88904268 14.37 1.1316 1.1287 15.78 2.45419969 14.2458772 4.75449530 3.98567817 4.29666836 4.02381019 4.83248161 5.58536560 2.64827500 11.2670449

C (X 10 2)

Sellmeier Coefficients

662.55 662.55 300 300 1200 100 20 20 2200 700 700 3000 300 300 300 700

E

0.57-25 0.42-4.0 0.42-3.6 0.42-3.6 0.36-5.1

0.55-18 0.7-11.5

0.55-10.5 0.45-2.4 0.36-1.4

0.45-4.0

0.64-12

(J1m)

Range

3.39 3.28 4.03 4.12 5.4 5.2 4.63 4.62 4.05 3.27 3.12 3.29 5.69 5.98 5.64 7.67

[124, *]

[15, *] [31, *]

[48, *] [50, 19]

[48, *] [31, *] [123, *]

[31, *]

[50, 51]

Absorp- Refs. tion band n and gap (eV) Eqn.

w

>
(l

~

N

:::d

~

g > ~

0

FK5

BK7

SF6

SSK4A K5

BaLF4 BaKl PSK3 SK4

Schott

BAH32 BAlA BALlI BAL23 BSM4 BSM9 BSM22 BSM24 NSL5 PBM3 PBH6 PBL26 S-BAL2 S-BAL14 S-BSL7 S-FPL51 S-FPL52 S-FPL53 S-FSL5

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

(OC)

Temp.

B 0.82387607 0.80996963 0.90159230 0.91813830 1.02432620 0.94601273 0.92219278 0.96932959 0.77615273 0.71735619 1.03775076 0.65645522 0.72988760 0.82264667 0.84014624 0.87986171 0.76802212 0.80625247 0.82393169

A

1.89298845 1.64229267 1.53163489 1.45898859 1.53418270 1.61343315 1.66142576 1.60142499 1.50879617 1.81718900 2.07890466 1.74817094 1.69375672 1.59915353 1.43131380 1.33782101 1.33336620 1.24672804 1.36459305

2.88317368 1.84821497 1.55142286 1.30700097 1.47483619 1.69493728 1.82037106 1.66620095 1.52773045 3.06533130 4.23019093 2.65692509 2.09938544 1.69901106 1.28897582 0.96071215 0.88491995 0.77204904 1.10838681

C (X 10 2)

D 1.85495930 1.32867267 1.25742434 1.53653204 0.97315045 1.36313318 1.35578530 1.17740089' 1.26816265 1.73341223 1.39213190 1.68327025 1.27069569 1.37033083 0.97506873 0.80365148 0.71925847 0.61817308 0.93306279

Sellmeier Coefficients

200 150 150 150 100 150 150 130 150 200 150 200 150 150 100 150 150 150 100

E

(pm)

Ref.

7.3 9.1 10.0 10.8 10.2 9.5 9.2 9.6 10.0 7.1 6.0 7.6 8.6 9.5 10.9 12.7 13.2 14.1 11.8

[125, *] [126, 127] [126, 127] [126, 127] [126, 127] [125, *] [125, *] [126, 127] [126, 127] [125, *] [126, 127] [125, *] [125, *] [125, *] [126, 127] [125, *] [125, *] [125, *] [126, 127]

fornand band gap (eV) Eqn.

Absorp.

A +B/(l - CI A2 ) +

Range

2=

0.36-2.33 0.33-2.33 0.31-2.33 0.31-2.33 0.33-2.33 0.36-2.33 0.36-2.33 0.33-2.33 0.31-2.33 0.36-2.33 0.36-2.33 0.36-2.33 0.36-2.33 0.36-2.33 0.31-2.33 0.36-2.33 0.36-2.33 0.36-2.33 0.28-2.33

Sellmeier Coefficients for Some Optical and Fiber Glasses; [n D /(1 - E / A,,2), A is i n ,urn]

Ohara

Glasses

Table 2.2:

w .......

~

~

n ~

~

~

n

~ ~

~

~

N

~

Ohara

(OC)

Temp.

150 150 90 150 100 120 70 100 100 100 100 100 100 100 100 150

1.65229731 1.61986479 1.28954539 1.28405587 1.00701091 1.03208380 0.88503243 0.91298262 0.92097109 0.92107106 0.99952574 0.90459781 0.90324053 0.94473185 0.94868688 0.50447475 2.35627833 2.08695053 1.90728779 2.11054490 2.04645940 1.58905079 2.12286099 1.10325933 1.15873913 1.09439467 1.09514652 1.31952169 1.48132649 1.07852766 1.19368978 2.61388311

1.05969452 1.03443101 1.11987423 0.60902200 0.65644946 0.78537672 0.77858837 0.78910058 0.78426280 0.79514557 0.79429689 0.95486678 0.88229977 0.81471590 0.76529922 0.35130064

1.90798862 1.79270360 1.77798367 1.65681095 1.63685176 1.53023177 1.70910663 1.31563132 1.33514985 1.30912262 1.31138256 1.40919164 1.49368555 1.28802423 1.35308250 1.91829737

and * - My fit.

E

D

C (X 10 2 )

B

A

Sellmeier Coefficients

AS - Alumino silicate, and V - Vycor glasses;

S-LAM2 S-LAM51 S-LALI0 S-NSL36 SSL6 ZSLI

FS - Fused silica,

LaF2

20 20 20 LaKI0 20 20 20 ZKl 20 KzFS6 --FS (Coming 7940) 26 FS (Coming 7940) 471 20 Fused silica 20.5 Si02 AS (Corning 1723) 28 AS (Corning 1723) 526 V (Coming 7913) 28 V (Coming 7913) 526 BGZA(4) (Fluoride) 25

Schott

Glasses

Table 2.2: (Continued)

0.36-2.33 0.36-2.33 0.36-2.33 0.36-2.33 0.31-2.32 0.31-2.33 0.36-2.33 0.23-2.26 0.23-2.26 0.21-2.33 0.24-0.55 0.36-2.26 0.36-2.26 0.26-2.26 0.26-2.26 0.40-4.9

(J1m)

Range

Ref.

8.1 8.6 9.0 8.5 8.7 9.8 8.5 11.8 11.5 11.9 11.9 10.8 10.2 11.9 11.4 7.67

[126, 127] [125, *] [126, 127] [125, *] [125, 127] [126, 127] [126, 127] [128, 129] [128, 129] [130, 129] [131, 129] [128, 129] [128, 129] [128, 129] [128, 129] [135, *]

band forn and gap (eV) Eqn.

Absorp.

w

~

>
~

g

00

w

nco

37.6279 0.104 38.8743 0.104 38.2908 0.104 40.6628 0.104 39.4112 0.104 15.3175 0.270 L =15914.2 AiP=24

0.104 0.104 0.104 0.104 0.104

37.6279 39.8834 40.5481 45.4648 48.7405

-50.3250 -53.7544 -57.6322 -60.5962 -64.4885

-50.3250 -43.7855 -39.7869 -34.8578 -32.3908 -53.5987

0.106 0.106 0.105 0.103 0.106

26.5232 25.5833 41.1111 40.9336 27.2883

-17.6270 -1.9969 -36.1213 -30.0274 -1.9436

H

9.3 9.3 9.3 9.3 9.3 9.3 9.6 9.6 9.6 9.6 4.3

11.4 11.4 11.4 11.4 11.4 4.6

10.4 10.4 9.6 5.5 10.4

11.4 11.4 11.4 11.4 11.4

11.6 11.6 11.4 14.0 11.6

Band gaps Isentropic Excitonic

J.lm Eig reV] Eeg reV] G

14.45 ["] 12.20 ["] 10.95 ["] 9.50 ["] 8.70 ["] 16.9 [155]

-3.01 -3.03 -2.97 -3.23 -3.34

-2.4 -2.5 -2.5 -1.8 -2.7

14.45 -3.01 12.20 -3.12 10.95 -3.03 9.50 -3.19 8.70 -3.05 -0.6 16.8

14.45 14.65 15.15 15.45 15.85

14.45 14.65 15.15 15.45 15.85

[153] ["] ["] ["] ["]

5.1 0.6 7.8 8.0 0.6

5.1 [16] 0.62 [16] 7.75 [16] 7.95 [16] 0.61 [12]

[154] [154] [154] [154] [154] [155]

[154] [154] [154] [154] [154]

[16, 26] [16, 26] [16, 26] [16, 26] [12, 25] [153]

Refs. for Expansion Coeff dEeg /dT a [10- 61°C] [10 -4eVI °C] n, dnldT Expt.[Ref.] Fit Fit andEq.

The experimental value of dEeg /dT for Si0 2 glass is -2.3±0.5 (10 - 4eVI °C) [31]; therefore the fit values of dEeg /dT for other glasses (except fluoride) are within the experimental accuracy.

1.46 Borosilicate-717 1.44 Silica-739 Soda lime-710 1.60 1.50 Lead silicate 1.435 Vitreous silica Soda-Lime-Silica 25Na20.pCaO.(75 - p)Si0 2 1.47 p=O 1.49 =5 1.505 = 10 1.52 = 15 1.535 =20 (25 - p)Na2O.pCaO.75Si02 1.47 p=O 1.48 =2.5 1.487 =5 1.491 =7.5 1.497 = 10 Fluoride BGZA (4) 1.45

Glasses

Table 3.2: (Continued)

w

\C

~

~CI5

n~

~

~

n

~ ~

~ ~

~ C/)

~

~

~

~

~ ~

~

~

140

CHAPTER 3. THERMO-OPTIC COEFFICIENTS

3.4.1 Nonlinear Crystals The nonlinear crystals, such as ammonium dihydrogen phosphate, (ADP), potassium dihydrogen phosphate, (KDP), and their isomorphs have been used as the primary group of workhorse crystals in nonlinear optics and electro-optical devices since the mid-1960s. The temperature dependence of the refractive indices of ADP and KDP crystals along with deuterated potassium dihydrogen phosphate, KD*P were measured by Phillips [5] in 1966. Data were taken between 150 and 298 K from 0.365 Jlm to 0.691 Jlm and were fitted to the following form

n(T) - n(298) == cf(n).(298 - T),

(3.31)

where f == n 2 + a · n + b, and T is the absolute temperature. In the same time, Yamazaki and Ogawa [35] reported their measurements between 100 and 300 K for KDP, and 150 and 300 K for ADP and KD*P only at three, and two wavelengths in the visible region, respectively. They used a fit of the form (3.32) where a, b, and c are some constants. In 1981, Onaka and Kawamura [36] measured the temperature variation of the refractive indices of KDP at eight wavelengths from 0.405 Jlm to 0.546 Jlrn. In 1982, Onaka and Takeda [37] reported the temperature dependence of the refractive indices for ADP at four wavelengths from 0.365 Jlm to 0.546 Jlrn. The temperature dependence was fitted to the following formula

Llno,e == no,e(T) - n o,e(298) == ao,e(298 - T)2

+ bo,e(298 -

T), (3.33)

where the constants ao,e and bo,e were determined from the experimental data using the least-square method. They reported some numerical values of thermo-optic coefficients for KDP and ADP crystals. In 1982, Ghosh and Bhar [38] reported double-pole Sellmeier fits to the indices of KDP, ADP, and KD*P and represented each of the four Sellmeier parameters as varying linearly with temperature. Simultaneously, data on thermo-optic coefficients for nine of the fsomorphs of KDP were reported

3.4. DERIVATION OF SELLMEIER COEFFICIENTS

141

by Barnes et ale [8] at five wavelengths in the visible region from 0.405 p,m to 0.633 p,m. There is a review paper entitled Electro-optic, Linear, and Nonlinear Optical Properties of KDP and Its Isomorphs by Eimerl [43] on nonlinear optical properties and room temperature Sellmeier coefficients. However, there is no physical explanation or a suitable model to account for the thermo-optic coefficients and their dispersions for these important nonlinear optical materials. So, we have used all the above experimental data and also some estimated values within the measured values allowable to analyze Eq. (3.29) above. The computed optical constants are shown in Table 3.1. The calculated values and the experimental data are cited in Table 3.3. Thus, the Sellmeier coefficients cited in Tables 2.1 and 3.1 can be used to calculate refractive indices for any wavelength within the transmission region at any operating temperature. It is worthwhile to see the characteristic behavior of dispersion of dn/dT, i.e., to estimate the dn/dT values beyond the measured wavelength region both in the UV and in the IR region. Hence, experimental data are plotted in Figs. 3.2 to 3.6 for ADA, CDA, CD*A, KDA, and RDA and in Figs. 3.7 to 3.10 for ADP, KDP, KD*P, and RDP crystals, respectively. Thermo-optic coefficients are negtive for all these KDP isomorphs crystals, except for ADA extraordinary polarization. Thermal expansion coefficients are dominant to contribute negative dn/dT to these crystals. Since the thermal expansion coefficient is lowest and minimum for ADA extraordinary polarization, the dn/dT values are positive. Similarly, it is the case for extraordinary polarization in ADP crystal, in which the dn/dT values are negative but the absolute value is very small. The dn/dT values asymptotically decrease from high negative to less negative values in the UV to near-IR region for most of the crystals. However, in some cases the opposite behavior is also observed. But, the thermo-optic coefficients are decreasing asymptotically from high positive values to less positive values for ADA (e) crystal; and from positive to negative values (the absolute value is within 1.5 ppm) in ADP (e) crystal. Since the dn/dT values are different in orthogonal polarizations in ADA and ADP crystal, it is possible to use these crystals efficiently in temperature-tuned optical devices.

CHAPTER 3. THERMO-OPTIC COEFFICIENTS

142

-140

ADA (0)

,.......,

U

Aig

0

......... \0

= 0.165 Jlm

-145

•

I

0 ,.,.,..-4 ~

,.-.....

~

-150

~ ==

"= "-" -155 == N

-160 0.4

0.5 0.6 Wavelength [Jlm]

0.7

50

ADA (e) Aig = 0.18 Jlm

,.......,

u 0

.........

\0

I

0

,.,.,..-4 ~

,.-. . . 40

•

~

~ ==

"= "-" N==

35 30

a..__.a...-

0.4

--.&..

