Physica status solidi: Volume 27, Number 1 May 1 [Reprint 2021 ed.] 9783112494943, 9783112494936

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plxysica status solidi

V O L U M E 27 . N U M B E R 1 . 1 9 6 8

Classification Scheme 1. Structure of Solids 1.1 Alloys. Metallurgy 1.2 Solid-State Phase Transformations 1.3 Surfaces 1.4 Films 2. Non-Crystalline State 3. Crystallography 3.1 Crystal Growth 3.2 Interatomic Forces 4. Microstructure of Solids 6. Perfectly Periodic Structures 6. Lattice Mechanics. Phonons 6.1 Mossbauer Investigations 7. Acoustic Properties of Solids 8. Thermal Properties of Solids 9. Diffusion in Solids 10. Defect Properties of Solids (Irradiation Defects see 11) 10.1 Defect Properties of Metals 10.2 Photochemical Reactions. Colour Centres 11. Irradiation Effects in Solids 12. Mechanical Properties of Solids (Plastic Deformations see 10)see 10.1) 12.1 Mechanical Properties of Metals (Plastic Deformations 13. Electron States in Solids 13.1 Band Structure. Fermi Surfaces 13.2 Excitons 13.3 Surface States 13.4 I m p u r i t y and Defect States 14. Electrical Properties of Solids. Transport Phenomena 14.1 Metals. Conductors 14.2 Superconductivity. Superconducting Materials and Devices 14.3 Semiconductors 14.3.1 Semiconducting Films 14.3.2 Semiconducting Devices. Junctions (Contact Problems see 14.4.1) 14.4 Dielectrics 14.4.1 High Field Phenomena, Space Charge Effects, Inhomogeneities, Injected Carriers (Electroluminescence see 20.3; Junctions see 14.3.2) 14.4.2 Ferroelectric Materials and Phenomena 15. Thermoelectric and Thermomagnetic Properties of Solids 16. Photoconductivity. Photovoltaic Effects 17. Emission of Electrons and Ions from Solids 18. Magnetic Properties of Solids 18.1 Paramagnetic Properties 18.2 Ferromagnetic Properties 18.3 Ferrimagnetic Properties. Ferrites 18.4 Antiferromagnetic Properties (Continued on cover three)

phys. stat. sol. 27 (1968)

Author Index V . M . AGRANOVICH

435

YA. O . DOVHYJ

M . M . AKSELROD

249

G . DRÄGER

K121 513

H . ALEXANDER

391 131

G. F . ALFREY

541

N . A . EISSA

M . AMIN

741

0 . V . EMELYANENKO

L . I . ANATYCHUK

101

H . W . EVANS

695

V . FEDOSEEV

751

I . K . ANDRONIK

45

G. APPELT

657

I . ARTZNER

K5

E . K . ARUSHANOV

45

M . J . ATTARDO

383

J . AUTH

653 77

A . AUTHIER Z . BACHAN K H . S . BAODASAROV G . BALDINI R . W . BALLUFFI R . BALZER D . L . BHATTACHARYA J . BIERSACK U . BIRKHOLZ H . K . BIRNBAUM A . BIVAS J . BLOK V . N . BOGOMOLOV D . N . BOSE J . BOSSE M . BRAUER J . L . BRIMHALL J . O . BRITTAIN H . A . BROWN E . L . BROYDA 0 . BRUMMER F . P . BULLEN H . BURKHARD P . BYSZEWSKI S. D. S. J. L.

CERESARA G . COATES M C K . COUSLAND L . CRAWFORD CSER

K25 KL

161

F . W . FELIX

529

H . J . FISCHBECK T . M . FITZGERALD

345 473

R . J . FLEMING

K57

H.FRIEDRICH W . FUHS

237 171

H . GABRIEL J . M . GALLIGAN W . GEBHARDT L . GEORGESCU M . GEORGIEV J . GINTER 1. G L A D K I H P . GÖRLICH U . GRADMANN A . GRAJA

301 383 713 125 K33 K103 131 109 313 K93

325 301 151 K89 185 K17 249 513 501 K43

P . GROSSE M . GRYNBERG A . G . GUREVICH

K149 255 K85

D . HAHN G. HARBEKE

K133 9

K25,

K125 517 K U 501 559 131

B . K . DANIELS

535 51 249 145 201

1. DEZSI H . - D . DIETZE W . DIEZ

E . FELDTKELLER

95 195 K165 427 139 413, K 6 9 701 K29 219,225 443

A. K. K. P.

S . DAVYDOV M . DEMCHUK DEUTSCHER DEVAUX

45

131 601,611 139

R . P . HARRISON W . HAUBENREISSER W . HEERING K . HENNIG P . R . HERCZFELD K . HIRSCHBERG V . HIZHNYAKOV J . HOLZMAN H . HORA

KL25 593

E . IGRAS G . ISCHENKO V . D . ISKRA I . IVANOV-OMSKII L . K . IVES

K153 K145 K69 K115, K161 671, 681 145 751

69

.

639 101 KL69 117

M . JAROS

K21

J . JAUMANN C. J E C H H . JENA

K43 573 639

Author Index

766 T . JOSSANG

579

D . D . MISHIN

K49

G. JUNGK

237

J . W . MORON

K37

G . B . KABIERSCH

593

J . MÜLLER

A. A. KAMINSKII

K1

R . J . MURPHY

H . KARRAS

109

A . MYSYROWICZ

H . K . MÜLLER

P . P . KESAMANLY

K169

L . KESZTHELYI

131

E . KIERZEK-PECOLD

K107

H . KILIAN

639

K I M YUNG

K161

J . KINEL

K37

M. KLESNIL

545

M. I . KLINGER R . KOEPP J . KOLODZIEJCZAK. B . T . KOLOMIETS L . A . KOLOSKOVA Y U . V . KONOBEEV G . KOOPMANS V . A . KOPCHIK G . KOTITZ L . KOZLOWSKI B . KRAHL-URBAN

479 .

.

K117 .K25, K107, K125 K15, K169 263 435 219,225 741 109 K37 K149

J . KBEJÒÌ E.W.KREUTZ H . RRONMÜLLER I . N . KRUPSKII V . S. KRYLOV D . KULGAWCZUK

545 KILL 371 263 K1 131

B . M. LEBED R . LEVY I . LICEA M. P . LISITSA A . P . LUBCHENKO P . LUKAS O. J . LÜSTE K. T. V. K.

MAIER N . MAMONTOVA G . MANZHELY MARINOYA

S . I . MASHAROV B . MASTEL S . D . MCLAUGHLAN D . B . MEADOWCROFT A . S H . MEKHTIEV J . MERTSCHING M. MESHII C. D. J. D. S.

MICHALK MICHELL MIMKES N . MIRLIN A . MIRONOV

K85 K29 K137,

K143 ' K81 K73 545 101 713 K15 263 K117

455 K89 695 535, 541 K169 345 185 K51 291 K133 443 K85

D . N . NASLEDOV V . V . NEGRESKUL H . NEUHÄUSER S . NIKITINE

723, 733 313 559 K29 45 K15 281 K29

H . PAGNIA

KILL

M. D . PAI V . I . PAKHOMOV W . F . PARKS

671 K129 K17

H. PEISL

K165

E . PETROVA A . POLICEC I . POLLINI

K33 K5 95

V . I . POPOV M . PORSCH J . K . POZHELA I . PRACKA

KL 359 757 K107

S . RADELAAR K . RANDER I . M . RARENKO R . RAUCH K . K . REPSHAS I . S. REZ

K63 301 101 109 757 KL29

G. K. B. A.

G . ROBERTS P . RODIONOV F . ROTHENSTEIN W. RUFFJR

J . SAK V . K . SAXENA J . SCHELM E . SCHMIDT T.SCHOBER M . SCHOTT K . SCHRÖDER D. A. G. D. K. B. B. V. A.

SCHWARZ SEEGER K . SEMIN SEYBOTH V . SHEVLYAGIN D . SILVERMAN S . SKORCHEV A . SKRIPKIN P . SMITH

G . SPINOLO

209 249 K5 117 521 427 413 57 195 201 601, 611 K77 371 K129 639 K85 473 K161 K169 291 95

J . T . STANLEY E . STERK

701 131

H . STILLER

269

Author Index F . STÖCKMANN

K69

R . STOCKMEYER

269

767

0 . G. VENDIK N.

K99

I. VLTRIKHOVSKLI

K 8 1

K25

K . M.VANVLIET

671,681

R . STRUIKMANS

225

R , V U H U Y DAT

K41

B . A . STRUKOV

741

H . STRAMSKA

J . STUKE

171

P . SÜPTITZ

631

W . SZYMANSKA T . C. TISONE B . S . TOSICI

K103 185

.

J . TÓTH I . M . TSIDILKOVSKI W . ULRICI

623 K47

W . WAIDELICH T . WARMINSKI J . H . P . VAN W E E R E N R . WEIDMANN

KILL,

K165 69

219, 225 631

K . WETZIG

K7

M. WUTTIG

701

R . B . MAKULA

623

249 333,489

L . I . ZARUBIN

101

A . P . ZHUKOV

K129

K . VACEK

K29

B . H . ZIMMERMANN

M . Y A . VALAKH

K81

M . ZVÄRA

639 K157

physica status solidi B o a r d of E d i t o r s P. A I G R A I N , Paris, S. A M E L I N C K X , Mol-Donk, V. L. B O N C H - B R U E V I C H , Moskva, W. D E K E Y S E R , Gent, W. F R A N Z , Münster, P. G Ö R L I C H , Jena, E. G R I L L O T , Paris, R. K A I S C H E W , Sofia, P.T. L A N D S B E R G , Cardiff, L. N É E L , Grenoble, A. P I E K A R A , Warszawa, A. S E E G E R , Stuttgart, F. S E I T Z , Urbana, 0 . S T A S I W , Berlin, M. S T E E N B E C K , Jena, F. S T Ö C K M A N N , Karlsruhe, G. S Z I G E T I , Budapest, J . T A U C , Praha Editor-in-Chief P. G Ö R L I C H Advisory Board M. B A L K A N S K I , Paris, P. C. B A N B U R Y , Reading, M. B E R N A R D , Paris, W. B R A U E R , Berlin, W. C O C H R A N , Edinburgh, R. C O E L H O , Fontenay-aux-Roses, H.-D. D I E T Z E, Saarbrücken, J . D . E S H E L B Y , Cambridge, P. P. F E 0 F I L 0 V, Leningrad, J . H O P F I E L D , Princeton, G. J A C 0 B S, Gent, J. J A U M A N N , Köln, E. K L I E R , Praha, E. K R O E N E R , Clausthal-Zellerfeld, R. K U B O , Tokyo, M. M A T Y A S , Praha, H. D. M E G A W , Cambridge, T. S. MOSS, Camberley, E. N A G Y , Budapest, E. A. N I E K I S C H , Jülich, L. P A L , Budapest, M. R O D O T , Bellevue/Seine, B. V. R O L L I N , Oxford, H. M. R O S E N B E R G , Oxford, R. V A U T I E R , Bellevue/Seine

Volume 27 • Number 1 • Pages 1 to 464, K1 to K84, and Al to A4 May 1, 1968

AKADAMIE-VERLAG•BERLIN

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Contents Page

Review Article G.

On t h e B a n d Structure of Anisotropic Crystals

HARBEKE

9

Original Papers I . K . ANDRONIK, E . K . ARUSHANOV, 0 . V . EMELYANENKO, a n d D . N . NASLEDOV

A.

S.

E.

SCHMIDT

DAVYDOV

Negative Magnetoresistance in p-CdSb

45

Theory of Urbach's Rule

51

Optical Properties of GeSi Alloys in the Energy Region from 13 eV

1

to

57

E . IGRAS a n d T . W A R M I N S K I

On Electron Mirror Microscopic Observations of the Details of p - n J u n c t i o n Regions in Si

69

Contrast of a Stacking F a u l t on X - R a y Topographs

77

A . AUTHIER

G . B A L D I N I , I . P O L L I N I , a n d G . SPINOLO

Optical Properties of a- and ß-TiCl 3 L . I . ANATYCHUK,

V . D . ISKRA,

O. J . LÜSTE,

95

I . M . RARENKO, a n d L . I . ZARUBIN

Anisotropy of Electroconductivity in CdSb P . GÖRLICH,

H . KARRAS,

101

G. KOTITZ, a n d R . RAUCH

Polarized Luminescence of X-Irradiated CaF 2 :Y and SrF 2 :Y Crystals and t h e Structure of their Luminescence Centres

109

L . K . IVES a n d A . W . R U F F J R .

L.

GEORGESCU

L . CSER,

and E.

Studies of t h e Effect of Annealing on Extended Dislocation Nodes in Silver-Tin Alloys

117

Nonlinear Thermodynamics of Thermoelectric Phenomena in Ionic Semiconductors

125

I . DEZSI,

I . GLADKIH,

L . KESZTHELYI,

D . KULGAWCZUK,

N . A. EISSA,

Mössbauer Study of Hyperfine Fields in Mn-Zn Ferrites

STERK

131

J . BIERSACK a n d W . D I E Z

Motion of Markers and Bubbles in Solids by Self-Diffusion in a Temperature Gradient

139

K . DEUTSCHER a n d K . HIRSCHBERG

M.

BRAUER

Thickness Dependence of the Quantum Yield of Cesium-Antimony Films

145

On the Behaviour of a Cylindrical Electron-Hole Plasma in a Magnetic Field with Transverse Component

151

Magnetic Domain Wall Dynamics

161

E . FELDTKELLER

W . FUHS a n d J . STUKE

Hopping Recombination in Trigonal Selenium Single Crystals T . C. TISONE,

. .

171

J . O. BRITTAIN, a n d M . MESHII

Stacking Faults in a C u - 1 5 a t % Al Alloy (I)

185

T . SCHOBER a n d R . W . B A L L U F F I

Direct Evidence for Dechanneling of 40 kV Gold Ions a t Planar Defects in Silver and Gold l

195

Contents

4

Page P . DEVAUX a n d

G. G.

M . SCHOTT

ROBERTS

Field Effect on Photocarriers in Copper Phthalocyanine Single Crystals

201

Electron Injection into a p-Type Semiconductor

209

J . H . P . VAN W E E R E N ,

G . KOOPMANS, a n d J .

J . H . P . VAN W E E R E N ,

R . STRUIKMANS,

BLOK

The Position of the Dislocation Acceptor Level in n-Type Ge . . . G . KOOPMANS, a n d J .

219

BLOK

Low-Field Magnetoresistance and Magnetoconductivity in n-Type Ge with Dislocations

225

H . FRIEDRICH a n d G. JUNGK

Generation-Recombination Statistics in Semiconductors — A Model with Two Centres

M . M. AKSELROD,

K . M . DEMCHUK,

I . M . TSIDILKOVSKI,

E . L. BROYDA,

and

K.

237

P.

RODIONOV

The Effect of Pressure on the Electron Effective Mass in InSb

.

249

M.

Influence of Uniaxial Stress on the Optical Properties of CdSe . .

255

V . G . M A N Z H E L Y , a n d L . A . KOLOSKOVA

263

GRYNBERG

I. N. KRUPSKII,

Thermal Conductivity of Solid Ammonia

R . STOCKMEYER a n d H . STILLER

Phonons, Torsons, and Rotational Diffusion in Adamantane

H.

NEUHÄUSER

.

. . .

