Semiconductors and Semimetals: v. 1: Physics of III-V compounds

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SEMICONDUCTORS A N D SEMIMETALS VOLUME 1 Physics of III-V Compounds

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SEMICONDUCTORS AND SEMIMETALS Edited by R . K . WILLARDSON 1BELL A N D HOWELL RESEARCH CENTER

PASADENA, CALIFORNIA

ALBERT C. BEER BATTELLE MEMORIAL INSTITUTE COLUMBUS, OHIO

VOLUME 1 Physics of 111-V Compounds

1966

ACADEMIC PRESS New York and London

COPYRIGHT @ 1966, BY ACADEMIC PRESS INC. ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS INC. I 1 I Fifth Avenue, New York. New York 10003

United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London. W . I

LIBRARY OF CONGRESS CATALOG CARDNUMBER: 65-26048 PRINTED IN THE UNITED STATES OF AMERICA

List of Contributors Numbers in parentheses indicate the pages on which the authors' contributions begin.

BETSYANCKER-JOHNSON, Boeing ScientiJic Research Laboratories, Seattle, Washington (379)

PETROS N . ARGYRES, Lincoln Laboratory,' Massachusetts Institute of Technology, Lexington, Massachusetts (1 59) FRANCOBASSANI,~ Argonne National Laboratory, Solid State Science Division, Argonne, Illinois (2 1)

W. M. BECKER,Physics Department, Purdue University, Lufayette, Indiana (265)

V. L. BONCH-BRUEVICH, Faculty of Physics, Moscow State University, Moscow, USSR (101)

T. H . GEBALLE, Bell Telephone Laboratories, Murray Hill, New Jersey (203) C. HILSUM,Ministry of Aviation, Royal Radar Establishment, Malvern, Worcestershire, England (3) E. 0. KANE,Bell Telephone Laboratories, Murray Hill, New Jersey (75) DONALD LONG,Honeywell Research Center, Hopkins, Minnesota ( 143) S . M. P U R I ,Stanford ~ University, Stanford, California (203)

E . H . PUTLEY,Ministry of Aviation, Royal Radar Establishment, Malvern, Worcestershire, England (289)

LAURA M. ROTH, Department of Physics, Tufts University, Medford, Massachusetts (159)

H . WEISS,Siemens-Schuckertwerke A. G., Erlangen, West Germany (315) 'Operated with support from the U S . Air Force. 'Present address: Istituto di Fisica, Universita di Messina, Messina, Italy. 'Present address : Department of Physics, University of Southampton, Southampton, England. V

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Preface The extensive research devoted to the physics of compound semiconductors and semimetals during the past decade has led to a more complete understanding of the physics of solids in general. This progress was made possible by significant advances in material preparation techniques. The availability of a large number of compounds with a wide variety of different and often unique properties enabled the investigators not only to discover new phenomena but to select optimum materials for definitive experimental and theoretical work. In a field growing at such a rapid rate, a sequence of books which will provide an integrated treatment of the experimental techniques and theoretical developments is a necessity. An important requirement is that the books contain not only the essence of the published literature, but also a considerable amount of new material. The highly specialized nature of each topic makes it imperative that each chapter be written by an authority. For this reason the editors have obtained contributions from ten to fifteen scientists to provide each volume with the required detail and completeness. Much of the information presented relates to basic contributions in the solid state field which will be of permanent value. While this sequence of volumes is primarily a reference work covering related major topics, the volumes will also be useful in graduate courses. Because of the important contributions which have resulted from studies of the 111-V compounds, the first few volumes of this series are devoted to the physics of these materials. Volume 1 reviews key features of the 111-V compounds, with special emphasis on band structure, magnetic field phenomena, and plasma effects. In Volume 2, the emphasis is on physical properties, thermal phenomena, magnetic resonances, and photoelectric effects, as well as radiative recombination and stimulated emission. Volume 3 is concerned with optical properties, including lattice effects, intrinsic absorption, free carrier phenomena, and photoelectronic effects. The editors are indebted to the many contributors and their employers who made this series possible. They wish to express their appreciation to the Bell & Howell Company and the Battelle Memorial Institute for providing the facilities and the environment necessary for such an endeavor. vii

viii

PREFACE

Thanks are also due to the US. Air Force Offices of Scientific Research and Aerospace Research, whose support has enabled the editors to study many features of compound semiconductors. The assistance of Rosalind Drum, Jo Ann Gibel, Eleanor Quinan, and Inez Wheldon in handling the numerous details concerning the manuscripts and proofs is gratefully acknowledged. Finally, the editors wish to thank their wives for their patience and understanding. December. 1966

R. K. WILLARDSON ALBERT C. BEER

Contents LISTOF CONTRIBUTOORS .

PREFACE .

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INTRODUCTION Chapter 1 Some Key Features of 111-V Compounds C. Hilsum I . Introduction . I1. Band Structure . 111. Effective Masses . IV. Effective Charge . V. Transport Properties VI . Futureprogress .

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BAND STRUCTURE Chapter 2 Methods of Band Calculations Applicable to 111-V Compounds Franc0 Bassani I . Introduction . . . . . . . . . . I1. Classificationof the Electronic States . . . . . . I11. Symmetry Functionsand MatrixElements . . . . . IV . Approaches to the Calculation of the Electronic States . . . V . Qualitative Features of the Band Structure and Preliminary Calculations VI . Conclusionsand Linesof Future Progress . . . . .

Chapter 3 The k p Method E . 0. Kane I . Introduction . .

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I1. Scope of Present Paper . . . . I11. The k . p Representation . . . . IV. Specific Diamond and Zinc Blende Symmetries Appendix . . . . . . .

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Chapter 4 Effectof Heavy Doping on the Semiconductor Band Structure V. L . Bonch-Bruevich I . Introduction . . . . . . . I1. Main Features of the Heavily Doped Semiconductors I11. Characteristicsof the Many-Electron System . . IV . Perturbation Theory . . . . . . V . Semiclassical Approximation . . . .

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Chapter 5 Energy Band Structures of Mixed Crystals of In-V Compounds Donald Long I. Introduction . . . . . . . . . . . 143 11. Energy Gap versus Composition . . 111. Other Band Parameters versus Composition

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MAGNETIC FIELD EFFECTS Chapter 6 Magnetic Quantum Effects Laura M . Roth and Petros N . Argyres .

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Chapter 7 Thennomagnetic Effects in the Quantum Region S . M . Puri and T. H . Geballe I. Introduction . . . . . . . . 11. Thermoelectric Power in High Magnetic Fields 111. Compound Semiconductors of 111-V Group . IV. Oscillations Induced by Optical Phonons .

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Chapter 8 Band Characteristics near Principal Minima from Magnetoresistance W , M . Becker I. Introduction . . . . . . . . . . . 265 11. Survey . . . 111. Experimental Measurements

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Chapter 9 Freeze-Out Effects, Hot Electron Effects, and Submillimeter Photoconductivityin InSb E. H . Putley I. Freeze-Out Effects

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Chapter 10 Magnetoresistance H. Weiss I. Introduction . . . . . Shape of Specimen and Electrodes . Inhomogeneities . . . . The Homogeneous Rodlike Semiconductor Influence of Pressure . . . .

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PLASMA EFFECTS Chapter 11 Plasmas in Semiconductors and Semimetals Betsy Ancker-Johnson .

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11. Plasma Oscillations . . . 111. Magnetoreflection and Absorption.

IV. V. VI. VII.

Wave Propagation PinchEffect . Instabilities. . Conclusions . . Notation .

AUTHORINDEX. SUBJECTINDEX.

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Semiconductors and Semimetals Volume 2 Physics of 111-V Compounds M . G. Holland, Thermal Conductivity S. I. Novikova, Thermal Expansion U . Piesbergen, Heat Capacity and Debye Temperatures G. Giesecke, Lattice Constants J. R . Drabble, Elastic Properties A . U. Mac Rae and G. W. Gobeli, Low Energy Electron Diffraction Studies Robert Lee Mieher. Nuclear Magnetic Resonance Bernard Goldstein, Electron Paramagnetic Resonance T. S. Moss, Photoconduction in 111-V Compounds E. AntonEik and J . Tauc, Quantum Efficiency of the Internal Photoelectric Effect in InSb G. W. Gobeli and F. G. Allen, Photoelectric Threshold and Work Function P. S. Pershan, Nonlinear Optics in 111-V Compounds M . Gershenzon, Radiative Recombination in the 111-V Compounds Frank Stern. Stimulated Emission in Semiconductors

Volume 3 Optical Properties of 111-V Compounds Marvin Hass, Lattice Reflection William G. Spitzer, Multiphonon Lattice Absorption D.L. Stierwalt and R. F . Potter, Emittance Studies H . R. Philipp and H . Ehrenreich, Ultraviolet Optical Properties Manuel Cardona, Optical Absorption above the Fundamental Edge E. Johnson, Absorption near the Fundamental Edge John 0. Dimmock, Exciton States in Semiconductors Benjamin Lax and John G. Mavroides, Interband Magnetooptical Effects H. Y. Fan, Effects of Free Carriers on the Optical Properties Edward D.Palik and George B. Wright, Free Carrier Magnetooptical Effects Richard H. Bube, Photoelectric Analysis B. 0. Seraphin and H . E . Bennett, Optical Constants

xii

Introduction

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CHAPTER 1

Some Key Features of 111-V Compounds C . Hilsum 1. 11. 111. IV. V. VI.