0.6

...I.-

0.8

---a._-----"

1

Wavelength [Jlm] Figure 3.2: 2n(dn/dT) versus wavelength for the ADA crystal: Curves are computed values from the optical constants G, H, and Eig from Table 3.1; solid circles are the experimental data [40, 8] at 33°C.

3.4. DERIVATION OF SELLMEIER COEFFICIENTS

143

-80 ....-....-----..------,,..--------,

•

CDA (0)

U -85 o

Aig

""""

=0.161

Jlm

•

\0

I

;:;

-90

~

~

~

~

-95

= .-0 = -100 "---"

N

-1 05

~.L...-

0.4

___a.

----.

0.5

--.I

0.6

0.7

Wavelength [Jlm]

-60

CDA (e)

~

U o

Aig = 0.187 Jlm

:0- -65

•

I

o

~ ~

~

:s2

-70

=

= 'C

N

-75

-80

•

I-.-.&.-----~----~-------.I

0.4

0.5

0.6

0.7

Wavelength [Jlm] Figure 3.3: 2n(dn/dT) versus wavelength at 33°C for the CDA crystal: Curves are the computed values from the optical constants G, H, and Eig from Table 3.1; the experimental points [40, 8] are shown as solid circles.

Figure 3.4: 2n(dn/dT) versus wavelength at 33°C for the CD*A crystal: Solid curves are the computed values from the optical constants G, H, and Eig from Table 3.1; the experimental points [40, 8] are shown as solid circles.

145

3.4. DERIVATION OF SELLMEIER COEFFICIENTS -110

r---.........------.------~-----

KDA(o)

u -115

Aig = 0.19 Jlm

o

.........

\0

I

o ::: -120 E-4 ~ ~ -125 ",-......,

"-"

c

N

-130

0.4

0.5

0.6

0.7

Wavelength [Jlm] -60

,...., U

o .........

-65

\0

r-or------....,...--------.--------..--. KDA(e) Aig = 0.21 Jlm

I

o

~

~ -70

~

~

•

I:

~ -75

c N

-80

...--oI...--llIIl..-

0.4

........

0.5

--I-

0.6

--I.---I

0.7

Wavelength [Jlm]

Figure 3.5: 2n(dn/dT) versus wavelength at 33°C for the KDA crystal: Solid curves are the computed values from the optical constants G, H, and Eig from Table 3.1; the experimental points [40, 8] are shown as solid circles.

146

CHAPTER 3. THERMO-OPTIC COEFFICIENTS

Figure 3.6: 2n(dn/dT) versus wavelength at 33°C for the RDA crystal: Solid curves are the computed values from the optical constants G, H, and Eig from Table 3.1; the experimental points [40, 8] are shown as solid circles.

3.4. DERIVATION OF SELLMEIER COEFFICIENTS

-130

r--r----....--•

G

••

o

.........

~

-150

o

~

147

..

..-----~---

ADP (0)

•

•

~

•

r--..

~

~

= -170 =

"C

• • • •

'-"

N

•

Aig

= 0.15 J.1m

•

-190 ------......-..---.-...----------0.3 0.4 0.5 0.6 0.7

Wavelength [J.1m]

G ~ I o

\0

2 ........-...---....-------....----~--- ..... ADP (e) Aig = 0.14 J.1m 0 t-----.......-----------------I

~

~

r--..

~

~

-2

= = -4

"C

'-"

N

• 0.3

0.4

0.5

0.6

0.7

Wavelength [J.1m] Figure 3.7: 2n(dn/dT) versus wavelength at 33°C for the ADP crystal: Solid curves are the computed values from the optical constants G, H, and Eig from Table 3.1; the experimental points [40,41,5,35,37,8] are shown as solid circles.

148

CHAPTER 3. THERMO-OPTIC COEFFICIENTS

-120 .----.----.~----...----...-----.-----. KDP(o)

u

Aig

= 0.11 J.lm

I

o

"'" ~ o

•

-130

•

,...-.4

L.......oI ~

~

~

,§ -140 "-'"

c9

• 0.25 0.3

0.4

0.5

0.6

0.7

Wavelength [J.lm]

-60 - .........-.....----.........----..~--.....--- ..... KDP (e) Aig

=0.14 J.lm

•

• • • •• •

• • • • 0.25 0.3

0.4

•• • •

• 0.5

0.6

0.7

Wavelength [J.lm] .Figure 3.8: 2n(dn/dT) versus wavelength at 33°C for the KDP crystal: Solid curves are the computed values from the optical constants G, H, and Eig from Table 3.1; the experimental points [40, 41, 5, 35, 36, 8] are shown as solid circles.

3.4. DERIVATION OF SELLMEIER COEFFICIENTS

149

-70 ~

U

Aig

0

.........

\0

I

KD*P(o) = 0.17 flm

•

-80

0

~

..........

,......... ~

~

c -90

~ "-'"

I:

N

• -100 0.4

0.3

0.5

0.6

0.7

Wavelength [flm]

-54

U -56

•

KD*P(e) Aig = 0.17 flm

0

•

.........

\0

t, -58 ~

..........

,.........

•

~ -60 ~

•

I:

~

"-'"

c -62

N

•• 0.4

•

• 0.5

0.6

0.7

Wavelength [flm] Figure 3.9: 2n(dn/dT) versus wavelength at 33°C for the KD*P crystal: Solid curves are the computed values from the optical constants G, H, and Eig from Table 3.1; the experimental points [40, 5, 35, 8] are shown as solid circles.

150

CHAPTER 3. THERMO-OPTIC COEFFICIENTS

-110 ,-,

U 0

•

"""I

\0

0 ",...-4

~ ~

-115

~

RDP(o)

•

'tS ==

"'-"

Aig

N==

=0.11

JlID

-120

0.5

0.4

0.7

0.6

Wavelength [JlID]

-74 ,-,

U 0

"""

RDP(e) Ajg =0.20 JlID

-77

\0 I

0

•

",...-4

~ -80

~

~ 'tS ==

•

"'-" -83

N==

•

-86 0.4

0.5

0.6

0.7

Wavelength [Jlrn] Figure 3.10: 2n(dn/dT) versus wavelength at 33°C for the RDP crystal: Solid curves are the computed values from the optical constants G, H, and Eig from Table 3.1; the experimental points [40, 8] are shown as solid circles.

3.4. DERIVATION OF SELLMEIER COEFFICIENTS Table 3.3:

Refractive Index and Thenno-Optic Coefficients of KDP Isomorphs at 33°e

Crystal Wavelength [Ref.] [ ,urn]

0.436 0.546 0.578 0.633 ADA(e) 0.436 [40] 0.546 [8] 0.578 0.633 ADP(o) 0.365 [40] 0.405 [41] 0.4047 [5] 0.4360 [35] 0.4916 [8] 0.5145 [37] 0.5460 0.5780 0.5893 0.6234 0.6330 0.6907 ADP (e) 0.3650 [40] 0.4047 [41] 0.4358 [5] 0.4916 [35] 0.4965 [8] 0.5145 [37] 0.5461 0.5779 0.5893 0.6234 0.6907 CDA(o) 0.405 [40] 0.436 [8] 0.546 0.578 0.633 ADA(o)

[40] [8]

151

Refractive index

1.59196 1.57888 1.57632 1.57263 1.53491 1.52413 1.52209 1.51924 1.54613 1.53961 1.53965 1.53578 1.53010 1.52875 1.52664 1.52477 1.52416 1.52245 1.52201 1.51951 1.49717 1.49159 1.48830 1.48391 1.48359 1.48249 1.48081 1.47937 1.47890 1.47763 1.47553 1.58992 1.58441 1.57202 1.56963 1.56625

dn/dT [( 10-6)/Kl Expt. Computed values (This work)

-48.5 -43.9 -45.3 -44.5 12.7 13.1 12.4 11.9 -59.0 -47.8 -47.0 -49.4 -52.9 -58.3 -52.3 -46.0 -50.2 -50.2 -50.8 -49.9 -0.10 -0.20 -0.40 -0.60 -0.70 -0.80 -0.90 -1.00 -1.10 -1.20 -1.40 -31.5 -30.5 -25.9 -27.6 -28.0

-48.59 -45.84 -45.36 -44.72 14.77 12.81 12.48 12.05 -54.88 -53.22 -53.23 -52.29 -51.11 -50.73 -50.31 -49.96 -49.85 -49.56 -49.49 -49.13 0.149 -0.259 -0.481 -0.756 -0.775 -0.838 -0.932 -1.010 -1.034 -1.097 -1.193 -31.38 -30.19 -27.77 -27.36 -26.80

Differ- Av. RMS ence Dev. Dev.

0.091 1.936 0.056 0.218 -2.067 0.294 -0.079 -0.149 -4.112 5.418 6.259 2.886 -1.771 -7.611 -1.992 3.958 -0.361 -0.628 -1.310 -0.729 -0.249 0.059 0.081 0.156 0.075 0.038 0.032 0.010 -0.066 -0.103 -0.207 -0.123 -0.308 1.873 -0.245 -1.200

0.57

0.97

0.65

1.05

3.1

3.8

0.098 0.12

0.75

1.01

152

CHAPTER 3. THERMO-OPTIC COEFFICIENTS

Table 3.3: (Continued) Crystal Wavelength [Ref.] [ Jim]

Refractive index

dn/dT [( 10-6)/Kl

0.405 0.436 0.546 0.578 0.633 CD*A(o) 0.405 [40] 0.436 [8] 0.546 0.578 CD*A(e) 0.405 [40] 0.436 [8] 0.546 0.578 0.633 KDA(o) 0.436 [40] 0.546 [8] 0.578 0.633 KDA(e) 0.436 [40] 0.546 [8] 0.578 0.633 KDP(o) 0.2537 [40] 0.3126 [41] 0.3650 [8] 0.4047 [5] 0.4358 [35] 0.5461 [36] 0.3653 0.4047 0.4078 0.4358 0.4916 0.5461 0.5791 0.6234 0.6907

1.56660 1.56157 1.55044 1.54837 1.54552 1.58522 1.57990 1.56808 1.56584 1.56456 1.55956 1.54858 1.54655 1.54376 1.58159 1.56924 1.56684 1.56341 1.52932 1.51938 1.51753 1.51498 1.56636 1.54112 1.52028 1.52337 1.51984 1.51155 1.52923 1.52337 1.52298 1.51984 1.51504 1.51155 1.50981 1.50777 1.50514

-18.9 -20.9 -21.2 -23.9 -25.6 -22.6 -22.6 -24.7 -23.1 -17.7 -15.1 -16.4 -17.1 -17.0 -36.4 -40.7 -39.8 -40.9 -23.1 -21.3 -25.1 -21.2 -45.4 -43.0 -43.6 -44.1 -42.3 -41.7 -40.1 -40.1 -43.1 -43.5 -42.5 -41.7 -41.1 -42.5 -42.0

CDA(e)

[40] [8]

Expt. Computed values (This work)

-18.85 -20.46 -23.27 -23.69 -24.23 -22.49 -22.90 -23.73 -23.88 -17.00 -16.94 -16.81 -16.78 -16.75 -36.53 -39.89 -40.38 -41.00 -25.35 -24.04 -23.80 -23.47 -46.25 -44.65 -44.11 -43.45 -43.21 -42.69 -43.85 -43.45 -43.42 -43.21 -42.90 -42.69 -42.59 -42.49 -42.38

Differ- Av. RMS ence Dev. Dev.

-0.053 -0.440 2.070 -0.208 -1.375 -0.108 0.296 -0.968 0.779 -0.696 1.845 0.409 -0.317 -0.252 0.125 -0.806 0.584 0.098 2.249 2.741 -1.299 2.275 0.848 1.653 0.511 -0.651 0.911 0.990 3.748 3.349 0.323 -0.289 0.400 0.990 1.495 -0.006 0.384

0.83

1.13

0.54

0.64

0.70

0.92

0.40

0.50

2.1

2.2

1.7

2.9

3.4. DERIVATION OF SELLMEIER COEFFICIENTS

153

Table 3.3: (Continued) Crystal Wavelength [Ref.] [ ,urn]

0.5460 0.6330 KDP(e) 0.2537 [40, 41] 0.3126 [8, 5] 0.3650 [35, 36] 0.4047 0.4358 0.5461 0.3653 0.4047 0.4078 0.4358 0.4916 0.5461 0.5791 0.6234 0.6907 0.4050 0.4360 0.5460 0.5780 0.6330 KD*P(o) 0.4047 [5] 0.4078 [40] 0.4358 [8] 0.4916 [35] 0.5461 0.5779 0.6234 0.6907 0.405 0.436 0.546 0.578 0.633 KD*P(e) 0.405 [40, 8] 0.436

Refractive index

1.51155 1.50737 1.51593 1.49432 1.48425 1.47930 1.47638 1.46984 1.48420 1.47930 1.47897 1.47638 1.47252 1.46984 1.46856 1.46712 1.46540 1.47927 1.47637 1.46984 1.46860 1.46685 1.51890 1.51850 1.51550 1.51110 1.50790 1.50630 1.50440 1.50220 1.51883 1.51554 1.50785 1.50632 1.50410 1.47754 1.47368

dn/dT [( 10- 6 )/Kl Expt. Computed values (This work)

-32.8 -39.4 -30.7 -31.1 -30.7 -29.7 -29.4 -29.9 -25.8 -24.7 -24.7 -24.2 -23.9 -23.9 -23.7 -23.7 -23.7 -31.5 -28.8 -29.0 -28.7 -25.4 -28.6 -28.6 -28.6 -28.6 -28.6 -27.1 -28.6 -27.1 -30.0 -33.7 -29.9 -30.0 -31.6 -18.6 -21.3

-42.69 -42.48 -32.46 -29.62 -28.33 -27.72 -27.36 -26.58 -28.33 -27.72 -27.68 -27.36 -26.89 -26.58 -26.43 -26.28 -26.10 -27.71 -27.36 -26.58 -26.44 -26.25 -29.57 -29.51 -29.07 -28.42 -27.99 -27.80 -27.57 -27.33 -29.56 -29.07 -27.99 -27.80 -27.53 -24.09 -24.38

Differ- Av. RMS ence Dev. Dev.