269

Untersuchungen zur aktiven Kristallänge im Fließbereich neutronenbestrahlter Kupfereinkristalle (II)

281

D . MICHELL a n d A . P . SMITH

The Nature of Oxide Layers on Single Crystals of Cadmium and Magnesium

H . GABRIEL,

J . BOSSE, and K .

U . GRADMANN a n d J .

D.

N.

BOSE

W . ULRICI

MÜLLER

313

Effect of Low Energy Electron and UV Irradiations on the Charge Condition of Etched Germanium Surfaces

325

Untersuchung dreiwertiger Übergangsmetallionen in Silberhalogeniden (I)

333

MERTSCHING

The Theory of Low-Field Oscillations of the Surface Impedance in Metals

345

Green's Function Calculation of Electro-Optical Effects in Semiconductors (I)

359

A . SEEGER a n d H . KRONMÜLLER

The Micromagnetic Equations of a Superconductor

371

J . M . GALLIGAN a n d M . J . ATTARDO

Radiation Hardening in Platinum

H . ALEXANDER

301

Flat Ferromagnetic, Epitaxial 48Ni/52Fe(lll) Films of Few Atomic Layers

H . J . FISCHBECK a n d J .

M . PORSCH

RÄNDER

Mössbauer Spectra in the Presence of Electron Spin Relaxation (II)

291

Elektronenmikroskopie eingefrorener Versetzungen

383 (II)

391

5

Contents

Page U . BIRKHOLZ a n d J . SCHELM

Mechanism of Electrical Conduction in ß-FeSi 2

413

V . K . SAXENA a n d D . L . BHATTACHARYA

Effect of Electron-Lattice Interactions on the Spin-Paramagnetism of Metals

427

V . M . AGRANOVICH a n d Y U . V . K O N O B E E V

Diffusion of Free Excitons in Molecular Crystals

435

V . N . BOGOMOLOV a n d D . N . M I R L I N

Optical Absorption by Polarons in Rutile (Ti0 2 ) Single Crystals . .

443

S . I . MASHAROV

Effect of Phonon Drag on the Kinetic Properties of Alloys . . . .

455

I . LICEA

Erratum

463

Short Notes K H . S . BAGDASAROV,

A . A . KAMINSKII,

V . S . K R Y L O V , a n d V . I . POPOV

Room-Temperature Induced Emission of Tetragonal Y V 0 4 Crystals Containing Nd 3 + B . F . ROTHENSTEIN,

K . WETZIG

D.

K1

A . POLICEC, a n d I . ARTZNER

Contributions to the Kinetics of Charging Nickel with Hydrogen

K5

Some Investigations Concerning the Position and the Melting Point of TaFe Lamellas in the System TaFe-TaFe 2

K7

Improvements in the Orientation Effect Observed b y Scanning Electron Microscopy Kll

G . COATES

B . T . KOLOMIETS,

T . N . MAMONTOVA, a n d V . V . N E G E E S K T J L

On the Recombination Radiation of Vitreous Semiconductors

. . K15

H . A . BROWN a n d W . F . PARKS

M.

JAROS

H . STRAMSKA,

The Magnetic Dilution Problem

K17

On the Covalency in Transition-Metal Impurity Covalent Complexes

K2I

Z . BACHAU,

P . BYSZEWSKI, a n d J . KOLODZIEJCZAK

Interband Faraday Rotation and Ellipticity Observed at the Absorption Edge in Silicon K25 K . VACEK,

A . MYSYROWICZ,

R . LEVY,

A . BIVAS, a n d S . N I K I T I N E

Luminescence of AgCl Excited by Ultraviolet Laser

K29

M . GEORGIEV a n d E . PETROVA

Investigation of the Drift of Photoelectrons in AgBr at Moderate Temperatures K33 J . KINEL,

L . K O Z L O W S K I , a n d J . W . MORON

Diffusional Relaxation Effects in Hydrogen Charged Silicon Iron K37 R. Vu

H U Y DAT

Dielectric Behaviour of Ferroelectric Mqnocrystalline Antimony Sulfo-Iodide with Electric Field Frequency, up to Microwave Range K 4 I H . BURKHARD a n d J . JAUMANN

The Determination of the "Gyroelectric" and "Gyromagnetic" Constants of Thick, Evaporated Layers of Iron by Means of Measurements in the Reflected Light only K43

6

Contents Page

J . TÖTH

The Temperature Dependence of Resistivity in Al-Ta Dilute Alloys K47

D. D.

The Effect of Dislocations on the Properties of Silicon Iron in Alternating Magnetic Fields K49

C.

MISHIN

Die magnetische Diffusionsnachwirkung in Ni-Fe-Ferriten mit geringen Co-Zusätzen K51

MICHALK

R. J .

Activation Energies and Emission Spectra of the Thermolumines-

FLEMING

cence of CaF 2 Crystals Gamma-Irradiated at 77 °K S.

U . BIRKHOLZ,

K57

The Kinetics of Short-Range Order in Some Ag-Au Alloys . . . . K 6 3

RADELAAE

W . H E E R I N G , a n d F . STÖCKMANN

Thermoelectric Power of Illuminated CdS

K69

A . F . LUBCHENKO

Theory of Urbach's Rule for Impurity Centres D . SCHWARZ M . YA. VALAKH,

ESR Study of Cu

2+

K73

in AgCl Crystals Additionally Doped with Se or S K77

N . I . VITRIKHOVSKII, a n d M . P . LISITSA

The Electron Effective Mass in CdS^Sei-s Crystals

K81

Pre-printed Titles of papers to be published in the next issue

:

Al

Contents

7

Systematic List Subject classification:

Corresponding papers begin on the following pages (pages given in italics refer to the principal subject classification) :

1.1 1.2 1.3 1.4 2 3.2 4 5 6 6.1 8 9 10 10.1 10.2 11 12.1 13 13.1 13.2 13.3 13.4 14.1 14.2 14.3 14.3. 1 14.3. 2 14.4 14.4.1 14.4. 2 15 16 17 18 18.1 18.2 18.3 19 20 20.1 20.2 20.3

K63 K7 69,291 291,313 K15 K21 161, K7, Kll Kll 263, 269, 455 131, 301 263, 455 139, K63 77, 109, 219, 225, 291, 333, 391, K73, K77 117, 185, 195, 281, 371, 383, K37, K49, K63 K33 195, 281, 325, 383 K37 301,443 9, 57, 249, 255, 345, 359, K25, K81 9, 201, 435 201, 325 45, 95, 171, 219, 237, 333, 435, K57, K73, K77 345, 455, K47 371 9, 45, 101, 219, 225. 249, 413 145 69 333 151, 209, 359 K41 125, 249, 413, 455, K69 171, 201, 237, K69 145 151, K25 427 161, 313, K5, K17, K37, K43, K49 131, K51 345, K21, K77 359, K25, K43 9, 51,57, 95, 145, 255, 333, 443, K43, K73. K81 K1 109, 255, K15, K29, K57

8 21 21.1 21.1.1 21.2 21.6 22 22.1 22.1.1 22.1.2 22.1.3 22.2.1 22.2.3 22.4. 1 22.4.2 22.4. 3 22.5 22.5.1 22.5.3 22.6 22.9 23

Contents 139, 345, 427, 455, K7, K47 161, 185, 281, K5 161, 313, K37, K43, K49 427 117, 195, 383, K63 9, 45, 51, 101, 125, 139, 145, 237, 359, 413, K15 9 9, 57, 151, 219, 225, 325, 391 9, 57, 69, 77, K25 9, 171, 209 Kll 249 9, K69, K81 255, K81 9, K21 95 333, K29, K33, K77 109, K57 9, 291, 443, K1 201, 269, 435 263

Contents of Volume 27 Continued on Page 467

Review Article phys. stat. sol. 27, 9 (1968) Subject classification: 13.1; 13.2; 14.3; 20.1; 2 2 ; 2 2 . 1 ; 22.1.1; 2 2 . 1 . 2 ; 22.1.3; 22.4.1; 2 2 . 4 . 3 ; 22.6 Laboratories

RCA,

Zurich

On the Band Structure of Anisotropic Crystals By G. HARBEKE

1.

Contents

Introduction

2. Interband

2.1 2.2 2.3 2.4

transitions

3. Measurement 4. Optical

4.1 4.2 4.3 4.4 4.5

and optical

constants

Theory of interband. transitions Density of states and critical points Selection rules for optical transitions Dispersion relations spectra

techniques and

interpretation

Nearly cubic crystals Crystals with layer structure Crystals with chain structure Biaxial crystals Exciton effects

5 . Induced

anisotropy

of cubic

crystals

5.1 Piezooptic effects 5.2 Electrooptic effects References

1. Introduction Considerable progress has been made during recent years in the study of the electronic band structure of crystalline solids. This is partly due to newly developed, efficient methods of calculation and the increasing use of fast computers. On the experimental side the measurement of the optical absorption and reflectivity spectra of many insulators, semiconductors, and metals — more recently also with external symmetry reducing parameters applied simultaneously — have in conjunction with the calculated band models provided a great deal of information about the dispersion of energy bands in fc-space. Whereas in the beginning the interest was concentrated on materials with cubic crystal structure, now increasing numbers of crystals of lower symmetry are also under study. I t is the purpose of the present review to summarize our knowledge of the band structure of anisotropic insulators and semiconductors deduced from their

10

G.

HARBEKE

optical spectra. In order to get an overall picture we concentrate on the intrinsic absorption region above the absorption edge up to energies where the valence electron plama resonances occur. Characteristic structure in the optical constants in this region is caused by the existence of critical points of high density of states in valence and conduction bands. In general an assignment of this kind is more difficult in crystals of lower symmetry but the experimentalist is given an additional parameter with the possibility of performing measurements using linearly polarized light under different orientations to the crystal axes and making use of the polarization dependence of the transition probability for electric dipole transitions. In this way the number of possible band models could be considerably reduced for all crystals investigated so far, and in some case a definite assignment has been proposed. A systematic comparison of the spectra of isoelectronic materials of the same or similar crystal structure also allows conclusions to be drawn about the band structure of some cubic crystals. This purpose is also served by experiments in which an optical anisotropy is imposed onto a cubic crystal by applying — preferably modulated — a uniaxial deformation thus yielding additional information about the symmetry of initial and final states involved in the optical transition. Such differential methods which also include electrooptic effects, have resulted in a higher accuracy in the determination of critical point energies. In the following chapter we introduce the joint density of valence and conduction band states and the selection rules governing the optical transitions, both of which determine the dispersion of the optical constants. A short description of the commonly used experimental methods is followed by the results of measurements on anisotropic crystals and their interpretation in Section 4. This section also includes a discussion of exciton effects. Section 5 finally contains some examples of imposed anisotropy. 2. Interband Transitions and Optical Constants 2tl

Theory

of interband

transitions

The transition probability for optically excited transitions of electrons results from the perturbation of the ground state by the time-dependent vector potential of the electromagnetic field of the incident light wave. The total Hamiltonian for a crystal electron is then H

= ^n{P

+

e A ) 2 +

F

where h is the number of transformations constituting the group, %j{R) and %i(R) are the characters of the i.r. for the transformation R, and the % ,y,z are the characters of the i.r. according to which the coordinates x, y, z transform. This rule holds both for single groups and for double groups of the Hamiltonian where the latter ones have to be used if the spin variable is also considered. I n this case one simply has to put the characters of the i.r. of the double groups into (17). Since in cubic crystals the coordinates x, y, z transform according to the same i.r., the matrix element of the transition is independent of the direction of polarization and the crystal is optically isotropic. Optical anisotropy is caused by the fact that x, y, z transform according to different i.r. resulting in a polarization dependent transition probability. The optical properties of anisotropic crystals are described by a tensorial complex dielectric function e = e1 + i e 2 or by the tensors and s 2 , respectively. The selection rules are, of course, only valid for a point of the Brillouin zone having the symmetry of the group chosen for the calculation according to (17). For the centre (k = 0) and symmetrically equivalent points this is the point group of the crystal. For all other points one has to find the group of the fe-vector consisting of all symmetry operations which leave k invariant or transform it into a wave vector k + Kj where K- is a vector of the reciprocal lattice. The fc-dependence of the selection rules can lead to the situation that the transition is forbidden a t a critical point (E0, k0) but allowed in the neighbourhood, e.g., along an axis in fc-space. In this case the matrix element e • M v c is proportional to jfe — fc0| [4] from the first term in its expansion around fe0. One thus obtains for e2 at an M 0 -type critical point after (9) and (11) x

e2 oc

|fc - fc0|* (E -

iQi/2

and hence in the parabolic approximation (10) £2OC

-L(E-E0fl*.

(18)

Up to this point only the interaction between the electrons and the electromagnetic radiation has been considered. If the interaction with lattice vibrations is taken into account in second-order perturbation theory, indirect, i.e. nonvertical transitions also become possible. The two basic selection rules are now fcc + q = ky and h a> = Ec — Ev + kB dq where q and kR Bq are the phonon wave vector and energy, respectively. The positive sign stands for the process of phonon emission where the phonon energy is delivered by the photon, the negative sign describes the absorption of a phonon with the energy being given up by the lattice. Of course, an indirect transition is allowed only if the symmetry properties of the initial and final state wave functions and of the operators describing the interaction of the electrons with the lattice and the radiation are such that the appropriate matrix element is different from zero. These conditions select phonons of certain symmetry for the transition. The energy de-

On the Band Structure of Anisotropic Crystals

15

pendence of e2 is then of the form [4] e 2 oc

(E ± k

0

B

-

q

E,Y ,

(19)

and in the case of indirect transitions which are forbidden at the critical point itself analogous to (18) e2 oc - L (E ± kB 0« -

E0)3 .

(20)

The absolute values of e2 for indirect transitions are, however, some orders of magnitude smaller than for direct transitions. They can therefore only be observed at energies where no direct transitions occur. 2.4 Dispersion

relations

The real and the imaginary parts of a complex quantity describing a linear relation between two amplitudes are connected by dispersion relations (KramersKronig relations) [6] provided some very general requirements are fulfilled [5]. The complex dielectric constant e(oi) describes the relation between the electric field E(a>) and the displacement vector D{w) in the form D(a>) = e(co) • E(a>). We therefore have the relations [7] oo

*(»> = l + - f 4 ^ d 71 J

0

CO

1

f —

l

, '

t

(21)

CO2

oo

0 That means that e1(w) can be calculated if the e 2 -spectrum given by (9) is known explicitly over the entire wavelength range, and vice versa. The relations, though, have a strongly "local" character since the values of e 2 close to w contribute most strongly to e^oj). This leads to some correlations discussed by Velicky [8], e.g., £j has a maximum (minimum) at frequencies where e 2 increases (decreases) very strongly and a maximum in e2 coincides with a steep decrease Of £j.

The same relations as (21) and (22) also hold for the complex refractive index

N = n + i k, e.g.,

oo

=

J o

u> * — a>*

(23)

and finally for the complex amplitude reflection coefficient r = \r\ elf = R1!2 civ with the intensity reflection coefficient B and the phase angle ~ 1 according to the Drude formula for frequencies above the valence electron plasma frequency cop:

Another possibility is to use the Ansatz R oc m x and to a d j u s t the exponent x such that the phase angle below and a t the absorption edge agrees with d a t a obtained b y absorption measurements. This method, however, is not very effective because of the local character of the Kramers-Kronig relations. F o r w -> 0 in (21) one obtains oo £l

(0) = 1 + - f ^ d c o ' n J to 0

.

