Principles of Electron Tunneling Spectroscopy: Second Edition: 152 [2 ed.] 0199589496, 9780199589494

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INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS SERIES EDITORS J. BIRMAN S. F. EDWARDS R. FRIEND M. REES D. SHERRINGTON G. VENEZIANO

CITY UNIVERSITY OF NEW YORK UNIVERSITY OF CAMBRIDGE UNIVERSITY OF CAMBRIDGE UNIVERSITY OF CAMBRIDGE UNIVERSITY OF OXFORD CERN, GENEVA

INTERNATIONAL SERIES

OF

MONOGRAPHS

ON

PHYSICS

153. R. A. Klemm: Layered Superconductors, volume 1 152. E. L. Wolf: Principles of electron tunneling spectroscopy, second edition 151. R. Blinc: Advanced ferroelectricity 150. L. Berthier, G. Biroli, J.-P. Bouchaud, W. van Saarloos, L. Cipelletti: Dynamical heterogeneities in glasses, colloids and granular media 149. J. Wesson: Tokamaks, Fourth edition 148. H. Asada, T. Futamase, P. Hogan: Equations of motion in general relativity 147. A. Yaouanc, P. Dalmas de Réotier: Muon spin rotation, relaxation, and resonance 146. B. McCoy: Advanced statistical mechanics 145. M. Bordag, G.L. Klimchitskaya, U. Mohideen, V.M. Mostepanenko: Advances in the Casimir effect 144. T.R. Field: Electromagnetic scattering from random media 143. W. Götze: Complex dynamics of glass-forming liquids—a mode-coupling theory 142. V.M. Agranovich: Excitations in organic solids 141. W.T. Grandy: Entropy and the time evolution of macroscopic systems 140. M. Alcubierre: Introduction to 3+1 numerical relativity 139. A. L. Ivanov, S. G. Tikhodeev: Problems of condensed matter physics—quantum coherence phenomena in electron-hole and coupled matter-light systems 138. I. M. Vardavas, F. W. Taylor: Radiation and climate 137. A. F. Borghesani: Ions and electrons in liquid helium 136. C. Kiefer: Quantum gravity, Second edition 135. V. Fortov, I. Iakubov, A. Khrapak: Physics of strongly coupled plasma 134. G. Fredrickson: The equilibrium theory of inhomogeneous polymers 133. H. Suhl: Relaxation processes in micromagnetics 132. J. Terning: Modern supersymmetry 131. M. Marin˜ o: Chern-Simons theory, matrix models, and topological strings 130. V. Gantmakher: Electrons and disorder in solids 129. W. Barford: Electronic and optical properties of conjugated polymers 128. R. E. Raab, O. L. de Lange: Multipole theory in electromagnetism 127. A. Larkin, A. Varlamov: Theory of fluctuations in superconductors 126. P. Goldbart, N. Goldenfeld, D. Sherrington: Stealing the gold 125. S. Atzeni, J. Meyer-ter-Vehn: The physics of inertial fusion 123. T. Fujimoto: Plasma spectroscopy 122. K. Fujikawa, H. Suzuki: Path integrals and quantum anomalies 121. T. Giamarchi: Quantum physics in one dimension 120. M. Warner, E. Terentjev: Liquid crystal elastomers 119. L. Jacak, P. Sitko, K. Wieczorek, A. Wojs: Quantum Hall systems 118. J. Wesson: Tokamaks, Third edition 117. G. Volovik: The Universe in a helium droplet 116. L. Pitaevskii, S. Stringari: Bose-Einstein condensation 115. G. Dissertori, I.G. Knowles, M. Schmelling: Quantum chromodynamics 114. B. DeWitt: The global approach to quantum field theory 113. J. Zinn-Justin: Quantum field theory and critical phenomena, Fourth edition 112. R.M. Mazo: Brownian motion—fluctuations, dynamics, and applications 111. H. Nishimori: Statistical physics of spin glasses and information processing—an introduction 110. N.B. Kopnin: Theory of nonequilibrium superconductivity 109. A. Aharoni: Introduction to the theory of ferromagnetism, Second edition 108. R. Dobbs: Helium three 107. R. Wigmans: Calorimetry 106. J. Kübler: Theory of itinerant electron magnetism 105. Y. Kuramoto, Y. Kitaoka: Dynamics of heavy electrons 104. D. Bardin, G. Passarino: The Standard Model in the making 103. G.C. Branco, L. Lavoura, J.P. Silva: CP Violation 102. T.C. Choy: Effective medium theory 101. H. Araki: Mathematical theory of quantum fields 100. L. M. Pismen: Vortices in nonlinear fields 99. L. Mestel: Stellar magnetism 98. K. H. Bennemann: Nonlinear optics in metals 94. S. Chikazumi: Physics of ferromagnetism 91. R. A. Bertlmann: Anomalies in quantum field theory 90. P. K. Gosh: Ion traps 87. P. S. Joshi: Global aspects in gravitation and cosmology 86. E. R. Pike, S. Sarkar: The quantum theory of radiation 83. P. G. de Gennes, J. Prost: The physics of liquid crystals 73. M. Doi, S. F. Edwards: The theory of polymer dynamics 69. S. Chandrasekhar: The mathematical theory of black holes 51. C. Møller: The theory of relativity 46. H. E. Stanley: Introduction to phase transitions and critical phenomena 32. A. Abragam: Principles of nuclear magnetism 27. P. A. M. Dirac: Principles of quantum mechanics 23. R. E. Peierls: Quantum theory of solids

Principles of Electron Tunneling Spectroscopy Second Edition E. L. Wolf Polytechnic Institute of New York University, USA

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Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c E. L. Wolf 2012  The moral rights of the author have been asserted Database right Oxford University Press (maker) First edition published 1985 Second edition published 2012 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by SPI Publisher Services, Pondicherry, India Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY ISBN 978–0–19–958949–4 1 3 5 7 9 10 8 6 4 2

In memory of Norman and Harriet

Preface to the First Edition Tunneling of electrons through classically forbidden barriers is a basic process in all matter at the atomic scale. Artificial tunneling structures, the topic of this book, contain barriers having an atomic or Angstrom length scale in one dimension, but, in the transverse directions have laboratory length scales of microns to millimeters. Overcoming the technical difficulties in preparing such uniform barriers in solid state structures and using these to reproducibly demonstrate the predictions of elementary quantum mechanics has been relatively recent, dating from the work of Esaki on semiconductor junctions in 1958, quickly followed by Giaever’s work on metal-insulator-metal thin film structures. The field of research opened up by these reproducible “macroscopic” tunneling barrier structures has had a major impact on condensed matter physics and on technology. Contributions to both basic physics and technology were hastened by the verification of the Bardeen, Cooper and Schrieffer superconducting energy gap by Giaever and by the brilliant theoretical prediction in 1962 by B. D. Josephson of dc and ac supercurrents in tunneling barrier diodes with both electrodes in the superconducting state. The Josephson effects, verified in the early 1960s, have led to major advances in detection of magnetic flux, current and voltage, and also to promising new families of logic devices for computer application. The importance of this new area of research to the advancement of physics and technology was recognized in 1973 by award of the Nobel Prize in Physics to Esaki and Giaever, and to Josephson. The present book is intended to review in a comprehensive fashion the various techniques, principally tunneling energy spectroscopies, for study of condensed matter, based on tunnel junctions, and also to review the solid state physics and materials properties that have been thus revealed. An effort has been made to provide an overall conceptual basis in which superconducting effects and the various normal state effects, including the vibrational excitation spectroscopy of organic molecules in tunnel barriers (IETS), can be uniformly treated. A rather large number of figures has been included, to illustrate all forms of energy spectroscopy that evolve from barrier tunneling, and to provide examples from as many as possible of the different physical situations or systems which have been studied. This material has been taken from what is by now an extensive and rather widely scattered literature, and has been quite rigidly organized, usually according to the physical mechanism by which the spectroscopic data are thought to arise. The conceptual organization will be evident from the Table of Contents, especially with reference to the introductory first chapter, particularly the discussion at the end of Chapter 1. This book has been made possible by assistance from many sources. Granting of leave with financial support from Iowa State University during 1981 and 1982, and hospitality extended to me as a visitor at the University of Pennsylvania, and later at the Yorktown Heights Research Center of IBM, were essential to getting the project started. The assistance, particularly, of Elias Burstein and E. Ward Plummer at Penn and of Charles Kircher and William Gallagher at IBM is gratefully acknowledged. Thanks are due also to Douglas vi

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Finnemore, Margaret Avery, Jerome Ostenson and Tar-Pin Chen for their assistance during my absence from Ames. Part of the manuscript was written at the Aspen Center for Physics, and all was typed with great efficiency and good humor by Ms. Lesley Swope of the Ames Laboratory—USDOE at Iowa State University. I am indebted to many Ames colleagues, and especially to John R. Clem, Douglas K. Finnemore and Bruce N. Harmon for discussions and for their encouragement and support during this project. Permission to reproduce figures from (at least one of) the authors listed in the figure captions is most gratefully acknowledged. Particular thanks for permission to include previously unpublished material go to G. B. Arnold, W. J. Gallagher, F. Gompf, Z. G. Khim, B. N. Taylor and D. G. Walmsley. I am also grateful to G. B. Arnold, F. Gompf, K. E. Gray, R. C. Jaklevic, D. G. Walmsley and others for providing original figures, enlargements, or other material. G. B. Arnold is particularly acknowledged as the source of many of the figures in Chapter 5. Special thanks are also due to B. D. Josephson for assistance in obtaining a copy of his Fellowship dissertation at Trinity College, University of Cambridge. Present and past members of our Tunneling Group have assisted in improving the manuscript: specific thanks are due to Hong-Jie Tao, Tar-Pin Chen, Mark Albers, Patricia Allen, Siyuan Han, Bret Hess, Quiming Li, Kwok-Wai Ng, Danny Shum, Anne Thomas, and Youwen Xu. Finally, I am most grateful to my family for their support during this project: specifically to Douglas for helping to establish an orderly relation between the innumerable figures and their captions, to David for searching and sorting among the unending illegible references, and to Carol for her patient and cheerful accommodation of the many inconveniences, which, happily, are now over. Ames, Iowa October, 1984

E. L. Wolf

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Preface to the Second Edition The Second Edition brings Principles of Electron Tunneling Spectroscopy up to date. Over twenty-five years, it is fortunate that the physics of tunneling has not changed, but has been validated, competence in its practice improved, and the extent of its domain extended. Important new developments include scanning tunneling spectroscopy (STS), Fourier-transform STS, high-temperature oxide (cuprate) and iron-based superconductors, and magnetic tunnel junctions as sensors in magnetic disk drives. The scanning function now frequently makes tunneling spectroscopy available with atomic spatial resolution! Similarly revolutionary has been the new physics of the oxide-based, electron-coupled, superconductors. These arise by doping of cuprate CuO2 planes, transforming antiferromagnetic Mott insulators into superconductors, with anisotropic nodal pairs and transition temperatures over 130 K. The ironbased superconductors also on the whole require doping from a parent magnetic state. The domain of physics now addressed by tunneling methods is broader, extending far beyond the Matthias rules of the earlier era. In the Second Edition of Principles of Electron Tunneling Spectroscopy, new material is added throughout, notably in new Chapters 7, 9, 10, 11, and Appendix B, to provide new concepts and information. The scanning tunneling spectroscopy (STS) methods are introduced in an extended Chapter 1, and more technically in a new version of Appendix B. Chapter 2 is expanded to include models of the spatial resolution afforded by the STM tip. Advances in Andreev tunneling spectroscopy are included in Chapter 7. Expanded treatment of the superconducting proximity effect appears in a revised Chapter 5, including wonderful new results on mesoscopic SNS junctions, likely inspired by Usadel’s simplification of the basic theory, now clearly revealing the spatial dependence of the local density of states in the superconducting phase. An entirely new Chapter 7 introduces high-T c electron-coupled superconductivity in copper oxide planes and in iron-based compounds. The basic ideas of the Mott-insulator to d-wave superconductor transition under hole doping are covered in some detail, because they differ greatly from ideas important in conventional superconductors, and are needed to understand the contributions of STS to the new field. Summary description of the most important of the new layered cuprate materials is included. In addition, an introduction to principal experimental methods of cuprate characterization, beyond the STS method, is included, and supplemented in Appendix B. Consequences of anisotropic pairing, e.g., that the pairing strength disappears along nodal directions, are described in various tunnel junction situations. This rather extensive background in Chapter 7 seems appropriate to prepare the reader for an account in Chapter 10 of the amazing and detailed results of STS on the cuprate superconductors. New Chapters 9 and 10 are devoted, respectively, to use of STM and STS to characterize single atoms and molecules, and (in Chapter 10) cuprate superconductors with comparison to manganite “colossal magnetoresistance” materials. Chapter 10 summarizes a large body of intense research, still going on, in which STS methods have been continuously refined ix

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to provide important insights. Chapter 10 includes results from complementary experiments, such as angle-resolved photoemission (ARPES) on cuprates, and also some STS results on the somewhat similar manganite materials that exhibit colossal magnetoresistance (CMR). (These materials start as Mott insulators and, under doping, turn into ferromagnets instead of superconductors. A tunneling feature known as the pseudogap occurs in both materials.) The intent has been to leave much of the original book intact, as the physics related specifically to tunneling has not changed, but to update it with newer examples. For instance, the earlier planar tunnel junctions exhibiting inelastic electron tunneling spectroscopy (IETS) are replaced by STS measurements of the same vibrational effects on single adsorbed molecules. The astounding advances in instrumentation are described, at least in outline (Appendix B). An attempt has been made to hold the text to about the same length as the original. The author thanks G. B. Arnold, John Zasadzinski, Kwok-Wai Ng, D. P. Shum, A. J. Millis, D. K. Finnemore, P.C. Canfield, I. Mazin, and J. C. Davis, for helpful discussion on content; Sonke Adlung, Clare Charles, and April Warman of Oxford University Press have been central in setting up and carrying out the production of this Second Edition, with additional thanks to Peter Sinclair for expert copy-editing. I thank Lorcan M. Folan and DeShane Lyew of the NYU-Poly Physics Department for assistance in several ways, and Manasa Medikonda for extensive careful assistance in preparing the new manuscript. Carol has been a constant source of support in this project. Brooklyn, New York September 30, 2010

E. L. Wolf

Contents 1 Introduction 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

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Concepts of quantum mechanical tunneling Occurrence of tunneling phenomena Electron tunneling in solid-state structures Superconducting (quasiparticle) and Josephson (pair) tunneling Tunneling spectroscopies The scanning tunneling microscope (STM): spectroscopic images Atomic spatial resolution in the scanning tunneling microscope Density of electron states (DOS) measurement in STM: STS Perspective, scope, and organization

2 2 6 9 13 15 16 16 20

Tunneling in normal-state structures: I

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2.1 Introduction 2.2 Calculational methods and models 2.2.1 Stationary-state calculations 2.2.2 Transfer Hamiltonian calculations 2.2.3 Ideal barrier transmission 2.3 Basic junction types 2.3.1 Metal–insulator–metal junctions 2.3.2 Metal–insulator–semiconductor junctions 2.3.3 Schottky barrier junctions 2.3.4 pn junction (Esaki diode)—direct case and the Si–Ge diode 2.3.5 Vacuum tunneling 2.3.6 Vacuum tunneling from a spherical STM tip 2.4 Dependence of J(V) and G(V) on band structure and density of states 2.4.1 Fermi surface integrals 2.4.2 Prefactors: wavefunction matching at boundaries 2.5 Nonideal barrier transmission 2.5.1 Approach to ideal behavior 2.5.2 Resonant barrier levels 2.5.3 Two-step tunneling 2.5.4 Barrier interactions 2.6 Assisted tunneling processes 2.7 Comments on the time for tunneling 2.8 Resolution obtained from a scanning tunneling microscope tip 2.8.1 Tersoff and Hamann’s model of STM resolution 2.8.2 C. Julian Chen’s atomic model of STM resolution

23 23 25 27 29 37 39 48 49 56 58 60 61 61 62 63 63 69 72 76 76 79 80 80 80

Spectroscopy of the superconducting energy gap: quasiparticle and pair tunneling

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3.1 Basic experiments of Giaever and Josephson tunneling 3.2 Superconductivity 3.3 Electron–phonon coupling and the BCS theory

82 85 93

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3.4 3.5 3.6

3.7 3.8

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3.3.1 The pair ground state 3.3.2 Elementary excitations of superconductors 3.3.3 Generalizations of BCS theory Theory of quasiparticle and pair tunneling Gap spectra of equilibrium BCS superconductors Gap spectra in more general homogeneous equilibrium superconductor cases 3.6.1 Strong-coupling superconductors 3.6.2 Gap anisotropy 3.6.3 Multiple gaps, two-band superconductivity 3.6.4 Excess currents, subharmonic structure 3.6.5 Effects of magnetic field 3.6.6 Magnetic impurities 3.6.7 Pressure effects 3.6.8 Interactions with electromagnetic radiation 3.6.9 Superconducting fluctuations Ultrathin-film and small-particle superconductors Transition from tunnel junction to metallic contact 3.8.1 Model of Klapwijk, Blonder, and Tinkham

Conventional tunneling spectroscopy of strong-coupling superconductors 4.1 4.2 4.3 4.4 4.5

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96 100 101 103 112 121 121 124 128 130 138 145 147 150 158 160 170 171

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Introduction Eliashberg–Nambu strong-coupling theory of superconductivity Tunneling density of states Quantitative inversion for α 2 F(ω): test of Eliashberg theory Extension to more general cases 4.5.1 Finite temperature 4.5.2 Anisotropy 4.5.3 Spin fluctuations 4.5.4 Electronic density-of-states variation 4.6 Limitations of the conventional method

173 173 177 178 182 182 185 187 190 194

Inhomogeneous superconductors: the superconducting proximity effect

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5.1 5.2 5.3 5.4 5.5 5.6

5.7 5.8 5.9 5.10

Introduction: continuity of the pair wavefunction Andreev reflection and specular SNS junctions Survey of phenomena in proximity tunneling structures Specular theory of tunneling into proximity structures McMillan’s tunneling model of bilayers The Usadel equations and diffusive SNS junctions 5.6.1 Reduction of Gor’kov’s equations by Eilenberger and Usadel 5.6.2 Application of reduced Gor’kov theory to tunneling problems 5.6.3 The experiment of Truscott and Dynes confirming the bound state in clean NS junctions 5.6.4 The experiment of le Sueur et al.: phase dependence of the density of states 5.6.5 Proximity effects in a ferromagnetic N layer, in an NS structure Proximity electron tunneling spectroscopy (PETS) Effects of elastic scattering in the N layer Proximity corrections to conventional results Further applications of proximity effect models

197 199 206 212 223 228 228 229 230 231 235 236 245 250 251

CONTENTS

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Superconducting phonon spectra and α 2 F(ω)

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6.1 Introduction 6.2 s–p band elements 6.3 Crystalline s–p band alloys and compounds 6.3.1 Crystalline s–p band alloy superconductors 6.3.2 s–p band compounds 6.4 Amorphous metals 6.5 Transition metals, alloys, and compounds 6.6 Extreme weak-coupling metals 6.7 Local-mode and resonance-mode superconductors 6.8 Systematics of superconductivity 6.9 Effects of external conditions and parameters on strong-coupling features 6.10 Eliashberg inversion of bismuthate and cuprate superconductor tunneling data

256 256 263 263 270 273 281 291 295 298 302 306

High-Tc electron-coupled superconductivity: cuprate and iron-based superconductors

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7.1 The discovery of cuprate superconductivity by Bednorz and Muller 7.2 The Mott antiferromagnetic CuO2 insulator and its doping to a metal 7.2.1 Paired holes in copper oxide planes 7.2.2 Hubbard and t–J models in two dimensions 7.3 Hole-doped cuprates Bi2212 and YBCO 7.3.1 Phase diagram for superconductivity in hole-doped cuprate 7.3.2 Crystal structures of common cuprates: I 7.3.3 Early tunneling measurements on hole-doped superconductors 7.4 Crystal structures of common cuprates: II 7.4.1 Range of Tc vs. number of copper oxide planes 7.4.2 Disorder sites and doping of cuprate superconductors 7.4.3 Comments on disorder and inhomogeneity in STS images 7.5 Andreev–St. James tunneling spectroscopy 7.6 Experimental signatures of nodal superconductivity 7.6.1 Specific heat at transition 7.7 Josephson junctions in d-wave cases 7.8 Further examples of non-BCS superconductors

312 313 313 316 317 317 318 319 325 325 325 327 328 328 330 332 335

Tunneling in normal-state structures: II

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8.1 Introduction 8.2 Final-state effects: I 8.2.1 Two-dimensional final states 8.2.2 Quantum size effects in metal films 8.2.3 Accumulation layers at semiconductor surfaces 8.2.4 Spin-polarized tunneling as a probe of ferromagnets 8.2.5 Julliere’s model of ferromagnetic tunnel junctions 8.2.6 Other bulk band structure effects 8.3 Assisted tunneling: threshold spectroscopies 8.3.1 Phonons 8.3.2 Inelastic electron tunneling spectroscopy of molecular vibrations 8.3.3 Inelastic excitations of spin waves (magnons) 8.3.4 Inelastic excitation of surface and bulk plasmons 8.3.5 Light emission by inelastic tunneling 8.3.6 Spin-flip and Kondo scattering 8.3.7 Excitation of electronic transitions

336 336 336 338 339 343 350 352 357 358 366 367 368 369 372 378

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8.4 Final-state effects: II 8.4.1 More general many-body theories of tunneling 8.4.2 Tunneling studies of electron correlation and localization in metallic systems 8.4.3 Phonon self-energy effects in degenerate semiconductors 8.4.4 Electron scattering in the Kondo ground state 8.5 Zero-bias anomalies 8.5.1 Giant resistance peak 8.5.2 Semiconductor conductance minima 8.5.3 Assorted maxima and minima in metals 8.5.4 The Giaever–Zeller resistance peak model

9 Scanning tunneling spectroscopy (STS) of single atoms and molecules 9.1 9.2 9.3 9.4 9.5

9.6 9.7

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Theory of observation of single atoms in STS and experiment Friedel oscillations in 2-D surface state 9.2.1 Effect of surface state: inference of wavevector 9.2.2 Fourier-transform STM/STS Quantum corrals 9.3.1 Elliptical corrals and focusing effects: quantum mirage Pair-breaking single adatoms on superconductors 9.4.1 Mn and Cr on Pb 9.4.2 Zn impurity atoms imaged in cuprate planes Spectroscopy of Kondo and spin-flip scattering 9.5.1 Introduction 9.5.2 Kondo spectroscopy of a single trapped electron 9.5.3 Spectroscopy of localized moments in Si:As Schottky junctions 9.5.4 Comparison of the two Kondo spectroscopy experiments STM spectroscopy of magnetic adatoms Molecules and their vibrational spectra

Scanning tunneling spectroscopy of superconducting cuprates and magnetic manganites 10.1

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384 384 389 394 401 407 407 409 411 414

419 419 422 425 425 426 427 429 430 431 432 432 433 435 436 436 443

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Gap imaging of optimally doped cuprates 10.1.1 Site dependence of apparent gap 10.1.2 Overdoped case 10.1.3 Anticorrelation of gap and zero-bias density of states 10.1.4 Internal proximity effect 10.2 Localized state at Zn impurity 10.3 Model for spectral distortions of noncuprate layers 10.4 Superlattice modulation in Bi2212 10.5 Fourier-transform STS (FT-STS) and application 10.6 Observations of charge ordering in cuprate superconductors 10.7 Relation of STS to angle-resolved photoemission spectroscopy (ARPES) 10.8 Evidence for electron-spin wave coupling 10.9 Colossal magnetoresistance: Mott transition in doped manganites 10.9.1 Introduction: mechanism of colossal magnetoresistance (CMR) 10.9.2 Pseudogap in manganite LSMO observed by ARPES 10.10 Relation of cuprates to ferromagnetic CMR manganites

447 447 449 449 449 452 456 458 460 460 464 467 470 470 472 473

Applications of barrier tunneling phenomena

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11.1 11.2

475 477

Introduction Josephson junction interferometers

CONTENTS

11.3 11.4 11.5

11.6

SQUID detectors: the scanning SQUID microscope 11.3.1 Establishing d-wave nature of cuprate pairing Josephson junction logic: rapid single-flux quantum devices 11.4.1 The single-flux quantum voltage pulse 11.4.2 Analog to digital conversion (ADC) using RSFQ logic Detection of radiation 11.5.1 SIS detectors 11.5.2 Josephson effect detectors 11.5.3 Optical point-contact antennas (high-speed MIM junctions) Tunnel-junction magnetoresistance sensors

Appendix A Experimental methods of junction fabrication and characterization A.1 Thin-film electrodes A.1.1 Evaporated films A.1.2 Film thickness measurement A.1.3 Substrate temperature A.1.4 Sputtered films A.1.5 Chemical vapor–deposited films A.1.6 Epitaxial single-crystal films A.1.7 Atomic layer deposition A.2 Foil and single-crystal electrodes A.3 Characterization of tunneling electrodes A.4 Preparation of oxide tunneling barriers A.4.1 Thermal oxide barriers A.4.2 Plasma oxidation processes A.5 Artificial barriers A.5.1 Totally oxidized metal overlayers A.5.2 Directly deposited artificial barriers A.5.3 Polymerized organic films A.6 Point-contact barrier tunneling methods A.6.1 Anodized metal probes A.6.2 Schottky barrier probes A.6.3 Deformable metal vacuum tunneling probes A.6.4 Analysis of point-contact data A.7 Characterization of tunnel junctions A.7.1 Initial characterization of junctions A.7.2 Derivative measurement circuitry

Appendix B Methods of scanning tunneling spectroscopy and competing approaches B.1 STM basics, tip production, and characterization; single atom tips B.2 Noise-free x, y, z translation; vibration isolation B.2.1 The cryogenic STM of Wilson Ho B.2.2 The 240-mK STM design of Pan, Hudson, and J. C. Davis B.3 Atomic force microscope; combination STM/AFM B.4 Scanning tunneling potentiometry and point-contact measurements B.5 Ballistic electron emission microscopy (BEEM) B.6 Scanning charge microscopy and spectroscopy B.6.1 Scanning single-electron-transistor electrometry B.6.2 Scanning subsurface charge accumulation microscopy: STM/SCAM B.6.3 Single electron capacitance spectroscopy B.7 Scanning Hall probe microscopy

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480 480 481 481 483 483 485 486 487 487

489 489 491 491 492 492 493 493 494 495 498 501 501 503 507 507 508 509 509 509 509 510 511 511 511 514

523 523 527 527 529 531 534 534 535 535 537 538 539

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Appendix C Tabulated results Table C.1 Table C.2 Table C.3 Table C.4 Table C.5 Table C.6 Table C.7

s, p elements Alloys and unusual phases: s, p elements d-band elements d-band alloys, oxides, and compounds f-band elements Metal overlayers for barrier formation Studies of Tomasch oscillations in thick superconducting films and of McMillan–Rowell oscillations in thick normal films Table C.8 Tunneling studies of superconductor phonons under hydrostatic pressure Tables C.9 Cuprate superconductors Table C.9a Gap values for Bi2 Sr2 CaCu2 O8+δ (Bi2212) Table C.9b Gap values for YBa2 Cu3 O7+δ Table C.9c Gap values for HgBa2 Can−1 Cun O2n+2+δ

542 543 544 545 546 548 548 548 548 549 549 550 551

References

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Index

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1 Introduction The several forms of energy spectroscopy of solids made possible by quantum mechanical tunneling of electrons between two metallic electrodes separated by a thin barrier are the subject matter of this book. With the advent of the scanning tunneling microscope (STM), local mapping of the energy spectroscopy is available in addition to the topography on an atomic scale. In addition to discussing techniques of tunneling spectroscopy that have evolved, we also attempt to collect and organize the wealth of information, primarily concerning the electronic and vibrational properties of metals, and notably superconductors and semiconductors, that tunneling has made available. Tunneling spectroscopies may be classified by the range of energies probed. These are, at most, on the order of metallic Fermi energies, or about 5 eV, in studies relating to aspects of electronic band structure. More typically, one measures in the range of a few to a few hundred millielectronvolts in studies of phonons and molecular vibrations. In the important applications of tunneling to determination of the excitation spectra of superconductors, a further characteristic scale is that of the superconducting energy gap, typically a few millielectronvolts. In the majority of cases, the object of study is the tunneling electrode, which may be a single crystal of metal or semiconductor but more typically is an evaporated polycrystalline or possibly amorphous metallic film. The energy resolution of tunneling spectroscopy is a few times the thermal energy kT, so that, for energy resolution of millivolts or less, measurements are usually made at liquid helium temperature. In the study of electrodes, one may be primarily interested in the normal-state properties of the given electrode material, in its particular superconducting properties, or in universal features of the superconducting state which are relatively insensitive to the choice of material. Indeed, a particularly fruitful application of tunneling has been to elucidate the properties of coupled superconductors, as first predicted by Josephson (1962a,b), in the superconductor–insulator–superconductor (SIS) configuration. An additional important application of tunneling is to quantitative determination of the electron–phonon spectral function α 2 F(ω) of many superconductors, from which a rather complete specification of their properties has been obtained. In other instances, the object of spectroscopic study is the insulating barrier, necessarily limited in thickness to the order of 100 Å to allow a measurable tunnel current. In one special case of this type of study, one measures the vibrational spectrum of a monolayer of molecules present at one edge of the insulator. Such spectra are obtained in a form similar to conventional infrared and Raman spectra. An important advantage of inelastic electron tunneling spectroscopy (IETS) is that comparable spectra may be obtained from samples containing orders-of-magnitude fewer molecules than are required in conventional optical techniques.

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2

I N T RO D U C T I O N

1.1 Concepts of quantum mechanical tunneling The basis for the several forms of electron tunneling spectroscopy is the inherently smooth behavior of the probability density ψ ∗ ψ even at points of finite discontinuity in the potential energy V (x). At a metal–vacuum interface, in particular, the wavefunction for an electron at the Fermi energy does not vanish outside the metal but rather decays exponentially as 1 1 exp(-κ x), where κ ∼ = (2m/2 ) 2 φ 2 and x is the distance into the vacuum; at metal–insulator interfaces, the electronic mass m and metal work function φ must be altered to reflect the properties of the solid-state insulator I. If, then, a metal–insulator–metal (MIM) junction is biased by an external voltage V , electrons in an energy range eV between the two shifted Fermi energies may elastically tunnel from one side to empty states opposite. Note that the most energetic of the injected electrons appear at an energy precisely eV above the Fermi energy of the positively biased electrode, in the limit T = 0. Similarly, eV is precisely the maximum energy (at T = 0) which can be given up in an inelastic process and yet allow the tunneling electron to appear in an unoccupied state of the positively biased electrode. Tunneling processes are fundamental to quantum mechanics, following directly from the nature of the solutions ψ(x) of Schrödinger’s equation and the probability interpretation of ψ ∗ ψ. The rate at which such processes occur is dominated by the exponential decay in the classically forbidden barrier region, where the potential energy V(x) or U (x) exceeds E x . The symbol U (x) for potential energy will often be used in cases where the potential barrier may be changed by an applied voltage: U (x, V ). For many purposes the rate can be adequately estimated from the Wentzel–Kramers–Brillouin (WKB) approximation D = exp(−2K )  x2 (E x ) κ(x, E x )dx K =

(1.1) (1.2)

x1 (E x )

 κ(x, E x ) =

2m ∗ [V (x) − E x ] 2

1 2

(1.3)

In Eqns. (1.1)–(1.3), E x is the kinetic energy; x1 , x2 are the classical turning points; and D is properly regarded as the fraction of the probability current k/m carried by an incident wave eikx which passes the barrier V (x) > E x entering (1.3). The probability current is defined as   ∂ψ i ∂ψ ∗ − ψ∗ j= ψ (1.4) 2m ∂x ∂x

1.2 Occurrence of tunneling phenomena The fundamental nature of the tunneling phenomenon makes it a pervasive feature in the behavior of small-mass particles in rapidly spatially varying potentials. These conditions apply to electrons in atoms, molecules, and solids. Tunneling phenomena are thus inescapably involved in the structure and behavior of matter viewed on an atomic scale. The first phenomena to be identified and convincingly explained in terms of tunneling allowed unequivocal detection of particles escaping into the vacuum by barrier tunneling from metastable states in matter. Three such phenomena were already identified in 1928: the natural

3

O C C U R R E N C E O F T U N N E L I N G P H E N O M E NA

decay of certain heavy nuclei by α-particle emission, the ionization of atomic hydrogen in a strong electric field, and, similarly, the emission of electrons from a cold, clean metal surface under the application of a strong electric field. As sketched in Fig. 1.1, the α particle which is subsequently observed outside the nucleus with a kinetic energy E k , typically a few MeV, was assumed (Gamow, 1928; Gurney and Condon, 1929) to occupy a metastable state at energy E inside the nucleus prior to decay. If we take the tunneling barrier as U (r ) = (Z − 2)2e2 /r for r > R, with R the nuclear radius, application of the WKB formula (1.2) gives 

2m α K = (Z − 2)2e2 2

1  2

b



R

1 E − r (Z − 2)2e2

1 2

dr

with b = (Z − 2)2e2 /E, m α the α-particle mass, and Z the atomic number. Evaluation gives an inverse lifetime (1/τα ) of ⎧ ⎡ ⎤⎫ 1  1  ⎨ 2 2 ⎬ 2 1 R R R 1 ⎦ − 2 = A exp(−2K ) = A exp −B E − 2 cos−1 ⎣ − ⎩ ⎭ τα b b b 1

with B = 2−1 (2m α ) 2 2(Z − 2)e2 (Gamow’s formula). For heavy nuclei, R/b = E/V (R) is reasonably small, so the cos−1 term is π/2, with slowly varying corrections. Neglecting the latter gives 1 −1 ln τα = − ln A + π B E k 2 2 which approximately describes τα values spanning 15 orders of magnitude. This expression provided a great success of the early quantum theory. U(r)

E

Ek

0 U0

b

r

R

Fig. 1.1. Decay of radioactive nucleus by tunneling of alpha particle through Coulomb barrier. Regularity between energy E k of emerging alpha particle and lifetime τα against decay substantiates tunneling model.

4

I N T RO D U C T I O N

–eFx

V(x)

yL

yF

x=b

x 0

x=L 2

Fig. 1.2. Hydrogen atom in an applied electric field decays as the electron tunnels through the Coulomb barrier to the classically allowed region at positive x.

A second application of tunneling concepts in 1928 was Oppenheimer’s treatment of the field ionization of hydrogen. A one-dimensional hydrogen atom at the center (x = 0) of a box with a field F = V /L is sketched in Fig. 1.2. For small values of F, there are two states at energy E 0 for the electron: the “localized atomic” state ψ L is peaked near the “nucleus,” while in the “free” states, ψ F , the electron oscillates between the classical turning point x = b and the boundary x = L/2. In Oppenheimer’s treatment for weak fields F, the atomic wavefunction ψ L is calculated by neglecting the field, and ψ F can be calculated as the eigenfunction at energy E 0 of the uniform field alone. Oppenheimer (1928) showed that the initial rate of transition w from state ψ L to the continuum of final states ψ F (in the limit of large L) can be calculated to good accuracy from the Golden Rule as w=

2π |M|2 ρ F (E) 

(1.5)

where |M|2 = |ψ F |H  (x)|ψ L |2 and H  (x) = −eF x is the perturbing Hamiltonian. Oppenheimer estimated the lifetime of the hydrogen atom against ionization in a field F of 103 V/m 10 as 1010 sec. The third early example of tunneling, which is closer to the subject of this book, is field emission of electrons from metals, treated in 1928 by Fowler and Nordheim. Again using (1.2) to estimate the decaying exponential behavior, assuming a triangular barrier as in Fig. 1.3, one finds for electrons at the Fermi energy E x = μ F

5

O C C U R R E N C E O F T U N N E L I N G P H E N O M E NA

E V (x)=m + f − eFx f

y (x)

m x=b

0

x

Fig. 1.3. Field emission of electrons from cold metal surface under the application of the strong electric field, as proposed by Fowler and Nordheim (1928), occurs by electron tunneling through the potential barrier formed by the work function and the electric field.



b

2K = 2 0



2m 2

1 2

√ 8π 2m φ 3/2 (φ − eF x) dx = 3 heF 1 2

Here, φ is the work function, F is the field strength, and b = φ/eF is the classical turning point. Thus, the transmitted fraction of the electron current incident upon the surface barrier at the Fermi energy is D = e−2K . The total emitted current is obtained by integration over the available electron energy states weighted by their thermal occupation, given by the Fermi function defined as    (E − μ F ) −1 (1.6) f (E) = 1 + exp kT Hence, since there are two electron states per k-space volume (2π )3 , with group velocity vx = −1 ∂ E/∂k x , the current density Jx is       1 ∂E 2e dk x dk y dk z Jx = (1.7) D(E x ) f (E)  ∂k x (2π )3 leading to the original result of Fowler and Nordheim (1928),   √ √ 8π 2mφ 3/2 4 μ F φ e3 F 2 exp − Jx = (μ F + φ) 8π hφ 3heF

(1.8)

6

I N T RO D U C T I O N

which successfully describes the measured field emission current, typically yielding straightline plots of ln(J/F 2 ) vs. 1/F.

1.3 Electron tunneling in solid-state structures Electron tunneling within a crystalline solid, where the Bloch states are of the form

r ) = u k ( r )ei k· r ψk (

(1.9)

with u k ( r ) a periodic function of r , occurs in the band gap, where k = iκ is imaginary, resulting in exponentially damped waves. The magnitude of the decay constant κ(E) at energy E depends upon the location of E within the band gap E g = E c − E v and the effective masses m ∗c , m ∗v which characterize the conduction and valence bands. In principle, the application of a uniform electric field to a homogeneous solid, as, for example, an intrinsic semiconductor at T = 0, should permit interband tunneling, or “internal field emission,” as illustrated in Fig. 1.4a. This effect was treated by Zener (1934) by a WKB method, yielding a transmission factor D = exp(−maE g w/42 ), with a the lattice constant, F the electric field, and the “barrier” width w = E g /|eF|. While this effect has probably never been observed in a homogeneous solid, it is closely related to tunneling effects observed by Esaki( 1958) in narrow semiconductor pn junctions. The schematic diagram of Fig. 1.4b shows Esaki’s structure, in which the width w is of order 100 Å, corresponding to a junction field F 3 × 107 V/m. The Esaki diode is degenerately doped p type on the left and n type on the right such that the metallic Fermi energies μ F are ∼ 0.05 eV. Application of a forward bias eV μ F , as illustrated, may be expected to allow a peak tunnel current, as all of the donated electrons in n can tunnel elastically into empty electron states (filled holes states) in the p-type region. One might expect further increase in forward bias to reduce the current, as there will become fewer available final electron states. This effect produces the characteristic and technologically useful negative-resistance region, first observed by Leo Esaki in 1958. The Esaki diode is an important invention, permitting construction of oscillators and many other circuits with wide applications. A further scientific importance of Esaki’s work was to provide the first example in which, as shown in Fig. 1.4c, the variation of the tunneling current as a function of the applied voltage V revealed spectroscopic information about the electronic structure of the solids involved. Clearly, the width in voltage of the current “hump” is at least a rough measure of the energy widths of the filled states on the n side and the empty states on the p side, i.e., the local Fermi energies as measured from the respective band edges. Any such spectroscopic aspect depends upon having highly conducting electrode regions such that the applied potential drops entirely across the thin barrier. One assumes that the tunneling process is elastic, leading to the injection of electrons on the other side whose energies extend precisely to the applied bias energy eV and that the tunneling process depends on the densities of states involved. A second type of spectroscopy is also evident in the results of Esaki (Fig. 1.4c) in the lower curve measured at 4.2 K. In this case, the points of maximum curvature of the I –V characteristic labeled as 18, 55, and 83 mV are simply related to phonon energies (arrows) of the silicon lattice. This case is an example of a tunneling threshold spectroscopy in which the tunneling process acquires a higher probability (and is said to be assisted) if

(a)

F=0

Ec

Eg

Ev

Ec = Ec0 – eFx

Ev yv (x)

W

(b) Ec Eg e

Ev mFP

eV = mFP Ec

P

n Ev

Fig. 1.4. (a) Schematic diagram illustrating intrinsic semiconductor in zero field (top) and at high electric field F = E g /ew (bottom). Zener interband tunneling occurs from valence band maximum to conduction band minimum through the semiconductor band gap, forming a barrier of width w. (b) Schematic of narrow tunneling (Esaki) pn junction between degenerately doped regions of a semiconductor single crystal. (c) Currents observed in narrow pn junction or tunnel diode by Esaki and Miyahara (1960) at three different temperatures. The arrows in the figure at 18, 55, 83, and 120 mV locate simple multiples of the zone boundary phonon energies in silicon, indicating that phonon emission is required in the tunneling process.

8

I N T RO D U C T I O N

(c) 80 298 K

60

4.2 K

40 0.018

I (mA)

50

80 K 0.120

0.055

0.083

70

30

20

10

0

0

0.02

0.04

0.06

0.08

0.10 0.12 Bias (V)

0.14

0.16

0.18

Fig. 1.4. Continued

some other mode of the system can be excited. As we shall see, the phonons are required in this case because the electronic transition in silicon from the lowest conduction band to the highest valence band involves a change in wavevector from the zone boundary to the zone center; conservation of crystal momentum within the single-crystal structure hence requires phonon emission or absorption to balance the change in wavevector incurred in the electronic transition. Many other types of excitations have been observed in this general fashion as onsets of new processes which, even to a small degree, alter the measured tunneling current; examples include excitation of plasmons, molecular vibrations, and Zeeman transitions of impurities in tunnel junctions. It is difficult to overemphasize the third importance of Esaki’s discovery: that for the first time a solid-state structure could be fabricated reproducibly in which the parameters of importance for the tunneling probability could be estimated, putting the calculation of a tunnel current on a semiquantitative basis. Earlier attempts to fabricate MIM structures, with insulators thin enough to allow tunneling to occur, had suffered from lack of reproducibility and difficulty in reliably estimating such relevant parameters as the thickness and the barrier height of the insulating layer. Esaki was awarded the Nobel Prize in 1973 for his invention and other contributions in this area of research. It also seems likely that Esaki’s success spurred renewed interest in the question of fabricating MIM thin-film structures in a similarly reproducible and quantifiable fashion. Such an effort was undertaken at the General Electric laboratory by Fisher and Giaever,

9

SUPERCONDUCTING AND JOSEPHSON TUNNELING

who by early 1959 (Giaever, 1974) had demonstrated reproducibility of currents measured in thin-film MIM tunneling structures. These were fabricated by allowing a thin, natural oxide layer to grow on an initially evaporated film, followed by evaporation of a second, crossing counterelectrode film (Fisher and Giaever, 1961). It was found that the results could be explained semiquantitatively within the framework of the elementary tunneling theory which had been developed much earlier (Frenkel, 1930; Frenkel and Joffe, 1932; Nordheim, 1932; Wilson, 1932), in part stimulated by early work on metal contacts (Holm and Meissner, 1932, 1933). An essentially unexpected development of great importance in 1960 allowed Giaever and Fisher to rule out appreciable current contributions in their MIM structures from mechanisms other than tunneling. This discovery was to initiate a long and fruitful connection between tunneling and superconductivity, and to demonstrate the simplicity and elegance of tunneling as a spectroscopic tool.

1.4 Superconducting (quasiparticle) and Josephson (pair) tunneling Giaever (1960a,b) discovered that MIM structures, in which the tunneling barrier was formed by exposing an evaporated film of Al or Pb to the atmosphere, leaving an oxide layer typically 20 Å thick (Fisher and Giaever, 1961), developed a pronounced nonlinearity in the 2

Relative slope dI / dV

1.5

1

0.5

0

0

1

2

3

4

5

Bias (mV)

Fig. 1.5. Original observation of the superconducting energy gap in the tunnel conductance dI /dV of an aluminum–oxide-lead junction at 1.6 K (after Giaever ,1960a). The plotted points are slopes of V –V characteristics. Such measurements were soon shown to be quantitatively in agreement with the BCS theory of superconductivity if one included the effect of thermal smearing in the counterelectrode.

10

I N T RO D U C T I O N

I –V curve when cooled well below the superconducting transition temperature Tc of one of the electrodes. The limiting behavior at very low temperature was zero current below an onset bias V = /e, where is half the energy gap in the superconducting electrode. Further, Giaever (1960a) (see Fig. 1.5) and Nicol et al. (1960) established that the slope of the I –V characteristic G(V ) = dI /dV , in the region of the nonlinearity, was directly related to the “density of states,” or of quasiparticle excitations, of the superconductor, as predicted by the 1957 theory of Bardeen, Cooper, and Schrieffer (BCS). The many important consequences of this discovery, for which Giaever was awarded the Nobel Prize in 1973 (Giaever, 1974), can be regarded as being of two types. First, as we have indicated, this observation for the first time unequivocally demonstrated that tunneling could be the origin of the entire current (to better than one part per thousand) in properly made MIM structures. This result spurred further experimental and theoretical work, which has led to a

(a)

Pb 0.3 K

N (E)

3.0

2.0

1.0

0

4

8

12

E/Δ

Fig. 1.6. (a) Normalized conductance of a tunnel junction involving lead at 0.3 K (after Giaever et al., 1962). Note the extremely sharp energy gap. The small deviations of the density of states from unity in the 4–10 mV range are due to the phonons of lead. (b) Illustration of the use of tunneling to determine the effective phonon spectrum α 2 F(ω) of a strong-coupling superconductor. The Pb phonons are revealed in detail by the analysis of McMillan and Rowell (1965). Curves A, B, and C, respectively, show the second derivative, first derivative, and effective phonon spectrum for lead.

11

SUPERCONDUCTING AND JOSEPHSON TUNNELING

(b) 1.20

0

5

10 0.2 A 0

1.15

1.10

B

–0.5

1.05

1.00 Pb

0.95 1.0 C

0.5

0

5

10

0

E (me V)

Fig. 1.6. Continued

considerably improved technology of tunneling structures and to much greater sophistication in understanding related phenomena. Second, a high-resolution energy spectroscopy based on tunneling was made available for detailed study of the phenomena of superconductivity. Giaever’s discovery of Fig. 1.5 led to a great deal of further experimental and theoretical work on superconductors, providing detailed verification of many features of the BCS theory, as well as providing evidence of anomalous behavior (Fig. 1.6a) in soft metals such as Pb and

12

I N T RO D U C T I O N

Hg, which could not be explained by the BCS theory. The anomalous appearance of structure in the normalized conductance σ (V ) = G(V )S /G(V )N in the bias range corresponding to phonon energies (visible from energy 4 to 8 in Fig. 1.6a) was rather quickly understood by extensions of the theory of superconductivity to strong electron–phonon coupling, due notably to Migdal (1958), Eliashberg (1960), Nambu (1960), and Schrieffer et al. (1963). In the hands of McMillan and Rowell (1965), this phonon-related structure became the basis for extraction of the effective phonon spectrum α 2 F(ω) (Eliashberg function) of the superconductor (Fig. 1.6b). This technique, involving an elegant combination of experiment and the strong-coupling theory, has provided the most sensitive probe of the superconducting state. The theoretical possibility of a supercurrent at zero voltage tunneling through a barrier between two superconductors was described in 1962 by B. D. Josephson (1962a, b). This further inherent feature of SIS tunneling is shown at V = 0 in the I –V curve of Nicol et al. (1960) (Fig. 1.7) but remained unexplained before Josephson’s work. Identifying features of this dc Josephson effect, which corresponds to tunneling of Cooper pairs of electrons, are a welldefined upper limit Jc of the V = 0 current density with a sensitive dependence on magnetic field, which, without shielding, may destroy the effect. The characteristic magnetic field dependence of the effect was confirmed by Anderson and Rowell in 1963. A second prediction of Josephson (1962a) was that the SIS structure exhibiting pair tunneling, when driven out of the zero-voltage state to a bias voltage V , should emit radiation of frequency ν = 2eV/ h. This ac Josephson effect was first directly observed by Giaever (1965) and by Yanson et al. (1965). Josephson also was awarded the Nobel Prize in 1973 for these theoretical discoveries, which have had many consequences.

Voltage mV

2

0

–2

–40

–20

0 Current mA

20

40

Fig. 1.7. The original published report of the current–voltage relation between two superconductors. The current, here plotted horizontally, rises at the sum energy gap of aluminum and lead, and also at V = 0, showing for the first time the Josephson supercurrent and quasiparticle current in the same tunnel junction. After (Nicol et al., 1960.)

13

T U N N E L I N G S P E C T RO S C O P I E S

1.5 Tunneling spectroscopies A common feature of the solid-state-barrier tunneling structures mentioned in the two preceding sections, and collected in Fig. 1.8, is that application of a bias voltage V leads to tunneling of electrons with a well-defined range of energies 0 < E < eV. This feature makes possible several forms of spectroscopy of the solid electrodes and of the barrier, with energy resolution set by kT. (Better resolution is possible when both electrodes are superconductors.) These forms include a spectroscopy of the superconducting state, which probes both the details of the energy gap structure and of the phonon spectrum that produces the paired-electron system. A second major area of spectroscopy, known as IETS for inelastic electron tunneling spectroscopy, involves measurement of inelastic excitations of the electrodes (usually in the normal state) and the barrier. An example is a threshold for phonon generation in an Esaki diode at a bias V = ωph /e, which is detected by an accompanying step increase in G(V ) and thus a peak in d2 I /dV 2 . In similar fashion, energies of plasmons, of spin waves, of spinflip (Zeeman) transitions of paramagnetic ions, and of a variety of other excitations occurring in or near the barrier in various tunneling structures have been measured. A related area of particular activity in IETS is the measurement of vibrational frequencies of molecules (Jaklevic and Lambe, 1966), including large organic molecules, incorporated in an MIM

(a)

(c) U(x)

UB

Ec

y(x)

n

Eg

mF

e

eV mF

P e

eV

t

L

(b) A

Ev

R

(d)

V Pb

Al

UB eV

e

Ec

mF

Eg

Glass V Metal

Ev N – semiconductor

Fig. 1.8. (a) In the metal–insulator–metal tunnel junction, the barrier U (x) between the two metal films is usually produced by oxidation of a film deposited on a glass slide. (b) Typical cross-stripe junction and elementary measurement circuit. (c) The tunneling pn junction or Esaki diode. (d) Metal-semiconductor contact or Schottky barrier junction.

14

I N T RO D U C T I O N

(a)

d2V/dI 2

O

0

800

1600 2400 ENERGY (cm–1)

3200

4000

(b) w

2w

2w

50 kHz

w

Ramp

L.I.A.

X

Y

Fig. 1.9. (a) Curve of d2 V /dI 2 reveals vibrational spectrum of phenolate anion at 2 K on aluminum oxide. (After McMorris et al., 1977.) (b) Schematic diagram of circuitry used for obtaining the second derivative d2 V /dI 2 .

tunnel junction by adsorption to the barrier oxide. A good example of such a tunneling spectrum is shown in Fig. 1.9a, while a typical circuit for obtaining such spectra is shown in Fig. 1.9b. This work provides information about the bonding of molecules to solid surfaces and about the interactions or reactions that adsorbed molecules may undergo. Interest in mechanisms of catalysis is a motivating factor in much of this work. A third class of spectroscopy in junction tunneling relates to the distribution of electron energy states either in the final electrode or, occasionally, in the barrier. Perhaps surprisingly (although theoretical and experimental reasons will be discussed), it has turned out that tunneling is usually not a particularly good diagnostic tool for measuring the normal density of states in a metallic solid. Nevertheless, spectroscopic observation of Landau levels and of size-quantized electronic states have been made, and energy band positions in semiconductors and metals have been measured. In addition, normal metal tunneling has provided useful

T H E S C A N N I N G T U N N E L I N G M I C RO S C O P E ( S T M ): S P E C T RO S C O P I C I M AG E S

15

spectroscopic information about fundamental electronic correlation effects in the metallic phase near the localization (Mott) transition.

1.6 The scanning tunneling microscope (STM): spectroscopic images The scanning tunneling microscope (STM) was announced by G. Binnig and H. Rohrer in 1982. The basic elements of their invention are shown in their own figure (see Fig. 1.10). The device is operated in ultra-high vacuum and requires atomically clean scanning tips and sample surfaces. Control over the relative positions of sample and tip was the great achievement of Binnig and Rohrer, who made adroit use of piezoelectric control elements in three dimensions coupled with feedback circuitry. The tunneling current between tip and sample at any fixed bias voltage is a rapidly decreasing function of the separation t, basically because the electron probability density 1 outside a metal falls off exponentially, approximately as exp ([–4π(2mϕ) 2 t/ h]) where h is Planck’s constant and ϕ the work function. The variation can also be thought of as exp[–t/a∗] where a∗ is an effective electron orbital radius, perhaps a tenth of a nm. Maintaining

PZ VP

Px

δ

Py

VT CU

S

B

A C

ΔS JT

Fig. 1.10. In this figure, a cylindrical tip is moved in three dimensions by piezoelectric elements px p y pz , which are actuated by control unit CU by means of piezo voltages such as Vp which controls tip height a. Tip is shown at height a above surface, and tunnel current is indicated as JT driven by tip voltage VT . Trajectory of tip is shown (dashed) as control unit senses magnitude of tunnel current and adjusts tip height a to maintain tunnel current at preset value by means of a feedback loop. The horizontal position, y in this figure, is advanced stepwise, and the dashed trajectory schematically shows a broadened image of the atomic step (labeled A) which appears in the motion of the tip. At location C, a region of reduced work function (or increased density of states) results in an apparent bump in the topography. (After Binnig et al., 1982b.)

16

I N T RO D U C T I O N

constant current as the tip is slowly scanned across the surface amounts to maintaining a constant separation t, and thus a topograph is generated. Binnig and Rohrer demonstrated their invention by displaying a detailed image of the complex 7 × 7 reconstruction of the Si [111] surface (Binnig et al., 1983a ,b), a great achievement and the opening of a new era in surface science, as well as earning for them a share of the Nobel Prize in Physics in 1986. From a technical side, the success in atomic-scale imaging demonstrated that piezoelectric elements under smooth changes in bias voltage can provide smooth motions on an angstrom scale, free of jumps. One might have feared erratic jumps as an analog of the well-known Barkhausen noise that occurs in the development of magnetization of a magnet under smooth change in ambient magnetic field. For a thorough treatment of the surface-science/microscopy techniques and applications of STM the reader is referred to Chen (1993).