INTRODUCTION . . . BANDSTRUCTURE . . EFFECTIVE MASSES . . EFFECTIVE CHARGE . . TRANSPORT PROPERTIES . FUTUREPROGRESS. .

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I. Introduction Since the discovery of the semiconducting properties of InSb in 1950, work on 111-V compounds has given us much information about those physical processes which cannot be studied on germanium and silicon. The particular features that attracted interest were the low effective mass, which revealed itself by a very high electron mobility, and the ionic component in the crystal binding. After the first exploratory research, nearly all of the work was limited to InSb, partly because this was an almost ideal vehicle for such studies, but also because it was easy to grow InSb single crystals. As a result the technology of InSb advanced to the stage that the best material made a year or so ago compared well in impurity content with the purest germanium and silicon. The technology of the other compounds fell far behind this, for the purification was very difficult and there was no great incentive to work in this field. The invention of the semiconductor laser and the discovery of the Gunn effect have caused a marked change in this situation. There is now an increased prospect of industrial application of these materials, and we may expect a growing interest in GaAs, Gap, and compounds or alloys of compounds which can emit visible or microwave radiation. It is useful at this stage to assess our present knowledge, and to point out areas where the evidence is insufficient or conflicting. These tasks are attempted in these volumes on Physics of 111-V Compounds, with a number of articles giving detailed treatments of specific aspects of the subject. This introductory article serves a rather different purpose. It is more genera1 in scope, and more speculative in its approach. It deals with four 3

4

C. HILSUM

of the most important features which have led to our interest in 111-V compounds, that is, band structure, effective masses, crystal binding, and transport properties. By presenting and linking these key topics from an over-all viewpoint, we set the stage for the articles which follow.

11. Band Structure As the interest in compounds shifts toward recombination, a knowledge of band structure becomes more important. The family of 111-V compounds has some members with a direct band gap and some with an indirect gap, and in order to predict the band structure of alloys of compounds it is necessary to know not only the energy of the lowest conduction band minimum, but also the energies of the higher minima. Fortunately we can derive much of the required information by combining the results of various experiments. The energies of two of the three important vertical transitions, X , - XI, and L3 - L , (Fig. I), can be measured or estimated by ultra-

i100)

GaP

FIG.1. Band structure of gallium compounds.

violet reflectivity, a technique pioneered by Philipp and Taft,’ and determinations of these energies have now been made for all the better known compounds.2 The energy difference between rI5and rl can also be found by optical techniques. However, unless we know the shape of either the conduction or valence band, we cannot draw the band structure.

’ H. R. Philipp and E. A. Taft, Phys. Rev. 113, 1002 (1959)

* M. Cardona, J . Phys. Chem. Solids 24, 1543 (1963).

1.

SOME KEY FEATURES OF III-V

COMPOUNDS

5

In a compound like Gap, where the lowest energy transition between the conduction and valence bands is the indirect one from T 1 5 - X 1 , we can deduce the relative positions of rl and X , from optical transmission or from the spectral response of the photosensitive barrier at a metal-semiconductor interface. We cannot find the position of L , in the same way. In the direct gap materials, GaAs and GaSb, the minima at X and L are not much higher than r, and their position can be determined by Hall effect measurements and by a study of the pressure dependence of resistivity. As an example we take the band structure of the three gallium compounds (Fig.1). The direct transitions have been established with good accuracy by a number of workers. The conduction band structure of GaSb has been studied by Sagar and Miller,3 and by Becker et al.,'' and the two sets of results are in good agreement. A recent investigation by Zallen and Paul5 on the optical properties of GaP has given the r15- X , and r15- rl energies with good accuracy, and the results agree well with those of Nelson et aL6 Spitzer and Mead7 obtain a slightly smaller value for r15- rl from measurements of photovoltage of GaP-Au surface barriers. We prefer to adopt Zallen and Paul's value, because the theory of the surface barrier photocell is not yet established well enough for the method to give an accurate determination. The direct transition rI5- rl in GaAs has been established by the transmission measurements of Sturge.* Ehrenreich' has calculated the energy difference between rl and X , from the high temperature behavior of the Hall coefficient, and he finds this to be 0.36 eV at low temperature. Spitzer and Mead give a slightly smaller value, 0.28 eV at 300°K. Kravchenko and Fan" are in disagreement with these results. They deduce an energy difference smaller than 0.05 eV from measurements of magnetoresistance, but their results are suspect because the samples they used were highly compensated. The band structure they propose would not be consistent with the results obtained for the electrical and optical properties of n-type GaAs and of GaAs-GaP alloys. We prefer Ehrenreich's method of deducing the energy difference, and find that, if his calculation is modified to include the effects of nonparabolicity of the conduction band, and of A. Sagar and R. C. Miller, J . Appl. Phys. Suppl. 32, 2073 (1961).

'W. M. Becker, A. K. Ramdas, and H. Y. Fan, J . Appl. Phys. Suppl. 32, 2094 (1961). R. Zallen and W. Paul, Phys. Rev. 134, A1628 (1964). R. F. Nelson, L. F. Johnson, and M. Gershenzon, Bull. Am. Phys. SOC.9, 236 (1964). ' W. G . Spitzer and C. A. Mead, Phys. Rev. 133, 872 (1964). M. D. Sturge, Phys. Reo. 127, 768 (1962). H. Ehrenreich, Phys. Reu. la, 1951 (1960). l o A. F. Kravchenko and H. Y. Fan, Proc. Intern. Con$ Semicond. Phys., Exeter, 1962 p. 737. Inst. of Phys. and Phys. SOC.,London, 1962.

6

C. HILSUM

interband scattering the energy difference is 0.33 eV. If we accept the result of Fenner's work'' on GaAs-P alloys, that this energy increases with temperature increase, the value at room temperature would be near 0.4 eV. This is rather too high to be in good agreement with Spitzer's value. Mead and Spitzer" have obtained clear evidence of direct and indirect transitions on AlSb and AlAs. For AlSb both the indirect transitions rI5- X , and rl,- L , require less energy than the rI5- rl transition, and so can be detected in the photovoltage measurements. We thus have a good picture of the band structure of GaSb, GaAs, and AlSb, and of GaP along the [lo01 direction. It is revealing to compare the shapes of the valence bands of these materials, and we do this by tabulating the energies the X , and L , maxima are below the r15 maximum. We include for comparison the values for Ge and Si and cr-Sn. TABLE I ENERGIES OF THE TWO VALENCE BAND AT x AND L BELOW r VALENCE MAXIMA BAND MAXIMUM. VALUESAPPROPRIATE TO 77°K Material

X5

GaSb GaAs

3.1 eV 3.3 3.0 2.9 3.3 3.2

GaP AlSb Ge Si a-Sn

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L3

1.2 eV 1.0 -

0.9 1.2 1.2 1.2

It is clear that the valence bands of all these materials are very similar. Cardona and G r e e n a ~ a y have ' ~ previously pointed out that the L , - rl energy difference at 300°K is similar for a number of zinc-blende semiconductors, but the low temperature values are in rather better agreement. The similarity of the X , - r,5energies is even more apparent, but this does not appear to have been reported. If we assume from this evidence that the main features of the valence band are determined by the crystal and the nature of the

' G. E. Fenner, Phys. Ret.. 134, A1 113 (1964). 12

C. A. Mead and W. G. Spitzer, Phys. Reti. Letters 11, 507 (1963). M. Cardona and D. L. Greenaway, Phys. Rev. 125, 1291 (1962). '"There is insufficient evidence to broaden this to include 11-VI compounds. The data for ZnTe confirm the rule, but HgTe and HgSe appear to contradict it.

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1.