(25)

Since we discuss here interband transitions e^O) means the real part of the dielectric constant in the low-frequency region below the absorption edge b u t above the lattice vibration region. I n this region there is no absorption in the ideal crystal and (25) thus gives a constant real part of the refractive index and e(0) E2 £j(0) which is equal to the static dielectric constant only if no strong lattice absorption bands are present. I t should be noted that our e(0) is identical to what is often termed s i n studies on infrared lattice vibrations since in this case it is the value above the frequency region of interest. The sum rule shows t h a t e(0) is given by e2{a>) over the entire spectral region. I t thus affords an independent check on the measured and extrapolated reflection spectrum as the value obtained for e(0) can be compared with the square of the long-wavelength refractive index. I t is also immediately obvious from equation (25) t h a t a high long-wavelength refractive index is to be expected if e2(a>) takes on high values a t low frequencies. The dispersion relations can also be applied to anisotropic crystals in a straightforward manner. I n this case they connect the corresponding components of the tensors C1(OJ) and £2(fo) or R(a>) and (f(h) where three valence electrons form covalent ~2 in equation (9), the joint density of states rises monotonically between 0 and 5 eV. I t should be noted t h a t the experiment has been performed with E _[_ c where the selection rules allow n - n and a - o transitions. In this case identical selection rules hold at all points of the Brillouin zone, for E \ \ c only mixed transitions of the tz-g or g-tz type are allowed. Experiments in order to determine the optical constants for E || c are under way [42], The compounds GaS and GaSe consist of a series of layers of the type S - G a -Ga-S, the crystal symmetry (D3 is equal to t h a t of graphite except for the lack of inversion symmetry in £)3 h . Bassani and Pastori [42] therefore were able to calculate the band structure with the above two-dimensional model and to interpret previous reflectivity measurements [44] qualitatively. GaSe has a direct optical gap r { ~ r z of 2.0 eV, the interband absorption in GaS starts with indirect transitions r { - P { at 2.6 eV [45]. A three-dimensional band calculation in the tight-binding approximation by Kamimura and Nakao [46] showed the effective masses in ^-direction to be so large that the bands have essentially two-dimensional character thus justifying the Bassani-Pastori model. There is, however, no detailed agreement concerning the relative order and the position of the bands. At the band edges of both materials as well as in the reflectivity spectrum of GaS up to 5.5 eV no polarization effects have been observed but there are pronounced differences for E || c and E J_c in the GaSe spectrum [47, 48]. Another group of layer structure compounds are Y - V I compounds of the type Bi 2 Te 3 with the layer sequence Te (2 >-Bi-Te (1, -Bi-Te (2) . There are two types of tellurium layers with different bonding; the Te n H,— ± : > —

1r*

6 0— i

!

A,

I):

.1; Ai

H

,

r r

±> AT II, — X>A* X, Aï

a ;

x^ A T

AÎ — i

AJ

A:

AT

X>

r r

T

II, A , t A5 1 > A6 A -L> II, X > A6 Ae

44, A5

A1 —\-+A2

A,,

Ar

A,

j

1>

Double group

Single group

r t

Ax

a2

allowed for E J_ c, —1-> forbidden

^

a2 A3

A j ^ A ,

A

A„ A

I, K : (Cs )

II, X Q-L x> Q, II, X >>Q2 fìl

Q3

«3 ß4

II, X • Q3 X, ß4

29

On the Band Structure of Anisotropic Crystals Fig. 12. Room temperature reflectivity of BisTeB between 0.2 and 11 eV for E || c and E ]_ c (Greenaway and Harbeke [50])

Bi 2 Te 3 [50] for E ±c in Fig. 12 indeed shows two triplets of similar line shape, Ev E2, E3 and Es, Ee, E7, with identical splittings of 1.4 eV between the first two and 1.0 eV between the last two components, respectively. For E \ \ c one finds an increase of the centre components and a decrease of the outer components. The latter are, however, not completely absent; there is probably a partial breakdown of the selection rules due to the quality of the surface which cannot be prepared by cleavage for measurements in this orientation. The quantitative difference in the spectra, however, seems to justify an assignment of the triplets to /"¡¡-/"i transitions. energy E (eV The region just above the absorption edge is shown in Fig. 13 in greater detail. These data have been taken at 77 °K but there is essentially no line narrowing at low temperatures. The structured maximum at 0.35 eV and the 1.4 eV maximum occur only for E c with the same distance as t h a t between the last components of the triplets. Only an assignment J e , (JT1, r t ) - J T i (of course, the parity can always be reserved) is compatible with the selection rules where T¿~ comes from a TY-state. The transition forbidden for E \\c in the single group is allowed in the double group for the component / " j \ - t \ but the oscillator strength is not sufficient to make it observable. The 0.25 eV maximum is common to both orientations, it is assigned to a critical point at A or D for this and other reasons. The E c spectra of a series of Bi 2 Te 3 -Bi 2 Se 3 alloys of different composition [50] exhibit the same features as Bi 2 Te 3 with the critical point energies being displaced to higher energies with increasing selenium content. Fig. 14 shows the energies of the reflectivity maxima E1 to E7 as well as the band gap E0 versus composition. We must conclude from Fig. 14 t h a t the band structure in the 78 ^

7 6

//

ELc Elle

is

/

/ Fig. 13. Reflectivity of Bi2Te3 at 77 "K between 0.18 and 1.5 eV for E || c and file (Greenaway and Harbeke [ 50])

\

'

— ^

/ " X 1 0.2

1 OA

1 0.6

1

10

12

Photon energy E (eV1 -

H

30

n

G. HAKBEKE

n

I ' ^

n (z;+zp r6(z;i

5 4

n

3 2

tA'j'Al! 0 piT

tn E || c and E J_ c (Greenaway and Harbeke [71])

6.0 6.5 W Photon energy EleV)

Fig. 22. Reflectivity of Cdl a between 5 and 7.5 eV, E JL c (Greenaway and Nitsche [74])

7.5 —

The line intensity is proportional to (n + l/2)~ 3 . There is no case reported so far where an observed series could be fitted to equation (29) but it should be noted that the intensity ratio of the first two lines is 27. Metastable excitons also exist in Cdl 2 [74]. Fig. 22 shows part of the intrinsic reflectivity spectra. At room temperature there are two resonances X1 and X2 separated by a symmetric antiresonance. The line widths of X1 and X2 are decreased at 77 ° K , the antiresonance is more pronounced and sattelite lines occur. The exciton structure has also been observed for E || c [71]. The excitons in Cdl 2 are thus not of two-dimensional character. The lines instead can be fitted to the formula for the three-dimensional Rydberg series with resulting binding energies of 0.17 eV for Xt and 0.41 eV for X2. We thus have the situation that two materials of the same structure and nearly equal atomic distances have entirely different exciton properties. It should be noted, however, that also the reflectivity spectra (and hence the band structure) do not show the similarity that is observed in other groups of materials with the same structure. This behaviour is probably due to the two extra outer p-electrons of Pb compared to Cd which will certainly lead to a different valence band structure. 5. Induced Anisotropy of Cubic Crystals 5.1

Piezooptic

effects

During recent years a number of differential methods have been increasingly used in the band structure studies. In this section we deal with piezooptic effects in interband spectra which have been mostly measured on cubic crystals. The change of symmetry induced by uniaxial deformation can result in splittings and energy shifts of edges or maxima in the e 2 -spectrum which, combined with the selection rules of the new system, give further information concerning the assignment to critical points. The optical properties are described by a complex, frequency-dependent second-rank tensor £ as in the case of anisotropic

38

G. H A B B E K E

crystals. The change of e2 under uniaxial strain can be expressed by (30) where the third axis is parallel to the stress axis. The components AgJ- and Aejl can again be calculated by means of the Kramers-Kronig relation from the quantities A-R-L and Ai?11 measured with linearly polarized light or from the corresponding changes in absorption. The experimental methods employ static [75], dynamic [76, 77] as well as combined static-dynamic [78] deformmations. The analysis of the fine structure measured in piezoabsorption at the absorption edge of Ge and Si, i.e. in the indirect interband transition regime, has been based on the known symmetry of the band extrema [79, 78], One then obtains the deformation potential constants of those extrema and the straininduced change of the binding energies of the associated excitons. At higher energies piezoreflection has been measured, e.g. at Ge at the first M 0 -type critical point at 0.8 eV and the first M r point at 2.1, 2.3 eV [80]. This doublet in the double group), had been assigned to Aa-A1 transitions (/14, A s - A e , A6-Ae i.e. to a critical point in [111] direction [22], The selection rules are thus those for the group C3„in Table 3. Stress along the [100] direction causes a shift of the maxima independent of the orientation of the electric light vector since all [111] bands are equally displaced and a relative change of the peak heights. For both peaks one finds compression

— 2 A Rmi 'compression

—

Assuming AR to be proportional to Ae2 over small frequency ranges, this result has been explained with the change of the matrix elements since in the perturbed system the group of the wave vector, A , becomes Cs [80]. Deformation along [111], on the other hand, causes a polarization dependence of the peak shift since the eight equivalent [111] axes are divided into two groups of six and two axes as in wurtzite (see Section 4.1). The expected splitting for could not be observed because of the small deformation but the results for both stress axes confirm the previous assignment. Fig. 23 shows the piezoreflection of Si at the first M r t y p e critical point. The peak shift depends on the polarization for stress along the [100] direction but not for stress along [111]. Analogous to the results for the M r point in Ge, it is concluded that here mainly transitions at points in [100] directions contribute to this maximum. According to Gerhardt [80, 81] we have A 6 -A 6 and A 7 -A 6 transitions. The line broadening of (about 10 meV compared to ii|j) in Fig. 23 is explained by the strain-induced splitting of the A7 and A 6 valence bands. The observed value is equal to the splitting of the Pg-state measured by cyclotron resonance under strain. Since the A7 -and Zl„-bands connect to r s it is concluded that the critical point is on the zl-axis but close to T , whereas previously the critical point had been placed at r . The result has been partly confirmed by more detailed pseudopotential calculations of Brust [22] and Kane [77] and by dynamic piezoreflectance measurements of Gobeli and Kane [77], The latter authors also discuss the line shapes in piezoreflection for the different types of critical points.

On the Band Structure of Anisotropic Crystals

39

Wavelength fnm/~ 365 370

Wavelength (nmi 365 370

3.50 3À5 3Â0 -—Photon energy E ie/J

m 3A0 135 Photon energy EleV!

Fig. 23. Room temperature piezoreflection of Si between 3.3 and 3.5 eV (Gerhardt [80])

Only a few piezooptic measurements have been performed on anisotropic crystals, e.g., on wurtzite CdSe by Grynberg [82]. No absorption edge shift could be observed for deformation perpendicular to the c-axis. U n i a x i a l pressure along the c-axis also does not shift the edge for JE 11 c b u t moves the E J_ c edge towards its position for E \ \ c. This implies t h a t t h e crystal field splitting between the _T9 valence b a n d and the upper _T7 valence band shown in F i g . 4 decreases with increasing pressure. T h e results permitted a determination of the difference of the deformation potential constants of the valence and conduction bands involved in t h e transitions. 5.2 Electrooptic

effects

I n general the s y m m e t r y of a crystal is not lowered b y an applied electric field b u t the field dependence of the wave functions can alter the s y m m e t r y and the components of the second-rank tensor e describing the optical properties. T h e t e r m "electrooptic e f f e c t s " for a long time m e a n t only field-induced changes of the long-wavelength refractive index b u t from equation (25) it is obvious t h a t t h e y are due t o changes in the £ 2 -spectrum a t and above the absorption edge. T h e theory of electroabsorption a t the edge given b y F r a n z [83] and K e l d y s h [84] predicts an exponential tail instead of t h e square root law of equation (11), for direct transitions. F o r crystals with an exponential absorption edge, which is in practice often the case due to absorption a t defects or by other interaction mechanisms, a parallel edge shift results which is proportional to the square of the field. Edge shifts of this kind have been measured in good agreement with theory in m a n y substances, e.g. in GaAs [85].

40

G.

HARBEKB

The Franz-Keldysh theory has been extended to energies above E0 b y Callaway [86] a n d Tharmalingam [87]. I n this range the field-induced absorption A K oscillates with a period of about 2 x 10~2 eV for F = 104 V/cm a r o u n d the zero line. Oscillations in A K have been observed in Ge at the indirect edge permitting an accurate determination of the energies k (9q of the phonons involved in the indirect transitions (see equation (19)) [88], as well as a t t h e direct edge [89]. I n the latter case, however, t h e authors claimed t h a t the line shape, its t e m p e r a t u r e dependence, a n d the position of the maxima relative to the band gap energy can be better explained b y field-induced exciton quenching. This effect has been theoretically t r e a t e d b y Duke a n d Alferieff [90], Electroreflection has proved to be a very useful m e t h o d where AR is mostly measured either in a field-effect arrangement with t r a n s p a r e n t electrodes [91] or a t a crystal-electrolyte interface [92]. Fig. 24 demonstrates, at t h e 3.4 eV m a x i m u m in p-Si [93], the ability of this differential technique to resolve a fine structure of several lines in a range where the static reflectance shows only one peak. For a discussion of the line shape in this region it is necessary to consider the influence of the field on direct transitions a t saddle points. The so-called duality theorem of Phillips [94] states t h a t Ae 2 at an Mx saddle point can be deduced from Ae2 a t a parabolic M 0 -point b y changing t h e signs of (E — E0) and Ae2. This s t a t e m e n t only holds for the longitudinal field direction, i.e. if the field is parallel to the direction of the negative reduced mass. The consequences for the line shape have been used in the analysis of the results in Fig. 24

41

On the Band Structure of Anisotropic Crystals

The line I I first is assigned to a metastable exciton because of its strong temperature dependence. I t is further concluded that there are two parabolic critical points at 3.33 and 3.41 eV in p-Si whereas in n-Si showing a different Ae 2 -pattern the second critical point isofMj-type. The reason for this "switching" of a critical point is as yet unknown, the behaviour has not been observed in measurements with the electrolyte method. As mentioned before the duality theorem only holds for the longitudinal case. The cubic symmetry is thus effectively reduced and the strength of the lines I I and I I I is indeed dependent on the crystallographic orientation of the re fleeting n-Si surface. This has been taken as evidence for the existence of an M r point at 3.41 eV [93]. Cardona et al. [96] have also demonstrated the anisotropy by measuring with linearly polarized light at a (100) plane. Recently the theory of electroreflection at saddle points has also been extended to the transverse case where the field is perpendicular to the direction of the negative reduced mass at an M r point or to the direction of the positive reduced mass at an M2-point [97,98]. The line shape analysis, however, is still very difficult because of the exciton quenching effects and the inhomogeneous field distribution in the experiment. I t should be said that the different shapes of Ae2 for all four types of critical points and the anisotropy effects certainly will be of great importance for the determination of position and type of critical points. Some measurements of electroabsorption and electroreflectance have been performed on noncubic crystals. Wurtzite CdS in absorption [99], for example, shows a minimum of A K at the energy of the exciton line A in Fig. 25. Temperature and field dependence of the minimum can be explaine d with the theory of exciton quenching [90]. Electroreflectance data obtained on hexagonal 30 CdSe [96] confirm the assignments from BaTiOj the previously measured reflectance 1=25% spectrum [30]. The electroreflectance spectrum of 25 polycrystalline selenium [100] shows Aii-minima at 1.85 and 2.25 eV, i.e. at the energies of the first reflectivity maxima for EL c and E || c, respectively [53, 55]. There is a rich structure 20 at higher energies but the interpretation has to await measurements with polarized light on single crystals. Fig. 25 shows reflectance and electro15 reflectance of tetragonal ferroelectric BaTiO s [101], The electrooptic effects in ferroelectrics are about two orders of magnitude stronger than predicted by the Franz-Keldysh theory; in this case they are directly correlated with the dielectric properties as was first shown for the electroabsorption of F i g . 25. a) R o o m t e m p e r a t u r e r e f l e c t i v i t y a n d l>) elect o r o r e f l e e t i v i t y of B a T i O s (Gahwiller [101])

W

k5

Photon energy

5.0 E(eV)—

42

G . HARBEKE

orthorhombic SbSI [102], I n contrast to the purely electronic case, here the electroreflectance curve is strongly broadened. The electric field in B a T i 0 3 causes a relative displacement of the oxygen and titanium sublattices resulting in an overall wave vector independent increase of the vertical separation of valence and conduction bands. References [1] G. F. BASSANI, in: Rendieonti della Seuola Internationale ,,E. Fermi",Corso XXXIV, Academic Press 1966 (p. 33). [2] F. SEITZ, The Modern Theory of Solids. McGraw-Hill Publ. Comp., 1940. [ 3 ] G . F . KOSTER,

[4] [5] [6] [7] [8] [9]

J . 0 . DIMMOCK,

R . G . WHEELER, a n d H . STATZ, P r o p e r t i e s of

the

Thirty-two Point Groups, M.I.T. Press, Cambridge 1963. T. P. MCLEAN, Progr. Semicond. 5, 53 (1960). J . S. TOLL, Phys. Rev. 104, 1760 (1956). R. DE L. KRONIG, J . Opt. Soc. Amer. 12, 547 (1926). F. STERN, Solid State Phys. 15, 299 (1963). B. VELICKY, Czech. J. Phys. 11, 787 (1961). H. Y. FAN, in: Methods of Experimental Physics, Ed. K. LARK-HOROVITZ and V. A. JOHNSON, Vol. 6B, Academic Press 1959 (p. 249).