1.7 Atomic spatial resolution in the scanning tunneling microscope The Scanning Tunneling Microscope invented by Binnig and Rohrer (see Fig.1.10) often achieves atomic resolution, an unexpected result in view of the relatively blunt apex of an etched tungsten tip. In retrospect, as we will discuss below, the tip is often decorated with a single atom, which can even be picked up from the sample surface by appropriate manipulation of tip voltage. In such a case, the physical process monitored is electron tunneling from that atom to the atoms of the specimen. This gives the highest spatial resolution presently available in any type of scanning sensor. The STM can image individual atoms, and in addition can provide spectroscopic measurements of the density of electron states (DOS) available on individual atoms. The energy levels for electrons in the specimen in principle define the bias voltages at which a tunneling current will flow. The topographic function (height measurement of a sample feature) is accomplished by using a servo-loop to maintain constant tunneling current, which effectively keeps constant height z between the tip atom and the surface atom.

1.8 Density of electron states (DOS) measurement in STM: STS At any x, y location, the servo-loop can be disabled to fix the tip height z and a conventional tunneling spectrum I –V or dI /dV acquired. A tunneling electron must reach a final state at the same energy, since tunneling is an elastic process. A measurement of the incremental current dI per increment in bias dV , available at bias V , namely [dI /dV ](V ), is thus a local measure of density of electron states (DOS), the number of states per unit energy. This may be called scanning tunneling spectroscopy (STS). It is found that an atomically sharp tip, providing atomic spatial resolution, can simultaneously provide spectroscopic energy resolution near the ideal limits expected from the Fermi distribution. For a normal metal tip, the energy resolution is 5.4 kB T , while if the tip is superconducting an improved energy resolution can be obtained. Some of these aspects are illustrated in the following figures, which depart from an historical account. In tunneling from a tip of effectively atomic size, one might fear a loss of energy resolution. An experimental answer is provided in the data of Pan et al. (1998), shown in Fig. 1.11. A sharp Nb tip is used in the upper panel of the figure to demonstrate ideal energy resolution, and the same tip is used in the lower panel to demonstrate atomic resolution.

(a)

8.6 K 7.0 K

Differential Conductance (a.u.)

5.0 K 3.0 K 1.6 K

380 mK

–8

–6

–4

–2 0 2 Sample Bias (mV)

4

6

8

(b)

Fig. 1.11. The two panels use the same Nb tip. In (a), ideal dI /dV gap spectra (the solid curves (a) are obtained from the BCS theory) of the superconductor tip Nb tunneling into normal metal gold, measured from 380 mK to 8.6 K. The lower image (b) of the semimetal NbSe2 taken with the same Nb tip also shows longer-scale modulations in the atomic positions arising from a charge density wave (CDW). NbSe2 is an example of a layered compound, similar in nature to graphite, which is favorable experimentally in that a clean surface can easily be prepared by sample cleavage (peeling off upper layers). (After Pan et al., 1998.)

18

I N T RO D U C T I O N

(a)

8 Nb tip

(b)

6 4

Pb

2

Si(111)

Pb

0 Mn

5Å

(e)

6 Normalized dI/dV

(c)

Cr

(d)

4 2

Mn

0

8

(f)

6 4 2

Cr

5Å 0 HI

LO

–6

–4

2 –2 0 Bias (mV)

4

6

Fig. 1.12. The schematic in panel (a) shows the superconducting Nb tip above the superconducting Pb layer, with 3 adatoms in view. The dI /dV characteristic in panel (b) measured at 0.4 K, shows nearly ideal peaks at the voltage corresponding to the sum of the energy gaps of Pb and Nb, about 1.52 meV. The energy resolution is about 0.1 meV at 0.4 K. Panels (c) and (e), respectively, show topographs of isolated Mn and Cr atoms on the Pb surface. The corresponding panels (d) and (f) show dI /dV spectra observed from the superconducting Nb tip directly above the magnetic atoms. (After Ji et al., 2008.) This figure is reproduced in colour in the colour plate section.

Further demonstrations of simultaneous atomic spatial resolution and ideal spectral resolution are shown in measurements of isolated magnetic atoms Mn and Cr deposited on a superconducting Pb surface, as observed with a superconducting Nb tip (Ji et al., 2008). The spectral features seen (see Fig. 1.12) are the superconducting sum gap of Pb and Nb, and sharply defined splitting of the sum gap when the tip is directly above the magnetic atom. The pair-breaking effect of the magnetic atom on the conventional superconductor Pb is localized to a region whose size is similar to the coherence length of Pb.

D E N S I T Y O F E L E C T RO N S TAT E S ( D O S ) M E A S U R E M E N T I N S T M : S T S

Lo

(c)

0.4

Height (Å)

Hi

(b) ----------------------------------------------

(a)

19

0.3 0.2 0.1 0.0 0

3

6 Distance (Å)

9

0

3

6 Distance (Å)

9

(d)

d2I/dV2 (nA/V2)

Hi

50

25

0

Lo

Fig. 1.13. Topographic and spectroscopic images at 13 K of single 18 O2 molecule on Ag(110) (aligned along x-direction, molecule fits into 4-fold hollow site of this surface). (a) The 0.12 nm × 0.12 nm topograph was taken with a CO-terminated tip at current 1 nA and tip bias 70 mV. (b) Line cuts of topograph. (c) Vibrational image (spatial distribution of STM–IETS intensity) obtained –76.6 mV sample bias, corresponding to the O–O stretch vibration of the oxygen molecule on Ag(110); data taken with bare metallic tip. Note that this vibrational energy is smaller than that for an isolated molecule. (d) Line cuts of the vibrational intensity image. The solid lines in the cuts align with the oxygen atoms of the molecule. (After Hahn et al., 2000.) This figure is reproduced in colour in the colour plate section.

These detailed spectra reflect the local destruction of superconductivity in the Pb by the magnetic pair-breaking effect on conventional singlet electron pairs. Clearly resolved (Fig. 1.12) are three distinct angular momentum states associated with the localized quasiparticle states in the superconducting Pb electrode, near the magnetic impurity, at energies inside the superconducting gap. Molecules deposited on solid surfaces have been observed by STM with the additional possibility of spectroscopically measuring vibrational energies of the molecule. This is the single molecule equivalent of the IETS depicted in Fig. 1.9 above. The measured quantity is d2 V /dI 2 , typically using a lock-in detector, recalling that d2 V /dI 2 = –r 3 d2 I /dV 2 with r = dV /dI . Since r is slowly varying with V, –d2 V /dI 2 represents as a peak the abrupt step in

20

I N T RO D U C T I O N

conductance 1/r associated with the additional tunneling path at threshold: electron tunneling plus excitation of the vibration mode. Normally the conductance increases at these thresholds, but a counterexample is given by the work of Hahn et al. (2000). Isotopic shifts of these vibrational frequencies have been measured. The interaction of the molecule with the surface in most cases under study is weak enough that the molecular parameters are not greatly altered. (In the data here shown, a strong interaction occurs, greatly weakening the molecular bond.) The initial question is what is the most favorable site and orientation of the molecule on the given surface? In the case of the oxygen molecule 18 O2 on Ag(110), Hahn et al.( 2000) identified a 4-fold hollow site. In their study (see Fig. 1.13), Hahn et al. deposited tiny amounts of CO on the Ag surface along with the oxygen molecules. They used the CO in topography mode to decorate the metal tip, thus achieving higher spatial resolution. These workers learned how to pick up a single CO molecule onto the bare metal tip, so that their topographic images are definitely taken using a single atom tip (assuming the CO orients reliably on the tip). Whether the tip retracts or extends (in the constant-current topographic mode) when located above the molecule depends on the electron density at that location. In some cases, the molecule will look like a stronger barrier than vacuum, so the tip will move down toward the molecule in response. A negative topographic image is thus possible. The spectroscopic image in Fig. 1.13c is a map of the second derivative signal, with the bias set at the threshold value. It is seen that this signal peaks with the tip located above each of the oxygen atoms. The data are surprising, however, in that the sign of the second derivative peak signals a reduction in conductance at threshold, a point addressed by the authors in their paper, and an indication of the wealth of information that can be obtained from such studies. Figure 1.14 shows second derivative spectra from a single molecule, identifying its isotopic composition. These data represent a tour de force in the technique of scanning tunneling spectroscopy. The apparatus that allowed this breakthrough is described below in Appendix B (Lee and Ho, 1999).

1.9 Perspective, scope, and organization These forms of spectroscopy and the physics that they reveal are the central themes of the present work, which is intended further to provide a reasonably comprehensive introduction to the rather wide range of topics in solid-state physics, and especially superconductivity, which have been probed by the tunneling spectroscopies. We do not take a strictly historical approach but rather emphasize developments since the survey of tunneling by Solymar in 1972. The reader is referred to Duke (1969) for a thorough history of tunneling in solids prior to 1969. Of several edited works, including the treatise “Superconductivity” (Parks, 1969) and the proceedings of the International Conferences on Low Temperature Physics and of the conferences on superconductivity in d- and f-band metals, the volume Tunneling Phenomena in Solids (Burstein and Lundqvist, 1969) and the volume on molecular spectroscopy (IETS) edited by P. K. Hansma (1982) are particularly useful. With regard to the Josephson effects and their consequences, which are not emphasized here, other sources include the monograph of Kulik and Yanson (1970) and the careful research-level review of Waldram (1976). Texts (Tinkham, 1975; van Duzer and Turner, 1981; Barone and Paterno, 1982), with strong emphasis on applications, are also recommended. More recent books that are useful include

P E R S P E C T I V E , S C O P E , A N D O R G A N I Z AT I O N

21

358 20 C2H2

d2I/dV2 (nA/V2)

0

1 266

–20

C2D2

2

1–2

0

100

200

300 Voltage (mV)

400

500

Fig. 1.14. Demonstration of isotopic identification in different species of acetylene on Cu. These spectra are obtained with a bare metal tip located directly above the molecule. These data represent a tour de force in the technique of scanning tunneling spectroscopy. As mentioned in Appendix B, the data were obtained in a carefully designed cryostat (on a vibration isolation table) attached to a helium dewar standing in the open in a laboratory room. (After Lee and Ho, 1999.)

Wiesendanger (1994), Ketterson and Song (1999), P. W. Anderson (1997), Alexandrov and Mott (1994), Kresin et al. (1993), Bennemann and Ketterson (2003), and Schrieffer and Brooks (2007). Some comments on the organization of the book may be useful. The treatment of tunneling in normal-state structures is divided into two parts. Chapter 2 introduces the subject in some detail from both the theoretical and the experimental points of view and is intended to provide enough coverage of tunneling in real structures to prepare for Chapters 3 through 7, which deal with tunneling applied primarily to the study of superconductivity. Chapter 2 thus stops short of treating the subject of many-body effects in normal-state tunneling, although it does include accounts of other theoretical advances. An example is the work of Feuchtwang in extending the WKB transmission calculation to the case of a barrier, coupling periodic potentials (Feuchtwang 1970). On the experimental side, Chapter 2 describes enough of the real-life pathology of tunneling structures, including the effects of various types of defects in the barrier, to forewarn the reader, who may be interested primarily in superconductivity, of various extraneous effects that may occur in practice. In the same vein, some emphasis is given to techniques for checking the degree of perfection (homogeneity, sharpness of interfaces, etc.) of tunneling barriers, although the systematic discussion of experimental techniques is deferred to Appendix A. The treatment of superconducting tunneling appears in Chapter 3, which introduces the theory of superconductivity and of superconductive tunneling in such a way as to emphasize the close relationships between quasiparticle and Josephson tunneling. This chapter summarizes the tunneling study of the superconducting energy gap, including cases of strong coupling and the effects of anisotropy, multigap structure external perturbations, ultrathin films and small-particle superconductors. Chapter 4 describes the basic method of McMillan and Rowell for determining the phonon spectral function α 2 F(ω) of superconductors and illustrates the method with the classic results for lead (McMillan

22

I N T RO D U C T I O N

and Rowell, 1965, 1969). This chapter also discusses the strong-coupling theory of superconductivity, i.e., the Eliashberg equations, allowing generalization to cases of anisotropy, spin fluctuations, and nonconstant density of electronic states, and finally notes the somewhat limited applications of the conventional phonon spectroscopy of superconductors. Chapter 5 is devoted to proximity tunneling methods as a means to extend the superconducting phonon spectroscopy to those metals, notably in the transition series, which had not been accessible to the conventional methods. Chapter 5 first surveys phenomena in bilayer junctions and traces the development of a quantitative spectroscopic technique for the study of both the N and S layers of C–I–NS proximity junctions. Chapter 6 then surveys strong-coupling superconductivity results, primarily the α 2 F(ω) functions, obtained to date by the various methods of tunneling spectroscopy. In this chapter, superconductors are separated into classes of s–p-band elements, alloys, and amorphous alloys; transition-metal related superconductors; and extremely weak coupling metals (studied by the proximity techniques); finally, the chapter provides a survey of what has been learned about the systematics of superconductivity from tunneling. Chapter 7 introduces the new area of electron-coupled superconductors of high transition temperature. The cuprate superconductors are emphasized if only because they have been studied much longer than the equally interesting iron-based superconductors. Chapter 8 returns to the study of normal-state structures, with emphasis on the possible spectroscopies. These can be broken down roughly into final-state effects and threshold spectroscopies, where the former subject again falls into two parts, roughly the simple one-body effects and the many-body effects. A separate survey of zero-bias anomalies is included in Chapter 8. Chapter 9 is devoted to STM/STS work on atoms and molecules, including excitation spectra. Chapter 10 is devoted STS results on new high-temperature superconductors and related materials. Chapter 11 describes some applications of tunneling phenomena. Appendix A on experimental topics emphasizes advances in preparation of electrodes and barriers that promise to extend the application of tunneling spectroscopy to a wider range of materials. Appendix B is devoted to methods of scanning tunneling spectroscopy, with some comparison to competing methods. Appendix C provides a compendium of material properties, including a good deal of information about superconductors as obtained by tunneling methods.

2 Tunneling in normal-state structures: I 2.1 Introduction In this chapter we are concerned with calculating the current I (V ) and conductance dI /dV = G(V ) in structures, as shown in Fig. 2.1, consisting of two metallic electrodes separated by a narrow potential barrier U (x). There are two basic approaches to such a calculation: a steadystate approach, as used originally by Gamow and by Condon and Gurney in treating the alpha decay problem mentioned in Chapter 1, and a transfer Hamiltonian approach indicated in the lower portion of Fig. 2.1, traceable to Oppenheimer’s original treatment of ionization of the hydrogen atom by electric field. Making simple assumptions about metals and the barrier, we will develop expressions for the current density J (V ) by using the two methods and show their equivalence. The consequences of making the model more realistic in various ways will be discussed, and the theoretical expectations for basic cases will be described and compared with some experimental results, including metal–insulator–metal (MIM), metal– insulator–semimetal, Schottky barrier, and direct pn junction cases. Within the framework of a one-electron model, the influence of electronic band structure and density of states on the current and conductance will be discussed, and the consequences of various types of barrier defects will be considered in some detail.

2.2 Calculational methods and models An extended wavefunction k , as indicated in Fig. 2.1, can be constructed from solutions of Schrödinger’s equation: −

2 ∂ 2 ψ(x) + U (x)ψ(x) = E x ψ(x) 2m ∂ x 2

(2.1)

where  = ψ(x) exp(ik y y + ik z z) within each of the three regions by matching the wavefunctions together at the electrode– insulator boundaries x = 0 and t such that the wavefunction ψ and its derivative dψ/dx are continuous at these points. For simplicity, we assume free-electron metals, i.e., U (x) = 0 in metals 1 and 2 and U (x) > 0 in the barrier region. A wave incident from the left, defined as eikx , suffers partial reflection with amplitude R at x = 0, is exponentially decaying, e−κ x , in the interval 0 < x < t, and emerges for x > t as Teiqx , where T 2 measures the probability of the particle penetrating the barrier. The probability current operator, (1.1), applied to the wavefunction in metals 1 and 2, respectively, will give hk/m 1 × (1 − R 2 ) and T 2 hq/m 2 , respectively, which must be equal. The same probability current must also be obtained within 23

T U N N E L I N G I N N O R M A L - S TAT E S T RU C T U R E S : I

24

(a)

EC U (x, V)

ψ

k

E = m1

Eg eV m2

E =0 I EV

E2 = 0

2 0

(b) ψ1(k)

1

U (x, V) ψ (q) 2

m1 eV m2 E =0 I

t 2

Fig. 2.1. (a) Steady-state wavefunction extending smoothly across the entire tunneling structure. (b) Separate wavefunctions ψ1 and ψ2 are, respectively, standing waves in the left and right regions of the structure. These approximate wavefunctions decay exponentially into the barrier and beyond into the opposite electrode. Such wavefunctions are regarded as initial and final states, respectively, in the tunneling Hamiltonian picture.

C A L C U L AT I O NA L M E T H O D S A N D M O D E L S

25

the barrier region. Using (1.1), one sees that this result is possible only if the wave reflected from x = t is included. The quantity of importance in calculating the tunneling current is D, (1.1), defined as the fraction of the incident probability current hk/m 1 that is transmitted; this fraction is T 2 × m 1 q/m 2 k. The assumption is made that the tunneling process is elastic, i.e., E, but not E x separately, is invariant. We also assume that components of a wavevector in the transverse directions k z , k y , referred to simply as kt , are conserved. i.e. kt = qt . In the important case of small transmission, the exact expression for D, the transmission factor, has been given (Harrison, 1961) by (2.2), where κ is the decay constant in the barrier region, defined in (2.3): D(E x ) = g exp(−2K )

(2.2a)

16kqκ 2 (k 2 + κ 2 )(q 2 + κ 2 )  t K = κ(x, E x )dx g=

0

 κ(x, E x ) =

2m 2

1 2

(2.2b) (2.2c) 1

[U (x) − E x ] 2

(2.3)

The last expression, (2.3), for a particle of fixed mass m in a potential U (x) may not be a good approximation to κ if the barrier, as in Fig. 2.1a, is regarded as the forbidden gap of an insulator, unless the energy of the particle lies close to the conduction or valence band edge of the insulator so that the usual electron or hole mass will be appropriate. The generalization of this approach to a more realistic three-dimensional picture has been discussed by Harrison (1961), who finds that Eqs. (2.2) are still valid. The prefactor g (2.2b), exact for a square barrier, results from the matching requirement on the wavefunctions at the two interfaces. This prefactor is unity in the usual WKB approximation, which is formally valid if variation of the potential U (x) is sufficiently slow in the vicinity of the turning points 0 and t. However, the exponent K is correctly given by (2.2c), independent of the steepness of the potential variation at the turning point. Note that the expression for g varies linearly with k and 1 1 q or, equivalently, with (E x ) 2 , predicting D(E x ) to vanish linearly in (E x ) 2 at the bottom of a conduction band. This behavior will obviously be noticeably altered depending upon whether g is taken as expression (2.2b) or as unity. However, if the electron energies in question are far removed from band edges, then the factor g has a slow variation with E, and its presence may be scarcely detectable in comparison with the typically exponential variation of D.

2.2.1 Stationary-state calculations To obtain the current density J (V ) in a junction between metals 1 and 2, we assume each is described by an equilibrium Fermi function. In notation suited to the case μ1 − μ2 = eV, we define    −1 E 1 − μ1 ≡ f (E) f 1 = exp +1 kT  −1   E 2 − μ2 +1 f 2 = exp ≡ f (E + eV) (2.4) kT

26

T U N N E L I N G I N N O R M A L - S TAT E S T RU C T U R E S : I

We conventionally measure energy E 1 = E from the bottom of the left-hand conduction band. Here, μ1 and μ2 are the Fermi energies measured from the bottom of the respective bands, and E 1 and E 2 are the total energies on the left and right, similarly measured. We take the convention that positive bias lowers by eV the Fermi level of the right-hand electrode, as shown in Fig. 2.1, and calculate the net current density J = J12 − J21 :      1 ∂E 2e 2 Df (E)[1 − f (E + eV)] (2.5) J12 = k dk d t x  ∂k x (2π )3      1 ∂E 2e 2 J21 = Df (E + eV)[1 − f (E)] d kt dk x  ∂k x (2π )3  ∞   2e J= dE [ f (E) − f (E + eV)] d2 kt D(E x , V ) (2.6) x (2π )2 h 0 To obtain J12 , one must integrate over all available k states in metal 1, weighting each k by the corresponding group velocity, vx = −1

∂E ∂k x

(2.7)

multiplying by the transmission factor D(E x ) and by the Fermi functions, guaranteeing that the initial state 1 is occupied and that the final state 2 is empty. In (2.5), the factor 2 represents the spin degeneracy, and the factor (2π )3 in the denominator gives the number of states per unit volume in k space. In arriving at (2.6) for the current density J under influence of applied bias voltage V, we employed (2.7) to obtain the integration variable E x . Other forms of the k-space integral (2.6) for J are possible, and will be discussed in Section 2.3. In connection with (2.5), it is appropriate to discuss some of the properties of D(E x ), which is explicitly a function of E x = E − E t = E − 2 kt2 /2m and implicitly a function of bias V , which distorts the barrier and hence influences the potential function U (x) = U (x, V ). Regarding D as calculated via (2.3) reveals that a maximum D, and strong preference in transmission, is given for electrons of a given total E that have a minimum value of kt , i.e., for kt = 0, E t = 0, and hence E x = E. This result can be given in terms of an angular dependence. Consider a Fermi surface electron, E = μ1 , and let φ be the angle between its wavevector of magnitude k F and the barrier normal, so that k x = k F cos φ and E x μ1 (1 − sin2 φ), and assume g = 1, which is strictly appropriate only to a slowly varying U (x). Consider the simple case of a square barrier U (x) = U0 , whose height from μ1 is defined as U B = U0 − μ1 . The WKB result is, approximately, for fixed total energy E = μ1 and varying tunneling angle φ,   μ1 sin2 φ (2.8) D exp(−2κt) exp −κt UB Taking values U B = 1 eV, t = 20 Å, μ1 = 5 eV, and m = m e , for example, one finds D D0 exp(−βφ 2 ) with β = 51 corresponding to transmission reduced to e−1 at an angle φ = 8o .

C A L C U L AT I O NA L M E T H O D S A N D M O D E L S

27

The effect of bias potential V on an ideal dielectric barrier of thickness t can be written as eVx (2.9) t which produces a trapezoidally shaped tunneling barrier. The dependence of D on bias voltage V can be treated by (2.9) in cases in which (2.3) is employed. As may be seen by inspecting Fig. 2.1, the effect of bias upon the barrier is such as to reduce the average barrier height for electrons tunneling in the forward direction (i.e., from the higher Fermi energy level μ1 ), and to reduce the probability for reverse flow from μ2 . The net effect is to increase the flow. U (x, V ) = U (x, 0) −

2.2.2 Transfer Hamiltonian calculations The transfer Hamiltonian approach, originating in the work of Oppenheimer in 1928 and extended to solid-state structures by Bardeen (1961), is motivated by the fact that a near-unity probability of reflection by the barrier occurs in a typical tunneling structure such as shown in Fig. 2.1a. In a real sense, then, the waves on the left side of the structure, rather than being traveling waves eikx are standing waves cos(kx) and the same is true on the right-hand side. The barrier can be regarded as separating the system into two nearly independent portions, and the weak residual coupling can be treated by a perturbing Hamiltonian H T , H = H1 + H2 + H T

(2.10)

which is then regarded as driving electron transitions from one side to the other in the usual Golden Rule calculation given by (2.11),   2π (2.11) w12 = |ψ2 |H T |ψ1 |2 ρ(E 2 )δ(E 2 − E 1 )  Equation (2.11) describes the transition rate w12 from a given state 1 (as on the left side in Fig. 2.1b) to a set of states of equal energy and density ρ(E 2 ) on the right. The explicit appearance in this formula of the density of states ρ(E) was useful to Bardeen (1961) in providing a mathematical explanation for the discovery of Giaever (1960a,b) that tunneling measured the density of excitations in the superconducting case. We first show that this formulation leads to the same results as the steady-state calculation. We will later see that the transfer Hamiltonian calculation can be generalized to cases where the tunneling transition is assisted, or proceeds by interaction of the tunneling electron with the barrier and also another mode of the system. From the treatment of Bardeen (1961) and Harrison (1961) in a WKB approximation, the wavefunction ⎧ 1 ⎪ w− 2 exp[i(k y y + k z z)] cos(k x x + δ), x < 0 ⎪ ⎨ 1 (2.12) ψ1 = 12 w − 2 exp[i(k y y + k z z)]g exp(−κ x), 0 < x < t ⎪ ⎪ ⎩ 0, x >t is taken as a standing wave normalized in the left electrode of width w. This is matched to an exponentially decaying wave in the barrier, which is assumed then to continue smoothly to zero in the right-hand region rather than oscillating. A similar expression applies for side 2, which again is a standing wave on the right and assumed to be zero on the left side of the

28

T U N N E L I N G I N N O R M A L - S TAT E S T RU C T U R E S : I

structure. In order to identify the matrix element (T ) for the operator H T , one substitutes an initial time-dependent wavefunction  bn (t)ψ2n e−i E 2n t/ (2.13) ψ(t) = a(t)ψ10 e−iE10 t/ + n

into Schrödinger’s time-dependent equation H ψ = i

∂ψ ∂t

(2.14)

This gives  T =

ψ1∗ (H − E 2n )ψ2n dτ ,

(2.15)

after manipulations making use of the fact that the wavefunctions ψ1 and ψ2 overlap only in the barrier region. It is established that the appropriate matrix element is essentially that of the quantum mechanical current density operator j, Tq,k = −iψq | j|ψk 

(2.16)

which must be evaluated within the barrier region, and in which we maintain our convention of q, k for wavevectors on the right and left. This expression was evaluated by Harrison (1961) to give for |Tq,k |2 |Tq,k |2 = δ(kt , qt )g exp(−2κ)(4πρ1x ρ2x )−1

(2.17)

within the WKB approximation, assuming conservation of the transverse wavevector. Here, ρ1x is a one-dimensional density of states on the left, assuming box normalization within the width w of the electrode:   w ∂ E −1 ρ1x = (2.18) π ∂k x In (2.17), recall that g is the prefactor required in the expression for D if wavefunction matching is employed as, for example, given by (2.2b); if g = 1, then the WKB approximation appropriate to a smoothly varying potential U (x) is assumed. Now that we have obtained the appropriate matrix element, it is straightforward to calculate the current J by summing transition rates from all filled states to all empty states and multiplying by 2e: J = 2e

 2π k,q



|Tq,k |2 [ f k (1 − f q ) − f q (1 − f k )]

(2.19)

This expression is equivalent to (2.6) as obtained in the steady-state method, with all of the density-of-states factors, excepting those contained in the prefactor g, having disappeared in the transformation from the sum to the integral over k space. Thus, apart from k dependence in the prefactor g, there is no explicit dependence of the tunnel current on the one-dimensional density of states. The generalization of this method to accommodate assisted tunneling processes is given in Section 2.6.

C A L C U L AT I O NA L M E T H O D S A N D M O D E L S

29

2.2.3 Ideal barrier transmission Clearly, the barrier is the crucial element in the tunneling experiment, providing the required decoupling between the electrodes so that carriers from one can be injected at a precise energy in the other. The central feature of the barrier tunneling process is the decaying wavefunction e−κ x in the classically forbidden region, which experimentally implies an extremely rapid variation of the resistance of a tunnel junction with thickness of the barrier. This feature is illustrated in Fig. 2.2, in which the characteristic exponential dependence of the resistance on the barrier thickness is obtained. Indeed, this dependence is one means of testing for the presence of a tunneling process. The purpose of the present section is to gain more detailed insight into transmission of particles through the most nearly ideal of the experimentally available barriers, which are typically thermally grown oxides and less frequently single crystals, as in Schottky barrier tunneling or in special cases involving layer compounds such as GaSe. The

Tunneling resistance (ohms)

IM

Ik

I

0

0.5

1.0

1.5

Polarizer change (degrees)

Fig. 2.2. The dominant role of the exponential decay of the wavefunction with increasing barrier thickness in setting the resistance of the junction is demonstrated in a logarithmic plot of the junction resistance, in ohms, at V = 0 vs. insulator thickness, measured ellipsometrically, with conversion 0.53◦ = 10Å. In both cases, the first metal film is aluminum and the insulator is aluminum oxide grown in a plasma discharge. The thickness range lies between 15 and 30 Å. The solid circles represent measurements made with a lead second electrode; the open circles represent measurements made with aluminum as the second electrode. (After Knorr and Leslie, 1973.)

30

T U N N E L I N G I N N O R M A L - S TAT E S T RU C T U R E S : I

questions involved are of two types. The first is a quantum mechanical problem: given a barrier potential U (x), what practical means are there for calculating the transmission of an electron at a given energy E? The second questions lie behind the first and involve more realistic methods to approximately describe the interaction and transmission of a tunneling electron through the barrier regarded as a periodic insulating solid. Finally, we investigate some of the practical dependences of transmission upon variables such as the energy of the tunneling particle, its tunneling angle φ with respect to the normal to the barrier, and the bias voltage V . We begin with a convenient method (Kane, 1969) for solving tunneling problems involving stepwise constant potentials V (x). Solutions of the one-dimensional, time-independent Schrödinger equation, Eq. (2.1), for constant potential V are the form ψ = αeikx + βe−ikx

(2.20) 1

where, if E > V , k is real, while for E < V , k = iκ, with real κ = [(2m/2 )(V − E)] 2 . In the frequently encountered case of a constant potential with steps at positions x p , taking barrier potential values V p for x p−1 < x < x p , specification of α p , β p , and k p is obtained by requiring that ψ and dψ/dx be continuous at each x p . Setting 2m/2 = 1, one finds that the two equations resulting from continuity of ψ and dψ/dx at x p are α p eik p x p + β p e−ik p x p = α p+1 eik p+1 x p + β p+1 e−ik p+1 x p

(2.21)

α p k p eik p x p − β p k p e−ik p x p = α p+1 k p+1 eik p+1 x p − β p+1 k p+1 e−ik p+1 x p

(2.22)

Multiplying the first equation by k p and adding and subtracting the two, one identifies the matrix equation     αp p α p+1 =R (2.23) βp β p+1 with R p given as the 2 × 2 matrix: ⎛ ⎜ Rp = ⎝

k p +k p+1 i(k p+1 −k p )x p k p +k p+1 −i(k p+1 +k p )x p 2k p e 2k p e k p −k p+1 i(k p+1 +k p )x p k p +k p+1 −i(k p+1 −k p )x p 2k p e 2k p e

⎞ ⎟ ⎠

(2.24)

(Kane, 1969). The problem of a plane wave k > 0 incident upon a single-step potential V2 > E 1 at x1 = 1 0 such that k2 = iκ2 = i(V2 − E) 2 is stated as     1 α2 = R1 0 β1 where β1 and α2 , respectively, are the amplitudes of reflected and transmitted (decaying) waves, as indicated by (2.20). Hence, one has 1 = R111 α2 =

k1 + iκ2 2k1 α2

β1 = R121 α2 =

k1 − iκ2 2k1 α2

Thus, in the forbidden region x > 0, one recovers ψ(x) = 2k1 (k1 + iκ2 )−1 e−κ2 x

(2.25)

C A L C U L AT I O NA L M E T H O D S A N D M O D E L S

31

The same problem with a square barrier of width t, as in Fig. 2.1, assuming 0 < k3 < k1 , as if the Fermi energy on the right were less than that on the left, is stated as     1 1 2 α3 (2.26) =R R 0 β1 To obtain the transmitted wave ψ3 (x), one evaluates α3 [(R1 R2 )11 ]−1

(R1 R2 )11 = R111 R211 + R112 R221

(2.27)

In general, then, there are two terms, so that a single exponential decay is not obtained. Supposing κt  1, however; the contribution R112 R221 is exponentially small compared with R111 R211 , and ψ3 (x) =

4ik1 κ2 e−κ2 t eik3 (x−t) (k1 + iκ2 )(iκ2 + k3 )

(2.28)

In the symmetrical case k1 = k3 = k, the exact result (2.27) for α3 , retaining both terms of (R1 R2 )11 , is α3 =

(k 2

2ikκeikt + 2ikκ cosh κt

− κ 2 ) sinh κt

(2.29)

(Baym, 1969). The fraction of the incident probability current k1 /m transmitted is α32 k3 16k1 k3 κ22 e−2κ2 t = T2 = 2 k1 (k1 + κ22 )(k32 + κ22 )

(2.30)

which provides a derivation of (2.2b). The prefactor of the exponential, designated as g, is seen to go to zero linearly in k1 and k3 . As has been noted before, this feature is absent in the WKB treatment, where g = 1, and suggests a sensitivity of g to details of the assumed interface potential. A generalization of this expression for the case of a trapezoidal barrier is given by Brinkman et al. (1970). Note that the incident wave eik1 x is transmitted as eik3 (x−t) . The loss in phase, k3 t, with respect to the wave that would occur if the barrier were absent, when the problem is cast in a wave-packet form, can make it appear that the particle spends no time in the barrier. A variety of one-dimensional well problems can be handled with Kane’s method. For example, the finite square well is described by k1 = k3 = iκ, k2 > 0, and the condition for a bound state is α1 = 0, β3 = 0, corresponding to only e+κ x for x < 0 and only e−κ x for x > t. The condition is then (R 1 R 2 )11 = 0, which reduces to k + iκ = eikt = e2iφ k − iκ

(2.31)

where φ = tan−1 (κ/k). Thus, tan(kt/2) = κ/k, which is the condition for the first bound state. Extensions of matching calculations to more realistic three-dimensional cases are discussed by Harrison (1961), Conley et al. (1966), Dowman et al. (1969), Feuchtwang (1970), and Leipold and Feuchtwang (1974). Unfortunately, such exact matching calculations can rarely be accomplished in realistic cases, except by numerical means, which is the reason for the great practical value of the WKB method. There is a considerable literature devoted to methods of approximating solutions of

32

T U N N E L I N G I N N O R M A L - S TAT E S T RU C T U R E S : I

the one-dimensional Schrödinger equation following the original work by Wentzel, Kramers, and Brillouin (for a review of the WKB approximation, see Schiff, 1955). Notable contributions to this literature have been made by Langer (1937) and by Miller and Good (1953), which have been commented on by Franz (1969). A central feature of WKB methods is evident in the expression for the approximate wavefunction ψ(x) = exp[i−1 S(x)]   1

p(x)− 2 exp ±i−1

x

 p(ξ )dξ

(2.32a) (2.32b)

as the appearance of the momentum to the inverse 12 power. This has the awkward consequence that the wavefunction becomes infinite at the edge of the barrier, i.e., at the classical turning point. The usual WKB wavefunction, (2.32b), is obtained by substituting the initial wavefunction (2.32a) into the one-dimensional Schrödinger equation −S 2 + iS  + p 2 = 0

(2.33)

(in units such that 2 /2m = 1), and then expanding the unknown function S(x) as powers of , which is assumed to be a small quantity: S(x) = S0 (x) + S(x) + · · ·

(2.34)

The method of Miller and Good can be paraphrased in the following way. Consider a soluble problem such as the harmonic oscillator for which the exact eigenfunctions corresponding to (2.32) are known. Of course, these functions also become large at the classical turning points for large quantum number n, as is familiar from elementary quantum mechanics, but the functions are finite at the classical turning points. Miller and Good (1953) established a mathematical transformation between the eigenfunctions of a soluble model system, such as the harmonic oscillator problem, and the problem at hand, and in this fashion obtained approximate wavefunctions well behaved at the two separate turning points in the typical well or barrier problem. The accuracy of their method for the calculation of the transmission coefficient    x2 −1 2 κ(x)dx (2.35) |T |MG = 1 + exp +2 x1

for an exactly soluble model barrier problem is illustrated in Fig. 2.3. Note that (2.35) agrees with (2.2) in the limit of an opaque barrier. The paper of Miller and Good provides expressions from which the error in applying the approximate solutions to a given situation can be estimated: as with the usual WKB method, the approximations are best for a slowly varying potential. If, however, one blindly applies (2.35) to the case of the square barrier, which was treated exactly (thick-barrier case) in (2.29), then the exponential factor is reproduced correctly but the prefactor g is missing. This result is believed to be the general result in applying the WKB method beyond its range of applicability (Landau and Lifshitz, 1958). It has been stated (Harrison, 1961) that application of the WKB method to determination of T 2 in solid-state structures generally gives such a result with g(E) = 1, i.e., devoid of any prefactor. However, an interesting extension of the WKB method to a barrier between regions in which the potential is specifically periodic, as in a true solid, leads to the reappearance of the prefactor (Feuchtwang, 1970), as indicated in

33

C A L C U L AT I O NA L M E T H O D S A N D M O D E L S 1.0

0.8

0.6

Approximate

Exact

T 0.4

0.2

0

2

3

4

5

6

W

Fig. 2.3. Tests of the accuracy of the WKB approximation for the transmission coefficient for a particular potential V (x). The abscissa in this figure is a measure of the energy of the particle. (After Miller and Good, 1953.)

|T |2F =

vr v |TB |2 ar qr a q

(2.36)

Here, vr and v , respectively, are the group velocities ∂E/∂q on the right and left, ar and a are the periods of the potential on the right and left, and the q’s are wavevectors. The remaining barrier factor TB is determined by the properties of the insulator and reduces to a single exponential dependence only if the barrier is thick. A different but related extension of the treatment of barrier transmission betwen two free-electron metals was given by Schnupp, (1967a,b), who treated the periodicity in a real crystalline barrier by an array of delta functions. This approach has been extended by Leipold and Feuchtwang (1974), who treat the barrier as a repetitive array of one-dimensional square well potentials. The motivation for this work is indicated in Fig. 2.4, which sketches the E(k) one would expect for a tunneling barrier assumed to be a crystalline solid. The straight-line portions of E vs. k 2 correspond to effective masses in the conduction and valence bands. The smooth connection of these sections through the middle of the energy gap suggests that the decay constant κ in the center of the gap will not be given reasonably by any formula such as (2.3) involving a single mass. In the work of Leipold and Feuchtwang, summarized in Fig. 2.5, the transmission factor as a function of energy for a finite array of square wells has been calculated and compared with that obtained on WKB-type approximations in which the function κ(E) was taken as that characteristic of the infinitely extended periodic array of square wells. An advance in the work of Leipold and Feuchtwang was in an application of an electric field, simulating a bias, to their periodic structure. This application makes possible a connection to the problem of interband tunneling first treated by Zener (1934) and later by Kane (1959a), although the biases used by Leipold and Feuchtwang are never so great as to permit tunneling through the upper (conduction) band. The formula of Zener that is tested is   x2  Im[k(E, x)] dx (2.37) |T |2z = exp −2 x1

34

T U N N E L I N G I N N O R M A L - S TAT E S T RU C T U R E S : I E

Slope =

K2

h 2mc∗

EC

K2

Eg

Ev

Slope =

h 2mv∗

Fig. 2.4. In the simple two-band model of an insulator, the wavevector squared k 2 becomes negative in the energy range between E c and E v , i.e., in the forbidden band gap, corresponding to decaying exponential waves. One can see from this plot that estimates of the decay constant κ can be in error in the middle of the gap where the solid line departs noticeably from the dashed portions, obtained by using the effective masses m c and m v for the conduction and valence bands, respectively.

and k is determined from an infinite repeating array of square wells. The dimensions of the square well potential, shown as it would appear under a bias eV in Fig. 2.5a, are well widths of 4 Å, a barrier width of 2 Å, and a well depth of 5.5 eV. The individual isolated wells would support bound states at 1.15 eV and 4.16 eV above the well bottom. In a repetitive array, the ground-state levels, as one would expect, broaden into bands that are identified as the valence and conduction bands of the solid represented by the array. The corresponding widths of the valence and conduction bands as determined by Leipold and Feuchtwang are 0.26 and 1.36 eV, respectively. Referring to Fig. 2.5b, these bands lead to regions of unity transmission coefficient separated by a region of low transmission coefficient that is interpreted as the gap of the corresponding repetitive array. In Fig. 2.5c, corresponding to an applied bias of about 2.5 V, great changes occur in the transmission coefficient as a function of energy. The increase by orders of magnitude of the transmission toward the center of the gap is interpreted as the lowering of the average barrier by the applied field. The sharp features in the low- and high-energy regions (corresponding to the valence and conduction bands, determined by the numerical exact calculation shown by the smooth curve) are interpreted as the splitting by the electric field of the individual resonance levels of the cells in the barrier. In comparing

35

C A L C U L AT I O NA L M E T H O D S A N D M O D E L S V (x)

(a) 6.5 eV

2Å

4Å

1Å

E = 0.0 eV

E = 0.00 eV

x E = −eV

(b) 1 −1

Numerical

−2

Modified WKB

10 10

10−3 10−4 log S(E)

10−5 10−6 10−7 10−8 10−9 10−10 10−11 10

−12

10−13 1.0

1.5

2.0

2.5

3.0

3.5 4.0 E (eV)

4.5

5.0

5.5

6.0

6.5

Fig. 2.5. In the heterojunction barrier model of Leipold and Feuchtwang (1974), the tunneling barrier is treated as a finite, repetitive array of cells. Each cell is a square well potential supporting two bound states (presumed empty) and lying, respectively, 2 and 5 eV above the zero of energy. (a) The potential V (x) resulting from application of bias is shown. (b) The transmission coefficient of the structure at zero bias is displayed as a function of incident electron energy. The two regions of large transmission correspond approximately to coincidence with the bound-state levels within the individual wells. The transmission is a slowly varying function of energy only for energies well away from the bound states. (c) The transmission as a function of energy for the structure under an applied bias. As before, the structure has 8 unit cells of width 6 Å. The bias assumed here is about 2.5 V. Here, as in (b), the smooth curve is obtained by exact calculation, while the dotted curve is obtained by Zener’s modified WKB approximation, Eq. (2.37). The sharp structures in the solid curve arise from the resonant transmission effects. (After Leipold and Feuchtwang, 1974.)

T U N N E L I N G I N N O R M A L - S TAT E S T RU C T U R E S : I

36

(c)

1 −1

10

Numerical Modified WKB

−2

10

10−3 10−4

log S(E)

10−5 −6

10

−7

10

10−8 10−9 10−10 10−11 10−12 10−13 1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

E (eV)

Fig. 2.5. Continued

Figs. 2.5b and 2.5c, it is clear that the Zener transmission factor (2.37), follows the trend of the exact calculation in the band gap reasonably well. This observation suggests that working backwards from an experimentally determined transmission factor to infer the decay constant, making use of (2.37), might not lead to an unreasonably large error. The results of Fig. 2.5 have been obtained from a barrier that is only eight cells thick, corresponding to an overall barrier width of about 50 Å. A close approximation to the E vs. k relationship of the infinite solid is obtained: for this purpose, a very thin sample on the order of 10 to 15 wells is entirely adequate. As Leipold and Feuchtwang (1974) pointed out, while significant errors occur, even in the center of the gap, between the exact and the WKB results based upon the infinitesample k(E) relationship, the trend of the data is predicted remarkably well. Further, the major change in the transmission evaluated at 3.2 V in Figs. 2.5b and 2.5c upon application of 2.5 V bias is consistent with a reduction of the average barrier height. Finally, however, the authors point out that the presence of a disordered, rather than crystalline, barrier drastically alters the transmission coefficient as a function of energy. Armed with this information, we will adopt the point of view that the transmission in practical cases will be well approximated by a WKB factor in which the decay constant is that of the solid forming the insulator. We remain skeptical of the WKB result that there are no prefactors but repeat that prefactors, which are generally slowly varying functions of 1 1 wavevector, or E 2 , are likely to play an appreciable role only if k or E 2 goes to zero, i.e., if one considers tunneling from the bottom of a band in one of the materials. In adopting this practical point of view, we shall go further and point out in a very rough way the dependences of D upon E x , the kinetic energy normal to the barrier, and the bias voltage V . In doing

BA S I C J U N C T I O N T Y P E S

37

so, we find it convenient to introduce φ(x) as the potential barrier measured from the Fermi energy, φ(x) = U (x) − μ W = Ex − μ

(2.38)

and similarly to measure the x component of kinetic energy in excess of μ by the symbol W in (2.38). For rough estimates, the average value φ¯ of the barrier  1 x2 φ(x)d x (2.39) φ¯ = t x1 can be used to observe that the application of positive bias V , which lowers the right-hand Fermi energy, will decrease the average value of the potential barrier by eV/2, eV (2.40) 2 a result due to Sommerfeld and Bethe (1933). When this result is inserted into (2.2) and (2.3), the transmission factor for g = 1 becomes   1/2   eV 2m ¯ D = exp −2 −W t φ− (2.41) 2 2 φ(V ) = φ¯ −

The lowest-order expansion of the square root term involving the energies gives     W eV exp , eV, W  φ¯ D exp(−2κ0 t) exp 2E 0 E0 κ0 =

2m φ¯ 2

1 ¯ 2 φ¯ (2 φ/2m) = E0 = κ0 t t

(2.42)

(2.43)

¯ W  φ. ¯ For small variations in V and W , D which we emphasize is valid only for eV  φ, then is seen to vary exponentially on a scale set by the parameter E 0 , which is typically a tenth of a volt if φ¯ is 1 or 2 eV.

2.3 Basic junction types We will typically consider junctions in which at least one electrode consists of a free-electron metal with a well-developed Fermi surface corresponding to Fermi energy μ. Supposing, in Fig. 2.1a, the left member of the junction to be such a free-electron metal, biased negatively by voltage V with respect to the electrode on the right, we may take the current density J as a function of V to be given by (2.6) and (2.7). Referring to Fig. 2.1a: at T = 0, from simply an energy point of view, electrons lying between μ and μ − eV are eligible to tunnel in that only these face empty states in the right-hand electrode. Referring now to a k-space picture of the left electrode in Fig. 2.6, the k-vectors for the electrons eligible to tunnel to the right lie between spherical shells characterized by energies μ and μ − eV. Since we consider only electrons moving to the positive x direction, only half of the volume included between the spherical shells need be integrated over in computing the current density. Note that the transmission factor D is constant for constant values of E x and is maximum for E x = μ (or

T U N N E L I N G I N N O R M A L - S TAT E S T RU C T U R E S : I

38

(a)

EF

(b)

(EF − eV ) kmin

kmin ky

ky

kmax

kmax

kx

kx kz

kz

Fig. 2.6. Schematic shows k-space location of electrons eligible to tunnel from left to right in a junction under bias such as shown in Fig. 2.1. (a) States of constant tunneling probability for k x < 1

kmin = [(2m/2 )(μ1 − eV)] 2 . (b) States of constant tunneling probability for kmin < k x < kmax , 1

where kmax = [(2m/2 )μ1 ] 2 . (After Floyd and Walmsley, 1978.)

W = 0). It is thus convenient to integrate over disc-shaped regions in performing the k-space integration, as indicated by shading in Fig. 2.6. For E x < μ − eV, however, the surface of constant E x becomes an annulus, Fig. 2.6a, corresponding to k x < kmin = [(2m/2 )(μ1 − 1 eV)] 2 . This breakup of the region of integration into two parts is reflected in (2.44) for the current density as a function of V :  μ  μ−E x 2eρt J (V ) = D(E x , V )dE x dE t h 0 μ−eV   μ−eV  μ−E x (2.44) + D(E x , V )dE x dE t 0

μ−eV−E x

where one assumes T = 0 and has defined a two-dimensional density of states ρt by d2 kt = 2π kt dkt =

2π m t dE t = (2π )2 ρt dE t 2

(2.45)

In obtaining this expression from (2.6) and (2.7), we have used the group velocity factor to express k x in terms of E x , and the integral over transverse wavevector kt has been transformed into an integration over the corresponding kinetic energy E t in (2.45) by adoption of a density of states ρt for transverse motion. The transverse integrals are performed, leading to    μ−eV  μ 2eρt eV J (V ) = D(E x , V ) dE x + D(E x , V )(μ − E x ) dE x h 0 μ−eV (2.46) valid for T = 0. Inspection of this equation, in which the first bracketed term is proportional to the voltage V , indicates that the current-voltage relationship will be linear for small V , a result first obtained by Sommerfeld and Bethe (1933).

39

BA S I C J U N C T I O N T Y P E S

2.3.1 Metal–insulator–metal junctions Evolution from the very early work of Frenkel (1930), Sommerfeld and Bethe (1933), and Holm and Kirschstein (1935, 1939) from the free-electron model represented by (2.46) to the quantitative treatment of MIM tunneling has required careful consideration of the transmission factors D in (2.46), even within the WKB approximation, neglecting the image force, and consideration of the effects of temperature, which have been neglected in (2.46). Quantitative developments which we will mention are due to Stratton (1962), drawing upon earlier work of Murphy and Good (1956), and to Simmons (1963a,b,c, 1964), making use of earlier work by Holm (1951). The developments of Simmons and Stratton applied to the case of a symmetrical barrier, in extension of (2.46), both arrive at a cubic V 3 correction term to the basic linear dependence of current density on voltage: J (V ) = αV + γ V 3 + · · ·

(2.47a)

G(V ) = α + 3γ V 2 + · · ·

(2.47b)

Equation (2.47b) corresponds in the symmetric junction case to a parabolic conductance G(V ) symmetric about V = 0. We shall find later that a frequently observed modification of this background conductance G(V ) is a parabola displaced from V = 0 as a result of a barrier profile which is not symmetric. The offset parabolic dependence of G(V ) is also obtained by numerical methods working from (2.46). The approach of Simmons to obtaining (2.47) is based on the use of the average barrier φ¯ given by (2.39). The approximate transmission factor of Simmons et al. (1963b) is given as D(E x ) exp[−A(φ¯ − W )1/2 ]   1 1 4πβt (2m) 2 = (16πβ 2 t 2 ρt ) 2 A= h

(2.48)

which is defined apart from the correction factor β, which is usually near unity. The use of (2.48) in (2.46) leads to Simmons’ basic result for the current density: J = J0 {φ¯ exp(−Aφ¯ 2 ) − (φ¯ + eV ) exp[−A(φ¯ + eV ) 2 ]} e J0 = 2π h(βt)2 1

1

(2.49)

The dependence of (2.49) comes entirely from the limits of integration in (2.46); when (2.40) is used in (2.49), Simmons’ expression becomes e J (V ) = 2π h(βt)2 

− φ¯ +

!

eV 2



   4πβt eV 1/2 1/2 ¯ (2m) φ− exp − h 2 " 1  1 eV 2 4πβt ¯ 2 (2m) φ + exp − h 2

eV φ¯ − 2  



(2.50)

T U N N E L I N G I N N O R M A L - S TAT E S T RU C T U R E S : I

40

16

Conductance (arb. units)

14

12 Tin positive 10

8

6 V1 4

2

0

1.2

0.8

0.4

0

0.4

0.8

1.2

Bias (V)

Fig. 2.7. Typical experimental conductance of an Al–I–Sn tunnel junction at 4.2 K. For voltages greater than 0.4 V, the plot is symmetric about V = −0.125 V. Note that the conductance is relatively flat in the region between ±400 mV. The conductance has an offset in its minimum, and the variation is generally parabolic at low biases and becomes more rapid at higher biases. (After Rowell, 1969b.)

which, for low voltages, reduces to the form (2.47), where α and γ are given by

α=

1 1 (2m) 2 # e $2 1 φ¯ 2 exp(−Aφ¯ 2 ) t h

(Ae)2 γ Ae2 = − 1 α 96φ¯ 32φ¯ 2

(2.51)

in the case that β = 1 (Simmons, 1963a,b). Although the analysis of Simmons presented thus far is intended to accommodate asymmetric barriers, it incorrectly predicts for these cases a parabolic dependence of G upon V centered at V = 0. A typical observed conductance spectrum is illustrated in Fig. 2.7, showing a small offset from V = 0. The method of Stratton (1962) differs in its treatment of the penetration factor D in the evaluation of (2.46):

41

BA S I C J U N C T I O N T Y P E S

 ln D(W ) = −2

2m 2

1  2

x1

1

[φ1 (x, V ) − W ] 2 dx

(2.52)

x2

= −[b1 + c1 (−W ) + f 1 W 2 + · · · ]

(2.53)

with φ1 (x, V ) = φ(x) −

eVx . t

(2.54)

Equation (2.52) is a restatement of (2.2) and (2.3) for an arbitrary barrier function φ1 that includes the effect of the applied field by (2.54). Motivated by the fact that the largest contributions to the tunneling current come from electrons whose E x value is about μ, i.e., W = 0, Stratton, following Murphy and Good (1956); expanded the integral in (2.53) in powers of W . The voltage-dependent expansion coefficients b1 and c1 of (2.53) are given by   1 8m x21 (φ1 ) 2 d x = b10 − b11 V + b12 V 2 + · · · (2.55) b1 (V ) =  x11   2m x21 c1 (V ) = (φ1 )−1/2 d x = c10 − c11 V + c12 V 2 + · · · (2.56)  x11 These coefficients in principle must be evaluated by integration of the indicated powers of φ1 , a function of x and V . However, useful results have been obtained by expanding, e.g., b1 (V ) as b10 − b11 V , etc. The result of Stratton’s analysis is J (V ) =

4π me exp(−b1 ) [1 − exp(−c1 V )] h 3 c12

(2.57)

written for T = 0 and further expanded as J (V )

4π me exp(b11 V − b12 V 2 )[1 − exp(−c10 V )] 2 h 3 c10

(2.58)

and J (V )

& 4π me % 2 2 V +(b11 − 12 c10 )V 2 + ( 12 b11 − 12 b11 c10 + 16 c10 − b12 )V 3 + · · · 3 h c10 (2.59)

Contact between (2.59) and the expected behavior is made since the coefficient of the quadratic term in V in (2.59) is zero for a symmetric barrier, leaving, as required, linear and cubic terms in the eV dependence of the current density J . An advantage of the Stratton method is that it leads more directly to the expected temperature dependence of the tunneling current density J (V, T ), which is    ∞ 1 + exp(−W/kT) 4π mkT = D(E ) ln (2.60) dE x J (V, T ) = e x 1 + exp[(−W − V )/kT] h3 0

T U N N E L I N G I N N O R M A L - S TAT E S T RU C T U R E S : I

42

The logarithmic function in the integrand replaces the limits of integration in (2.46), thus generalizing that result for finite temperature T . This expression leads to a useful prediction, π c1 kT J (V, T ) = J (V, 0) sin(π c1 kT )

1 + 16 (π c1 kT )2 + · · ·

(2.61)

for the ratio of the current at a temperature T normalized to the current at the same voltage at T = 0; namely unity plus a small term in T 2 . This fact, which was first found by Murphy and Good (1956), has also been derived by Simmons (1964). Some refinements and comparisons between these two methods have been offered by Hartman (1964). To summarize briefly, as shown in Fig. 2.8, the results of the approximate analytic analyses of Simmons, Stratton and others: in the WKB model of tunneling between normal metal films, the current-voltage relationship is linear, with small cubic corrections at low voltage, and becomes exponential at high voltage. In the low-voltage region, the observed offset to 1.7

d = 15 Å

1.6

f=2V

Conductance (arb. units)

1.5 f12 = 3.5, 0.5 1.4 3.0, 1.0 1.3 2.5, 1.5 1.2 2.0, 2.0 1.1

1.0

200

100

0 V (mV)

100

200

Fig. 2.8. Calculated conductance vs. voltage curves for a metal–insulator–metal junction, illustrating the effect of different barrier heights on the two sides in producing the offset in the conductance minimum. The successive curves shown have been normalized to fixed conductance at V = 0 and have been shifted vertically for clarity. (After Brinkman et al., 1970.)