9.891 3.075 1.760 2.4 -1.482 -2.366 -1.984 -2.042 -3.320 2.528 3.016 2.976 3.158 2.993 2.680 2.734 2.577 2.396 -3.788 -1.444 -2.420 -2.261 0.847 0.967 1.4 0.912 0.470 -0.178 -0.612 0.696 -1.025 0.228 -0.438 -4.634 -1.910 -2.205 -4.067 5.492 3.5 3.076

2.5

2.0

3.8

154

CHAPTER 3. THERMO-OPTIC COEFFICIENTS

Table 3.3: (Continued) Crystal Wavelength [Ref.] [ ,urn]

RDA(o)

[40] [8] RDA(e)

[40] [8] RDP(o)

[40] [8] RDP(e)

[40] [8]

0.546 0.578 0.633 0.436 0.546 0.578 0.633 0.436 0.546 0.578 0.633 0.405 0.436 0.546 0.578 0.633 0.405 0.436 0.546 0.578 0.633

Refractive index

1.46830 1.46707 1.46534 1.57341 1.56136 1.55902 1.55568 1.53421 1.52409 1.52222 1.51964 1.52075 1.51733 1.50917 1.50749 1.50502 1.48989 1.48689 1.48002 1.47889 1.47684

dn/dT [( 10- 6 )/Kl Expt. Computed values (This work)

-19.5 -25.2 -20.3 -30.9 -36.2 -33.8 -33.7 -19.7 -23.4 -21.7 -23.5 -36.9 -38.6 -37.2 -37.2 -37.2 -26.7 -27.6 -25.4 -28.0 -28.9

-24.76 -24.80 -24.86 -31.41 -33.98 -34.36 -34.86 -21.71 -22.72 -22.86 -23.02 -37.64 -37.55 -37.34 -37.31 -37.26 -25.46 -26.48 -28.02 -28.21 -28.44

Differ- Av. RMS ence Dev. Dev.

5.255 -0.397 4.557 0.509 1.1 -2.224 0.564 1.156 2.008 1.08 -0.678 1.156 -0.485 0.742 0.4 -1.050 0.143 0.106 0.059 -1.244 1.1 -1.116 2.617 -0.206 -0.464

1.3

1.23

0.6

1.4

3.4. DERIVATION OF SELLMEIER COEFFICIENTS

155

The nonlinear crystals, such as lithium niobate, LiNb0 3 (LNO), lithium iodate, LiI0 3 (LIO), and lithium tantalate, LiTa03 (LTO) have been used as the second group of workhorse crystals in nonlinear optics [45] since the mid-1960s. So far there has been no satisfactory fonnulation of the dispersion of thermo-optic coefficients for these crystals except for LNO. However, the temperature dispersion equation [4] of LNO cannot be interpreted in a physically meaningful model and cannot be used to assess many high-power nonlinear-optical devices (NLOD's). Temperature stability and tunability [46, 47] calculated from the different measured thermo-optic coefficients [48-52] of these crystals by different workers often widely differ from those of the experiments. The thermo-optic coefficients have been analyzed recently [27] and are shown in Table 3.1. The experimental values and the computed curves are shown in Figs. 3.11 and 3.12 for LNO and LIO, respectively. Some experimental data and computed values are shown in Table 3.4. Although there are only four data points for LIO crystal, the fit was excellent. The error bars for the LIO crystal are calculated by the accuracy of the dn/dTmeasurement, which is ±1.0 x 10- 6 /°C. It is interesting to note that the fitted gap Eig is close to the reflectivity peak in the electronic absorption spectra. From the fitted constants G and H, the thermal expansion coefficient a and the temperature coefficient of the band gap are computed and are also shown in Table 3.1. These values are compared with the experimental values and agree reasonably well within the experimental accuracy. The dn/dT values are negative in LIO crystal. The first factor, i.e., the contribution from the thermal expansion coefficient, is dominant in this crystal. We have used 6 eV as the value for the excitonic band gap of LIO crystal, to calculate dEegl dT from the evaluated constants, since there is no reported value of it at this moment. Because the thermo-optic coefficient is not constant up to 500°C, we have fitted (2n, dn/dT, tX) data at 50°C increments from 25 to 500°C for LNO [4, 49, 52] and LTO crystals [51] in the same form as in Eq. (3.29) by keeping G as a constant, since the thermal expansion coefficient is approximately constant [53] from room temperature to 500°C. The optical constant H and the fitting gap Eig are computed at the above temperature interval for these two crystals. The constants are then plotted against temperature and, interestingly, the variation is a quadratic function of temperature. However, the variation of fitting gap Eig is negligibly small. On the other hand, H is

156

CHAPTER 3. THERMO-OPTIC COEFFICIENTS

800

LNO

U

600

0

""'"

\0

I 0

400

~

'---J

Polarization

,-.......

~

.-c 200 1: .-c '-'" c: a

e o

N

-200

a

1

2

345

Wavelength [Jlrn] Figure 3.11: 2n(dn/dT) versus wavelength for LNO crystal at room temperature: The solid curves are the computed values and the points are the experimental values; open circles [48], open triangles [4], crosses [49], filled triangle [52], filled circles [48], pluses [4], open squares [49], and filled square [52] (after Ref. 27 with permission from OSA). strongly dependent on temperature. The optimum fitted equations for LNO and LTO crystals are ForLNO: Ho(T) = 159.75384 + 0.093435T - 0.000047759T 2 ,

(3.34)

= 179.66405 + 0.383470T + 0.000042605T 2 •

(3.35)

He(T) ForLTO:

= 199.11300 + 0.033678T + 0.00024602T 2 , He(T) = 130.17849 + 0.574140T + 0.00082996T 2 • Ho(T)

(3.36) (3.37)

These equations indicate that the variations of excitonic band gap with temperature are not linear in the stated temperature region.The temperature

3.4. DERIVATION OF SELLMEIER COEFFICIENTS

157

-200 - r - - - - - - - - - - - - - - - . LID Polarization

u o

~-----e

~ -300 I

o

~-----o

E:'

"0 -....

c -400

~ ,..

N

-500 -t--~-~---,.---,.----r---r---~~ o 234

Wavelength [Jlrn]

Figure 3.12: 2n(dn/dT) versus wavelength for LIO crystal at room temperature: The solid curves are the computed values and the points are the experimental values; filled and open circles [50] (after Ref. 27 with permission from OSA).

variations of excitonic band gap (dEeg / dT) are calculated at different temperatures and are plotted in Fig. 3.13. The behavior is almost linear for both polarizations. The excitonic band gaps (Eeg ) at different temperatures are also shown in the same figure. The band gap decreases asymptotically with temperature. The optical constants cited in Table 3.1 along with Eqs. (3.34) to (3.37) can be used to calculate thermo-optic coefficients at any temperature and wavelength. These thermo-optic coefficients along with the room temperature refractive indices calculated from the room temperature Sellmeier coefficients in Table 2.1 can be used either to formulate the temperature-dependent Sellmeier equations or to assess the currently available NLOD's more accurately, which are described in the next chapter. Refractive indices of KNO were measured by Zysset et ale [58] in the orthorhombic phase at six different temperatures between 22 and 180°C in the wavelength range from 0.4 to 3.4 J1m with the maximum experimental error ±2.5 x 10- 4 . They formulated 18 sets of Sellmeier coefficients by using a two-term oscillator model with an IR correction term for six tem-

158

CHAPTER 3. THERMO-OPTIC COEFFICIENTS

>QJ

4.50

bt:l

e - ray

Q)

~

-2 ~

U

4.45

0.

-3

rn

bQ

-r

"d 4.40

(:)

t::

rn

Ecg

-4 ~~ ~I "d"d

u

.~

t:: 4.35

LNG

0 .....,

~

(0)

~

.~

Eeg (e)

U

>< ~ 4.30

°

"'QJ" >

a

100

200

300

400

500

5

Temperature (OC) Figure 3.13: A schematic plot of the variations of the excitonic band gap with temperature (dEeg / dT) and the band gap (Eeg ) versus temperature for LNO crystal (after Ref. 27 with permission from OSA).

peratures along x, y, and z axes, respectively. For unified notations, x, y, and z axes have been used instead of a, b, and c axes. These Sellmeier coefficients are not able to analyze the formation of three temperature dependent Sellmeier coefficients along the three optic axes. Zysett et ale [58] tried to analyze the thermo-optic coefficients in terms of the quadratic polarization optical effect induced by the spontaneous polarization of the orthorhombic phase. However, it was noticed that the basic assumption of the temperature-independent quadratic polarization optic coefficients is not fulfilled. Also, the comparison between the quadratic polarization optic coefficients of KNO crystal from optical, electro-optical, and from the indirect calculation of the temperature dependence of the refractive indices is not explained quantitatively and even qualitatively. We have used their experimental [58] and calculated values of refractive indices for 22 different wavelengths between 0.4 and 3.4 J.Lm to compute thermo-optic coefficients at five temperatures. The optimum fitted constants G and H are shown in Table 3.1 along with the other parameters at 22°C. The other physical parameters are compared with the experimental values and agree reasonably well within the experimental accuracy. The

3.4. DERIVATION OF SELLMEIER COEFFICIENTS

159

experimental values and the computed curves are shown in Fig. 3.14 along two optic axes x and z, respectively. Thermo-optic coefficients along the y axis are negative and in this case the contribution from the thermal expansion coefficient is dominant. It is interesting to note that the fit gap Eig is slightly above the reflectivity peak [60] of this crystal. The comparison between the experimental data and the calculated values of thermo-optic coefficients are shown in Table 3.4. Since the thermo-optic coefficient is not constant up to 180°C, we have fitted [2n(dn/dT), A] data at four other measured temperatures, i.e., at 50, 75, 100, and 140°C by considering separate constant dn/dT at the stated temperature interval. In the fitting analyses, the same form of Eq. (3.29) was used by keeping G as a constant, since the thermal expansion coefficient is approximately constant [59] from room temperature up to 180°C. As we have noticed, the fitting isentropic band gap is invariant under temperature like previous LNG, LTG, etc. crystals. The fitting and experimental points at 140°C are also shown in Fig. 3.14 along the x and z axes, respectively. Henceforth, the optical constants H, computed at the above mentioned five temperatures, are expressed as a quadratic functions of temperature as shown below.

Hx(T) == 143.04726 + 0.38005T + 0.0014781T 2 ,

(3.38)

Hy(T) == 11.92604+ 0.76106T - 0.00412735T 2 ,

(3.39)

Hz(T) == 241.79441 + O.96897T + O.0035473T 2 ,

(3.40)

and where T is the temperature in degree centigrade. The analyses indicate that the variation of excitonic band gap with temperature is not linear within the stated temperature region. Therefore, the value of E eg and dEeg / dT are calculated and shown in Fig. 3.15. The variation of dEeg/dT is almost linear and E eg is decreasing asymptotically against temperature. The above optical constants are used to calculate refractive indices at any operating temperature and henceforth to assess the currently available nonlinear optical devices precisely. These are described in the next chapter. Gettemy et ale [50] reported the variation of refractive indices of KTP crystal to determine the average power limit of nonlinear interaction at four wavelengths from 0.532 to 1.32 J.Lm from -160 to 35°C. They quoted the least squares fit of dn/dT values at 35°C with the experimental accuracy of ±5.5 x 10- 6 JOC. Kato [61] analyzed the above experimental values to

CHAPTER 3. THERMO-OPTIC COEFFICIENTS

160

800

KNO (x)

,.-.,

U

0

""-

b

Aig

= 0.282 Jlrn

600

~

~

~

~

400

~

= 200 =

140

a

22

'C

"-"

N

a

0.5

1

1.5

2

2.5

3

3.5

Wavelength [Jlrn]

1200 KNO (z)

U 1000

Aig

= 0.2526 Jlrn

~

~

8

800

~

~

~

~

= 'C

600

140

"-"

~

•

400 22

o

0.5

1

1.5

2

2.5

3

3.5

Wavelength [Jlrn] Figure 3.14: 2n(dn/dT) versus wavelength for KNO crystal: Solid curves are the computed values; points are the experimental values [58] for x and z axes; temperatures are indicated along the curves as 22°C and (2) 140°C, respectively (after Ref. 28 with permission from AlP).

3.4. DERIVATION OF SELLMEIER COEFFICIENTS

4.11

-1

0

"">

-2

QJ

4.10

~

4.09

C\S

-3

~ t

4.08

~

L..-....J

""

0.0

~

b.O ~ C\$

~

Eeg (z)

-4

4.07

"d

Q)

~

'"0

0

~

r--,

>

r--,

U

161

.u...-. ~

0

-5

4.06

"d

....-.u

• +-J

>o ), in a positive uniaxial crystal; and

t1T

=

2.78345AI

1rL c I dnle((J)/dT + dnlo/dT - 2dn2o/dT

,

I

(4.17)

4.1. NONLINEAR OPTICAL LASER DEVICES

267

for Type-II ( e + 0 => 0 ), or III ( 0 + e=>o ), in a positive uniaxial crystal, where Al is the fundamental wavelength, and dnl/dT and dn2/dT are the thermo-optic coefficients of the fundamental and second harmonic, respectively, with the respective polarizations as indicated by 0 or e for ordinary and extraordinary waves. L c is the length of the crystal. All other symbols have their usual significance. Coherence length, Lcoh is an important parameter in NOLD. It is defined by the following equation for non-phase matchable materials Lcoh

==

Al , 4(n2 - nl)

(4.18)

where Al is the fundamental wavelength, and nl and n2 represent, respectively, the refractive indices at fundamental and second harmonic wavelengths. In the case of phase matchable materials the corresponding polarizations of the fundamental and second harmonic waves should be taken into account with the PM angle. Since the refractive indices are temperature dependent, Lcoh is also temperature dependent. The coherence length for any optical material gives a measure of dispersion. The dispersion is larger near the band edge; it falls nearly to zero near the middle of the transmission range, and it rises again near the infrared cutoff. Therefore, a peak is expected in the characteristic curve of coherence length versus transmission range. The coherence length is the traversed distance over which the desired frequency of electromagnetic wave is generated. The walk-off angles are also important in NLOD. This is due to the birefringence of the anisotropic materials, the ordinary and extraordinary polarized waves, having the same frequency propagate with different velocities along the same propagation direction. Therefore, the ordinary and extraordinary waves of finite size will not overlap completely throughout the length of the nonlinear optical material. The extraordinary wave is a walk-offof the axis of the ordinary wave. The angle p is called the walk-off angle and is defined for uniaxial crystals as tan p == O.5(n e (8)2)[1/n; - l/n~] sin(28)