[ 1 0 ] M . B . ROBIN, W . A . KUEBLER, a n d YOH-HAN PAO, R e v . sci. I n s t r u m . 3 7 , 9 2 2 ( 1 9 6 6 ) . [ 1 1 ] S . J . CZYZAK, W . A . BAKER, R . C. CRANE, a n d J . B . HOWE, J . O p t . S o c . A m e r . 4 7 ,

240 (1957). [12] D . G . THOMAS a n d J . J . HOPFIELD, P h y s . R e v . 1 1 6 , 5 7 3 ( 1 9 5 9 ) . [13] L . P . BOUCKAEKT, R . SMOLUCHOWSKI, a n d E . P . WIGNER, P h y s . R e v . 5 0 , 5 8 ( 1 9 3 6 ) .

[14] J . L. BIRMAN, Phys. Rev. 115, 1493 (1959). [15] M. BUJATTI, Phys. Letters (Netherlands) 24, A36 (1967). [16] M . CARDONA, M . WEINSTEIN, a n d G . A . WOLFF, P h y s . R e v . 1 4 0 , A 6 3 3 ( 1 9 6 5 ) . [17] M . CARDONA a n d G . HARBEKE, P h y s . R e v . 1 8 7 , A 1 4 6 7 ( 1 9 6 5 ) .

[18] G. HARBEKE, in: Festkorperproblcme, Ed. F. SAUTER, Vol. ILL, Vieweg, Braunschweig 1964 (p. 13). J . TAUC, Progr. Semicond. 9, 87 (1965). [19] J. C. PHILLIPS, Solid State Phys. 18, 55 (1966). [20] M. CARDONA a n d D. L. GREENAWAY, P h y s . Rev. 131, 98 (1963). [21] M . CARDONA, P h y s . R e v . 1 2 9 , 1 0 6 8 ( 1 9 6 3 ) .

[22] D. BRUST, J . C. PHILLIPS, a n d G. F . BASSANI, P h y s . R e v . L e t t e r s 9, 94 (1962).

D. BRUST, Phys. Rev. 139, A489 (1965). [23] M. L. COHEN and T. K. BERGSTRESSER, Phys. Rev. 141, A789 (1966). [24] T. C. COLLINS, R. N. EUWEMA, and J . S. DE WITT, Proc. Internat. Conf. Semiconductor Phys., Kyoto 1966 (p. 15). [25] E. O. KANE, J . Phys. Chem. Solids 1, 82 (1956). [26] M. CARDONA, J. Phys. Chem. Solids 24, 1543 (1963). [27] J . J . HOPFIELD a n d D . G. THOMAS, P h y s . R e v . 122, 35 (1961).

[28] E. GUTSCHE and H. LANGE, Proc. Internat. Conf. Semiconductor Phys., Paris 1964 (p. 129). [29] E . GUTSCHE a n d E . JAHNE, p h y s . s t a t . sol. 1 9 , 8 2 3 ( 1 9 6 7 ) .

[30] M. CARDONA, Solid State Commun. 1, 109 (1963). [31] T. K . BERGSTRESSER a n d M. L. COHEN, P h y s . L e t t e r s (Netherlands) 23, 8 (1966). [ 3 2 ] U . ROSSLER a n d M . LIETZ, p h y s . s t a t . sol. 1 7 , 5 9 7 ( 1 9 6 6 ) .

[33] H . GOBRECHT a n d J . W . BAARS, p r i v a t e communication.

[34] [35] [36] [37]

P. J . LIN and L. KLEINMAN, Phys. Rev. 142, 478 (1966). M. CARDONA and D. L. GREENAWAY, Phys. Rev. 133, A1685 (1964). P. J . LIN and J . C. PHILLIPS, Phys. Rev. 147, 469 (1966). M. S. DRESSELHAUS and J . G. MAVROIDES, Phys. Rev. Letters 14, 259 (1965).

[ 3 8 ] E . O . KANE, P h y s . R e v . 1 4 6 , 5 5 8 ( 1 9 6 6 ) .

On the B a n d Structure of Anisotropic Crystals

43

[ 3 9 ] W . J . CHOYKE. D . R . HAMILTON, a n d L . PATRICK, P h y s . R e v . 1 8 3 , A 1 1 6 3 ( 1 9 6 4 ) . G . ZANMARCHI, P h i l i p s R e s . R e p t . 2 0 , 2 5 3 ( 1 9 6 5 ) .

[40] B. E. WHEELER, Solid State Commun. 4, 173 (1966). [41] R. R. HAERING and S. MROZOWSKI, Progr. Semicond. 5, 273 (1960). [42] G . F . BASSANI a n d G . PASTORI, NUOVO C i m e n t o SOB, 9 5 ( 1 9 6 7 ) . [ 4 3 ] E . A . TAFT a n d H . R . PHILIPP, P h y s . R e v . 1 3 8 , A 1 9 7 ( 1 9 6 5 ) .

[44] F. BASSANI, D. L. GREENAWAY, and G. FISCHER, Phys., Paris 1964 (p. 51). [45] J . L. BREBNER and G. FISCHER, Proc. I n t e r n a t . 1962 (p. 760). [46] H . KAMIMURA and K . NAKAO, Proc. I n t e r n a t . 1966 (p. 27). [47] M. A. NIZAMETDINOVA, phys. stat. sol. 19, K i l l

Proc. I n t e r n a t . Conf. Semiconductor Conf. Semiconductor Phys., Exeter Conf. Semiconductor Phys., K y o t o (1967).

[48] N . A . GASANOVA, G . A . AKHUNDOV, a n d M . A . NIZAMETDINOVA, p h y s . s t a t . sol. 1 7 ,

K115 (1966). [ 4 9 ] J . A . DRABBLE, P r o g r . S e m i c o n d . 7, 4 5 (1963). [ 5 0 ] D . L . GREENAWAY a n d G . HARBEKE, J . P h y s . C h e m . S o l i d s 2 6 , 1 5 8 5 ( 1 9 6 5 ) .

[51] T. S. Moss, Optical Properties of Semiconductors, Butterworths, London 1961. [52] D . J . OLECHNA a n d R . S . KNOX, P h y s . R e v . 1 4 0 , A 9 8 6 ( 1 9 6 5 ) . [ 5 3 ] J . STUKE a n d H . KELLER, p h y s . s t a t . sol. 7, 189 ( 1 9 6 4 ) .

[54] R . SANDROCK and J . TREUSCH, Solid State Commun. 3, 361 (1965); phys. stat. sol. 16, 4 8 7 (1966).

[55] S. TUTIHASI and I. CHEN, Solid State Commun. 6, 255 (1967); Phys. Rev. 158, 623 (1967). [56] J . TAUC a n d A . ABRAHAM, C z e c h . J . P h y s . 1 5 , 7 3 0 ( 1 9 6 5 ) .

[57] G. BALDINI, Phys. Rev. 128, 1562 (1962). [58] J . E . E B Y , K . J . TEEGARDEN, a n d D . B . DUTTON, P h y s . R e v . 1 1 6 , 1 0 9 9 ( 1 9 5 9 ) .

[59] R . S. KNOX, Solid State Phys. (Suppl.) 5, (1963). [60] [61] [62] [63]

M . CARDONA a n d G . HARBEKE, P h y s . R e v . L e t t e r s 8 , 9 0 ( 1 9 6 2 ) . G . HARBEKE, Z . N a t u r f . 1 9 a , 5 4 8 ( 1 9 6 4 ) . J . C. PHILLIPS, P h y s . R e v . L e t t e r s 1 2 , 142 ( 1 9 6 4 ) . C. B . D U K E a n d B . SEGALL, P h y s . R e v . L e t t e r s 1 7 , 19 ( 1 9 6 6 ) .

[64] J . HERMANSON, Phys. Rev. Letters 18, 170 (1967). [65] B . VELICKY and J . SAK, phys. stat. sol. 16, 147 (1966). [66] K . P. JAIN, Phys. Rev. 139, A544 (1965). [67] U . FANO, P h y s . R e v . 1 2 4 , 1 8 6 6 ( 1 9 6 1 ) . [68] Y . TOYOZAWA, M . INOUE, T . INUI, M . OKAZAKI, a n d E . HANAMURA, P r o c . I n t e r n a t .

Conf. Semiconductor Phys., K y o t o 1966 (p. 133); J . Phys. Soc. J a p a n 21, 1850 (1966). [69] D. C. REYNOLDS, C. W. LITTON, and T. C. COLLINS, phys. stat. sol. 9, 645 (1965); 12, 3 (1965). [70] J . L. BREBNER and E. MOOSER, Phys. Letters (Netherlands) 24, A274 (1967). [71] D. L. GREENAWAY and G. HARBEKE, Proc. I n t e r n a t . Conf. Semiconductor Phys., K y o t o 1966 (p. 151). [72] For references see M. R. TUBBS and A. J . FORTY, J . Phys. Chem. Solids 26, 711 (1965). [73] H . I. RALPH, Solid State Commun. 3, 303 (1965). [ 7 4 ] D . L . GREENAWAY a n d R . NITSCHE, J . P h y s . C h e m . S o l i d s 2 6 , 1 4 4 5 ( 1 9 6 5 ) . [75] H . R . PHILIPP, W . DASH, a n d H . EHRENREICH, P h y s . R e v . 1 2 7 , 7 6 2 ( 1 9 6 2 ) . [76] W . E . ENGELER, H . FRITZSCHE, M . GARFUNKEL, a n d J . J . TIEMANN, P h y s .

Letters 14, 1069 (1965). [77] G . W . GOBELI a n d E . O. KANE, P h y s . R e v . L e t t e r s 1 5 , 142 ( 1 9 6 5 ) .

[78] [79] [80] [81] [82]

I. BALSLEV, Phys. Letters (Netherlands) 24, A113 (1967). I. BALSLEV, Phys. Rev. 143, 636 (1966). U. GERHARDT, phys. stat. sol. 11, 801 (1965). U. GERHARDT, Phys. Letters (Netherlands) 9, 117 (1964). M. GRYNBERG, Proc. I n t e r n a t . Conf. Semiconductor Phys., Paris 1964 (p. 135).

Rev.

44

G.

HARBEKE

: On the Band Structure of Anisotropic Crystals

[83] W . FRANZ, Z . N a t u r f . 1 3 a , 4 8 4 (1958).

Zh. eksper. teor. Fiz. 34, 1 1 3 8 ( 1 9 5 8 ) ; Soviet Phys. theor. Phys. 7, 7 8 8 ( 1 9 5 8 ) . [85] T. S . Moss, J . appl. Phys. 32, 2 1 3 6 ( 1 9 6 2 ) . [86] J . CALLAWAY, Phys. Rev. 130, 549 (1963); 134, A998 (1964). [ 8 7 ] K . THARMALINGAM, Phys. Rev. 1 3 0 , 2 2 0 4 ( 1 9 6 3 ) . [88] A. FROVA and P. H A N D L E R , Phys. Rev. 137, A1875 (1965).

[84] W . L. KELDYSH,

J.

exper.

[98] Y . HAMAKAWA, F . GERMANO, a n d P . HANDLER, P r o c . I n t e r n a t . Conf. S e m i c o n d u c t o r

Phys., K y o t o 1966 (p. 111). [90] C. B . DUKE a n d M . B . ALFERIEFF, P h y s . R e v . 1 4 5 , 5 8 3 (1966).

[91] B. 0 .

SERAPHIN

and R. B.

HESS,

Phys. Rev. Letters 14, 138 (1965).

[92] K . L . SHAKLEE, M. CARDONA, a n d F . H . POLLAK, P h y s . R e v . L e t t e r s 15, 8 8 3 (1965).