BA S I C J U N C T I O N T Y P E S

43

the parabolic dependence of the conductance of G(V ) is not obtained (it is obtained by exact numerical methods, as we will see), but the temperature dependence appears to be correctly predicted. Turning to the experimental side, one can say generally, as was observed by Fisher and Giaever (1961), that the I –V relationship is linear at low voltage and exponential at high voltage in tunneling through thermally grown oxides. On a quantitative basis, however, the early experiments were based upon thermally oxidized films, which are known to be amorphous and rather sensitive to preparation conditions. One straightforward source of difficulty is simply obtaining a good measurement of the thickness t of the oxide film that may be only 20 Å. The use of too large a value of t then could lead the researcher to an unreasonably small value of the mass (Stratton, 1962). Still, the general features, and particularly the temperature dependence, obtained by early workers (Hartman and Chivian, 1964; Pollack and Morris, 1965) left no doubt that tunneling was being observed. Because more extensive results involving single-crystal barriers, as in metal–semiconductor Schottky barriers and tunneling through thin-film crystal films, have since been put in quantitative agreement with the theory, it seems most reasonable to attribute discrepancies in the early thermally oxidized tunnel barrier literature to variations in the preparation of the films involved. The work of Giaever on superconductive tunneling, through the observation of the superconducting gap in tunnel junctions involving oxides of aluminum, tin, and lead made clear that, at least at 4 K, tunneling could be the only appreciable mechanism of charge transfer. In an accumulating body of evidence in the conductance spectra G(V ) of such junctions, the lower-voltage ranges, less than ±0.1 V, indicate the weak parabolic behavior of (2.47), as shown in Fig. 2.7. The important work of Brinkman et al. (1970) examines, within the framework of (2.46) but including trapezoidal barriers, the influence of the prefactor g, which we have thus far ignored; also, the image force corrections. This work has answered many of the questions left by the early analyses, most notably the origin of the offset minimum in the conductance. Figure 2.8, calculated numerically by using the WKB approximation with the prefactor ignored for a barrier of thickness 15 Å and average height φ¯ of 2 V but with varying degrees of asymmetry in the barrier parameters, demonstrates the offset of the parabolic minimum in conductance. These calculations have assumed the free electron mass and have ignored the image correction. The curves as plotted do not show the expected changes in conductance level which has here been normalized to the same value at V = 0. The barrier asymmetry has been included in the calculation by adopting (2.54) in which φ(x) is originally taken as a trapezoidal barrier with heights φ1 and φ2 on the two sides. Numerical calculations were extended to include an exact prefactor g, a generalization of (2.2b) to unequal barrier heights; comparison of the conductance curves is shown in Fig. 2.9, in which the lower curve is obtained with the prefactor. In both cases, the thickness is 15 Å, the trapezoidal heights ψ1 , ψ2 on the two sides are 1 and 3 V, respectively, and the Fermi energies μ are 10 V. It can be seen here that the exact prefactor (see Eq. 9 of Brinkman et al., 1970) has very little influence on the appearance of the conductance. While the conductance curves calculated in the previous two figures have the right shapes and magnitudes of offset to agree reasonably with experiment, the results have been obtained by using rather small values of thickness t, quoted as 15 Å, less than experimental estimates which range typically from 20 to 30 Å. Two plausible origins of this discrepancy are the utilization of the full electron mass rather than the possibly reduced effective mass in the barrier oxide (see Fig. 2.4) and neglect of the image force correction to the barrier potential, which might be appreciable for a thickness of only 15 Å.

T U N N E L I N G I N N O R M A L - S TAT E S T RU C T U R E S : I

44

1.5

d = 15 Å

1.4

f=2V I

Conductance (arb. units)

f1 = 3 V f2 = 1 V

1.3

EF1 = EF2 = 10 V 1.2

1.1

1.0 II 0.9 200

100

0

100

200

V (mV)

Fig. 2.9. Calculated conductance vs. voltage curves, using barrier parameters d = 15Å, ψ1 = 3 V, ψ2 = 1 V, with differing assumptions concerning the boundaries of the junction. In model I (upper curve), the boundaries are assumed to be diffuse (g = 1), whereas in model II, the boundaries are assumed to be sharp and the exact prefactor included in the calculation. The conductances are normalized at V = 0 and have been offset to show the curves more clearly. (After Brinkman et al., 1970.)

Inclusion of the image force for an electron tunneling into vacuum from a pure metal surface is essential to calculate the correct probability of field emission; and this effect was treated carefully in the early work (Sommerfeld and Bethe, 1933). The modification of the trapezoidal barrier φ(x) by the image force is a strong function of the assumed dielectric constant, as indicated in Fig. 2.10 (after Simmons, 1963a). The effect of the image force is to round the edges of the barrier and to slightly lower the average barrier height. The two φ(x) curves shown in Fig. 2.10 include, respectively, the exact image term and an approximation shown in the last term in (2.62): φ(x, V ) = φ1 + λ = e2 ln

x 1.15λt 2 (φ2 − eV − φ1 ) − t x(t − x)

(2.62)

2 8π εε0 t

(2.63)

45

BA S I C J U N C T I O N T Y P E S

Δf = 60/Ks

0

0.2s

Barrier width (Å) 0.4s 0.6s

0.8s

S

20/Ks

40/Ks

60/Ks

80/Ks

100/Ks

120/Ks

140/Ks

True barrier Approx. barrier

Fig. 2.10. The effect of the image force on a typical trapezoidal barrier profile. The rounding of the barrier which occurs very close to its edges is frequently ignored, its effects being absorbed in effective barrier height and thickness parameters. (After Simmons, 1963.)

This form was used by Brinkman et al. (1970) to obtain the conductance curve shown in Fig. 2.11, assuming a dielectric constant ε of 4.0. The effect of the image term is Is judged to be minor by these authors. The main changes are that the conductance offset is slightly reduced and that the rate of increase of conductance with voltage is made somewhat more steep. Hence, in most of their work and in the work of later researchers, the image term has been omitted. The use of WKB relations based on (2.46) is illustrated finally in Fig. 2.12, after Walmsley et al. (1977), who have actually fit data from Al–I–Pb tunnel junctions (solid line) with calculated values (dots) corresponding to a trapezoidal barrier, as indicated, with parameter values ψ1 = 1.57 V, ψ2 = 4.7 V, and t = 11.6Å; these parameters rather closely match the measured zero-bias conductance, given as 42.4mmho mm−2 . The first definitive measurement of the energy-dependent κ(E) from tunneling through a crystalline insulator was reported by Kurtin et al. (1970) (Fig. 2.13) in their study of GaSe. The barrier transmission was treated within the WKB approximation with no prefactors, as given in (2.12). The temperature dependence of the Fermi function was included, and the integration over transverse k was done by a method of steepest descents (Kurtin et al., 1971). The key to this work was development of a technique for peeling single-crystal films of the layer compound GaSe down to thicknesses on the order of 100 Å. A series of independent measurements established not only the thickness but also the individual barrier heights φ1 and φ2 , specifying the trapezoidal barrier profile as in (2.62), neglecting the image correction. Figure 2.13a shows a configuration of a Cu–GaSe–Au tunnel junction whose thickness is known to be 83 Å. These measurements established that the band gap of the GaSe is 2.0 eV,

5.0

δ = 20 Å f1 = 1.0 V f2 = 3.0 V

Conductance (arb. units)

4.0

3.0

2.0

1.0

400

300

200

100

0

100

200

300

400

V (mV)

Fig. 2.11. Calculated conductance for metal–insulator–metal junction having thickness d = 20Å, with ψ1 = 1 V, ψ2 = 3 V, and including the effect of the image force. The image force tends to round the edges of the barrier, increasing the overall transmission. Inclusion of the image force slightly reduces the asymmetry of the conductance and leads to a conductance that increases more rapidly at higher bias. (After Brinkman et al., 1970.) 4.0

Normalized conductance

f2 f1

3.0

d 2.0

1.0 −800

0 Bias (mV)

+800

Fig. 2.12. The solid line is the measured conductance of an Al–Al2 O3 –Pb tunnel junction, while the points are obtained from calculation, assuming sharp interfaces and the following parameters: ψ1 = 1.57 V, ψ2 = 4.07 V, and d = 11.6Å. The calculation accurately reproduces the numerical value of the junction conductance at V = 0. (After Walmsley et al., 1977.)

47

BA S I C J U N C T I O N T Y P E S

(a)

(b) 2.0

10 t = 83 Å 0

10

10

−2

10

−3

10

−4

Cu

Au

1.0

0.5

Experiment Theory VCu < 0

VCu > 0 10−5

0

0.2

0.4

E (eV)

−1

2

J (A / cm )

1.5 10

0.6 0 V (V)

0.2

0.4

0.6

0.03

0.01

0.02 2

−k (Å)

0

0

−2

Fig. 2.13. (a) Current density J as a function of applied voltage V for a tunnel junction having a singlecrystal GaSe barrier layer of 83 Å thickness. The electrodes in this case are copper and gold. The plotted points are experimentally determined, while the solid line is calculated from the theory, using the deduced E vs. k relationship. (b) Inferred E vs. k 2 relation for the band gap of GaSe. The evident agreement between theory and experiment confirms that the displayed E(k) relation is an accurately determined property of GaSe. Note that the E(κ) is determined in the gap from the valence band edge to 1.5 V above this energy. In the region near the valence band edge, the result is consistent with an effective mass m ∗ /m e = 0.07. (After Kurtin et al., 1971.)

the low-frequency dielectric constant ε = 8, and the optical dielectric constant ε = 7, while internal photoemission measurements determined separately the interfacial barrier heights on the copper and gold sides, as measured from the valence band of the GaSe, to be 0.68 and 0.52 eV, respectively. A further important assumption is necessary to obtain the purely trapezoidal profile, namely the lack of band bending, as could result from Debye charge layers at the interfaces. This assumption is confirmed by a low value of about 3 × 1014 cm−3 of trapped- and free-carrier density in the specimen. The dielectric constant and the rather large thickness may be taken as justification of the neglect of the image correction and adoption of the simple trapezoidal barrier shown. Figure 2.13a shows the dependence upon bias of both signs of the current measured at 77 K, verified to differ only slightly from the values at room temperature. As tunneling occurs, a trajectory through the forbidden gap of the GaSe is followed, which corresponds to a constant energy relative to the external Fermi levels but a changing energy with respect to the band edge of the GaSe. This trajectory is obviously substantially changed by the application of biases ranging from ±0.6 V. Thus, a substantial portion of the GaSe energy gap from near the valence band edge to approximately 1.5 V above the valence band edge has been sampled in measurements interpreted in terms of (2.2a) and (2.2c) and leading finally to the dependence shown in Fig. 2.13b for the variation of −k 2 = κ 2 as a function of energy in GaSe at low energies (i.e., near the valence band edge). The limiting slope of this plot is constant and corresponds to the valence band effective mass m ∗ = 0.07.

48

T U N N E L I N G I N N O R M A L - S TAT E S T RU C T U R E S : I

2.3.2 Metal–insulator–semiconductor junctions The characteristic feature of a semimetal, a small value of the Fermi degeneracy μ, should bring into an easily measurable range of bias voltages the condition when the bottom of the conduction band crosses the Fermi energy of the opposite electrode. At eV = μ, one would expect to see a noticeable change in the tunneling conductance, because beyond this bias no further change occurs in the number of electrons able to tunnel. Further change in the conductance for eV > μ could be attributable to the change in the barrier shape alone. An estimate of the expected behavior can be obtained by the use of (2.46), which we have rewritten as    μ  μ−eV 2eρt (2.64) g D(E x )dE x + D(E x )g(E x + eV − μ)dE x eV J= h 0 μ If we neglect the voltage dependence of D, then the first term in (2.64) will vary smoothly through the condition eV = μ and hence will contribute only a constant to the derivative dJ/dV . Therefore, we concentrate on the second term in (2.64) and examine dJ/dV near μ = eV, which gives 2e2 ρt dJ = dV h



μ−eV μ

g D(E x )dE x + smaller terms

(2.65)

because the integrand evaluated at the upper limit vanishes, and we are neglecting the voltage dependence of D. The band edge behavior in the conductance g(E) then is estimated by assuming a simple form (2.66) gD(E x ) = D0 E xn exp

Ex E0

(2.66)

for the transmission factor. In this equation the possibility of a prefactor is included by the factor E xn . Examination of the V -dependence of (2.65) reveals, in addition to smoothly varying terms, a term which disappears at the band edge eV = μ as the first power of μ − eV (taking g = 1) and quadratically in μ − eV, assuming g is proportional to E x . Thus, for eV μ and n = 0, 1, one has G(V )

2e2 ρt D0 (μ − eV)n+1 θ (μ − eV) + other terms h

(2.67)

The presence of the prefactor, linear in E x , will make the variation of the conductance rather smooth and the eV = μ effect difficult to observe in the presence of the other smoothly varying terms. This result is consistent with the early conclusion of Harrison (1961) that band structure effects in normal metal tunneling would be difficult to observe and is also consistent with the rather weak effects which have been seen in tunneling studies of bismuth and Bi–Sb alloys (Chu et al., 1975). Stronger effects related to band structure are observable when different bands have significantly different transmission factors, as in semiconductors in comparison of zone center and zone boundary conduction bands, or in transition metals comparing wavefunctions of predominantly s vs. predominantly d character.

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BA S I C J U N C T I O N T Y P E S

2.3.3 Schottky barrier junctions The direct metal contact to a heavily doped semiconductor produces a tunnel junction, as illustrated in Fig. 2.14, which is both rather easy to make and susceptible to detailed characterization. The barrier in this case is the single-crystal semiconductor depletion region. Charge from this region transfers to the metal or to the interface states, leaving a uniform space charge, i.e., a region in which the electrostatic potential varies quadratically with distance. The thickness of this region can be controlled by the concentration of donor or acceptor impurities in the semiconductor. Schottky barrier junctions can be made by cleavage of a semiconductor crystal in the presence of an evaporating metal (e.g., Wolf and Compton, 1969), producing an abrupt and contamination-free interface between metal and crystalline semiconductor. In tunneling studies made at low temperatures, it is important that the semiconductor have sufficient doping density to lie on the metallic side of the Mott transition, so that, as indicated in Fig. 2.14, a metallic Fermi energy μ typically in the range 10–100 mV is obtained. This condition is required in practice to ensure that the applied voltage drops across the depletion region barrier and not across the bulk of the semiconductor crystal, especially at 4 K where the resistance of a nonmetallic semiconductor becomes exponentially large. A complicating feature of Schottky barrier contacts is that as the free-carrier concentration in the semiconductor is increased with impurity concentration N D , the thickness of the barrier is reduced −1

proportionally to N D 2 (n-type case), with a consequent decrease in the resistance level of experimental devices. The resistance can also be adjusted to some extent by choosing different contact metals to vary the barrier height U B , which also affects the zero-bias thickness of the barrier. A second complication is that the thickness of the Schottky barrier is a function of bias voltage. In the n-type case shown in Fig. 2.14, at a bias V equal to the barrier height U B , the thickness of the barrier is zero. The tunneling problem posed by the parabolic depletion layer barrier has been solved exactly (Conley et al., 1966), including the effect on the conductance of the bias variation of barrier thickness. The conductance spectra of Schottky barrier contacts also permit determiE

UB

μF

e eV μF n+

EC

Fig. 2.14. Schematic band diagram of Schottky barrier contact to a degenerate (metallic) n-type semiconductor. Conventional forward bias in metal–semiconductor contacts is that sign which reduces the height of the barrier seen from the semiconductor.

T U N N E L I N G I N N O R M A L - S TAT E S T RU C T U R E S : I

50

nation of the band edge (i.e., the Fermi degeneracy μ) referred to in the previous section. The conductance G(V ) falls to a minimum at a bias eV equal to μ, as shown (in resistance) in the calculated dV /dI curves for Schottky barrier contact on n-type germanium (Fig. 2.15). In this figure, the arrows locate the μ values calculated independently from the known properties of the semiconductor crystals. The calculation of Conley, Duke, Mahan, and Tiemann (1966) (CDMT) is based upon matching of exact eigenfunctions in the metal, the semiconductor barrier, and the semiconductor bulk, i.e., following (2.7) in which D is obtained from the exact eigenfunctions and hence includes the proper prefactor. The barrier potential ψ(x) used by Conley et al. is e2 N D (d − x)2 + eV − μ, 2ε   2ε(U B + μ − V ) 1/2 d= eN D

φ(x) =

0 |4, 2 > and |2, 7 > (see text). (After Crommie et al., 1993.)

Q UA N T U M C O R R A L S

427

of Schrodinger’s equation for a particle in a hard-wall container. The details of the situation are not all understood, but the basic picture is that the electrons are confined to the surface by a strong intrinsic energy barrier in the perpendicular direction arising from the Cu band structure at the (111) surface, and the strong scattering of these 2-D electrons by the Fe atoms. The circular standing-wave patterns are caused by interference of incident and scattered surface state electrons, an example of the textbook problem of an electron trapped in a round twodimensional box. The eigenstates inside are given as ψn,l (r, ϕ) = c Jl (kn,l r )eilϕ , where Jl is the lth order Bessel function, l is the angular momentum quantum number, c is a constant, ϕ the azimuthal 2 /2m ∗ . These states are referred to as ψ (r, φ) = |n, l>, angle, and the energy is E n,l = 2 kn,l n,l and the density of states, fitted to the height of the ripples seen in the topograph, is proportional 2 . to ψn,l The data in Fig 9.9b show that the apparent height oscillations can be well fit by linear combinations of the expected eigenstates within a circular hard-wall box. The spectroscopy dI /dV was also measured, for example, at the center of the box, and the peak energies were in agreement with |n, 0 > energies of the model, consistent with the fact that nonzero l eigenstates have nodes at the center of the ring. Repeating the dI /dV measurement 9 Å away from the center of the ring showed new peaks corresponding to expected energies for nonzero angular momentum values l. The principal discrepancy with the hard-wall model, assuming a fit value of effective mass, was the line width, corresponding to a lifetime for the trapped carrier of about 3 × 10−14 s. This rather short lifetime was not understood. Crommie et al. suggested mechanisms, including scattering from the surface into the bulk, scattering of electrons past the assumed hard-wall boundary, or possibly inelastic scattering for this rather short lifetime. Elliptical corrals of Co atoms on Cu(111) have been constructed by Manoharan et al. (2000) as described in Fig. 9.10. The property of the ellipse is to focus rays emanating from one focus onto the other focus. Co atoms are magnetic, as are Fe atoms, and have been found to have a Kondo temperature of 53 K on the Cu(111) surface, higher than that of Fe. The Kondo scattering and resulting magnetically screened state for T < TK , which will be more fully described below, induces an increased electron density of states in the vicinity of the Co atom. A Kondo impurity is a magnetic moment, which moment is screened by slight adjustments in the locations of a cloud of electron moments nearby. The results are to screen, for T < TK , the magnetic moment of the Co as seen from a large distance, and to enhance the local density of opposite-spin electrons near the impurity. The process is similar in principle to Debye screening of a charged impurity in a plasma, but in this case the screened quantity is magnetic moment rather than a charge. The length scale of this screening is measured as about 10 Å by STS, an analog of the Debye length. The Kondo effect produces a “Fano resonance” anomaly (for Co a deep sharp depression) in the tunneling dI /dV , offset by 1 meV from zero bias (see Fig. 9.10c, left panel), which we will describe in more detail later. This effect offers more flexibility in the appearance of images, by choice of the bias point V . In the topographs of Fig. 9.10a, the images of the Co atoms taken at V = 5 mV appear as 0.8 Å protrusions.

9.3.1 Elliptical corrals and focusing effects: quantum mirage The remarkable observation of Manoharan et al. (2000) was that a Co impurity placed at one focus of an elliptical corral would induce a peak in the density of states at the other focus, even

428

S C A N N I N G T U N N E L I N G S P E C T RO S C O P Y ( S T S )

(a)

(b) Fig. 9.10. (a) Topograph of quantum ellipse of Co atoms with one Co at the left focus. Co is found to have TK = 53 K on this Cu(111) surface and as such induces a peak in the density of electronic states nearby. (b) Density-of-states mapping of identical region, with bias voltage set to peak of Kondo resonance anomaly induced under the Co atom on the left, but evidently refocused by the quantum container to the right focus. (Comparison with Fig. 9.10c, left panel, shows that the bright points in Fig. 9.10b are basically points of reduced conductance.) This is evidence that a contribution to the STM spectroscopy of the magnetic adatom Kondo resonance comes from the metal conduction electrons, separate from the d-states of the magnetic atom. (After Manoharan et al., 2000.) (c) (Left panel) Voltage scan of density of states above Co atom at left focus (solid). Dashed, tip displaced from atom. (Right panel) Same measurements (note dI/dV scale change) at right focus, where there is no atom. A densityof-states dip is observed, although considerably weakened, as if there were a Co atom, suggesting that the electronic correlations induced by the Co atom, rather than the atom itself, are a noticeable source in the STS observation of the Kondo resonance of magnetic adatoms. (After Manoharan et al., 2000.)

with no atom at the second focus. The word “quantum mirage” was applied to this situation. The electron correlations produced at one focus reappear at the other focus, suggesting that a portion of the STS observation of the Kondo impurity is via its effect on the nearby electrons, with little or no contribution from the magnetic atom itself. The focusing property of the ellipse is that the path from one focus to the other, via specular reflection from the boundary, has the same path length and propagation time for all paths. While this effect, depending on 2-D surface electrons, certainly exists, we will see later evidence that the larger part of the Kondo anomaly actually arises from 3-D bulk electrons under the magnetic atom. The Kondo

PA I R - B R E A K I N G S I N G L E A DATO M S O N S U P E R C O N D U C TO R S

(c)

429

(d) Left focus: atom

4

Right focus: no atom

0.5 Off focus Dx = +5 Å

Off focus Dx = –5 Å 0.4

dl / dV (a.u.)

3

0.3 2 0.2

1 0.1 On focus Dx = 0 Å

On focus Dx = 0 Å 0 –20

–10

0

10

20

0.0 –20

–10

Sample bias (mV)

0

10

20

Sample bias (mV)

Fig. 9.10. (continued)

resonance effect and TK for Co are found to be larger on the Cu(100) surface, with no surface state, than on the Cu(111) surface. Manoharan et al. (2000) found that changing the corral atoms from Co to nonmagnetic CO molecules did not change the essential features of their remarkable observations. They also confirmed that nonmagnetic S atoms or CO molecules at the left focus did not produce the mirage effect at the right focus. They speculated that the spin correlations induced under the magnetic Co are indeed replicated at the second focus.

9.4 Pair-breaking single adatoms on superconductors Single atoms have been observed to have dramatic spectral effects when deposited on superconducting surfaces. In a conventional phonon-coupled superconductor with singlet pairing, a magnetic impurity is a pair breaker because it can flip one spin, turning the singlet into a triplet configuration. While these effects have been inferred from long-standing indirect measurements, the effects have recently been strikingly confirmed at the single atom level by Ji et al. (2008). These workers have deposited single magnetic atoms, Mn and Cr, on the conventional superconductor Pb, as illustrated in Fig. 9.11 (see also Chapter 1, Fig. 1.12). The dI /dV spectra have been taken at 0.4 K and the energy resolution is enhanced by using a superconducting Nb tip. The spectra in Fig. 9.11c focus on Cr adsorbates (upper curve), far away from any atom; arrows locate peaks attributed to delocalized impurity bands within the superconducting gap! The coherence length in Pb is about 30 nm.

S C A N N I N G T U N N E L I N G S P E C T RO S C O P Y ( S T S )

430

(a)

(c) 5Å

24 22 20

Pb

Cr dimer I

(b) 5Å

Normalized dI/dV

18 16 Cr atom

14 12 10

Cr dimer I

8 6 4

Cr dimer II

2 Cr dimer II

0 –6

HI

LO

–4

–2

0 2 Bias (mV)

4

6

Fig. 9.11. (a) Topographic image of Cr dimer I with 6.6-Å interatomic distance. (b) The topographic image of Cr dimer II with 3.0-Å interatomic distance. (c) Typical dI /dV spectra taken on Pb (away from adsorbates), an isolated Cr atom, Cr dimer I, and Cr dimer II. The curves are offset vertically for clarity. All STS were acquired at a set point of V = 10 mV and I = 0.2 nA at 0.4 K. (After Ji et al., 2008.)

9.4.1 Mn and Cr on Pb The second curve in Fig. 9.11c is taken on a Cr atom, and reveals three separate intragap states in the electron channel (positive bias, below the main peak) and similarly in the hole channel (negative bias, above the main peak). These peaks occur at 0.17, 0.53, and 1.03 meV relative to the main peaks, and are assigned, respectively, to scattering events having angular momenta l = 0, 1, and 2. While the positions of the electron-like and hole-like resonances are symmetric with respect to the zero bias, their amplitudes are asymmetric as a result of the broken symmetry under the particle-hole transformation (Ji et al., 2008). Evidently, these peaks arise from the interaction between the magnetic atoms and the superconducting host, since they disappear completely when a 3-T magnetic field was applied perpendicularly to quench the superconductivity in the Pb film. The multiple peak structure has its origin in the different angular momentum channels, (l = 0, 1, 2 . . . ) in the scattering and is directly related to the magnetic moment of the impurity. It is inferred that partial waves beyond l = 0 are involved in these peaks. Turning to the spectra of the distant and near-neighbor dimers of Cr (3rd and 4th curves in Fig. 9.11c), it is seen that the distant-dimer spectrum is similar to that of the single Cr, while the close-spaced dimer gives no pair-breaking effect; the spectrum is identical to the curve on the bare Pb surface. The interpretation is that the near-neighbor

PA I R - B R E A K I N G S I N G L E A DATO M S O N S U P E R C O N D U C TO R S

431

dimer atoms are bound antiferromagnetically, which makes the dimer act as a nonmagnetic impurity with the total spin zero. The bound state induced by a nonmagnetic impurity on a conventional s-wave superconductor such as Pb lies essentially at the gap edge with much smaller amplitude and cannot be resolved by STS.

9.4.2 Zn impurity atoms imaged in cuprate planes A different but analogous situation occurs in the cuprate superconductors, where a nonmagnetic impurity serves as pair-breaker of the triplet pair. Figure 9.12 shows STS spectra taken near to a Zn impurity in BSCCO, showing a sharp near-zero bias, not present when the tip is away from the Zn impurity. The intense quasiparticle scattering resonance is at about −1.5 meV. The Zn impurity is not imaged directly because it is two layers below the surface, where it substitutes for Cu. The peak location centers a region of about 15-Å radius in which the superconducting peaks are strongly suppressed. More detailed imaging studies, below, show a four-fold symmetric quasiparticle “cloud” aligned with the nodes of the d-wave superconducting gap function.

2.5 Zn

Differential Conductarce (nS)

2.0

away from impurity On center of Zn atom

30Å b

1.5

a 1.0

0.5 a

0.0 –100

–50

0 Sample Bias (mV)

50

100

Fig. 9.12. Spectroscopy above a Zn atom in BSCCO cuprate superconductor with Zn doping 0.6% and Tc = 84 K. Inset shows lattice arrangement, orientation of the triplet pair wavefunction, and location of single Zn impurity. (After Pan et al., 2000a, Balatsky et al., 2006, and Fischer et al., 2006.)

432

S C A N N I N G T U N N E L I N G S P E C T RO S C O P Y ( S T S )

9.5 Spectroscopy of Kondo and spin-flip scattering This topic is not heavily based on imaging capability, but is a strong example of the use of tunneling to determine the energy dependence of electron scattering processes. Two quite different situations lead to rather similar results.

9.5.1 Introduction The resistivity of metals containing small amounts of magnetic impurities contained a puzzling − log T anomaly in the resistivity: ρ ∝ − ln(T ). This a small but strongly temperature dependent addition to the elastic scattering component, conventionally presented as ρ = m(1/τ )/ne2 , where the scattering rate is (1/τ ) = v F /λ. (We can associate this with the term g1 in Eq. 8.39.) The additional temperature-dependent scattering was explained straightforwardly by Kondo (1964), who used perturbation theory and invoked the interaction of the conduction electron spin with the localized spin on the magnetic moment, −2J s · σ. Kondo found two extra terms scattering the electron from the localized spin: one corresponds to spin-flip scattering, which we will call g2 . Kondo’s famous discovery was of the g3 anomalous term, proportional to − ln{[ε2 + (k B T )2 ]/D 2 }, where ε = E − E F (see Eq. 8.35). The physics of this anomaly arises from a virtual intermediate state involving flipped conduction electron and localized spins, described above in connection with Eq. (8.34). These two Kondo terms are scattering processes in transport that increase the resistivity. Because of the spins, a magnetic field H has a strong effect on these terms, introducing a threshold energy gμ B H . In the context of tunneling spectroscopy, it has been learned that magnetic moments at the edge of a tunneling barrier can scatter electrons across the barrier, as assisted tunneling processes that add to the tunnel conductance. The details were presented by Appelbaum (1967) and Appelbaum and Shen (1972). The beauty of the tunneling experiment is that the energy dependence of the processes, including the magnetic field splitting, can be seen. We emphasize that the experiments we will now describe are forward scattering or assisted tunneling experiments in which the tunneling electron is initially on one metal electrode, scatters elaborately from the moment, and goes to a well-defined state in the other electrode. The final state of the electron is not on the magnetic moment. The concept is that additional channels for the tunneling transfer are opened up with the interactions, similar in concept to the additions in conductance seen when tunneling occurs simultaneously with excitation of a vibration on an interposed molecule. (See Fig. 1.9.) A much larger g-shift, −2J N (0) = −2Jρ, is well known (Yosida, 1957), see Eq. (8.46), in the circumstance of the localized donor electron coupled with conduction electrons by −2J S · σ. It is perhaps less well known (Korringa, 1950) that the coupling also produces a level broadening (see Eq. 8.47). As is discussed in connection with Eq. (8.49) above, the formation of the localized moments, based on hydrogenic donor states, is predicted by the Anderson model (Anderson, 1961, 1966), which inherently gives the coupling J needed for Kondo scattering, via the Schrieffer–Wolff transformation (Schrieffer and Wolff, 1966). The observed g-shift for As moments in Si is about −0.8 (Wolf and Losee, 1970); a g-shift −0.1 has more recently been reported in beautiful spectroscopic measurements of Kondo scattering on single trapped electrons by Ralph and Buhrman (1994). The numerical values of Jρ, controlling the g-shift, of course, depend on the physical situation of the magnetic moment.

S P E C T RO S C O P Y O F KO N D O A N D S P I N - F L I P S C AT T E R I N G

433

9.5.2 Kondo spectroscopy of a single trapped electron Ralph and Buhrman (1994) (RB) used an elegant nano-fabrication method (see RB, Fig. 2, not shown here, and see also Ralph et al., 1995) which is summarized in Fig. 3.52. A nanoconstriction is built by evaporations onto both sides of a thin insulating membrane through which a nanometer-size hole had been drilled by electron beam lithography (Appendix A). Here RB deposit Cu metal on both sides, producing a lithographically engineered constriction of diameter about 3 nm. Their study of the resulting devices inspired the model sketched in Fig. 9.13. Data which surely indicate Kondo scattering are shown in Fig. 9.14. The RB data in Fig. 9.14 show an assisted Kondo conductance about 0.5 e2 / h, consistent with the maximum of 2e2 / h, for a Kondo moment. The data in Fig. 9.14 are extracted from RB Fig. 2, where they represent a small fraction of an elastic background conductance. RB Fig. 2 also shows a − ln T dependence of the peak, showing that TK is less than 50 mK. These data (Fig. 9.14) have excellent spectral resolution, consistent with a base temperature of 50 mK, resolution 25 times better than was available in earlier work presented below for comparison (Wolf and Losee, 1970) (WL). The data here show clearly the anomalous Kondo scattering conductance peak and its magnetic field splitting. RB infer g = 1.9, or a g-shift −0.1, a coupling parameter Jρ = −0.05 from the magnetic field splitting. RB conclude that their data are qualitatively but not satisfactorily described by the Appelbaum extension of Kondo perturbation theory, mentioned above. RB state that they do not find in their data evidence for the Korringa broadening which WL identify as an inherent feature deriving from the g-shift and coupling parameter. In this connection, since the broadening scales as (Jρ)2 , the predicted broadening in their case would be reduced relative to WL by a factor (0.41/0.05)2 = 67. This factor is larger than the compensating factor-25 change in temperature, which might make difficult identification of Korringa broadening in the data of RB.

eV/2

EF ~

e0

Fig. 9.13. Sketched model of assumed single trapped electron spin in resonant tunnel barrier produced in a metal constriction. The single trapped spin is analyzed in this work via the Anderson (1961, 1966) model and the Schrieffer and Wolff (1966) transformation. (Schrieffer and Wolff map the Anderson parameters to the “s–d exchange coupling” −2J S · σ, which was shown by Kondo (1964) to lead to the anomalous scattering peak.) The authors (RB) identify situations in which purely resonant tunneling occurs, and other situations, of interest here, in which Kondo and spin-flip assisted tunneling is observed, with clear spectroscopic signatures in the voltage- and magnetic field-dependences of the tunnel conductance. (After Ralph and Buhrman, 1994.)

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0.4

0 Teslo

0.2

0.0 0.4

0.85 Teslo

ΔG(V) (e2 / h)

0.2

0.0 0.4

1.70 Teslo

0.2

0.0 0.4

2.55 Tesla

0.2

0.0 –1.0

–0.5

0.0 V (mV)

0.5

1.0

Fig. 9.14. Tunnel conductance measurements (points) at low temperature in micro-constriction producing a single trapped spin as shown in the previous Fig. 9.13. The authors interpret these data within the Schrieffer and Wolff (1966) transformation of the Anderson (1961, 1966) localization model, following some aspects of the Appelbaum (1966) and Appelbaum and Shen (1972) extensions of Kondo (1964) scattering theory. The data surely show the anomalous Kondo scattering peak, and the field splitting is interpreted by the authors as indicating a g = 1.9, thus Jρ = −0.05. The data suggest a large ratio of anomalous Kondo scattering (g3 ) to spin-flip scattering (g2 ). Solid curves are models as described in the RB paper. The calibration of the conductance in units of the conductance quantum, e2 / h, make credible the assumption that a single electron spin produces the observed features. (After Ralph and Buhrman, 1994.)

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The RB data wonderfully show the Kondo peak and its detailed splitting with field. The conductance scale in units of G 0 suggests at the outset that a single channel conductor might be observed, not a tunnel junction. However, RB establish that their data do reflect assisted tunneling. Ostensibly similar constrictions (however produced differently) in ferromagnetic Fe, Ni, and Co (Calvo et al., 2009) fail to show the anomalous peak, but show material-dependent line shapes that are parameterized in a Fano model, with energy measured in units of kT K , making use of the dimensionless Fano free parameter q. Calvo et al. suggest that the width of the main V = 0 resonance, of variable line shape (q) is set by TK . The sharpness of the RB conductance peak, in light of the discussion of Calvo et al. (2009), would suggest a low TK for the RB device. This is consistent with RB’s estimate of 30 mK. Calvo et al. suggest TK values 90 K, 120 K, and 280 K, for their one-atom-constricted wires of ferromagnetic Fe, Co, and Ni, respectively. These authors briefly discuss calculations of the size of the magnetic moments in various cases, to suggest a wide range of possible values. One of the features which is not clearly addressed by RB is the ratio of the Kondo conductance to the expected spin-flip conductance, but surely this ratio is larger in the RB data than in the WL data, which we now summarize.

9.5.3 Spectroscopy of localized moments in Si:As Schottky junctions The localized moments in MIM junctions are thought to originate typically from the unpaired d-electrons of isolated transition metal ions. In contrast, the moments in the Schottky barrier case are reasonably believed to occur inherently by electron localization at the Mott transition, which occurs in a narrow range at the inner edge of the depletion region (see Fig. 8.26). The Schottky barrier conductance peak has been studied extensively in junctions formed by cleaving single crystals in high vacuum in the presence of rapidly evaporating metal, so that an atomically clean metal–semiconductor contact is formed. The bulk semiconductor is metallic with the Fermi degeneracy of about 20 meV, but it forms at the surface a depletion barrier about 0.85 eV and about 84 Å in thickness. The silicon crystals are typically free of transition metal impurities at the level of 1 ppm. The donor atoms are typically As or P that have five valence electrons, and give up four of these, which covalently bond into the silicon lattice. The remaining electron is weakly bound in a hydrogenic fashion by the additional positive charge on the donor nucleus. It is, perhaps, an underappreciated beauty of semiconductor physics that shallow donor properties can be quantitatively scaled from the hydrogen atom. Such an impurity inherently provides an unpaired electron spin magnetic moment (Wolf, 2006, p. 145 in Wolf, 2009). Because of the large permittivity and small effective mass, the Bohr radius scales to a ∗ ∼ 20 Å and the binding energy becomes 53 meV for arsenic. The localized electron spin is unpaired, producing a paramagnetic moment, and unquestionably can give Kondo scattering. Electron scattering experiments have shown that the cross section σx for spin exchange collisions with a conduction electron at low temperature is about 10−12 cm2 . This cross section is large and close to that obtained by scaling the corresponding result for the hydrogen atom. The g-value of the isolated donor electron spin has been measured as 2.00. More precisely, of course, the g-value for the free electron is well known to be shifted by 0.00232 by interac-

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tion with vacuum fluctuations. The larger shifts predicted by Yosida (1957) with coupling Jρ have been mentioned above. Discussion of this type of junction has been given in Section 8.3.6. The background elastic conductance for the junction (shown in WL, Fig. 15) is well understood within the theory of Conley et al. (1966), and the assisted conductance is a small addition, ∼ 10%. On this basis, the situation is suitable for analysis by the perturbation theory approach of Appelbaum (1967). The considerable broadening of the features in Fig. 9.15 are attributed, first, to the relatively high temperature and, second, to the large Korringa broadening implied by the large g-shift and large coupling constant (Jρ = −0.41). The experimental curves in Fig. 9.15 are semiquantitatively in agreement with the Appelbaum extension of Kondo’s theory as displayed in Fig. 24a of the WL paper. The ratio of the spin-flip to anomalous terms has been adjusted to fit, and seems larger than the theory would suggest.

9.5.4 Comparison of the two Kondo spectroscopy experiments Both RB and WL experiments show clearly the Kondo peak spectroscopy including its splitting with applied H field. The RB experiment indicates a single conductance channel, one single spin, with conductance directly as 0.5 units of the conductance quantum, e2 / h = 1/(25.78 k). The assisted conductance in WL is 0.13 of the background conductance ∼ 0.095 S, thus about 0.0124 S, or about 319 conductance quanta. The WL paper argues (p. 3676, 2nd column) that the density of eligible moments in the “reserve region” of the barrier is ∼ 0.8 × 1011 /cm2 , which, based on the junction area, gives 1.9 × 107 moments. From this point of view, only a small fraction, 1.6 × 10−5 , of the estimated eligible localized moments in WL actively conduct. More appropriately for a tunnel junction, this factor should be interpreted as a barrier transmission probability. In the RB experimental data, on the other hand, there is no discernible spin-flip conductance. Somehow the experiment sees only the anomalous scattering term g3 , with no perceptible broadening in the field-split peaks. As RB state, comparison of their data with Appelbaum’s theory (1967) is unsuccessful, confirmed by absence in the data of spin-flip assisted conductance. Further, inspection of the actual splitting of the data peaks in the 2.55-T data (lowest panel, Fig. 9.14), suggests to the present author a g-value scarcely less than 2.00. This would lower the estimate of coupling parameter Jρ, to make the Korringa broadening too small to see, consistent with the data, and to remove any conflict with the WL data. Finally, the RB data credibly involve a single electron spin, while the WL data certainly involve an ensemble of spins, undoubtedly with a spread of parameter values, likely adding to the broadening of features. Both of these experiments, containing isolated electron spin moments, likely within the purview of Kondo’s theory, do indeed show his anomalous peak and its splitting with magnetic field. We turn now to the STM spectroscopy experiments, which beautifully show magnetic adatoms and a Kondo–Fano resonance phenomenon rather than the anomalous Kondo scattering peak.

9.6 STM spectroscopy of magnetic adatoms Madhavan et al. (1998) (see also Li et al., 1998; Manoharan et al., 2000) deposit isolated Fe adatoms on a clean metal surface, image the individual atoms, and measure the STM

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10 Au|Si : As H⊥I

8

b

6

c

4

ΔG(e)/G (%)

1.25°K

a

d

2

e

0 f –2 h –4

i

–6 –8 –10

a c

ΔG(e)/G (%)

4

4.2°K

d

2

e 0

f

–2

g

–4

h i

–6 –3

–2

–1

0

1

2

3

Bias mV

Fig. 9.15. Kondo scattering spectroscopy. Measured tunnel conductance curves for Au Schottky barrier tunnel junction on Si:As at 1.25 K and at 4.2 K.The data are symmetrized around V = 0 and a background subtracted. Curves labeled a to i correspond to magnetic fields H = 0, 1.88, 3.75, 5.63, 7.5, 9.38, 11.25, 13.12, and 15 T. Rather similar theoretical curves for the 1.25K data are shown in Fig. 24a of the WL paper, with parameter values g = 1.18(Jρ = −0.41) and Korringa broadening  = π(Jρ)2 (gμ B H + k B T ), following Kondo’s paper (1964) as adapted to tunnel spectroscopy by Appelbaum (1967) and Appelbaum and Shen (1972). In fitting the shape of the H = 0 Kondo peak, (Eq. 8.35), k B T was replaced by 2.12 k B T and D fit as 6 meV. In the fit to these data (not shown, see WL, Fig. 24a) the ratio of spin-flip to Kondo scattering g2 /g3 was larger than the theory predicts. (After Wolf and Losee, 1970.)

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12 Å

2.5

9Å 5Å 2.0 dl / dV (arbitrary units)

3Å 2Å 1.5 0Å –3 Å 1.0

–4 Å –7 Å –10 Å

0.5

–12 Å 0.0 –100

–50

0

50

100

Sample voltage (mV)

Fig. 9.16. STM conductance data taken on a line through the location of a single magnetic Co adatom on an atomically clean Au(111) surface. The characteristic Fano (1961) line shape is seen, with no trace of a Kondo peak. (After Madhavan et al., 1998.)

conductance at locations directly above and displaced laterally from the adatom. The spectra are shown in Fig. 9.16. The data curves as seen can be fit quantitatively with a modified version of the Fano theory (Fano, 1961; Plihal and Gadzuk, 2001). Co as a dilute impurity in Au is known as a Kondo system, as briefly described in Section 9.6 above. The Kondo temperatures for the adatoms always exceed the Kondo temperature TK , so the magnetic atom has formed a Kondo state. This is a polarized electron cloud surrounding the local moment producing a many-body singlet state. The spin-polarized electrons are attracted to the moment, to raise the local electron density, and to screen the magnetic moment from the bulk. The characteristic energy for the system is k B TK = k B T0 exp(− 12 Jρ). The effect is similar in principle to Debye screening of a bare charge in a plasma. The size of the polarization cloud is measurable by STM spectroscopy, and seems about 0.5 nm from inspection of Fig. 9.16. The adatom situation here differs from the situations in the RB and WL experiments, where the measurements were made above the relevant Kondo temperatures TK and in which the role of the moment was scatterer, not final state.

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The title of the Madhavan et al. (1998) paper is “Tunneling into a single magnetic atom: Spectroscopic evidence of a Kondo resonance.” The atom and the locally perturbed substrate, in some combination, form the final state for tunnel transitions. The calculation shown in Fig. 9.2 is a zero-order reminder of conduction electron density being pulled into an adatom. The measurements of Madhavan et al. are not taken in an assisted tunneling situation. More precisely (see Ujsaghy et al., 2000; Plihal et al., 2001a), an interference occurs between two tunneling channels, one direct channel through the resonance εK localized on the magnetic impurity and a second channel into the conduction band of the substrate. [The conduction electrons in cases including Cu(111) and Au(111) are surface-state electrons, but scattering of the surface-state electrons can occur into the three-dimensional electronic states of the metal.] Ujsaghy et al. (2000) chose to neglect direct tunneling into the d- or f-level of Co or Ce, as too deeply localized in the atomic core, and to focus on the effect of the d- or f-state on the local density of states. Justification for this assumption is given by the experiment of Manoharan et al. (2000), in which the conduction local density of states, modified by a Kondo atom in one focus of an elliptic quantum corral, was spatially separated from the d-level by imaging it into the other focus, where it still did show a Fano line shape (see Fig. 9.10a,b,c above). As a practical matter, the line shapes in this and following experiments are well fit to the original Fano expression, which, in the present context, is d I /d V ∝ (q + ε)2 /(1 + ε2 ).

(9.2)

Here, ε = (eV − εK )/k B TK , q is the dimensionless Fano line-shape parameter, and εK is the energy of the Kondo resonance, which may be shifted slightly from zero. This line shape (9.2) is a negative Lorentzian function for q = 0, a positive Lorentzian for large ±q, and is most asymmetric when q = ±1. It is clear that the width of the feature scales with TK . The possible role of the surface state has been addressed by Knorr et al. (2002), who studied Co on Cu(111), which is a surface state, and on Cu(100), which is not a surface state. This distinction is made clear in Fig. 9.8 above, in which Friedel oscillations signal the surface state. The STS curves of Knorr et al. (2002) for the two cases are shown in Fig. 9.17. A major change in the line shape is evident by switching surfaces, and the fit values for q are shown in Table 1 of Knorr et al. (2002). Knorr et al. (2002) conclude that the change in q comes simply from the difference in the local density of states (at the Co) in the two surfaces, rather than from the presence/absence of the surface state. These authors fit their data to find that the Kondo temperature scales linearly with the host electron density at the magnetic impurity. The change in DOS can be directly traced to the larger number (4) of nearest Cu neighbors to the Co on the (100) surface, than (3) on the (111) surface. Quantitative analysis of the tunneling spectra indicates that the Kondo resonance is dominated by the Cu bulk electrons, whose local density differs in the two cases. It was remarked that if the resonance were to arise entirely in a two-dimensional system, the lateral decay distance would surely be quite different from the 3-D case, no such difference is seen. More recently, Neel et al. (2007) have studied the change in the spectrum as the tip, held directly above Co on Cu(100), moves in from a tunneling regime to a contact regime, shown in Fig. 9.18.

S C A N N I N G T U N N E L I N G S P E C T RO S C O P Y ( S T S )

440

Co/Cu(111)

dl/dV (normalized)

1.3

1.2

1.1

1.0 Co/Cu(100) –40

–20

0 bias (mV)

20

40

Fig. 9.17. Atom differential conductance dI /dV spectra for Co/Cu(100) (x points) and Co/Cu(111) (solid dots). The solid curves fitted to these normalized curves are the Fano line shape (see text) and fit parameters are shown in Table 9.1. In moving the Co from the (100) to the (111) face, one sees a large change in shape (Fano q goes from 1.13 to 0.018, with TK going from 88 K to 53 K). [The lateral decay distance (see Fig. 4 of Knorr et al., 2002), about 0.5 nm, however, is almost unchanged, which means the nature of the relevant electron states is not changed, i.e., both must be bulk states.] The differences in these two spectra come only from the larger number of nearest-neighbor Cu atoms, 4 for the Co on the 4-fold hollow site of Cu(100) vs. 3 for the Co on the 3-fold hollow site of Cu(111). (After Knorr et al., 2002.) Table 9.1. Mean Fano line-shape parameters and Kondo temperatures TK from fits of Eq. (9.1) to scanning tunneling spectra of ten different Co adatoms on Cu(100) and Cu(111). n is the number of nearest-neighbor Cu atoms

TK [K] n q E [meV]

Co/Cu(111)

Co/Cu(100)

Co in bulk

54 ± 2 53 ± 5[6] 3 0.18 ± 0.03 1.8 ± 0.6

88 ± 4

∼ 500[9]

4 1.13 ± 0.06 −1.3 ± 0.4

12

These authors find that the distinct change in resonance dI /dV characteristic upon tip– atom contact, and reaching the conductance quantum value numerically, is a broadening, modeled as an increase in TK from 78 K to 137 K. These authors find that the change in q-value from 1.2 to 2.1 arises by a slight shift of the d-band level in the atom due to the proximity of the tunneling tip. These authors report detailed numerical calculations of the densities of spin-up and spin-down d-states on the Co and how they shift (changing q) with approach of the tunneling tip. It appears that a sound understanding of the problem of the magnetic adatom has been achieved.

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Dz (Å)

Fig. 9.18. (a) Variation of tunnel conductance of tip directly above Co adatom shows distinct transition from tunneling regime (right portion, characterized by a 3.5-eV barrier) to atom contact (left portion). Insets show detail of transition from contact to tunneling regimes and rendering of topograph of the Co atom. (b) Abrupt broadening of the Fano-type resonance curve occurs between curves 3 and 4, changing fit value TK from 78 K to 137 K and Fano q from 1.2 to 2.1. (After Neel et al., 2007.)

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442

(a)

low

(c)

(b)

high

low

high

(d) H

Side

Cu

C

C

H Cu

Top

Fig. 9.19. (A) STM image of a C2 H2 molecule on the Cu(100) surface at 8 K. Acetylene appears as a dumbbell-shaped depression on the surface with a maximum depth of 0.23 Å. The stable, clean metal tips necessary to perform IETS (inelastic electron tunneling spectroscopy) on single molecules rarely yielded atomic resolution of the Cu(100) surface. The imaged area, 2.5 nm square, was scanned at a sample bias of 100 mV and tunneling current 10 nA. (B) The molecule as in (A) was transferred to the tip by means of a voltage pulse (0.6 V, 100 nA, 1 s) and the same area was scanned. The atomicresolution image has a corrugation of 0.009 Å. This corrugation is sensitive to the nature of the tip and the tunneling parameters. Copper atom spacing is 2.55 Å. The image was scanned at a sample bias of 10 mV and a tunneling current of 10 nA. (C) The atoms in (B) were fitted to a lattice. The lattice is here shown on top of the image (A). (D) Schematic drawing showing side and top views of the molecule’s orientation and suggested adsorption site. The adsorption-site determination assumes that the transfer of the C2 H2 molecule to the tip did not change the position of the tip’s outermost atom. The dashed line shows the outline of the STM image shape. The dumbbell-shaped depression in STM images may result from π bonding to the Cu atoms perpendicular to the C–C axis, reducing the local density of states for tunneling. This would cause the axis of the dumbbell shape to be perpendicular to the plane of the molecule. (After Stipe et al., 1998.) This figure is reproduced in colour in the colour plate section.

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443

The basic problem of STS of the magnetic adatom now seems solved. Still, there may be an opportunity for perhaps more exciting, temperature- and magnetic field-dependent spectra in engineered experiments where the magnetic moment is controllably isolated from the conduction electrons, e.g., by encasing moments in a small molecule.

9.7 Molecules and their vibrational spectra The image in Fig. 9.20 resulted from several steps, working with a W tip above an Ag 110 surface initially containing 5 Fe adatoms and 5 deposited CO molecules. The Fe atoms image as protrusions and the CO molecules image as depressions. The W tip positioned above a CO will pick the CO up if the tip voltage is increased to 250 mV and the tunnel current ramped from 0.1 nA to 10 nA. The tip then carries the CO above an Fe atom, which resides at a 4-fold hollow site on Ag 110. The Fe–CO bond is then formed by adjusting the tip voltage to −70 mV and ramping the current to 10 nA and then ramping the bias to −4 mV. This forms the FeCO molecule and leaves the W tungsten tip bare. An additional step forms the species Fe(CO)2 . Thus is demonstrated chemistry at the single atom level. Not shown are vibrational spectra d2 I /dV 2 showing peaks on the order of 20 nA/V2 at voltages in the range of 220 mV to 280 mV arising from C–O stretch vibrations of FeCO and Fe(CO)2 . The authors further identify isotopic variations (12 C,13 C,16 O,18 O) of the CO vibrations in these molecular species. Finally, the authors in Fig. 9.21 show improved spectral resolution of the same scanned area as in Fig. 9.20, by picking up a CO molecule onto the tip and scanning at 22 mV bias and 2.5 nA tunnel current.

Fe CO

FeCO

Fe(CO)2

Fig. 9.20. 6.3-nm square image on Ag 110 surface obtained with 70 mV bias, 0.1 nA tunnel current at 13 K. This image shows Fe and CO species deposited and manipulated using a W tip on a silver 110 surface. On the surface, initially 5 single Fe atoms and 5 CO molecules were deposited at 13 K. (After Lee and Ho, 1999.) This figure is reproduced in colour in the colour plate section.