SOME KEY FEATURES OF 111-V COMPOUNDS

7

atoms simply perturbs this slightly, we can make an estimate of the indirect energy gaps in the remaining compounds. The final list of energy gaps is given in Table TI. The figures in brackets are estimates based on the invariant band, taking rI5- X , as 3.2 eV and rI5- L , as 1.1 eV. TABLE I1 ENERGIES ABOVE rI5VALENCE BANDMAXIMUM OF THREE PRINCIPAL CONDUCTION BANDMINIMA(77°K) Material GaSb GaAs GaP AlSb AlAs AIP InSb InAs InP

rl 0.80 eV 1.52 2.88 2.20 3.00 >4 0.22 0.43 1.40

x, 1.2 eV 1.85 2.32 1.65 2.25 3 (1.0)

(1.6) (1.8)

L,

0.88 eV 2.0 (2.7) 1.95 3 -

(0.81 (1.5) (2.1)

These data can be used to predict the properties of alloys of 111-V compounds. As an example we can attempt to find the alloy with the maximum direct energy gap. Experiments on this topic have been restricted to the GaAs-GaP system, but we see from Fig. 2 that the InP-GaP system is marginally better. The AlAs-GaAs system appears to be slightly less favorable than the other two systems illustrated. It is quite likely that the InPAIP system is the best, with an energy gap of 2.1 eV with 25% AIP, but the determinations for AlP are rather imprecise, and it is not known if the system is practicable. All such speculations are based on linear extrapolations of energy transitions throughout the alloy system, and before we accept the results we should carefully scrutinize the evidence on this point. Much of this evidence is summarized by Long in Chapter 5 of this volume. The systems on which we have the most data are those of Si-Ge, and GaAs-Gap. Here it is clear that there is no detectable departure from linearity. There are several systems where the lowest energy transition throughout the composition range is a direct transition at the center of the zone, and of these the InAs-InP system seems to show a linear dependence, whereas the InSb-GaSb and InAs-GaAs systems show a slight departure from linearity. The InAs-InSb system has a minimum energy gap for the

8

C . HILSUM

AIAs-GaAs SYSTEM COMPOSITION 30% AlAs 1.96 e V ENERGY GAP

GoP-Gabs 36% GaP 1.99 eV

G a p - InP 41% GaP 2.01 eV

alloy made up of equal parts of the two compound^'^ (Fig. 3). This type of behavior cannot be considered exceptional because it has also been observed for ZnSe-ZnTeI5 and InAs-CdSnAs,. l 6 We must conclude that, though the linear dependence of energy transition on composition is fairly common, an appreciable proportion of alloy systems do not behave in this way. Linear extrapolation is simple, and will normally be used when there is no experimental evidence to guide us. But we should remember that the method is based on convenience rather than proof, and weigh the predictions accordingly.

III. Effective Masses The electron effective masses are now well established for most of the

111-V compounds. The values for InSb and GaSb were accurately determined some years ago, and there now seems general agreement on the value for InAs. The results for GaAs have gradually decreased over the years, and the latest Russian” determinations, 0.068- are 5 % lower than those previously quoted. Russian work” also shows that the effective electron mass in InP is lower than that in GaAs, in contrast with earlier

’* J. C. Woolley and J. Warner, Can. J. Phys. 42, 1879 (1964). S. Larach, R. E. Shrader, and C. T. Stocker, Phys. Reu. loS, 587 (1957). S . Mamaev, D. N. Nasledov, and V. V. Galavanov, Fiz. Tuerd. Tela 3, 3405 (1961) [English Transl. : Soviet Phys.-Solid State 3, 2413 (1962)l. Yu. I. Ukhanov and Yu. V. Mal’tsev, Fiz. Tuerd. T e f a5,2926 (1963) [English Transl.: Soviet Phys.-Solid State 5, 2144 (1964)l. 18F. P. Kesamanly, E. E. Klotyn’sh, Yu. V. Mal’tsev, D. N. Nasledov, and Yu. I. Ukhanov, Fiz. Tuerd. Tela 6, 134 (1964) [English Transl.: Soviet Phys.-Solid State 6, 109 (1964)l.

l5 l6

1. SOME KEY FEATURES OF

111-V

COMPOUNDS

9

0 lnAs

lnSb

FIG.3. Energy gap at 300°K for alloys of InAs-InSb. (After Woolley and Warner, Ref. 14.)

results. Values for GaP and A1Sbl9 obtained from the Faraday effect are now available, but we should note that these are values for the X minimum, and not for the r minimum as for the other compounds. For GaSb the effective mass in the L minimum can be deduced from the dependence of Hall coefficient on temperature, and for GaAs a similar procedure gives the mass in the X minimum. The masses in the r minimum for AlSb and GaP have been calculated by Ehrenreich” from Kane’s theory. All these masses are given in Table I11 together with hole masses. TABLE 111 EFFECTIVE MASSESOF CARRIERS IN 111-V COMPOUNDS Compound lnSb InAs InP GaSb GaAs GaP AlSb

Electron mass

0.014m 0.022 0.067

0.047

0.09 l9

2o

(r)

0.36 ( L ) 0.068 0.35 (X) 0.12 (r) 0.39 (X)

(r)

Hole mass 0.4m 0.41

0.23 0.5

0.4

T. S. Moss, A. K. Walton, and B. Ellis, Proc. Intern. Conf: Semicond. Phys., Exeter, 1962 p. 295. Inst. of Phys. and Phys. SOC., London, 1962. H. Ehrenreich, J . Appl. Phys. Suppl. 32, 2155 (1961).

10

C . HILSUM

The heavy hole effective masses have not been established with much accuracy. For InAs the value generally adopted is 0.41q deduced by Matossi and Stern2 from infrared absorption measurements, and a similar calculation2* for InSb gives 0.4m. This latter value is in poor agreement with the first estimate for hole mass in InSb, 0.189 obtained by cyclotron resonance, but is confirmed by the latest cyclotron resonance determinations, for Bagguley et also find the hole mass to be 0.4m. Zwerdling et d Z 4 deduced a value of 0.18m from magnetoabsorption. The larger value is in good agreement with the hole mass deduced from the known intrinsic carrier concentration at room temperature, and we believe it to be the better determination. The hole mass in AlSb has been found from the dependence of mobility on impurity concentration2’ and that for GaAs from measurements of thermoelectric power.26 Theoretical calculations9 for GaAs give a slightly higher value. The only values reported for hole mass in GaSb are in poor agreement with each other. Leifer and DunlapZ7 deduced a value of 0.39m from Hall coefficient and resistivity data, and Ramdas and Fan28 found m* to be 0.23~1from an analysis of infrared reflectivity measurements. The first calculation is certainly not accurate, because it assumed GaSb has a simple conduction band. For this reason we give the smaller value, but we feel it must be treated with caution. The measurement was made on a single impure sample. We note that the two well-established determinations of hole mass are near 0 . 4 9 and none of the values are very different from this. We have found already that the valence band has much the same shape in all the compounds, so where no determination of hole mass has been reported, we assume it to be 0.4m. IV. Effective Charge

One of the fundamental differences between the 111-V compounds and the elemental semiconductors is the crystal binding. In the 1950’s there was much discussion about. the nature of the binding in the compounds, F. Matossi and F. Stern, Phys. Rev. ill, 472 (1958). F. Stern, Proc. Intern. Con$ Semicond. Phys., Prague, I960 p. 363. Czech. Acad. Sci.. Prague, 1961. 2 3 D. M. Bagguley, M. L. Robinson, and R. A. Stradling, Phys. Letters 6, 143 (1963). 24 S. Zwerdling, W. H. Kleiner, and J. P. Theriaulf Proc. Intern. Car$ Semicond. Phys., Exeter, 1962 p. 455. Inst. of Phys. and Phys. Soc., London, 1962. 2 5 F. J. Reid and R. K. Willardson, J . Electron. Control 5, 54 (1958). 2 6 J. T. Edmond, R. F. Broom, and F. A. Cunnell, Rept. Meeting Semicond., Rugby. 1956 p. 109. Phys. Soc., London. 1956. H. N. Leifer and W. C . Dunlap, Phys. Rev. 95, 51 (1954). A. K. Ramdas and H. Y. Fan, Bull. Am. Phys. SOC. 3, 121 (1958). 21