Phys. Rev. 140, A 1 7 1 6 ( 1 9 6 5 ) . [94] J . C. PHILLIPS, Phys. Rev. 146, 584 (1966). [95] J . C. P H I L L I P S and B . O . S E R A P H I N , Phys. Rev. Letters 1 5 , 107 (1965). [ 9 6 ] M . CARDONA, F . H . P O L L A K , and K. L . S H A K L E E , Proc. I n t e r n a t . Conf. Semiconductor Phys., K y o t o 1966 (p. 89). [97] D. E. ASPNES, Phys. Rev. 147, 554 (1966). [98] F. A Y M E R I C H and G. F. B A S S A N I , Nuovo Cimento 48B, 358 (1967). [99] B. B. SNAVELY, Solid State Commun. 4, 561 (1966). [100] J . H . CHEN, Phys. Letters (Netherlands) 9, 516 (1966). [101] CH. GAHWILLER, Solid State Commun. 5, 65 (1967). [102] G. HARBEKE, J . Phys. Chem. Solids 24, 957 (1963). [93] B . O . SERAPHIN,

(Received

October

11,

1967)

Original

Papers

phys. stat. sol. 27, 45 (1968) Subject classification: 14.3; 13.4; 22 A. F. lojfe Physico-Technical Institute, Academy of Sciences of the USSR, and Faculty of Physics, Kishinev State University (b)

Leningrad

(a),

Negative Magnetoresistance in p-CdSb By I . K . A n d e o n i k (b), E . K . A e t j s h a n o v (b), 0 . V . E m e l y a n e n k o (a), a n d D . N . N a s l e d o v (a) Negative magnetoresistance is observed a t low temperatures in cadmium antimonide crystals with hole concentration S; 5 x 1016 c m - 3 . The results of t h e investigation of t h e field, angular and temperature dependences in the temperature range 2.2 to 77 °K of t h e magneto-resistance of copper, silver, germanium, and tin doped samples with carrier concentrations ranging from 1016 to 1018 c m - 3 are described. I t is shown t h a t the negative magnetoresistance is related probably to impurity band conduction. I t is noted t h a t t h e regularities observed, namely the quadratic dependence of the negative magnetoresistance on magnetic field strength a t weak fields and saturation a t strong fields, the linear change of the negative magnetoresistance with temperature and the isotropy of the effect have t h e same character as in A I V and A n l B v crystals and conform qualitatively Toyozawa's theory. I n Kadmiumantimonidkristallen wird bei niedrigen Temperaturen und Löcherkonzentrationen über 5 x 1016 cm" 3 eine negative Magnetowiderstandsänderung beobachtet. E s wurden die Ergebnisse der Untersuchungen der Feld-, Winkel- und Temperaturabhängigkeit der Magnetowiderstandsänderung an Kupfer-, Silber-, Germanium- und Zinn-dotierten Proben bei Ladungsträgerkonzentrationen von 1016 bis 1018 c m - 3 im Temperaturbereich 2,2 bis 77 °K beschrieben. E s wird gezeigt, daß die negative Magnetowiderstandsänderung wahrscheinlich mit einer Störbandleitung verbunden ist. E s wird bemerkt, daß die beobachteten Regelmäßigkeiten: die quadratische Abhängigkeit der negativen Magnetowiderstandsänderung von der magnetischen Feldstärke bei schwachen Feldern und Sättigung bei starken Feldern, die lineare Änderung der negativen Magnetowiderstandsänderung mit der Temperatur u n d die Isotropie des Effekts den gleichen Charakter wie in den A I V - und A I I T B V -Kristallen besitzen und qualitativ mit der Toyozawaschen Theorie übereinstimmen.

1. Introduction The negative magnetoresistance, that means the decrease of the crystal resistance in a magnetic field, has been observed in a great number of semiconductors, namely, germanium, silicon, ATIIBV compounds. We have detected and investigated the negative magnetoresistance in CdSb doped with copper, silver, germanium, and tin. Single crystals were prepared by the method of zone recrystallization in argon atmosphere. The impurities have been introduced into CdSb purified to a charge carrier concentration of 1015 cm - 3 at 77 °K. From the homogeneous part of the ingot, specimens were cut along the crystallographic c-axis [001] in the form of parallelepipeds having the dimension 10 X 2 X x 2 mm 3 . The parameters of the specimens investigated are given in Table 1. It is known that CdSb crystals are anisotropic [1], As we need only the true sequence and not the rigid values of the hole concentration in our samples, the latter was defined according to the common formula p = 1/e R.

46

1 . K . ANDRONIK. E . K . ARUSHANOV, 0 . V . E M E L Y A N E N K O , a n d D . N . NASLEDOV

Table 1

T = 77 °K Specimen No.

IP 2p 3p 4p 5p 6p 7p 8p 9p

Impurity v

Cu Cu Sn Ge Ag Cu Cu Cu Cu

Carrier concentration

Carrier mobility

= J _ (cm" 3 )

(cm 2 Vs)

3.3 X IO16 4.5 x IO16 5.4 X IO16 1.3 x IO17 2.0 X IO17 2.2x10" 2.9 X IO17 1.2 x IO18 3.7 x IO18

2880 2800 2410 1910 1880 1840 1800 1270 1100

eR

2. Results The negative magnetoresistance has been detected at low temperatures in all specimens with concentration greater than 5 x l 0 1 6 c m ~ 3 . First of all let us consider the dependence of the magnetoresistance on magnetic field (Fig. 1). For small fields the negative magnetoresistance is proportional to the square of the magnetic field strength H. As H increases the magnetoresistance saturates in heavily doped specimens (p 10 18 c m - 3 ) , tends to saturation at less 1 7 degree of doping (p = (1 to 3 ) x l 0 c m 3 ) , and changes in sign to positive one in purer specimens (p = 5.4 X 10 l a c m - 3 ) . In the purest specimens (p < 5.4 X X 10 16 c m - 3 ) the effect is positive for all H. Similar dependences on the magnetic field have been observed in A I l I B v compounds [2], Thus it is possible to consider the magnetoresistance as consisting of two effects: firstly, of the negative magnetoresistance showing a square law at weak H and reaching satur ation at strong H, and, secondly, of the positive magnetoresistance depending on H quadratically at all fields and prevailing over the negative one for sufficiently strong H (in pure specimens in our example). At a rotation of the specimen around the axis coinciding with the current direction the negative magnetoresistance is practically isotropical (the change is not over 2 0 % ) , whereas the positive magnetoresistance (large H) changes by several times (Fig. 2). The positive magnetoresistance depends on the angle in the same way as at higher temperatures in undoped p-CdSb [3]. The isotropy of the negative magnetoresistance has been observed before when studying crystals of A l n B v compounds [2], The isotropy of the effect in anisotropic p-CdSb crystals indicates its independence of the structure of the crystal. The isotropy of the negative magnetoresistance and anisotropy of the positive one give us the possibility to eliminate the negative magnetoresistance in pure form. The comparison of dependences on magnetic field taken at two orientations of the specimen corresponding to the maximum and minimum of the positive magnetoresistance reveals the fact that the negative magnetoresistances in both cases coincide at H = 2 kOe (Fig. 1, curves 3 p and 3 p ' ) , and thus they do not contain any positive component.

Negative Magnetoresistance in p-CdSb

47

The concentration dependence of the negative magnetoresistance at H = 2 kOe and T = 4.2 °K is given in Fig. 3. The negative magnetoresistance appears for p > 5 X 1016 c m - 3 . The absolute value of the magnetoresistance increases with increasing concentration, thus reaching the maximum at p = (2 to 3) X X 1017 cm - 3 , and then decreases. The typical temperature dependence of the magnetoresistance shown in Fig. 4 as well as the concentration dependence are given for a weak field of H = 2 kOe (it should be noted that the negative magnetoresistance for heavily doped specimen was taken at H = 10 kOe). The absolute value of the magnetoresistance decreases with increasing temperature and becomes positive at high temperatures. The dependence of the negative magnetoresistance on temperature is almost linear. The temperature range for the occurrence of the negative

H2(k0e2)

-

Fig. 2. Angular dependence of the magnetoresistance for the specimen 3p. The positive magnetoresistance was obtained for H = = 10 kOe, the negative one for H = Z kOe. T = 4.2 °K

Fig. 1. Dependence of the magnetoresistance on magnetic field strength at 4.2 ° K . Here and hereafter the numbers of the plots correspond to the specimen numbers

p(cm~3)- Fig. 3. Dependence of the magnetoresistance on the hole concentration in different specimens

Fig. 4. Dependence of magnetoresistance and Hall constant on temperature ^ ^

48

I . K . ANDRONIK,

E . K . ARUSHANOV,

O . V . E M E L Y A N E N K O , a n d D . N . NASLEDOV

magnetoresistance increases with increasing carrier concentration. This causes that the temperature of change of the negative magnetoresistance into the positive one follows in all specimens the temperature maximum of the Hall effect, but remains always below it. 3. Discussion

First of all it should be noted that such accessory factors as illumination, specimen heterogeneity, different kinds of contact didn't affect the value of the negative magnetoresistance in our experiment. The kind of impurity is not of great importance in the effect investigated. This can be proved by the fact that the identical character of the phenomena studied is observed at similar concentrations of different impurities. As it was stated before the negative magnetoresistance is observed at temperatures below those corresponding to the maximum of the Hall effect. That is in the temperature range where the impurity band conduction plays the main role. The negative magnetoresistance is connected in all probability with this mechanism. This assumption is confirmed by the concentration dependence of the magnetoresistance: the negative magnetoresistance has a maximum at a concentration p = (2 to 3) X 10 17 c m - 3 where the impurity band conduction is most important (presence of the maximum in the temperature dependence of the Hall effect at equal R from both sides [4]) and drops in the direction of high and low concentrations. The magnetoresistance is positive in pure specimens when hop conduction is considerable. The negative magnetoresistance does not vanish at high concentrations, despite the fact that the impurity band overlaps the main one. This is because the peculiarities of the conduction characteristic for the impurity band [5] remain the same in these specimens. Of all available theories on the negative magnetoresistance, Toyozawa's theory is the only one that does not contradict our data. The theory considers scattering on "localized spins" of partially isolated impurity atoms for impurity band conduction. The magnetic field decreases scattering by regularizing localized spins, and leads to the rise of the negative magnetoresistance. The regularities observed are as follows: the square dependence of the negative magnetoresistance on H at weak fields and saturation at strong ones, linear change of the negative magnetoresistance with temperature, independence of the effect of rotation of the specimen. As far as the qualitative form is concerned they conform to the conclusions of Toyozawa's theory [6], The dependences which we obtained when studying the negative magnetoresistance in doped p-CdSb are similar to the results of measurements of the negative magnetoresistance in crystals of the type A I V , A m B v . The fact that the negative magnetoresistance is observed in crystals of compound structure like cadmium antimonide leading to the same regularities as in diamond-like semiconductors indicates the considerable community of this effect for semiconductors. In all probability the negative magneto-resistance in p-CdSb as in other crystals is connected with impurity band conduction. 4. Conclusions

1. The negative magnetoresistance was observed at low temperatures (below 3 to 30 °K) in p-CdSb crystals doped with elements of the first and fourth group, that is copper, silver, germanium, and tin.

Negative Magnetoresistance in p-CdSb

49

2. According to the experiment the negative magnetoresistance in doped p-CdSb is in all probability connected with impurity band conduction. 3. All experimental results, namely, isotropy of the negative magnetoresistance, its linear dependence on temperature, square dependence at weak H, and saturation at strong, have the same character as for crystals of the type A I V , A m B v and conform to Toyozawa's theory as far as the qualitative form is concerned. Keferences [ 1 ] I . K . ANDRONIK a n d M. V . KOT, F i z . t v e r d . T e l a 2, 1 1 2 8 ( 1 9 6 0 ) . [ 2 ] O. V . EMELYANENKO, T . S. LAGUNOVA, a n d D . N . NASLEDOV, P h y s i c a l P r o p e r t i e s o f

Semiconductors A m B v and A m B V I , Proc. III. All-Union Conf., Baku 1965 (to be published).

[ 3 ] M . MATYAS a n d A . MULLER, Czech. J . P h y s . B 1 6 , 1 0 6 ( 1 9 6 6 ) . [ 4 ] O. V . EMELYANENKO, T . S. LAGUNOVA, D . N . NASLEDOV, a n d G. N . TALALAKIN, F i z .

tverd. Tela 7, 1315 (1965).

[ 5 ] O. V . EMELYANENKO, D . N . NASLEDOV, a n d Z . SH. OVSUK, F i z . i T e k h . P o l u p r o v . 1 , 1094 (1967). [ 6 ] J . TOYOZAWA, J . P h y s . S o c . J a p a n 1 7 , 9 8 6 ( 1 9 6 2 ) .

(Received November 21,

4

physica 27/1

1967)

51

A. S. DAVYDOV: Theory of Urbach's Rule phys. stat. sol. 27, 51 (1968) Subject classification: 20.1; 22 Institute of Theoretical

Physics, Academy of Sciences of the Ukrainian

S8E,

Kiev

Theory of Urbach's Rule By A . S. DAVYDOV

A general theory of the long-wavelength edge of the fundamental optical absorption in crystals is developed which explains the empirical Urbach rule. Pa3BHTa 3JieMeHTapHan TeopiiH aJiHiiiioBOJinoBoro Kpan nepBoii nojiocw norjiomeHHH cBexa B upHCTajuiax, oôiHCHHiomaH BMnnpHiecKoe npaBHjio Y p 6 a x a .

1. Introduction When investigating the coefficient of light absorption in silver halides, Urbach [1] discovered (in 1953) the empirical formula g (eu, —ai) X(W)

=

X(OJ0)

e

k T

(1.1)

determining the dependence of the absorption coefficient on photon energy 1 ) ft) and crystal temperature T in the range to ft>0- When the values of the parameter a are close to unity, (1.1) gives a good description of the long-wavelength tail of the absorption band over a large frequency and temperature range. Later it became clear that (1.1) gave a remarkable accurate expression for the change (4 to 6 orders) of the absorption coefficient with frequency in the longwavelength tails of the first bands of the intrinsic absorption (excitons and localized excitations) and the impurity absorption for crystals of many types. In order to explain the temperature dependence of the absorption coefficient, one has sometimes to introduce a weak temperature dependence of the parameter a. For instance, Matsui [2] showed that formula (1.1) gives a good description of the change of the absorption coefficient of anthracene in the interval 105 to 10- 1 cm- 1 if at the temperatures 79, 200, 253, 293, and 346 °K a is chosen equal to 0.630, 1,238, 1.400, 1.472, and 1.533, respectively, for the first absorption band of the a-component of the spectrum, and 0.727, 1.361, 1.500, 1.539, and 1.637, respectively, for the first band of the b-component. Mahr [3] also notes a weak dependence of the parameter a on temperature. References to earlier experimental works can be found in Knox's book [4], Many attempts have been made to provide a theoretical basis for the formula (1.1) (the so-called Urbach's rule) (e.g., see [5 to 9]). However, there is no satisfactory theory of this rule up to date. Knox [4] remarks that "Urbach's rule remains a challenge to theorists as of the present writing. Further complicating the picture is the appearance of Urbach-rule absorption in impurity centres." The units are chosen so that h = 1. 4*

52

A . S . DAVYDOV

The present paper suggests a simple theory of Urbach's rule. The principal idea is that the long-wavelength edge of the absorption band (and so Urbach's rule) is due to quantum transitions from vibrational sublevels of the crystal lattice to the level of the first electron excitation. In our opinion, the reason why the preceding theories have not been successful in explaining Urbach's rule is that they tried to connect this rule with the change of the first band of the crystal electron excitation. For instance, in order to justify Urbach's rule attempts were made to find exciton states whose levels are located much lower than the energy of the exciton generated in the direct transition. Agranovich and Konobeev [9] tried to explain the long-wavelength edge of the absorption band by introducing polaritons (mixed exciton-phonon states). They found a rather weak absorption in the region below the edge of the exciton absorption band. However, the frequency dependence found by them essentially differs from that predicted by Urbach's rule. 2. Theory of the Long-Wavelength Edge of the Absorption Band We suppose that the crystal contains a small number of similar impurity centres, having no internal vibrational sublevels. Let y)0(r), y>f{r), E 0 , and E j be the wave functions and the energies of the ground and excited electron states of an impurity molecule, respectively. If one ignores the variation of the frequencies of the normal vibrations of the crystal lattice caused by the transition of the impurity molecule to an excited state, then in the adiabatic approximation the crystal states with the energies + £ ( » .