Fig. 9.21. Image obtained with CO tip, of the same configuration as in Fig. 9.19. Now the Cu lattice is resolved, and the Fe(CO)2 molecule is resolved; 2.5 nm square, 22 mV bias, and 2.5 nA current. Using CO as a tip all species appear as protrusions. The FeCO image looks similar to the Fe(CO)2 image because the measurement induces frequent flips of the FeCO molecule. This image is not a standard topograph but rather an image of dz/dy, where y is the scan direction (top to bottom), which enhances resolution of features. (After Lee and Ho, 1999.)

(b)

(a)

Height (Å)

0.4 0.3 0.2 0.1 0.0 0 Low

High

(c)

4

8 12 Distance (Å)

16

(d)

[110]

Low

High

[001]

Fig. 9.22. A single O2 molecule is bonded to Ag(110), 2.5-nm square area. (a) Topograph with bare tip at 70 mV bias and 1 nA current. (b) Cross sections of (a) taken along [001] (solid) and [110] (dashed) directions. (c) Atomically resolved STM topograph obtained with CO-terminated tip at 70 mV sample bias and 1 nA tunnel current. The larger circles are protruding oxygen atoms and the smaller perpendicular circles are imaged depressions. Grid lines are drawn through the surface Ag atoms. (d) Schematic diagram in which the larger circles are silver, the smaller circles are oxygen atoms. This site is referred to as the four-fold hollow site on the Ag(110) surface. (After Hahn et al., 2000.)

M O L E C U L E S A N D T H E I R V I B R AT I O NA L S P E C T R A

(a)

(b)

445

(c)

Tip

C6H5l

C6H5

(d)

C6H5

l (e)

l

C6H5

2C6H5

(f)

C12H10

Fig. 9.23. Schematic of single-molecule STM organic chemistry on a copper surface. The reaction is the copper-catalyzed transformation of iodobenzene C6 H5 I to make C12 H10 . This reaction is known as the Ullman reaction, here carried out at 20 K, with the usual thermal activation energy replaced by pulses of energy from the tip. The first pulse is applied in panel (a) to dissociate the I, leaving a phenyl group, C6 H5 . The second pulse is applied in panel (e) to cause two adjacent phenyl molecules to bind. In panels (c), (d), and (f) the tip is used to move a molecule. (After Hla et al., 2000.)

Inelastic electron tunneling spectroscopy measurements (see Fig. 1.13) indicate O–O stretch vibrations at 82.0 (76.6) meV for 16 O2 (18 O2 ) in this chemisorbed state. The feature is anomalous in representing a decrease in conductance at the vibrational threshold, and is referred to as resonant inelastic electron tunneling. These vibrational energies are lower than for oxygen molecules in vacuum as a consequence of electron transfer between molecule and metal surface. Chemical reactions have been carried out step by step on single crystal surfaces of Cu by Hla (2000), as shown in Fig. 9.22. The upper and lower panels show molecules lined up along a mono-atomic step on the surface, which offers stronger bonding than does a flat surface. The starting molecules are iodine-substituted benzene rings, known as iodobenzene. In the first action by the tip, a pulse breaks off the iodine, leaving a phenyl ring. The tip is then used to pull the iodine out of the way, and to pull two phenyl rings closer together. The second tip action provides energy to allow the two adjacent phenyl rings to fuse and form C12 H10 , biphenyl, the desired product. Finally, in panel (f), pulling on one end of the biphenyl is demonstrated to move the whole molecule.

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This experiment shows a new use of the STM tip, in adding energy, either to dissociate a molecule into parts or to cause two molecules to react, to overcome a barrier to the reaction. In this experiment, a prepared Cu surface was employed, cut at a slight angle to produce monoatomic steps. Step-edge locations offer stronger binding to an atom or molecule, because it is attracted to two surfaces, rather than to one.

10 Scanning tunneling spectroscopy of superconducting cuprates and magnetic manganites In this chapter, we pay particular attention to the role of imaging in advancing understanding of electron-coupled superconductivity. The development of a variable temperature lattice tracking high-resolution STM instrument has been important in this area. Using such an instrument the same atom can be observed from 300 K down to cryogenic temperature. High-resolution topographs and spectral images have become susceptible to Fourier analysis (Fourier-transform scanning tunneling spectroscopy, FT-STS) to make comparison with k-space information more directly available from angle-resolved photoemission measurements.

10.1 Gap imaging of optimally doped cuprates The state of the art in imaging gap properties has advanced to now allow imaging of the same atomic region over a wide temperature range. This rather recent advance has been important. On the one hand it confirms in general terms the early indications of “inherent inhomogeneities in the tunneling spectra of Bi2 Sr2 CaCu2 O8−x crystals in the superconducting state” typified by the work of Howald et al. (2001). Second and more important it has recently lent strong weight to the origin of the superconductive pairing from the properties of the underlying normal state. A thorough review of gap imaging results through 2004 has been given by Fischer et al. (2006). We therefore focus on results since that time.

10.1.1 Site dependence of apparent gap Bi2 Sr2 CaCu2 O8+δ crystals have been studied recently in a variable temperature lattice tracking instrument by Pasupathy et al. (2008), based on slightly earlier instruments described in Appendix B.2.2. These samples were cleaved in ultrahigh vacuum at low temperature and scanned at high resolution, producing large data files that were searched to provide statistics on various properties of interest. Some of the information is reproduced in Fig. 10.1. The inset to panel (C) in this figure shows a typical pairing-gap map (300 Å square) for an optimally doped Bi2212 sample at 40 K. Equations similar to (7.1) are fitted to local spectra to extract parameters such as pairing gap, mapped in inset. The field of view in the inset is 30 nm × 30 nm; it is immediately seen that the gap values vary substantially on a length 10% or less of this field, so on a scale of 3 nm the values change drastically. Figure 10.1c gives a histogram of fitted gap values, and then shows spectra, with complete temperature dependence typical of regions having gap values as indicated by the arrows. 447

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S U P E R C O N D U C T I N G C U P R AT E S A N D M AG N E T I C M A N G A N I T E S (a)

dI/dV (pS)

150

60 K 70 K 80 K 90 K 95 K 100 K 105 K

100

50

0 –200

OPT-D –100

0 Voltage (mV)

100

200

(b) 60 K 70 K 80 K 90 K 95 K 100 K 105 K

dI/dV (pS)

150

100

50

OPT-E 0 –200

(c)

–100

0 Voltage (mV)

100

200

0.25

Probability (D)

0.2

0.15 35 OPT-D

0.1

D (mV)

80

OPT-E

0.05

0

0

20

40 60 D (mV)

80

100

Fig. 10.1. (a) and (b) Spectra taken at two different atomic locations on optimally doped sample (Tc = 93 K, OPT) at various temperatures. These spectra remain voltage- and temperature-dependent well above the resistive Tc . Hole-like excitations, at negative bias, exceed in strength electron-like excitations at positive bias. (c) Histogram of gap values observed in the OPT sample, indicating locations OPT-D and OPT-E. Inset shows typical pairing gap map (300 Å square) for an OPT sample at 40K. Equations similar to (7.2) are fitted to local spectra to extract parameters such as pairing gap, mapped in inset. (After Pasupathy et al., 2008.)

It is difficult to fault this experiment, nor the straightforward analysis. The more useful approach seems to be to think of its implications and to see how these measurements can describe a system whose specific-heat jump, for example, is reasonably conventional, as described by Loram et al. (2004) and Loram and Tallon (2009). In their data analysis, Pasupathy et al. (2008) have defined the gap operationally as the voltage of a peak in dI /dV at positive bias. A wide range of points in the k space contribute to these data,

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449

measured as a function of position along the surface of the sample. Strong variations in the conductance curves remain at both of the exemplary locations, at 105 K, higher than the resistive Tc = 93 K of the optimally doped sample. In another report of superconductivity above the resistive Tc , Wen et al. (2009) have measured specific heat in underdoped Bi2 Lax CuO6+δ (Bi2201) crystals and find evidence for residual superconductivity in the normal state far above Tc .

10.1.2 Overdoped case Turning to measurements on an overdoped sample of Tc = 62 K, data in Fig. 10.2 reveal a strong anticorrelation between the local gap strength (panel C) and the local zero-bias conductance (panel B). Locations with large gap are likely to have a small value of dI /dV . The correlations evident in Fig. 10.2 are quantified in Fig. 10.3. Curves of panel (a) show that the local gap and the local density of states at zero bias both have correlation lengths between 10 and 20 Å. These are short correlation lengths, but are not greatly different from the coherence length 15 Å quoted for the ab plane.

10.1.3 Anticorrelation of gap and zero-bias density of states The surprising, and clear, result is strong anticorrelation of the local gap strength and the local density of states at V = 0. A related striking relation between the local gap strength and the voltage position of the maximum dI /dV in negative bias is shown in Fig. 10.3c. The authors conclude from these data that the strength of pairing is controlled by properties of the normal state of the material. The dip seen beyond the basic coherence peak does not occur in a BCS superconductor, but does occur in a variety of forms of phonon-coupled superconductor, where it has been successfully interpreted by Eliashberg theory and used to extract the effective spectrum of phonons, α 2 F(ω) and the related electron–phonon coupling parameter, λ. A summary of the dip feature at positive and at negative bias is provided in Fig. 10.4b. It is seen that the strength of the feature is similar at all locations. In particular, in the right panel (c) it is shown that there is no correlation between the strength of the dip feature and the local pairing gap. This is incompatible with the idea that the dip is a measure of interaction strength that underlies the pairing, in which case there would surely be a positive correlation between the strength of the feature and the local gap. This is well known from long experience with the Eliashberg theory as a tool for fitting the dip feature and extracting consistent gap and Tc values.

10.1.4 Internal proximity effect Observation of a proximity effect in evolution of superconductivity in overdoped Bi2212 has been reported by Parker et al. (2010). A large data set obtained on a 2.5 nm × 2.8 nm region in a sample of resistive Tc = 65 K has been analyzed to provide maps of gap and local pairing temperature for measurement temperatures between 50 K and 76 K. The authors find evidence that nanoscale regions distinguished by inhomogeneity can influence each other by the proximity effect. A set of respresentative dI /dV curves is shown in Fig. 10.5. The authors demonstrate the influence of nearby regions on the temperature at which a local gap closes. In the following, a circled region is identified as a region of weak

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(a)

1100

dI/dV (pS)

800

400

offset 0 –400

–200

0

200

400

Voltage (mV)

(b)

160

T = 93 K, normal state dI / dV (V = 0)

290

dI / dV (pS)

(c)

25

T = 50 K, superconducting gap map

40

Δ (mV)

Fig. 10.2. Spatial variations and anticorrelations in overdoped Bi2212 sample, p = 0.24, Tc = 62 K. (a) At 94 K all positions along 250-Å line show normal-state dI /dV curves (offset for clarity) with strong electron-hole asymmetry and noticeable variation in voltage position of peak (hump) at negative bias. (b) Map of zero-bias conductance at 93 K, scale range is 160 pS to 290 pS. (c) Map of local superconducting gap on the same 250-Å square, taken at 50 K, values ranging from 25 mV to 40 mV. This gap map is strongly anticorrelated (see Fig. 10.3) with normal conductance map in (b), but shows no correlation with mapped strength of boson-like dips in conductance, as seen, e.g., in Fig. 10.1a, at ±80 mV. The authors state that this rules out boson coupling as origin of pairing, but ties pairing to properties of normal state. (After Pasupathy et al., 2008.) This figure is reproduced in colour in the colour plate section.

superconductivity, and a square region (to the right in Fig. 10.6b) is identified as a region of strong superconductivity. Within each region are localities of similar gap value, as measured at 50 K. What is demonstrated is that these regions of similar local gap at 50 K see gap closure at noticeably different temperatures, evidently influenced by their proximity to differing neighbors. As seen in the two panels of Fig. 10.7, taken at 62 K and 66 K, respectively, superconductivity has completely disappeared in the (circled) weak superconducting environment, while it is still thriving in the (square) strongly superconducting environment. The authors go forward to quantify the proximity effect they have observed.

G A P I M AG I N G O F O P T I M A L LY D O P E D C U P R AT E S

(a)

451

1

Correlation

0.5

0

–0.5

–1

0

20

40

60

Distance (Å)

(b)

–160

Hump position (mV)

–180

–200

–220

–240

–260

–280 15

20

25

30

35

40

Δ (mV)

Fig. 10.3. (a) Upper curve: angle-averaged autocorrelation of the 50 K gap map, Fig. 10.2c. Middle curve, autocorrelation of the 93 K zero-bias conductance map Fig. 10.2b. Lower curve: Cross correlation of zero-bias conductance map and gap map. Gap and conductance are strongly anticorrelated (0.75). All correlation lengths are about 15 Å. (b) Relation of hump feature position (Fig. 10.2a, negative-bias region) to size of gap 0 . (After Pasupathy et al., 2008.)

The authors define an “excess pairing temperature,” δT p = T p – < T p >, where < T p >is the average value of T p for gaps with the same size. An image and a correlation plot relating to the excess pairing temperature are shown, respectively, in the left and right panels of Fig. 10.8. The uppermost curve in Fig. 10.8 (right panel) shows that the correlation length for the transition temperature is larger than that for the gap value. The lowest curve, dash-dotted in this figure, has a peak about 1 nm, which indicates that local values of pairing temperature are significantly affected by gaps in a 1-nm environment.

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S U P E R C O N D U C T I N G C U P R AT E S A N D M AG N E T I C M A N G A N I T E S (a) 1.6

1.4

R

1.4

R

1 1.2

0.8

0

V–Δ (mV)

80

1

0.8 0 (b) 0.08

100

V–Δ (mV)

RMS deviation

0.06

0.04

0.02

10

15

20 25 Δ (mV)

30

35

Fig. 10.4. The strength of the boson-like dip feature in the conductance does not correlate with gap strength. (a) The positive-bias conductance ratios are referenced to the local gap at different locations, showing that the magnitude of the dip–hump feature is similar in all locations. (Inset shows negative-bias behavior.) (b) The rms deviation of the conductance ratios from both bias directions from the d-wave model over the energy range from 20 to 120 mV bias. No correlation is seen between the magnitude of the deviations and the size of the gap. (After Pasupathy et al., 2008.)

The authors note that other indications of an inhomogenous superconducting transition in Bi2212 have come from Nernst measurements (Wang et al., 2006) and from muon spin resonance experiments (Sonier et al., 2008). It is not clear whether measurements such as these allow prediction of the resistive transition of the material. A discussion was provided at the end of Chapter 7 on limits on transition sharpness that are provided by specificheat measurements. The present authors suggest that their work contributes to research that might allow interface superconductivity to occur at temperatures higher than available in bulk materials.

10.2 Localized state at Zn impurity One of the most striking images resulting from STS in cuprate superconductors is that obtained of the localized state in the gap at the site of a pair-breaking Zn impurity in

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Δ (mV)

20

86 K

Differential Conductance (pS)

200

80 K

0

74 K

150

10

0

50 100 Temp. (K)

50

100

66 K 62 K 50 K

100

30 K

50

0 –100

–50

0 Bias (mV)

Fig. 10.5. Representative dI /dV spectra at the same point at different temperatures for overdoped Bi2212 sample of resistive Tc 65 K, using lattice-tracking variable temperature STS technique. The curves are fit by d-wave formula similar to (7.1). Local gap has closed by 78 K. Large data set from 2.5 nm × 2.8 nm region is searched, local gap is assigned as voltage of maximum in dI /dV for positive bias; see inset. Local pairing temperature T p is defined as the highest temperature for which a maximum in dI /dV can be observed for positive bias. (After Parker et al., 2010.) (a)

(b) 50 K

5 nm >76 K

Tp

40mV

Δ

, where < T p > is the average value of T p for gaps with the same size. This is the same area as represented in Figs. 10.6 and 10.7. Bright regions in this plot have excess pairing temperature up to 10 K with respect to dark regions.(Right panel) Correlation functions on the 2.5 nm × 2.8 nm region. Upper solid curve, correlation function for pairing temperature T p . Middle dashed curve, correlation function for pairing gap . Lower dash-dot curve, cross correlation between excess pairing temperature δT p = T p – < T p > and pairing gap . Peak in this curve near 1 nm means that values of T p are significantly affected by gap values 1 nm away, which is attributed to a proximity effect. Note that the autocorrelation of the pairing temperature is greater than that for the gap for lengths larger than about 1 nm, and is nearly twice as large as that for the gap on scales between 2 nm and 6 nm. The pairing temperature distribution is smoother than the gap distribution. (After Parker et al., 2010.)

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(a) 9

N (r,Ω0)

6

3

0 –2 –1 x/

(b)

ξ0

0 1

N (r,–Ω0)

1 2

6

0

–1

–2

y / ξ0

3

0 –2

–1 x/ ξ0

0 1

1 2

0

–1

–2

y / ξ0

Fig. 10.9. The local density of states N (r, ω) of a d-wave superconductor for (a) ω = 0 and (b)ω = –0 as a function of position r = (x, y) around a repulsive impurity (c = 0.1) placed at r = 0. Note streaks extending to corners of the plane. The impurity resonance energy is taken as 0 = 0.03 0 and ξ0 /l F = 6. The maximum values of N (r, ±0 ) are approximately 8.3 and 2.9 (in units of N F ). The shaded profile shows the projection of N (r, ±0 ) on the vertical plane. (After Salkola et al., 1996.)

It appears that the data of Pan et al. (2000a) and the theory of Salkola et al. (1996) mentioned above are in striking agreement. From their measurements, 0 = –1.5 meV, 0 = 45 meV, Pan et al. estimate the strength parameter c (see above) and from this estimate the phase shift as 0.48π, close to the unitarity limit 0.5π . The image in Fig. 10.11 is consistent with a disk of obliterated superconductivity of dimension about 15 Å, described by others as a “Swiss cheese” model (Basov et al., 1998; see also Uemura, 2004). Figure 10.11 affords a direct measure of the coherence length, in agreement with the correlation lengths that were found by Parker et al. (2010) above.

10.3 Model for spectral distortions of noncuprate layers In performing STS to determine properties of the cuprate layer, as in the previous sections, the BiO and SrO2 layers intervening between the tip and the cuprate layer in Bi2212 have for the most part successfully been ignored, as if they were homogeneous regions in which

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2.5

Differential conduct (nS)

2.0

1.5

1.0

0.5

0.0 –200

–100

0 Sample bias (mV)

100

200

Fig. 10.10. Two tunneling spectra from nearby locations Zn-BSCCO crystal. The spectrum marked by arrows is taken with tip laterally displaced from location of Zn impurity. The spectrum with prominent peak at V = –1.5 mV is obtained with tip situated just above Zn impurity, at exactly the center of a bright scattering site. It shows both the intense scattering resonance and that the suppression of both of the superconducting coherence peaks. (After Pan et al., 2000a.)

the wavefunction from the tunneling tip decays. Topographs indeed image the BiO layer, and reveal distortions that we will describe below. In doing spectroscopy, since the scale of these layers is the same as the scale of the cuprate layer, one cannot exclude spectral distortions in the measured properties from the discrete nature of the intervening layers. Fortunately, these effects, which have been addressed by Yang et al. (2006) seem not to be large. A case in which a distortion effect has been identified is the Ca2−x Nax CuO2 Cl2 (NaCCOC) cuprate superconductor which has shown a “checkerboard” charge-ordering pattern (Hanaguri et al., 2004; Shen et al., 2005; Kohsaka et al., 2008). A related discussion for Bi2212 has been given by Hashimoto et al. (2006). For further details of the barrier tunneling process as it applies to this situation, the reader is referred to the excellent review of Fischer et al. (2007).

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1.00 nS 60 Å 0.10 nS

0.01 nS

30 Å b

a

Fig. 10.11. Spectral image of the resonant peak; dI /dV image (note logarithmic intensity scale) is taken at V = –1.5 mV (see Fig. 10.10). Note that in this image (see geometry in lower panel) the nodes of the d-wavefunction are aligned along x, y directions. Correspondingly, the brightest four nearest-neighbor atoms are along nodal directions. The weaker outer features, including the 3- nm long “quasiparticle beams” are oriented along the gap maxima, at 45◦ angles in this picture. (After Pan et al., 2000a.) This figure is reproduced in colour in the colour plate section.

10.4 Superlattice modulation in Bi2212 A feature of the Bi2212 surface is an incommensurate modulation of the surface height, described in Fig. 10.13, which has been shown recently to produce a small corresponding modulation in the strength of the local superconducting gap. A mismatch in dimensions of the BiO and CuO2 planes causes Bi2212 to have a superstructure modification that is incommensurate with the CuO2 lattice. The modulation, which

STM

C1 3pz

Ca(Na)

O 2py Cu 3dx2–y2 Fig. 10.12. Schematic demonstration of the interference of two possible tunneling paths from the STM tip to the Cu–O–Cu hybrid state in Ca2−x Nax CuO2 Cl2 when the tip is midway between two Cl atoms. (After Yang et al., 2009.)

c a λSM = 26Å ≈ 4.8 unit cells

Fig. 10.13. (a) Bi2212 surface showing three superlattice peaks running horizontally, “x” in this topograph locates Zn impurity in Fig. 10.11. Imaged is Bi atom; Zn atom is directly beneath. (After Pan et al., 2000a.) (b) Sketch of superlattice modulation in Bi2212. c-axis labeled in (b) is perpendicular to page in (a); scale for wavelength 26 Å approximately agrees. (After Slezak et al., 2008.)

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S U P E R C O N D U C T I N G C U P R AT E S A N D M AG N E T I C M A N G A N I T E S

has wavelength 26 Å oriented at 45◦ to the Cu–O bond direction, slightly affects the local superconducting gap, as was shown by Slezak et al. (2008). The smaller the vertical spacing the larger the local gap, and the important spacing is believed to be the spacing between the CuO2 plane and the apical oxygen atom above that plane. This is consistent with the report of Eisaki et al. (2004) that disorder near the apical oxygen site was deleterious to superconductivity; see Fig. 7.12 and related text.

10.5 Fourier-transform STS (FT-STS) and application Fourier-transform studies of atomic-scale spatial modulations in the density of states were first carried out on Bi2212 by McElroy et al. (2003). Interference patterns in the density of states on two-dimensional metal surfaces were shown in Chapter 9 and by Hoffman et al. (2002b) on Bi2212. In this technique, dI /dV = g(r, ω), with ω = eV, is spatially mapped at each bias voltage, V . The resulting maps may show interference patterns with characteristic wavelengths λ corresponding to wavevectors q = 2π/λ. In the FT-STS method, the characteristic q values are found by taking a Fourier-transform g(q, ω) of the spatial map g(r, ω). Characteristic scattering vectors q in superconductors like Bi2212 connect regions of high |∇ k E(k)|−1 , which are shown as circles. Contours of constant quasiparticle energy are banana shaped, as shown in Fig. 10.14a. Because each “banana” exhibits its largest rate of increase with energy |∇ k E(k)|−1 near its two ends, one can imagine that the largest contributions arise at the eight locations, at the ends of the four Fermi arcs, as circled in Fig. 10.14a. Applying this method to their data on Bi2212, McElroy et al. (2003) were able to identify locations of the Fermi arcs in the Brillouin zone, which compared well with locations observed by angle-resolved photoemission (ARPES), in Fig. 10.15b. The first step in the process is to map the energy associated with the main scattering vectors, as shown in the left panel of Fig. 10.15a. The authors were able to go on from the data shown in Fig. 10.15 to construct a plot of the pairing gap strength as a function of the angle ϕ in the k x , k y plane such that the nodal line (zero gap) occurs at ϕ = 45◦ . The largest gap value, at angles near 15◦ and near 75◦ , positions symmetric about the nodal line, was about 30 meV. These data were shown to agree well with data from a similar Bi2212 crystal obtained from ARPES measurements. An application of the Fourier-transform technique is illustrated in Fig. 10.16, Hoffman et al. (2002a). The curve represents a cut along a diagonal direction of an image of g(q, ω), with a subtraction technique applied to isolate effects arising from an applied magnetic field of 5 T. The plot shows a clear peak arising from the periodic Bi atoms, and also the interesting field-induced effect. The method used for this particular plot will be described in Section 10.6 below.

10.6 Observations of charge ordering in cuprate superconductors The superconducting state arises from the localized Mott antiferromagnetic insulating phase illustrated in Fig. 7.4. The phase diagrams as shown in Fig. 7.1 contain antiferromagnetic and superconducting phases, as well as several connecting regions, which have been the subject of continuing

C H A R G E O R D E R I N G I N C U P R AT E S U P E R C O N D U C TO R S

(a)

(–1,1)

(0,1)

461

(1,1)

q1 q7 q2 q3 ky (π / a)

q4

q6 q5 (1,0)

(–1,1)

(1,1) kx (π / a)

(b)

(–2,2)

(0,2)

qy (π / a)

q5

(2,2)

q4

q6

q3

q7

q2

(–2,0)

q1

(2,0)

qx (π / a)

Fig. 10.14. Elements of Fourier-transform scanning tunneling spectroscopy FT-STS. The expected wavevectors of quasiparticle interference patterns in a superconductor with electronic band structure like that of Bi2212. (a) Solid lines indicate the k-space locations of several banana-shaped constant quasiparticle energy contours (CCE) as they increase in size with increasing energy. As an example, the octet of regions of high |∇ k E(k)|−1 are shown in circles. The seven primary scattering q-vectors interconnecting elements of the octet are shown as labeled. (b) Each individual scattering q-vector from this set of seven is shown as an arrow originating from the origin of q-space and ending at a point given by a circle. The end points of all other inequivalent q-vectors of the octet model (as determined by mirroring each of the original seven in the symmetry planes of the Brillouin zone) are shown as solid circles. Thus, if the quasiparticle interference model is correct, there would be sixteen inequivalent local maxima in the inequivalent half of q-space detectable by FT-STS. (After McElroy et al., 2003.)

research. The idea that a magnetic field applied to a superconducting sample might lead to a transition into a magnetic ordered state was investigated by Hoffman et al. (2003). The surprising result of “checkerboard ordering” surrounding the vortex cores in a Bi2212 sample at 5 T is shown in Fig. 10.17. It is important to note that these regions are larger than the core diameter, which is expected to be essentially the coherence length; see for example Fig. 3.28.

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(b)

(a) q5

ARPES

q3 q2.6

1.2

ky (π / a0)

|q| (π / a0)

FT-STS

0.8

1.6

q7 0.8

q1

0.4

5

10

15

20 25 ω (meV)

30

0.6

0.4

0.2

35

0.2

0.4 0.6 kx (π / a0)

0.8

Fig. 10.15. Measured dispersion of all sets of q-vectors, and the FT-STS derived locus of scattering. (a) A plot of the magnitude of q1 to q7 excluding q4 . (b) The locus of the scattering k as determined from the data of (a). The line is a model and the gray band is data from ARPES measurements. (After McElroy et al., 2003.)

k-vector [2π/a0] 35

0.25

0.50

1.00

Bi atoms

30 25 PS[S112(x,y,5)]

0.75

vortex-induced LDOS

20 15 10 5 0 0.00

0.05

0.10

0.15

0.20

0.25

k-vector [Å–1]

Fig. 10.16. Fourier spectrum (“power spectrum” based on bias voltages in range 1mV to 12 mV) showing strong peak at the wavevector corresponding to the Bi atoms on 56-nm square region of Bi2212 surface. Also evident is peak “vortex-induced LDOS” arising from charge localization surrounding vortex cores in 5-T magnetic field; see below. (After Hoffman et al., 2002a.)

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463

Fig. 10.17. “Checkerboard “localization observed surrounding seven vortices in spectral image obtained at 5 T on a 560-Å square region of Bi2212. The vortex core is much smaller, dimension set by the coherence length, about 1.5 nm. Inset shows topographic image at twice the spatial resolution, showing atomic resolution and supermodulation pattern. See text for definition of imaged quantity. This image is source of spectrum shown in Fig. 10.16. (After Hoffman et al., 2003.)

The localization surrounding the vortex cores evident in the spectral image of Fig. 10.17 has been emphasized by a subtraction of zero-field spectra from 5-T spectra in the following way, based on the local density of states dI /dV = g(r, ω). This image is based on an integrated density of states: S1 12 (x, y, B) =

12  1

g(E, x, y, B) −

12 

g(E, x, y, 0)

1

where the sum runs over bias (E = eV) values 1 mV to 12 mV in increments of one mV, and x, y values are contained in a 56- nm square. The voltage range is chosen to emphasize the contribution of the vortices. It is seen that the checkerboard pattern in the integrated density of states is oriented along the Cu–O bonds. The periodicity of the checkerboard pattern is (4.2 ± 0.4)a0 and the decay length is ≈ 30Å, larger than the coherence length. The Fourier transform (power spectrum) of S1 12 (x, y, 5) was the basis for the linear plot shown in Fig. 10.16. The approximate 4-unit cell (4a0 ) periodicity observed here was later seen in several other experiments. Other effects of vortices have been examined by Hanaguri et al. (2009). Inherent localization at B = 0 is found in Bi2212 as the doping level is reduced. An analysis of this effect has been recently provided by Kohsaka et al. (2008), the source of Fig. 10.18. Earlier work relating to localization and Fourier transforms has been reported by Wang and Lee (2003), Capriotti et al. (2003), and Nunner et al. (2006). As shown in Fig. 7.1, and in Figs. 7.6 and 7.7, the “pseudogap” excitations occur at energies larger than the superconducing gap for doping levels less than ≈ 0.16 The distinction and relation between the two phases in a Bi2212 sample at p = 0.08 is illustrated in Fig. 10.18. The left panel is a spectroscopic image Z = g(r, V )/g(r, –V ) taken at eV larger than the superconducting gap, thus arising in the “pseudogap” regime. (In relation to Fig. 10.15a, eV exceeds 30 meV, and in relation to Fig. 10.15b, the region of the Brillouin zone is beyond the diagonal line defined by k x = k y = π/a.) The ratio defined in Z is found to accentuate the spatial structure of interest, the 4a0 squares. The right panel describes the

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Fig. 10.18. Two spectral images of the same sample, differing by tip voltage of the STM. The left panel has axes measured in nm, the right panel, a Fourier transform, has axes measured in wavevector, units π/a0 . The pseudogap excitations (left) are separated from the quasiparticle wave-like excitations of the superconductor (right) approximately by the diagonal line in the Fermi surface between k = (0, ±π/a0 ) and k = (±π/a0 , 0). (After Kohsaka et al., 2008.)

same sample but typifies the nature of the electronic states closer to the nodal line, as shown in Fig. 10.15b. The right-hand panel is a Fourier-transform image (at energy 16 meV in the same p = 0.08, Tc = 45 K sample) based on the 8-point interference model described above in connection with Fig. 10.15. The states at energy 16 meV are traveling-wave states whose interferences generate the pattern nicely quantified by the Fourier transform in the right panel. The contrast between spatial localization and phase-space localization is the contrast between the superconducting state and the pseudogap state, here simultaneously existing in the same sample. The spatial scale of the localization can be related to the Fermi surface as will be mentioned next. Prominent charge ordering was reported by Hanaguri et al. (2004) in the lightly holedoped Ca2−−x Nax CuO2 Cl2 (Na-CCOC). An image showing the charge ordering is reproduced in Fig. 10.19b. Hanaguri et al. performed Fourier-transform STS on this material, including the image shown, to reveal wavevectors including 2π/4a. The paper of Shen et al. verifies in the technique of angle-resolved photoemission spectroscopy that the wavevector 2π/4a is dominant in the effect. Momentum distribution curves (MDC) are shown in Fig. 2c,d of Shen et al., (2005), measuring the q-values in agreement with the spatial dimension 4a0 .

10.7 Relation of STS to angle-resolved photoemission spectroscopy (ARPES) A useful ARPES measurement of the temperature-dependent opening of superconducting gap in underdoped Bi2212 was carried out by Lee et al. (2007), whose results are shown in Figs. 10.20 and 10.21. The superconducting gap in the ARPES measurement is difficult to find because it occurs in a restricted portion of the Brillouin zone, but, once found, its distinct temperature dependence and k-space location differentiate it in a definitive fashion from the pseudogap features. An additional observation of Lee et al. was photoemission from thermally excited quasiparticles (see Section 3.3.2), a resolved band of electrons emitted at energy above the Fermi energy.

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R E L AT I O N O F S T S TO A R P E S

(a)

(b) (π,π)

q = 2π / 4a

Γ

24 mV

+45 ° 0°

4a θ

–45 °

Fig. 10.19. Identification of wavevector transitions in Brillouin zone of lightly hole-doped Ca2−x Nax CuO2 Cl2 (Na-CCOC) cuprate superconductor that are associated with checkerboard order observed in this material. (a) Sketch of the low-energy excitations as related to the Brillouin zone for x = 0.1 sample. The shaded regions along the nodal directions represent quasiparticles similar to those depicted in Fig. 10.15 for Bi2212. The hatched regions, in the antinodal portions of the Brillouin zone, were measured in ARPES (see Fig. 2c,d of Shen et al., 2005), and correspond to q = 2π/4a, where 4a is the cell dimension evident in spectral map shown in (b). Panel (b) is STM dI /dV map (analogous to Fig. 10.18b above) taken at 24- mV bias and 100- mK temperature, exhibiting 4a0 × 4a0 ordering as labeled in panel (b). (After Shen et al., 2005; data from Hanaguri et al., 2004.)

70 K Z hn

82 K

θ

87 K

Electron energy analyzer

e

92 K

Y

φ 97 K

–0.1

X

Crystal

0 E – EF (eV)

Fig. 10.20. (Left) Temperature-dependent electron distribution curves (EDC) in underdoped Bi2212 sample of Tc = 92 K, obtained along a line in the Brillouin zone, parallel but displaced slightly from the nodal line (see inset to Fig. 10.21), using 7- eV photons. The short bars indicate the peak of the thermally populated upper Bogoliubov (see Eq. 3.23) band, whose position at 70 K is marked by a bar for reference. In a superconductor, the basic excitations are linear combinations of electrons and holes; the thermal population of these states allows electrons to be collected from locations shifted above the Fermi energy by the superconducting gap energy. (After Lee et al., 2007.) (Right)Angle-resolved photoemission spectroscopy (ARPES) measurement. (After Zhou et al., 2007.)

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Δk (meV)

15

10

TC

(π,π)

5 C G 0 0

20

40

60 T (K)

80

100

Fig. 10.21. Temperature-dependent superconducting gap measured by ARPES along k-space direction marked as “C” in the inset. Data from underdoped Bi2212 sample of Tc = 92 K, obtained along a line in the Brillouin zone, parallel but displaced slightly from the nodal line (see inset to Fig. 10.21), using 7-eV photons. (After Lee et al., 2007.)

The angle-resolved photoemission spectroscopy (ARPES) measurement (see Fig. 10.20) allows measurement of energy and momentum of an excitation, according to the rules (Zhou et al. (2007) E B = hν–E kin –ϕ 1

K ll = (2mEkin ) 2 sin θ where hν is the photon energy, E kin is the measured kinetic energy of the photoemitted electron, E B is the binding energy of electrons in the material, kll is the momentum of electrons in the material parallel to the surface, and ϕ is the work function. K ll = kll + G is the projected component of momentum of electrons in the sample surface, where G is a reciprocal lattice vector. K ll can be calculated from the measured kinetic energy and collector angle θ ; see Fig. 10.20. Therefore, by measuring the intensity of the photoemitted electrons as a function of their kinetic energy at different emission angles, the energy and momentum of the electrons in the sample surface can be measured directly. The angle information is not available in STS, but, of course, STS is a local probe, while ARPES necessarily integrates information from a large area of sample surface. The Bogoliubov electron-hole excitations (Eq. 3.23), the basic quasiparticles of superconductivity, are found to exist only in restricted regions of the Brillouin zone; one is labeled C in Fig. 10.21. Near the diagonal of the Cu–O bond direction (nodal direction) the authors

E V I D E N C E F O R E L E C T RO N - S P I N WAV E C O U P L I N G

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found a gap that opens at Tc and has a BCS-like temperature dependence accompanied by the appearance of the Bogoliubov quasiparticles, a classical signature of superconductivity. The dispersion of this feature with temperature is key to identifying its superconducting origin. This observation demonstrates that the near-nodal gap is related to superconductivity, because the Bogoliubov quasiparticles exist only in the coherent superconducting state. This is in contrast with the pseudogap near the Cu–O bond direction (antinodal region), which is temperature independent to well above the superconducting transition temperature Tc .

10.8 Evidence for electron-spin wave coupling The dip in the conductance beyond the coherence peak has been suggested as arising from a spectrum of bosonic excitations, perhaps spin fluctuations, analogous to phonons in the conventional strong-coupled superconductor such as Pb. The conclusion of Pasupathy et al. (2008) (see Fig. 10.4) is that the strength of the dip feature does not correlate with the gap strength in a survey of locations. However, Pasupathy et al. do not speculate on the source of the dip feature, which has been addressed in work which we now describe. Zasadzinksi et al. (2001, 2006) have identified the dip feature labeled as  in Fig. 10.22 with a “resonance mode” of magnetic origin that has been observed in neutron scattering (reviewed by Tranquada, 2007) and in far-infrared optical measurements (Huang et al., 2004). See also Kirtley and Tafuri (2007), Eschrig and Norman (2000) and Abanov and Chubukov (2000). The data of Zasadzinksi et al. (2001) show a strong variation of the energy  (their analysis does not deal with the strength of the dip as opposed to the energy at which it occurs) with hole doping. The other measurements of the resonance mode energy in Bi2212 are similarly doping-dependent, although Huang et al. (2004) find that the mode disappears at p = 0.23, while it remains in the tunneling data at that doping. Zasadzinksi et al. (2006) have fit tunneling data as shown in Fig. 10.23 to generate a spectrum of bosonic modes which, within the Eliashberg theory, match the conductance data. The resulting α 2 F(ω) peak positions are in line with the decrease of the resonance energy going away from optimal doping. In this interpretation, the dip feature appears through energy dependence of the gap function, which is plotted in Fig. 3b of the cited paper. In principle, the α 2 F(ω) functions shown in Fig. 10.23 account for the pairing, based on exchange of boson excitations analogous to phonons but magnetic in nature. The subject seems still open, and it may be useful to mention an alternative model, even if the model is speculative. A model for the dip feature in tunneling into superconducting cuprate has been advanced by Abanov and Chubukov (2000). This model is based again on the assumption of strong interaction between electrons (holes) and magnetic excitations. As suggested in the left panel of Fig. 10.24, due to Abanov and Chubukov (2000), the electron that tunnels from a normal metal can emit a propagating spin wave if eV = + res , where res is the minimum frequency for spin excitations. After emitting a spin wave, the electron falls to the bottom of the band, which leads to a sharp reduction in the current and produces a drop in dI /dV , to be identified with the dip feature above the coherence peak. The process shown, however, looks like a threshold process, which, in many circumstances (see Section 8.3), as for example in emission of phonons, leads to an increase in conductance.

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48 over

under

Ω 100

40

Tc

2.5

16

1.5

60

2

2Δ limit

40

Ω/Δ

Ω (meV)

24

Tc (K)

80

32

1

20

0.5

8

0

0 0.05

0

10

0.1

20

30 40 Δ (meV)

50

60

0 0.3

0.25

0.2 0.15 hole concentration

Fig. 10.22. Energy of dip feature  in dI /dV (solid circles) and Tc (open circles) with hole concentration in Bi2212. , identified as resonance mode energy, obtained from 17 junctions over a wide doping range from underdoped with Tc 74 K to overdoped with Tc = 48 K. Inset shows / versus measured . (After Zasadzinksi et al., 2001.)

4

Normalized Conductance

3

4

Tc–62 K

4

1

3

#1

2

0 –200 –100 0 100 200 V(mV)

SIS break junction Δ = 10.5 meV

2 Fit

Boson spectral function

3 α2 F(ω)

5

2 #2

1 1 0 0 –200 –150–100 –50

0 50 100 150 200 V(mV)

0

20

40

60

80

100

120

140

160

ω (meV)

Fig. 10.23. (Left panel) Normalized SIS break-junction conductance (dots) and d-wave Eliashberg fit (solid line) for overdoped Bi2212 with energy gap10.5 meV. Inset “A” set of normalized break-junction tunneling conductances on three different overdoped Bi2212 crystals. These have energy gaps in the range 17–19 meV and estimated Tc = 60 K. (Right panel) Fitted α 2 F(ω) functions for heavily overdoped sample (#2) and near optimum sample (#1). The energies at which these fitted functions peak are in good agreement with the resonance excitation energies for overdoped and optimum samples and in agreement with the dip positions  plotted in Fig. 10.22. (After Zasadzinksi et al., 2006.)

E V I D E N C E F O R E L E C T RO N - S P I N WAV E C O U P L I N G

(a)

normal

469

superconducting

(b) dI/dV 2 2 -d I/dv Ωres

Δ

Ω

eV

–ω

eV Δ

1

Δ

eV

Fig. 10.24. (Left panel a) Schematic diagram for the dip feature in SIN tunneling conductance with strong spin–fermion coupling. The electron which tunnels from a normal metal can emit a propagating spin wave if eV = + res , where res is the minimum frequency for spin excitations. After emitting a spin wave, the electron falls to the bottom of the band, which leads to a sharp reduction in the current and produces a drop in dI/dV . (Right panel b) Schematic form of SIN tunnel conductance for strong spin–fermion interaction. In these diagrams, the authors identify res with the resonance mode seen in neutron scattering near 41 meV. (After Abanov and Chubukov, 2000.)

The argument for an increase in conductance is that at the threshold energy two different channels are available, an elastic process with no excitation plus an assisted process. On the other hand, strong interaction between a tunneling carrier and a phonon in the case of Si:B, as shown in Fig. 8.36, led indeed to a decrease in conductance (in negative bias), and the effect was described as a self-energy mechanism. Several similar cases have been observed and treated; see Section 8.4.3 and Table 8.2, Eqs. (8.59)–(8.66). Abanov and Chubukov (2000) it seems are treating a similar case; they assert (but see Kivelson and Fradkin, 2007) that spin waves are freely propagating, analogous to phonons, once the sample is in the superconducting state, while these excitations are heavily damped in the normal state. A detailed discussion of spin wave interactions in cuprate superconductors has been given by Uemura (2004). Uemura finds an analogy between the resonance mode discussed above and the roton excitation in superfluid 4 He, and suggests a “microscopic model of charge motion resonating with antiferromagnetic spin fluctuations.” Uemura suggests that the resonance mode is important in determining the Tc in the underdoped region and proposes phase separation in the heavily overdoped case where the specific-heat data (see Fig. 7.16) is clearly anomalous. Inhomogeneity evident in Fig. 10.2 for an overdoped sample may fall short of phase separation, but it might fit the need that is found by Uemura (2004). An essay entitled “How optimal inhomogeneity produces high temperature superconductivity” is offered by Kivelson and Fradkin (2007). Although the energy and doping dependences of the resonance mode are seen in the tunneling data, Tranquada (2007) more recently finds it lacks clear physical description.

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105

Magnetization [μB / Mn – ion]

4

3

3T 104

0.3 T 0T 1T

2

2T 5T 1

0

103

Resistivity [μ ohm • cm]

470

9T

102 0

100

200 Temperature [K]

300

Fig. 10.25. Temperature dependence of the Mn magnetic moment and electrical resistivity of an La0.7 Ca0.3 MnO3 single crystal. The dc magnetization was measured in external fields of 0.3 and 3 T; the resistivity curves correspond to zero field (top curve) and fields of 1, 2, 5, and 9 T (bottom curve). (After Fath et al., 1999.)

10.9 Colossal magnetoresistance: Mott transition in doped manganites Jin et al. (1994) reported a thousand-fold change in resistivity at 77 K in magnetoresistive La–Ca–Mn–O films. A negative magnetoresistance was observed in films or perovskite-like La0.67 Ca0.33 MnOx films grown by laser ablation on LaAlO3 substrates and heat treated in oxygen at temperatures between 700◦ C and 900◦ C. According to these authors, these compounds exhibit ferromagnetic ordering in the ab planes (Mn–O layers), separated by nonmagnetic La(Ca)–O layers and antiferromagnetic ordering along the c-axis. The mixed Mn3+ –Mn4+ valence state in these compounds is responsible for ferromagnetism and also metallic conductivity.

10.9.1 Introduction: mechanism of colossal magnetoresistance (CMR) The interplay of magnetization and resistivity in this system was studied in single crystals of La0.7 Ca0.3 MnO3 , as depicted in Fig. 10.25, and also in thin films. The ferromagnetic transition near 230 K is a dominant feature in the effects. The CMR effect is based on a metal–insulator transition between a ferromagnetic metal at low temperature and paramagnetic insulating state at high temperature. Scanning tunneling spectroscopy applied to this system by Fath et al. (1999) suggested a spatially inhomogeneous metal–insulator transition, which could arise from a phase separation, was involved in the large change in resistance. Earlier tunneling work on this system by Wei et al. (1997) showed pronounced dI /dV peaks identified as exchange-split spin-polarized density-of-states peaks for itinerant band electrons. The imaging of Fath et al. (1999)

471

C O L O S S A L M AG N E TO R E S I S TA N C E

(b) 400

H=0

Frequency [arb. units]

H = 40 KOe 300

200

100

0 0.00

0.03

0.06 0.09 dl / dU [nA / V]

0.12

Fig. 10.26. (a) (Left panel 5 T, right panel 9 T) Image of slope of I–V curve at bias 3 V, 0.61 μm × 0.61 μm on thin film of LCMO measured at T just below Tc . Scale bar is 100 nm in right panel. (After Fath et al., 1999.) (b) Left curve is at H = 0, right curve is at 4 T, statistics of dI /dV values at V = 0, 267 K, from a scanned LCMO thin film (scanned dimensions 250 nm × 250 nm), grown by a layer-bylayer method on MgO(100). The image on which this was based showed neither phase separation nor evidence for percolative transport. (After Koster et al., 2008.)

(Fig. 10.26a) shows the change in conductivity as the magnetic field is increased from 5 T to 9 T at a temperature just below the phase-transition temperature on a thin-film sample. In this image, dark is metallic, light is insulating, the scale of the image is 0.61μ m × 0.61 μm. The authors suggest that the image reveals coexistence of metallic and insulating phases, as well as intermediate cloud-like regions. It is suggested that the phase separation and related percolative transport are the basis for the high magnetoresistance, an idea reinforced in work of Ward et al. (2008). These workers argued that if a confining geometry becomes sufficiently small, coexistence may be impossible; only a pure phase of one or the other will fit. On the other hand, STS measurements on a similar system La0.75 Ca0.25 MnO3 by Koster et al. (2008), grown on MgO(100) in two different methods, did not show phase separation but a more homogeneous change from insulating to metallic character with increasing magnetic field. The change in dI /dV with magnetic field is summarized by histograms collected from a scan in Fig. 10.26b. The authors mention that they consider their films to be strain free. The

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phase separation evident in the scan from Fath et al. was taken at a temperature below the ferromagnetic phase transition, while the data of Koster et al. came from 267 K, well above the Curie temperature.

10.9.2 Pseudogap in manganite LSMO observed by ARPES A similar colossal magnetoresistive manganite, La1.2 Sr1.8 Mn2 O7 (LSMO), grown as single crystals by a floating-zone method were studied by angle-resolved photoemission spectroscopy by Manella et al. (2005). The spectra have some similarity to those in cuprate superconductors doped in the pseudogap regime that intervenes between the Mott insulator and the superconductor phase at low doping. Mannella et al. found a “nodal quasiparticle” at 15 K, well below the ferromagnetic Tc = 120 K in LSMO, and corresponding to a large effective mass about 3.3 m e . The technique of angle-resolved photoemission spectroscopy has been described above as a means of measuring the density of states in cuprate superconductivity; see for example Lee et al. (2007). The quasiparticle peak shown (Fig. 10.27) (Manella et al., 2005) is regarded as small but well defined. The tiny intensity in the quasiparticle peak is taken as evidence that electron–phonon interaction is so strong that the band theory based on the usual metallic state fails. A large effective mass, given as 3.3 m e , leads to a prediction of a large resistivity, which is observed. The estimate, based on m ∗ = 3.3 and a scattering time estimated as 3.28 fs gives ρ = 0.66 × 10−3 cm, which is 0.33 of the measured in-plane resistivity (this material is regarded as a bilayer and is highly anisotropic). It is suggested that conduction, rather than metallic in the usual sense, comes from a highly correlated liquid of lattice polarons. The polaron liquid turns into a ferromagnetic metal below the Curie temperature. The electron–phonon coupling leading to polarons of high mass seems to be part of the colossal magnetoresistance phenomenon, but this feature is not important in the cuprate superconductors. The inset to Fig. 10.27a shows on an expanded scale the “peak–dip–hump” structure familiar from tunneling measurements on cuprates; see e.g., Fig. 10.4b. The EDC curves are obtained with varying wavevectors k along the nodal trajectory shown in panel (b). As seen in panel (a), the broad “hump” peak marked “H” moves rapidly in energy with changing wavevector k. The data are described as a small quasiparticle peak (labeled), which crosses the Fermi energy at wavevector k F ≈ 0.37Å−1 ; and a rapidly dispersing broad peak, the “hump”. The data were collected at 15 K and 20 K using photons of energy 42 eV with light-polarization vector lying in the sample plane. The temperature dependence of similar data showed that the quasiparticle peak weakens near 90 K or about 0.75Tc and disappears at the Curie temperature TC = 120 K. At the Curie temperature, a metal– insulator transition occurs between a low-temperature ferromagnetic-metallic ground state and a high-temperature paramagnetic-insulating state. The weight of the quasiparticle peak diminishes as the trajectory is shifted away from the nodal line, and entirely different curves are seen along the second trajectory at the top of panel (b), termed the antinodal trajectory. It appears (Shen et al., 2005) that the antinodal features are related to electron coupling to distortions of the MnO6 octahedra in the manganite material.

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473

(a) 15 K (b) H

–0.2

0 Quasiparticle

–1.2

–0.8

–0.4

Γ

0

Fig. 10.27. (Left panel) Electron distribution curves near the gamma point (center of zone, along a nodal direction as indicated by the 45◦ line in the right panel. Inset shows detail. (Right panel) Brillouin zone of LSMO showing 45◦ line in nodal direction along which electron distribution curves were obtained. (After Manella et al., 2005.)

10.10 Relation of cuprates to ferromagnetic CMR manganites The authors Manella et al. (2005) point out that “nodal direction” in manganite LSMO does not have direct meaning as it does in the superconducting case, where it denotes the location of the node in the superconducting order parameter. But the terminology is useful because of the similarity to the cuprates. The progression in spectra, moving away from the nodal line toward the antinodal line (see Fig. 4 of Manella et al., 2005) is quite similar to the progression of similar data seen in the underdoped cuprate superconductor Ca2−x Nax CuO2 Cl2 (Hanaguri et al., 2004; Shen et al., 2005). Samples studied in Fig. 10.28 were single crystals of typical dimensions 1 mm × 1 mm × 0.1 mm grown by a high-pressure flux method. The samples at x = 0.12 and x = 0.10 had superconducting Tc values of 13 K and 22 K, respectively, while the x = 0.05 sample was not superconducting. The authors note similarity in these features to those observed in colossal magnetoresistive manganite La1.2 Sr1.8 Mn2 O7 (LSMO). The similarity of cuprate Na-CCOC and manganite LSMO in these measurements are regarded as instructional in relation to one of the key puzzles in the development of superconductivity. From the Mott insulator, what is the nature of an intermediate “pseudogap” phase that intervenes and leads finally to the superconducting phase? The surprising observation is of “pseudogap” features in LSMO, in which superconductivity does not occur, which are quite

474

S U P E R C O N D U C T I N G C U P R AT E S A N D M AG N E T I C M A N G A N I T E S

x = 0.12 x = 0.10 x = 0.05

Photoelectron Intensity (arb. units)

0º (nodal)

+45º (antinodal)

–0.4

–0.2 Energy (eV)

0.0

Fig. 10.28. Electron distribution curves as a function of Fermi surface location, nodal direction at top, antinodal direction at bottom, for lightly hole-doped Ca2−x Nax CuO2 Cl2 (Na-CCOC) cuprate superconductor with three doping levels, x = 0.12 (upper curves), 0.10 (middle curves), and 0.05 (lower curves). Photoemission data at 15 K using photon energy 25.5 eV. (After Shen et al., 2005.)

similar to those in superconducting Na-CCOC. This observation suggests that a “pseudogap phase” is not necessarily related to superconductivity. On the other hand, Wen et al. (2009) have recently measured the specific heat in underdoped Bi2 Lax CuO6+δ (Bi2201) crystals and find residual superconductivity in the normal state far above Tc . The STS gap maps shown at the beginning this chapter (Figs. 10.1–10.8) also show residual superconductivity above Tc . The lesson may be that the pseudogap is a more common phenomenon in metallic oxides, but in the cuprates, in contrast to the manganite LSMO, it is indeed related to superconductivity.

11 Applications of barrier tunneling phenomena 11.1 Introduction In the fundamental sense, a large fraction of all electronic processes on the microscopic level, where wavefunctions and quantum mechanics are essential, may be regarded as involving tunneling. For example, electronic conduction in a band is a process of repetitive tunneling from classically allowed regions (near the nuclei) through Coulomb barriers in the interstitial regions, where the wavefunctions decay exponentially. In this sense, applications of tunneling phenomena encompass much of chemistry and condensed matter physics. However, the domain of barrier tunneling—as implicitly associated in this book with geometries in which only one dimension is microscopically small, providing the barrier V (x), while in the other two dimensions the structure is macroscopically planar—is greatly restricted. The difficulty in experimentally realizing the required regularity in the microscopic barrier V (x) over the two macroscopic dimensions of solid-state structures is reflected, at least in part, in the award of the Nobel Prize in physics to Esaki, Giaever, and Josephson in 1973. Restricting consideration to the barrier tunneling structures that have been considered in the preceding chapters, we may already point out many diagnostic applications of tunnel junctions. These applications have been largely in the category of deduced normal-state and superconducting properties of materials, principally metals and semiconductors. Phonon spectra and energy gaps are obvious examples. A more fundamental group of applications already discussed has been use of Josephson tunneling structures and the basic Josephson relation ω = 2eV to examine the exactness of pairing e∗ /e = 2 in superconductivity; assuming this result, to measure with improved accuracy the ratio 2e/ h, as shown in Fig. 11.1; and assuming only constancy of e/ h, to standardize voltage measurements against frequencies. The use of tunnel junction devices, composed of single or multiple junctions, to perform specific functions has been mentioned occasionally but remains to be systematically discussed. Examples of tunnel junction devices as electronic circuit elements include the Esaki diode, as an oscillator or nonlinear element, and the Josephson junction, as a magnetically actuated bistable element (switch) from which families of computer logic circuits have been synthesized. The latter applications exploit the magnetic field dependence of the maximum Josephson supercurrent shown in Fig. 11.2. Examples of tunnel junction devices as sensors include the use of normal-state and superconducting junctions as light or microwave detectors, the use of superconducting junctions as phonon generators and detectors, and, especially, the use of pairs of Josephson junctions as magnetic interferometers or in the SQUID (superconducting quantum interference device). The latter class of devices has allowed measurements of magnetic field (and magnetic field gradient), current, and voltage at unprecedented sensitivity. Further superconducting junction 475

476

A P P L I C AT I O N S O F BA R R I E R T U N N E L I N G P H E N O M E NA

0.0050

1 ppm

2e/h –483.5900 (MHz / mV69 NBS)

0.0045

Petley & Morris Denenstein et al.