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1. SOME KEY FEATURES OF

111-V

COMPOUNDS

11

and several workers tried to decide if the binding were stronger or weaker than in germanium and silicon. None of this need concern us here. We are mainly interested in establishing the charge difference between the 111 atom with its surrounding electron cloud and the V atom with its cloud, because this plays a major role in determining the transport properties. The simple picture we can use, say for GaAs, starts with the Ga3+ atom and its three valence electrons in proximity to the As5+ atom and five valence electrons. If no charge transfer took place the bonding would be neutral. The eight valence electrons could, however, all congregate around the As atom, forming Ga3+and As3- ions. The crystal would be ionic. A third possibility is for one electron to pass from the As atom to the Ga atom. Tetrahedral bonds could be formed as in germanium, and we would say the crystal binding was covalent. It is important to realize that covalence of the bond does not imply neutrality of the atoms. A charge transfer of one electron is large, and such a covalent compound would behave in a number of ways exactly like a strongly ionic crystal. We can give a semblance of a numerical scale to the charge transfer by postulating that we can arbitrarily divide the crystal, with each 111 atom and an electron cloud in a sphere, and the V atom and an electron cloud in a neighboring sphere. We can then define an effective charge e*, as the total charge within the 111 atom sphere. Ideally e* would be zero for neutral binding, - e for covalent binding, and + 3e for ionic binding. There are two objections to this procedure. It assumes that there will be some natural boundary between the two spheres so that it is clear where one sphere ends and the other begins. Though this might well be the case for an ionic crystal, where the electrons would probably cluster closely around one of the atoms, so that the position of the boundary would be easy to determine, it might not be so simple when the binding is covalent. The common picture of covalent binding conceives an electron pair bridging the gap between nearest unlike atoms, and obviously the effective charge would then depend on the exact position chosen for the boundary. This very real objection has lost some force recently, for X-ray diffraction experiments have shown that the electron clouds are quite closely concentrated around the atoms (Fig. 4). The second objection is that the model we are using is a static one. Any attempt to measure the effective charge must distort the electron distribution so that the dynamic effective charge is not necessarily the same as the static one, and may be quite different during different types of experiment. The simplest method of measuring effective charge uses far infrared reflectivity. The shape and position of the reflectivity maximum give the low frequency and high frequency dielectric constants and the transverse optical mode frequency. The square of the effective charge can then be

12

C. HILSUM

Calculated. But it is not possible to obtain from this experiment the sign of the effective charge, i.e., to decide in which direction the charge transfer takes place. In fact, there are very few experiments which reveal this sign, and we might well conclude that for most physical effects it is immaterial

Sb

\

\

In SITE 3.45e Sb S I T E 4.55e

InSb A

DISTANCE A

FIG.4. Approximate representation of electron density distributions along I 1 111 directions in InAs, GaAs, and InSb. (InAs and GaAs after Sirota and Olekhnovich, Refs. 30 and 31 ; InSb after Attard and Azaroff, Ref. 29.)

which way the charge is transferred. The important thing is the of the dipole. The X-ray diffraction experiment gives both the and sign of e*, and Attard and AzaroffZ9 have calculated it to for InSb. Sirota and O l e k h n o ~ i c h ~ have ~ * ~obtained ~ results

magnitude magnitude be -0.45e for charge

A. E. Attard and L. V. Azaroff, J . Appl. Phys. 34, 774 (1963). N. Sirota and N. M. Olekhnovich, Dokl. Akad. Nauk S S S R 136, 660 (1961) [ En glish Transl.: Proc. Acad. Sci. U S S R , Phys. Chem. Sect. 136, 97 (1961)]. 3 1 N. N. Sirota and N. M. Olekhnovich, Dokl. A h d . Nauk S S S R 136, 879 (1961) IEnglish Transl.: Proc. Acad. Sci. U S S R , Phys. Chem. Sect. 136, 137 (1961)J

29

'' N.

1.

SOME KEY FEATURES OF 111-V

COMPOUNDS

13

distribution on InAs and GaAs, but do not quote a value for e*. It appears from their results, however, that e* is negative for these compounds too. The piezoelectric effect should in principle give the magnitude and sign of e*, but it can only be performed with accuracy on high resistivity semiconductors. Semi-insulating GaAs has a high resistivity, and two measurements have been made on it. H a m b l e t ~ nmeasured ~~ the dc piezoelectric effect, and found that e* was negative. Zerbst and B ~ r o f € k amade ~ ~ an ac measurement and obtained a value for e* of -0.51e. The values obtained for e* are tabulated in Table IV. It is interesting to note that the values obtained by piezoelectric effect and X-ray diffraction are in good agreement with the infrared reflectivity values.34*3 5 TABLE IV EFFECTIVE CHARGES OF 111-V COMPOUNDS Effective charge Compound From reflection (sign indeterminate)

GaSb InSb GaAs AlSb InAs GaP InP

From other methods

0.33e 0.42 0.48 0.48

- 0.45e - 0.5 1

0.56 0.58 0.68

The negative sign for e* indicates that negative charge is being passed from the V atom to the 111 atom. It might be thought that the natural way for the charge to pass is actually in the opposite direction, since the V atoms are more electronegative than the I11 atoms, and should therefore attract electrons. However, P a ~ l i n gshows ~ ~ that the partially ionic character of the covalent bond caused the 111 atom to be charged positively. Electron transfer reduces this charge, and partially restores neutrality. K. Hambleton (private communication). M. Zerbst and H. BorolTka, Z . Naturforsch. 18, 642 (1963). 34 M. Hass and B. W. Henvis, J . Phys. Chem. Solids 23, 1099 (1962). 35 C. Hilsum, P r o p . Semicond. 9, 135 (1965). 36 L. Pauling, “The Nature of the Chemical Bond,” p. 432. Oxford Univ. Press, London and New York, 1960. 32

33

14

C . HILSUM

In the past, several authors20*37-39 have attempted to predict spinorbit splitting in 111-V compounds by averaging between the splitting of the two elements which make up the compound. The average is based on the time a valence electron spends on each site, which is, of course, a measure of e*. The time an electron spends on the I11 atom is given by (3 -e*)/8, and the various estimates for this have ranged from 30 to 37%. The implicit assumption is that e* is positive, whereas we know that this is not so. The correct value is between 42 and 46%, depending on the particular compound, and it is interesting to see how this more accurate estimate affects the predicted value of spin-orbit splitting. Recently Herman and co-workers4' have given values of atomic spin-orbit splitting for a TABLE V SPIN-ORBIT SPLITTING PREDICTION

Predicted Ao" Experimental A.

Compound

1 ~~

AlSb GaSb GaAs GaP InSb InAs InP

~~

0.56 eV 0.63 0.3 1 0.12 0.75 0.44 0.25

2 ~~

0.66 eV 0.71 0.34 0.11 0.80 0.44 0.20

+

0.60 eV 0.70 0.35 0.13 0.81 0.43 0.2 1

1. Time-sharing: A" = A,1,(3 - e*)/8 Av(5 + e*)/8. 2. Variation across periodic table (Herman4'): A, = A 111 .-;+ A"

'3.

number of elements, together with the corrections to be used when the elements are incorporated into a solid. Herman also predicts the splitting in 111-V compounds by an average for the two elements, but he chooses his average as an increase in contribution with progression across the periodic table. The question of charge transfer does not arise, and .is not included. A comparison between the spin-orbit predictions by the two methods is shown in Table V. In nearly all cases, Herman's prediction is the J. C. Phillips, Proc. Intern. Con$ Semicond. Phys., Prague, 1960 p. 405. Czech. Acad. Sci., Prague, 1961. 3 8 M. Cardona, J . Appl. Phys. Suppl. 32, 2151 (1961). 3 9 J. Tauc, Proc. Intern. Conj: Semicond. Phys., Exeter, 1962 p. 333. Inst. of Phys. and Phys. SOC., London, 1962. 40 F. Herman, C. Kuglin, K. Cuff, and R. Kortum, Phys. Rev. Letters 11, 541 (1963). 37

1. SOME KEY FEATURES OF 111-V COMPOUNDS

15

better, and we conclude that the agreement with experiment observed by previous workers on the basis of time sharing is fortuitous. V. Transport Properties

Those 111-V compounds which have a direct energy gap have a low electron effective mass, and we can expect them to have a high electron mobility. The limit to mobility is likely to be set by polar scattering because all the compounds have an appreciable effective charge. For electrons in compounds with an indirect gap, and for holes, the effective mass is much larger, and acoustic scattering which leads to a mobility proportional to (m*)-5 / 2 , becomes relatively more important than polar scattering, which gives an (wI*)-~’’mobility dependence. However, none of the indirect gap materials show an electron mobility as high as that of Si, and we anticipate that even in these high mass materials polar scattering will be dominant. Similar considerations apply to hole mobilities except for GaSb. This compound has only a small effective charge, and polar scattering is therefore not as strong as in the other compounds. Calculations show that for holes acoustic scattering will have an appreciable effect at room temperature, and may be the dominant process at slightly higher temperatures. Calculations were made some years ago of the polar mobility in the 111-V compound^.^^ Better values of the physical constants are now available, and a recalculation has been made (Table VI). The agreement with experiment is very good in most cases. We should note that the standard relation for polar mobility is used in these calculations and the value given for InSb must be modified to include effects due to carrier screening and the nonparabolicity of the conduction band. When this is done the agreement with experiment is good. The observed electron mobility in AlSb is low, but pure n-type material has never been prepared. This is also true of p-type InP and Gap. As we have noted, in GaSb, acoustic scattering of holes cannot be ignored, and if we include an allowance for this, the predicted mobility would be much nearer the observed value. The only major disagreement is for electrons in GaSb. Apparently the limit here is not set by polar scattering. Ehrenreich” has estimated that interband scattering has the effect of halving the calculated mobility, but this is still not enough to explain the low values observed. It is certainly possible that defects and compensated impurities are reducing the mobility more than we expect, but it is more likely that interband scattering is playing a larger role than Ehrenreich suggests, and is the dominant scattering process. 41