+ y )

a n d

fy

+

2

(
A1 [53]. Gerhardt's conclusions [23] from the pressure dependence of E2 were consistent with the latter possibility. The electroreflectance experiments [32], regardless to the difficulties in interpretation, showed t h a t the structure in this region is influenced by more t h a n one transition. Recently, Herman et al. [12, 13] deduced from careful band structure calculations t h a t this peak cannot correspond to / ^ s -T15, because their value for this transition is 2.8 + 0.1 eV and also the theoretical pressure coefficient does not agree with experiment. According to this, the E2-maximum is associated with transitions near the (111) axis in an extended region of Brillouin zone. Contrary to the earlier paper of Brust and Bassani [51] (exchange transition A3 —> A1 with -» ri5), Herman et al. [12, 13] interpret the break near 80% (Fig. 8) as a change of conduction band profile in [111] direction induced by cross-over of r 2 and Our experimental results, t h a t is the fast change of the shape of maxima and A-R/AT versus composition, support the idea dealing with the different influences of various transitions or the role of variable extended region of reduced zone. I t is also necessary to consider the extrapolation of Ea. The shoulder E3 between 2.9 and 3.7 eV for Ge can arise according to Brust [8] from the transitions As A1 (Mx and M2) and A s -T15 (M0). This conclusion is confirmed by more detailed calculations carried out by Higginbotham et al. [15], who quote -»- /" 15 (M0) and the critical point M0 in [100] direction. The weak structures in e 1 and e2 have been found by Potter [54] between 2.6 and 3.1 eV, and from 3.3 to 3.64 eV. The first he ascribed to the i ^ s As transition and the second to the critical point Mj. A rather complicated shape in this region show also Ghosh's [55] electroreflectance spectra for Ge. From our experimental results it is seen t h a t the shoulder becomes narrower with composition and t h a t the extrapolated centre has a tendency to go to the E2region for Si-rich samples. For more precise conclusions it would be useful a detailed study of the reflectivity of GeSi alloys for composition from 60 to 100% and the energy range between 2.5 and 3.5 eV. The most prominent maximum Ei was from the beginning related to two quasi-degenerate singularities Mj (X4 -> X1) and M2 ( £ 2 £3) [8J- Kane [14] has calculated e2 for Si as function of k and he has shown t h a t the contributions of these transitions, especially of X4 Xv are comparatively small, and t h a t the maximum is caused by accumulation of transitions over an appreciable p a r t of the Brillouin zone. Recently, Higginbotham et al. [15] have confirmed Kane's conclusions also for Ge in a similar way. The main contribution is given by transitions near the critical point a t (7/8, 7/8, 3/8) in the /¿-space. The nearly constant temperature change and also the constant dependence of the position of Ei on composition confirms the similar character of the Brillouin zone for all GeSi alloys in the corresponding region. On the other hand, there are some differences, e.g. in the shape (compare E'± and Ex in R and Ei in e2) and in the absolute change of R and e2. The maximum Ei with the shoulder E'4 in R for Si-rich alloys reminds of the M J M 2 singularity but, unfortunately, in e2 we did not find a similar structure. Both extrema Et and E± for Si have been resolved in the electroreflectance experiment [56]. The maximum E5 for Ge according to Brust's interpretation [8] occurs due to the critical points M3 (L'z L3) and

Optical Properties of GeSi Alloys

65

Mj (0.56, 0.56, 0.39) in the fc-space. Usually, the main argument for the L's -> L3 transition has been the splitting of this maximum by spin-orbit interaction [44], what has not been found in our data. Higginbotham et al. [15] take into consideration for this extremum also the transition As -> A2. The maximum Ee for Ge has been found by Pajasova [40], and as a possible interpretation she has quoted the transition As A'% according to [52], or Z^s ri according to Herman. For Si the situation is similar. Brust [8] has suggested the transition L5 La, Cardona and Pollak [52] As —> A'2 and _2J3 Our experimental results confirm the existence of both maxima in the whole composition range, and so it is necessary to interpret them for all GeSi alloys in the same way. The results giving the change of the energy band structure in some important points or directions are summarized in Fig. 8. As usual we put for E = 0. The spin-orbit splitting of the valence band at J25 up to 74% is according to Braunstein's [18] absorption measurements on p-GeSi, and for Si according to [57] and [58]. The indirect transition 5 Lx and so the position of Ll up to 15% was obtained from [17]. The line for L r was extrapolated to Si, and from our experimental data as E1 we have fixed L'3. From the transition 5 -> [17] one can determine il> up to 13%. According to the electroreflectance measurements of Cardona et al. [33] it is possible to extrapolate to 4.05 eV for Si. For the position of X 4 we have used Herman's et al. [13] results for determining .Xj from Et. The value for X j was reduced by a small correction of 0.2 eV [14]. The position of L3 corresponds to the experimental results for Ey In Fig. 8B are shown transitions with changeable k opposite to Fig. 8A where fc is fixed. The construction of this diagram is similar to Fig. 8B. The position of A l m i n has been determined from absorption measurements [17], A3 for Ge from the energy band structure calculated by Brust [8] and extrapolated to Si similar as ¿3 (here can be involved serious uncertainties), a n d ^ 2 also from theoretical calculations [52], The lines Alt and the spin-orbit splitting of A 3 correspond to the experimental values E2, E4, and There can arise some doubts about such a construction, mainly because we use theoretical results of different authors, but in this time we do not see a better way if we want of interpret optical spectra in the manner mentioned above. In the diagrams the dashed lines express the inter- and extrapolated results similarly as transitions with a less certainty. The energy band structure for GeSi alloys has been computed only by Bassani and Brust [51], but their results for Ge and Si are not consistent with the last ones of Herman et al. [12, 13], and the discussion would be only a repeat of the previous one. For pure Ge and Si it is more reasonable to discuss instead of [51] the energy band structure of Brust [8] as it has been calculated with the exception of the transition Aa A1 having been determined theoretically in the whole composition range in his and Bassani's work [51]. Their conclusions on A3 -*• A1 are in qualitative agreement with experiment. For further and more complete interpretation it is necessary to compute in detail e2(a>, k) for Ge, Si, and GeSi alloy mainly near 80%. The experimental results will be the criterion for theoretical calculations of the energy band structure of these materials. Acknowledgements

I am indebted to Prof. Dr. J . Tauc, Dr. F. Lukes, Dr. B. Velicky, and J . Sak for many valuable discussions, to Prof. Dr. J . Tauc and Dr. Z. Trousil for lending the Ge Sisamples, to L. Pajasova for lending the far uv experimental arrangements, to 5

physica 27/1

66

E . SCHMIDT

J. Dvor&k for the computation of optical constants, and to J. Kubena and V. Orel for the determination of the lattice parameters. I also thank to Prof. Dr. F. Herman for sending reprints and preprints of his papers. References [1] H. R . P H I L I P P a n d E. A. T A F T , Phys. Rev. 113, 1002 (1959). R . P H I L I P P and E . A . T A F T , Phya. Rev. 1 2 0 , 3 7 ( 1 9 6 0 ) . [ 3 ] J . C. P H I L L I P S , J . Phys. Chem. Solids 1 2 , 208 (1960).

[2] H .

[4] J . TAUC a n d E . ANTONCIK, P h y s . R e v . L e t t e r s 5, 2 5 3 (1960).

[5J J . TAUC and A. ABRAHAM, Proc. I n t e r n a t . Conf. Semiconductor Physics, Prague 1960 (p. 375). [ 6 ] J . T A U C and A . A B R A H A M , J . Phys. Chem. Solids 2 0 , 1 9 0 ( 1 9 6 1 ) . [7] D . BRUST, J . C. PHILLIPS, a n d P . BASSANI, P h y s . R e v . L e t t e r s 9, 9 4 (1962).

[8] [9] [10] [11]

D. BRUST, Phys. Rev. 134, A 1337 (1964). M. CARDONA, Proc. I n t e r n a t . Conf. Semiconductor Physics, Paris 1964 (p. 181). J . C. PHILLIPS, Solid State Phys. 18, 55 (1966). J . TAUC, Progress in Semiconductors, E d . A. F . GIBSON, Vol. 9, Hey wood, London 1965 (p. 87). [ 1 2 ] P . H E R M A N , R . L . K O R T U M , C. D . K U G L I N , and R . A. SHORT, Quantum Theory of Atoms, Molecules, Solid State, Academic Press, New York 1966 (p. 381). [ 1 3 ] F . H E R M A N , R . L . K O R T U M , and C. D . K U G L I N , I n t e r n a t . J . Quantum Chem., Slater Symposium Issue, E d . Per-Olov LOWDIN, Interscience Division of J o h n Wiley and Sons, New York 1967. [14] E. O. KANE, Phys. Rev. 146, 558 (1966). [ 1 5 ] C. W . H I G G I N B O T H A M , F . H . P O L L A K , and M . CARDONA, Solid State Commun. 5 , 5 1 3 (1967).

[16]

E.

R.

JOHNSON

and S.

M . CHRISTIAN,

Phys. Rev.

95,

560 (1954).

[17] R . BRAUNSTEIN, A . R . MOORE, a n d F . HERMAN, P h y s . R e v . 1 0 9 , 6 9 5 ( 1 9 5 8 ) . [18]

R.

BRAUNSTEIN,

and

Phys. Rev.

130, 869 (1963).

Proc. I n t e r n a t . Conf. Semiconductor Physics, Prague (p. 3 7 1 ) . M. CARDONA, J . appl. Phys. Suppl. 32, 2151 (1961). F . L U K E S and E. S C H M I D T , Proc. I n t e r n a t . Conf. Semiconductor Physics, Exeter 1 9 6 2 (p. 389). A . A B R A H A M , J . T A U C , and B . V E L I C K Y , phys. stat. sol. 3 , 7 6 7 ( 1 9 6 3 ) . U. GERHARDT, phys. stat. sol. 11, 801 (1965). R. Z A L L E N and W . P A U L , Phys. Rev. 1 5 5 , 7 0 3 ( 1 9 6 7 ) . M. CARDONA and H . S . SOMMERS, Phys. Rev. 122, 1382 (1961). F . L U K E S and E . S C H M I D T , Proc. I n t e r n a t . Conf. Semiconductor Physics, Paris 1964 (p. 197). G . B . D U B R O V S K I I and V . K . S U B A S H I E V , Fiz. tverd. Tela 5 , 1 1 0 4 ( 1 9 6 3 ) . B. BATZ, Solid State Commun. 4, 241 (1966). C. N. BERGLUND, J . appl. Phys. 37, 3019 (1966).

[19] F . LUKES

E . SCHMIDT,

1960

[20] [21] [22]

[23] [24]

[25] [26] [27]

[28] [29]

[30] W . E . ENGELER, [31] [32]

H . FRITZSCHE,

M . GARFINKEL,

and

J . J . TIEMANN,

Phys.

Rev.

Letters 14, 1069 (1965). G . O. G O B E L I and E. 0 . K A N E , Phys. Rev. Letters 15, 1 4 2 ( 1 9 6 5 ) . B . 0 . S E R A P H I N and N . B O T T K A , Phys. Rev. 1 4 5 , 6 2 8 ( 1 9 6 6 ) .

[ 3 3 ] M . CARDONA, K . L . SHAKLEE, a n d F . H . POLLAK, P h y s . R e v . 1 5 4 , 6 9 6 ( 1 9 6 7 ) . [34]

S. H .

GROVES,

C.

[ 3 5 ] J . G . MAVROIDES,

R . PIDGEON,

and

J . FEINLEIB,

M . S. DRESSELHAUS,

Phys. Rev. Letters 17,

R . L . AGGARWAL,

and

643 (1966).

G . F . DRESSELHAUS,

Proc. I n t e r n a t . Conf. Semiconductors Physics, K y o t o 1966 (p. 184). [36] D. C. LANGRETH, Phys. Rev. 148, 712 (1966). [37] B . V E L I C K Y and J . S A K , phys. stat. sol. 1 6 , 147 (1966).

Optical Properties of GeSi Alloys [ 3 8 ] Y . T o Y O Z A w a , M . INOUE,

67

T . I N U I , M . OKAZAKI, a n d B . HANAMURA, P r o c . I n t e r n a t .

Conf. Semiconductor Physics, K y o t o 1966 (p. 133). [ 3 9 ] E . SCHMIDT, a n d F . L U K E S , Ces. Cas. F y s . A 1 7 , 2 6 8 ( 1 9 6 7 ) .

[40] L . PAJASova, Solid S t a t e C o m m u n . 4, 619 (1966). [ 4 1 ] J . P . DISMUKES, L . EKSTROM, a n d R . J . P A F F , J . p h y s . C h e m . 6 8 , 3 0 2 1 ( 1 9 6 4 ) .

[42] M. CERNOHORSKY, P r a c e B r n e n s k e z a k l a d n y CSAV 81, 77 (1959). [ 4 3 ] H . R . P H I L I P P a n d H . EHBENBEICH, P h y s . R e v . 1 2 9 , 1 5 5 0 ( 1 9 6 3 ) .

[44] H . EHRENREICH, H . R . PHILIPP, a n d J . C. PHILLIPS, P h y s . R e v . L e t t e r s 8, 59 (1963). [45] S. ROBIN-KANDABE a n d J . ROBIN, Proc. I n t e r n a t . Conf. Semiconductor Physics, P r a g u e 1960 (p. 379). [46] E . SCHMIDT, P h y s . L e t t e r s (Netherlands) 21, 640 (1966). [47] F . LUKES, Czech. J . P h y s . B10, 742 (1960). [48] F . LUKES, Czech. J . P h y s . B10, 317 (1960). [49] D . M. ROESSLER, B r i t . J . appl. P h y s . 16, 1119 (1965); 17, 1313 (1966). [50] R . F . POTTER, P h y s . R e v . 150, 562 (1966). [51] F . BASSANI a n d D. BRUST, P h y s . R e v . 131, 1524 (1963). [ 5 2 ] M . CARDONA a n d F . H . POLLAK, P h y s . R e v . 1 4 2 , 5 3 0 ( 1 9 6 6 ) . [ 5 3 ] D . BRUST, P h y s . R e v . 1 3 9 , A 4 8 9 ( 1 9 6 5 ) .

[54] R . F . POTTER, Bull. Amer. P h y s . Soc. I I , 12, 320 (1967). [55] A. K . GHOSH, Solid S t a t e Commun. 4, 565 (1966). [56] A. K . GHOSH, P h y s . L e t t e r s (Netherlands) 23, 36 (1966). [ 5 7 ] S . ZWERDLING, (1960).