(4.19)

for a propagation angle 8 with respect to the optic axis. In general, p is of the order of a few degrees for 8 = 45 0 • But for NCPM, p is negligibly small. Since p is related to the refractive indices of both polarizations and these refractive indices are temperature-dependent, the walk-off angle

268

CHAPTER 4. APPLICATIONS

is also temperature-dependent. In some cases, the value is not negligibly small. The walk-off has a detrimental effect on NLOD to achieve higher conversion efficiency. There are many books on nonlinear optics [7, 8]. However, there is no theoretical estimation of the phase-matched temperatures or explanation of temperature-tuning characteristics of various NLODs. As there are no accurate Sellmeier equations to predict the same at different temperatures, the temperature effect has been analyzed in various NLODs by using the previous analyses of refractive indices at room temperature and thermo-optic coefficients for both ordinary and extraordinary polarizations in anisotropic and biaxial crysytals. Conversion of tunable dye laser radiation of the visible ~pectrum to the coherent tunable UV radiation has been obtained via SHG in the nonlinear crystals ADP and KDP since the begining of nonlinear optics. Quantum Technology Inc. [9], has collected data on the wavelengths that can be phase-matched with NCPM configurations in various KDP isomorphs. NCPM has the advantage of a larger angular acceptance bandwidth and a vanishing walk-off angle. Temperature tuning has an advantage over the angle tuning by phase matching. The angle tuning is very critical and thus requires every time setting of the crystal. On the other hand, only the ambient temperature of the crystal is required to vary to have a temperaturetuned NCPM. These crystals are uniaxial and negative (no > n e ). Type I NCPM requires that the fundamental beam should be polarized in the XY plane, and furthermore, it should propagate in this plane at an angle of 45° to the XY axis. The generated second harmonic is then polarized along the Z axis. A crystal oven, the temperature of which was controlled by proportional temperature controller, was used to heat the crystals above ambient; a thermoelectric cooler was used down to - 20°C. A special cryostat cooled by liquid nitrogen was used for much lower temperatures. A laser-pumped dye laser provided the fundamental radiation. The three dyes used were crystal violet, rhodamine 6G, and sodium fluorescein. A NCPM of the important 1.06 11m transitions to Nd:YAG and Nd:Glass is achieved in two other KDP isomorphs, CDA and CD*A. CDA phase matches 1.064 /-lm at 42°C and 1.073 /-lm at 61°C. CD*A (99% deuterated) phase matches 1.064 11m at about 110°C depending upon the duration level. The damage threshold of the KDP isomorphs is quite high (about 200-300 MW/cm2 peak powers), with somewhat higher values for the arsenates than for the

4.1. NONL/NEAR OPT/CAL LASER DEV/CES

269

Figure 4.1: The phase matching characteristics of the fundamental and second harmonic waves in a negative uniaxial crystal, CDA at 33°C.

phosphates. Figure 4.1 shows a schematic room-temperature NCPM characteristic in a CDA crystal for SHG of 1.0535 /-Lm laser. At this NCPM temperature, the refractive index ( no) for the pump wavelength is exactly the same as the refractive index ( n e ) for the second harmonic. There is no theoretical estimation of these NCPM temperatures. Here, we have calculated the NCPM temperatures and the temperature bandwidths for several nonlinear interactions for these crystals by using the room temperature Sellmeier equations, Table 2.1 and thermo-optic Sellmeier coefficients, Table 3.1. The temperature bandwidths vary from less than one °C cm to 25°C cm. The temperature bandwidth is maximum for CDA and CD*A crystals and is minimum for ADA and ADP crystals. These are shown in Table 4.1 and the agreement is satisfactory within the experimental accuracy of the thermo-optic coefficient and the refractive index measurement. Figure 4.2 shows the coherence length versus wavelength characteristic curve for ordinary refractive indices of ADP and KDP crystals at 33°C. It exhibits a maximum near 1.3 /-Lm and 1.42 /-Lm for KDP and ADP, respectively. The maximum coherence length in these ferroelectric crystals is of

CHAPTER 4. APPLICATIONS

270 Table 4.1:

Crystal

Comparison of the NCPM Temperatures and Temperature-Bandwidths L1TL c for SHG (Type I) in KDP Isomorphs Crystals

Fundamental Wavelength ( j1m)

ADP AD*P KDP KD*P RDP ADA AD*A KDA RDA RD*A CDA

CD*A

0.510 0.525 0.554 0.517 0.532 0.560 0.518 0.522 0.528 0.531 0.538 0.627 0.635 0.568 0.586 0.619 0.578 0.592 0.625 0.598 0.603 0.679 0.684 0.695 0.695 0.717 1.050 1.064 1.073 1.078 1.034 1.062 1.064

NCPM Temperatures Expt. Calculated (OC) eC)

-30 30 120 -30 30 120 20 100 -30 30 120 20 80 -30 30 120 -15 30 120 20 100 -10 20 100 20 100 20 42/48 61 100 20 100 90-112

-31 [Z] -25 -27 [Z] 30 124 [Z] 123 -23 30 120 25 [Z] 20 95 [Z] 90 o [P] -10 40 [P] 30 140 [P] 124 50 82 -25 31 123 -14 31 126 18 98 -25 16 100 19 105 -20 32 61 100 10 70 80

Temp. BW(L1TLc ) Expt. Calculated (OC em) (OC em)

Ref. for Expt.

0.41 0.43 0.44 0.43 0.43 0.43 2.1 2.1 1.8 1.77 [10] 1.8 2.0 1.3 1.3 0.3 0.35 0.4 0.39 0.41 0.45 2.2 2.2 2.2 2.2 2.3 1.2 1.2 20.3 21.9 23.0 23.7 5.8 6.0 6.0 [11] 6.0

[9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9] [9]

Z- Sellmeier from Zemike's data, P- Sellmeier from Philipps's data; Table 2.1

4.1. NONLINEAR OPTICAL LASER DEVICES

271

17.5 15 12.5 ,.....,

e

10

:::l

~

~

eCJ

~

7.5 5 2.5 0 0.4

0.6

0.8

1

1.2

1.4

Wavelength [flm] Figure 4.2: Coherence length for second harmonic generation in ADP and KDP crystals (ordinary refractive index at 33°C).

the order of 17 J-Lm, whereas for ZnGeP2 this value is 80 J-Lm [65]. This large value for ZnGeP2 is due to a wide transmission range and consequent low dispersion. This low value of coherence length in these ferroelectric crystals is due to large dispersion and the narrow transmission region. The variation of coherence length with temperature, Le., dLcoh/dT, is an important parameter. The characteristic curves are shown in Fig.4.3 for ADP and KDP crystals for ordinary refractive indices. Barnes and Piltch [64] calculated the dLcoh/ dT values for CdTe. The variation with temperature of the coherence length is very small in ADP and KDP and is almost one fifteenth of the magnitude as that of CdTe. The variation of coherence length versus temperature is positive for these ferroelectric crystals because of negative thermo-optic coefficients. dLcoh/dT is determined by two factors; dispersion (n2 - nl) and (dn2/dT - dnl/dT). Both these factors change independently from a large to a lower value as one passes from band edge to the region of transmission cut-off. There is a similar behavior of exhibiting a maximum and a minimum in ADP and KDP to that of CdTe. This type of characteristic curve can be calculated from the Sellmeier coefficients as cited in Tables 2.1-2.3 and Tables 3.1-3.2 for any optical material. There are three types of phase-matched interactions which can lead to a

CHAPTER 4. APPLICATIONS

272

,...., 1.75 U ~ 1.5 :::1. 1.25

ADP (0)

~

~

o

~

~ ,.--. ~

1 0.75

~

0.5

-=8

0.25

~

0

~

0.4

0.6

0.8

1

1.2

1.4

Wavelength [J.1m] Figure 4.3: Derivative of coherence length with respect to temperature dLcoh/dT versus wavelength in ADP and KDP crystals (ordinary refractive index).

vanishing phase mismatch as denoted by Eimerl et al. [12]. Table 4.2 lists the experimental and calculated PM angles for various NLOD's in BBO crystal. The calculated PM angles are compared with the other calculated and observed values of Kato [13] and Eimerl et al. [12] for various interactions by using 1.064 p,m laser and its harmonics. The temperature sensitivities of the PM angles are also calculated by using Sellmeier coefficients in Table 2.1 and optical constants G and H in Table 3.1, i.e., with the consideration of dispersion. Eimerl et al. [12] calculated the same by using an average, dispersionless values for the thermo-optic coefficients. These values are compared with the experimental data. It is shown that the present work has agreed well between the measured and the computed values for fJ and dfJ/ dT. Therefore, the dispersion of thermo-optic coefficients, particularly its values at the UV edge, assesses the PM characteristics more accurately. Recently, LBO crystal is important for various NLODs. NCPM Type-I SHG ( fJ == 90°, ¢ == 0°) is useful for a wide range of applications due to its higher conversion efficiency. In this configuration, the fundamental

273

4.1. NONLINEAR OPTICAL LASER DEVICES

Table 4.2:

Interactions

Comparison of the Calculated and Observed PM Angles and Temperature Sensitivities of BBO for Various Parametric Three-Wave Interactions Using a 1.064-j.1m Laser and its Harmonics

(Type)

PM angle (()) in degree Expt. [13]

1(0 + 1(0 1(0 + 2(0

2(0 + 2(0 1(0+3(0 1(0 + 4(0 2(0 + 3(0

(I) (II) (I) (II) (Ill) (I) (II) (I) (II) (I) (II) (I)

22.8 32.9 31.3 38.5 59.8 47.5 81.0 40.2 46.6 51.1 57.2 69.3

Expt. Calc. [12]

[12] 22.7 32.4 31.1 38.4 58.4 47.3

22.9 31.9 31.1 37.8 58.9 47.4 82.1 40.2 46.1 51.1 56.9 69.4

Calc. [24] 22.8 32.0 31.2 38.0 58.9 47.4 81.6 40.2 46.2 51.1 56.9 69.5

dfYdT [ j.1radl DC]

Calc. [12] 14.1 20.9 19.2 24.1 50.0 31.2 191.0 24.3 28.8 29.9 35.2 65.0

Expt. [12] 9.9 20.9 17.3 26.2 73.4 43.7

Calc. [24]

+ 1 12.8 + 2 20.5 + 1 19.6

+ 2 27.1

+ 3 64.0 + 1 38.8 280.0 30.2 39.3 46.7 60.0 100.0

wave is polarized parallel to the Z-axis and is propagated in the XY-plane of the LBO crystal. The temperature-tuned NCPM phase-matched characteristics are explained thoroughly near room temperature by using the room temperature Sellmeier coefficients and the optical constants G and H, cited in Table 3.1. At the higher temperature region, however, it is not possible to explain the same. As we have analyzed, most of the semiconductors and nonlinear crystals exhibit the constancy of dn/dT throughout a small operating temperature region. Such constancies have been observed by Eimerl et al. [12] from 20 to 80°C in BBO and by Velsko et al. [14] from 16 to 35°C in a LBO crystal. However, the thermo-optic coefficients, dn/dT are not constant at the higher temperature crystals such as LNO, LTO [15], and KNO [16]. Since the thermal expansion coefficient is not constant along the Y-axis for LBO, the experimentally measured NCPM SHG temperatures, [17-19] for the fundamental wavelengths from 0.95 to 1.31 jlm, the Sellmeier coefficients, and the computed optical constants G and H at 20°C

CHAPTER 4. APPLICATIONS

274

are used to calculate the G's at these NCPM temperatures. The optical constants G's are now expressed as a function of temperature T and shown below along the Y-axis

G y (T)

== 372.170 - 2.199 x 10- 1 T + 1.1748 x 10- 3 T 2 - 2.05077 X 10- 6 T 3 ,

(4.20)

where T is the temperature in degree centigrade. It is worthwhile to examine Eq. (4.20) at higher temperatures. As the temperature increases, the value of G decreases and it corresponds to the shift of the thermal expansion coefficient from more negative to the less negative values. This behavior was observed experimentally [20] in measuring thermal expansion coefficients along the Y-axis of LBO. Again, the NCPM temperatures [18, 19] in the fundamental wavelength region of 1.40 -1.75 /lm, the room temperature Sellmeier coefficients, and the optical constants G and H are used to calculate different values of H along the Z-axis. These values are then expressed as a nonlinear function of temperature T as

Hz(T)

== 410.66123 + 1.667 x 10- 1 T - 5.1887 X 10- 4 T 2

+ 5.56251 X 10- 7 T 3 •

. (4.21)

As the temperature rises, the value of Hz increases for LBO, similar to the characteristics of the LNO crystal [15]. The temperature-tuned NCPM SHG is calculated throughout the transmission region as shown in Fig. 4.4. The bold solid curve is based on this formulation by using Tables 2.1 and 3.1 and Eqs. (2.16), (3.29), (4.1), and (4.20-4.21). The tuning range is varied from 0.95 to 1.75 /lm for the fundamental wavelength by changing the temperature from -40 to 320°C. The solid and dashed curves are made the same by using the other room temperature Sellmeier [14], [21] and the average constant thermo-optic coefficients for the X-and Y-axes, and a wavelength-dependent thermo-optic coefficient for the Z-axis of the LBO crystal as analyzed by Velsko et al. [14]. The calculated tuning range is 0.98-1.82/lm for a temperature change from 0 to 294°C. Both curves are unable to explain the NCPM negative temperatures. We have observed NCPM Type I travelling-wave SHG by using a laser diode having a wavelength of 1.48 /lm and an output power of 100 mW in a LBO crystal at 50°C. Some computed and experimental NCPM temperatures are cited in Table 4.3. The experimental points

275

4.1. NONLINEAR OPTICAL LASER DEVICES 350

i

r--"1

U 0

~

\

250

.~

.~~

Q) ~

\

::3

~

ro ~

Q)

~

0

LBO SHG

150

E Q) ~

~ ~ U

50

Z

Type II

-50 0.9

1.1

1.3

Wavelength

1.5

1.7

[~m]

Figure 4.4: The NCPM temperatures versus fundamental wavelengths for Type I and II SHGs in LBO: The bold solid curve is calculated using optical constants in Tables 2.1 and 3.1, and Eqs. (2.16), (4.1), (4.20) and (4.21). The solid and dashed curves are also calculated using the room temperature Sellmeier coefficients of Velsko et at. [Ref. 14], Mao et at. [Ref. 21], and thermo-optic coefficients of Velsko et al. [Ref. 14], respectively. Experimental data are denoted as open circles [Ref.18], squares [Ref. 17], and solid circles [Ref. 19]; and open circles [Ref.19] for Type I and II SHGs, respectively, (after Ref. 24 with permission from AlP).