C. Hilsum, Proc. Phys. SOC.(London) 76, 414 (1960).

16

C. HILSUM

TABLE VI PREDICTED AND OBSERVED MOBILITIES FOR 111-V COMPOUNDS”

Compound

ec*

InSb

0.16e

InAs

0.22

InP

0.27

GaSb

0.13

GaAs

0.19

GaP

0.24

AlSb

0.19

Calculated polar mobilities (cm2/v-sec)

130,000 (electron) 850 (hole)

38,000 470 6,600 460 33,000 3,000 10,400 520 750 720 860 830

Observed mobilities (cm’iv-sec)

78,000 750 33,000 460 4,600 150 4,000 1,400 8,800 400 300 100 200 550

“The second column shows the Callen effective charge deduced from the Szigeti effective charge given in Table IV.

Though the mobility in the rl conduction band minimum in GaAs is in accord with simple theory, it has been suggested that the mobility in the X , minima is extremely low. High pressure resistivity measurement^^^ show that this mobility is a factor of 600 less than the mobility in the rl minimum, i.e., it is less than 10 cm2/V-sec. Under the pressures used for these experiments the band structure of GaAs is similar to that of Gap, with the X , minimum several tenths of an electron volt below the rl minimum. However, the mobility in the X I minimum in GaP is well over 100 cm2/v-sec, though the impurity content is high. Moreover GaP is more polar than GaAs. We would therefore expect the corresponding mobility in GaAs to be 500 cm2/V-sec or more at atmospheric pressure. It is not clear why there should be a drastic decrease of mobility with pressure, and it is possible that the number of carriers is decreasing. Recent results by Sladek4j on Hall effect in GaAs at high pressure indicate that on some 42

”

W. Howard and W. Paul (unpublished results quoted in Ref. 20). R. J. Sladek, in “Physics of Semiconductors” (Proc. 7th Intern. Conf.), p. 545. Dunod, Paris, and Academic Press, New York, 1964.

1.

SOME KEY FEATURES OF 111-V COMPOUNDS

17

samples both the mobility and the number of carriers can decrease with pressure increase. If this were also the case for the samples used by Howard Ehrenreich's analysis would need modification. and

VI. Future Progress Our knowledge of the physics of a material must always be restricted by the technology available to us. Work on 111-V compounds excluding InSb has made only halting progress in recent years, because no new ideas in material preparation were very successful. Marginal improvements were

5

0

1

'

0

0

1

I

I 0

RESISTIVITY

(\X

-x.

tt

HALL

-I

10-2 1

x-x-x-x-

4

CONSTANT

12

81

0

16

3

1000

To K

FIG.5. Electrical properties of a GaAs epitaxial sample. (After Knight et a/.. Ref. 46.)

reported from time to time, resulting mainly from the use of purer starting materials or of more exotic crucibles. Any technique which makes it simpler to grow and zone-refine crystals containing a volatile component will obviously have some impact, and we note with interest a method of growing InAs single crystals used successfully by Mullin et ~ 1 . 4In~ the normal method of crystal growth the whole reaction tube is held above 500°C so that no arsenic escapes from the melt. In the new method the escape of arsenic is frustrated by a pool of molten boric oxide which floats on the melt. A single crystal can be pulled by the Czochralski technique 44

J. B. Mullin, B. W. Straughan, and W. S. Brickell, J . Phys. Chem. Solids, 26, 782 (1965).

18

C. HILSUM

on the end of a seed which is dipped through the boric oxide. If this technique is used the walls of the apparatus can be kept at room temperature, and the volatile impurities which penetrate the oxide can be swept away in a gas stream. The same technique has been used with some success on GaAs, but here boric oxide may not be the best suppressant. Another useful technique which has recently been reported involves heat treatment of a grown crystal of GaAs. The material is prepared by the horizontal Bridgeman process in an atmosphere of 1 atm of arsenic vapor and 0.1 atm of Ga,O vapor.45The material is then semi-insulating. Heat treatment at about 700°C removes acceptors (presumably defects) and lowers the resistivity. Electron mobilities as high as 38,000 cm2/volt sec at 77°K have been measured on crystals prepared in this way. Vapor growth techniques have always shown promise for 111-V compounds, but early progress was slow. Recently, however, Knight et have obtained excellent results with the reproducible deposition of GaAs epitaxial layers, with electrical properties considerably better than those of pulled crystals4' The total impurity concentrations of these layers were in the low l O I 5 cm-3 range, and the room temperature mobilities as high as 8800 cm2/volt sec. At low temperatures the mobility rises to 55,000 cm2/volt sec. The characteristics of a good layer are shown in Fig. 5. Further work by Whitaker and B01ger~~ using a similar vapor phase technique produced layers with mobilities up to 110,000 cm2/volt sec at 56"K, and total impurity concentration near 6 x 10'4cm-3. This epitaxial material is much purer than any GaAs reported previously, and it opens the way to work on the physics of this material unhampered by the high impurity content which has previously limited our investigations.

ACKNOWLEDGMENTS I am grateful to T. McLean, N. March E. Paige, and W. Paul for valuable discussions, and to K. G. Hambleton, J. R. Knight, and J. B. Mullin for the communication of unpublished

results. J. M. Woodall and J. F. Woods, Solid State Commun. 4, 33 (1966). J. R . Knight, D. Effer, and P. R. Evans, Solid-state Electron. 8,178 (1965); D. Effer, J . Electrocheni Soc. 112, 1020 (1965). 4 7 W. K. Liebman and G. Kampschulte, Solid-State Electron. 9, 828, (1966). 48 J. Whitaker and D. E. Bolger, S o M State Commun. 4, 181 (1966). 45

46

Band Structure

This Page Intentionally Left Blank

CHAPTER 2

Methods of Band Calculations Applicable to 111-V Compounds* Franco Bassanit I . INTRODUCTION

. . . . . . . . . . . . . . . . 21

11. CLASSIFICATION OF THE ELECTRONIC STATES 1 . Generol Discussion . . . . . .

. . . . . . . . . . . . . . . .

23 23 24 25 28

. . . . . . .

29 29 31 32 31

2 . States of ihe Simple Group . . . . . . 3 . Siates of the Double Group and Time Reversal 4 . Comparison with Diamond Latiice . . . .

111. SYMMETRY FUNCTIONSAND MATRIXELEWNTS

5. General Procedure . . . . . . . . 6 . Symmetrized Combinations of Bloch Functions 7 . Symmetrized Combinations of Plane Waves . 8 . Matrix Elements . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

IV. APPROACHES TO THE CALCULATION OF THE ELECTRONIC STATES 9 . Introductory Remarks . . . . . . . . . . . 10. Tight-Binding Approach . . . . . . . . . . . I I . Orthogonalized Plane Wave Approach . . . . . .

. . . .

. . . .

. . . . . .

. .

46 46 52 55

V . QUALITATIVE FEATURES OF THE BAND STRUCTUREAND PRELIMINARY CALCULATIONS . . . . . . . . . . . . . . . . 61 12. Tight-Binding Approach . . . . . . . . . . . . . 61 13. O P W Approach . . . . . . . . . . . . . . . 64 14. Spin-Orbit Splitting . . . . . . . . . . . . . . 10

VI . CONCLUSIONS AND LINWOF FUTURE PROORESS .

. . . . . .

13

.

I Introduction The available evidence indicates that all the electronic properties of solids can be explained by quantum mechanics when one takes into account the Coulomb interactions among the nuclei and the electrons of the

* Based on work supported in part by the U.S. Atomic Energy Commission and in part by the Consiglio Nazionale delle Ricerche.

t On leave of absence from Argonne National Laboratory to the University of Messina while the work was being completed.