K . J . BUTTON, B . LAX, a n d L . M . ROTH, P h y s . R e v . L e t t e r s 4 ,

[58] T . STAFLIN, J . P h y s . Chem. Solids 27, 65 (1966). (Received

5'

November 13,

1967)

173

E . IGRAS a n d T. WABMINSKI: Electron Mirror Microscopic Observations

69

p h y s . s t a t . sol. 27, 69 (1968) Subject classification: 1.3 a n d 14.3.2; 22.1.2 Institute

of Physics,

Polish Academy of Sciences,

Warsaw

On Electron Mirror Microscopic Observations of the Details of p-n Junction Regions in Si By E . IGBAS a n d T . W A B M I N S K I

E l e c t r o n m i r r o r microscopic observations of lithium p - n a n d p - i - n junctions in Si show t h a t t h e c o n t r a s t of t h e j u n c t i o n b o u n d a r y strongly decreases a f t e r being i r r a d i a t e d w i t h electrons. Simultaneously t h e process of s a t u r a t i o n w i t h lithium of t h e micro-cracks present on t h e surface in t h e neighbourhood of t h e j u n c t i o n line is observed. T h e observed process is i n t e r p r e t e d in t e r m s of t h e field e x t r a c t i o n of l i t h i u m ions f r o m t h e deeper lying positions i n t o t h e surface. T h e electric field responsible for t h e e x t r a c t i o n of lithium ions originates f r o m t h e electrons impinging on a high resistive solid polymer film f o r m e d on t h e crystal i n t h e course of o b s e r v a t i o n . A similar process, t h o u g h w i t h slower r a t e , was observed on p - n junctions o b t a i n e d b y t h e diffusion of a l u m i n u m i n t o n - t y p e silicon. B e o b a c h t u n g e n m i t einem Elektronen-Spiegelmikroskop a n L i t h i u m - p - n - u n d p - i - n Übergängen in Silizium zeigen, d a ß d e r K o n t r a s t d e r Übergangsgrenze n a c h E l e k t r o n e n b e s t r a h l u n g s t a r k ansteigt. Gleichzeitig wird ein Sättigungsprozeß d e r Mikrorisse m i t L i t h i u m a n d e r Oberfläche in der N ä h e der Übergangslinie b e o b a c h t e t . Dies wird d u r c h einen E x t r a k t i o n s p r o z e ß tiefer liegender Lithiumionen zur Oberfläche d u r c h ein elektrisches Feld e r k l ä r t . D a s zur E x t r a k t i o n notwendige Feld r ü h r t v o n E l e k t r o n e n her, die auf einen festen P o l y m e r - F i l m a u f t r e f f e n , d e r im Verlauf der B e o b a c h t u n g gebildet wird. E i n ä h n licher P r o z e ß , jedoch m i t geringerer Geschwindigkeit, w u r d e a n p - n - Ü b e r g ä n g e n erhalten, die d u r c h Diffusion von A l u m i n i u m in n-leitendes Silizium gebildet w u r d e n .

1. Introduction It is well known that the electron mirror microscope can be used as a tool for investigating p-n junctions [1, 2]. In the papers [3] and [4] it was established that the electron mirror microscope can be used to study the processes of impurity segregation on crystal surface defects. These papers describe the temperature segregation of lithium around defects in silicon crystals. The present work was done during the investigation of the uniformity of p-n and p - i - n junction fronts in silicon. In the course of these examinations it was found that if electron mirror microscopic observation of a given region containing a p-n junction is carried out for a long time, then strong changes of the electrical topography in the junction region are observed. This paper aims at investigating the nature of these changes. It has been found that they are due to the segregation of ionized donors. The segregation is caused by an electric field perpendicular to the crystal surface. It is supposed that this field originates from the electron beam charging a thin and high resistive polymer film formed on the surface. Such field can extract positively charged ions.

70

E . IGRAS a n d T . W A R M I N S K I

2. Experimental The p - n junctions were prepared by diffusion of lithium into p-type silicon of resistivity 5 and 400 Qcm and by diffusion of aluminum into 50 Qcm n-type silicon. The diffusion was carried out in high vacuum. Lithium was diffused at 400 °C during 10 to 20 minutes. Aluminum was diffused at 900 °C for 6 hours. The p - i - n junctions were made by the drift of lithium ions in an electric field of 200 V/cm at temperatures ranging from 25 to 100 °C. 3. Observations of Lithium Junctions I t was found t h a t the contrast of the boundary between the p- and n-type regions is weakening during the time of observation. The junction boundary

Vig. 1. L i t h i u m j u n c t i o n in Si visible d u r i n g t h e first seconds of o b s e r v a t i o n

Fig. 2. T h e s a m e region of t h e c r y s t a l s u r f a c e as in Fig. 1 visible a f t e r f i v e m i n u tes of i l l u m i n a t i o n w i t h electrons. T h e c o n t r a s t of t h e j u n c t i o n b o u n d a r y s t r o n g l y decreased a n d " j u n c t i o n s a t e l l i t e s " a p p e a r e d

Electron Mirror Microscopic Observations of p - n Junctions in Si

71

which is usually very sharp in the first moment of observation, is diffusing in time and, depending on the intensity of the illuminating electron beam, it vanishes almost completely after about 2 to 10 minutes. Moreover in the junction region of mechanically polished crystals structures having the forms of elongated spots, needles, and straight or curved lines appear. Their contrast is increasing during the time of observation, i.e. the time of illumination of the surface by electrons. In the further description we shall call these structures "junction satellites". Fig. 1 shows a p - i - n junction photographed during the first seconds of observation. Fig. 2 illustrates the same region of the crystal surface after 5 minutes of illumination by the electron beam. On Fig. 3, 4, and 5 another region of the crystal surface is visible on which the process of growth of the "junction satellites" is shown. Fig. 3 shows the topography of the surface after about 5 minutes of illumination with electrons. The contrast of the initially sharp p - n junction boundary has decreased and the first " p - n junction satellites" (shown by arrows) are appearing. Fig. 4 and 5 show this region after 10 and 15 minutes from the beginning of the illumination, respectively. New " p - n junction satellites" are visible and their contrast is growing with time. I t should be said here t h a t although the majority of electrons from the velocity spectrum of the electron beam is reflected from equipotentials near the surface of the sample and takes part in image formation, some of them have enough energy to impinge on the surface. The intensity of the electron beam from the electron gun was of the order of mA. Unfortunately it is difficult to tell what part of the electron beam was impinging on the crystal surface. On Fig. 1 to 5 it is easily seen t h a t the contrast of the junction line diminishes with the time of illumination with electrons. At the same time new details of the surface potential relief appear. On the p-type side of the junction boundary the above mentioned "junction satellites" are becoming visible. They are clearly visible when the crystal is slightly reverse biased. The number of these "junction satellites" and their contrast increase in the time of observation. The process of the formation of the "junction satellites" was observed only on mechanically polished surfaces. On chemically polished surfaces a decrease of the contrast of the initial junction boundary was only observed. The appearance of the "junction satellites" and the decrease of the contrast of the initial p - n junction boundary can be explained by changes of lithium concentration on the surface region which was illuminated with electrons. These changes are caused by extraction of positive lithium ions by an electric field. Lithium ions from the regions below the surface are extracted. The electric field is formed by the electron charge which gathers to some extent on a thin insulating layer formed from' silicone oil polymerized on the crystal surface due to the action of electrons. The formation of such insulating layers under the action of an electron beam striking the substrate in a vacuum apparatus when silicone oil in a diffusion pump is used, is a well known phenomenon [5, 6]. The bombarding electrons cross-link the oil molecules adsorbed on the substrate to form a solid polymer film. The film has a high resistivity, so it can be charged by electrons impinging on the surface. In our diffusion pump the silicone oil was Dow Corning DC-704, a tetramethyltetraphenyltrisiloxilane. The depth of penetration of the electric field into the crystal should depend upon surface defects. For example, this depth should be higher in regions with microcracks.

72

E . IGRAS a n d T . WARMINSKI

I t means that field extraction of lithium in microcrack regions is carried out from deeper layers of the crystal than in regions free from cracks. Therefore after a sufficient time of the action of the electric field the microcrack regions should have high concentration of lithium and should convert themselves to strongly n-type regions. The cause of the observed inhomogeneous concentra tion of lithium impurity on the surface region may also have its origin in a higher mobility of L i + ions along microcracks. The disappearance of the contrast of the normal p - n junction boundary can be explained in the following w a y : The extraction of lithium ions by the electric field takes place not only in microcrack regions but on the whole surface of the crystal illuminated with electrons. As a result a thin surface layer of silicon enri:bes with lithium and converts itself into n-type. The thickness of this

Fig. 4

Electron Mirror Microscopic Observations of p - n Junctions in Si

73

Fig. 5 F i g . 3, 4, and 5. The process of field extraction of l i + ions on the s a m e surface region containing a junction. Fig. 3 corresponds to the situation after five minutes of illumination with electrons, Fig. 4 and 5 correspond to the situation a f t e r ten a n d fifteen minutes of illumination, respectively

surface layer is governed in some way by the depth of penetration of the electric tield. The formation of such n-type thin surface layer surrounding the p-n junction leads to the situation in which the junction becomes more and more diffused. In the electron mirror microscope we observe it in the form of the contrast drop of the junction boundary. In the electron mirror microscope the junction boundary is observed with good contrast if the crystal is slightly reverse biased. In our case the junction boundaries and described "junction satellites" were clearly visible only when the crystal was reverse biased. Under the conditions of forward biasing the contrast of the above details was poor. The latter proves that visibility of the "junction satellites" under the condition of reverse biasing implies the existence of a thin n-type layer lying on a p-type part of the silicon crystal. In other words, the "junction satellites" are clearly visible if the crystal is reverse biased, because a thin n-type cover connects them with the n-type part of the crystal. As a final result in this

Fig. 6. A qualitative illustration of the change of lithium concentration at the surface layer with microcracks. The change is caused b y inhomogeneous field extraction of L i + ions. The thick curve corresponds to the initial s t a t e before the action of the electric field, the thin curve corresponds to the final state, i.e. after the process of extraction. N is the concentration of lithium, A'0 the concentration of acceptors in p-type Si, a n d x the distance

74

E . IGRAS a n d T . WARMINSKI

%

mi

jjm. • jrra Bfl H e

A

Fig. 7. The map of microcracks with field-extracted lithium on the region of a Si crystal which was pressed with a small ball

region we have to do with a thin n-type layer decorated with thicker n-type inclusions along microcracks introduced by mechanical polishing. This can be illustrated with the aid of Fig. 6. The field extraction of lithium ions in the microcrack regions was observed not only in the neighbourhood of drifted junctions but also around diffused ones. Fig. 7 shows an example of a microcrack configuration in the vicinity of the diffused junction. The microcracks were produced artificially by pressing a small steel ball into the surface region near the junction of a chemically polished crystal. During the initial period of observation of the reverse biased crystal only a straight line of the junction boundary was visible. After about 10 minutes of observation the initial junction boundary practically vanished and due to the intense extraction of lithium ions by the electric field a compound configuration of microcracks visible on Fig. 7 was developed. The visibility of the "junction satellites" weakens if the temperature is increased. At temperatures of about 150 °C they disappear completely on the electron mirror microscopic image, and appear again with growing contrast if the temperature is decreased. The above temperature behaviour of the "junction satellites" is the same as for the case of typical p-n junctions. 4. Observation ol p—n Junctions Formed by Diffusion of Aluminum into n-Type Silicon These junctions were observed in the electron mirror microscope with the aim to reveal the possible impurity segregation on microcracks as in the case of lithium p-n junctions. The extraction by the electric field of ionized phosphorus donors was expected. Contrary to lithium ions the field extraction of the phosphorus ions was extremely slow. Only small changes in the immediate vicinity of the junction boundary were observed. They were visible after about one hour of illumination of the selected region containing the p-n junction. This is illustrated on Fig. 8 and 9. On Fig. 8 the image of the "fresh" aluminum p-n junction is visible.

Electron Mirror Microscopic Observations of p-n Junctions in Si

75

Fig. S. p - n junction made by diffusion of A1 into n-type Si. The topography of the junction region is visible

Fig. 9. The same region as on Fig. 8 visible after one hour of illumination with the electron beam. The micro-regions with field-extracted positive phosphorus donors are visible (black spots near the junction boundary)

Fig. 9 shows the same junction region as on Fig. 8 but after one hour of continuous observation (illumination with electrons) in the electron mirror microscope. The differences in details near the junction boundary on these two figures are clearly visible. The black spots which adhere to the junction boundary visible on Fig. 9 are on the p-type side of the crystal and are microcrack regions with field-extracted positive phosphorus ions. They were visible only on mechanically polished surfaces. The reason due to which the process of field extraction in microcracks in silicon of ions from the fifth column of the periodic table is uncomparably slower than in the case of lithium ions can be easily understood if we remind the ex-

76

E. IGKAS and T. WARMINSKI: Electron Mirror Microscopic Observations

tremely high differences of mobilities of these two types of ions. At a given temperature the mobility of ionized donors from the fifth column of the periodic table is extremely slow as compared with the mobility of ionized lithium donors. Acknowledgement

The authors want to express their gratitude to Prof. Dr. L. Sosnowski for helpful discussions. References [1] E. IGRAS, Bull. Acad, polon. Sci. I X , 5, 403 (1961). [ 2 ] E . IGRAS a n d T . WARMINSKI, p h y s . s t a t . sol. 9 , 7 9 ( 1 9 6 5 ) . [ 3 ] E . IGRAS a n d T . WAHMINSKI, p h y s . s t a t . sol. I B , 1 6 9 ( 1 9 6 6 ) . [ 4 ] E . IGRAS a n d T . WARMINSKI, p h y s . s t a t . sol. 1 9 , K 6 7 ( 1 9 6 7 ) .

[5] R. W. CHRISTY, J . appl. Phys. 35, 2179 (1964). [6] H. T. MANN, J . appl. Phys. 35, 2173 (1964).

(Received January 2, 1968)

A. AUTHIER: Contrast of a Stacking Fault on X - R a y Topographs

77

phys. stat. sol. 27, 77 (1968) Subject classification: 10; 22.1.2 Laboratoire de Minéralogie-Cristallographie,

associé au C.N.R.S.,

Faculté des Sciences de Paris

Contrast of a Stacking Fault on X-Ray Topographs By A . AUTHIER

The contrast of the image of a stacking fault on a section topograph is discussed in detail. The intensity distribution of the X-rays reflected by a crystal containing a stacking fault is the sum of three terms : the first one is due to interferences between waves which have suffered no interbranch scattering, the second is due to interferences between waves which have suffered interbranch scattering, the third term is a cross term due to interferences between waves of both types. The latter is shown to be the predominant term when the crystal is absorbing. This is confirmed by experiment. The derivation has been done using Kato's spherical wave approach and the stationary phase method. The expressions are given for an absorbing crystal and an asymmetrical orientation of the reflecting planes and the stacking fault with respect to the crystal faces. There is good agreement between the calculated intensity distribution and t h a t observed in the image of a stacking fault. On discute en détail le contraste de l'image d'une faute d'empilement sur une topographie en pose fixe. L'intensité des rayons X réfléchis par un cristal contenant une faute d'empilement est la somme de trois termes. Le premier est dû à des interférences entre des ondes dont le point caractéristique est resté sur la même branche de la surface de dispersion. Le deuxième est dû à des interférences entre des ondes dont le point caractéristique a sauté d'une branche à l'autre au passage de la faute d'empilement. Le troisième est dû à des interférences entre des ondes de ces deux types. On montre que c'est le terme prépondérant de la distribution d'intensité lorsque le cristal est absorbant. Ce résultat est confirmé par l'expérience. Le calcul a été fait dans le cadre de la théorie des ondes sphériques de Kato et en utilisant la méthode de la phase stationnaire. Les expressions sont données pour un cristal absorbant et pour une orientation asymétrique des plans réflecteurs et du plan de faute par rapport aux faces du cristal. — Il y a bon accord entre la distribution d'intensité calculée et celle observée sur une image de faute d'empilement. 1. Introduction There are t w o m a i n w a y s t o calculate the contrast of a stacking f a u l t on a n X - r a y topograph. One is t o use a generalized f o r m of t h e d y n a m i c a l t h e o r y such as T a k a g i ' s t h e o r y [1], W e shall discuss this p o i n t in a separate paper [2] . A n o t h e r w a y is t o use t h e s t a n d a r d f o r m of the d y n a m i c a l t h e o r y a n d m a k e use of a plane w a v e e x p a n s i o n of t h e incident w a v e which, for X - r a y s , is usually a spherical w a v e . Such a t r e a t m e n t was introduced b y K a t o [3] t o e x p l a i n t h e P e n d e l l ô s u n g fringes o b s e r v e d on section a n d t r a v e r s e p a t t e r n s b y K a t o and L a n g [4], W h e n a w a v e - f i e l d crosses a f a u l t plane, it decouples i n t o its t w o c o m p o n e n t s , a refracted a n d a reflected w a v e . These, in turn, e x c i t e t w o w a v e - f i e l d s each in the s e c o n d half of the crystal (Fig. 1). Those e x c i t e d wave-fields w i t h a tie-point l y i n g o n t h e same branch of t h e dispersion surface as t h a t of t h e e x c i t i n g field are called normal fields. L e t JL' be t h e total a m p l i t u d e at a g i v e n p o i n t p of such wave-fields. T h e e x c i t e d w a v e

78

A . ATJTHIER Fig. 1. Wave-field multiplication a t a f a u l t plane in a crystal. K : incident plane wave; 1,2: wave fields excited in p a r t I of t h e crystal; 01, 02, Al, h'Z: wave components of these fields in t h e incid e n t a n d reflected directions; 1(01) . . . 2(A2): wave fields excited in p a r t II of t h e c r y s t a l ; n : normal to t h e e n t r a n c e surface; / : n o r m a l to t h e f a u l t plane

\\

P

fields with a tie-point on the other branch of the dispersion surface are the new fields. Let =

exx> (—2 n i k r) \ n r

(2.1)

79

Contrast of a Stacking Fault on X-Ray Topographs

can be expanded in the following way, according to Kato : *(r)

-I- OO exp ( — 2 n i K • r)

=

i £ f f .