0.0040

+s

0.0035

–s

0.0030

0.0025

0

10

20

30

40

50

60

Days

Fig. 11.1. Illustration of the use of Josephson effect in determination of the fundamental constant ratio 2e/ h. Here experimental values of 2e/ h as a function of time of reported observation are displayed (after Finnegan et al., 1970) in comparison with earlier results.

devices which we have mentioned earlier (Section 7.4) are Gray’s transistor (1978) and the QUITERON (Faris et al., 1983). A different type of application of the tunneling process based on its short range is the scanning tunneling microscope (Binnig and Rohrer, 1982; Binnig et al., 1982a,b,). Several other scanning or imaging devices based on the tunneling process will be mentioned. The aim of this chapter is somewhat limited: to point out only the basic concepts for the digital and analog applications (principally based on the SQUID) of Josephson junctions, and for tunnel junctions as detectors, for these topics have already been extensively reviewed; and to describe only a relatively few novel and possibly promising applications from the broad remaining range of diagnostic and device applications of barrier tunneling phenomena.

477

J O S E P H S O N J U N C T I O N I N T E R F E RO M E T E R S 30 Maximum dc Josephson current vs. Magnetic field Sn-Sn 1.2 K

25

Current (mA)

20

15

10

5

0

0

1

2

3 4 5 Magnetic field (Gauss)

6

7

Fig. 11.2. The maximum supercurrent in a Josephson tunnel junction oscillates as shown with applied magnetic field, in the same Fraunhofer diffraction pattern behavior shown earlier, and can be used as the basis for switching elements for computers. In this case, a control current may provide the magnetic field that switches the junction. (After Langenberg et al., 1966.)

11.2 Josephson junction interferometers The concept of the Josephson junction interferometer is illustrated in Fig. 11.3, while its first experimental realization, by Jaklevic et al. (1964, 1965), is illustrated in Fig. 11.4a. As first mentioned in Section 3.1, the supercurrent density J in a single Josephson junction is given by J = J0 sin φ, where J0 is the maximum supercurrent density and φ is the phase difference between the two superconductor electrodes. In the geometry of Fig. 11.3a, the total current I = Ia + Ib is the sum of those through a and b junctions. The phase difference ψ between points 1 and 2 (see Fig. 11.3a) must be the same over either path a or b ; thus, using Eq. (3.4), one has 2π φ = φa + 0

 a

= φb + 2π A · dr 0

 b

A · dr

(11.1)

where 0 = h/2e and A is the magnetic vector potential. But this equation can be rewritten, letting ψb − ψa = ψ, as φ =

2π 0

 b

− A · dr

 a

  = 2π A · dr A · dr 0 r

(11.2)

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A P P L I C AT I O N S O F BA R R I E R T U N N E L I N G P H E N O M E NA

(a)

I

1 Γa

Γb B

2

(b)

I

–3Φ0 / 2

–Φ0 / 2

Φ0 / 2

3Φ0 / 2 ΦL

Fig. 11.3. (a) Schematic diagram of Josephson junction interferometer. (b) Response of interferometer current to increasing magnetic flux  L threading the loop.

where  is the closed-loop path b –b . Hence, using Stokes’s theorem, one finds    L 2π B · dA = 2π φ = 0 0

(11.3)

where  L is the magnetic flux threading the area A L of the large loop , neglecting flux from the current I itself. We arbitrarily set ψa = δ + ( ψ/2), which requires ψb = δ − ( ψ/2), and neglect the flux cutting the individual junctions compared with . Then      π L π L + sin δ − I = Ia + Ib = I0 sin δ + 0 0 Hence, the maximum current Imax , reached when δ = π/2, is 2 2 2 π  L 22 Imax = 2I0 22cos 0 2

(11.4)

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J O S E P H S O N J U N C T I O N I N T E R F E RO M E T E R S

b c f (a) a

f B

d e (b)

Josephson current (arb. units)

A

B

–600 –500 –400 –300 –200 –100

0

100

200

300

400

500 600

Magnetic field (mG)

Fig. 11.4. (a) Schematic of junction interferometer employed by Jaklevic et al. (1965). A magnetic field B is applied parallel to the long direction of the substrate. (b) Experimental traces of the maximum Josephson current vs. magnetic field showing interference and diffraction effects. The field periodicity is 39.5 mG for A and 16 mG for B. In both cases, the junction separation w is 3 mm and the junction width is approximately 0.5 mm. (After Jaklevic et al., 1965.)

where I0 = AJ 0 , A being the junction electrode area. In a more complete treatment, (11.4) is multiplied by 2    2 2 π  j −1 22 π j 2 (3.104) 2 2sin 2 2 0 0 where  j is the smaller flux through each junction. The response of the interferometer current I to increasing flux  L   j , (11.4) (see Fig. 11.3b), is oscillatory, with its first minimum at  L = 0 /2. This function can be an extremely rapid function of magnetic field if the loop

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A P P L I C AT I O N S O F BA R R I E R T U N N E L I N G P H E N O M E NA

area A L is large, as follows from the small size of the flux quantum, 0 2.07 × 10−7 G · cm2 . If the area A L is 1 cm2 , e.g., one period of the current oscillation in (11.4) occurs for A L B/0 = 1, or B 10−6 G. This effect was first observed in structures of smaller loop area, as shown in Fig. 11.4, by Mercereau and co-workers at the Ford Scientific Laboratory. The modulation of the interference pattern by the (sin x)/x single-slit diffraction envelope is also discernible in these data. We have seen in Section 7.6, Fig. 7.18, a description of Josephson junctions in the d-wave case, and the design and use of corner-mounted junctions for a YBCO-Pb dc superconducting quantum interference device (SQUID).

11.3 SQUID detectors: the scanning SQUID microscope In Fig. 11.4 are shown rapid oscillations of current as magnetic flux is varied through a loop containing two Josephson junctions. To make a practical magnetic field sensor from such characteristics, the dc SQUID detector (Clarke, 1966; Zimmerman and Silver, 1966) uses a servo loop to maintain a constant value of the flux through the loop, as shown in Fig. 11.5. The limiting sensitivity of such a dc SQUID sensor system is estimated (Kleiner et al., 2004) as 10−6 φ0 . This is usually described as the most sensitive sensor of magnetic field, and is easily adapted to make sensitive measurements of electrical current and voltage. One of the applications of the SQUID is to make a scanning magnetic field microscope, the scanning SQUID microscope. One version is sketched in Fig. 11.6, in which a sensor coil is directly inserted into the dc SQUID loop.

11.3.1 Establishing d-wave nature of cuprate pairing The use of a corner SQUID detector by Wollman et al. (1993) to differentiate the nature of superconducting pairing was described in Section 7.6. In Fig. 7.19 (Tsuei et al., 1996), evidence was presented for trapping of half-flux quanta in rings containing a d-wave Josephson IB

Ys Lp

lock-in Detector

Amplifier

Li

Integrator

L

100-k Hz Cabillater

Rf

Vo

Fig. 11.5. Typical circuit based on loop L of two Josephson junctions to measure magnetic field, falling on pickup coil Lp at upper left. Flux is inductively coupled into the SQUID loop. The Josephson junctions in this loop are shown as ideal elements shunted by capacitance and resistance. An ac method with a lock-in detector, integrator, and flux feedback maintains an operating point (this is called a fluxlocked-loop) on a rapidly varying I –B characteristic such as shown in Fig. 11.4. (After Kirtley and Wikswo, 1999.)

481

JOSEPHSON JUNCTION LOGIC

Fig. 11.6. Adaptation of Josephson junction SQUID device to make a scanning microscope sensitive to local magnetic field. Josephson junctions are shown as crosses, all wires are superconducting. The pickup coil dimension may be as small as 4μm. In some designs, the sample need not be refrigerated, using a cryostat with a thin window at the bottom. (After Kirtley and Wikswo, 1999.)

junction. These experiments were influential in establishing the d-wave nature of cuprate superconductive pairing.

11.4 Josephson junction logic: rapid single-flux quantum devices The basic switching process in the (unshunted) Josephson junction circuit (Likharev and Semenov, 1999) occurs as a jump from the superconducting (V = 0) state into the voltage state, as illustrated in the Fig. 11.7, panel (b). Switching out of the supercurrent state occurs when the applied current (in panel (a) this is shown as Iin + Ib ) exceeds the critical Josephson junction current Ic . As long as the Josephson junction ( “J” in the figure) is superconducting, all current flows through it and no current flows through the load resistor, R L . The output voltage is thus zero. After the junction switches to its voltage state, at V ≈ 2 (T )/e, this voltage appears at the output of the device, representing a logical one. This form of Josephson-junction superconducting logic has evolved from extensive earlier work [see Anacker (1980), Matisoo (1980)] into a related but subtler scheme, based on resistively shunted Josephson junctions, leading to the I (V ) shown in the figure, panel (c). The shunting resistor parallel to the Josephson junction has a small value (perhaps 1 to 8 ) so that the current I remains nearly constant after switching, but a transient voltage pulse [illustrated in Fig. 11.7, panel (d)] is carried along a transmission line (panel (a) in the figure), to provide a voltage pulse at the output.

11.4.1 The single-flux quantum voltage pulse The voltage pulse from transit of a single flux quantum through the Josephson junction is ∫ V (t)dt = 0 = h/2e = 2.07 mV ps (RSFQ pulse)

(11.5)

This is just Faraday’s law, V = –d0 /dt. These are the “Single Flux Quantum” pulses, which form the basis for the “RSFQ” logic and memory schemes. This result for the “RSFQ” pulse can be understood also using the Josephson junction relation dψ/dt = (2e/)V (t) (the ac Josephson effect)

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(a)

(b) Iout

P

Iin J

Rl = P

Current

Ib + Iin Io

Ib

Ib

“O” Iout 1 2 Δ (T)

0

Voltage

e

(d) Voltage

(c)

Vc

J

0

Ic Ib Phase

Current

Io+Iin

0

Vc

Voltage

2π J 0

10 Time (τo)

Fig. 11.7. The simplest form of the Josephson junction logic. (a) Equivalent circuit and schemes of its operation. The input current is the signal plus the bias, and when the sum exceeds the critical current for the junction, it switches (passes one flux quantum) and the current (abruptly) is sent to the load resistor. How this happens is different when using (b) an underdamped (unshunted) and (c) and (d) an overdamped (shunted) junction. The situation in (b) was found eventually to be detrimental, in that it was “latched,” slow to release. The use of the “shunted junction” [panel (c)], avoids the latching, but now the only message of the junction switching is the voltage pulse, sent down the transmission line, as seen in panel (d). This is the present state of the art, keeping track of the SFQ pulses. (After Likharev and Semenov, 1999.)

where φ is the difference of the superconducting phase across the junction. The SFQ voltage pulse corresponds to an advance of this phase difference by 2π . In terms of the angle φ, the bias current, near to the critical value, as shown in the Fig. 11.7, corresponds to an angle a bit less than π/2 (from the relation I J = I J 0 sin φ, where sin φ is nearly unity). There is a well-known analogy between the current-biased Josephson junction and a pendulum whose deflection angle is φ. The arrangement in Fig. 11.7, panel (a), corresponds to a pendulum biased to an angle ϕ close to π/2, nearly horizontal, with the idea that the incoming pulse will kick the pendulum “over the top” to advance by 2π . Since the (equivalent) pendulum is overdamped, it will make only one turn, leading to the single SFQ pulse. According to calculations Likharev and Semenov (1999), the time scale for the SFQ pulses varies with the linear scale L of the junctions, with a time unit of 4 ps for L = 5 micrometers down to 0.5 ps for L = 0.7 micrometers. The numerical calculations [leading to waveforms as shown in panel (d)] are advanced, and based on the assumption that the characteristic impedance of the transmission line [panel (a)] matches the load resistance R L . The essential test of the scheme is whether the output SFQ pulse, when presented to another biased junction, can change that junction into the voltage state. The answer is yes, in that RSFQ circuits have demonstrated counting at 750 GHz (Chen et al., 1998).

483

D E T E C T I O N O F R A D I AT I O N SQUID Current Bias

Single Flux Quantum (SFQ) Pulse Train

R Counter with destructive read-out

Analog Input

DC SQUID

Output Interface

Sampling Clock

Fig. 11.8. This is a schematic of a sensitive V /F analog to digital converter (ADC) based on the V -flux transfer characteristics of the SQUID and followed by an RSFQ binary counter. The train of SFQ pulses coming from the dc SQUID device is counted by an RSFQ counter. (After Mukhanov et al., 2004.)

11.4.2 Analog to digital conversion (ADC) using RSFQ logic One strong advantage of the RSFQ scheme is its fast operation, with 750 GHz counting as demonstrated by Chen et al. (1998). A more recent diagram for an analog to digital converter, based on the RSFQ scheme (Mukhanov et al., 2004), is shown in Fig. 11.8. It is stated that complete analog-to-digital systems with digital sampling rates ∼ 20 GHz have been demonstrated. The most immediate applications of such systems seem to be in acquisition of radar signals, i.e., directly digitizing the incoming radio frequency voltage. The ADC converters are large systems, with up to 10 000 junctions. The conventional and reliable junction technology uses the Nb/Al2 O3 /Nb “trilayer method” (Gurvitch et al., 1981; see also Wolf et al., 1979), with Josephson critical current densities on the order of a few kA/ cm2 in junction sizes on the order of several micrometers. The “trilayer method” involves depositing Al onto Nb, then oxidizing the Al (followed by evaporation or sputtering of the Nb counterelectrode), since Nb itself does not produce a suitable tunnel barrier (Wolf et al., 1979). Since damped junctions are needed, as mentioned above, the junctions require resistive shunts, which are typically made using a metallic layer such as Mo or TiPd. The circuits also require inductors, which are usually Nb striplines, separated from a ground plane by an SiO2 insulator.

11.5 Detection of radiation There are long histories and extensive literatures associated with photodetection by both normal-state tunnel junctions and by superconducting tunnel junctions. The normal junction prototype is probably the classical metal–semiconductor square-law detector in the form of a metal “cat whisker” point contact on a galena (PbS) crystal used as an rf detector in a “crystal radio.” Operation of this detector, like that of its primary present-day successor, the conventional tungsten–silicon microwave diode (which, however, is not a tunneling device), depends upon nonlinearity in the I –V relation, characterized conventionally by S=

d2 I /dV 2 dI /dV

(11.6)

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This nonlinearity parameter is closely related to the figure or merit of the direct detector, the classical current responsivity Ri =

Idc S = Pa 2

(11.7)

where Idc is the change in dc output for an increment Pa in absorbed power. Note that for a device obeying the ideal exponential law,     eV I = I0 exp −1 kT the S parameter is just e/kT. A second requirement for an rf detector is a low internal shunt capacitance C such that dI /dV /ωC  1 at frequencies of interest. This condition, favored by the small contact area, had—at least until the advent of the edge junction of Heiblum et al. (1978) and Havemann (1978)—best been satisfied in the mechanically unstable point-contact type of device. A second, related mode of detection is heterodyne detection, mixing the signal at ω S , with a local oscillator signal at ω L0 . The desired output is (usually) the lower difference frequency ω S − ω L0 , termed the intermediate (IF) frequency, at which subsequent amplification at reduced bandwidth can be accomplished. Clearly, the mixing effect still depends on large nonlinearity S. The case of direct detection is formally obtained as mixing to zero frequency with ω S = ω L0 . A classical figure of merit for heterodyne detection is the available conversion efficiency, ⎤2 ⎡ L −1 ≡

⎢ ∂ I (V, PL ) ⎥ R D PL ⎥ =⎢ ⎣  1 ⎦ 8 PS 2 ∂ PL

(11.8)

where R D = [dI /dV ]−1 and PL and PS are, respectively, the available powers (assuming perfect matching) at the local (IF) and signal (S) frequencies. Here, L −1 > 1 (conversion gain) can occur classically only if a negative resistance appears in the I –V characteristic. Conversion gain can be important in attaining low-noise mixing, because the overall noise temperature of the heterodyne receiver can be shown to be TIF (11.9) L −1 Here, TD and TIF , are, respectively, the noise temperatures of the detector (mixer) and IF amplifier. Since it is common for the term in TIF to be dominant, a large value of L −1 may be the only way to obtain a low overall noise temperature TR . In tunneling devices, the nonlinearity S required in both direct and heterodyne detection can arise in several different ways. The typical weakly quadratic behavior of the background conductance G(V ) = G 0 + αV 2 of the MIM tunnel junction, as studied by Brinkman et al. (1970), offers a minimum level of S, which can be increased by choosing a tunnel barrier I of low height. Such low barriers form, as we have noted, rather easily with transition metals, with oxides such as NbOx (Walmsley et al., 1979; Brunner et al., 1982), which make them less than fully suitable for tunneling spectroscopy; this feature is of advantage in TR = TD +

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application to detection. An extremely useful, but still poorly characterized, tunneling device, which presumably involves such a low barrier, is the Ni–W freestanding, optical antenna, point-contact optical detector (Dees, 1966; Sanchez et al., 1978). This device is used as a heterodyne detector of laser light up to about 200 THz frequency (λ = 1.5 μ m) (Baird, 1983; Pollock et al., 1983). A second tunneling mechanism for obtaining larger curvature S, and corresponding large detective responsivity Ri , is to use a Schottky tunnel diode of very small contact area, which may be akin to the historical cat-whisker detector mentioned earlier. However, the frequency response of this device appears inherently to have a much lower cutoff, possibly due to the nature of conduction in the series-spreading resistance region (necessarily a region of relatively low carrier density n < 1020 cm−3 ) connecting the small-area Schottky barrier tunnel contact to the doped semiconductor. A rather extensive body of information on this type of device had accumulated consequent to its use with a superconducting counterelectrode for near-millimeter wave detection [“super-Schottky detector”; see, e.g., Silver et al. (1978)], where it appears to be effective up to about 30 GHz. If cryogenic operation is not an impediment, the use of one or two superconducting electrodes offers the possibility of vastly enhanced nonlinearity S, in NIS or SIS devices operated, respectively, near biases /e and ( 1 + 2 )/e. Considerable advance, both experimental and theoretical, has occurred, specifically for the SIS quasiparticle mixer biased at the sum gap edge (e.g., see Richards and Shen, 1980; Phillips and Dolan, 1982).

11.5.1 SIS detectors The subject of SIS quasiparticle photodetection has followed an interesting and surprisingly slow line of development (at least until 1978). The original proposal of SIS photodetection (Burstein et al., 1961) was couched in the quantum terms of “photo injection” of carriers (quasiparticles) by optical excitation across the energy gap of the superconductor, the quasiparticles being subsequently collected by low-voltage bias, eV < 2 . The proposed quantum mechanism of Burstein et al. (1961) imposed a long-wavelength detection limit λmax ≤ hc/2 , estimated as λmax ≤ 3.9 mm for Al–I–Al junctions and λmax ≤ 0.46 mm for Pb. The next relevant development, again understood in quantum terms, was the splitting of the SIS sum-gap edge by an imposed microwave field (Dayem and Martin, 1962; Tien and Gordon, 1963; Miller and Dayem, 1967). In spite of this history, however, until Richards, (1978), experimental work had largely been limited to the NIS case in the form of the super-Schottky detector (McColl et al., 1973; Silver et al., 1978); these NIS results and projected work on SIS quasiparticle detectors (Richards, 1978) were both being considered in wholly classical terms. It appears possible that discovery of the Josephson effect and the original application of that effect to rf detection (Grimes et al., 1966, 1968) diverted attention for about fifteen years from the SIS quasiparticle detector. Later, the research and development emphasis returned to the SIS quasiparticle detector, following key theoretical work of Tucker and Millea (1978) and Tucker (1979, 1980). This work showed (unexpectedly, in the recent context) from a purely quantum mechanical approach, expanding upon the original work of Tien and Gordon (1963), that the S–I–S junction as quasiparticle heterodyne detector is, in fact, capable of conversion gain L −1 > 1. With the subsequent observations of this effect, the S–I–S gap edge–biased quasiparticle detector became the most promising superconducting

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rf detector. These devices have been developed for heterodyne detection at 115 GHz in radio astronomy, where a detector noise temperature TD approaching the quantum limit T = hν/k can be expected (Shen et al., 1980; see also Phillips and Dolan, 1982). This surpasses the performance of all other detectors, including cooled Schottky diodes, super-Schottky diodes, InSb bolometer mixers (see Phillips et al., 1981), and Josephson mixers (Taur and Kerr, 1978; Richards and Shen, 1980; Taur, 1980). It has also been established in general by the Tucker analysis (applied to direct detection, rather than heterodyne detection) that, in the case of sufficiently strong nonlinearity S, a quantum regime is entered in which unit quantum efficiency (one charge per absorbed quantum) can be attained. Thus, in the SIS case, the circle has finally all but returned, in 1983, to the original quantum picture of Burstein et al. (1961), with most of the details properly filled in. The Josephson mixer, on the other hand, after intense and extensive study, has been established as being inherently noisy (Taur et al., 1974; Tucker, 1980) and probably incapable of attaining the quantum noise limit. The quantum limit of detection is attained in general for nonlinearity S satisfying e S > (11.10) 2 ω (Tucker and Millea, 1978; Tucker, 1979, 1980). In this case, the responsivity is given by Rω :   e I (V + ω/e) − 2I (V ) + I (V − ω/e) (11.11) Rω = ω I (V + ω/e) − I (V − ω/e) This expression can be compared with (11.6) and (11.7) by noting that the limits (derivatives) have been replaced by finite differences. If the value of ω is effectively larger than the width of the corner in the SIS I –V characteristic at eV = 1 + 2 , so that I (V + ω/e)  I (V ) while I (V − ω/e) I (V ), then Ri takes on the quantum limiting value Ri = e/ω (one charge per absorbed photon). A similar change is required in the value of detector resistance, R D = (dI /dV )−1 , in (11.8) for the conversion efficiency, the new result being RD =

2ω I (V + ω/e) − I (V − ω/e)

(11.12)

These quantum limit modifications apply generally, but they have first been fully observed, including the negative-resistance effect, in the SIS quasiparticle mixers (Kerr et al., 1981; McGrath et al., 1981; Smith et al., 1981). As pointed out in the 1981 references above, the important implication of the observed negative resistance is that a properly matched mixer should have unlimited conversion efficiency (gain) L −1 , allowing a low-noise temperature via (11.9).

11.5.2 Josephson effect detectors In the original work of Grimes et al. (1966, 1968), analysis of rf detection by Josephson junctions was based on the known sensitivity of the Josephson current to magnetic flux, i.e., to electromagnetic radiation. In fact, the V = 0 current is depressed, and current steps (Shapiro, 1963) occur at voltages eV = nω. For detection purposes, a nonhysteretic junction is used, with a current bias point such that the voltage state exists. The effect of rf radiation is to depress the I –V curve qualitatively as shown in connection with dc SQUID operation. This

T U N N E L - J U N C T I O N M AG N E TO R E S I S TA N C E S E N S O R S

487

depression effect is utilized, with a current bias established, to extract a voltage signal dependent on the amplitude of the applied rf. A thorough review of the extensive work following this initial demonstration has been given by Richards (1977); a comparison of the resulting performance with that expected from SIS mixers is offered by Richards and Shen (1980). Other detector applications of the Josephson effect, in a parametric amplifier and a temperature sensor in a composite bolometer detector, are also discussed by Richards (1977).

11.5.3 Optical point-contact antennas (high-speed MIM junctions) The original device of Dees (1966) was a fine tungsten-wire cat whisker in contact with a polished metal plate, used for detection of submillimeter waves. Although the detailed mechanism of detection was (and still is) unclear, the extremely low capacitance (∼ 10−2 pF) and low spreading resistance were essential factors in an extremely low RC constant, on the order of 10−13 s. The device has evolved (further references to earlier work are given by Heiblum et al., 1978; Sanchez et al., 1978) into a fairly standard configuration, which is an ultrafine W wire with an etched point (comparable in dimensions to a conventional field emission tip), in contact with a flat Ni anode. Mechanical stress, or possibly electrical pulsing, is used to establish a contact of somewhat enhanced mechanical stability through the layers of oxide on both members of the structure. The resulting barrier, in some cases, is believed to be only 7–10 Å in thickness (Liu et al., 1979). In use as an infrared optical mixer, the W wire whisker is freestanding, and angular measurements have revealed the expected antenna patterns at relatively large wavelengths. Sanchez et al. (1978), e.g., show results obtained with a whisker antenna of length 1.95 mm as a detector of 311 and 337 μm, radiation, analyzed within conventional antenna theory. At wavelengths of the same order, support can be given for a detection mechanism involving tunneling. Several such mechanisms are discussed by Heiblum et al. (1978). At optical wavelengths, however, clear evidence has been given by Elchinger et al. (1976) that photoemission over the oxide barrier, and not a nonlinear I –V rectification mechanism, is dominant. The useful cutoff of the present devices appears to be around 1.5μm; however, see also Pollock et al. (1983). In spite of serious attempts to analyze the operative mechanisms in these useful devices in the cited works of Heiblum et al. and Sanchez et al., a clear picture of all aspects of their structure and operation is still lacking; see, e.g., Feuchtwang et al. (1981). Efforts to produce fast MIM antenna detectors based on the edge-junction geometry have been reported (Heiblum et al., 1978; Yasuoka et al., 1979), but the devices appear thus far not to have improved upon the performance available in the W–Ni whisker antennas. This whole area is of considerable interest from both practical and fundamental points of view, and it may eventually benefit from the edgejunction fabrication techniques further developed in connection with the Josephson computer programs.

11.6 Tunnel-junction magnetoresistance sensors The ferromagnetic tunnel junction has been described above in Section 8.2.5. The performance of a practical magnetic tunnel junction sensor for general use (Shen et al., 2006) is shown in Fig. 11.9. The construction of the device is sketched in the inset. The sensitivity to magnetic field arises in the “free layer” ferromagnetic film, a soft ferromagnet whose magnetization M is controlled by the ambient field B. Metal film thicknesses in the 100 to

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250 V+ Ru Gold

Ta

200

CoFeB SiO2

MgO(001)

V

MR: 236 % Ravg: 3.84 kΩ

Ru IrMn

TMR Ratio (%)

CoFe

150

Tn SiO2

100

50

0 –1500

–1000

–500

0 500 Magnetic Field (Oe)

1000

1500

Fig. 11.9. Performance of a magnetic tunnel junction MTJ field sensor. Inset shows diagram of thin-film device. (After Shen et al., 2006.)

1000 Å range can be deposited by evaporation or sputtering. The lateral dimensions of the sandwich can be patterned lithographically, so that this unit can be very small. By convention, the magnetoresistance ratio is defined as TMR = (RAP –R P )/R P . The active portion of the device shown in the inset to Fig. 11.9 comprises a sandwich of two CoFeB ferromagnetic films joined by an ultrathin tunnel barrier layer labeled “MgO(001).” The lower CoFeB layer has its magnetization direction pinned by the fixed magnetization of the IrMn layer. The magnetization M of the upper CoFeB layer, the “free layer,” reorients with the ambient magnetic field, B. The tunneling resistance is lower when the magnetizations M are parallel than when they are antiparallel. It is seen that the current flow is perpendicular to the planes of the magnetic layers, while the applied B field is in the plane of the magnetic layer. The overall thickness of the MTJ sandwich can be in the nanometer range. The average resistance of this particular device of Fig. 11.9 is about 3.8 k, and the magnetoresistance ratio at maximum is 236%. The hysteresis in the two curves is related to the fact that directions of magnetic free-layer domains are being switched as B changes. The devices can be fabricated using conventional semiconductor methods, making them inexpensive in large quantities. The devices such as that shown in Fig. 11.9 can have active areas of a few microns or smaller (Shen et al., 2006).

APPENDIX A

Experimental methods of junction fabrication and characterization The steps in obtaining a tunneling spectrum and full characterization of a superconductor, including the Eliashberg function α 2 F(ω), can be summarized as those of junction fabrication and characterization, measurement of tunneling spectra [e.g., I (V ), dV /dI , and d2 V /dI 2 ], and analysis to obtain σ = dI /dV, d2 I /dV 2 , the reduced conductance σ/σBCS − 1, and, finally, the desired α 2 F(ω), S (E), Z S (E), and Z N (E) functions. The present survey of experimental methods concerns the fabrication and measurement steps. The discussion is oriented primarily toward the goal of obtaining superconductive and IETS phonon spectra. Emphasis on recent advances in experimental methods seems appropriate, since several good sources on conventional methods are available. The basic reference in many respects is the experimental section of the review of McMillan and Rowell (1969). An introductory article by Giaever (1969) in the Burstein–Lundqvist volume is also recommended. The book by Solymar (1972) contains sections on the fabrication of both quasiparticle tunnel junctions and Josephson devices. The experimental tunneling review of Coleman et al. (1974) is a recommended source, and there is also useful, experimental information in the IETS reviews of Hansma (1977, 1982) and of Weinberg (1978). Apart from the interesting point-contact tunneling schemes, to be considered separately, producing a tunnel junction involves three steps: obtaining the first electrode, producing a sufficiently thin but uniform and continuous barrier (without degrading the surface of the first electrode), and providing the second electrode without damaging the barrier. Typically, the electrodes are deposited as thin films, but several useful methods are available allowing one of the electrodes to be a single crystal. The tunneling barrier is typically produced by thermal or glow discharge oxidation of the first electrode: important exceptions include Schottky barriers; the oxidation of subsequently deposited ultrathin metal proximity layers such as Al, Mg, Eu, Lu; and deposit of artificial barriers such as Si. The counterelectrode again is usually a deposited thin film.

A.1 Thin-film electrodes The deposit of thin films of metals (possibly intermetallic compounds) is usually accomplished by high-vacuum evaporation from thermal or electron beam sources or by sputtering in an inert atmosphere; reactive sputtering and chemical vapor deposition (CVD) are also viable methods. A general summary of such methods is given by Larson (1974); more extensive sources are Mayer (1955), Holland (1956) (vacuum evaporation), Chopra (1969), and Maissel and Glang (1970). Sputtering and CVD deposition methods and film-etching (removal) methods are emphasized in the edited volume of Vossen and Kern (1978). Apparatus used in tunneling studies illustrates the state of the art in evaporation (in Fig. A.1) and in sputtering (in Fig. A.2). 489

RATE SENSORS

SUBSTRATE, Ts

RATE CONTR.

RATE CONTR.

E-BEAM POWER

E-BEAM POWER

ELECTRON BEAM SOURCES

VACUUM CHAMBER

VAC PUMPS

Fig. A.1. Essential features of an electron beam coevaporation system used in fabrication of A–15 superconductor films. An additional thermal source (not shown) is sometimes used finally to deposit ∼ 10Å of Si to improve the insulating properties of the subsequently grown oxide barrier. (After Hammond, 1975.) MAGNETRON HEAD

BAFFLE

SUBSTRATE TABLE

NITROGEN TRAP

SUBSTRATES

Fig. A.2. Schematic features of a three-magnetron sputtering apparatus used by Rowell. Rotation of substrate table permits growth of multilayer structures, which can be covered finally with a thin deposit of Al. Split magnetron sputtering sources or a less-restrictive baffle permit alloy growth. The extended nature of the sputtering sources and the possibility of variable angles of incidence on the substrate with rotation may promote coverage of microscopically rough metal films with Al. (After Geerk et al., 1982.)

T H I N - F I L M E L E C T RO D E S

491

A.1.1 Evaporated films The diffusion-pumped electron beam coevaporation system of Fig. A.1 (after Hammond, 1975) is used to prepare A–15 structure superconductor films (e.g., Nb3 Sn, Nb3 Al) by simultaneous evaporation of the constituents from separate electron beam sources onto heated substrates. In electron beam-heated sources, a focused high-energy (∼ 1 keV) electron beam locally heats and melts the charge, held in a water-cooled hearth. Very simple electron beam– heating schemes have also been described for deposition of films of Nb and Ta (Neugebauer and Ekvall, 1964) and also for simply melting wires of such refractory materials to produce small single-crystal substrates (Gärtner, 1976). Since in these schemes only the charge itself becomes extremely hot, undesirable thermal release of gases (e.g., from the boat or basket, in thermal evaporation) is substantially reduced. Electron beam evaporation is therefore especially suitable for refractory metals, such as the transition metals. Tantalum, e.g., (melting point, 2996◦ C), requires a temperature of 2590◦ C to reach a (sublimation) vapor pressure of 10−4 torr. In such a case, evaporating the charge by heating it in a boat made of a different metal is virtually impossible, although self-heating a free Ta foil with a large current could be used to produce an appreciable Ta flux by sublimation. Typically, however, electron beam guns are operated with the charge in a locally molten state (e.g., T > 2500◦ C for Nb). The turbulent motion of the molten charge tends to produce fluctuations in the rate of evaporation, which must be accurately controlled if codeposition with a uniform and predictable stoichiometric ratio is required. In this case, modified ion gauge tubes sampling the particle flux have been developed (Hammond, 1975) as evaporation rate monitors. The outputs from these rate monitors are compared with preset values and the differences used in a feedback circuit to correct the evaporation rate through the power supplied to the electron beam source. This control system has been described by Hammond (1975).

A.1.2 Film thickness measurement The scheme of Hammond (1975) described above is a deposition rate monitor, the output of which could be integrated to obtain the film thickness. A more common method of determining deposit thickness is with a quartz crystal microbalance. This device is an exposed wafer of quartz with an evaporated metal electrode that is incorporated as the frequency-determining element (typically 5 MHz) of an oscillator circuit. The resonant frequency is electronically counted to the nearest hertz during deposit of metal in order to detect the small frequency change resulting from mass loading of the oscillating quartz crystal. The deposit thickness is expressed typically by an empirical law as t 2 f /ρ, where t is the thickness of the film in angstroms, ρ is its mass density in grams per cubic centimeter, and f is the frequency change in hertz. Calibration of such devices can be achieved by accumulating a thick deposit: for t 104 Å or more, t is determined by weighing the substrate before and after deposit; in thicknesses t 103 Å, it is determined by measuring the deposit thickness by using an optical interferometer (a typical commercial instrument is the Angstrometer, marketed by Varian Associates) or by the mechanical deflection of a stylus drawn over the edge of the film. Typical of commercial instruments of the latter type is the Dektak, manufactured by Sloan Technology Corporation. Other methods suitable for thin films include measurement of the film resistance (typically t ≥ 100 Å before electrical continuity occurs), optical reflection

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or absorption (Mayer, 1955), or shift in the polarization of reflected light in the method of ellipsometry (see, e.g., Knorr and Leslie, 1973).

A.1.3 Substrate temperature The morphology of deposited metal films is influenced by several parameters, including the temperature of the substrate. In general, a very low temperature inhibits mobility of the deposited atoms, limiting the growth of microcrystals, and, in some cases, creating an amorphous deposit. High temperatures aid growth of microcrystals. Thus, Pb deposited onto a 4.2- K substrate may result in an extremely smooth amorphous film [see Fig. 6.16 after Knorr and Barth (1970)], while Pb deposited on a substrate held at 77 K [Fig. 8.1, after Jaklevic et al. (1971)] is known to be composed of crystallites of typical dimension comparable to the deposit thickness with the [111] direction normal to the substrate [but with unspecified azimuthal orientation; such a film is said to have [111] texture]. The variation in individual crystallite thickness in the [111] direction is about 10% in the average thickness range ∼ 100 to 1000 Å. The intermediate substrate temperature 77 K thus produces a rather smooth but polycrystalline film, and higher substrate temperatures would be expected to produce larger crystallites and a rougher film, whose roughness would increase with thickness. The typical dull appearance of thick Pb films is an indication of such conditions. The scale of temperature required to achieve microcrystals of a given size depends on the metal being deposited (and also on other factors, including the evaporation rate). Thus, for the growing of good films of transition metals (Nb or Ta) or transition metal compounds (Nb3 Sn, Nb3 Al), substrate temperatures in the range 800 − 1000◦ C are frequently found necessary. A useful guide to the degree of order in a grown film is the residual resistance ratio (RRR), usually defined as the resistance at 300 K divided by the resistance at 4.2 K (or just above Tc , if the metal is a superconductor). The RRR is essentially the electron mean free path at 4.2 K expressed as a multiple of its 300-K value, which is usually phonon limited. Experimental values typically range from unity to a few hundred. Heating of substrates is conveniently accomplished with quartz–iodine tungsten lamps, whose radiation is focused by reflectors mounted inside the vacuum chamber. Higher temperatures can be achieved with radiation from bare W filaments and if care is taken to limit the thermal conduction from the substrate via its mounting. Cooling of substrates to near 77 K is possible by passing liquid nitrogen through the substrate holder using thermally insulated vacuum feedthroughs.

A.1.4 Sputtered films A convenient way to avoid the extreme temperatures needed to thermally evaporate refractory metals is to use dc or rf sputtering, usually in a purified argon atmosphere and typically at pressures in the range 10−3 to 1 torr. In sputtering, the source or target material is biased negatively at −V0 , and the desired flux of atoms to the substrate is generated by positive-ion collisions onto the target at energy eV 0 from the ionized gas. Advantages of this method include the ability to produce films of compounds having nearly the same stoichiometry as the target, even though the constituent metals may individually possess very different vapor pressures (which would make thermal evaporation of the compound difficult to control), and convenience in controlling the rate of sputter deposition, by varying

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the target voltage or inert gas pressure. Conditions optimizing the production of highly ordered and pure films of refractory metals are roughly those maximizing the sputtering rate and minimizing the sputtering gas pressure (to reduce impurity incorporation from water and other impurities in the inert gas) consistent with kinetic energies of arriving atoms on the substrate not exceeding a few electron volts at optimum growth temperature (to avoid damaging the growing film). One solution of the experimental problem posed by the conflicting requirements is achieved in dc magnetron sputtering. Here, a magnetic field converging toward the source (target) traps electrons in the target region, greatly increasing the supply of ions and hence the ejected flux in this vicinity. This enhancement is still consistent with low target voltage V0 (500 − 1000 V), which facilitates achieving a nearly thermalized particle flux, and relatively low sputtering gas pressure (2 − 4 × 10−3 torr). The magnetron sputtering apparatus shown schematically in Fig. A.2, capable of producing good sputtered films of refractory metals and intermetallic compounds, is of this type. An excellent source of information on sputtering configurations is the edited volume of Vossen and Kern (1978). In special cases, reactive sputtering (see, e.g., Thornton and Penfold, 1978) may be used to advantage. For example, nitrides such as the cubic superconducting compounds NbN (Keska et al., 1971; Komenou et al., 1971) and VN can be sputtered by using N2 (or N2 –argon mixtures) as a reactive sputtering atmosphere and Nb as the target.

A.1.5 Chemical vapor–deposited films In a typical procedure for chemical vapor deposition (CVD) of a metallic film, reactant gases flow into a quartz tube in a tubular furnace, which is adjusted to provide an axial temperature gradient, and the substrates are located in the cooler region of the furnace. The reaction leading to deposit of metal or a metallic compound is frequently hydrogen reduction of a gaseous chloride: e.g., the reaction 5H2 + 2NbCl5 → 2Nb + 10HCl could be used to deposit Nb. Many variations are possible (see, e.g., Kern and Ban, 1978, and references therein), and excellent films and crystals have been achieved. Notably, superior Nb3 Sn films can be made in CVD by hydrogen reduction of NbCl4 − SnCl4 mixtures at 900 − 1200◦ C (Enstrom et al., 1970). Films of Nb3 Ge have been similarly prepared by H2 reduction of chlorides typically at 900◦ C (Newkirk et al., 1976). As a further example, Oya et al. (1974) have reported epitaxial deposit of NbN films on single-crystal MgO substrates at about 1000◦ C by reaction of gaseous NbCl5 with NH3 and H2 in an open-flow system. Finally, carbides such as NbC (Caputo, 1977) and VC can be produced similarly, with the carbon introduced via CCl4 , CH4 or other simple hydrocarbons.

A.1.6 Epitaxial single-crystal films Epitaxial films are grown upon single-crystal substrates of suitable lattice constant; growth, at a suitable temperature, occurs with long-range order stabilized by and commensurate with that of the substrate. An early review of this phenomenon is that of Pashley (1967); more recent sources are Arthur (1976), Mathews (1975), and Cullen et al. (1975). Some idea of the flexibility in achieving epitaxial relationships is indicated by the fact that Nb and Ta can be grown (Durbin et al., 1982) in four different single-crystal film orientations on four different faces of single-crystal sapphire (Al2 O3 ). The possibility of growing single-crystal

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Nb–Ta superlattices starting from individual epitaxial films on MgO and Al2 O3 has also been demonstrated (Durbin et al., 1981). Not surprisingly, many properties of epitaxially grown films, due to the higher degree of crystalline order, are vastly superior to those of their polycrystalline counterparts.

A.1.7 Atomic layer deposition Complete atomic layer control over growth processes is desired in nanotechnology. Epitaxial growth by molecular beam epitaxy is one well-known example of such a process. A more general approach to producing an epitaxial growth of more complicated layers, such as metal oxides, has been developed based on self-limiting surface chemistry. This approach has been demonstrated in cases where a binary chemical vapor reaction can be separated into two halfreactions. As schematically illustrated in Fig. A.3, a first species A must be found which epitaxially grows on the substrate of interest. The completed epitaxial layer of A is then flooded with species B, which reacts with all available A sites, completing one epitaxial molecular layer. This process is then repeated. A celebrated success of this approach is in the “high kappa” oxides now routinely grown on Si surfaces to make field effect transistors. The problem was that the scaling of native SiO2 was leading to layers so thin that tunnel-gate leakage occurred. A requirement of the gate insulator is to allow a sufficiently high capacitance to pull into the conductive channel an adequate density of mobile carriers under the bias available in the device technology. By going from SiO2 , κ = 3.9, to a heavy metal oxide such as hafnium dioxide, of larger permittivity

A

Desorbed Products

+

Desorbed Products

Repeat

+

B

Fig. A.3. Schematic depiction of the atomic layer deposition process using self-limiting surface chemistry in an AB binary reaction sequence. In each step, gases are released. Epitaxial growth of compound layers is demonstrated. (After George et al., 1996.)

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κ = 20 − 22, a thicker layer, sufficient to block tunnel leakage, retains adequate capacitance per unit area. Reactions that are mentioned by George et al. (1996) include the following: HfCl4 + 2H2 O → HfO2 + 4HCl 2Al(CH3 )3 + 3H2 O → 3Al2 O3 + 6CH4 The gaseous products are released when the surface is held at a suitable temperature. It would seem possible to supply the first A species in some cases in an inert carrier gas at suitable temperature, following chemical vapor deposition (CVD) methods, rather than in high vacuum from an oven, as would be used in molecular beam epitaxy (MBE).

A.2 Foil and single-crystal electrodes Unprecedented fully crystalline Fe/MgO/Fe tunnel junction structures have been fabricated by Wulfhekel et al. (2001). These devices have been essential in adoption of the magnetic tunnel junction as a basic disk storage–reading device, leading to miniaturization of hard drives. The authors found that MgO(001) will form on Fe(001), (Fig. A.4), and, again, that Fe(001) can be grown to complete a tunnel junction. The images of Fig. A.4 make clear that MgO can be made to grow epitaxially on Fe(001), to form an (001) oriented crystal with reliable crystal orientation relative to that of the Fe. This produces an interface of extremely high specular quality so that the transverse momentum of the tunneling particle can be conserved. This type of epitaxy has been key to making highperformance magnetic tunnel junction devices. The authors characterized their structures with STM and AFM, and concluded that the MgO buried layer was indeed acting as expected as a tunneling barrier. They reported a few regions in their devices of high conductance, which they attributed to isolated defects in the grown MgO layer. An intermediate between the typical thin film and a true single-crystal electrode is the metal foil. Foils of pure (zone-refined) transition metals of thickness in the range 40 − 100 μm produced by cold rolling of vacuum-melted material are convenient as self-supporting tunnel junction electrodes. As obtained from the manufacturer, these foils typically have small grain (a)

(b)

Fig. A.4. Low-energy electron diffraction LEED spots taken at indicated electron energies on. (a) Clean surface of Fe(001) whisker. (b) Same, after deposition of 5 ML of MgO(001). (After Wulfhekel et al., 2001.)

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sizes and substantial amounts of dissolved atmospheric gases. However, by the simple procedure of rigorous heating in ultrahigh vacuum, the gases can be driven off. Recrystallization of the initially microscopic grains constituting the flat foil into very large, flat crystallites (lateral dimensions of millimeters) can be readily achieved by simple heating for bcc metals and quite possibly in other cases. An easy method of heating a foil is to pass a high current (I < 100 A for 40 μm × 1 cm × 3 cm foil might allow 2000 ◦ C to be reached) through it; smaller samples can be heated by electron bombardment. Optical pyrometry is convenient in monitoring the temperature of such processes, a main practical consideration being to avoid melting the foil. These methods have been described by Shen (1970, 1972a), by Strongin (1972), and, more recently, by Wolf et al. (1980b). A particularly convenient vacuum system, capable of achieving extremely high pumping rates for the copious condensable gases (N2 , O2 ) released by a heated foil has been described by Shen (1972c): one version of this system is sketched in Fig. A.5. This type of system, which has an ultimate pressure near 10−10 torr, has been found especially useful in preparing proximity junctions, where an atomically clean interface between the foil and a second metal (proximity) layer is required. The second (N) layer is evaporated in situ onto the recrystallized surface as soon as its temperature has fallen to a suitable value. A particularly useful characterization method, including those with such overlayers, for foils is with a scanning electron microscope (SEM) in a channeling mode, by which the crystalline orientation of the individual crystallites can be determined. Possible epitaxial relationships can be found by channeling inspection of the opposite side of the underlying foil (S) and comparison of the orientations of the N and S crystals. For a recent description of this method, see Joy et al. (1982). The counterelectrode in a foil-based junction is evaporated through a mask after the tunneling barrier is grown and after suitable insulation (masking) of the foil has been applied to define the tunnel junction area. In many cases, the foil specimens mentioned above can be single crystals over the tunnel junction area. However, the orientation of the crystalline axis in recrystallized foils is usually not easily specified before the tunnel junction fabrication is completed. Single-crystal electrodes of known orientation have been used in a variety of experiments, although seldom in full spectroscopic studies (see, however, Lykken et al., 1970). Extensive tunneling studies have been carried out by using degenerate (metallic) semiconductor singlecrystal tunnel electrodes. An example is the Esaki diode junction, which can be generated by diffusing a high concentration of p-type impurities into a degenerate n-type semiconducting material, or vice versa. For purposes of tunneling spectroscopy, both thermally grown oxide barriers and direct metal–semiconductor (Schottky barrier) contacts have been used. In the Schottky barrier, the transfer of electrons (in n-type material) from donor impurities to the metal–semiconductor interface leaves behind a distributed, positive space charge barrier (Conley et al., 1966; Conley and Mahan, 1967; Steinrisser, Davis, and Duke, 1968). Schottky barriers of a variety of metals deposited onto vacuum-cleaved Si containing 5 × 1018 to 2 × 1019 donors cm−3 provide a well-understood tunneling system in which the Si acts as a single-crystal electrode and also as a single-crystal barrier (Wolf and Losee, 1970). Oxidized barriers on degenerate InAs and several other semiconductors have been employed in elegant tunneling studies (Tsui, 1971a) of the two-dimensional electron gas. In the context of tunneling, single crystals of metals have been employed primarily in studies of the anisotropy of the superconducting energy gap. Metals studied include Al

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L C

Q I K F Sh

A

C W

M

D

S N2

S Ti

Fig. A.5. Schematic diagram of an ultrahigh-vacuum chamber (after Shen, 1972c) used for preparing Al and Mg proximity layers on transition metal foils. Here, (K ) represents a Type 304 stainless steel keg (acid container), with several 1 12 -in i.d. and one 4-in i.d. flanged tubulation. Filaments of Mo–Ti are used to deposit titanium metal onto the walls, maintained at 77 K by immersing (K ) in an insulated container (D) filled with liquid N2 . The foil (F), shielded from Ti by shield (S), is heated slowly to near its melting point by current from copper feedthroughs (C). The pressure during this process, monitored by the ion gauge (I ) and quadrupole mass spectrometer (Q), is typically 10−9 torr. Aluminum is evaporated onto the cooled foil from the source ( A), monitored by a quartz microbalance (M). Deposit on the foil is controlled by the shutter (Sh) positioned (not to scale) by the linear motion feedthrough (L). (After Wolf et al., 1980b.)

(Blackford, 1976), Ga (Gregory et al., 1971), Pb (Blackford and March, 1969), Nb (Bostock et al., 1976; Schoneich et al., 1979), Sn (Zavaritskii, 1964, 1965; Blackford and Hill, 1981), and Re (Ochiai et al., 1971). The procedure employed by Zavaritskii (1964, 1965) for Sn was to crystallize the molten metal between glass plates. Blackford (1976) (see also Blackford and March, 1969; Blackford and Hill, 1981) has had success in crystallizing vacuum-melted bulk samples of Al, Pb, and Sn, held in shape before crystallization simply by surface tension. In a subsequent annealing step, recrystallization occurs, and flat facets of several crystallographic orientations form, which may be sufficiently large to accommodate a tunnel junction. The orientation under individual tunnel junctions can be determined by Laue back-reflection x-ray analysis. This type of crystallization procedure becomes more difficult with metals of high melting point. For example, following earlier work by MacVicar and Rose (1968), Schoneich et al. (1979) report

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annealing an Nb single crystal (4.5 cm in length, 0.6 cm diameter) inductively to 2300◦ C (measured on a pyrometer) in a UHV chamber at 1 × 10−10 torr. A relatively simple approach to vacuum melting and small crystals of Nb has been described by Gärtner (1976). An Nb wire is mounted in a UHV chamber, biased positively to accelerate electrons provided by a nearby tungsten filament. The tip of the Nb wire melts and recrystallizes in an approximately spherical shape. Such a small sample is likely to be of higher purity than the larger samples because outgassing from adjacent portions of the vacuum chamber is minimized and because a smaller sample cools more rapidly, decreasing the time during cooling in which it is hot enough to redissolve impurities from the residual gas. Cleavage of metal crystals to form single-crystal tunneling electrodes has scarcely been reported, except in the context of layered materials that can be peeled apart. These include NbSe2 (Clayman, 1972; Frindt, 1972; Morris and Coleman, 1973) and Bi8 Te7 S5 (Lykken and Soonpaa, 1973). Cleavage of a wider range of metals becomes possible when the metal is cold; while this fact has been exploited, e.g., in LEED and photoemission work (Palmberg, 1967; Baker and Blakely, 1972), it has apparently not been used in connection with tunneling experiments.