21

22

FRANC0 BASSANI

crystal. Solutions of the general equations are impossible to obtain, but one can adopt simplified models within the general framework and obtain physically significant results. First of all one uses the Born-Oppenheimer approximation : the motion of the nuclei is separated from the motion of the electrons and the electronic states are defined by keeping the nuclei in their lattice positions and considering only the wave functions of the electrons. A many-electron problem results, which still cannot be solved. The independent particle model is then commonly adopted because of its success in atomic and molecular problems. It consists in separating the electronic wave functions in products of one-electron wave functions. Hartree-Fock equations for the electrons of the crystal result. The eigenvalues are occupied by the available electrons up to the Fermi level, and the higher states are available for extra electrons and optical transitions. Since the self-consistent Hartree-Fock equations are still too difficult to solve one must introduce further simplifications to obtain numerical results. The “band approximation” consists in constructing a model crystal potential which can be written as the sum of atomic-like potentials at the lattice sites; this potential is the same for all the electrons of the crystal, and the eigenvalues of the Schrodinger equation which is obtained are the energy bands Edk) where n is a band index and k is the wave vector. The various methods of band calculations which have been suggested differ in the model potential adopted and in the procedure which is followed for solving the Schrodinger equation. In spite of the approximations involved, physically significant results are obtained with the band approximation, and very often the numerical results can be compared with experimental data on the cohesive energy, the optical properties, and the transport properties. We believe this is due to the fact that the general features of the electronic states are determined by the symmetry of the lattice and can be brought out in every case by using a suitable method of band calculations and a reasonable crystal potential. The corrections introduced by the requirement of self-consistency, by correlation effects, and by relativistic effects may be of importance in some cases and will generally improve the agreement with experimental data but are not expected to alter drastically the over-all band structure. In the present paper we will try to expose the qualitative features of the band structure of the 111-V compounds as they result from applying the tight-binding approach and the orthogonalized plane wave approach within the “band approximation.” To make the most use of symmetry we shall confine the detailed analysis to three points of the reduced zone. An extension to the other points can be made along the same lines. The detailed calculations which have been performed to date are also reviewed ;

2.

METHODS OF BAND CALCULATIONS

23

they are still of a preliminary nature and quantitative improvements are likely to come shortly. In Part I1 we indicate how the electronic states of the zinc blende crystal can be defined from the symmetry of the lattice. In Part I11 we indicate how to derive symmetry-adapted basic sets of functions and expressions for the matrix elements between them. In Part IV we describe the methods for the calculation of the energy bands which seem best suited to the 111-V compounds, namely, the tightbinding approach and the orthogonalized plane waves approach. The pseudopotential approximation and the perturbation approach to the empty lattice are also described. In Part V we present a qualitative analysis of the band structure and refer to the numerical results so far obtained. In Part VI we indicate possible improvements on the calculation of the band structure. No attempt is made to give complete references or to review the general methods of band theory; we confine ourselves as far as possible directly to 111-V compounds.

11. Classification of the Electronic States 1 . GENERAL DISCUSSION

The zinc blende lattice for the 111-V compounds is constituted by two fcc sublattices, formed with atoms of group V and group 111 elements, respectively. In the following we will call the first type A and the second B and refer in general to a compound with the zinc blende structure. The two sublattices can be defined by the basic translational vectors: a/2 (1, 1,0), a/2 (0,1, l), a/2 (1,0,1) and are displaced by a vector d = 4 4 (1,1,1) with respect to one another. Since the translational symmetry of the lattice is that of the fcc sublattices, the values of the wave vector k on which energies and wave functions depend can be defined within a volume of the reciprocal space which is obtained by bisecting with perpendicular planes the reciprocal lattice of basic vectors 2n/a (1, 1, I), 2n/a (1,1, l), 2n/a (1, T, 1). The reduced zone (1st Brillouin zone) is displayed in Fig. 1, and the symmetry points and symmetry lines are indicated. Besides the translations, the other symmetry operations which transform the zinc blende lattice into itself are the 24 operations of the group &. At every value of k the operations of & which send the vector k into an equivalent vector form the small group G(k), and the irreducible representations of G(k) permit a classification of the electronic states at every value of k. The electronic states are represented by wave functions which are

24

FRANC0 BASSANI

t'

/

FIG.1. Reduced zone for the zinc blende lattice. Standard notations are used for the symmetry points and the symmetry lines. Only half the points on the surface are to be considered, the others being obtained by the addition of reciprocal lattice vectors. Points which can be transformed one into the other by symmetry operations of the lattice have the same energy spectrum. For instance there are three X-points and four L-points in the reduced zone.

partner functions in such irreducible representations. These general concepts are described in the fundamental paper by Bouckaert et a/.' ; the irreducible representations of G(k) for the zinc blende lattice at all points of interest in the reduced zone have been derived by Parmenter2 and by Dres~eihaus.~ 2. STATESOF

THE

SIMPLE GROUP

The symmetry properties of the lattice are discussed by Koster? but for convenience we report in Table I the symmetry operations of the of the zinc blende lattice. Sometimes it is convenient to point group choose a lattice point as the origin, but in most cases it is more convenient to choose the midpoint between atoms A and B. In Table I we define the operations of symmetry with both choices of origin. In Table I1 we report the characters of the irreducible representations of G(k) at the points T[k = 01, L[k = 2x/u (i,i,i)] and X[k = 2x/a (1,0, O)] and along the lines A and A which connect r with L and with X . The compatibility relations between the irreducible representations are also given. L. P. Bouckaert, S. Smoluchowski, and E. Wigner, Phys. Rev. 50, 58 (1936). R. H. Parmenter, Phys. Rev. 100, 573 (1955). G. Dresselhaus, Phys. Rev. 100, 580 (1955). G. F. Koster, Solid State Phys. 5, 173 (1957).

2.

TABLE I SYMMETRY OPERATIONS OF THE POINTGROUPT,

Class

c42

Operation R defined for the point origin

{SZ,lO)

f

j

x x

j z y i

{IS,’ 10) {164$}

{I&:lO} {164$’} {I&; 10) {I~~YIO} ICZ

ZINCBLENDELATTICE’

z

y X l j x z -

Conventional origin {R]T}-’ - r

X

f

a14

x

P

+ aJ4

j j

f a14

+ a14 y

z Z f a14 i aJ4

+

y

P -k a14 i i- uJ4 j + u/4 x Z f uJ4 z j n/4 f i- aJ4 E a14 i a14 y Z i- a14 j f a14 x z j a14 X a14

+

X

Z

Y

x i z

z j j

y x i

{ISZY*lO}

j i x

i z y x i j

{I~zx#}

Y

X

Z

j i- a14 2 a14 x Y

{Ibx$4

Z

Y

X

2

Y

X

{I&yElo~

X

Z

Y

X

2

Y

{&:yZIO}

Z

X

Y

Z

X

Y

{&XYZIOI

Y

Z

X

Y

Z

X

z

x

j

z

j

Z

X

{I~ZXYIOI ~ I L Z l O }

c3

OF THE

Point origin R-‘r

{62XlO)

{SZYlOI IC4

25

METHODS OF BAND CALCULATIONS

{&&:lo}

{Ss:jilo) {63,ji10) {GY$} {&XYZlO}

z x y j z a i x j y z x

+

+

j+ a14

+ + i+ a14 y Z + a14

j f aJ4

X

Z

+

z

2

+ a/4

%. -k a14 j f aJ4 i f a14 x

+

f a14 .t a14 y j f aJ4 z X a14 i t a14 x j aJ4 y i -k a14 2 f a14

2

+ +

In the first column the class notation is given. In the second column the operations are indicated; for example I S 2 , indicates a rotation by 2n/2 about the axis whose direction cosines on the three axes are in the ratio 0: 1 :f, multiplied by the inversion. In the third and fourth columns the coordinate transformations are given for the origin at a lattice point and the origin at the midpoint between the two sublattices, respectively.

3.