[17-19] are also shown in Fig. 4.4. The computed values based on this dispersion model interpret the experimental values to within ±4.0°C. Recently, Gontijo [22] measured the temperature dependence of ~n, i.e., the refractive index difference between the fundamental and second harmonic ( ~n == nz,w - n y,2w) in a NCPM Type-I configuration of SHG generation by using a 1.064-JLm laser. It was found that d(~n)/dT is dependent on both wavelength and temperature. We have calculated the same at different temperatures and this is shown in Fig. 4.5. The nonlinear equation is written as

276

CHAPTER 4. APPLICATIONS

Wavelength [J1,m] 0.8

(V')

1.0

1.2

1.4

1.6

1.8

0.1

14

-0.1

12

-0.3

0 rl

10 -0.5

~ ~

~

8

-0.7

~

E u U

°

~

u

'-""

"99.5% in the spectral range between 1.050 and 1.150 /lm,

294

CHAPTER 4. APPLICATIONS

and a transmission higher than 90% at 0.532 /lm.The output mirror had a reflectivity Rr = 80% in the spectral range between 1.050 and 1.150 /lm \Vith a transmission Ts > 90% at 0.532 /lm. A frequency-doubled Q-switched Nd:YAG laser was used as a pump (A p = 0.532 /lm) with a repetition rate of 5 Hz and pulse width of 25 ns. The pump was propagating along Yaxis with polarization parallel to the Z-axis. The generated signal and idler waves were propagated along the same direction with polarizations parallel to the X-axis. The nonlinear-optical coefficient d l13 was used for the NCPM interaction. Stable oscillations of both signal and idler waves were observed for a pump-beam intensity of 10 MW/cm2 • The OPO was tunable in the spectral range from 0.960 /lm for the signal wave to 1.2 /lm for the idler wave by changing the temperature of the crystal from 188.2 to 195.5°C. The schematic temperature-tuning curve based on this model of thermo-optic coefficients for this OPO with the experimental data as open circle [36], is shown in Fig. 4.14. A conversion efficiency of 10% was reported and an intensity ratio of 0.5 between idler and signal waves were measured in the spectral range 1.13-0.997 /lm.

4.1.3 Sum Frequency Generation Sum frequency generation (SFG) is used for effective conversion of IRradiation to the visible- or UV-region, where highly sensitive detectors are available. This scheme is used to convert the CO 2 laser (A = 10.6 /lm) into the visible- or near-IR region with a conversion efficiency of 30 to 40% by mixing with dye or Nd:YAG laser as a pump in a nonlinear crystal. Here only the temperature-tuned SFG will be discussed. ADP is useful for a temperature-tuned SFG. The analyses of thermo-optic coefficients are useful to verify the temperature-dependent optical devices, even at the room temperature, to get the information about the temperature stability. Table 4.8 shows some SFG schemes by using ADP and some other nonlinear crystals. Stickel and Dunning [68] demonstrated a NCPM Type-I SFG of 0.2400.242 /lm by mixing the SH ofmby laser and IR-dye laser in a temperaturetuned -20 to 80°C KDP crystal. Similarly, a NCPM Type-I SFG [69] is obtained by mixing the stimulated Raman Scattering (SRS) of 1.064 /lID with radiation at 0.220 /lm, which is generated by mixing the OPO radiation with SRS Stokes wavelengths produced by the fourth harmonic of Nd:YAG laser in nitrogen atmosphere, Le., the wavelength of 0.283 /lm in

4.1. NONLINEAR OPTICAL LASER DEVICES

295

1.3 - , - - - - - - - - - - - - - - - - _

KNOOPO 1.2

0.9

-;------.----~---.,.__--~

185

190

195

200

205

Figure 4.14: The NCPM temperature-tuning range for a KNO OPO, pumped at 0.532 p,m in a Type-I interaction along the X-axis (e--+oo, 0=0°, =90°): The solid curve represents a calculated tuning range using constants from Tables 2.1 and 3.1 and Eqs. (2.16), (3.29), (3.38) - (3.40), (4.1), (4.26), and (4.27); open circles are the data of Biaggio et al. [Ref. 36].

this crystal. An output energy of 40 p,J was generated. ADP crystals have been widely used to make a NCPM temperature-tuned Type-I SFG device. There are various schemes [70-76] for NCPM mixing as shown in Table 4.8. Normally, argon, ruby, and Nd:YAG lasers have been used to mix with various dye lasers to generate high power UV radiation. Massey and Johnson [76] used Nd:YAG laser to mix with the SH of dye laser (0.246-0.302 p,m) and generated the shortest wavelength of 0.208 p,m in ADP crystal. In this scheme, a NCPM Type-I SFG was achieved by keeping the ADP crystal at -120°C. The experimental and calculated temperature changes for the same tuning are shown in Table 4.8. It is noticed that the calculated temperatures differ with the experimental values. The calculated values vary sensitively with the generated wavelengths, and there is no experimental

296

CHAPTER 4. APPLICATIONS

Table 4.8: Comparison of the NCPM Tuning Ranges for Several SFGs in KDP Isomorphs and Other Crystals Interacting lasers Ap, Ai ( ,urn)

NL Type Expt. Tuning Power, crys- of Inter- range (,urn) energy, tal actions CE (%)

0.3471, dye KDP 0.220, 1.39 KDP 0.477-, dye ADP 0.3638, dye ADP 0.351, dye ADP 0.4131, dye ADP 0.3471, dye ADP 0.532, dye ADP 1.064, SHdyeADP 1.064, SHdyeADP THO of 0.586 BBO 0.5145,2.7-4.5 LNO 0.6764, 1.064 KNO 0.6943, 1.064 KNO 0.19,1.9

I (ooe) I (ooe) I (ooe) I (ooe) I (ooe) I (ooe) I (ooe) I (ooe) I (ooe) I (ooe) I (ooe) I (ooe) I(ooe) I (ooe)

LBO I (CPM)

0.240-0.242 0.190 0.252-0.268 0.243-0.247 0.243 0.2475 0.240-0.248 0.246-0.249 0.222-0.235 0.208-0.214 0.1953 0.43-0.46 0.4136 0.4201 0.1727

Temp.Variations Refs. Expt. Calculated for (OC) (OC) expt.

1MW -20 to 80-56 to 70 [68] 20 I1J 20 0 [69] 8 mW 20 15 [70] 4 mW 20 14 [71] 0.3 mW 8 5 [72] -103 -58 [73] 1 MW -20 to 80 -21 to 74 [68] 3 I1J -120 to 0 -103 to -78 [74] 10% -120toO 24 to 94.5 [75] 1.7I1J-120toO -61to-24[76] 5% -178 -183.5 [77] 180-400 175-380 [78] 0.1 mW -4 -6 [79] 7.9 W 27.2 28.0 [79]

-50

nJ

200

190

[80]

accuracy on the measured wavelengths. Due to the extreme sensitivity of dOpm / dT to small changes in the dn/dT values, particularly at the band edge region for NCPM Type-I SFG, this large discrepancy could be accounted for by relatively small errors in the extrapolated thermo-optic coefficients. A SFG Type-I scheme has been used by Lokai et ale [77] to generate the shortest wavelength of 0.1953 J-lm in a BBO crystal at -178°C by mixing the l:lnpolarized output of an excimer laser-pumped dye laser, 0.586 J-lm with its second harmonic. The calculated temperature is -183.5°C and agrees well with the experimental value. Similarly, a temperature-tuned, NCPM Type-I SFG was demonstrated by Abbas et ale [78] by using a LNO crystal. They used an intense cw argon-ion laser as a pump to mix with the IR-radiation from 2.7 to 4.5 J-lm in a phase-matched condition by varying the crystal temperature from 180 to 400°C. The generated signal is

4.2. OPTICAL-FIBER COMMUNICATION SYSTEMS

297

tuned from 0.43 to 0.46 /-lm. The present theoretical model on thermo-optic coefficients explains these experimental parameters with the consideration of the uncertainties in measuring refractive indices. Baumert and Gunter [79] used a KNO crystal for NCPM Type-I SFG of a cw Nd:YAG laser (1.064 /-lm) of power 2.1 W with the pump radiation of krypton laser (0.6764 /-lm) of power 26.2 mW at a temperature of -4°C. The generated signal wavelength is 0.4136 /-lm and the power was over 0.1 mW. They also used a Q-switched ruby laser having peak power of 20 kW to mix with the Nd:YAG laser to generate a signal of wavelength 0.4201 /-lm at 27.2°C. The present model is capable of explaining the NCPM temperatures. Recently, Seifert et ale [80] have demonstrated the generation of tunable femtosecond pulses as low as 0.1727 /-lm by a phase-matched «(J =90° , 4> = 73°) SFG in a LBO crystal. They used a commercial tunable fs Ti:sapphire laser having wavelength 0.760 to 0.815 /-lm as a pump. The idler pulse (1.9 /-lm) is mixed with the fourth harmonic of 0.760-/-lm pump in a LBO crystal which was kept at 200°C to produce a VUV pulse at 0.1727 /-lm. Our calculated phase-matched temperature agrees well with this experiment.

4.2

Optical-Fiber Communication SysteDls

Silica glass ( Si0 2 ) is the basic optical-fiber material, and it has been extensively used to make various kinds of optical fibers [81], amplifiers [82], and fiber lasers [83] with suitable doping materials since the late 1960s. Fiber lasers and amplifiers are the first step to replace the older electronic regenerators/repeaters and to lead into the all-optical, complete photonics age in the future. The refractive index, its dispersion, chromatic dispersion, and its variation as a function of temperature are important characteristics of the silicabased glasses, which are necessary for the evaluation of optical fiber transmission system designs using such glasses made with optical fibers. In general, they are measured at discrete wavelengths in the transmission range of the glasses, and their interpolation/extrapolation technique is state-of-theart. A knowledge of the refractive index as a function of temperature is necessary to evaluate chromatic dispersion at different temperatures because it plays a vital role in the optical fiber communication system. In Chapter Two, the room temperature refractive indices are fitted in a

298

CHAPTER 4. APPLICATIONS ~

A

1.2 1

D •

•

0.8 • c 0.6

•

r.I'.J.

.• ~

• •

• •

• •

~

• •

•

• •

• •

• •

•

C •

•

•

•

•

•

B

•

•

~

e ~

0

U

0.4 0.2 0 • • -100

•

•

-50

•

0

50

100

Temperature [Oel Figure 4.15: A typical plot of the temperature-dependent Sellmeier coefficients (A, B, C, D) for Si0 2 glass (after Ref. 85 with permission from IEEE).

two-pole Sellmeier formula for three kinds of silica glasses-Si0 2 , aluminosilicate, and Vycor glasses [84]. Here, we arrive at relations of Sellmeier coefficients using temperature and smoothed thermo-optic coefficients, which are evaluated in Chapter Three. Two other aspects are also considered from the temperature-dependent Sellmeier coefficients [85] viz. chromatic dispersion and zero dispersion wavelengths at different temperatures, and are compared with the experimental values. The Sellmeier coefficients at any temperature T are computed from the room temperature Sellmeier equation and the smoothed dn/dT values by calculating refractive indices from Eq. [4.1]. We have evaluated the Sellmeier coefficients from the calculated refractive indices of Si0 2 , aluminosilicate, and Vycor glasses from -100°C to 100°C at 20°C intervals. All these Sellmeier constants (S) are then separately plotted against temperature. Strangely enough, all are found to fit nicely into straight lines. A typical plot of the coefficients for the fused silica is shown in Fig. 4.15. Constants of the straight lines are obtained by least-square analysis of the data and are shown in Table 4.9. These optical constants are used to calculate refractive indices at any operating temperature.