A.3 Characterization of tunneling electrodes It is often desirable to characterize an electrode film that is the object of study in tunneling measurements by other techniques of surface chemical analysis, structural analysis, etc. In this section, we are primarily concerned with methods that can be used in situ to characterize a film (or single-crystal surface) before the tunneling barrier is formed. We have seen, especially in connection with normal-state tunneling, that differing interface conditions, related to the properties of only a few monolayers near the interface, can have profound effects. The same sensitivity to conditions at an interface, again relating essentially to surface conditions in interfacial layers, occurs in proximity tunneling. In this case, specular transmission at the NS interface of a C–I–NS junction leads to results (de Gennes and SaintJames, 1963; Arnold, 1978) differing from those expected (McMillan, 1968b) if a weak barrier or simply diffuse scattering occurs at the NS interface. Also, in cases of high Tc , shortcoherence-length superconductors, serious reduction in the gap parameter can occur in even a thin disordered or contaminated electrode surface region just below the tunneling barrier, generally diminishing the superconducting (E) and related properties observed by tunneling. It has even been suggested (Rowell, 1980) in A–15 superconductors such as Nb3 Ge, where ξ v F /π 70 Å (for v F 107 cm/s and 3 meV), that relaxation of even a perfect crystal near its surface could lead to measurable reduction in the surface pair potential. The surface properties of a tunneling electrode are thus often of interest as a guide to whether or not subsequently obtained tunneling spectra will truly reflect the bulk properties of the material under study. A typical experimental apparatus for analyzing surface properties is shown in Fig. A.6. A basic tool for surface chemical analysis, Auger spectroscopy (reviewed, e.g., by Chang, 1971), is available to identify atoms in the first few atomic layers of the surface. The same instruments, an electron gun, and an electron energy analyzer of the cylindrical mirror (CMA) type described by Palmberg (1974), are used to perform reflection electron energy loss spectroscopy (ELS). A comprehensive review of ELS is given by Froitzheim (1977); for an example of an application to transition metal surfaces, see Schubert and Wolf (1979). The

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Ar

IG

RG A

W

P

W

S E

UV

D I L CMA

Fig. A.6. Schematic diagram of a vacuum chamber, operated with base pressure 10−10 torr, for cleaning, analyzing, and depositing N layers on foil samples (S). Foil (S)can be rotated to large window (W ) for pyrometer observations of outgassing (heating) by currents up to 80 A. As shown, (S) is at the focal point of the electron gun (E), argon-ion sputter gun (I ), windowless helium resonance lamp (UV), and double-pass, cylindrical mirror, electron energy analyzer (CMA). Rotation of (S) to position D allows deposition of Mg, Al, or other metal on (S), monitored by a quartz microbalance (not shown). Residual gas analyzer (RGA) and ion gauge (IG) monitor pressure, while leak valve (L) permits introduction of gases. (After Schubert and Wolf, 1979.)

information available from ELS includes plasmon energies, which are in the simplest case a 1 measure of the “free” electron concentration Ne , through the relation ω p = [4π Ne e2 /m] 2 . On a more sophisticated level, the quantities measured in ELS are the volume and surface loss functions, defined, respectively, as Im(−1/˜ε ) and Im[−1/(˜ε + 1)], where ε˜ = ε1 + iε2 is the complex dielectric function. These quantities depend sensitively on the bonding and collective electronic properties, in contrast to the essentially atomic information provided by Auger spectroscopy. Thus, ELS can be especially useful if the electrode is a compound that may or may not be metallic, depending not only upon the stoichiometry actually achieved (which can be estimated from Auger spectra) but also on the crystalline phase. A useful addition to a system such as that shown in Fig. A.6, in fact, would be reflection highenergy electron diffraction (RHEED) (Pashley, 1956), from which the lattice parameter and crystal structure of a polycrystalline deposit can be determined in situ. An additional probe of electronic properties very close to the surface (∼ 10 Å) available in the system of Fig. A.6 is ultraviolet photoemission spectroscopy (UPS), which, in simplest terms, probes the density of

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filled electron states below the Fermi level. This measurement also is extremely sensitive to surface contamination such as O or N. Useful reviews of UPS are given by Plummer (1975) and Feuerbacher and Fitton (1977). Other surface spectroscopies that might be applied in a system of the type shown in Fig. A.6 include x-ray appearance potential spectroscopy (APS) (Park et al., 1970; Park, 1975) or inverse photoemission spectroscopy (IPES) (see, e.g., Dose et al., 1981; Woodruff and Smith, 1982). These methods probe empty states above the Fermi level and hence are complementary to the photoemission spectroscopy. Other useful methods of analysis of films or foils generally require transfer of the sample to a different apparatus; these include the usual x-ray methods, transport measurements, etc. We will briefly describe two methods, SEM channeling and Rutherford backscattering, which are particularly useful in characterizing metal tunneling electrodes. Electron channeling, recently reviewed by Joy et al. (1982), allows determination of the crystal structure of a film or other solid surface on a microscopic scale in a scanning electron microscope (SEM). The patterns observed in the SEM arise from modulation of the backscattered electron intensity when the angle θ of incidence (varying in the usual scanning of the beam) crosses the Bragg angle given by 2d sin θ = nλ. For channeling measurements in an SEM at a 30-keV beam energy, corresponding to a typical Bragg angle of 40 mrad, the electron beam is usually reduced in its angle of convergence (defocused) to between 0.5 and 3.0 mrad. Characteristic channeling patterns are then visible in the usual SEM display of scattered intensity vs. position, revealing Bragg planes and, thus, the symmetry of the microcrystal being scanned. The usual SEM topographic image of precisely the same region can also be observed simply by increasing the beam convergence cone angle to more than the Bragg angle, ∼ 40 mrad. A microscopic crystal structure identification is thus possible, which can usefully be applied to the region of a tunneling electrode specifically facing the tunneling barrier. The technique of Rutherford backscattering is appropriate to diagnosing the composition depth profile of a layer or superimposed layers on the thickness scale of 0.1 to 1 μm, presumed homogeneous over a cross-sectional area of 1 cm2 or so. In this technique, monoenergetic α particles (m = 4u), e.g., at E 0 = 1.8 MeV (usually from a van de Graaf generator), undergo elastic nuclear reflection against individual nuclei (mass M) in the film. The energy spectrum of the 180◦ backreflected α particles is measured: if the film is composed entirely of mass M atoms, then the highest reflected α-particle energy will be Er 0 = E 0 (1 − m/M)2 /(1 + m/M)2 . This expression is derived directly from conservation of energy and momentum in the 180◦ elastic collision event. Measurement of Er 0 thus determines M. The energies of α particles reflected from mass M nuclei at depth x into the sample are found to be systematically shifted to lower energies in amounts predictable from tabulated energy-loss functions, −dE/dx. A homogeneous film of mass M and thickness t then produces a reflected energy spectrum extending from Er 0 to ∼ Er 0 − 2t (dE/dx), a characteristic step-function shape from which t can be deduced. Computer analyses of such spectra permit accurate modeling of film composition vs. depth, as done, e.g., for Nb3 Ge by Testardi et al. (1975). Reviews of this method have been given by Nicolet et al. (1972) and by Mayer and Ziegler (1974). A further technique useful for characterizing the degree of local order in a material is EXAFS [extended x-ray absorption fine structure; see, e.g., Kincaid and Eisenberger (1975) and references therein], from which, in principle, the radial distribution function of neighbors around specific atoms in a structure can be obtained. For example, application of this method to sputtered Nb3 Ge films is reported by Brown et al. (1977).

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A.4 Preparation of oxide tunneling barriers From an experimental point of view, the tunneling barrier is usually the most critical junction element, as well as the most difficult to characterize. Its small thickness, ∼ 15 to 100 Å, requires high dielectric strength and tends to limit independent chemical analysis to the rigorous methods of surface physics. Large changes in electron barrier transmission may result from defects providing localized electron states, from fluctuations in thickness, as well as from defects such as pinholes in evaporated barrier films. While the thermally grown oxide is the most widely used barrier, single-crystal barriers, either a surface depletion layer of a semiconductor (Schottky barrier) or a wafer of layer compound, are alternatives that have been used in tunneling studies. Such barriers permit more complete characterization than do thermal oxide barriers, whose beauty lies in their convenience.

A.4.1 Thermal oxide barriers From the early work of Fisher, Giaever, and their colleagues, as described by Giaever (1969), it was established that the simplest of approaches to oxidation of Al, Cr, Ni, Mg, Nb, Ta, Sn, and Pb, “oxidation in ordinary air at room temperatures,” can lead to insulating oxide barriers suitable for tunneling measurements of the superconducting gap. According to Giaever (1969), Cu, La, Co, V, and Bi are “difficult,” while Ag, Au, and In were rated as “impossible” for the purpose of growing thermal oxide tunnel barriers. A useful collection of oxidation conditions found suitable in several cases is available in the review of Coleman, Morris, and Christopher (1974). In approaching the conditions appropriate to oxidize a given metal, one should first obtain a highly resistive “capacitor junction,” from which a capacitance estimate of the oxide thickness can be obtained and used to guide changes in the oxidation time or temperature. Inability to obtain a high-resistance junction with even a thick oxide is a pessimistic indication of the prospect of making a thin (∼ 20 − Å) barrier with good properties. A different experimental approach to monitoring oxide growth to obtain a suitable barrier thickness is through measurement of the optical properties of the growing film. Figure A.7 (Sixl et al., 1974) illustrates an in situ evaporator-cryostat equipped with a light source, monochromator, and detector. This apparatus is used for monitoring thicknesses of ultrathin Al, Pb, and In metal films through their optical absorption in a parallel geometry as described by Mayer (1955). The same apparatus is utilized to monitor deposit of ∼ 20 − Å insulator (tunnel barrier) films, using a method of interference colors (Heavens, 1965). A sensitive optical instrument to monitor film thickness is the ellipsometer (see, e.g., Passaglia et al., 1964; Smith, 1969), which precisely measures rotation of the plane of polarization of linearly polarized light incident upon the growing film. This method can be made extremely sensitive to changes in film thickness. For example, Knorr and Leslie (1973), using an automated instrument (Ord, 1969) with an He–Ne laser source, state that the plane of polarization of the reflected light can be measured to ±0.01◦ , while a 10-Å film leads to a change in polarization angle of 0.53◦ . These numbers imply that a change in average film thickness of ∼ 0.2 Å is detectable. Identification of precisely what film it is whose thickness has changed, however, is not always possible. For example, simple surface adsorption of gas or water vapor might easily give a large signal in such an instrument, and this signal could be confused with growth of the desired stoichiometric oxide film.

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A

Tungsten lamp

A Filter

B

ZEISS M4 QII

C

Cryostat Shutter A

Evaporation sources

Bellows

Fig. A.7. Specialized apparatus for deposition of ultrathin and controllable amounts of three different materials on a cryogenic substrate, with means for determination of the thickness of the deposit. A substrate is indicated at point C, a liquid-helium temperature shield is at B, and liquid-nitrogen temperature shields are indicated as A. Bellows are employed to allow positioning of the selected evaporation source, minimizing the solid angle of room-temperature radiation as seen by the cold substrate. (After Sixl et al., 1974.)

The physics of the growth of thin, self-limiting oxide layers on certain metals is understood in outline, although details of the oxidation conditions and particularly the defect structure of the initial metal can have a large influence in specific cases. An excellent source of information on mechanisms is the book by Fromhold (1976). A large body of experimental information and interpretation is also available in other books, including those of Kofstad (1966) and Samsonov (1973). The most successful tunnel barriers, as listed by Giaever (1969), are examples of oxide films that grow at a rate decreasing with thickness (self-limiting). This property has the desirable consequence of producing uniformity of film thickness, for in its thinner regions, the oxide will grow faster, thus tending to correct their thickness deficiency. The self-limiting, low-temperature growth processes are empirically described by two logarithmic laws for the thickness d : d = K 1 log(t + t0 ) + C1 and 1/d = C2 − K 2 log t, where d is the thickness and t is the time. At high temperatures, a parabolic growth law, d 2 = K 3 t + C3 , is known, which is associated with rate limiting by a thermal diffusion process. Another well-known limiting case is linear growth, d = K 4 t + C4 , which is associated with situations in which some feature of the surface, independent of d, limits the rate. The cases of self-limiting oxide growth, of primary interest in connection with barriers for tunneling, has been associated with rate limiting by the tunneling of electrons from the oxide–metal interface to the outer oxide surface, where incident oxygen molecules must be transformed into negative ions. This electron tunneling current associated with film growth is expected to vary with d as exp(−2κd) and is the central feature of a model first advanced

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by Mott, (1939, 1940) (see also Fehlner and Mott, 1970). A later proposal of Mott (1947), expanded upon by Cabrera and Mott (1949), described rate limitation by nonlinear, fieldassisted diffusion. This proposal was based on the assumption of rapid electron equilibrium across the growing film (established either by tunneling or thermionic emission), with rate limitation by the necessary diffusion of the negative oxide ions through the oxide to the oxide–metal interface. This diffusion, moreover, was assumed strongly affected by an internal electric field resulting from the electron equilibrium mentioned above—hence the term nonlinear diffusion. Numerical investigations of more general models based on the earlier work cited above have been carried out by Fromhold and Cook (1967) and are discussed by Fromhold (1976). The results from the generalized theory show rate limiting, at first controlled by nonlinear diffusion (Mott and Cabrera theory) and later by electron tunneling (as first proposed by Mott) and leading again to a logarithmic growth law. It is certainly to be expected that the growth rates will generally be increased by crystalline imperfections. The role of other parameters is not as easily guessed, but it is known experimentally that humidity is one parameter that can profoundly influence the oxidation rate. For example, oxidation of pure (freshly evaporated) aluminum in situ in an ultrahigh-vacuum environment with pure O2 gas is generally reported to proceed extremely slowly at room temperature; the rate certainly increases with increasing humidity. Other factors, including the orientation, can be important. For example, the initial stages of oxidation of pure singlecrystal Al have been reported as anisotropic, with an initially stable chemisorbed O2 state Al(111), reported Flodström et al. (1978), and a transition from an initially amorphous oxide on Al(111) transforming to a crystalline oxide at elevated temperatures (Lynn, 1980). A remarkable increase in the rate of oxidation of Pb with increasing humidity is shown in Fig. A.8a (after Garno, 1977). Garno’s paper is recommended for a discussion of practical steps in growing reproducible oxide barriers—and also in applying evaporated counterelectrode films without damaging the grown oxide—to produce excellent SIS I − V characteristics at 1.2 K, as shown in Fig. A.8b. A summary of the recommendations of Garno for making good Pb–I–Pb junctions includes the use of heat shielding during the counterelectrode evaporation, scrupulous use of only filtered oxygen as input to the gas humidifier and furnace (anhydrous CaSO4 filters were used), and use of the lowest possible oxidation temperature to avoid irreproducible results and defects. Other “chemical” influences, besides humidity, on the rate and/or quality of oxidation have been reported and suggest further research. One of these is the reported beneficial effect of acetic acid vapor in air oxidation of In (Hebard and Arthur, 1977). A second material reported to oxidize better in the presence of acetic acid vapor is Nb3 Sn (Rudman et al., 1979). In principle, similarly grown surface-insulating layers of nitrides or other compounds should provide workable tunnel barriers on some metals. Apart from brief reports of AlN grown on Al (Lewicki and Mead, 1966; Shklyarevskii et al., 1974), there seems to be almost no literature on this approach. The possibility of using “reactive” excited-state N2 (e.g., see D’Silva et al., 1980) to grow nitride layers might be worth investigation.

A.4.2 Plasma oxidation processes In plasma processes, oxygen is supplied to the surface of the oxidizing metal in the form of negative ions with additional kinetic energy deriving from the discharge, itself typically driven by voltages between a few hundred to 1000 V. There are two principal variants of the

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plasma oxidation process: using a dc voltage to sustain the discharge (Miles and Smith, 1963) and using an rf voltage (Greiner, 1971). These processes assist in oxidation in three ways: by providing ions, rather than molecules, which are needed in the growing oxide layer and thus accomplishing one possibly limiting step; by adding some control over the electric field, assisting diffusion across the oxide layer; and by providing a kinetic energy larger than kT, making the growth process less sensitively dependent upon temperature and possibly avoiding need for heating the sample. An important practical advantage is that the plasma oxidation, which usually does not require heating, can be done in situ with no need to expose the sample before or after oxidation to the laboratory atmosphere. This improved control over the atmosphere during oxidation is a great advantage, e.g., in IETS, to avoid molecular contamination (see, e.g., Walmsley, 1980), and in the oxidation of easily contaminated surfaces, such as Nb (e.g., see Hohenwarter and Nordman, 1982). The conditions employed by Miles and Smith (1963) included an Al wire cathode, negatively biased relative to the baseplate (at ground potential) by between a few hundred and 1000 V, with a shield to block any direct line-of-sight path between this Al cathode and the oxidizing surface. It is essential that the film being oxidized lie in the negative-glow region of the discharge. The pressure of O2 was ∼ 50 μm, with oxidation times varying between a few seconds and tens of minutes. In most applications of this glow discharge oxidation technique, particularly in IETS, the metal film is left at ground potential. However, a discussion of the application of a specific bias potential to the Al film, as is applied in liquid-phase anodization, is given by Miles and Smith (1963). Variations on this dc method achieved by using an O2 – helium atmosphere and by use of water vapor instead of O2 have been reported by Magno and Adler (1976). An interesting and important use of an rf plasma to provide precisely repeatable oxide thicknesses for Josephson junctions was reported by Greiner (1971). In the rf plasma, consisting of Ar with an admixture of O2 with again peak voltages in the range 200–1000 V, + − competing processes of oxidation (by O− 2 or O ions) and sputtering (by Ar ions) occur in the portions of the rf cycle corresponding, respectively, to oxide positive and negative. Thus, following Greiner (1971), the overall rate dx/dt (where x is the oxide thickness) is assumed to be given by the difference of two terms: dx = dt



dx dt



 − oxidation

dx dt

 sputter

The point here is that (dx/dt)oxidation is a rapidly decreasing function of depth [corresponding to the same self-limiting behavior as that in thermal oxidation, which was observed experimentally in dc plasma growth by Miles and Smith (1963)], while (dx/dt)sputter is essentially independent of oxide thickness and is easily controlled by varying the Ar partial pressure, voltage, etc. Hence, the ratio of the two can be used to set the limiting oxide thickness. In more detail, following Greiner (1971), suppose the oxidation rate decays with thickness as K exp(−x/x0 ), where K and x0 are process-dependent parameters, while (dx/dt)sputter = R is a constant. The balance of the two terms produces a limiting thickness x(t) = x L , given by the condition (dx/dt)total = 0 = Kex/x0 − R, or x L = x0 ln(K /R). The implication that the limiting oxide thickness in the rf sputter–oxidation process can be preset easily with process parameters was demonstrated by Greiner, (1971). Increasing O2 partial pressure

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(a) 10

98% humidity

Conductance (Ω × cm2)–1

102

103 70% humidity

104

50% humidity

105

0

10

20

30 Time (min)

40

50

60

Fig. A.8. (a) Influence of humidity on the rate of growth of oxides on Pb film. (b) Illustration of I − V ˙ for a good Pb–PbO–Pb junction. Shielding the substrate from the full heat of characteristics at 1.2K the evaporation source, filtering of the dry oxygen before introducing it into the oxidation chamber, and maintaining the temperature of the oxygen flow near room temperature were found important in achieving a high yield of Pb–PbO–Pb junctions of excellent quality. (After Garno, 1977.)

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(b)

3

Pb - I - Pb 1.2 K

I (mA)

2

1 × 103

× 102 0

0

1

2 Bias (mV)

3

4

Fig. A.8. Continued

increases tunnel junction resistance, while increasing rf power reduces the resistance. The latter dependence indicates a greater sensitivity of sputtering rate R than of oxidation rate to the incident ion energy. The time constant τ of the oxide thickness equilibration can be deduced from the growth law x(t), obtained by integration of dx/dt (above):     K K xi /x0 −Rt/x0 − −e x(t) = x0 ln e , for x > 0 R R where xi is the initial oxide thickness. Hence, the time constant is τ = x0 /R. Typical parameters are x0 = 1.5 − 3.5Å, while the sputtering rate R is typically about 0.1 Å/ s for power density 0.1 W/cm2 when the O2 pressure is on the order of 10−2 torr. These parameters give τ on the order of 1 minute. Use of rf oxidation in making Nb Josephson junctions is described

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by Raider and Drake (1981). One of the useful variants of the above procedure, in which the vacuum chamber is filled with the oxygen or argon/oxygen mixture, is to keep the chamber pressure low and to introduce only a beam of the ionized gas mixture. This task can be done by introducing gas through a leak valve mounted on the flange of a commercial argon sputter gun, for example. The use of such a reactive ion–beam oxidation technique in fabrication of Nb Josephson juntions has been described by Kleinsasser and Buhrman (1980). One of the advantages of this method is that, since the ion beam here is focused on the sample alone, the oxidizing surface can be kept free of contamination from material that may be released from contaminated chamber walls, etc., if the glow discharge fills the entire chamber. Earlier comments on sputtering as a means for generating tunnel junctions on Nb have been given by Keith and Leslie (1978).

A.5 Artificial barriers The term artificial has become associated with a barrier not grown on the electrode material but deposited artificially in some way onto the electrode. Historically, there has been great skepticism (e.g., see Giaever, 1969) that such an approach can produce pinhole-free barriers in the relevant thickness range.

A.5.1 Totally oxidized metal overlayers At least one counterexample to Giaever’s pessimistic view of 1969 is now known to exist (see Fig. A.9). This barrier, convincingly pinhole-free, was produced on sputtered Nb by subsequent, in situ, sputter deposit of a thin (∼ 20 − Å) layer of Al, which was then totally oxidized. While this Al overlayer technique was actually invented very early in the history of superconductive tunneling spectroscopy by Adkins (1963) and used by Hauser, (1966) and by Hauser et al. (1966) (and probably by others), legitimate concern about definitive interpretation of the early results took two forms. First, one could question the continuity of the initial 20-Å deposit, leading to the real possibility of pinholes, or at least thickness fluctuations, in the resulting oxide. Secondly, the question of the influence on tunneling characteristics of any metallic Al left by incomplete oxidization of the Al metal was not answerable until recently. The answer to the first question—or, rather, demonstration of experimental conditions under which continuity is convincingly achieved—has come in the course of the work reported in Chapter 5. See, specifically, Wolf et al. (1980b) for demonstration of foil-based “aluminum overlayer” samples, and the tunneling work of Rowell et al. (1981) and the XPS study of Kwo et al. (1982) for sputtered films. The question of the influence of the remaining unoxidized metal has also been answered, again, in the course of the development of the PETS proximity tunneling methods (Chapter 5), originating in the paper of Arnold (1978). Other metals that evidently will make continuous layers and hence continuous artificial barrier oxides on clean refractory metals are Mg (Burnell and Wolf, 1982) and rare earth metals Er and Lu (Umbach et al., 1982). General considerations and tabulated instances relevant to the “wetting” or epitaxy of the typically soft artificial barrier metal on the hard S-electrode metal are discussed by Biberian and Somorjai (1979). Thermodynamic considerations at the surface of a two-phase system are carefully discussed by Miedema and den Broeder (1979). However, the understanding of the relevant conditions is regarded as still incomplete. Another question of practical importance, independent of the equilibrium arrangement of a thin layer

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Nb / A艎 – OXIDE – Ag 10 K 42 Å

2%

21.5 Å

dv / dI (ARB UNITS)

7.2 Å

3.6 Å

1.8 Å

t=0

–20

–10

0 VOLTAGE (mV)

10

20

Fig. A.9. The dV /dI spectra of sputtered Ag-oxide-Al/Nb tunnel junctions measured at 10 K. Monolayer thicknesses t of Al sputtered on an Nb surface immediately after Nb deposition are seen to alter the subsequently formed oxide barrier, which implies uniform coverage of the Nb microcrystals by Al. (After Rowell et al., 1981.)

of metal A on metal B, is the time and temperature regime such that interdiffusion of the two metals is of no importance. These considerations are also important when the objective is a metallic overlayer to form an NS bilayer for proximity tunneling.

A.5.2 Directly deposited artificial barriers A different type of artificial barrier, also depending upon oxidation of an initial deposit, is that obtained with ∼ 20 − Å deposit of amorphous silicon (Rudman and Beasley, 1980). This method has been used in quasiparticle tunneling studies on several A–15 superconductors (Moore et al., 1979; Kihlstrom and Geballe, 1981; Kwo and Geballe, 1981). The use of hydrogenated amorphous Si barriers in Josephson junctions on Nb is reported by Kroger et al. (1979) and by Kroger et al. (1981).

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A variety of other materials have been used as artificial barriers. These include evaporated amorphous carbon (MacVicar, 1970), Ge (Giaever and Zeller, 1968b, 1969; Ladan and Zylbersztejn, 1972), CdS (Giaever and Zeller, 1968b, 1969; Lubberts, 1971), CdSe (Lubberts, 1971; Josefowicz and Smith, 1973; Rissman, 1973), and InSb, as well as Ge, by Keller and Nordman, (1973). In addition, ZnO, CdO, and ZnS have been mentioned by Giaever and Zeller (1969) and Te barriers by Seto and Van Duzer (1971) and also by Cardinne et al. (1971), while cryogenically deposited Al2 O3 has been used by Moodera et al. (1982). The narrowband gap oxides In2 O3 and Bi2 O3 have been used by Aspen and Goldman (1976), while we have already mentioned the use of SiO and naphthalene by Sixl et al. (1974) and formvar by Mancini et al. (1979). Finally, wax is used by Beuermann and El Haffar (1981).

A.5.3 Polymerized organic films The use of polymerized benzene barriers is described by Magno and Adler (1977a,b). Earlier reports of organic barriers deposited by the Blodgett and Langmuir (1937) technique are given by Miles and McMahon (1961), Simpson and Reucroft (1970), Léger et al. (1971), and Mann and Kuhn (1971).

A.6 Point-contact barrier tunneling methods The use of a preoxidized wire or Schottky barrier probe in physical contact with, or extremely closely spaced from, a counterelectrode of interest is an appealing idea that has received attention at several times, most recently in terms of the scanning tunneling microscope of Binnig et al. (1982a,b). The objective in these experiments is to observe tunneling, either through a Schottky barrier, oxide layer, or vacuum, and is different from the barrierless, metallic point-contact spectroscopic techniques (Yanson, 1974; Jansen et al., 1977; Jansen et al., 1980; Yanson et al., 1981).

A.6.1 Anodized metal probes The early use of anodized (oxidized) tips of Nb, Ta, and Al to make tunneling measurements of the energy gaps of Nb3 Sn, V3 Si, and V3 Ge was reported by Levinstein and Kunzler (1966). The contact diameter of the junctions was estimated to be less than 10 μm. The tips were etched to a conical shape, heavily anodized, and then brought into mechanical contact with the counterelectrode under liquid helium. It is not clear whether tunneling occurred through a thick anodic oxide layer (presumably of low barrier height) or whether mechanical contacting was used to break away portions of the oxide, leaving a conventionally thin barrier. Levinstein and Kunzler reported that, within limits, the contact resistances could be varied by changing the pressure, controlled by an externally operated screw.

A.6.2 Schottky barrier probes The idea of using the Schottky barrier layer of an etched degenerate semiconductor point (von Molnar et al., 1967) apparently was stimulated by the brief report of Rowell and Chynoweth (1962) of freezing Hg directly onto degenerate silicon and observing tunneling

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through the intrinsic Schottky barrier layer. [This scheme was revived, using clean In probes, by Wolf (1968) and also by others.] The GaAs Schottky barrier probes were more fully described by Thompson and von Molnar, (1970), and the technique was also independently assessed by Tsui (1970c). Singlecrystal p-type GaAs with carrier concentration of ∼ 2 × 1019 cm−3 is typically ground to a conical point mechanically and then etched to remove surface damage, as verified by observation of sharp Laue spots (Tsui, 1970c). The Schottky barrier tunneling properties of p-GaAs, well known from the work of Conley et al. (1966) and Conley and Mahan (1967), were verified in the probe configuration using gold as a counterelectrode (Thompson and von Molnar, 1970) and with Pb by Tsui (1970c). Tsui reported, however, that even with a freshly etched and polished Pb crystal with a mirror-like finish, the “sharp point is needed to penetrate the contaminations of the freshly prepared Pb surface.” Such an embedding procedure also is no doubt helpful in overcoming the sensitivity to vibration that is usually a problem with point-contact methods. While rather nice gap structures and repeatable second-derivative d2 I /dV 2 Pb phonon spectra were obtained, Tsui (1970c) found that the phonon spectra were systematically shifted to higher energy in a fashion consistent with the known effects of high pressure on the Pb phonon spectrum (e.g., Svistunov et al., 1981). Local damage in the region under the tip also seems likely. An in situ sputter-cleaning procedure for the surface opposite the GaAs probe was devised by Thompson and von Molnar (1970). This consisted essentially of a local helium plasma discharge cleaning, arranged with the probe withdrawn slightly from the surface (all immersed in liquid helium), and connected to a small capacitor charged from 3 to 6 V. The probe was then moved back toward the sample until an arc was struck. Evidence is given by Thompson and von Molnar (1970) that this method is effective in sputter cleaning the surface to the extent of improving the gap characteristics, but few phonon spectra resulting from such methods have been published. One problem inherent in this scheme, of course, is that no annealing of the sputter-etched surface, as frequently found beneficial in UHV cleaning, is possible. However, interesting results using this method on intermediate-valence materials have been given by Güntherodt et al. (1982). Improved control over the motion of the tip is available with piezoelectric drives, although this technique is useful only if the level of microphonic vibrations is initially low enough to make this improvement noticeable. In this connection, the improved method of suspension of the point-contact tip (Binnig et al., 1982a,b), is especially significant. An additional important advance in vacuum tunneling has been reported by Moreland et al. (1983) and by Moreland and Hansma (1983).

A.6.3 Deformable metal vacuum tunneling probes With improved stability of suspension and the sensitivity of the piezoelectric drive (∼ 2 Å/V), Poppe (1981) has reported vacuum tunneling determination of superconducting gap characteristics of a cleaved ErRh4 B4 crystal. In this case, the presumably hard and clean ErRh4 B4 cleaved surface is initially contacted with an electropolished Ag or Au tip, which has no inherent barrier and which is reportedly soft enough to deform to match the contour of the local ErRh4 B4 surface. The tip is then retracted to a vacuum spacing of ∼ 100 Å and tunneling spectra obtained. Presumably, wide-voltage conductance spectra would help to discriminate

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between the vacuum tunneling mechanism and an alternative possibility, Schottky barrier tunneling through a conjectured charge compensation layer on the ErRh4 B4 surface.

A.6.4 Analysis of point-contact data The problem of separating various tunneling and nontunneling current contributions at temperatures up to 300 K in point-contact measurements on nonsuperconducting organic metals such as (CH)x (Leo et al., 1982) and TTF–TCNQ (Leo, 1981) has been approached by a mechanical modulation technique described by Thielemans et al. (1979). However, the mechanical modulation seems not to have been utilized in a revised point-contact spectrometer design (Leo, 1982), which also seems inadequate in its approach to the important mechanical stability problem when compared with the design of Binnig et al. (1982a,b). The useful analysis suggestion of Thielemans et al. (1979) is to separate the contributions in “tunneling” point-contact currents arising, in fact, from tunneling, thermionic emission, and resistive shunts or micro-shorts by the markedly more rapid dependence, ∼ e−2κt , of the tunneling current contribution upon barrier thickness, t. Thus, an approach is used in which the pointcontact pressure is sinusoidally modulated by an ac voltage on a piezotransducer, with lockin detection of the current. Stability of the local oxide sample surface configuration under such modulation would appear to require that a purely elastic mechanical operating point be maintained.

A.7 Characterization of tunnel junctions We have already mentioned various techniques, including surface analysis, channeling, and others, which may be used to characterize the component films making up a tunnel junction. The purpose of the present section is to describe several primarily electrical measurements other than the tunneling derivative spectra that can be performed to characterize the completed structure. The techniques for obtaining dI /dV and d2 V /dI 2 spectra will be considered separately.

A.7.1 Initial characterization of junctions Initial tests of a set of completed tunnel junctions to estimate their quality, the absence of shorts, approximate barrier thickness, and so forth are worth mention. A simple measurement of the zero-bias resistance using a low-voltage ohmmeter, to avoid damaging barriers, (typical of such devices available commercially is the Fluke 8024A), should give values varying inversely with the cross-sectional area of the junctions on a given substrate. Further, the resistance should increase slightly as the junctions are cooled to 77 K. The opposite behavior is a sure indication that shorts dominate the measured conductance. Measurement of the junction capacitance will, in principle, determine the average barrier thickness if the area and dielectric constant are known. In practice, this measurement may be complicated by the typically high shunt tunnel conductance, so that an ac method at perhaps 100 kHz [which can be done by using the bridge type of tunneling measurement circuit, originally described by Adler and Jackson (1966), if care is taken to subtract the capacitance of cables, etc., in the measurement circuit] may be necessary. The other point is that, in the presence of any thickness fluctuations, the average barrier thickness inferred

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from the capacitance may well be larger than that typical at the thinner regions dominating the measured conductance. A much higher capacitance than expected may signal the presence of metallic conductivity in the nominal barrier insulator, with tunneling occurring through a Schottky barrier at its interface with one of the electrodes. Such a situation was diagnosed by Magerlein (1983) in Pb–In–An alloy Josephson tunnel junctions for which the oxide barrier grown on the Pb–In–An alloy consists largely of In2 O3 . Magerlein concluded that the In2 O3 layer was likely a degenerately doped semiconductor, with a Schottky barrier occurring at its interface with the electrode metal. The variation of the capacitance with bias V is of interest in the Schottky barrier case, in which 1/C 2 (V ) vs. V ideally gives a linear plot, with the Schottky barrier height VB occurring as the 1/C 2 = 0 intercept and the donor-center density N D given (see, e.g., Sze, 1969) by the slope through the relation d(C −2 )/dV = −2/(qεN D ). Estimates of the barrier heights in the more usual case of a “trapezoidal” barrier can be obtained by modeling the conductance G(V ), as discussed in Chapter 2, and also by examining the current–voltage J (V ) relation over a large voltage range, if this experiment can be done without danger of damaging the devices in question. Two approaches to determination of barrier heights from the J − V data, where J is the current density, will be mentioned. In the method of McBride et al. (1974), a plot of log(J/V 2 ) vs. 1/V (Fowler and Nordheim, 1928) is used initially to obtain the product Sψ 3/2 , where S is the effective barrier thickness and ψ is the effective barrier height. Then, a plot of the experimental log J vs. V data is compared with calculated J − V curves, the latter obtained assuming a trapezoidal model in which the product Sψ 3/2 is constrained to be constant but S and ψ are individually varied. It was found by McBride et al. (1974) that separate determinations of S and ψ are possible in this fashion. This method, when applied by McBride et al. (1974) to Al − AlOx barriers, gave ψ = 2.62 ± 0.12 eV and S = 20 Å for a junction resistance of 1300/cm2 . A related approach is to plot d(log |J± |)/dV vs. V (Gundlach and Hölzl, 1971; Gundlach, 1973), which yields peaks when ±eV = ψ ≷, ψ ≷ being the higher (lower) of the barrier heights. A further variant is a plot of (d2 J/dV 2 )/J vs. V. A second analysis used to determine barrier heights is based on the temperature-dependent ratio J (V, T1 , T2 ) =

(J (V, T1 ) − J (V, T2 ) J (V, T2 )

(Simmons, 1964), which again gives maxima vs. V at ±eV = ψ ≷. This analysis is based on the fact that the temperature dependence of the tunnel current is greatest when eV = ψ, the barrier height. If difficulties are experienced in deducing ψ1 and ψ2 from such methods, one possible recourse, useful if the barrier height is not too large (and is independent of barrier thickness), is to make the barrier sufficiently thick to eliminate the tunnel current, and to measure the thermionic (over the barrier) current as a function of T for fixed bias V . If this thermionic range is attainable, plots of log(J± ) vs. 1/T can be analyzed to give the barrier heights ψ ≷ (for ± biases) as the usual activation energies from the slopes d(log J )d(1/T ). An example of this method for a metal–semiconductor system is given by Lubberts et al. (1974), who also confirmed their results by internal photoemission measurements (Fowler, 1931; Schermeyer et al., 1968; Sze, 1969).

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Barrier thickness uniformity over a large area of oxidized electrode was investigated by Miles and Smith (1963) by evaporating an array of small tunnel contacts, measuring these individually, and plotting contours of equal junction resistance. A rather sophisticated method of assessing barrier uniformity, based on the Fraunhoferlike dependence of the Josephson current Ic (B) on magnetic field B, was described by Dynes and Fulton (1971). Treating a rectangular junction area of sides a, b, with B parallel to the b direction, one should obtain for an ideal junction 2 2 2 sin 1 βα 2 2π(2λ + d)B 2 2 Ic (B) ∝ 2 1 2 2 , where β = 2 βα 2 0 2 where λ is the penetration depth, d is the barrier thickness, and 0 is the flux quantum. Deviations from this behavior are analyzed in terms of a current profile  b/2 J (x) J (x, y) dy −b/2

obtained by a Fourier-transform procedure. The even and odd parts of J (x) are given by Dynes and Fulton (1971) as    ∞ 1 Je (x) = dβ Ic (β) cos θ (β) + βa cos βx 2 −∞    ∞ 1 J0 (x) = dβ Ic (β) sin θ (β) + βa sin βx 2 −∞ where β θ (β) = 2π



∞

−∞

db

ln Ic (b) − ln Ic (β) β 2 − b2

The resulting profiles from the data of Fig. A.10 from an Sn − SnO2 − Sn junction at 1.46 K have been shown in Fig. 2.21. The results indicate that, for junctions with plasmagrown oxides, the current density is greatest at the center of the barrier, but the variation, ∼ 10%, is remarkably small considering the extreme sensitivity of J (x, y) to barrier thickness. In situ scanning methods are also possible to infer properties of tunnel junctions. Reports have included use of scanned optical illumination on superconducting metal films (Chi et al., 1982) and on Al–I–Al junctions (Gilmartin, 1982) and of electron scanning with the tunnel junction mounted on a 4.2 K cold stage of an SEM (Epperlein et al., 1982; Eichele et al., 1983) while monitoring a current or voltage. Subtle changes in the microscopic condition of the electrode in a superconducting tunnel junction may lead to important effects. One example is illustrated in Fig. A.11 (Hebard, 1973), which shows pronounced changes in the phonon spectra of initially superimposed Au/Pb bilayers (due to interdiffusion and compound formation.) Demonstration of such measurements as a diagnostic for film interdiffusion has been given also by Donovan-Vojtovic et al. (1976), also dealing with soft metals in which such effects are possible at room temperature. If one is dealing with bilayers of such metals, it may be necessary to fabricate and measure the structures in situ at low temperature. Methods of doing so have been described, e.g., by Woolf and Reif (1965), Garno (1978), and Bermon and So (1978a,b). In the case of

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Sn – I – Sn

IC (H) (mA)

750

500

250

0

0

2

4

6

8

H (Gauss)

Fig. A.10. Use of the Josephson critical current Ic (H ) dependence upon magnetic field to infer the uniformity of the current density in a tunnel junction. The Ic (H ) pattern at 1.46 K for an Sn − SnO2 − Sn junction is shown. The central peak is reduced by a factor of 5; the circles indicate the positions and heights for the maxima of a true Fraunhofer pattern. (After Dynes and Fulton, 1971.)

transition metals, such interdiffusion effects are negligibly slow at room temperature Geerk et al. (1982). A more subtle property of a superconducting electrode that may noticeably affect tunneling derivative spectra is a state of nonuniform strain. This condition may result, in a polycrystalline evaporated film, from differential contraction in cooling to 4.2 K, if the expansion properties of the film and the substrate differ. One way to minimize such strains is to use the same material comprising the films of the junction also as the substrate. In this fashion, by using a single crystal of Pb as the substrate for an evaporated Pb–I–Pb junction, Banks and Blackford (1973) have obtained and displayed (Fig. A.12) better resolution of Pb phonon features (ω2 , ω3 in the figure) in d2 V /dI 2 than can be obtained by using glass substrates. The evident sensitivity of phonon features to strain is also reflected in the well-known sensitivity of phonon features in transition metal tunnel junctions to disorder or contamination in the region of the electrode immediately beneath the tunnel barrier (or proximity N layer). The distorting effect of voltage drops in thin normal-state electrodes on the energy of observation of gaps, phonon spectra, etc., has been emphasized and analyzed by Osmun (1980).

A.7.2 Derivative measurement circuitry One can identify three distinct measurement tasks in connection with superconductive tunneling and IETS (inelastic electron tunneling spectroscopy (of molecules)), for which the requirements, and hence the optimum experimental circuits, differ considerably. First, measurements in the superconducting gap region |eV| ≤ 1 + 2 of S–I–N and S–I–S junctions usually encounter large changes in conductance, including the possibility

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Pb

Harmonic signal

AuPb3 + Pb

AuPb3

AuPb2

2

4

6 8 10 Bias (mV)

12

14

Fig. A.11. Second-derivative spectra obtained from Pb films overlayed with Au after degrees of aging (top to bottom) show changes indicative of alloying and, finally, formation of intermetallic compounds AuPb3 and AuPb2 . (After Hebard, 1973.)

of negative values of dI /dV . In good junctions with large gaps at low temperatures, one can expect an increase of dI /dV in excess of 103 in crossing eV = 2 . At the same time, resolution of the gap edge in SIS junctions may require voltage resolution on the order of 10 μV (see Fig. 7.7). For study of the negative-resistance region beyond the difference gap eV ≥ | 2 − 1 | in S–I–S junctions, a low-impedance voltage-biasing source (e.g., Blackford, 1971) is required. Secondly, quite different requirements apply to obtain accurate measurements of the tunneling density of states (dV /dI )N /(dV /dI )S = σ in the phonon region. Since σ differs from unity by at most a few percent and itself must be accurate to a percent, one needs stability and resolution in measurements of dV /dI to one part in 104 or better, together with voltage resolution of the order of 100 μV, to preserve details of van Hove singularities in phonon spectra. Since the phonon information is present most directly in the second derivative d2 V /dI 2 , one should preferably measure either this function or the closely related quantity

516

E X P E R I M E N TA L M E T H O D S O F J U N C T I O N FA B R I C AT I O N

w7 w1

d2I / dV 2 (arb. units)

w5

w2

w 3 w4 w6

0

2

4

6

8 Bias (mV)

10

12

14

16

Fig. A.12. Effect of strain on phonon structure observed in tunneling. The peaks labeled ω2 and ω3 in this figure are rarely if ever resolved in films of lead deposited upon glass. In this case, improvement has been obtained by depositing the Pb–PbO–Pb junction on a single-crystal Pb substrate, thereby avoiding differential expansion effects in cooling to cryogenic temperature. (After Banks and Blackford, 1973.)

d2 I /dV 2 = −(d2 V /dI 2 )/(dV /dI )3 . Various forms of the Kelvin double-bridge circuit are in use for these precision measurements. The third case, which does not differ greatly in its requirements from the case of superconducting phonon spectroscopy, is in IETS, where one needs primarily high sensitivity and high resolution in measurement of d2 I /dV 2 or d2 V /dI 2 . The highly accurate (1/104 ) measurement of the first derivative is usually not necessary in IETS, roughly because there is no theory of accuracy equivalent to Eliashberg theory to use in full analysis of IETS spectra. In this application, bridge circuits are usually considered unnecessary, and sharply tuned filters for rejection of the first harmoic and enhancement of the second harmonic are used to obtain the most detailed spectra. In IETS, then, sensible compromises on circuit design are those favoring a high signal-to-noise ratio and permitting better energy resolution at the expense of minor distortion of the scale of d2 I /dV 2 (but not of the voltage measurement). A.7.2.1 Gap region I − V measurements The I − V measurements of S–I–S and N–I–S junctions are accomplished with series connection of the (four-terminal) junction, a current-measuring resistance Rm , and a voltage source V of low internal resistance R V . Circuitry for this purpose is described, e.g., by Adler and Jackson (1966). However straightforward this experiment may seem, great care must be exercised to faithfully reveal the Josephson current, possible Fiske mode structures,

C H A R AC T E R I Z AT I O N O F T U N N E L J U N C T I O N S

517

as well as the cusp and negative-resistance region at eV ≥ | 2 − | and the sum-gap edge eV 1 + 2 . The features of interest have energy widths on the microvolt level; therefore, shielding and grounding must exclude pickup of extraneous signals on this level. A low source resistance R V is needed to observe the negative-resistance regions. In order to observe the maximum Josephson current, one must rigorously exclude flux quanta from the junction. Magnetic shielding is necessary, and it may be desirable to provide for easy warming of the junction above Tc to expel trapped flux. On the other hand, if subgap structure is of interest, the availability of a small magnetic field may be desirable to simplify the spectra by quenching the Josephson current and possible Fiske modes. Such a magnetic field should be provided by an air core solenoid or equivalent near the sample, since it is essential to avoid residual magnetic fields and hysteresis. In this kind of adjustment and subsequent measurements, an oscilloscope display of the I − V curve is a great convenience. We will return to the question of gap-region derivative measurements after a discussion of resistance and conductance bridge circuits. A.7.2.2 Harmonic detection and bridge circuits for derivative measurements The most common approach to obtaining derivative spectra is the dV /dI measurement with sinusoidal current modulation. Constant-current conditions are usually approximated by applying ac and dc biases through series limiting resistors R L whose values are large compared with the typical junction resistance R J . The measured quantity is V (I ), obtained from the separate pair of potential terminals of the four-terminal cross-strip junction. Usually, the current I is swept slowly [I0 (t)] with the addition of a sinusoidal component, I (t) = I0 (t) + (δ I )(cos ωt). The principle of harmonic detection is easily seen by Taylor series expansion of the V (I ) relationship about the value I0 in powers of δ I = I − I0 : 2 2  d V 22 1 d2 V 22 δ I cos ωt + (δ I cos ωt)2 + · · · dI 2 I0 2 dI 2 2 I0 2 2   d V 22 1 d2 V 22 δ I cos ωt + (δ I )2 (1 + cos 2ωt) + · · · = V (I0 ) + dI 2 I0 4 dI 2 2 I0 

V (I ) = V (I0 ) +

Synchronous detection at ω and 2ω, respectively, thus provides signals proportional to dV /dI and d2 V /dI 2 , evaluated at V (I0 ), with the modulation amplitude δ I sufficiently small. The problem of maintaining stability in measurement of dV /dI to one part in 104 or better is usually solved by incorporating the junction in a bridge circuit, matching the junction resistance R J with a decade resistor R S , balanced at R J = R S for the bias region of interest. The signal to the lock-in amplifier (synchronous detector) is then the difference of the voltages across R J and R S . This balance condition also provides a null of the first harmonic signal, which is useful if the second harmonic is to be measured. In Fig. A.13a is sketched the basic double-Kelvin bridge circuit of Rogers et al. (1964), adequate for precision dV /dI measurement of four-terminal tunnel junctions. In this circuit, the bridge unbalance voltage Vω is given by 2     δI d V 22 Vω = cos ωt RS − β dI 2 I0

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E X P E R I M E N TA L M E T H O D S O F J U N C T I O N FA B R I C AT I O N

(a) R R V 1

2

RS

R

Tunnel junction

Sweep

R

(b) R

6

ri

R C

rv

Junction ri dc 5

Rb

Cb

rv

R

R

4

3 ac

1

2

Fig. A.13. In bridge-type measurement circuits, choice of the balancing resistor (R S ) at any given bias voltage enables one to eliminate the first-harmonic contribution to the voltage V , which goes to the lockin detector. This facilitates determination of the second harmonic at high gain. A feature of the resistance bridge circuit (a) is also that large resistors (R) on both sides of the junction reduce the sensitivity of the measurement to changes in the resistance of the leads to the samples, which may occur in variation of temperature. (After Rogers et al., 1964; Adler and Jackson, 1966.) (b) Conductance bridge circuit of Moody et al. (1979) described in the text. The role of the capacitor (C) is to avoid significant errors in the junction voltage measurements with the bridge substantially off balance.

C H A R AC T E R I Z AT I O N O F T U N N E L J U N C T I O N S

519

where the circuit parameter β has the form β = 1 + R S /R B + 2R S /Z and Z is the input impedance of the amplifier (or transformer, if this device precedes the amplifier). The important function of the large series bridge resistors R B  R J in series with both current terminals of the junction is to make the results insensitive to changes in the junction lead resistances R L . The latter typically effectively include the resistance of leads into a dewar, which may vary as the liquid helium level changes and, more importantly, as portions of the connecting leads and films may change from the superconducting to the normal state during the necessary measurement of the normal-state spectra. The improvement brought about by this circuit is best understood by thinking of the series current leads R L to the junction as small parts of the large bridge resistors R B . Rogers et al. (1964) show, in fact, that the change δσ in σ = (dV /dI ) N /(dV /dI ) S due to a change in lead resistance δ R L (for a balanced bridge) is given by δσ = (δ R L /2)(R P /R B R S ), where R P is the resistance of the potential lead to the junction. This value can be kept small with large values, 104 or 105 , of R B . The circuit originally described by Rogers et al. (1964) was refined by Adler and Jackson (1966) and by Adler and Straus (1975) and also adapted to act as a conductance bridge by Rogers (1970). The disadvantages of the resistance bridge are, first, its unsuitability for measuring in the gap region |eV| < 1 + 2 where R J becomes very large and, second, the complication of having subsequently to numerically convert measurements of d2 V /dI 2 to the more fundamental current derivative d2 I /dV 2 . These disadvantages would not be present if a true conductance bridge could be used. In such a circuit, in direct analogy to the resistance bridge, constant-voltage dc and ac biases would be required, and the current in each arm of the bridge would be measured as the voltage across a measuring resistor Rm . In order to satisfy constantvoltage conditions, however, one would have to maintain Rm  R J and also R L  R J . Since these conditions are not easily met, the circuits in use for direct measurement of dI /dV and d2 I /dV 2 resort to feedback circuitry to maintain a constant voltage across the tunnel junction. This experiment was first attempted by Rogers (1970) (see also Blackford, 1971) in a resistance bridge circuit such as that of Fig. A.13a by connecting an operational amplifier input between the sample (R J ) and the balancing resistor R S . The output of this amplifier was connected through a feedback impedance Z F to the upper terminal of the sample R J and used to maintain bridge balance by adjusting the current through the sample R J . The in-phase output Vω of the operational amplifier was monitored to record the conductance, from the equation (Rogers, 1970)   dI δV cos ωt − R S−1 + R −1 Vω = Z F F dV Here, δV is the voltage modulation amplitude, R S is the balancing resistance, and R F is the resistive part of the feedback impedance Z F . This analysis neglects the role of lead resistances R L ; and as later pointed out by Blackford (1971), these will enter the equation as additions to R S . Hence, the bridge balance is potentially affected by the liquid helium levels, etc. This same feature is apparently retained in the subsequent and more completely analyzed conductance bridge circuit of Hebard and Shumate (1974) (cf. Eq. 8 ff, of the latter paper), which does achieve an important improvement in noise relative to the circuits of Rogers (1970) and of Blackford (1971) (which also avoids the lead-resistance drift problem). The fully evolved conductance bridge circuit shown in Fig. A.13b is taken from the paper of Moody et al. (1979), which is recommended also for a review and comparison of the several versions of the conductance bridge circuit. In this circuit of Moody et al. (1979), the firstharmonic voltage measured by the preamplifier is given by

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E X P E R I M E N TA L M E T H O D S O F J U N C T I O N FA B R I C AT I O N

 Vω =

RB R B + RS



 δVR S

dI − R S−1 dV



and is, hence, free of lead-resistance drift effects. The noise analysis presented by Moody 1 et al. (1979) indicates a best-noise voltage value of 2nV Hz− 2 , and the resulting resolution in conductance with 10% unbalance of the bridge is 4 × 10−5 using 35 − μV rms modulation and a 1-s time constant. This result appears to be only slightly inferior to the performance reported for the Hebard–Shumate circuit. The second-derivative signals in all of the bridge circuits are available as the 2ω component of the bridge output and are free of ω content at balance. A.7.2.3 Dedicated second-derivative circuits utilizing filter networks The original electronic circuit of Thomas and Rowell (1965) solved the problem of measuring specifically the superconducting second derivative, at high resolution, by use of a rejection– selection filter network. This circuit borrowed from earlier designs of Hall et al. (1960) and of Thomas (1963), which used electromechanical servomechanisms; the first purely electronic circuit to be reported is apparently that of Patterson and Shewchun (1964). The Thomas– Rowell circuit contains two meshes: an input mesh feeds the fundamental signal ω to the junction and simultaneously suppresses the transmission of any second-harmonic content from the source, while the second mesh feeds the second-harmonic signal generated by the junction to the detector and simultaneously suppresses the first-harmonic output. The second mesh also transforms the impedance level from the low value of the junction to the high value of the preamplifier input. This circuit provides high resolution and low noise, but it has the disadvantages of being limited in resistance level of junctions to be measured and of making only two-terminal connections to junctions. This disadvantage leaves the possibility of errors in measurement of the electrode potential difference if the leads or film contacts should be appreciably resistive or nonlinear in their behavior. This basic approach has been carried forward in several circuits designed for IETS and described, e.g., by Lambe and Jaklevic (1968) and Weinberg (1978). One of the simplest and best of these circuits, judging from the resulting spectra (e.g., Figs. 1.9 and 10.3), is shown in Fig. A.14 (Walmsley, 1983; Walmsley et al., 1983) and will be described in some detail. The main problems in obtaining excellent, high-sensitivity d2 V /dI 2 spectra are to avoid extraneous signals from pickup or ground loops; to avoid generation or feedthrough of the second harmonic from any source other than the nonlinearity of the tunnel junction I − V characteristic; to effectively filter out the first-harmonic signal before the amplifier input, and to obtain optimum impedance matching from the junction to the amplifier or preamplifier. The need for high sensitivity stems from the weakness of the nonlinearity due to molecular vibrations in the tunnel barrier ( G/G 10−2 ) and from the necessity of using small equivalentvoltage modulation (δV 1 mV rms in IETS) to preserve energy resolution of the vibrational structure. Since the second-harmonic signal is proportional to (δV )2 , this requirement is a serious one. The choice of 50 kHz as the fundamental frequency, with detection at 100 kHz, follows Lambe and Jaklevic (1968), who noted qualitatively that junction noise falls off with increasing frequency, while frequencies above about 100 kHz lead to capacitive currents which cause increasing difficulty. The 100-kHz range is also optimum from the point of view of noise figures in typical amplifiers.

521

C H A R AC T E R I Z AT I O N O F T U N N E L J U N C T I O N S

200

510 1k

50 MP 120 C 37.5 V dc battery

22

2.2 k

510 k

5.7 k

100 k

V 2k

10 k 51 k

20 k

2.5 nF ref

4.7mF

10 k

Oscillator 3310 A

10 k

2 mH

1 mH 5 nF

1 50 nF

250 pF

4.7mF

0.2 mH 0.05 mH

5

4

6

3

7

2

10 mH

8

Tunnel junctions

5001 Recorder 9503 ref

Y

7004 B

X

Fig. A.14. Tuned filters in the second-harmonic circuit of Walmsley (1983), designed for IETS measurements, block the second harmonic in the junction modulation input and also block the first harmonic in the junction output to the preamplifier. Earlier circuits of this type have been described by Thomas and Rowell (1965) and by Lambe and Jaklevic (1968). (After Walmsley et al., 1983.)