STATES OF THE

DOUBLE GROUPAND TIMEREVERSAL

In the presence of spin-orbit interaction the symmetry operations of the small group G(k) are increased by the addition of a new operation Q, which is a rotation by 2x and has the effect of changing the sign of the spin functions. One obtains a “double group” G(k), whose irreducible representations for the zinc blende lattice at all points of interest are derived and discussed by Parmenter’ and Dre~selhaus.~ For convenience we give in Table 111

26

FRANC0 BASSANI

TABLE I1 IRREDUCIBLE REPRESENTATIONS OF THE SIMPLE GROUPG(k) r, A, L, A. and X"

Point

I E

r

rl r12

rrs r2 5

Point A or L

I

E

61C4

61C,

8C3

1 1 2

1 -1 0 -1 1

1

-1 0 1 -1

1 1 -1 0 0

-1 -1

I

162,3, 16,,,, 162y3 63xyr,6&

1 1 1 1

1

1 1

-1

-1 -1

1

rl r2

1 -1 -1

-1

Compatibility relations

Point X Xl x2 x3

x4 x,

~~

E 1

1 1 1 2

A1 A2

XI

r12 A , + A2 r15 A 1 + A 3 + A 4

1

Tzs Az ~

POINTS

3c42

1

1 2 3 3

r2

AT THE

+ A3 +

x,

X3 X4 A4 X, A3

A1 A2 A, A2

+ A4

~

62,

1 1 1 1

-2

a,

62, 1

1 -1

-1 0

16;; Id,,

1

I62YJ62,Z

1

-1

-1

-1 1

1 -1

0

0

"The compatibility relations at the points r and X are also given. The irreducible representations A3 and A4 are degenerate by time reversal.

the irreducible representations of the double group at the points r, L, and X, and along the symmetry lines r - L and r - X (A and A, respectively). Some of the states of the simple group split in the presence of spin-orbit interaction in a way which can be easily derived by considering the product of the irreducible representations of the simple group G(k)by the representation of the spin functions D(l/');the representations which result will decompose into the sum of irreducible representations of the double group. Such decomposition is indicated in Table IV for the points reported in the previous tables.

TABLE 111 ADDITIONAL IRREDUCIBLEREPRESENTATIONS OF THE DOUBLE GROUP G(k) AT THE POINTS r, A, L, A , and X"

r

E

E

3c42,3c42

r6

2

-2

0

J

l-7

2

-2

0

-fi

l-8

4

-4

0

0

A or L

E

A,,

L4

1

As, Ls

1

A63 L6

2

A

E

A5

2

X

E

x6

2

-2

0

0

x 7

2

-2

0

0

61C4

c4

61

-Jz 0

61C2,61c2

8C3

8C3

0

1

-1

0

1

-1

0

-1

1

81

Compatibility relations

-J

-fi

2

0

Jz

0

"The compatibility relations at the points r and X are also given. The irreducible representationsL, and L5 are degenerate by time reversal, but

A, and A5 are not.

E3

Y

28

FRANC0 BASSANI

TABLE IV DECOMPOSITION OF THE

GROUPINTO DOUBLE GROUP

STATES OF THE SIMPLE

THE STATES OF THE

Another symmetry operation to be considered is the time reversal operation T ; it must be considered separately because it is not a linear operator. The effect of time reversal is to introduce additional degeneracies between otherwise separated irreducible representations. Wigner' has given the general rules for obtaining the additional degeneracies due to time reversal. Herring6 has shown how the rules can be applied to the case of the small group of the vector k in a lattice. A simple test can decide which irreducible representations are degenerate by the time reversal. For every irreducible representation of the small group of k one considers three cases depending on the sign of the quantity Xsx(S2), where 31 indicates the character and S the operation which sends k into -k + h, h being a vector of the reciprocal lattice. When spin is included, if the sum is positive (case a) the representation appears twice, if the sum is negative (case c) there are no additional degeneracies, if the sum is zero (case b) there is another irreducible representation which also belongs to case b and is degenerate with the first. If spin is not included, the conclusions drawn from case a and case c are exchanged. When there are no operations of the point group which send k into -k, case b is always obtained but there is no additional degeneracy between bands. Time reversal makes E(k) = E(-k). The application of Herring's test to the irreducible representations of Table I1 and Table I11 indicates that the states A3 and A4 of the simple group and L , and L, of the double group are the only states to be degenerate by time reversal. 4. COMPARISON WITH DIAMOND LATTICE A few remarks are in order about the results of the symmetry analysis on the zinc blende lattice. The basic difference between this lattice and the diamond lattice of the group IV semiconductors is the absence in this case ti

E. Wigner, Gottingen Nnchr. p. 546 (1932). C . Herring Phys. Rev. 52, 361 (1937).

2.

METHODS OF BAND CALCULATIONS

29

of the inversion symmetry. The point group of the diamond lattice is the product group of the point group of the zinc blende lattice times the parity group E, I. Consequently, at the points k where the inversion symmetry is an operation of the point group of the diamond lattice, all irreducible representations of the zinc blende lattice originate two irreducible representations of the diamond lattice of the same order, one even and the other odd under inversion (points r and L). Along line A, where the inversion symmetry does not change the group G(k), the irreducible representations are the same for the 111-V semiconductors as for the group IV elements. Along the line A the number of operations of G(k)is doubled in the diamond symmetry; each of the irreducible representations A1 and A, originates two onedimensional irreducible representations ; A3 and A4 correspond to the two-dimensional irreducible representation A, of the diamond lattice. A special case where a difference in the degeneracy occurs is the point X because in the diamond lattice a lattice translation belongs to G(k), whereas this is not the case for the zinc blende lattice. The irreducible representations XI and X , go into the two-dimensional irreducible representation X, of the diamond lattice; similarly X 2 and X, go into X 2 of the diamond lattice; the irreducible representation X 5 corresponds to both X , and X , of the diamond lattice. The notations of Herring,’ of Bouckaert et d.,’ and of Elliot’ are used for the diamond lattice. When we include spin-orbit interaction and time reversal we see that at a general k point the levels are singly degenerate in the zinc blende structure and the time reversal operation does not introduce additional degeneracy ; Kramers’ theorem is still satisfied in the sense that the eigenvalue at k is equal to the eigenvalue at -k. On the contrary, in the diamond structure the states are at least doubly degenerate by time reversal because of the inversion symmetry. The difference manifests itself in the A direction, where the irreducible representations A, and A5 of the zinc blencle structure are degenerate by time reversal in the diamond structure. At the point X the two two-dimensional irreducible representations of the double group in the zinc blende structure go into a four-dimensional irreducible representation of the diamond structure.’

III. Symmetry Functions and Matrix Elements 5. GENERAL PROCEDURE Every eigenfunction which is a solution of the Schrodinger equation in the zinc blende lattice will belong to a row of an irreducible representation

’C. Herring J. Franklin Inst. 233, 525 (1942); see also W. Doring and V. Zehler, Ann. Physik 13, 214 (1953).

* R. J. Elliot, Phys. Rev. 96, 280 (1954).

30

FRANC0 BASSANI

of the group G(k); the other independent eigenfunctions of the same eigenvalue will belong to the other rows of the same irreducible representation. If the functions are chosen to be orthonormal the irreducible representation is unitary. One sees that from the eigenfunctions one can generate an irreducible representation, and, vice versa, if one has a set of degenerate eigenfunctions one can choose appropriate combinations which belong to the rows of a given irreducible representation. Any such functions belonging to the irreducible representations a and p and to columns i and j satisfy the well-known relations :

independent of i. Equations (la) and (lb) apply to the exact eigenfunctions and to all symmetrized functions of a basic set. The exact eigenfunctions belonging to a given row of a given irreducible representation can be expanded in functions of the basic set which belong to the same row of the same irreducible representation. To construct symmetry-adapted basic functions is straightforward once one knows the irreducible representations of G(k). One sees how such functions transform under the operations R of the group, and one applies the result of group theory :

where W is any function of the set,ORW= W(R-'r), M;#7) is the matrix element of column i and row i in the irreducible representation a, and Sa,iis a symmetrized basic function. The basic functions can also be used to construct the matrix elements Myi(R) if they are not already known. To obtain all independent functions belonging to the irreducible representation a one makes use of the formula

where xa(R) is the character of the irreducible representation a. Then one chooses linear combinations of such functions which are orthonormal ; these are then partner functions in the unitary irreducible representation CL.All matrix elements M?'(R) can be obtained by considering the way such functions transform, and formula (2) can then be applied.

2.

31

METHODS OF BAND CALCULATIONS

6. SYMMETRIZED COMBINATIONS OF BLOCHFUNCTIONS

The simplest and physically most significant basic set of functions is obtained from Bloch sums of atomic orbitals of the type 1

@,(kr) = -1exp[ik J

N

- (R, + s)]u,(r

- R, - s),

(4)

V

where c labels an atomic state, R, is a lattice translation, and s is the vector which gives the position of the atom in the unit cell. The result of a symmetry operation on a Bloch function (4)can be obtained from the transformation properties of atomic orbitals; by definition, we have

This equation can be written in a more convenient form by substituting for {KIT)- the explicit expression {K'J - R-'T}, and by multiplying both terms of the dot product by the pure rotation R. We obtain

1

x -

4% v

exp[iRk -(R,

+ t)] u,[R-'(r

- R,

- t)],

(5b)

where t indicates a lattice point in the unit cell, which is defined by (R(.t}s= t + R,. Equation (5b) proves that the Bloch functions corresponding to degenerate atomic orbitals transform into one another when {Rlz} belongs to G(k); the symmetrized Bloch functions can then be obtained by applying formulas (2) and (Sb). In the zinc blende lattice there are two sets of functions of this type, one on each of the two fcc sublattices. Using the method of Section 5, it is possible to obtain symmetrized combinations of Bloch functions at all the symmetry points. For example, let us consider at the point r the functions involving atomic states s, p , and d. Let us label the two sublattices with the two types of atoms A and B. From the s atomic functions we have the two Bloch sums

32

FRANC0 BASSANI

every symmetry operation sends I ) ~ ~ ,and , 1,9:,~,~ into themselves so that both belong to the irreducible representation rl.From the p atomic functions we have six Bloch sums, three of them on lattice A and three on lattice B. One can choose linear combinations of the type

Ph)

= un,1,o(r).