K,

dKr

(2.2)

dK„

with \K\ = k = 1 jX; Kz is the direction in which the propagation is being studied. This expression is interpreted as a sum of plane waves with wave number k. We shall assume that each plane wave is diffracted independently by the crystal, as if it were alone, and that the total amplitude of the electrical displacement at point p in the crystal is given by D

= ^

f f

2

[3>o j +

2>ki e x p ( -

2 n i h - r)]

X

exp { - 2 ti i [K • rs + Koj • (r - r B )]}

_

1,2.

y

K,

(2.3)

rs is the position vector of a point at the surface of the crystal and r the position vector of p. K 01 and K o i are the wave vectors of the refracted waves excited in the crystal by a plane wave with wave vector K. Expressions for 2)0j and 2)/,j are given by the classical dynamical theory: Ai h_

^4>, = D h i =

•Xn! -XK

(2-4)

±

-

x

with ^o;

= i \ c \ k

i/J> Vh

{ n ± 1 1

+

rj•) =

Po/P,.

(2.5)

P 0 j is the projection of the tie-point P^- on L 0 T 0 , asymptotic to the sphere centered in 0 and with radius n k , n index of refraction (Fig. 2) ; À0 sin 2 0 — — X o V* V

=

2

n

n

(2.6)

Vo

C is equal to 1 or cos 2 0, depending on whether the polarization lies normal or parallel to the plane of incidence; %0, %h, are the (000), (h k I), (h k I) Fourier coefficients of the electric susceptibility respectively; Ad is the departure from Bragg's law of the incident plane wave with wave vector K = O M (Fig. 2); y0, yh are the cosines of the angles between the normal n to the face of the crystal and the incident and reflected directions respectively.

80

A. AUTHIER

Fig. 2. Dispersion surface for a single crystal. If the incident wave is spherical, two wavefields travel along any given p a t h ; their tiepoints P t and P 2 conjugated

fM

/ Thanks to the continuity of the tangential component of wave vectors, we can write the phase factor in (2.3) K • rB + K0j

• (r -

rs) •-= K • r +

MPTcZ,

where d is the depth of point p in the crystal. Expression (2.3) can be integrated directly. This leads to Bessel functions, complex in the absorbing case. An asymptotic approximation can also be obtained using the stationary phase method (e.g. Born and Wolf [9]). I t gives a physical interpretation of the result and it has been shown by Kato that the resulting amplitude can be considered as the sum of the amplitudes of two waves having as tie-points two conjugate points and P^ of the dispersion surface (Fig. 2). Their interferences give rise to the well known Kato or so-called Pendellosung fringes. 2.2 Crystal containing

a stacking fault — plane wave case

We shall assume that the crystal is divided into two parts by a fault plane, the second part being shifted by u with regard to the first part. The expansion of the electric susceptibility in Fourier series becomes, in part I I of the crystal, Z=2'z"exp(-2

with

nih-r)

d =

(2.7)

2nh.u.

On crossing the fault plane, each of the four waves 2)0j, 2)hj excited in the first half of the crystal by an incident plane wave will excite two wave fields in the second half. Sixteen waves are thus created. We shall call their amplitudes •®0 j'(hj) = Doj' •&oj'(Oj) =

, 2>oj >

^hj'(hj)

= Dhy 3)hj ,

-®a/(0J) = Dhy

(2.8)

S)\j.

j = 1, 2 denotes the branch of the dispersion surface to which belongs the tiepoint of the exciting wave field, j' = 1 , 2 denotes that of the excited field; I , I I denotes the part of the crystal where the wave is created; h or 0 means that the wave is reflected or refracted in the particular part of the crystal where it is generated. Thus both .0Oy(hj) and 3)hj'(oj) have wave vectors in the reflected direction with regard to the wave incident on the crystal taken as a whole.

81

Contrast of a Stacking Fault on X - R a y Topographs

j

M / \N/>

Fig. 3. Wave-field multiplication observed in reciprocal space — plane wave case. Tie-points are P j and P * in part I of the crystal; excited tie-points in part I I are P * and P * , P * * and P p t n and / are the normals to the entrance surface and fault plane, respectively

. /

\

Fig. 4. X - r a y paths-spherical wave case. BiCi fault plane, Ap normal fields, Aq^ new fields, A' focusing point of new fields, 1 mid-point of BC, 1' mid-point of B 2 C 2

According to the definition given in the introduction, wave fields with j' =)= j are new fields. Let us now call P* and P* the tie-points of the two wave fields excited in part I of the crystal. They lie on a same normal n to the entrance surface. They excite in part I I couples of tie-points P{ and P* 1 , P* and PJ 1 ; each couple lies on a same normal / to the fault plane (Fig. 3). Let y'0 and y'h be the cosines of the angles between this normal and the incident and reflected directions, respectively. The following relations between the coordinates of these four tie-points normal to the asymptotes of the dispersion surface can easily be shown: Yh

i =

Yo

VI ¿02,

X1 2 —¿A

Yo Yh v i ¿x 0u2 — - ; — ¿ 0 2 ;

X 2 —¿A

Yo Yn V I 11 ¿Y0 1 — —— ¿ 0 1 ,

Y n 1 —¿A

YhYo

YhYo

u

Yh Yo

Y

I

¿01 >

Yh V I 7 ¿01 Yo

(2.9)

:

VI r ¿02 •

Yh

Vo

The four waves 2>oy(hj) have been excited by a wave the wave which originates at the reciprocal point H. To obtain their amplitudes D0j. one should replace in (2.4) the distances X 0 j from the tie-points to the asymptote L 0 T 0 by the distances X h j to the asymptote to the sphere centered in H. The full expressions of the eight waves emitted in the reflected direction are calculated from (2.8), (2.4), (2.9), and (2.7) and given in (A 1) in terms of and XJ 2 . 2.3 Crystal containing

a stacking fault — spherical

wave

case

Let us now substitute the amplitudes for the plane wave case in the expansion (2.3) of the incident wave. In order to obtain a physical interpretation of the derivation, we shall integrate by the stationary phase method and obtain an asymptotic expression; it is possible to show that the total amplitude of the 6

physica 27/1

82

A . AUTHIER

reflected wave at point p of the exit surface of the crystal is the sum of eight terms. Four correspond to waves having travelled along A p (Fig. 4); they are normal waves. We have expressed their amplitudes in terms of a parameter Y describing the position of p on the exit surface: y

=

iE = 1 C

* , ]/l + i?2

where 1 is the middle of the base B C of the Borrmann triangle. One obtains:

1(0 1)

Un u, + Uh

=

o1(A 1) = U uh n

exp i ( ö - * +

-axj

2 yd (1 - 72)1/4

exp

(

2

0

««)

(2.10)

2 -j/rf (i - y2)1/!

J.0 uh exp [i{d + 0 - Oj)] U0 + U„ 2]/d(l - 72)i/4

•®A 2(0 2) =

exp [t

3>,0 2(h 2)

J

t7«

- o4)]

+ C^A 2 y'd (1 - F 2 ) 1/4

with uh=74(

Yh

i

I 7 ),

n

7o

0 =

/l /L '

(2.11)

.

A is a complex quantity. We shall call t~l and r _ 1 the real and imaginary parts of A~\ respectively, t is the Pendellosung fringe spacing; d is the thickness of the crystal. to ai are related to the imaginary parts of the coefficients of the amplitudes for the plane wave case ((A 1) and (A 2)). They give rise to shifts of the fringes but are in general very small and can be neglected. VYoYh = 0„ ' 8 n sin 2 6

1/2

C

X exp — 2 n i\h • r

AK,. z Kzz

kXod.

1 + 1 7o Yh

Y ( 1 + 1)1 \Yo Yh!J

2

—1. n)

The four other terms in the development of the total amplitude at point p correspond to new wave fields the tie-point of which has jumped from one branch of the dispersion surface to the other. Their path is A q in the first half of the crystal, q p in the second (Fig. 4). Let and C, be the intersections of the fault plane with the sides of the Borrmann triangle, B x A' and Cj A' two straight lines parallel to A Cj and A Bj respectively. I t has been shown that q p goes through A' [6], Let C2 and B 2 be the intersections of B j A' and A' with the exit surface. The amplitudes of the new waves are given in terms of a parameter Z' defined in

Contrast of a Stacking Fault on X-Ray Topographs

83

triangle A' B 2 C2 in the same way as Y in A B C : = YÏ Y C,

Z'

where 1' is the middle of B 2 C 2 . Z' is related to Y by Z'

Yo y'h + d ' A'— d Yo —d y'h +

Yh y'o Yh Yo

d

=

dA'

1

A

(2.12)

d K ' is the distance between A' and the entrance surface. One obtains: _ j>* 2(01) - ±

d The stacking fault is nearer the exit surface and, according to (2.14), IF is positive. For an absorbing crystal, the most important terms are proportional to cosh (0 i + Wt) and sinh (0 i + ; the shape and positions of the fringes due to the oscillating terms are given by 0r -

=

- F-io^T] = const .

(3.8)

The fringes are constructed by taking the intersections of the two series of hyperbolae: /.r0 xh = const,

\!x'0 x'h = const

corresponding to the two sets of fringes previously described. Fig. 5 illustrates how it is done. 3.4.2 d.y < d The stacking fault is nearer the entrance surface and Wt is negative. The most important terms are now cos (0 r + WT) cosh (0; - Wi)

and

sin (&T + lFT) sinh (0l - ¥{) .

The shape and positions of the fringes is also given by (3.8) and the same geometrical construction.

86

A.

Fig. 5. Fringes due to the interferences of normal and new wave-fields. Hyperbolae turning upwards: normal fields; hyperbolae turning downwards: new fields; flat, thick fringes; interferences between normal and new fields

AUTHIER

Fig. 6. Fringes observed on a section pattern, a ) Interferences between new fields alone, b) Interferences between normal and new fields. Notice that the number of fringes is double in the second case ( f l a t fringes). I G : dynamic image of the entrance surface; F H : dynamic image of the exit surface; F G ; direct image of the fault plane

Fig. 6 b shows equal intensity fringes to be therefore expected on a section pattern of an absorbing crystal containing a stacking fault. The two extreme fringes are straight and lie on the dynamic images of the traces of the fault plane on the entrance and exit surfaces. They correspond to = — W and to = W respectively. When the crystal is not absorbing, the term between curly brackets in (3.5) becomes +

2 sin ~ cos ¡& T —

cos

+ ^j

• ^he

fringes

described in this section are only visible when the crystal is sufficiently absorbing. 3.5 Comparison

of the fringes due to the new wave fields and those due to interferences between new and normal wave-fields

The two types of fringes are compared in Fig. 6a and 6b. a) Their geometrical localization on a section pattern is the same. I t can be noticed that the dynamical image of the trace of the stacking fault on the entrance surface is normal to the axis of the section pattern, since it lies in a single plane of incidence. On the other hand, that of the trace on the exit surface is simply obtained by projection and is, in the general case, oblique. In both cases, as well, there is a maximum of the energy distribution along the direct image of the fault. b) The fringe spacing is double in the case of interferences between normal and new wave fields. The total number of fringes is equal to d/t, and it is very easy to distinguish them from those due to interferences between new wave fields alone. c) The shape of the two sets of fringes are very different: those due to the new wave fields are hyperbolic, while the other ones are not and become flat near the entrance or exit surfaces.

Contrast of a Stacking Fault on X-Ray Topographs

87

Fig. 7. Section pattern of a fault plane in dolomite (already published in [14]). 100 reflection, M o K « ; crystal thickness 1.21 mm (174 x )

3.6 Contrast

of the first

fringe

Let us now assume, for simplicity, that the trace of the stacking fault on the plane of incidence is parallel to the faces of the crystal and that y0 =

yo,

yn =

y'h-

We shall call Z\ and zn the distances of the fault plane from the entrance and exit surfaces, respectively. We therefore have d =

Zi +

zn,

2 zi =

•

If zc decreases towards zero, equations (2.12) and (2.14) show that Z' Y and that W -»• — 0. In the same way, when Zu tends towards zero, Z' — Y and V

->

0.

In both cases, it can be easily checked that the total intensity tends towards the normal expression for a perfect crystal without a stacking fault. To get the contrast of the first fringe, we expand Of ZT or 2JJ. Stopping at the first order, we obtain sinh2 3>i + cos2 1 = Mol:

M

Zj sin

2Y (1

_

ô | I sinh sir 2 tf>i

COS —

I

ô Y

¿(1 ± Y)

¿ y 1 - Y2

+ 2 71

+ /2 + I 3 in power series

cos 2 0T \

t

sinh 2 0:

+ 2 n sin —

2 Y2 . ô sin 2 0. + cosh 2 sin — =—

(3.9)

Y2 )l/2

The upper sign corresponds to the entrance surface (j = I). a) The crystal is highly absorbing, and much thicker than one Pendellôsung distance. Expression (3.9) reduces to

I

sinh2

+ cos2 (& r - — ) \ 4/ d (i -

r2)i/2

sin si ô +

l ±-

Z1

Ytd

sinh 2 0-. \

(3.10)

88

A . AUTHIER

The sign of the contrast of the first fringe is opposite for the entrance and exit surfaces, depends on the value of the phase difference d introduced by the stacking fault, but it is independent of the thickness of the crystal. b) The crystal is non-absorbing but much thicker than one Pendellosung fringe thickness. The expression (3.9) becomes

g

J

=

{

M°.|2 M I1 " 1

c o g ,

d V

(

0

'

(i- -r

Z]

~ f ) y r«)i/* V -

-

2

i

J

a

s•n r2 ±

8

—

r

r

°

o

s

2

"

( 3 1 1 )

The sign of the contrast is the same for both surfaces, is independent of the value of the phase difference, depends on the thickness of the crystal.

l

1.4:1:1.5 ^33 ' ' ' ^22 a

C22 > ffU > ff3i

°33'- ll = 2.1; O33