299

4.2. OPTICAL-FIBER COMMUNICATION SYSTEMS

Table 4.9: The Linear Fitted Constants (m = Slope, c t = Intercept), of the Least Squares Analyses of the TemperatureDependent Sellmeier Coefficients (8 ) of FS, AS, and V Glasses (after Ref. 85 with permission from IEEE) Sellmeier (S) Fitted Coefficients Constants

A

m x 106 C1

B C D

m x 105 C 1 x 10 m x 107 C 1 x 100 m x 107 C1

E

Fused Silica (FS) Si02 6.90754 1.31552 2.35835 7.88404 5.84758 1.10199 5.48368 0.91316 100.0

Alumino Silicate (AS)

VycorGlass

24.95380 1.41294 -0.11466 9.50465 12.24700 1.32143 11.60740 0.90443 100.0

45.70720 1.27409 -1.47194 8.27657 12.35900 1.06179 12.58560 0.93839 100.0

(V)

(S = mT + C1, T is the temperature in degree centigrade)

4.2.1

Chromatic Dispersion and Zero-Dispersion Wavelength

Chromatic dispersion plays an important role in an optical fiber communication system, manifested through the wavelength dependence of the refractive index n(A) as in Eq. (2.16) by the following relation (4.28)

and

V(,x)

l/(cn)[ -4/ A5 {BC 2 /(1 - C/ ,x2)3

+ A(dn/dA)2 + 3n(dn/dA)],

+ DE 2 /(1 - E / ,x2)3} (4.29)

where

and c is the speed of light. Chromatic dispersions are computed for these glasses at different temperatures by using the temperature-dependent Sellmeier coefficients and

300

CHAPTER 4. APPLICATIONS

are shown in Fig.4.16 for Si0 2 and alumino-silicate glasses at 26°C. The dispersion characteristics are not linear for the whole spectral region. The temperature dependence of chromatic dispersion at 1.53 Ilm is computed for Si0 2 glass and it is -1.5 x 10- 3 ps/(nm.km.K). This value is exactly the same as that of experimental one [86]. The wavelength at which the chromatic dispersion is zero is called the zero-dispersion wavelength and is denoted as Ao. The dispersion behavior near the zero dispersion wavelength is shown in Fig. 4.17 for Vycor glass at -100 o e and 100°C, respectively. The dependency is almost linear as shown. The zero-dispersion wavelengths are also computed for these glasses at different temperatures. The zero-dispersion wavelengths are 1.273, 1.393, and 1.265 Ilm at 26°C for Si0 2 , aluminosilicate, and Vycor glasses, respectively. Interestingly, the temperature dependency of zero-dispersion wavelength is linear and dAo/dT is 0.025 nm/K for Si0 2 • This study agrees well with the experimental [86] values of 0.029 ± 0.004 nm/K and 0.031 ± 0.004 nmIK for two dispersion-shifted optical fibers within the experimental accuracy. dAo/dT is 0.03 nmlK for both aluminosilicate and Vycor glasses. Zero-dispersion wavelength as a function of temperature T is shown in Fig. 4.18 for Si0 2 glass. This analysis implies that dAo/dT is dominated by the material of the core-glass of the optical fiber instead of the optical fiber design. This is one of the fundamental optical properties of the glass itself. The Ohara and Schott optical glasses [87-88] are useful in many optical and opto-electronic studies. The room-temperature Sellmeier coefficients [89] as cited in Table 2.2 can be used to faithfully interpolate and extrapolate the refractive indices beyond the measured wavelength region. The Abbe number (Vd) is calculated from (4.31) where nj, nd and n c represent refractive indices at wavelengths 0.4861, 0.5876, and 0.6563 Ilm, respectively. The Abbe numbers of these optical glasses lie between 50 to 70 as shown in Table 4.10 for some selected Schott and Ohara optical glasses. The material chromatic dispersions (V) of these glasses also play a vital role in the optical communication systems, femtosecond technology, and other optical devices. They are shown through the wavelength dependence of the refractive index as shown in Eq. (4.28). Chromatic dispersions [89] are calculated for these glasses by using Sellmeier coefficients of Table 2.2 and are shown in Fig. 4.19 from the 0.7

4.2. OPTICAL-FIBER COMMUNICATION SYSTEMS

301

~

e

--- 30 ~

5

20

~ 10 VJ. c.

~

0

~

-10

I------,.:::.--~;...-.-------___,

.s= -20 ~ ~ -30 VJ.

Q

-40

-'--_~_ _...._ 1.7 1.3 1.4 1.5 1.6 Wavelength [J.lrn]

L..a..-_---L_ _....a..-_---L_ _

1.1

1.2

Figure 4.16: The characteristic behavior of dispersion for Si0 2 (a) and aluminosilicate (b) glasses at 26°C from 1.1 to 1.7 p,m (after Ref. 85 with permission from IEEE).

---e ~

1

~

e=: 0.5

'-" ........

VJ.

C.

~

---

~

0

'-"

....5 -0.5

.. .. VJ.

~

~

-1

Q

1260

1264

1268

1272

1275

Wavelength [flm] Figure 4.17: The characteristi~ behavior of dispersions for Vycor glass at -100°C and +100°C (the upper and lower lines, respectively), (after Ref. 85 with permission from IEEE).

302

CHAPTER 4. APPLICATIONS

1274 1273 ~

g 1272 1271 1270 -100

-50

0

50

100

Temperature [Oel Figure 4.18: Zero dispersion wavelength AQ as a function of temperature T for Si0 2 glass, (after Ref. 85 with permission from IEEE).

to 1.8 J-lm wavelength region. The dispersion characteristics are not linear and increase from high negative values to small positive ones for these glasses. These values are calculated for three optical windows Le., at 0.85, 1.33, and 1.55 J-lm and are shown in Table 4.10 for some Schott glasses and SSL6 Ohara glass at 20°C. The zero-dispersion wavelength is varied from 1.28 to 1.58 J-lm for these optical glasses. Since the zero-dispersion wavelength lies at 1.884 for SF6 glass, it is useful for compressing laser pulses in the femtosecond region. Similarly, the dispersion characteristics are calculated for the high-index tellurite glasses [90] by using Sellmeier coefficients [91] of Table 2.3 and are shown in Fig. 4.20. The dispersion characteristics are not linear and are increasing from high negative values to small positive ones as the wavelength increases. This nonlinear behavior is due to the variation of dispersion of refractive indices. Since the Si0 2 glass possesses less dispersion in the visible region, the zero-dispersion wavelength lies at 1.273 J-lm. The pure Te02 glass has zero dispersion at 1.688 J-lm, whereas, by using a modifier, this value shifts toward the longer wavelength side, Le., at around 2 J-lm as shown in Table 4.10. Therefore, the chromatic dispersions are neg-

303

4.2. OPTICAL-FIBER COMMUNICATION SYSTEMS Table 4.10: Optical Properties of Some Selected Schott, Ohara, Binary Tellurite pM mO n-(100-p)Te0 2 and S i O2 Glasses at Room Temperature Glasses Abbe number Chromatic dispersions [ps/(nm.km)] at

BaLF4 BaKl PSK3 SK4 SSK4A SK5 LaF2 BK7 FK5 LaK10 KzFS6 ZK1 SSL6 SF6 pMmOn 20Na2O 10Al 20 3 25ZnO 20BaO 15BaO 10Sb20 3 20Mo0 3 20W0 3 10Nb2Os 20Tl2O TeO 2 Si02

Ao

(v d)

0.86,um

1.33 ,urn

1.55,um

(,urn)

53.69 57.59 63.51 58.58 55.12 59.52 44.67 64.15 70.41 50.37 48.43 57.99 51.19 25.35

-151.9 -140.2 -114.2 -145.5 -159.4 -119.9 -242.2 -104.3 -86.6 -196.3 -162.7 -126.2 -138.1 -471.1

-14.80 -13.35 -1.87 -10.90 -16.65 -7.37 -33.46 1.00 4.67 -12.62 -4.40 -8.21 -5.96 -88.60

6.28 6.36 17.10 10.90 5.36 10.73 -3.28 19.53 21.34 17.33 22.72 10.85 15.78 -40.30

1.474 1.468 1.348 1.429 1.487 1.409 1.581 1.320 1.281 1.412 1.361 1.414 1.382 1.884

18.5 20.0 18.7 18.3 18.1 17.5 15.4 16.4 16.4 14.5 17.7 69.0

-805.1 -781.6 -864.0 -890.7 -899.1 -972.8 -1123.2 -1070.2 -1062.8 -1196.3 -864.5 -83.8

-166.90 -159.30 -174.60 -183.70 -182.00 -198.30 -230.20 -216.10 -208.60 -244.10 -122.20 5.30

-91.20 -84.20 -93.10 -100.10 -97.70 -106.90 -127.10 -116.00 -108.30 -136.00 -35.70 21.70

2.221 2.121 2.149 2.214 2.169 2.179 2.274 2.170 2.086 2.324 1.688 1.273

CHAPTER 4. APPLICATIONS

304

o

--G--

-(100)

BaLF4

.......-........ SF6

~

:> "'--' Q -(300)

U

-(400)

····0····

BaKI

----6.----

PSK3

- - -111- --

SK4

_._.+_._.

SSK4A

---$---

K5

- - v--

LaF2

_.;;1-.-

BK7

--+--

FK5

................

LaKIO

••• -e:- ••••

KzFS6

---~--_.

ZKI

- - -c>- -. SSL6

-(500) -+-----'-----....------...------~ 1.85 1.05 1.45 0.65

Wavelength (f.!m)

Figure 4.19: The characteristic behavior of chromatic dispersion for some selected Schott and Ohara optical glasses at 20°C (after Ref. 89 with permission from OSA).

305

4.3. THERMO-OPTIC SWITCH

150 ,...-, ~

8

~

8

c

'--" .........

-50

-250

o 20B30

-450

lO AI 203 20]\.10°3 A 201120 .6. 20Na20 • )(

CI.2

~

~

~

;,

'--"

c

....,.0 .... CI.2

aJ ~ CI.2

~

-650 -850 -1050 -1250

•

15BaO

o lO Nb 20 3

N

~

~

IQ

•

lOSbZ03 20W03 252nO

~ 4

0.75

1.25 1.75 2.25 Wavelength [flm]

2.75

Figure 4.20: The characteristic behavior of chromatic dispersion for binary tellurite glasses pMmOn(100-p)Te02 at 23°C (after Ref. 91 with permission from ACS). ative at the three optic windows for binary tellurite glasses. We believe this demonstrates that binary tellurite glass prisms are better than SF6 glass prisms for compressing laser pulses in the femtosecond region.

4.3

TherDlo-Optic Switch

Thermo-Optic (TO) switches based on optical waveguides are efficient to reroute transmitted laser from one port to other by changing the refractive index of the waveguide only by means of varying the ambient temperature surrounding the waveguide. As we have seen, there are many optical materials which possess thermo-optic coefficients of different values within the range of 5x 10- 3 to 1.0x 10- 6 • Therefore, by a judicious choice of the optical materials and by making a waveguide, an efficient TO switch can be made. TO switches have some beneficial characteristics such as low trans-

CHAPTER 4. APPLICATIONS

306

... ..... ...

.'.0 .'.' .'.' '. .'.'

Figure 4.21: A thermo-optic switch (8 x 8 optical matrix), based on thermo-optic effect (after Ref. 92). mission loss, low fiber-coupling loss, high stability, high producibility, and and they are suitable for large scale integration. This switch will lead to the all-optical age of an OFCS's switching system. Recently, NTT [92] of Japan has developed a 8 x 8 Optical Matrix Switch (OMS) based on a silica-based waveguide on a silicin substrate. This waveguide operates as an active device by using the phase-control technique that is based on the thermo-optic effect of the waveguide material and a Mach-Zehnder interferometer (MZI). A product of this switch is shown in Fig. 4.21. The basic principle of it is explained by the initial investigation of the 2 x 2 TO switch [93]. Figure 4.22 shows (a) a top view (b) a cross-sectional view along a line Q - U of a 2 x 2 TO switch. This switch has a MZI configuration with two 3-dB directional couplers (DC) with a 50% coupling ratio and two waveguide arms of the same length. TO phase shifts are accomplished by using thin film heaters at both arms of

307

4.3. THERMO-OPTIC SWITCH 3dB directional coupler

4

2

-JJ

3

10

~14

Thin-film phase shifter (a) Top view Thin-film phase shifter

Q ~J~ . _~_-~_ U Cladding-

0

o---Core

_ _ Si substrate (b) Cross-sectional view along line

Figure 4.22: Configuration of a 2 switch (after Ref. 93).

Q- U

x 2 Mach-Zender interferometer (MZI)

the waveguide. The normalized optical output power of the MZI are

13 /10 == (1 - 2k)2 cos2(~cP/2)

+ sin2(~cP/2),

14 /10 == 4k(1 - k) cos2(~cP/2),

(4.32) (4.33)

where ~cP is the phase difference between the waveguide anns of the MZI and is expressed by the following equation. (4.34) Ais the wavelength and k is the DC coupling ratio. Here the coupling ratios are assumed to be the same at both arms. Lht is the heater length and ~n the effective refractive index difference between the waveguide arms. In the off-state (~n = 0), Eqs. (4.32) and (4.33) are reduced to

13 /10 == (1- 2k)2,

(4.35)

14 /10 == 4k(1 - k).

(4.36)

308

CHAPTER 4. APPLICATIONS

The optical signal light passes through the cross path. However, some part of the optical signal remains in the through-path except when k =0.5. The switching action from the cross-state (path 1-4 or 2-3) to the bar-state (path 1-3 or 2-4) is achieved by using a half wavelength optical path-length difference between the two waveguide arms, which is obtained by tuning the TO phase-shifter. In the on-state (LhtLln == A/2, the Eqs. (4.35) and (4.36) are (4.37) /3//0 == 0,

/4/ /0 == 1.

(4.38)

These equations do not depend on k. The temperature change for switching at the waveguide core is about 13° for Lht =5 mm at A =1.31 /lm for dn/dT = 10.Ox 10- 6 / o C. A combination of flame hydrolysis deposition (FHD) and reaction ion etching (RIE) techniques are used to fabricate this switch on a Si substrate. A waveguide with a low Ll (relative refractive index difference between the core and cladding Ll =0.3%) is normally used for the switch. The core size is 8 /lm x 8 /lm and the cladding thickness is 50 /lm. A Cr thin-film heater is used for the TO phase-shifter. Its size is almost 5 mm long and 50 /lm wide. The switch is approximately 30 mm long and 3 mm wide. The output optical characteristics of this 2 x 2 TO switch are shown in Fig. 4.23 as a function of the phase-shifting driving power. The insertion loss is 0.5 dB and the switching electric power is about 0.4 W. The switching response time is 1 to 2 msec.

4.4

Optical-Fiber TeDlperature Sensor

The thermo-optic coefficients of the optical glasses made into optical fibers are used as a sensing element for measuring temperatures and other sensing devices. The phase of the guided laser will change as the ambient temperature of the sensing fiber varies. A polarimetric fiber sensor using a frequency stabilized semiconductor laser has been demonstrated [94] for remote temperature measurement. They measured the temperature change from the phase delay in between two orthogonally polarized modes in a polarization maintaining fiber. The measurement accuracy was less than 0.OO5°C. The construction of this sensor is simple. A leading fiber (long length) and a sensing fiber (short length) are both polarization maintaining

309

4.4. OPT/CAL-F/BER TEMPERATURE SENSOR 1.0~

CD

~r-

--,~_ _

0.8

~

o

0-

~ 0.6

"5 o (ij .~

C. o

0.4

Q)

.2:

Cti 0.2

CD

a:

0.5

1.0

Phase-shifter driving power (W)

Figure 4.23: A characteristic response of the thermo-optic 2 switch, (after Ref. 93).

x 2 MZI

fibers, and are connected together by a polarization rotating coupler. This coupler is used to rotate the polarization axes of the sensing fiber by 45 0 with respect to the axes of the leading fiber. The endface of the sensing fiber is polished and some metallic alloy is used to make an in-built mirror. A semiconductor laser was used to launch a linearly polarized light into the leading fiber; and there is only one polarization mode. Two polarization modes are equally excited at the coupler, i.e., at the input end of the sensing fiber. A phase delay occurs after reflection from the mirror. This phase delay is due to the modal birefringence of the fiber. The phase delay ¢ is related to the modal birefringence, B f as (4.39) where 1I is the laser frequency, L f is the length of sensing fiber, and c is the velocity of light. The effective optical-path difference between two modes is B f L f. The phase delay in the sensing fiber is measured by the interference between the two modes with an analyzer and a photodiode. The sensing fiber induces a phase delay between two modes by changing the temperature.