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E X P E R I M E N TA L M E T H O D S O F J U N C T I O N FA B R I C AT I O N

The circuit of Walmsley (1983) (Fig. A.14) deals with these requirements in a simple and effective fashion. Ground loops and sources of pickup related to the ac mains are minimized by making only one earth connection (no connections to the third pin of the usual ac plug) and by using batteries for the sweep voltage circuit. The second harmonic is filtered from the oscillator output by use of 100-kHz reject and 50-kHz pass filters in the line to the sample. The pickup of airborne signals (including 60-Hz magnetic signals) by the ferrite cores of these filters is simply but effectively dealt with by maintaining a high-level signal from the oscillator into the filters (the HP 3310 A oscillator can produce a 30-V pp signal) sufficient to dwarf pickup signals, followed by a pick-up-free resistive potential divider to reduce the level to that appropriate for the junction modulation. Walmsley (1983) includes a 10 −  resistor at the oscillator to avoid overload (harmonic) distortion of the oscillator at resonance; the 4.7 − μF capacitor to block dc current from the sweep supply through the inductors (which could lower the resonant circuit Q values by polarizing the ferrite inductor cores); and the 10 −  level-reducing divider resistor at the output of the filter, in series with the junctions. Optimum transfer of the second-harmonic (2ω) signal generated by the junction to the high-impedance (108 ) preamplifier input (a Brookdeal-Ortec 5001 in this particular circuit) is accomplished by a step-up transformer whose secondary circuit is tuned to resonance (for each sample) at the second-harmonic frequency 2ω. Walmsley (1983) points out that tuning of the secondary circuit to resonance with the 250-pF capacitor (which is done by using the 5001 as a high-impedance input, rather than with an oscilloscope directly) is possible only if the cable to the preamplifier is short (perhaps 25 cm) so that its capacitance is low. The resonant filter in the transformer primary circuit blocks any 50-kHz signal from the preamplifier, so that it can be operated at high gain on the 2ω signal. It may be useful to quote Professor Walmsley’s (1983) comments on the construction of the high-Q filter and transformer elements that form the heart of this system and on the operation of the system: Every effort was made to optimize the quality factor of the coils in the resonant circuits. The three single coils were wound on ITT ferrite cores type C22 following the manufacturer’s advice on wire diameter. Specifically, the 1 mH inductance has 60 turns of 26 SWG lacquer insulated copper wire, the 2 mH inductance has 85 turns of the same wire and the 0.2 mH inductance has 28 turns of 24 SWG wire. The cores are 2 mm in diameter and 13 mm high with an inductance factor, A L , of 250 nH per turn. The transformer is on a type C26 core (26 mm diameter, 16 mm high) with an A L of 400 nH per turn. The 0.05 mH primary has 10 turns of 24 SWG wire and the secondary has 155 turns of 34 SWG wire to give a self inductance of 10 mH. Typical quality factor values were 50 to 150. The ferrite cores have tuning rods but they do not offer good resolution and it is convenient to have 100 pF trim capacitors on the 2.5 nF and 5 nF components. The circuits cannot be tuned with a standard 1M ohm input impedance oscilloscope; the 9001 preamplifier performs well as a buffer in this task. In recording spectra, a lock-in time constant of 3 sec and sweep time of 40 minutes for 0 to 500 mV offers a fair compromise between requirements of noise reduction and signal lag as against overall time for data acquisition. Switching between samples such as from terminals 4 and 5 to 3 and 6 should always be done with the dc bias set to zero as otherwise the samples may be destroyed.

Tests of performance of the circuit are given in the article by Walmsley et al. (1983).

APPENDIX B

Methods of scanning tunneling spectroscopy and competing approaches B.1 STM basics, tip production, and characterization; single atom tips The original diagram depicting the STM and its operation was shown as Fig. 1.10, above (Binnig et al. 1982b). In this section, we describe more fully a few of the essential elements in the STM. For details the reader is referred to the excellent texts by C. Julian Chen (1993) and R. Wiesendanger (1994). We here provide only a few more details that may help the reader in understanding the descriptions of the most successful STM designs that follow. A generic outline of the electronics needed to keep the tip-sample spacing constant is shown in Fig. B.1. The tip-sample tunneling current under fixed voltage bias is taken as the measure of tip-sample spacing. (Errors, false “topographic” features, may occur if there are local variations in the work function of the surface or in the density of states near the Fermi level. Both effects alter the tunnel current for a given bias voltage. An example of a densityof-states peak making an apparent peak in topography is shown in Fig. 9.7, the ripples are to be understood as quantum states in a two-dimensional infinite potential well.) Following Section 1.6, the current is exponentially dependent on the tunneling gap here depicted “s.” In the feedback loop, the exponential dependence is removed by use of a logarithmic amplifier, and the output voltage of that amplifier is then a measure of the tip-sample spacing. This voltage is compared to a preset voltage, and the difference between the actual and preset voltages here, the error signal, is minimized by the feedback electronics. The final result is a “voltage to z-piezo”. This voltage is plotted as the topograph, the tip/z-piezo height needed to keep the tunneling current to its preset value. The set current is normally in a picoamp Voltage to z-piezo Feedback Electronics

z-piezo

Tunneling Gap s Z

IT

VT

e

Error Signal

Vp

VL –

ZT X

Tunneling Current

Current Amplifier

Logarithmic Amplifier

Set Point for Current

Fig. B.1. Schematic of typical STM control system. Topographic image is represented by voltage applied to (deflection of) the z-piezo as controlled by feedback electronics to minimize the difference between the set tunnel current and the actual tunnel current. This approximately keeps the tip-sample spacing constant.(After Chen, 1993.) 523

524

M E T H O D S O F S C A N N I N G T U N N E L I N G S P E C T RO S C O P Y

PIEZOELECTRIC

X-oc ELECTRODE

Z Y-oc

Y-dc

TIP X-dc

Fig. B.2. Sketch of cylindrical tube scanner. Metallizations on inner and outer surfaces allow radial electric fields to be applied separately in the four quadrants. Attendant distortions allow displacement of tube end, carrying tunneling tip, in three dimensions. For many purposes, this device is more convenient than the tripod device shown in Fig. 1.10.(After Binnig and Smith, 1986.)

. to nanoamp range. In the early days of STM it took some adjustment to realize that a single atom can rather easily pass a current of one nA, since it requires an electron hopping frequency through the atom of only f = 10−9 /(1.6 × 10−19 ) = 6.25 GHz. As an atomic frequency this is small, as seen by comparison to the orbital frequency for the n = 2, l = 1 state of the electron in hydrogen, which is 1.64 × 1015 Hz. The typical scanner (such as that in Fig. B.2) is made of a ceramic called PZT-5H, in a class of lead zirconate titanate piezoelectric materials. Near 45% zirconate the piezoelectric coefficients, typified by d31 = S1 /E 3 , defined as the strain (fractional distortion) in the x direction divided by the electric field in the z direction, (a negative number) and d33 = S3 /E 3 , defined as the strain (fractional distortion) in the z direction divided by the electric field in the z direction (a positive number), are large, on the scale of 2 to 6 Å/ V at room temperature, and falling by perhaps a factor of four at 4.2 K. The change in length x of a given rectangular slab element of length L and thickness h, for example, with a voltage V across the thickness, will be x = d31 V L/ h = KV, which defines a piezo constant K ≡ x/V for a given geometry. For example, with L = 20mm, h = 2mm and d31 = –2.74 Å/ V, so we have a piezo constant K = 28.4 Å/ V. For a tube scanner the calculation is more involved, but one ends up the piezo constants K of similar sizes for three directional displacements. The resonant frequency of the scanner is a figure of merit but can be above 10 KHz for small tubes used in leading STM designs, as we will see. A critical element in any control circuit is the current amplifier directly connected to the tip, which must convert the small current, on the scale of 0.01 nA to 50 nA, to a voltage. Two basic current amplifier circuits are illustrated in Fig. B.3. In each of these, the operational amplifiers are very high gain devices with extremely high input impedance and very low output impedance. In the first feedback circuit (Fig. B.3a), the feedback resistor RFB couples back to the input a current, which will null the signal input current I . The output voltage VOUT that accomplishes this is proportional to the signal input current, i.e., I = VOUT /RFB . The second circuit is a voltage amplifier with a shunt resistor on the input to convert the input

S T M BA S I C S , T I P P RO D U C T I O N , A N D C H A R AC T E R I Z AT I O N

CFB

(a)

525

(b) R1

RFB

IIN –

VOUT

IIN

+

+ CIN

–

CIN

R2 VOUT

RIN

Fig. B.3. Two basic types of current amplifiers. (a) Basic elements of feedback nanoammeter are operational high gain amplifier and feedback resistor (RFB ). Typical value for (RFB ) is 100 M. Undesirable stray capacitances (dashed lines) at input, and around feedback resistor, limit performance. (b) An electrometer used as a current amplifier. The voltage at the input resistor is amplified by the circuit, which consists of an operational amplifier and the resistors (R1 ) and (R2 ). The parasitic input capacitance limits the frequency response. (After Chen, 1993.)

current to a voltage. Discussions of these circuits with attention to methods to improve the frequency response are given by Chen (1993). The tunneling tip is most frequently produced by etching a wire to a point, followed by various cleaning procedures. These tip preparation methods were refined long ago by Erwin Muller (Muller and Tsong, 1969) and his co-workers in the development of the field emission microscope followed by the field ion microscope. A summary of this research is given by Tsong (1990). Often, however, simple methods for making tips are productive. In the work of Naaman et al. (2001), focusing on methods of making reliable superconducting STM tips, PtIr tips were initially cut from a 0.25-mm diameter wire. The tips were selected for quality by imaging the surface of a highly ordered pyrolytic graphite single crystal in an STM at room temperature. Naaman et al. (2001) state that the tips with the best resolution were selected, and placed in a bell-jar evaporator with the tip axis pointing toward the evaporation sources. A 500-nm thick layer of Pb was deposited, followed, without breaking vacuum, by a thin layer of Ag, in thicknesses 3 nm to 6 nm. Results from a 3.0-nm Ag-coated Pb/PtIr tip are shown in Fig. B.4. Note that strong-coupling superconducting features, marked “T” and “L,” from the phonon spectrum of Pb are visible, indicating that the Ag film exhibits superconducting proximity effects as described above in Sections 5.4–5.6. STM tips, beyond cutting a wire, are more frequently prepared by electrochemical etch of a W or PtIr wire to a point. For example, Stipe et al. (1998) state in connection with beautiful experiments on Cu(100) surface: “Tips were made by etching polycrystalline tungsten wire in potassium hydroxide, KOH. Tips were cleaned in ultrahigh vacuum UHV by repeated cycles of sputtering from field emission in a Ne atmosphere and annealing with an electron beam heater. Tips were sharpened just before the experiment by bringing the end of the tip in contact with the clean copper surface. Thus, the last atom on the tip is likely to be copper rather than tungsten.” We have already shown in Fig. 9.19 images from the work of Stipe, the left panel showing a single acetylene molecule on an unresolved copper surface. The right panel shows the same

526

M E T H O D S O F S C A N N I N G T U N N E L I N G S P E C T RO S C O P Y

1.6 T = 2.0 K 1.4 T

Normalized Conductance

1.2

L

1.0

Pt0.8 Ir0.2

0.8

0.6

Pb (5000 Å)

Ag (30 Å)

0.4

0.2 Au Film 0.0

0

2

4

6 8 V (mV)

10

12

14

Fig. B.4. Tips can be made by cutting wire to a point, followed by trial imaging on a reliable sample surface such as pyrolytic graphite. Reliable superconducting tips, the interest of Naaman et al. (2001), were achieved by coating a selected PtIr tip with Pb and then Ag. The data shown reveal the expected superconducting phonon effects (see Fig. 4.1) as seen through the proximity effect in the Ag film, shown in Fig. 5.10. (After Naaman et al., 2001.)

copper surface at atomic resolution. To produce the enhanced image, it is stated in the paper that the acetylene molecule was picked up, and then served as tip to provide clear atomic resolution of the copper surface. Similarly, in the text just previous to Fig. 1.13, Hahn et al. (2000) describe picking up a CO molecule onto a metal tip to provide higher resolution. The C60 molecule has also been attached to a metal tip. In this case, a reciprocity between tip and sample imaging has been demonstrated. If a sharp Au tip is used to image a C60 molecule standing alone on a gold surface, the same image is observed as is observed by a C60 -decorated tip in imaging a gold adatom (isolated gold atom) on an underlying Au surface. In this work, an array of C60 molecules is imaged on the left with an Au tip and on the right with a C60 tip. The underlying surface is Au(111), and the Au tip is actually an etched W tip, cleaned with Ar ion bombardment, and finally indented into the Au surface to pick up a Au atom. In the upper left of the field are three Au adatoms and a cluster marked β consisting of two or three Au adatoms. These features are better resolved in the right panel, where the tip is C60 , with the artifact that the Au adatoms appear to be C60 , really representing images of the tip itself. This confirms that the image of the field of C60 molecules in the lower right

N O I S E - F R E E X, Y, Z TRANSLATION ; VIBRATION ISOLATION

(a)

527

(b)

Fig. B.5. STM images (I = 10 nA, V = 2.5V , 14 × 11 nm2 ) of Au(111) partially covered with C60 molecules (lower right) obtained with (a) metal and (b) C60 tip over the same area. Gold adatoms (α) and a small gold cluster (β) of two or three adatoms are discernable. (After Schull et al., 2009.)

of the right panel, a constant-current topograph, is obtained by tunnel currents between C60 molecules, one on the tip and the other on the Au film. The procedure used with the metal tip to pick up a C60 is described: the tip is placed over a target molecule and the voltage was varied from 2 to 0.01 V and back with the current held at 100 nA. The success of the procedure was recorded (see black arrow in lower right of Fig., B.5b where one can see a missing C60 ).

B.2 Noise-free x, y, z translation; vibration isolation The central task in making a successful STM, in addition to the control system and tips as discussed above, is to provide a stable mounting structure so that a sharp tip can be controllably and stably moved to within angstroms of the atoms on a surface to be imaged. Coarse approach of the tip to the sample must be reliably provided, difficult because the useful operating spacing range is set by a few multiples of the Bohr radius. A successful design must provide isolation against vibration and electrical noise, smooth controlled motion of the tip, and compensation of the system against temperature changes.

B.2.1 The cryogenic STM of Wilson Ho Success in surmounting these challenges is illustrated in the work of Stipe et al. (1999) as indicated in Fig. B.6a. Here is shown the low-temperature portion of this very successful STM. The tip and sample are mounted on an Mo base plate, which is suspended on three Inconel springs; this assembly can be seen in the bottom third of Fig. B.6a. In the drawing, the locations of 4 piezoelectric cylindrical scanning tubes and molybdenum sample holder are indicated. The motion of the tip and sample is controlled in this portion of the apparatus, which is very rigid to minimize vibrations. The central vertical coaxial piezoelectric tube scanner (of the type shown in Fig. B.2) allows smooth angstromlevel displacements of the tip in x, y and z directions, necessary to map the locations of atoms on the sample surface. The sample is rigidly mounted to a molybdenum sample holder above the tip. The molybdenum sample holder is in turn supported by three piezoelectric tubes, similar to the central piezo, each having the same capability of controlled distortion in x,

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(a) Scale 2.5 cm

Continuous Flow L-He / N2 Cryosrat

Sample at 8 K to 350 K

Outer Shield

Cold Tip

Inner Shield

Electrical Feedthroughs

Inconel Springs (3) Inner Access Door

W Balls (3) and Tip

Mo Sample Holder Piezotubes (4)

Mo Base Plate

Hole for Dosing Clamping Screw

Sapphire Laser Window

Samariamcobalt Magnets

Fig. B.6a. Details of the low-temperature STM of Wilson Ho and collaborators. (After Stipe et al., 1999.)

y, and z directions. These tubes have sensitivity to applied voltages of 200 and 30 Å/V in lateral and vertical directions, respectively, at room temperature, which fall by a factor four in cooling to 8 K. Following the design of Besocke (1987), the three piezo tubes allow for coarse motion in the vertical z direction so that the sample and tip may safely approach within the tunneling distance, which is only a few angstroms. This is accomplished in a sophisticated fashion. The tips of the three piezo tubes (which support the molybdenum sample holder) are controlled to slightly distort in the sense of rotation around the z-axis in synchronism and thus

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to rotate the sample holder around the vertical direction. Since the sample holder is merely resting on ball bearings attached to the top of each piezo, the three tubes can be controlled to snap quickly back to their original positions. In this step, the forces are large enough to produce sliding of the ball bearings on the molybdenum sample-holder surface. The inertia of the sample holder leaves it in the rotated condition, while the tubes have returned to the original condition. This stick-slip motion has the result of stepwise rotating the sample holder. The further design feature is that the support surfaces on the upper sample holder are tilted at 2.5◦ from the horizontal, so rotation has the effect of moving the sample in the vertical direction in coarse steps that would be several hundred angstroms. It is stated that the overall range of the electronically controlled motion is 4μm. The central sample and tip assembly is all made of rigid (large Young’s modulus) ceramics and metals having similar and small coefficients of thermal expansion. The authors state that the same sample region can be kept in view in the wide temperature range from 8 K to 80 K. The stiffness and high vibrational frequencies of the tip-sample structure are desirable to avoid unwanted motion between the tip and sample. In a good design the vibrational modes of the tip and sample structure are pushed to higher frequencies while the noise coming from the environment is filtered by suspending the structure, e.g., on a soft spring. In the instrument shown, three Inconel springs support the mass of the sample holder and scanner, leading to a spring system of resonant frequency 2 Hz. This system will not transmit vibrations of frequency above 2 Hz. The desirable outcome is that the noise approaching the sensor region is confined to frequencies below 2 Hz, while allowed vibrations in the motion of the sample and tip assembly are available only above 10 kHz, as the authors state. This is an excellent degree of vibrational isolation. A further aspect of the vibrational isolation is that the whole assembly is in vacuum, eliminating sound waves. Samarium cobalt magnets located on the scanner system close to the copper radiation shield provide eddy current damping of any noise motions that may persist. This particular device is cooled by a continuous-flow liquid helium cryostat allowing temperatures between 8 and 80 K. This method of cooling does not introduce excessive vibrational noise. Figure B.6b shows the overall arrangement of this very successful scanning tunneling microscope built by Wilson Ho at Cornell University. It is remarkable that this whole system, which has produced some of the finest second-derivative data, revealing inelastic vibrations of individual molecules, did not require a separate soundproof room, but was mounted on a laboratory floor. In Fig. B.6b is seen that the cryostat containing the vacuum chamber and the sample assembly is simply mounted on an air-suspension table that is a common laboratory item. The liquid helium cryostat can be wheeled up and connected to this assembly. This is a remarkably efficient and probably relatively inexpensive scanning tunneling microscope capable of the finest spatial resolution on the atomic scale and also capable of determining the vibrational frequencies of individual molecules. The gravity-based stick-slip motion that is used in the coarse approach in this design may be a weakness in the sense that it will only operate in the vertical orientation.

B.2.2 The 240-mK STM design of Pan, Hudson, and J. C. Davis The second low-temperature STM design we describe (Fig. B.7) was developed at the Lawrence Berkeley Lab by J. C. Davis and his collaborators, S. H. Pan and E. W. Hudson (Pan

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(b) L–He Transfer Line Scale 30 cm e–Beam Heater Flange UHV Chamber

Load–Lock Attachment

STM

Carousel Flange

Air Floated Table

Ion Pump

L–He Dewar

Turbo Pump

Fig. B.6b. Overall view of the low temperature STM of Wilson Ho and collaborators. (After Stipe et al., 1999.)

et al., 1999). This is an extremely successful device, which facilitates application of a large (7 T) magnetic field at much lower temperatures, 240 mK, as attained in a 3 He refrigerator. (It is noted that such cooling is free of vibrational noise.) As we see in Fig. B.7, the STM head, containing tip and sample; is rigid and compact, 1.5" diameter and 1.75" length. The active parts are encased in a solid Macor cylinder, enclosing the tip piezo unit, which is rigidly mounted into a hole in a triangular sapphire prism (labeled “6” in Fig. B.7). An essential difference from the Ho design, beyond the temperature and magnetic field capabilities, is the simpler and more robust z-dimension coarse-approach system. Again, the tip is mounted vertically at the top of a tube scanner, mounted rigidly into the triangular sapphire prism. The coarse approach is now described as a friction motor. Six deflectable legs (“shear piezo stacks”) here frictionally clamp the triangular sapphire prism within the strong encasement of the Macor outer body of the STM unit shown in Fig. B.7. (In the drawing only 4 of the 6 legs are visible; in the text it is stated that only 4 legs are driven in the approach method). The coarse approach is accomplished by sliding (deflecting), one at a time, in sequence, each of the four active legs, while the other five legs immobilize the central prism unit. After the 4th leg has been slid, relaxation of the four active legs moves the unit one coarse step in the z direction. (Note that this relaxation step, which apparently

ATO M I C F O R C E M I C RO S C O P E ; C O M B I NAT I O N S T M / A F M

531

(a)

3

4 8

7

6

9 (b)

1 2 3 8

4

6

7 5

Fig. B.7. STM head designed by S. H. Pan, E. W. Hudson, and J. C. Davis. (After Pan et al., 1999.)

slides the two inactive undeflected legs, can be done slowly, so that the tunnel current can be monitored, and used as a signal to stop if the tip gets into tunneling range.) The maximum coarse-step size is stated to be 200 nm at room temperature and 30 nm at 4 K and below. In this STM head, the tip (at the top of the coaxial tube scanner, glued to the sapphire prism) and the sample holder are always tightly bound within the strong Macor structure (Macor is a machinable ceramic material). This is a very reliable approach system within a small and rigid assembly. At the top of Fig. B.7 one can see that the sample holder (labeled “2”) can be inserted from elsewhere in the vacuum system to this location, secured by means of spring clips. For example, the sample may be moved from a station where cleavage to produce a new surface is accomplished, to the top of the scanner as shown here.

B.3 Atomic force microscope; combination STM/AFM A combination cryogenic AFM/STM has been designed, built and described by H. le Sueur (2007, 2008), and outstanding results from this system have been shown above (Section 5.6.4, Figs. 5.23– 5.26). This device has a lower base temperature, 30 mK, than the cryogenic STM systems that we have described above. This change, in conjunction with a decision to do spectroscopy only in fixed locations, has allowed spectroscopic measurements of unprecedented energy resolution. The AFM mode, unusual in a dilution refrigerator instrument, images the topography, equally, of insulating and conducting samples, with readout

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M E T H O D S O F S C A N N I N G T U N N E L I N G S P E C T RO S C O P Y (b)

m (a)

z–1

k0 k support connecting wire

z0 m0

tuning fork tip

Sample

k

m

z1

δm

δk

Fig. B.8. (a) Sketch of tip-sample configuration in the cryogenic combination AFM-STM device of le Sueur et al. (2008). The tuning fork is quartz, a piezoelectric material, and is the essential element of the AFM readout, which is run as a high-Q oscillator. (b) Interaction of the tip with the surface, modeled as incremental spring and mass elements, “δk” and “δm,” respectively, shifts the resonant frequency of the oscillator, forming the basis of scanning AFM detection. (After le Sueur et al., 2008.)

based on a piezoelectric quartz tuning fork. (Aspects of the device are shown in Fig. B.8.) The AFM feature facilitates locating the sample regions to be studied by STM. (The STM is of course limited in its application to conducting samples.) The author points out that the large energy dissipation of the common optical readout of AFM tip deflection greatly complicates its use within a dilution refrigerator. Here the tip is mounted on a quartz (piezoelectric) tuning fork with a Q at least 105 at room temperature. The tuning fork is driven into oscillation electrically, and its frequency is monitored. The tip-sample interaction influences the resonant frequency of the tuning fork. Van der Waals forces exist between tip and sample, and these forces increase as the spacing decreases. The interaction can be modeled as a small, added spring constant, whose strength varies rapidly with tip-sample spacing and shifts the resonant frequency. A given frequency shift can approximately be associated with tip-sample spacing. So a feedback loop controlling the z-piezo height so as to maintain a constant frequency shift is a way of creating a topograph, an AFM image. Demonstration of the simultaneous AFM and STM capabilities at a temperature 30 mK are demonstrated in Fig. B.9. The image and tunneling density of states relate to the 300-nm Ag proximity wire connected in a U-shaped Al superconducting structure, as described in Section 5.6.4 above. Variation of the superconducting phase along the Ag proximity wire was accomplished by varying the magnetic flux through the loop, driven by current A through a coil. Calibration of this phase was carried out as shown in Fig. B.10.

ATO M I C F O R C E M I C RO S C O P E ; C O M B I NAT I O N S T M / A F M

(a)

533

(b)

G (normalized)

300 nm

3.0 2.5 2.0 1.5 1.0 0.5 0.0 –400

–200

0 V (μV)

200

400

500 nm

Fig. B.9. (a) AFM image taken at 30 mK of the 300-nm Ag proximity wire structure described in Section 5.6 above. The U-shaped electrode is made of 60-nm thick, 20-nm wide aluminum, and the silver wire inserted across the U is 30 nm thick, 50 nm wide and 300 nm long. The structure was fabricated on an insulating Si/SiO2 substrate by standard electron-beam lithography and double angle evaporations. (b) Density of states measured at various positions in the structure, with zero phase difference between both ends of the normal wire. The positions where the different curves of the graph were measured are indicated on the AFM image by arrows. (After le Sueur et al., 2008.)

phase difference

dl/dV (normalized)

–π

π

–π

1.0

π

500 nm wire 0.5 0.0

300 nm wire

–0.1

0.0 Coil current (A)

0.1

Fig. B.10. Phase dependence of the differential conductance in the middle of two Ag wires (300 and 500 nm long) measured by varying the magnetic field perpendicular to the loop. The phase calibration is obtained from this measurement. (After le Sueur et al., 2008.)

The dI /dV is measured at the midpoint of the Ag wire. This plot serves to calibrate the phase difference ϕ as contained in Figs. 5.23– 5.26. It is stated that three or four multiples of 2π are at most available, because, as the loop is tiny, the magnetic field needed to get one flux quantum in the loop is a large fraction of the critical field of the thin Al film. It is stated that the cross section of the Al wire is eight times that of the Ag wire. Other aspects of this device are explained in the extensive Ph.D. thesis of H. le Sueur (2008), available in English as referenced.

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B.4 Scanning tunneling potentiometry and point-contact measurements A variable-temperature scanning tunneling potentiometry system, stemming from the pioneering work of Muralt et al. (1986), has been described by Bozler and Beasley (2008). This device combines angstrom-scale STM imaging with 130 nV potentiometric sensitivity on scales down to 2 nm. The device can cover a scan size up to 15 μm. A practical point-contact apparatus designated a “Cantilever-Andreev-Tunneling rig” has been described by Park and Greene (2005). The device makes use of piezoelectric bimorphs under analog voltage control to adjust the tip-sample contact. The article contains a recipe for etching long slender metal tips, which are suggested as suitable for making point-contact measurements. This device is not a scanning device, but is useful, typically, for making preliminary measurements on a new superconducting material for which procedures for cleaning surfaces and/or preparing tunnel barriers are not available.

B.5 Ballistic electron emission microscopy (BEEM) A scanning device that is an elaboration of the STM, with a third electrode, was pioneered in the work of Bell and Kaiser (1988). A review of recent versions of the BEEM device is given by Narayanamurti and Kozhevnikov (2001). The initial application of BEEM was in mapping the Schottky barrier height Vs existing at a metal–semiconductor interface, for example Au on Si. The method has been refined to allow deeper barrier features to be identified (Vh in Fig. B.11, right panel). To extract the voltage parameters from the characteristics a model must be adopted. In leading models the critical current near threshold varies as Ic ∼ (Vt –Vs )α , where α is 2 (in the Bell–Kaiser model) or 2.5 in the more refined treatment of Ludeke and Prietsch (1991). A recent example of BEEM research is that of Bobisch et al. (2009), who measured barrier heights of 2–3 nm thick Bi(111) films grown epitaxially on n-type Si substrates. (b) STM tip It, Vt

Au layer

Vt

Schottky barrier Buried heterostructure

Vh Vs

Vh Vt

Vs

Ic Tip

Collector

Semiconductor Back contact

BEEM current

(a)

Base

Semiconductor

Fig. B.11. (a) left panel Schematic of BEEM experimental setup, applied to a sample with a buried heterostructure. Some of the electrons injected by tunnel tip pass without scattering out the back of the Au electrode into the underlying sample of interest. These ballistic electrons may, depending on their energy, overcome internal barriers in the underlying structure to be collected at the back contact. The imaging reveals variations in the energy height of barriers in the underlying sample. (b) right panel Energy band diagram showing tip-vacuum-metal film electron injector, and (dark) underlying structure of interest. The generic I –V characteristic is sketched, identifying voltages Vs and Vh (see labels in band diagram). The features of the I –Vt are expected to vary in x, y as the tip scans, and various images may be formed from the characteristics. (After Narayanamurti and Kozhevnikov, 2001.)

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The effective heights of the Schottky barrier are 0.58 eV for the Bi/Si(100) − (2 × 1) while 0.68 eV is found for the Bi/Si(111) − (7 × 7) reconstructed surface. (As we have seen above, spectroscopy in the form of measurement of the ballistic emission current vs. the tip-film voltage is needed.) These workers observed strong increases in the ballistic electron emission current at step edges of the epitaxial Bi(111) films, revealed in the BEEM microscopy, in the latter but not the former case. To explain this difference they confirm the conservation of lateral electron momentum at the metal–insulator interface, in these cases.

B.6 Scanning charge microscopy and spectroscopy Mentioned above was a scanning tunneling potentiometry system, suitable for mapping at high voltage resolution potential gradients on the surface of a conducting sample carrying a current (Bozler and Beasley, 2008). This device ideally has STM spatial resolution based on the extent of an atomic wavefunction on the tip. Detection and mapping of surface and subsurface charge distributions has been addressed and accomplished in several ways. A Gauss’s law relation, E = σ/ε0 , suggests diverging surface electric fields that will accompany a charge distribution σ (x, y), which in some cases of interest may be subsurface. If a small metal tip characterized by an area A is scanned nearby, it will develop a voltage δV = δ Q/C, where δ Q = Aσ = Aε0 E, and C is the tip’s capacitance with respect to the surface. If the scanned tip is insulated and neutral it will become electrically polarized with –σ facing the sample surface. Such a device could be described as a scanning electrometer. If a small ac voltage is applied, the experiment may be carried out measuring a small ac current. The term surface charge accumulation microscopy has been suggested (Tessmer et al., 1998), which we may abbreviate as SCAM. A spectroscopic aspect is available when the sample has discrete electron levels, which may be pulled through the Fermi level of the probe by a gate electrode at voltage V0 . In this case, with ac voltage excitation, the corresponding capacitive ac current can peak as individual levels in the sample cross the Fermi level. This situation is referred to as capacitance spectroscopy, C(V0 ) = dQ ac (V0 )/dV . The spatial resolution of all such devices is lower than that of the STM, related to larger tip sizes and r −2 decay of electric field from a point rather than exp(–βr ) decay of a wavefunction. Such devices however may have single electron charge resolution.

B.6.1 Scanning single-electron-transistor electrometry A single-electron-transistor scanning electrometer (SETSE), capable of mapping static electric fields and charges with 100-nm spatial resolution and a charge sensitivity of a small fraction of an electron charge e, was developed and reported by Yoo et al. (1997). (This device is a form of SCAM as previously described.) The single electron transistor (SET) (Fulton and Dolan, 1987; Kuzmin and Likharev, 1987) is a submicron-sized tunneling device whose current flow is governed by the Coulomb blockade effect. It consists of a small metal island connected to metal source and drain leads by two small tunnel junctions. The current tunnels through the junctions at a rate determined by the island’s electrostatic potential with respect to the source and drain. This potential is in turn controlled by the electric field the island experiences from external sources. The SET current is periodic, with charge e, in island charge. The device is turned on when the net island charge (including continuous induced

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M E T H O D S O F S C A N N I N G T U N N E L I N G S P E C T RO S C O P Y

ISET

Vb

Y X

Drain

Source

Island Tunnel junction 100 nm

Si dopants 2DEG

Fig. B.12. Construction of scanning single-electron-transistor electrometer tip. SET island is at the tip, facing the surface to be scanned, with source and drain electrodes up the sides of the tip, coupled to the island by tunnel junctions. The application of the SETSE/SCAM device in the figure is to mapping subsurface charge in a 2-D electron gas. (After Yoo et al., 1997.)

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537

tip Q(a.c.) PtIr z x y 1-20 μm

scan area

contact 2 mm

GaAs 900 Å doped AlGaAs AlGaAs GaAs

electron gas

Fig. B.13. Schematic of device that can be operated as subsurface charge accumulation microscope, STM/SCAM, with ac charge detector attached to the PtIr STM tip. The device can also be run as an STM and as a Kelvin probe (Shockley et al., 1963) imager of static surface charge. In the SCAM mode, the tip is connected to a high-electron-mobility transistor (HEMT) of 0.01e ac sensitivity at cryogenic temperature. In this mode, the tip is scanned at about 5 nm constant height. Charges are induced on the tip as the ac drive signal changes the electron density in the subsurface electron gas. The spatial resolution is approximately given by the depth of the two-dimensional gas below the surface. The small tip-sample capacitance and the small ac drive voltage (1–4 mV) ensure that the measurement does not perturb the electron gas system. (After Tessmer et al., 1998.)

surface charge) is an odd multiple of e/2. One sees, e.g., that at that charge Q = e/2, adding one charge –e can lead to the same electrostatic energy Q2 /2C of the island, and thus electron hopping occurs with no Coulomb barrier. Small islands and cryogenic operation are needed for the Coulomb blockade: e2 /2C > k B T . The device described by Yoo et al. (1997) (and sketched in Fig. B.12) senses electron charge on the 100-nm-size island held in proximity to the sample surface at sensitivity ∼ 0.01 e. Because all the geometrical parameters are known, one can assign a quantitative interpretation to the SETSE/SCAM signal. This device does not require application of electric field between tip and sample. One can see that its spatial resolution is set by the island size, on the order of 100 nm.

B.6.2 Scanning subsurface charge accumulation microscopy: STM/SCAM This SCAM device is similar to the scanning capacitor mentioned at the introduction to Section B.6. In Fig. B.13 its operation as an ac charge detector is illustrated, with the intent of mapping charge distributions in a subsurface two-dimensional electron gas. The ac charge Q(ac) induced on the insulated tip is read by an electrometer sensitive to 0.01 electron charge (below). Tessmer and his group (Urazhdin et al., 2001) have described a more recent version of such an STM/SCAM system, in which the STM portion uses the Besocke (1987) coarseapproach design mentioned in Section B.2 above. The new system also uses the HEMT

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M E T H O D S O F S C A N N I N G T U N N E L I N G S P E C T RO S C O P Y

(a)

(b) Cr GaAs QUANTUM WELL

EF

(c)

++

n + + + ++

1 μm

AlGaAs GaAs

Vo

n+ GaAs CX

(d) CS

CT

RG HEMT

Fig. B.14. (a) Conduction band profile of “tunnel capacitor” device. Gate voltage V0 (right side) corresponds to top electrode in (b). Heavily doped n+ GaAs on the left, with band edge below the Fermi level E F . The tunnel capacitor exists between n+ GaAs on the left and discrete levels in the quantum well, whose energy distribution is the subject of the study. The Schottky barrier in the right portion of (a), between the well and the gate electrode, is high and does not support tunneling. Ac bias on n+ GaAs induces single electron transfers across tunnel barrier into quantum-well states, depending on their average occupancy as controlled by gate bias V0 . This bias-dependent ac charge on the quantum-well states also appears on the gate by induction, and the ac electrometer reads this ac charge. (b) Schematic cross section of device, top here corresponds to right side in panel (a). The cap structure serves to confine the active mobile two-dimensional electron gas laterally to a region about a micrometer in diameter. (c) TEM picture top of device, shows 2-micron Cr–Au gate electrode, which connects to HEMT. (d) The HEMT electrometer, input shunt capacitance 0.2 pF, which has proved very useful, as well, in later research. A bridge scheme plus the metal film resistor RG = 20M brings the high-electron-mobility cryogenic transistor to a suitable operating point. The ac charge readout is from the low-impedance output of the HEMT, which is actually on the same chip in the dilution refrigerator as the tunnel capacitor device. A 10-T magnetic field can be applied. (After Ashoori et al., 1992.)

electrometer design, which was pioneered in the earlier work of Ashoori et al. (1992), to which we now turn.

B.6.3 Single electron capacitance spectroscopy Productive lines of research are based on, or have drawn from Tessmer et al. (1998), and the pioneering work of Ashoori et al. (1992). In this work, capacitance C(V0 ) = dQ ac (V0 )/dV was first used to identify electron levels. The initial work was not done on a scanning basis, but the method is adaptable to scanning. (An update on this line of research has recently been given by Dial et al., 2007). This important original paper Ashoori et al. (1992) is fairly

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S C A N N I N G H A L L P RO B E M I C RO S C O P Y

(a)

(b)

E

D C

A 0

5

5 meV B 10 0 5 MAGNETIC FIELD (Tesla)

10

Fig. B.15. Gray-scale images of the sample capacitance dQ/dV as a function of energy (gate bias V0 ) and magnetic field. The vertical axis corresponds to –600 mV at the bottom of the figure and –500 mV at the top. Curve “A” is a theoretical result for field dependence of a central state, and curve “E” represents the field dependence of the lowest Landau level, hf /2. (After Ashoori et al., 1992.)

detailed and requires some explanation. Aspects of the device, which is termed a “tunneling capacitor,” are shown in Fig. B.14. Figure B.15 below shows pioneering energy-level spectroscopy measurements, with the tunnel capacitor device as described. Shown are Landau level structures in the twodimensional electron gas at applied magnetic fields up to 10 T. This line of research, pioneered and extended by Ashoori and co-workers, has proved important in understanding quantum Hall states, as reviewed by Dial et al. (2007). It appears that the methods here can be carried over to a scanning probe design.

B.7 Scanning Hall probe microscopy We have described in Chapter 11 above the physics and applications of the scanning SQUID microscope, which has outstanding magnetic field resolution but spatial resolution limited by the dimension of a pickup coil, perhaps 5 μ m. The readout of such a device is rather complex, and of course the device has to operate below the critical temperature Tc of the superconductor used in the Josephson junctions. (One might mention, however, that some scanning SQUID microscopes put the pickup coil inside a cryostat, but located close to cryogenic windows, so that a room temperature sample can be imaged.) For a discussion of magnetic field mapping using the magnetic force microscope (MFM), an adaptation of the atomic force microscope

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M E T H O D S O F S C A N N I N G T U N N E L I N G S P E C T RO S C O P Y Bx(x,y)

z(x,y)

Scan Signals Electronics

Tip Bias

PZT Tube

VH

Bx(x, y)

PC

VH =

IH Bx(x, y) ne

IH 1-2’

Sample Hall Probe

Fig. B.16. Schematic diagram of scanning Hall probe microscope. A PZT-5H piezotube is used for the scanner with a 10-μm scan range at 4.2 K. This device simultaneously operates as an STM. The Hall probe is manufactured in a GaAs/Al0.3 Ga0.7 heterostructure containing a two-dimensional electron gas with carrier density 2.7 ×1011 cm−2 and carrier mobility 3 ×105 cm2 /Vs at 4.2 K. The Hall bars of width 1 μm are microfabricated with optical lithography, at a location about 13 μm away from the corner of a deep mesa etch, which is coated with gold. The corner of the etch, not the corner of the chip, serves as the tunnel tip. (After Oral et al., 1996.)

–3

0 Bz [mT]

3

Fig. B.17. Magnetic image of stripe domain structure in (Ga1−x Mnx ) As using scanning Hall probe microscope. Mn concentration is about 4.3%. Measurement temperature is 20 K, and scanned region is 4.8 μm square. (Horizontal axes lie in [100] directions.) Magnetic field Bz is seen to reverse sign across the two boundaries. The magnetic stripe direction is close to [110], consistent with magnetocrystalline anisotropy known for this material. (After Shono et al., 2000.)

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541

(AFM), the reader is directed to the excellent book of Wiesendanger (1994). One might also point out that single electron spin detection has been achieved in a further adaptation of the AFM, the magnetic resonance force microscope (MRFM), as reviewed recently by Berman et al. (2006). The scanning Hall probe microscope, is a relatively simple device with unquestionably linear detection of field strength. It is a practical device, with higher spatial resolution than the scanning SQUID microscope, and has now improved field sensitivity. Without going into the history of this device, a version of greatly improved field sensitivity, utilizing a 2-D electron gas in the Hall bar, has been described by Oral et al. (1996). Their device can be operated simultaneously as a scanning tunneling microscope. The basis for this device is the Hall coefficient RH = 1/ne (in the case of a single carrier of density n), which enters the basic expression for the Hall field EH = RH J × B. This reduces, in a mapping context, to an operating expression VH = IH B(x, y)/ne, for an electron conductor with density of carriers n. The spatial resolution is set by the size of the Hall bar, which can be around 1 micrometer. The system of Oral et al. (1996) is shown in Fig. B.16. Briefly, this√instrument (Oral et al., 1996) has a magnetic field sensitivity ∼ 1.1 × 10−3 G/ Hz at 77 K. The Hall bar is located at the corner of GaAs chip, the carriers residing in a two-dimensional electron gas. The situation of the mobile carriers is a similar to that shown in Fig. B.13. As seen in Fig. B.16, the chip carrying the Hall bar and also serving as the STM tip, is located on a conventional cylindrical tube scanner, which is stated to have scanning range of 10 μm at 4.2 K (larger at higher temperatures). While the default mode of operation is simultaneous STM/SHPM, faster magnetic scans can be achieved by raising the tip 0.2 to 0.5μm and scanning at fixed height. The device can be operated in the range 4 K to 300 K, and has strongly improved field sensitivity by choice of the two-dimensional electron gas for the Hall bar. A recent research article based on the SHPM is that of Shono et al. (2000). The authors find the wide temperature operating range to be an advantage in mapping the development with temperature of magnetic domain structure in the (Ga1−x Mnx ) As system. Their image at 20 K is shown in Fig. B.17.

APPENDIX C

Tabulated results Summaries of superconducting parameters for several classes of superconductors are contained in Tables C.1–C.5. Many of the Tc and θ D values were obtained from the extensive compilation of Roberts (1976). Many of the values of ωlog ≡ lnexp ω, especially those in Table C.2, devoted largely to s–p-band alloys, are from the paper of Allen and Dynes (1975b). Additional sources, including tabulated values of superconducting functions [e.g., (E), Z (E)] are the two compilations of strong-coupling tunneling results due to J. M. Rowell, W. L. McMillan, and R. C. Dynes and available from these authors. The estimated values of λ for many weak-coupling metals are obtained by use of McMillan’s Tc formula, Tc =

1.04(1 + λ) θD exp 1.45 λ − μ∗ (1 + 0.62λ)

(C.1)

(McMillan, 1968a). In general, the intent has been to provide the best present estimates of parameters rather than a full history of work on each superconductor. Excellent sources of tabulated information on superconductors, not restricted to tunneling results, in addition to the compilation of Roberts (1976), include the books by Grimvall (1981) and Vonsovsky et al. (1982); as well as the earlier monograph by Savitskii et al. (1973). Table C.6 lists primarily reports of metals used as an initial overlayer subsequently oxidized to produce a tunneling barrier. Table C.7 provides a partial list of references to tunneling work on thick N/S or S/N bilayers in which the oscillatory (Tomasch and McMillan– Rowell) phenomena were of interest. Excellent and more complete reviews of this area of research have been given by Nédellec et al. (1976) and by Nédellec (1977). A companion review to the latter (also in French) is that of Gilabert (1977) on the proximity effect. Table C.8 gives a partial listing of tunneling results on superconductivity under hydrostatic pressure. Finally, Table C.9 summarizes parameters for cuprate superconductors. This compilation is based on that of Fischer et al. (2007). Tables C.9A, C.9B, C.9C, respectively, list parameters for leading cuprate families Bi2212, YBCO and HgBa2 Can−1 Cun O2n+2+δ.

542

Table C.1. s, p elements Tc (K)

θD (K)

Be Mg Al

0.026 ≤ 0.3 mK 1.18

1390 400 420

Cd Zn Ga

0.52 0.85 1.08

Am–Ga

8.56

Sn

3.72

In Tl

Material

ωlog (meV)

ω (meV)

√

ω2  (meV)

209 310 325

λ

2 /k B Tc

0.23a 0.29 ± 0.03 0.38a

3.53, 3.50

0.38a 0.38a 0.40a

3.2 3.63

4.74

6.63

8.70

1.62

4.5

195

8.53

9.48

10.43

0.72

3.7

3.41 2.38

109 79

5.86 4.48

6.81 5.0

7.67 5.51

0.805 0.795

3.68 3.6

Pb

7.196

105

4.83

5.20

5.55

1.55

4.67, 4.30

Am–Pb Am–Hg Hg(α)

7.2 3.90 4.15

1.6

4.65 4.4 4.61

a

72

2.5

Value obtained using McMillan’s Tc formula (McMillan, 1968a).

3.27

4.22

Tunneling reference

Burnell and Wolf (1982) Blackford and March (1968) Hubin and Ginsberg (1969) Kumbhare et al. (1969) Yoshihiro and Sasaki (1968) Wühl et al. (1968) Chen et al. (1969) Leslie et al. (1970) Rowell et al. (1969) Zavaritskii (1965) Dynes (1970) Dynes (1970) Clark (1968) Townsend and Sutton (1962) McMillan and Rowell (1965, 1969) Dynes and Rowell (1975) Ziemba and Bergmann (1970) Ziemba and Bergmann (1970) Hubin and Ginsberg (1969)

Table C.2. Alloys and unusual phases: s, p elements ./

0

Tc

θD

ωlog

ω

(K)

(K)

(meV)

(meV)

(meV)

Tl0.9 Bi0.1

2.3

4.14

4.74

In0.5 Tl0.5 In0.9 Tl0.1 In0.57 Tl0.43 In0.67 Tl0.33 In0.07 Tl0.93 In0.73 Tl0.27

2.52 3.28 2.60 3.26 2.77 3.36

4.57 5.43 4.57 4.91 4.22 4.74

β–Ga In0.17 Tl0.83

5.90 3.19

In0.27 Tl0.73 Pb0.4 Tl0,6 Pb0.6 Tl0.4 In2 Bi Sb2 Tl7 Pb0.8 Tl0.2 Bi2 Tl Pb0.9 B0.1 Pb0.6 Tl0.2 Bi0.2 Pb0.8 Bi0.2 Pb0.7 Bi0.3 Pb0.65 Bi0.35 Am–Pb0.45 Bi0.55

3.64 4.60 5.90 5.6 5.2 6.8 6.4 7.65 7.26 7.95 8.45 8.95 7.0

Material

ω2

λ

2 /k B Tc

5.34

0.78

3.58

5.51 6.46 5.51 5.86 4.83 5.77

6.29 7.41 6.38 6.81 5.43 6.63

0.83 0.85 0.85 0.90 0.89 0.93

— — — — — —

7.5 3.88

9.31 4.74

11.1 5.43

0.97 0.98

—

3.62 4.14 4.31 3.96 3.19 4.31 4.05 4.31 4.14 3.96 4.05 3.88 2.50

4.57 4.79 4.87 4.91 4.14 4.84 4.57 4.80 4.57 4.44 4.48 4.31 3.27

5.43 5.31 5.34 5.77 5.00 5.27 5.08 5.20 4.96 4.88 4.87 4.74 4.05

1.09 1.15 1.38 1.40 1.43 1.53 1.63 1.66 1.81 1.88 2.01 2.13 2.59

— 4.06 4.25

4.37 4.67 4.80 4.70 4.86 4.78

Tunneling reference

Dynes (1970) Allen and Dynes (1975b) Allen and Dynes (1975b) Allen and Dynes (1975b) Allen and Dynes (1975b) Allen and Dynes (1975b) Allen and Dynes (1975b) Dynes (1970) Allen and Dynes (1975b) Allen and Dynes (1975b) Dynes (1970) Allen and Dynes (1975b) Allen and Dynes (1975b) Dynes and Rowell (1975) Dynes and Rowell (1975) Rowell et al. (1978) Rowell et al. (1978) Dynes and Rowell (1975) Rowell et al. (1978) Dynes and Rowell (1975) Dynes et al. (1969) Allen and Dynes (1975b) Dynes and Rowell (1975) Dynes and Rowell (1975) Allen and Dynes (1975b)

Table C.3. d-band elements Material

Tc (k)

θD

ωlog

ω

./ 0 ω2

(K)

(meV)

(meV)

(meV)

Ag Cu W Ir Ti Zr Ru Os Mo Re Ta

0 >0 0.015 0.1125 0.40 0.61 0.49 0.66 0.92 1.70 4.47

390 420 425 290 550 500 460 415 258

V

4.49 5.40

383

Tc Nb

5.35 7.77 9.25

276

9.2

276

Am–Mo Am–Nb a b c

8.8 5.2

17.9

11.37

12.06

12.75

10.9

14.8

12.7

17.1

18.8

λ

2 /k B Tc

0.11 0.13 0.28a 0.34a 0.38a 0.41a 0.38a 0.39a 0.41a 0.46a 0.69

— —

Wilson (1979) Wilson (1979)

3.59 3.66

0.73 0.60a

3.72 3.5

0.82b

3.51

Ochiai et al. (1971) Shen (1970) Townsend and Sutton (1962) Wolf et al. (1981a) Noer (1975) Robinson and Rowell (1978) Zasadzinski et al. (1982)

0.82a

3.89

14

15.3

1.04c

3.91

12.0 10.8

14.0 12.5

0.9 0.8

3.7 3.7

Value obtained by using McMillan’s Tc formula (McMillan, 1968a). A λ value of 0.88 is estimated after correction for spin fluctuations (Burnell et al., 1982). A λ value of 1.09 is estimated after correction for spin fluctuations (Burnell et al., 1982).

Tunneling reference

Townsend and Sutton (1962) Bostock et al. (1976) Robinson et al. (1976) Wolf and Zasadzinski (1978) Arnold et al. (1980) Khim et al. (1981) Kimhi and Geballe (1980) Kimhi and Geballe (1980)

Table C.4. d-band alloys, oxides, and compounds Material

Tc (K)

θD (K)

ωlog (meV)

253

9.4

Nb0.6 Ti0.4 Nb0.75 Zr0.25 Nb0.80 Zr0.20

11.0

Mo0.6 Re0.4 NbN

12.2 15

V3 Ga

15.9

310

V3 Si

17.0

530

NbN0.65 C0.35 Nb3 Sn

17.5 18.3

290

ω (meV)

√

ω2  (meV)

λ

2 /k B Tc

1.31

4.1

9.8 11.3

12.8

1.12

Nb3 Al NbAl0.23 NbTax Nb3 Al0.8 Ge0.2

17.5 18.9 16.4 21

3.8

∼ 1.67

4.3

10.8

13.1

15.0

1.78

4.37

9.5

11.4

13.5

1.7

4.3

Tunneling reference Hulm and Blaugher (1961) Wolf et al. (1980a) Hulm and Blaugher (1961) Dietrich (1964) Hulm and Blaugher (1961) Horn and Saur (1968) Komenou et al. (1968) Flükiger and Jorda (1977) Zasadzinski et al. (1980) Blaugher et al. (1969) Hauser et al. (1966) Pessall and Hülm (1966) Matthias et al. (1954) Shen (1972b) Moore et al. (1976) Allen and Dynes (1975b) Wolf et al. (1980a) Sahm and Pruss (1969) Flükiger and Jorda (1977) Hertel et al. (1982) Matthias et al. (1967) Dew-Hughes (1975) Gregory et al. (1973)

(continued)

Table C.4. Continued Material

Tc (K)

θD (K)

ωlog (meV)

ω (meV)

√

ω2  (meV)

λ

378

2 /k B Tc 4.2

Tunneling reference

Nb3 Ge

23

NbGe0.23 Cs3 C60

19.8 38

Csx Rb y C60 Potassium-doped C60 Ba1−x Kx BiO3 (BKBO) Nd2−x Cex CuO4−y , (NCCO) PbMo6 S8 Nb3 Al MgB2 Ca2−x Nax CuO2 Cl2(Na − CCOC) CaAlSi CaC6

33 18 30 23

Tanigaki et al. (1991) Hebard et al. (1991) Huang et al. (1990) Huang et al. (1990)

15 16.8 39 28 7.8 11.5

Mo1−x Rex YNi2 B2 C

12.3 15.2

Niu and Hampshire (2004) Mondal et al. (2008; 2009) Eskildsen et al. (2002) Shen et al. (2005) Kuroiwa et al. (2007) Bergeal et al. (2006), Kurter et al. (2007) Shum et al. (1986) Weber et al. (2008)

14

1.64 ± 0.2

4.35

Gavaler (1973) Testardi et al. (1974) Rowell and Schmidt (1976) Kihlstrom and Geballe (1981) Ganin et al. (2010)

Table C.5. f-band elements Material

Hf U(α) Th La(α) La(β) La(dhcp)

Tc

θD

ωlog

ω

./ 0 ω2

(K)

(K)

(meV)

(meV)

(meV)

0.128 0.68 1.38 4.88 6.00 4.95

λ

2 /k B Tc

165 151 139

Tunneling reference

3.47 3.8

Haskell et al. (1972) Lou and Tomasch (1972)

3.75

Wühl et al. (1973)

Table C.6. Metal overlayers for barrier formation Metals

References

Al

Adkins (1963); Wolf et al. (1980b); Rowell et al. (1981); Kwo et al. (1982) Burnell et al. (1982) Kwo et al. (1983) Umbach et al. (1982)

Mg Y Eu, Lu

Table C.7. Studies of Tomasch oscillations in thick superconducting films and of McMillan–Rowell oscillations in thick normal films Metal (s)a

References

Zn/Pb Pb/Ag Pb/Cu Ag/Pb Al/Pb In/Ag Sn In/Pb Pb Cu Cd/Pb Sn/Pb In

Rowell (1973); Wong et al. (1981) Lykken et al. (1971); Haywood and Mitchell (1974) Nédellec (1977) Rowell and McMillan (1966); Khim and Tomasch (1979) Khim and Tomasch (1978) Nédellec (1977) Tomasch (1966b); Nédellec (1977) Tomasch (1966a) Tomasch (1965) Nédellec et al. (1971) Colucci et al. (1974) Chaikin (1975) Tomasch and Wolfram (1966)

a

The first-mentioned metal faces the tunneling barrier.

Table C.8. Tunneling studies of superconductor phonons under hydrostatic pressure Material

References

Pb Pb, Tl Nb, Ta PbInx La

Franck et al. (1969); Svistunov et al. (1981) Galkin et al. (1971) Revenko et al. (1980) Hansen et al. (1973); Wright and Franck (1977); Svistunov et al. (1981) Wühl et al. (1973)

549

TA B U L AT E D R E S U LT S

Tables C.9. Cuprate superconductors Table C.9a. Gap values for Bi2 Sr2 CaCu2 O8+δ (Bi2212) (after Fischer et al. 2007) Values indicated without parentheses are given explicitlyin the publications. Values within parentheses were estimated Tc 1− T max

max = 92 K. Where c using the generic formula p = 0.16 ± 82 .6 (Presland et al., 1991) with Tc max the sample is stated to be optimally doped, yet Tc  = Tc , p is set to [0.16].