310

CHAPTER 4. APPLICATIONS

Ball et ale [95] have used an active fiber sensor to measure temperature. In this sensor, a rare-earth doped fiber is used as a fiber laser instead of using the fiber grating. This fiber laser acts as an active Bragg-grating sensor. The effective refractive index of the fiber controls the Bragg wavelength. A laser diode is used to pump the fiber laser. The response of the active sensor was measured by a grating spectrum and a Fabry-Perot optical spectrum analyzer. In measuring temperature, the fiber laser was wound onto a thin copper cylinder, which was kept in a hot plate. A calibrated thermocouple was used to measure temperature with a resolution of 1°C. They have correlated their measurements with the fiber expansion and thermo-optic coefficients. The wavelength varies linearly with both the applied strain and temperature and is given by (4.40) where f is the applied strain, 8t\ is the wavelength shift, and ~T is the temperature change in degrees centigrade. At the operating wavelength of an erbium-doped fiber laser (about 1.53 Jlm), the laser wavelength changes at a rate of 0.011 nml°C. This resolution is not sufficient. Therefore, some semiconductor waveguides having large thermo-optic coefficients should be employed to make an efficient high-resolution temperature sensor.

4.5

Thermo-Optic Modulators

OFCS will require the realization of low-cost guided-wave active and passive components. Crystalline silicon (c-Si) is a suitable candidate for such applications in the near-infrared wavelength region. Recently, hydrogenated amorphous silicon (a-Si:H) has been used to make a planar waveguide [96] with the crystalline silicon as a substrate by a low-temperature plasmaenhanced chemical vapor deposition (PECVD) technology. A schematic cross section of the planar waveguide is shown in Fig. 4.24. It consists of a rv3 Jlm-thick undoped a-Si:H core layer, between two undoped a-SiC:H cladding layers. For good ohmic contacts, a heavily B-doped a-Si:H top layer has been deposited on the top. Planar waveguides of various lengths were fabricated by substrate cleaving. Waveguides with lengths of 0.62 and 1.65 Jlm were realized [96]. A Fabry-Perot-like characteristic was observed when these waveguides were heated and the transmitted intensity versus temperature was plotted. A periodic amplitude modulation of the

4.6. REFERENCES

311

L

a - SiC:H, 150 nm

a - SiC:H, 150 nm

a - Si:H, 350 nm

a - SiC:H, 250 nm

c - Si, 200 nm 1

2

3

4

Refractive index

Figure 4.24: A schematic diagram of the thermo-optic modulator based on waveguide structure (after Ref. 96 with permission from the author and IEEE). transmitted light was due to the thermo-optic effect in an etalon cavity. Thermal transient simulations predict that a modulation rate of 3 MHz is achievable. The thermo-optic effect was already proposed [97, 98] for realizing light modulators in silicon. Thermal phenomena are inherently slow processes. In spite of that, modulators with operation frequencies near 1 MHz were integrated on silicon chips using the standard VLSI technology [97]. This was possible due to the high thermal conductivity of silicon. But the silicon-on-silicon waveguides to date are very lossy. Therefore, some suitable highly thermal conducting and low-loss optical waveguide is to be investigated for realizing a good thermo-optic modulator.

312

4.6 1. 2. 3. 4. 5. 6 7. 8. 9. 10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

CHAPTER 4. APPLICATIONS

REFERENCES P.A. Franken, A. E. Hill, C. W. Peters, andG. Weinreich, Phys. Rev. Lett. 7,118 (1961). J. A. Giordmaine, Phys. Rev. Lett. 8, 19 (1962). P. D. Maker, R. W. Terhune, M. Nisenoff, andC. M. Savage, Phys. Rev. Lett. 8,21 (1962). J. A. Giordmaine and R. C. Miller, Phys. Rev. Lett. 14, 973 (1965). N. Bloembergen, Nonlinear Optics (Benjamin, New York 1965). F. Zemike and J. E. Midwinter, Applied Nonlinear Optics (John Wiley, New York, 1973). R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, CA, 1992). Y. R. Shen, The Principles of Nonlinear Optics (John Wiley, New York, 1984). Data Sheet 704, (Quantum Technology Inc., 1990). Y. S. Liu, W. B. Jones, andJ. P. Chernoch, Appl. Phys. Lett. 29, 32 (1976). Quantum Technology, Inc. Data sheet 708, (1990). D. Eimerl, L. Davis, S. Velsko, E. Graham, andA. Zalkin, J. Appl. Phys. 62, 1968 (1987). K.Kato, IEEE J. Quantum Electron. QE-22, 1013 (1986). S. Velsko, M. Webb, L. Davis, and C. Huang, IEEE J. QuantumElectron. 2 7, 2182 (1991). G. Ghosh, Opt. Lett. 19, 1391 (1994). G. Ghosh, Appl. Phys. Lett. 65,3311 (1994). T. Ukachi, R. Lane, W. Bosenberg, and C. L. Tang, J. Opt. Soc. Am. B 9, 1128 (1992). K. Kato, IEEE J. Quantum Electron. 30, 2950 (1994). S. Lin, B. Wu,F. Xie, andC. Chen,J. Appl. Phys. 73, 1029(1993);Appl. Phys. Lett. 59, 1541 (1991); 60, 2032 (1992). L. Wei, D. Guiquing, H. Qingzhen, Z. An, and L. Jingkui, J. Phys. D. Appl. Phys. B 2, 1073 (1990). H. Mao, B. Wu, C.Chen, D. Zhang, andP. Wang, Appl. Phys. Lett. 62, 1866 (1993). I. Gontijo, Opt. Commun. 108, 324 (1994). J. T. Lin, J. L. Montgomery, and K. Kato, Opt. Commun. 8 0, 159 (1990). G. Ghosh, J. Appl. Phys. 78,6752 (1995). M. S. Webband S. P. Velsko, IEEEJ. Quantum Electron. 26, 1394(1990). G. Ghosh, IEEE Photon. Technol. Lett. 7, 68 (1995).

4.6. REFERENCES 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

313

W. P. Risk, R. N. Payne, W. Lenth, C. Harder, and H. Meier, Appl. Phys. Lett. 5 5, 1179 (1989). J. C. Baumert, F. M. Schellenberg, W. Lenth, W. P. Risk, and G. C. Bjorklund, Appl. Phys. Lett. 5 1, 2192 (1987). T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckhardt, Y. X. Fan, R. L. Byer, andRe S. Feigelson, Appl. Opt. 26,2390 (1987). T. Kishimoto, K. Imamura, and M. Itoh, in Digest of Annual Meeting (Japan Society of Applied Physics, Tokyo, 1991), paper 29p-PB-11. V. M. Garmash, G. A. Ennanov, N. J. Pavlova, andA. V. Tarasov, SOY. Tech. Phys. Lett. 12, 505 (1986). R. A. Stolzenberger, C. C. Hau, N. Payghambarian, J. J. E. Reid, andRe A. . Morgan, IEEE Photon. Technol. Lett. 1, 446 (1989). K. Kato, IEEE J. Quantum Electron. 27, 2043 (1991). W. Wang and M. Ohtsu, Opt. Commun. 102, 304 (1993). B. Zysset, I. Biaggio, andP. Gunter, J. Opt. Soc. Am. B 9,380 (1992). I. Biaggio, P. Kerkoc, L.-S. Wu, and P. Gunter, J. Opt. Soc. Am. B 9, 507 (1992). H. Tsuchida, Opt. Rev. 3, 309 (1996). J. C. Baumert, P. Gunter, andH. Melchior, Opt. Commun. 48,215 (1983). A. Hammerich, D. H. McIntyre, C. Zimmermann, and T. W. Hansch, Opt. Lett. 15, 372 (1990). M. K. Chun, L. Goldberg, and J. F. Weller, Appl. Phys. Lett. 53, 1170 (1988). W. J. Kozlovsky, W. Lenth, E. E. Latta, A. Moser, andG. L. Buna,Appl. Phys. Lett. 5 6, 2291 (1990). P. Gunter, Appl. Phys. Lett. 34,650 (1979). L. S. Wu, H. Looser, and P. Gunter, Appl. Phys. Lett. 56, 2163 (1990). G. J. Dixon, Z. M. Zhang, R. S. F. Chang, and N. Djeu, Opt. Lett. 13, 137 (1988). I. Biaggio, H. Looser, and P. Gunter, Ferroelectrics 94, 157 (1989). M. Poulin, C. Latrasse, M. Tetu, andM. Breton, Opt. Lett. 19, 1183 (1994). G. Ghosh, M. End>, and H. Yajima, in Digest of Annual Meeting (Japanese Soc. Appl. Phys. Kyushu, 1996), paper 29p-PB-ll. G. C. Bharand G. Ghosh, IEEE J. Quantum Electron. QE-16, 838 (1980). G. A. Massey, J. C. Johnson, and R. A. Elliott, IEEE J. Quantum Electron. QE-12, 143 (1976). G. A. Massey and R. A. Elliott, IEEE J. Quantum Electron. QE-I0, 899 (1974).

314

CHAPTER 4. APPLICATIONS

51.

G. A. Massey and J. C. Johnson, IEEE J. Quantum Electron. QE.15, 201

52. 53. 54.

G. Ghosh andG. C. Bhar, IEEE J. Quantum Electron. QE.18, 143 (1982). Y. Tanaka, T. Koshida, and S. Shionoya, Opt. Commun. 25, 273 (1978). W. J. Kozlovsky, C. D. Nabors, R. C. Eckardt, andRe L. Byer, Opt. Lett. 14,

(1979).

66 (1989). 55.

D. C. Gerstenberger and R. W. Wallace, J. Opt. Soc. Am. B 10, 1681

56.

G. Banfi, R. Danielius, A. Piskarskas, P. Trapani, P. Foggi, andRe Righini, Opt. Lett. 18, 1633 (1933). M. Ebrahimzadeh,G. J. Hall, andA. I. Ferguson, Opt. Lett. 18, 278 (1993). S. Lin, J. Huang, J. Ling, C. Chen, and Y. R. Shen, Appl. Phys. Lett. 59,

(1993).

57. 58.

2805 (1991). 59. 60. 61. 62. 63. 64. 65. 66.

F. Hanson and D. Dick, Opt. Lett. 16, 205 (1991). Y. Cui, M. H. Dunn, C. J. Norrie, W. Sibbett, B. D. Sinclair, Y. Fang, and J. A. C. Terry, Opt. Lett. 1 7, 646 (1992). Y. Tang, Y. Cui, and M. Dunn, Opt. Lett. 1 7, 192 (1992). G. Robertson~ M. J. Padgett, and M. H. Dunn, Opt. Lett. 19, 1735 (1994). K. Kato, IEEE J. Quantum Electron. 18, 451 (1982). N. P. Barnes and M. S. Piltch, J. Opt. Soc. Am. 67, 628 (1977). G. D. Boyd, E. Buehler, and F. G. Storz, Appl. Phys. Lett. 18, 301 (1971). G. D. Boyd, H. M. Kasper, and J. H. McFee, IEEE J. Quantum Electron.

QE-7, 563 (1971). 67. 68. 69 70. 71.

G. D. Boyd, E. Buehler, F. G. Storz, andJ. H. Wernick, IEEEJ. Quantum Electron. QE-8, 419 (1972).. R. E. Stickel, Jr. and F. B. Dunning, Appl. Opt. 1 7, 1313 (1978). Y. Takagi, M. Sumitani, N. Nakashima, and K. Yoshihara, IEEE J. Quantum Electron. QE-21, 193 (1985). E. Liu, F. B. Dunning, and F. K. Tittel, Appl. Opt. 21, 3415 (1982). B. Couillaud, T. W. Hansh, and S. G. Mac Lean, Opt. Commun. 50, 127

(1984). 72. 73. 74.

H. Hemmati andJ. C. Bergquist, Opt. Commun. 47,157 (1983). R. P. Mariella, Jr., Opt. Commun. 29,100 (1979). G. A. Massey, M. D. Jones, and J. C. Johnson, IEEEJ. QuantumElectron.

75.

M. D. Jones and G. A. Massey, IEEE J. Quantum Electron. QE·15, 204

76.

G. A. Massey and J. C. Johnson, IEEE J. Quantum Electron. QE·12, 721

QE·14, 527 (1978). (1979). (1976).

4.6. REFERENCES 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87.

88. 89. 90. 91.

92. 93. 94. 95. 96. 97. 98.

315

P. Lokai, B. Burghardt, and W. Muckenheim, App!. Phys. B 45, 245 (1988). M. M. Abbas, T. Kostiuk, andK. W. Ogilvie, App!. Opt. 15,961 (1976). J. C. Baumert and P. Gunter, App!. Phys. Lett. 50, 554 (1987). F. Seifert, J. Ringling, F. Noack, V. Petrov, and O. Kittelmann, Opt. Lett. 19, 1538 (1994). B. Benoow and S. S. Mitra, Fiber Optics: Advances in Research and Developments (Plenum, New York, 1979). E. Desurvire, Erbium-Doped Fiber Amplifiers (Wiley, New York, 1994). M. J. F. Digonnet, Ed, Rare-Earth Doped Fiber Lasers and Amplifiers (Dekker, New York, 1993). J. H. Wray and J. T. Neu, J. Opt. Soc. Am 59, 774 (1969). G. Ghosh, M. Enoo, and T. Iwasaki, IEEE J. Lightwave Techno!' LT-12, 1338 (1994). K. S. Kim and M. E. Lines, J. App!. Phys. 73, 2069 (1993). Schott Optical Glass Catalog (Schott Glass Technologies Inc., Duryea, Pa, 1992). Ohara Optical Glass Catalog (OHARA Inc., Kanagawa, 1990; 1995). G. Ghosh, App!. Opt. 3 6, 1540 (1997). H. Takebe, S. Fujino, K. Morinaga:, J. Am. Ceram. Soc. 77, 2455 (1994); and private communication. G. Ghosh, J. Am. Ceram. Soc. 78,2828 (1995) Technical Data Sheet (NTT Elec. Tech. Corp., Ibaraki, 1996). M. Okuno, N. Takato, T. Kitoh, andA. Sugita, NTTReview 7, 57 (1995). H. Tsuchida, Y. Mitsuhashi, and S. Ishihara, IEEE J. Lightwave Techno!' LT-7, 799 (1989). G. A. Ball, W. W. Morey, and P. K. Cheo, IEEE Photon. Technol. Lett. 5, 267 (1993). G. Cocorull0, F. G. Della Corte, I. Rendina, A. Rubino, and E. Terzini, IEEE Photon. Techno!' Lett. 8, 900 (1996). G. Cocorullo, M. Iodice, and I. Rendina, Opt. Lett. 19,420 (1994). G. Cocorullo, M. Iodice, I. Rendina, and P. M. Sarro, IEEE Photon. Technol. Lett. 7, 363 (1995).

Chapter 5

Future Technology The refractive index of any optical material at an ambient temperature T for any electromagnetic wave within the transmission region of the material, having wavelength A and intensity I, can be expressed by the following equation:

dn n('x, T) = n(,X, rt) + dT (T - rt)

+ n2 1

(5.1)

where rt is the room temperature, dn/dT is the thermo-optic coefficient, and n2 is the intensity dependent nonlinear coefficient of the optical material. Measured values of n2 are found to vary in the range of 2.2rv3.4x 10- 20 m 2 tw [1] for silica glass/fibers. Suppose a Ti:Sapphire femtosecond laser pulse of average power 2.0 W with a beam diameter of 0.8 mm is focused onto a silica or a BK7 glass. The value of I is 4 x 106 W/m 2 • Thus, the intensity dependent nonlinear part is of the order of 8rv 13 X 10- 14 . On the other hand, the contribution due to the thermo-optic coefficient is of the order of 10- 6 / o C. So, the effect of fluctuation from ambient temperature in micro °C is greater than that of the intensity contribution. Therefore, in bulk materials, the contribution of thermo-optic coefficients is not negligible and it is a dominating factor, in particular, for femtosecond pulse transmission. The values of thermo-optic coefficients vary from 10- 3 to 10- 6 for semiconductors to optical glasses. In some materials, these values are negative. Several other kinds of optical glasses having large n2 values have been used to make optical fibers [4-6]. A lead silicate fiber has n2 value of 0.2x 10- 18 m 2tw [4]. Some chalcogenide optical fibers [6] possess large n2 values, twice that of lead silicate fiber.

317

CHAPTER 5. FUTURE TECHNOLOGY

318

Therefore, the future technology based on engineering the refractive indices of the optical materials is exciting. The foregoing analyses will help to design either a highly temperature sensitive or a temperature-immune system/device by a judicious choice of the optical materials. In the case of optical fiber communication systems (OFCS), by utilizing wavelength division multiplexing (WDM), we can transmit many lasers in a single fiber. But the chromatic dispersion and the thermo-optic coefficients are posing problems for the present THz communication systems. Therefore, some suitable systems/devices are to be incorporated within this OFCS to nullify this effect. On the other hand, some low-cost temperature sensors can be made with some suitable optical materials which possess large thermo-optic coefficients. Although most scientists believe that the temperature effect is slow, we believe this effect will create a new scenario by a prudent choice/ investigation of optical materials. Also, the wavelength of the laser propagating through the optical materials is of prime importance since the temperature effect is not at all the same throughout the transmitting wavelength region. All optical switches based on thermo-optic coefficients and the intensity of the transmitting laser will provide us with the momentum to achieve an all-optical communication age. By combining thermo-optic, electro-optic, and magneto-optic phenomena, new systems or designs can also be made.

REFERENCES 1. 2. 3.

4. 5. 6.

R. H. Stolen, W. A. Reed, K. S. Kim, and K. W. Quoi, Proc. Symp. Opt. Fiber Measurements (NIST, Boulder, Colorado, 1992), p. 71. K. S. Kim, R. H. Stolen, W. A. Reed, andK. W. Quoi, Opt. Lett. 19,257 (1994). T. Sato, Y. Suetsugu, M. Takagi, E. Sasaoka, and M. Nishimura, OptoElectronics Conf., Chiba, Japan, July 12-15, 1994; Symp. Opt. Fiber Measurements, Boulder, Colorado, Sept. 13-15, 1994. M. A. Newhouse, D. L. Weidman, and D. W. Hall, Opt. Lett. 15, 1185 (1990). M. Asobe, K. Suzuki, T. Kanamori, and K. Kubodera, Appl. Phys. Lett. 60, 1153 (1992). M. Asobe, T. Kanamori, and K. Kubodera, IEEE Photon. Technol. Lett. 4, 362 (1992).

Subject Index Boldface numerals indicate tables. Italic numerals indicate figures.

A Abbe number definition, 300 values of BaLF4, 303 BaKl, 303 BK7, 303 FK5, 303 KzFS6, 303 LaF2,303 LaKI0,303 PSK3, 303 SF6, 303 Si02,303 SK4, 303 SK5, 303 SSK4A, 303 SSL6, 303 Tellurite glasses, 303 Te02,303 ZKl, 303 Abhisaricas, 1-2 B

Band gaps absorption, 14, 119-127, 1633,124 excitonic, 123-126, 129-139,

124, 244 isentropic, 124-127, 129-139, 124 lattice, 129-137, 124

c Coherence length, 267 ADP, 271, 272 CdTe, 283, 284 KDP, 271, 272 ZnGeP2, 283, 284 D

Difference frequency generation KTP,279 Dispersions (chromatic and zero) definition, 299 values of BaLF4, 303, 304 BaKl, 303, 304 BK7, 303, 304 FK5, 303, 304 fs 7940,301,302 KzFS6, 303,304 LaF2, 303,304 LaKI0, 303,304 PSK3, 303,304 319

320

Index SF6, 303, 304 Si02, 303, 305 SK4, 303, 304 SK5, 303,304 SSK4A , 303, 304 SSL6, 303,304 Tellurite glasses, 303, 305 Te02, 303, 305 vc 7913,301 ZKl, 303,304

o Optical-fiber communication systems, 297-305 Optical parametric oscillators, 283-294, in ADP, 286 CDA,286 CD*A, 286,286 KD*P, 286 KNO, 286,295 LBO, 291,288-290, 293 LNO, 286 LNO:MgO, 286

R Refractive Index Cauchy formula, 11 Characteristic curves, 13 Hartmann formula, 11 Herzberger equation, 11 Lorentz and Lorentz relation , 118

Measurement, 6-9 Schott formula, 10 Sellmeier equation, 12-15 Sellmeier coefficients and

values of as1723, 32, 103 ADA, 16,34 AD*A, 16,35 ADP, 16,36-37 AD*P, 16,38 Ag3AsS 3 ,16 AgBr,16 AgCI, 16,39 AgGaS 2,16 AgGaSe2,17 AgInSe2, 17,40-41 Ag3SbS 3,17 AlAs, 17,42 AIN, 17 ALON,17 AL20 3,17 AIP04, 17, 43 BaF2,18 BAH32; 31 BAL4, 31, 103 BALlI, 31 BAL23, 31, 107 BBO, 17 Ba(COOH)2, 1 7 BaTi03, 18,45-46 BeO, 18 BeSO(BeS04,4H20),18 BOO, 18 BSO, 18 BGZA(4),32

Index

321 BK7,104 BNN,18,44 BP,18 BSM4,31 BSM9, 31,104-105 BSM22,31 BSM24,31 C (diamond), 18 CaC03, 18,47-50 CaF2, 18, 19, 51 CaMo04, 19,52-53 Cawo4, 19, 53-54 CDA, 19,55 CD*A, 19, 56 CdF2,19 CdGeAs2,19 CdGeP2,19 CdS, 19,20 CdSe,20 CdTe,20 CO(NH4)2, Urea, 20 CsBr,20 CsCl,20 CsI,20 CuAlSe2,20, 57-58 CuCl,20 CuGaS 2,20 CuGaSe2, 20, 59-60 CuInS 2,20, 61-62 C 12H220 11 ,20, 21 DLAP,21 ~7940, 32, 105-106 GaAs,21 GaN,21 GaP, 21, 63

GaSb,21 GaSe,21 Ge,21 Hf02: Y20 3,22 HgGa2S4,21 HgS,21 a-HI03,21, 22, 63-65 InAs,22 InP,22 InSb,22, 66 KBO,22 KB*O,22 KBr,22 KCl, 22, 66-67 KCl:Ki, 22, 67-68 KDA,22, 69 KDP,22, 23, 69-71 KD*P, 23, 72-73 KDP:Rb,23, 73 KF,22 KI,23 KLN,23 KM,23 KNO,23, 74-75 KRS5,29 KRS6, 29, 92-93 KTA,23, 24 KTP,24 KzFs6,32 LaF3,24 LAP, 24 LBO, 24, 75 LFM,24 LiF, 24 LIO,24, 25

322

Index LNO,25, 76-79 LNO:MgO,25 LTO, 25, 79-80 LiYF4,25 MAP, 25 MDB, 25, 26, 81 mNA,26 MgA1 20 3,26 MgF2,26 MgO,26 NaBr,26 NaCI,26 NaCOOH,26 NaP, 26 NaI,27 NaN0 3,27, 82 NSL5,31 PbF2,27 PBH6,31, 107-108 PBL26, 31, 106 PBM 3,31 PbMo04,27 PbS, 27 PbSe,27 PbTe,27,82 POM,27 RDA,27, 83 RD*A, 27, 83 RDP,27, 84 RD*P, 27, 28, 85 RTP,28 Ruby, 28, 85-86 Se,28, 86 Si,28, 86 ~-SiC, 28

a-SiC, 28 Silica, 32 a-Si02 (Quartz), 28, 87 S-BSL7,31, 104 SF6, 31,107-108 S-FPL51,31 S-FPL52,31 S-FPL53,31, 108-109 S-FSL5,31 S-LALI0,32 S-LAM2,32 S-LAM51,32 S-NSL36,32 SrF2,28, 88 SrMo04,28, 29, 89 SrTi03,29, 90 SSL6,32 Te,29 Te02,29 Tellurite glasses, 33 Ti02 , 29, 91 TI3AsSe 3,29, 91-92 TIBr,29 TICI,29 vc7913, 32, 109 YAG,29, 93-94 Y20 3,29 ZnO, 30,95 a-ZnS, 30, 96-97 ~-ZnS, 30, 96 ZnS,30, 94 ZnSe,30, 98 ZnSiAs 2,30 ZnTe,30 ZnW04,30, 99-100

323

Index Zr0 2:12%Y203' 30, 101· 102 ZSL1,32 Zernike's formula, 11

s Second harmonic generation ADA, 270 AD*A, 270 ADP,270 AD*P, 270 BBO,273 CDA, 269,270 CD*A, 270 KDA, 270 KDP,270 KD*P, 270 KNO, 280, 282 KTP,279 LBO, 275,276-278 RDA, 270 RD*A, 270 RDP,270 Sum-frequency generation, 294-297 ADP, 296 BBO, 296 KDP, 296 KNO,296 KTP, 279 LBO, 291,296 LNO, 296

T Temperature bandwidth, 266 Temperature sensor, 308-310 Temperature tuning,275, 276294 Thermo-optic coefficients definition, 115 measurement, 115 expt. equation, 116 dispersion relations Clausius-Mossotti, 117 Prodhommes' model, 119, 125 Schott glass 11 9 Semiempirical equations, 121 Sellmeier equation, 126 Tsay's model, 120-121 Sellmeier coefficients and values of as1723, 138,237,239 ADA, 129,142,151 ADP, 129,147,151 Agel, 129, 216 AgGaS 2, 129, 183 A120 3, 129, 192, 198 ALON, 129, 192, 198 BAL4, 138, 250 BALlI, 138,250 BAL23, 138,250

BaF2, 129·130,204, 216 BBO, 130,167,171 BeO,130, 193, 198 BGO, 130, 198

324

Index BGZA(4), 139,206, 216 BK7, 138, 249,252 BSM4, 138,251 BSM24, 138,251 C (diamond), 130,234 CaC03, 130, 191, 198 CaF2,130,205,217218 CaMo04, 130, 199 CaO, 246 OlW04, 131,194,198 CDA, 131,143,151, 152 CD*A, 131,144,152 CdF2, 131,206,218 CdGeP2, 131, 180, 183-185 CdS, 131,234 CdSe, 131,234 CdSiP2, 131, 182 CdTe, 131,228,234, 236 CsBr, 131,207,218219 CsI, 132, 207, 219 CuGaS 2, 132,181, 185-187 DLAP, 132, 169, 171 fs7940, 238, 239241 Ge,132,228,229-231 GaAs, 132,228,234 GaN, 132,234 GaP, 132,234

GaSe, 132, 171 Hf02:Y203,132, 196, 199 HgS, 132, 171 InAs, 132, 228,234 InP, 132, 235 KBr, 132, 208,219220 KCI, 133, 209, 220 KDA, 133, 145, 152 KDP, 133, 148, 152, 153 KD*P, 133, 149, 153 KI, 133, 21 0, 220 KNO,133, 160, 171173 KRS5, 137,216,224225 KTP,133, 173-175 KzFS6,253 LBO, 134, 166, 175 LiF2, 134,221 LIO, 134,157,175 LiYF4, 134, 210,221 LNO, 134, 156, 17-5177 LTO, 134, 177-178 MgAI 20 4,134, 199 MgF2,134, 215,221222 MgO, 134, 197, 199 NaCI, 135, 212, 222223 NaF, 135, 213, 223224

Index

325 NSL5, 138,251 PBH6,138,249,253 PbMo04,135, 199 PbS, 135, 235 PbSe, 135, 228,235 PbTe,135, 228,235 RDA, 135, 146, 154 RDP, 135, 150, 154 a-Si, 136, 234 Si,136, 228,231,233, 235