Tc (K) (85)

p

p (meV)

2 p k B Tc

0.13

45 ± 12

12.3

Matsuda et al. (2003)

9.0

Nakano et al. (1998)

6.8

Chen and Ng (1992)

85

(0.13)

33

85

(0.19)

25

85

(0.19)

36.7

10.0

Reference

Hoffman et al. (2002a)

85

(0.19)

35

9.6

Maki et al. (2001)

85

(0.19)

26

7.1

Nakano et al. (1998)

86

(0.13)

45

12.1

Sakata et al. (2003)

86

(0.19)

35 ± 5

9.4

Kaneko et al. (1999)

86.5a

(0.18)

45 ± 15

12.1

Howald et al. (2003a,b)

87

(0.19)

32

8.5

Hudson et al. (1999)

87

(0.19)

26

7.4

Ichimura and Nomura (1993)

87

(0.19)

17b

87

(0.19)

31–40c

–

–

40

88

(0.14)

50

88

[0.16]

32 ± 2

4.5

Suzuki et al. (1999b)

∼ 9.3

Suzuki et al. (1999a)

–

Hudson et al. (2000)

13.2

Matsuura et al. (1998)

8.4

Oda et al. (1997)

88

(0.18)

28 ± 2

7.4

Kitazawa et al. (1997)

89

0.16

40 ± 10

10.4

Matsuda et al. (2003) McElroy et al. (2005)

89

0.19

33 ± 1

8.6

(89)

0.18

35.6

9.3

Lang et al. (2002)

(89)

0.18

36 ± 1

9.4

McElroy et al. (2005)

(89)

0.18

35 ± 7

9.1

Matsuda et al. (2003)

90

[0.16]

54

13.9

Matsuda et al. (1999a,b)

90

[0.16]

39

10.1

Wolf et al. (1994)

92

(0.16)

35

92

(0.16)

43.7

8.8

De Wilde et al. (1998)

11.0

Hoffman et al. (2002a)

7.6

Iavarone et al. (1998)

92

[0.16]

30 ± 2

92.2

[0.16]

41.5

92.3

[0.16]

29 ± 4

7.3

93de

(0.16)

25–55

∼ 10.0

10.4

Renner et al. (1998b) Renner and Fischer (1995) Cren et al. (2000)

As above, with Zn, Ni, Co, or Pb substitutions: 84 f 85g –h –j 68k

(0.13) (0.19) – – (0.22)

45 20 50i 50 40 ± 20

12.4 5.5 – – 13.7

Pan et al. (2000b) Hudson et al. (2003) Zhao et al. (2000) Cren et al. (2001) Kinoda et al. (2003)

a

Bi2.1 Sr1.9 CaCu2 O8+δ . Tunneling direction in ab plane varies between [110] and [100]. I|| [110]. d Thin film. onset = 93 K by dc resistivity and 55 K by ac susceptibility. e Bi Sr 2 1.98 Ca1.38 Cu2.28 O8+δ , Tl f Bi Sr Ca(Cu 2 2 1−x Znx )2 O8+δ . g Bi Sr Ca(Cu 2 2 1−x Nix )2 O8+δ . h Bi 1.83 Sr1.8 Ca(Cu1−x Cox )2 O8+δ . i T measured at 66 K. j Bi 1.95 Pb0.5 Sr2 CaCu2 O8+δ . k Bi Pb Sr CaCu O 1.4 0.6 2 2 8+δ . b c

Table C.9b. Gap values for YBa2 Cu3 O7+δ (after Fischer et al. 2007) Tl (K)

p

p (meV)

2 p k B Tc

85b 90a

(OP) ?

15c 14 ± 2d

4.1 3.6

90 90a 90 90 >90a 91 91 92a 92a 92 92 92a 92a 92 92 92 92 92a 92a 92 92 92 92.9

OP OP OP OD (OP) (OP) (OP) OP OP OP OP OP OP OP OP OP (OP) OP OP OP OP OP OP

28e ∼20 28 20 ∼17.5d 24–32 ∼25 f 20g 20 30 ± 8 20 7.5d ∼21g 28 ± 2 20 ± 2 18 18 20d 30 ± 10g 19 ± 4 29 ± 3d 27 ± 4g 18

7.2 5.2 7.2 5.2 4.5 7.1 6.4 5.0 5.0 7.6 5.0 4.4 5.3 7.1 5.0 4.5 4.5 5.0 7.6 4.8 7.3 6.8 4.5

a

Thin film. Measured by high-field magnetic susceptibility, Tl = 92 K by resistivity. c I|| ab plane. d Tunneling on (100) plane. e Tunneling on (110) and (100) planes. f Tunneling on electrical field etched surface. g Tunneling on (110) plane. b

Reference Kirtley et al. (1987) Hoevers et al. (1988) Tanaka et al. (1994) Ueno et al. (2001, 2003) Wan et al. (1989) Shibata et al. (2003a,b) Miller et al. (1993) Murakami et al. (2001) Murakami et al. (2001) Alff et al. (1997) Born et al. (2002) Edwards et al. (1992) Edwards et al. (1995) Koinuma et al. (1993) Koyanagi et al. (1995) Kugler et al. (2000) Maggio-Aprile et al. (1995) Maggio-Aprile et al. (2000) Maki et al. (2001) Nantoh et al. (1994a) Nantoh et al. (1995) Wei et al. (1998b) Wei et al. (1998b) Wei et al. (1998b) Yeh et al. (2001)

551

TA B U L AT E D R E S U LT S

Table C.9c. Gap values for HgBa2 Can−1 Cun O2n+2+δ (after Fischer et al. 2007) Tc (K) Hg1201 (n = 1) Hg1212 (n = 2) 123b Hg1223 (n = 3) 125c 97a

133a 135a

p

p (meV)

2 p k B Tc

OP OP (OP) (OP) OP

33 50 ∼ 24d ∼ 38e 75

7.9 9.4 4.5 6.6 12.7

Reference Wei et al. (1998a) Wei et al. (1998a) Rossel et al. (1994) Rossel et al. (1994) Wei et al. (1998a)

a

Polycrystal. Thin film. Melted sample. d = 15 meV using Dynes fit. e = 24 ± 2 meV using Dynes fit. b c

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Index 3.0-nm Ag-coated Pb/PtIr tip, 525 3dx 2 −y 2 hole with S = 1/2, 314 A-15 compounds, 190, 289, 292, 508 Abrikosov-Gor’kov (AG) theory, 145–147 ABZ theory, 386–389, 399, 405 ac Josephson effects, 12, 153, 156, 481 Acetylene molecules, IETS study of, 475–476 Ag, 295, 363 normal-state phonon structure of, 363 superconductivity of, 295 Ag electrodes, 371, 411 Ag:Fe, 407 Ag host metal electrodes, 411 AgMn backed by Pb, 147, 250 Al, 295, 363 normal-state phonon structure, 363 in proximity with Pb, 295 Al alloys, 412 Al-Al2 O3 -Bi thin-film junctions, 353 Al-Al2 O3 -Pb junctions, 45, 142, 151, 303, 338, 366 Al:D, 295 Al films, 160, 163, 165–167, 415, 430 Al:H, 295 Al host metal electrodes, 411 Al-I-Al junctions, 377, 485, 513 Al-I-Pb junctions, 209, 364 Allen-Dynes plot, 287, 300–01 Tc formula, 176, Alloy superconductors, crystalline s-p band, 263–373 Alloying, systematic variation with, 264–267 Alloys; see also specific alloys d-band, tabulated results on, 545–547 tabulated results on, 542, transition metal, 281–289 Al-Nb system, 219 Alpha particle emission, 3 Al system, granular, 390 Aluminum-aluminum oxide-aluminum tunneling, 64 Amorphous alloys, 278 Amorphous metals, 273–280 Bi, 275 Cu, 274–278 isotropic, 277–298 Sn, 275–276

Anderson’s model for magnetic moment, 377 Andreev reflection, 170, 171, 199–210 oscillations in tunneling spectra, 206–212 postponed, 223 Anisotropy, 185–187 gap, 124–128 Anodized metal probes, 509 Antennas, 487 optical-point-contact, 487 whisker, 487 Apical oxygen, disorder sites, 325, 460 Appelbaum, Brinkman and Zawadowski theory, 385, 386, 403 Appelbaum-Kondo theory, 79, 372 Arnold’s proximity tunneling theory, 213–19, 232 Arnold’s elastic scattering theory, 246–7 ARPES (angle-resolved photoemission spectroscopy) on BSCCO, 464–467 Measurement of BSCCO energy gap, 466–467 Bogoliubov quasiparticles observed, 467 Relation to STS, 464–467 Artificial barriers, 507–509 Assisted tunneling, 76–79, 137, 336, 357–383 excitation of electronic transitions, 378–383 inelastic electron tunneling spectroscopy of molecular vibrations, 366–367 inelastic excitation of spin waves (magnons), 367–368 inelastic excitation of surface and bulk plasmons, 368–369 light emission by inelastic tunneling, 369–372 phonons, 358–365 processes, 28, 76–79 spin-flip and Kondo scattering, 372–378 threshold spectroscopies, 357–383 Atomic layer deposition, 494–495 HfCl4 , 495 “high kappa oxides”, 494 crystalline Fe/MgO/Fe tunnel junction, 495 Au, 194, 230–232, 338–339, 352, 353, 357, 403, 407, 421–422 normal-state phonon structure, 363 Au:Co system, 404, 407 Au:Cr, 407 Au electrodes, 371 583

584

INDEX

Au:Fe, 407 Au host metal electrodes, 411 Au:Mn, 407

Bridge circuits for derivative measurements, 517–520 BSCCO see Bi2212

Ballistic electron emission microscopy (BEEM), 534–535 Bell–Kaiser model, 534 Barrier height measurement, internal, 534 Band-edge measurement, 47, 48, 50, 53, 54, 337–341, 343, 345, 353, 356, 357 Bardeen, Cooper and Schrieffer see entries beginning BCS Barriers, tunneling artificial, 507–509 crystalline, 6, 33–49 directly deposited artificial, 508–509 discrepancies in early literature on, 43 formation of, metal overlayers for, 507, 548 heights of, determination of, 512–513 interactions in, 76 microshorts in, 138, 252–254, 511 oxide, 63–65, 67, 68, 72, 75, 208, 220, 368, 496, 501–511 pinholes in, 63, 501, 507 preparation of, 501–507 sharpness or abruptness of interface, 64, 68 thickness uniformity of, 64, 69, 502, 513 totally oxidized metal overlayers, 507–508 trapezoidal, 29, 34, 47–51, 236, 553 Barrier tunneling phenomena, applications of, 475–488 introduction to, 475–477 Josephson junction interferometers, 477–480 logic applications of Josephson devices, 481–482 radiation, detection of, 483–487 scanning tunnel microscope, 476, 509 SQUID detectors, 480–481 Barrier tunneling results, anomalous, 346, 348, 349 BCS superconductors, equilibrium, gap spectra of, 112–121 BCS (Bardeen, Cooper, and Schrieffer) theory, 10, 82, 83, 91–103, 112–119, 122, 130, 133 central equation of, 99 direct test of, 118 electron-phonon coupling and, 93–103; see also Electron phonon coupling and BCS theory generalization of, 101–103 Bi, 278, 319, 325–328, 364, 534–535 BiO planes, in BSCCO, 327 BiSb alloys, 353 Bi8 Te7 S5 , 498 Bi thin-film junctions, 353 Bogoliubov equations, 174, 203, 212 Bogoliubov-Valatin operators, 101 Bosonic excitations see electron-spin coupling, 467

Ca2−x Nax CuO2 Cl2 (Na-CCOC), 457, 464, 465, 474 Charge ordering in, 464 Capacitance, junction, 480 Caroli et al. theory, 385, 387 CDMT calculation, 51, 53 Charge ordering in cuprate superconductors, 460–464 In BSCCO vortex cores, 496–497 In BSCCO pseudogap underdoped regime, 498 in Ca2−x Nax CuO2 Cl2 (Na-CCOC), 464 Chemical vapor-deposited films, 493 C-I-NS junctions, 236, 238, 291 CIS junctions, 196 Clem’s model of anisotropy, 124 Co, 411, 413 Co films, 348 Cohen, Falicov, and Phillips method, 105, 107, 108 Colossal magnetoresistance: Mott transition in doped manganites, 470–473 La0.67 Ca0.33 MnOx films, 470 La0.7 Ca0.3 MnO3 crystals, 470 La0.75 Ca0.25 MnO3 , STS measured, 471 La1.2 Sr1.8 Mn2 O7 (LSMO), pseudogap measured in ARPES, 472–473 Compounds, metallic superconducting d-band, tabulated results on, 590–591 group III-group V, zero-bias anomalies in, 409 transition metal, 281–291, 492, 495, 497, 514 Conductance bridge circuit, 517–519 Conductance minima, 409–410, 414 Conductance minimum zero-bias anomaly, 167, 390, 407, 410 Conductors, electronic super see Superconductors Conventional tunneling Hamiltonian theory see Transfer Hamiltonian model Conventional (McMillan-Rowell) tunneling spectroscopy of strong-coupling superconductors, 173–196 anisotropic case, 185–187 difficulty in preparing CIS junctions, 196 as method for determining Eliashberg function for metal, 178 Eliashberg-Nambu strong-coupling theory, 173–177 Eliashberg theory, test of, 178–182, 192 electronic density-of-states variation case, 190–194 extension to general cases, 182–194 finite temperature case, 185 interposed layers/proximity effects, 196

INDEX

introduction to, 173 limitations of, 194–196 methods of, 173 PETS vs, 236–245 proximity effects, 196, 235 quantitative inversion for Eliashberg function, 178–182 spin fluctuation case, 187–190 tunneling density of states, 177 Cooper limit density-of-states expressions, 215 Corner-mounted Josephson-junction SQUID, 333, 480 Correlation of gap strength with zero bias conductance, not with “bosonic dip”, 452 Cos term, Josephson current, 110, 119 Coulomb barrier, 3 Cryogenic STM/AFM of le Sueur, 228, 531–33 Cryogenic AFM, 531 unprecedented energy resolution, 531 piezoelectric quartz tuning fork, 532 Scanning tunneling potentiometry, 534 Cryogenic STM of Pan, Hudson, and J. C. Davis, 529–531 7T , 240 mK, 530 Friction motor coarse approach, 530–531 Sample transfer in vacuum, 531 of Wilson Ho, 527–529 coarse approach design of Besocke, 528 stick-slip motion, 529 three Inconel springs, 529 Crystal structures of common cuprates, 325–328 Crystalline metallic alloy of In-Sn, 267 of In-Tl, 264, 267 of Pb, 263 phonon energy linewidth of, 281 Crystalline solid, electron tunneling within, 6–9 Crystalline tunneling barrier, energy-dependent κ(E) from tunneling through, 45 Cs3 C60 , with 38 K Tc non-BCS, 335 Cu, 274, 279, 280 Cu electrodes, 371 CuO5 pyramids, 325 CuO6 octahedra, 325 Cuprate electron-coupled superconductors, 140, 325–328 Bi2 Sr2 CaCu2 O8+δ (BSCCO or Bi2212), 325–328 YBa2 Cu3 O7−δ (YBCO or Y123), 325–328 Cuprate superconductor compilation of Fischer et al., 549–551 Current amplifiers, 525 Current density in three-dimensional structure, 25 Dayem–Martin Steps, 154 d-band alloys, tabulated results on, 546–547

585

d-band compounds, tabulated results on, 546–547 d-band elements, tabulated results on, 545 d-bands, 356–357 dc Josephson effect, 12 Decay constant in barrier region, definition of, 25 Dedicated second-derivative circuits utilizing filter networks, 520–522 Deformable metal vacuum tunneling probes, 510–511 de Gennes boundary condition, 207, 252 de Gennes quasiparticle bound states, 203, 207–210, 218, 231 Density of states, specific heat coefficient relation to, 287 Derivative measurement circuitry, 514–516 Dimensionality of potential, 71 Disorder sites and doping of cuprate superconductors, 325–327 Dopant striations, effect of, 51–52 d-wave superconductivity, gap function dx 2 −y 2 , 335, 431, 449, 454, 458, 567, 480 Elastic scattering in N layer, 245–250 Electromagnetic radiation, 150–157 Electron correlation and localization in metallic systems, 389–394 Electron-coupled superconductivity, 310–312, 317–318, 447 Electron-doped superconductor, 310, 313, 317 Electron-hole (Andreev) reflection, 199, 203 Electron-pairing, exactness of, 156, 475 Electron-phonon coupling and BCS theory, 93–103 determination from phonon linewidth, 281 elementary excitations of superconductors, 100, 101 generalization of BCS theory, 101–103 pair ground state, 96–100 Electron scattering in Kondo ground state, 401–407 Electron tunneling see Tunneling, electron Electron-spin wave coupling in cuprates, 467, 469 Electronic transitions, excitation of, 378–383 Eliashberg function see Superconducting phonon spectra and α 2 F(ω)/Eliashberg function Eliashberg theory, 147 strong-coupling, 173–177, 192, 194, 199 test of, 178–182 Eliashberg–Nambu strong-coupling theory of superconductivity, 173–177 Energy gap see Gap, energy Epitaxial single-crystal films, 493–494 Equilibrium superconductors BCS superconductors, 112–121 general homogeneous, 121–160

586

INDEX

Esaki diode, 6, 13, 56–58, 358, 360, 475–6 Esaki’s work, 6 Evaporated films, 491 Even conductance, definition of, 363 Families of cuprates, structures, chemistry and properties, 325–327 f-band elements, tabulated results on, 548 Fe films, 343 Fe–CO bond, FeCo and Fe(CO)2 molecules, 443–446 C–O stretch vibrations of, 443 O–O stretch vibrations oxygen on silver surface, 445 Fermi arcs, 313, 460–62 Fermi function, definition of, 5 Fermi Golden Rule calculation for transition rate, 78 Fermi surface integrals in tunneling, 61–62 Ferromagnets, spin-polarized tunneling as probe of, 343–350 Feuchtwang theory, 385, 386, 387 Field emission current, 5, 59, 63 Film thickness, measurement of, 491–492 Filter networks, dedicated second-derivative circuits utilizing, 520–522 Final state effects in normal state tunneling, 336–357, 384–406 ABZ theory, 385–386 accumulation layers at semiconductor surfaces, 339–343 Caroli et al. theory, 385, 387 electron correlation and localization in metallic systems, 389–394 electron scattering in Kondo ground state, 401–407 Landau levels, 339, 341, 346, 378 more general many-body theories of tunneling, 384–389 other bulk band structure effects, 352–357 phonon self-energy effects in degenerate semiconductors, 394–401 spin-polarized tunneling probe of ferro-magnets, 343–350 transfer Hamiltonian model of, deficiencies in, 384–387 Fiske steps, 150, 154, 517 Fluctuation superconductivity, 158–160 Flux quantum, 85–88, 92, 154, 232, 480–483, 513 Foil electrodes, 495–498 Foil technique, 237–238 Forward bias, definition of, 398 Fourier-transform STS (FT-STS), 460–461 Fowler-Nordheim tunneling expression, 66 Fraunhofer pattern critical current density in planar Josephson junction, 332, 475, 477, 512

Friedel oscillations, 387, 422–425, 439 Friedel oscillations seen as “ripples” in topography, 425, 427 quantum corrals, 426–429 quantum mirage, 427–429 g-shifts, 377 GaAs, 399, 400, 410 GaAs-Pd Schottky junctions, 409 Galkin, D’yachenko, and Svistunov inversion method, 173, 178, 182 Gallagher compound resonances, 210, 214 Gallagher theory, 210 Gamow’s formula, 3 GaP, 409 Gap anisotropy, 124–128 Gap region measurement of, 516 Gap spectra of foil-based junctions, 228, 238 Gärtner’s technique of junction fabrication, 498 GaSb, 401 Ge1−x -Au alloy system, 390 Ge pn junction, 358–367 Giaever tunneling, 9, 10, 43, 63, 82–85 Giaever-Zeller resistance peak model, 408, 414–418 Giant resistance, peak, 407–409, 414–418 Golden Rule calculation, 27, 78 half-integer number of flux quanta, 334 Harmonic detection, 517–520 Heterodyne detection, 484, 486 Hg, 123, 194, 257, 259, 262 High-pressure tunneling measurement, 303 High-speed MIM junctions, 487 Host metal, choice of, for local moment, 411 HTC compilation of Fischer et al., 549–551 Hubbard-like cuprate Hamiltonian model, 316–317 Hybridization (covalent bonding), 315 Hydrogen, field ionization of, 3–5 Hydrostatic pressure effects, 147–150, 284, 303, 548 on Pb and Pb alloys, 264–266, 278, 304–306 on superconducting energy gap, 149–150 superconductor phonons under, 548 Ideal barrier transmission, 29–37 Kane method involving stepwise constant potentials, 30, 31 Leipold and Feuchtwang approach, 31–36 Miller and Good method, 32 in practical cases, 36 questions involved, 30 WKB method, 31, 32 IETS see Inelastic electron tunneling spectroscopy Image force, 39, 43–46, 51, 58

INDEX

Impurity band in phonon spectrum, 53, 148, 263–265, 429 InAs, 55, 339, 345–346, 410 Induced pair potential, 201, 217, 272 Inelastic (real phonon) effects, 365 Inelastic electron tunneling spectroscopy (IETS), 1, 13, 358, 366, 442, 445, 489, 504, 514, 516, 521; see also entries beginning IETS applications/uses of, 504 bridge circuits for derivative measurement, 517–520 dedicated second-derivative circuits utilizing filter networks, 520–522 gap region I–V measurements, 516–517 harmonic detection, 517–520 light emission by, 369–372 measurement tasks, 514 of molecular vibrations, 366–367 Inhomogeneity in BSCCO, controversy over, 330–332, 447–452 InSb, 409, 410 In-Sn alloy, 267–268 In-Tl alloy, 264 Inhomogeneous superconductors, 197–255 Andreev reflection, 199–206 BCS parameter (NV)N for Ir, 251 elastic scattering in N layer, 245–250 McMillan’s tunneling model of bilayers, 223–228, 234, 238, 247, 249, 251 proximity effect models, 251–255 proximity tunneling structures, survey of phenomena in, 206–212 specular theory of tunneling into proximity structures, 212–223 Internal proximity effect, BSCCO, 449–452 local values of pairing temperature, 451 correlation length for the transition temperature, 451 Inversion methods, superconducting phonon, 218 Iodobenzene, 445 Iron-based electron-coupled superconductors: “pnictides”, 140, 141, 310–311 LaFeAsO1−x Fx , 311 NdFeAsO1−x Fx , 311 SmFeAsO1−x Fx , 311 La1−x Fex AsO, 311 Ba0.6 K0.4 Fe2 As2 , 311 BaFe1.8 Co1.2 As2 , 140 Isotope effect, 93, 129, 295, 298 Josephson’s application of Cohen, Falicov, and Phillips method, 105–110 Josephson coupling energy, 110 Josephson effects, 20, 98, 197 Josephson effect detectors, 486–487

587

Josephson junction interferometers, 477–480 Josephson junction logic RSFQ, 481–483 resistively shunted Josephson junctions, 481 single-flux quantum voltage pulse, 481 analog to digital converters ADC, 483 Josephson penetration depth, 154 Josephson radiation, cos  term, 110, 119, 120 in microwave cavity, 150, 156 self-detection of, 150, 153 with second electromagnetically coupled tunnel junction, 154 Josephson self-coupling (JSC) mechanisms, 137 Josephson tunneling, 9–11, 82–86, 111 Josephson tunneling supercurrent, 82, 111 Junction fabrication and characterization, experimental methods of, 489–522 characterization of tunnel junctions, 511–522 characterization of tunneling electrodes, 498–500 foil electrodes, 495–498 Gärtner’s technique, 498 point-contact barrier tunneling methods, 509–511 single-crystal electrodes, 495–498 thin-film electrodes, 489–495 Junctions, tunnel, 475–488, 511–522 basic types of, 37–60 barrier heights, determination of, 512 barrier thickness uniformity, 513 capacitance, 484, 511, 522, 535, 537, 538 characterization of, 511–522 derivative measurement circuity, 514–522 gap region I–V measurements, 516–517 harmonic detection and bridge circuits for derivative measurement, 517–520 in situ scanning methods, 513 initial characterization of, 511–514 metal-insulator-metal junctions, 39–47 metal-insulator-semiconductor junction, 48 pn junction (Esaki-diode)-direct case, 56 Schottky-barrier junctions, 49–56; see also Schottky-barrier junctions vacuum tunneling, 58–60, 510–511 zero-bias resistance, 511 Junction tunneling of single-crystal oriented electrodes, 63, 495–498 Kane’s method for step-potentials, 30, 31, 69 Kinetic energy parallel to barrier, definition of, 57 Kondo and spin-flip scattering spectroscopy, 432–436 Fe, Ni, Co constrictions, 435 Single trapped electron, 433–435 Si:As local moments, 435–436 Kondo ground state, electron scattering in, 401–407 Kondo scattering, 372–378

588

INDEX

Kondo tunneling process, 79 Korringa relation of magnetic resonance, 376 La, 283–286 LaBaCuO system of Bednorz and Muller, 310, 312 Landau levels, 399–346, 352 Lattice tracking STS study of BSCCO, 447–456 Anti-correlation of gap and density of states at V = 0, locally observed, 449–450 Electronic origin of pairing: inapplicability of bosonic models, 447, 449–450, 452 Leakage current, source of, 63 Leipold and Feuchtwang method, 31–36 Light emission by inelastic tunneling, 369–372 Local magnetic moments, 76, 372–377, 401, 411, 413, 432 Local-mode superconductors, 256, 295–297 Localization effects, 411 Logarithmic amplifier, 523 Logic applications of Josephson devices, 481–483 McMillan-Rowell spectroscopy see Conventional tunneling spectroscopy McMillan’s Tc formula, 542 Magnetic field effects, 138–145 Magnetic Force Microscope MFM, 539 Magnetic impurities, effects of, 145–147, 252, 255 Magnetic resonance, Korringa relation of, 376 Magnetic Resonance Force Microscope MRFM, 539 Magnons, 367–368 Many-body theories of tunneling, 384–389 Meissner effect, 88, 138 Metal-insulator-metal junctions see MIM junctions Metal-insulator-semiconductor (MIS) junctions, 48 Metallic systems, electron correlation and localization in, 389–394 Metal overlays for barrier formation, 548 Metal-oxide-semiconductor (MOS) tunnel junctions, 65, 341 Metals amorphous, 256, 273–280 assorted conductance maxima and minima in MIM junctions, 411–414 field emission of electrons from, 5 having s and p valence electrons, 256–263 transition, 281–291 Metal vacuum tunneling probes, deformable, 510–511 Mezei-Zawadowski theory, 403–405 Mg, 194, 201, 294, 339, 343 normal state phonon effects, 363 Mg-deposited films, 128 Mg electrode, 341 Mg-SiO2 – p-Si junction, 341 Microshorts, 138, 511

Miller and Good method, 32 MIM (metal-insulator-metal) junctions, 8–10, 13, 39, 56, 385, 401, 435, 484, 487; see also MIM tunneling fabrication of, 8 high-speed, 487 phonon self-energy, origin of, 385 MIM tunneling, 39–47 conductance curves, 43–48 Simmons approach, 39 Stratton method, 39–41 summary of approximate analytic analyses of, 42 MIS (metal-insulator-semiconductor) junction, 48 Mo, crystalline, 280, 284 Molecular vibrations, IETS of, 366–367 MOS (metal-oxide-semiconductor) junction, 65 MTJ see Tunnel-junction magnetoresistance sensor Multiparticle (MPT) mechanisms, 137–138 Nanoscale chemical inhomogeneity, 327 Nb anomalous early results, 281 comparative tunneling study of Nb and Nb0.75 Zr0,25 , 287–291 double gap structure in SrTiO3 doped with, 129–132 superconductive parameters of, 301 Nb compounds, 281, 289, 299, 492, 547 Nb-Cu wires, 302 Nb-NbOx -PbBi junctions, 133 Nd2−x Cex CuO4+δ (NCCO) (electron doped), 306–309, 316 Ni, 413 Ni films, 348, 356 NIN case, 103 NiO tunneling barrier, 367 NIS case, 104, 114, 118, 158, 485 Nambu strong-coupling theory of superconductivity, 173–177 Nobel prize Bednorz and Muller, 91, 310, 312–313 Binnig and Rohrer, 60 Esaki, 8, 475 Giaever, 10, 475 Josephson, 12, 475 Nodal line, 460, 464–466, 472, 473 Nodal pairing (d-wave) tunneling analysis, 322–323 Nonideal barrier transmission in normal-state tunneling, 63–76 approach to ideal behavior, 63–69 barrier interactions, 76 resonant barrier levels, 69–72 test for ideal behavior, 63–64 two-step tunneling, 72–75 Normal-state tunneling, 21, 336, 498

INDEX

assisted tunneling see Assisted tunneling assorted maxima and minima in metals, 411–414 band structure and density of state, 61–63 basic junction types, 37–60 calculated methods and models, 25–41 final state effects see Final state effects in normal-state tunneling G(V), 61–63 introduction to, 23 J(V), 61–63 nonideal barrier transmission see Nonideal barrier transmission in normal-state tunneling prefactors, 36, 62–63 stationary-state calculation, 25–27 transfer Hamiltonian calculations, 27–28 wavefunction matching at boundaries, 62–63 zero-bias anomalies see Zero-bias anomalies NS tunneling, 203, 219 Oppenheimer’s calculation, 4 Optical point-contact antennas, 487 “optimally disordered” BSCCO, 327 Oscillations, Rowell-McMillan, 207–211 Oscillatory effects and interface sharpness, 64–68 Overlayers, in barrier formation, 548 Oxide tunneling barrier, 64, 501–507 effects of humidity, 503, 505 preparation of, 501–507 plasma oxidation processes, 503–507 thermal, 501–503 Pair-breaking single adatoms, 429–431 Mn and Cr on Pb, 430–431 Pair tunneling, 9–12, 82–172; see also Superconducting energy, gap, spectroscopy of Josephson, 9–12 theory of, 103–111 Paramagnons see spin fluctuations Pb Al in proximity with, 295 crystalline alloy of, 263, 266, 287 determining strong-coupling superconductor functions of, 178 external effects on strong-coupling features, 302–306 gap spectra in, 121–160 local-mode superconductors, 256 neutron scattering, 242, 259–264, 286–291, 360 normal-state phonon effects, 365, 385 proximity tunneling structures, 206–212 Pb-Al junction, 147, 185 Pb alloys, 148, 263–266, 278, 302, 306, 544 PbBi alloy, 122, 131 Pb compounds, 266, 295, 515

589

Pb:D system, 295 Pb films, 125, 126, 129, 167, 206, 227, 273, 302, 338 Pb:H, 295 Pb-In alloys, 303–306, 512 PbIn-I-Pb junctions, 133Pd:D, 295 Pd:H, 295 PETS see Proximity electron tunneling spectroscopy Phase diagram for electron-coupled superconductivity, 312, 317–318 Phonon-assisted transition, concept/term, 358 Phonons assisted tunneling: threshold spectroscopies, 357–383 impurity band in spectrum of, 264–266, 429 insulating barrier layer of tunnel junction, 363 sensitivity of detection in superconductive tunneling, 514 spectra, determined by tunneling and neutron scattering, 364 virtual, 365, 398 Phonon self-energy Combescot and Schreder work on, 385, 399 in degenerate semiconductors, 394–401 origin of normal metal, 401 semiconductor, 393 Phonon softening, 287, 289 Photoemission measurement of spin-polarization, 345 Photon-assisted (quasiparticle) tunneling, 151; see also Quasiparticle tunneling Picking up a CO molecule, 526 Piezoelectric coefficients, 524 Plasma oxidation processes, 503–507 Plasmons, 368, 369, 371 pn junctions, 56 Point-contact barrier tunneling methods, 509–511 Point-contact “Cantilever-Andreev-Tunneling rig”, 534 Point-contact spectroscopy, 186, 283 Polarization, 59, 60, 63 Pressure effects, 147–150 Probability current, definition of, 2 Proximity bilayer problems, alternative approach to, 228 Proximity coherence length, definition of, 204 Proximity effects, 196, 235–236 Proximity electron tunneling spectroscopy (PETS), 236–245 accuracy of, 238, 244 accuracy of step-pair-potential approximation, 217 application to superconductivity, 238 BCS-like behavior vs, 238–240 coherent reflections from rear surface of film two, 214–215

590

INDEX

Proximity electron tunneling spectroscopy (PETS) (cont.) compound geometrical resonances, 214–215 Cooper limit density-of-states expression, 215 density of states at tunneling surface, 177 elastic scattering in N-region, 247 “empirical model,” 228 fully specular (Gallagher) case, 223 Gallagher case, 223 idealized step-pair potential bilayer, 215 McMillan tunneling model vs, 223, 226–228, 238, 247 N1 (E) measurement, 213 NT (E) in gap region, 218, 226 NS bilayer problem, 212 NS tunneling, 203, 219 phase angle, 214 phonon features, 219–221 phonon structures in NT (E), 219–223 reflection r2 , 222–223 results using foil tunneling, 238 sampling depth, 242 self-consistency of pair potentials between N and S, 214, 217, 226 tight-binding model calculation, 221 Wolfram calculation, 210, 212, 213 Proximity tunneling structures specular theory of tunneling into, 212–223 survey of phenomena in, 206–212 PtIr tips, 525 PZT-5H piezoelectric, 524 Quantum mechanical tunneling, concepts of, 2 Quasi-particle and super-currents, connection between, 110–111 Quasi-particle bound state see Zn impurity in Bi2212: intense quasiparticle scattering peak Quasiparticle tunneling, 82, 101, 106, 108, 110, 111, 113, 121; see also Superconducting energy gap, spectroscopy of multiphoton, 151 theory of, 103–111 Radiation, detection of, 483–487 Radioactive decay by alpha particle emission, 3 Rare-earth metals, 350 Real-intermediate-state tunneling, 73–75 Real phonon (inelastic) effects, 365 Resistance bridge, disadvantages of, 519 Resistance peak, giant, 407–409, 414–418 Resonant barrier levels, 69–72 Resonant tunneling, 74, 75, 433 rf detector, 483, 484, 486 Riedel singularity, 120

Rigid-muffin tin approximation (RMTA) method, 281 Rowell-McMillan oscillations, 207 RSFQ see Josephson junction logic Rutherford back scattering, 500 Scanning charge microscopy, spectroscopy, 535–539 surface charge accumulation microscopy (SCAM), 537–538 capacitance spectroscopy, C(V0 ) = dQac (V0 )/dV , 535 single-electron-transistor scanning electrometer (SETSE), 535 STM/SCAM system of Tessmer, HEMT electrometer of Ashoori, 537–538 “tunneling capacitor” of Ashoori, 538–539 Landau level spectroscopy, 539 Scanning Hall probe microscopy SHPM, 539–541 Combination SHPM/STM, 541 2-D electron gas Hall bar, 541 Scanning SQUID microscopy, 480–481, 541 Scanning tunneling microscope (STM), 15–20, 476, 509, 523–531 atomic spatial resolution, 16, 18, 80 feedback circuitry, 15, 524–552 piezoelectric control elements, scanners, 15–16, 523–524 scanning density of states spectroscopy (STS), 16–20 scanning vibrational spectroscopy, 21–22 Topograph, constant current mode, 15–16, 527 tripod scanner, 16 tube scanner, 524 Vibration isolation table, 530 7×7 reconstruction of the Si [111] surface, 16 Scanning tunneling spectroscopy (STS) of superconducting flux lattice, 138–141 MgB2 , 139–140 BaFe1.8 Co1.2 As2 , 140–141 Schottky-barrier junctions, 49–56 CDMT calculation, 51 estimating resistance level of, 52 experimental importance of dopant striations, 51 ferromagnetic barrier fluctuations, 356 GaAs-Pd, 409 GaSb under hydrostatic pressure, 352–355 Ge, 352–363 In-EuS, 356 single-crystal semiconductor, 353 two-step tunneling, 72–75, 408, 415 Schottky barrier probes, 509–510 s-p band elements, tabulated result on, 543, 544 Self-energy (virtual phonon) effects, 365 SEM Channelling, 496 Semiconductor model

INDEX

of superconductors, 104, 105, 109–110 of SNS junctions, 252 Semiconductors accumulation layers at surfaces of, 339–343 amorphous, 414 conductance minima, 409–410 phonon energies of various, 363 self-energy effects, 385 Semimetals, 353 Sensors, 487–488 Shapiro steps, 150 Sharpened STM tips, with single atom decoration, 525–527 CO, 526 Cu, 525 C2 H2 , 20 C60 , 526 CO on Ag surface, 520 C2 H2 on Cu surface, 442 C60 on Au surface, 526 Si:B system, 390 Simmons’ theory, 39 Single-crystal electrodes, 495–498 Single-crystal films, epitaxial, 493–494 Si:P, 389 SIS case, 103–104, 120, 486 SIS detectors, 485–486 S-I-S junctions, 254, 270, 514 Skimming orbits, 352 Small metal particles, capacitor model, 75, 418 Small-particle superconductors, 160–170 Sn, 256, 258, 259, 269, 343 normal-state phonon effects, 363 SN bilayers, tunneling into, 223 Sn single crystals, 126–127 SNS junctions, 156, 199–206, 228–236, 251 Sn-SnO2 counterelectrode-barrier combination, 353 Sn-SnOx -Sn junctions, 133 Solid-state structures, electron tunneling in, 6–9 Specific heat coefficient, density of states relation to, 287 Specific-heat jump, sharpness of, 328–332 Bi2212, 328–332 Tl2201, 330–331 Comparison Bi2212 vs. YBCO, 332 Spectroscopy, literature of, 13 Spectrum of bosonic excitations, 306, 321, 467 Specular theory, application of, 212 Spin flip, 372–378 Spin fluctuations, effect on superconductivity, 187–190, 295, 301, 310 Spin-glass problem, 147 Spin polarization, 60, 63, 343

591

Spin-polarized tunneling as probe of ferromagnets, 343–350 Spin waves, inelastic excitations of (magnons), 367–368 Sputtered films, 492–493 SQUID detectors, 480–481 SQUID microscope image, Tl2201 implies d-wave, 334 SrO plane, in BSCCO, 327 SrTiO3 , 129 Stationary-state calculation, 25–27 STM tips, 18, 429, 525–527 Superconducting Nb, 18, 429 Superconducting Ag/Pb, 526 Etched tungsten, PtIr, 525 Cut wire, 525 STS of magnetic adatoms on Cu, 438–441 Fano model of Kondo resonance, 439–441 Fano parameter q, 438–441 Stratton theory, 39 Strong coupling superconductivity, 121–124 external conditions and parameters, 302–306 N(E) variation in, 192–194 proximity structure theory and, 212–223; see also Proximity electron tunneling spectroscopy Strong-coupling Eliashberg theory, 192 Strong-coupling superconductors, 121–124 conventional tunneling spectroscopy see Conventional tunneling spectroscopy of strong-coupling superconductors Strong-coupling theory, tunneling density of states, 177 Subharmonic structure, 130–138 Subharmonic subgap structure in S-I-S tunnel junctions, 252 Substrate temperature in film deposition, 492 Superconducting electrode, nonuniform strain on, 514 Superconducting energy gap, spectroscopy of, 82–172 BCS theory of, 83–103, 118, 124, 130 electromagnetic radiation, 150–157 electron-phonon coupling and BCS theory, 93–103; see also Electronphonon coupling and BCS theory equilibrium BCS superconductors, 112–121 equilibrium superconductor case, more general homogeneous, 121–160 evidence for, 85, 91, 93, 121, 128, 129, 130, 140 excess current, 130–138 gap anisotropy, 124–128 Giaever tunneling, 82–85 Josephson tunneling, 82–85 magnetic field effects on, 138–145 magnetic impurities, 145–147

592

INDEX

Superconducting energy gap, spectroscopy of (cont.) methods of determining gap parameter values, 118 multiple gaps, 128–130 pressure effects, 147–150 strong coupling, 121–124 subharmonic structure, 130–138, 252–255 superconducting fluctuations, 158–160 T dependence, 118–121 voltage dependence, 119–121 Superconducting excitations, density of, 96 Superconducting flux lattice, 140–141 Superconducting phonon inversion methods, 173, 178–182 Superconducting phonon spectra and α 2 F(ω)/Eliashberg function, 256–309 amorphous metals, 273–280 comparison with neutron scattering method, 293 crystalline s-p band alloy superconductors, 263–273 density of electronic states, effect of variation of, 16–20, 426, 428 external conditions and parameters, 302–306 extreme weak-coupling metals, 291–295 s-p band elements, 256–263 strong-coupling features, 302–306 transition metal alloys, 281–291 transition metal compounds, 289–291 transition metals, 278–291 Superconducting proximity effects, 525 Superconducting tunneling, 21, 64, 168, 195, 355, 364–365 final state effects, 358 Giaever on, 43 Superconductivity, 85–93, 298–302, 312–313, 328–332 coherence property of, 92–93 electron pairing, 92 exactness of, 156, 475 experimental features of characterizing, 85 fluctuations, in 158 fundamental characterization, 85 gapless, example of, 145–150 isotope effect, 93, 129 materials and circumstances of, 92 Meissner effect, 88 minimum film thickness for, 162 quasiparticle excitation spectrum of, 96 spin fluctuations, effect of, 187–190, 295, 300 systematics of, 298–302 temperature dependence of thermodynamic critical field, 89 tunneling studies of, 48, 237, 256, 548 universal features of, 1 Superconductor phonons under hydrostatic pressure, 548

Superconductors A-15, 91, 190, 250, 289, 292, 490, 498, 508 alloy, 263–273 BCS, gap spectra of equilibrium, 112–121 crystalline alloy, 263–273 dirty, 102 elementary excitations of, 100–101 Eliashberg-Nambu strong-coupling theory of, 173–177 gapless, 103, 120, 145 general homogeneous equilibrium, gap spectra of, 121–160 inhomogeneous see Inhomogeneous superconductors local-mode, 197–255 macroscopic quantum-state picture of, 88 magnetic impurities, effects of, 146 PETS application to, 238 semiconductor model, 103–105, 109–110, 252 small-particle, 160–170 strong-coupling see Strong-coupling superconductors superlattices, 236, 270, 494 tabulated summaries of superconducting parameters for classes of, 542–551 thermodynamics of, 158, 298 two-band, 128–130 types of, 89 ultrathin-film see Ultrathin-film superconductors weakly interacting, 92 Super-currents, quasi-particle and, connection between, 85, 110 superlattices, 215–16, 223, 236, 270, 494 Superlattice modulation in Bi2212, 458–460 Super-Schottky detector, 485 Swihart modes, 150 Systematics of superconductivity, 295–297 Ta, 283, 300–302 Tabulated superconductor properties, 542–551 Temperature dependence of thermodynamic critical field, 89 Tc formulas, 176, 287, 542 Thermal oxide barriers, 501–503 Thermodynamics of superconductors, 237, 298 Thick normal films, McMillan–Rowell oscillations in, 548 Thick superconducting films, Tomasch oscillations in, 548 Thin-film electrodes, 489–495 characterization of, 498–500 chemical vapor-deposited films, 493 epitaxial single-crystal films, 493–494 evaporated films, 491 film thickness measurement, 491–492 sputtered films, 492–493 substrate temperature, 492 textured, 492

INDEX

Thouless theory, 100 Threshold spectroscopies, 357–383; see also Assisted tunneling Tl2201 system, Tl2 Ba2 CuTl6+δ , 331 Tight-binding model calculation, 221 Tl, 123, 161, 194 Tl:In, 295 TMR see Tunnel-junction magnetoresistance sensor Tomasch oscillations in thick superconducting films, 548 Transfer Hamiltonian theory/model/approach, 27–28 of conductance peak, 376–377 deficiencies of, 384 to include assisted tunneling, 76 Transition metal alloys, 284–289 Transition metal compounds, 289, 492 Transition metals, 128, 281–291 gap anisotopy, 128 superconducting phonon features, 263–273 Transmission resonances, 62, 69 Trap states, 73 Triplet pairing, 316, 431; see also d-wave superconductivity Tsui tunneling technique, 339–343 Tube scanner, 524 Tunnel-junction magnetoresistance sensor (MTJ sandwich), 487–488 TMR = (R A P − R P )/R P , 488 free layer of soft ferromagnet, 488 CoFeB soft ferromagnetic films, 488 MgO tunnel barrier, 488 Tunneling advantage of, 261 applications of, 1, 2–6 channels, SIS case, 106, 110–111 concepts of, 2 first phenomena identified and explained, 2–4 in normal-state structure see Normal-state tunneling in semiconductor accumulation layers, 339–343 in solid-state structures, 6–9 superconductivity relations to, 9–12 vacuum, 58–60, 510–511 Tunneling barriers see Barriers, tunneling Tunneling current, definition of, 25 Tunneling density of state, 96, 177 Tunneling electrodes, characterization of, 498–500 Tunnel junction devices, 475–476 Tunnel junctions see Junctions, tunnel Tunneling measurement of distribution of electron energy states, 14 of spin-polarization, 343–350 Tunneling method, advantage of, 261–262 Tunneling model of bilayers, McMillan’s, 223–228 Tunneling spectroscopy

593

classification by range of energies, 1 forms of, 13–15 two-step tunneling, 72–75 Tunneling threshold spectroscopy, 6 Two-step tunneling, 72–75 “underdoped,” “optimally doped,” and “overdoped,” 317 Ultrasonic attenuation, 128, 200 Ultrathin-film superconductors, 160–170 conductance minimum zero-bias anomaly, 167 extra peaks, 161–162 minimum film thickness for superconductivity, 162 V, 234–235, 243–244 Vacuum tunneling, 58–60, 510–511 Variable temperature lattice tracking high-resolution STM, 447, 453 Virtual phonon effects (self-energy), 365 Virtual phonon emission, 395 Wavefunction matching at boundaries, 62–63 Weak-coupling metals, extreme, 291–295 Whisker antennas, 487 WKB approximation, 2, 25, 27–35, 39, 43, 45, 53, 59, 62 CDMT calculation relation to, 51 WKB method, 6, 31–32 Wolfram theory, 210–213 Xe atom on Ni(110), 419–421 nudging an atom with an STM tip, 419 Xe on Pt(111), 421 YBCO see Y123 YBCO–Au–Pb Josephson junction, 332–333 Zawadowski–Appelbaum–Brinkman (ABZ) theory, 386 Zeeman transitions, 8, 13, 376–378 Zener tunneling, 33–36, 56 Zero-bias anomalies, 407–418 Giaever–Zeller resistance peak model, 414–418 giant resistance peak, 407–409 in normal-state tunneling, 336, 498 semiconductor conductance minima, 409–410 term, 407 Zero-bias resistance, junction, 511 Zhang–Rice singlet, 315–316 Zittartz, Bringer, and Müller–Hartmann theory, 147 Zn, 93, 186–7, 194, 207–210, 431, 452–60, 548, 550 Zn impurity in Bi2212: intense quasiparticle scattering peak at -1.5 meV, 431, 452–457

(a)

8 Nb tip

(b)

6 4

Pb

2

Si(111)

Pb

0 Mn

5Å

(e)

6 Normalized dI/dV

(c)

Cr

(d)

4 2

Mn

0

8

(f)

6 4 2

Cr

5Å 0 HI

LO

–6

–4

2 –2 0 Bias (mV)

4

6

Fig. 1.12. The schematic in panel (a) shows the superconducting Nb tip above the superconducting Pb layer, with 3 adatoms in view. The dI /dV characteristic in panel (b) measured at 0.4 K, shows nearly ideal peaks at the voltage corresponding to the sum of the energy gaps of Pb and Nb, about 1.52 meV. The energy resolution is about 0.1 meV at 0.4 K. Panels (c) and (e), respectively, show topographs of isolated Mn and Cr atoms on the Pb surface. The corresponding panels (d) and (f) show dI /dV spectra observed from the superconducting Nb tip directly above the magnetic atoms. (After Ji et al., 2008.)

Lo

(c)

0.4

Height (Å)

Hi

(b) ----------------------------------------------

(a)

0.3 0.2 0.1 0.0 0

3

6 Distance (Å)

9

0

3

6 Distance (Å)

9

(d)

d2I/dV2 (nA/V2)

Hi

50

25

0

Lo

Fig. 1.13. Topographic and spectroscopic images at 13 K of single 18 O2 molecule on Ag(110) (aligned along x-direction, molecule fits into 4-fold hollow site of this surface). (a) The 0.12 nm × 0.12 nm topograph was taken with a CO-terminated tip at current 1 nA and tip bias 70 mV. (b) Line cuts of topograph. (c) Vibrational image (spatial distribution of STM–IETS intensity) obtained –76.6 mV sample bias, corresponding to the O–O stretch vibration of the oxygen molecule on Ag(110); data taken with bare metallic tip. Note that this vibrational energy is smaller than that for an isolated molecule. (d) Line cuts of the vibrational intensity image. The solid lines in the cuts align with the oxygen atoms of the molecule. (After Hahn et al., 2000.)

(a)

(b)

1 .8

250 nm

.6 .4 .2 0 250 nm

250 nm

Fig. 3.27. Conductance images of fluxons in magnesium diboride in magnetic field. (After Eskildsen et al., 2002.)

2

1 p

p

0

p

0

0

p

0

p

0

0 dl/dV (normalized)

2

1 p

p

0

p

0

0

0 2

1 p 0

–200

p

0 0

200

–200

p

0 0

200 –200 V (μV)

0 0

200

–200

0

200

Fig. 5.25. Tunneling dI /dV measured spectra at center of 300-nm proximity wire at 35 mK (see Fig. 5.24) as a function of superconducting phase (insets) equivalent to current density. Note the excellent energy resolution. (After le Sueur et al., 2008.)

(b)

(a)

Ni

[0 0

1]

Al

[11– 0]

LOW

Au

HIGH

(c)

Fig. 9.4. From single atoms to linear chains of 20 atoms. (a) Individual Au atoms adsorbed on NiAl(110) appear as protrusions in constant-current STM topographic images. (b) By lateral manipulation, atoms are brought together to form linear chains along the [001] direction. A schematic shows an individual Au atom and a six-atom chain. (c) STM topographic images from a single Au atom to a linear chain of twenty atoms, arranged in single atom increments from left to right. The images are cut from twenty separate scans, each taken with a sample bias voltage between 2.0 and 2.5 volts and a tunneling current between 1.0 and 1.5 nA. Each chain has an apparent height between 2.4 and 2.7 Å. (After Wallis et al., 2002.)

(a)

low

(c)

(b)

high

low

high

(d) H

Side

Cu

C

C

H Cu

Top

Fig. 9.19. (A) STM image of a C2 H2 molecule on the Cu(100) surface at 8 K. Acetylene appears as a dumbbell-shaped depression on the surface with a maximum depth of 0.23 Å. The stable, clean metal tips necessary to perform IETS (inelastic electron tunneling spectroscopy) on single molecules rarely yielded atomic resolution of the Cu(100) surface. The imaged area, 2.5 nm square, was scanned at a sample bias of 100 mV and tunneling current 10 nA. (B) The molecule as in (A) was transferred to the tip by means of a voltage pulse (0.6 V, 100 nA, 1 s) and the same area was scanned. The atomicresolution image has a corrugation of 0.009 Å. This corrugation is sensitive to the nature of the tip and the tunneling parameters. Copper atom spacing is 2.55 Å. The image was scanned at a sample bias of 10 mV and a tunneling current of 10 nA. (C) The atoms in (B) were fitted to a lattice. The lattice is here shown on top of the image (A). (D) Schematic drawing showing side and top views of the molecule’s orientation and suggested adsorption site. The adsorption-site determination assumes that the transfer of the C2 H2 molecule to the tip did not change the position of the tip’s outermost atom. The dashed line shows the outline of the STM image shape. The dumbbell-shaped depression in STM images may result from π bonding to the Cu atoms perpendicular to the C–C axis, reducing the local density of states for tunneling. This would cause the axis of the dumbbell shape to be perpendicular to the plane of the molecule. (After Stipe et al., 1998.)

Fe CO

FeCO

Fe(CO)2

Fig. 9.20. 6.3-nm square image on Ag 110 surface obtained with 70 mV bias, 0.1 nA tunnel current at 13 K. This image shows Fe and CO species deposited and manipulated using a W tip above a silver 110 surface. On the surface, initially 5 single Fe atoms and 5 CO molecules were deposited at 13 K. (After Lee and Ho, 1999.) (a)

1100

dI/dV (pS)

800

400

offset 0 –400

–200

0

200

400

Voltage (mV)

(b)

160

T = 93 K, normal state dI / dV (V = 0)

290

dI / dV (pS)

(c)

25

T = 50 K, superconducting gap map

40

Δ (mV)

Fig. 10.2. Spatial variations and anticorrelations in overdoped Bi2212 sample, p = 0.24, Tc = 62 K. (a) At 94 K all positions along 250-Å line show normal-state dI /dV curves (offset for clarity) with strong electron-hole asymmetry and noticeable variation in voltage position of peak (hump) at negative bias. (b) Map of zero-bias conductance at 93 K, scale range is 160 pS to 290 pS. (c) Map of local superconducting gap on the same 250-Å square, taken at 50 K, values ranging from 25 mV to 40 mV. This gap map is strongly anticorrelated (see Fig. 10.3) with normal conductance map in (b), but shows no correlation with mapped strength of boson-like dips in conductance, as seen, e.g., in Fig. 10.1d, at ±80 mV. The authors state that this rules out boson coupling as origin of pairing, but ties pairing to properties of normal state. (After Pasupathy et al., 2008.)

1.00 nS 60 Å 0.10 nS

0.01 nS

30 Å b

a

Fig. 10.11. Spectral image of the resonant peak; dI /dV image (note logarithmic intensity scale) is taken at V = –1.5 mV (see Fig. 10.10). Note that in this image (see geometry in lower panel) the nodes of the d-wavefunction are aligned along x, y directions. Correspondingly, the brightest four nearest-neighbor atoms are along nodal directions. The weaker outer features, including the 3- nm long “quasiparticle beams” are oriented along the gap maxima, at 45◦ angles in this picture. (After Pan et al., 2000a.)