Because of the properties of spherical harmonics, p x , p,,, and p , transform as x, y, and z under symmetry operations. Consequently the first three are and the last three partner functions of the irreducible representation rI5, on sublattice B are also partner functions in the same irreducible repreFrom the d atomic functions one can form combinations sentation r15. which transform as x2 - y2, 3z2 - r2, xy, xz, yz on each sublattice. The first two combinations on each sublattice transform as r I 2the , last three combinations are partner functions of the irreducible representation In this sense one often names the crystal states after the atomic functions of smaller angular momentum which belong to them; for instance, the T1 is an s-like state, rI5is a p-like state, and rI2, rZ5 are d-like states. If the two sublattices have the same kind of atoms (A = B) we go back to the case of the diamond lattice by considering the symmetric and the antisymmetric combinations of functions on sublattices A and B. These now belong to the symmetric and antisymmetric irreducible representations. The construction of symmetrized functions for the zinc blende lattice up to functions with I = 4 has been given by many authors9; we report in Table V the symmetrized combinations for the points r, X , and L of the Brillouin zone. 7. SYMMETRIZED COMBINATIONS OF PLANEWAVES A complete basic set of great importance for expanding valence and conduction states is the set of plane waves. For any value of the wave number k this set consists of all plane waves with wave number k + h, where h indicates a reciprocal lattice vector. In fact, because of Bloch’s theorem all eigenfunctions can be written Y(k, r) = exp(ik r) uk(r),

-

where uk(r) is a periodic function with the periodicity of the basic lattice; the periodic function can be expanded in plane waves with reciprocal For a summary of the expressions of the cubic harmonics see D. J. Bell, Rev. Mod Phys 26, 311 (1954).

TABLE V SYMMETRY-ADAPTEDCOMBINATIONS OF BLOCHFUNCTIONS UP TO AT THE

ATOMIC FUNCTIONS

BLOCH FUNCTIONS

s.4

.

_''

r;

r rl

POINTS r, A s L. A. and X a

I =2

A (or L)

sA

'1

sA

A

Al

X

sA

'I

'A

34 d

a

h

I

-a

z

I

+

-3

4a

U

FRANC0 BASSANI 4

-

d

I

4

-

8

d

I

I

0 8

=

I

4

d

m

-a

I

'p

m

-a

I

4

T i

4

w

m

I

4

Q

*

.p"

u.

I

a

4

-2

+

-a

d

-

I

d

2 'p"

L

N

-

m

h

+

-U

m

h

+

a"

A1

d3~2-r2,B

A2

dy2-z2,B

x3

x4

d3x2-r2.B

dy‘-z‘.B

N

r s m a In the first column basic atomic functions are given with their normalization factors. Their angular parts are related to spherical harmonics in the following way (except for normalization factors):

1

=

Y,,,,(e,

2

U

UY

8

c~);

W

=

Y1.1 +

=

Y1,-1;

( - - ~ N Y ~-, ~y1,-]1;

>

3 c)

:=

yz 7 -

2 r2 =

Y1.0;

(-i)[Y2,2

zx

(-i)[Y2,1

- Y2,-11~

7=

Y2.1

-

+ Y2.-

Y ~ , - ~ I ;

1

;

In the second column the Bloch functions on the two sublattices are indicated and in the other columns the appropriate combinations belonging to the irreducible representations are given. Partner functions are indicated with braces.

F

c)

36

FRANC0 BASSANI

lattice vectors h as wave numbers ; consequently,

the plane waves are normalized on the lattice volume NQ R being the volume of the unit cell. With the rules of Section 5 one can construct symmetrized combinations of plane waves (SCPW) S,",'(k,r). These belong to the rows of the irreducible representations of G(k), and their use greatly reduces the number of unknown coefficients in the expansion (6). The result of a symmetry operation on a plane wave can be obtained directly from Table I by using the definition

-

O{Rlrjexp[i(k + h) r]

= exp[i(k

+ h) - { Rlz)

-

lr]

(74

~

This equation can be written in a different form by using the explicit expression {R-lI - R-%} for {RJz}-', and then multiplying both terms of the dot products by the rotation R. We obtain O{RJ+) exp[i(l

+ h) - r] = exp[

-

i(k

+ h) - R - '23 [i(k + h)

= exp[ - iR(k

R-'r]

+ h) -r] exp[iR(k + h) .r].

(7b)

This proves that operation (7a) sends a plane wave into another with the same k and the same Ik + h( when {RIT}belongs to G ( k ) ;formulas (2) and (7b) can then be applied to obtain the S>i. At the point r for instance k = 0, and the sets of plane waves to be used are defined by the reciprocal lattice vectors (0),

(lll),

(200), (220),

(311),

(400),

..., etc.

The symbol (1 11> indicates the eight reciprocal lattice vectors

and similarly for the other sets. From these sets one can then form linear combinations of plane waves which belong to the irreducible representations at r by applying formulas (3) and (7a). From the set (111) and the characters of the irreducible representations rl and r15, one obtains the following independent combinations by applying Eq. (3).

f1= +{(I 1 1) - (1 i i) - (i 1 i) - (ii I)},

2.

1

.f

37

METHODS OF BAND CALCULATIONS

4--{(~

-Jlz

1 .f - -{3(TTT)

"fi

1 I)+ 3(iTT)-(iiT)-(Tii)},

+ (T 1 1 ) + ( 1 i 1) + (1 1 i)},

1

f, = -{(i i i)+ 3(i 1 1) - (1i 1) - (1 1i)},

J12

The combinationsf, and f2 or their symmetric and antisymmetric combinationsf, + fz andJ; - f2 belong to r,. In the case of the three-dimensional r15 we must choose combinations of the remaining functions which are orthonormal ; we obtain

1

--'f

Jz'

- f5}1

-{f4

*

+ f519

-f8>>

-{f6

+ f8).

1

1 $f3

-

f419

- f7},

-{f3

Jz

1

1

1

-'f

J3'

Jz

The first three functions are partners of rI5and so are the last three functions ; they determine the matrices of the irreducible representation. Formula (2) can then be applied to the other sets of plane waves to obtain the olher combinations belonging to rI5. The symmetrized combinations of plane waves at the points of interest are given in Table VI for the lowest sets of plane waves.

8. MATRIX ELEMENTS From Eqs. (1) we know that the only nonvanishing matrix elements of the crystal Hamiltonian are those involving functions with the same k which belong to the same row of the same irreducible representation of C@).

TABLE VI. SOMESYMMETRIZED COMBINATIONS OF PLANE WAVES (SCPW)

( I I I)

(ITi)

(TIT) (TiI) (ii T)

(111) ( i i i ) (I I

T)

I -I -I -I I -I -I -I

,

I I -I I

-i -i

I I

-1

I I I

( I 0 0) (TOO)

I I

( I 0 2) (1 2 0 ) ( I 0 2) (I 10)

I I I I I I I I

AND

X

OF THE

REDUCED ZONE'

, -i

I

-i

-I

I

( T O 2) (1 2 0 ) (T 0 2 ) ( i2 0 )

,

I I I I I I I I

I I I

-1

-i -1

I

, I

I -1

J

--I

b

-i -1

-i

(2 1 I) I

0

0 0

I

0

(2 I T ) (27 I) I?i i)

I

0

0 0 I 0 0

I

I

( I i i) ( I T I)

I

0

0

I

I -I

I

-I

I I I

( 2 I i) ( 2 I I)

I I I I I

( 3 0 0) ( 3 0 0)

I I

(1 22)

I I I I I I I I

I

I

--I

I

i

I I

I I

I

- 1

-I -I

I

-i -i -i

I

-i

I 0 0 I I 0 0 I

I

-I -I

-i

-i

b 0 i -i

0 0 -1

-I

--I

0

-i -i

-I I

--I

0

0

I

0 0

8

,

I

--I

-i

SYMMETRY POINTSr, L,

I

-1

--I

AT THE

(I 2 I) (I 22) (I 2 I)

( iI ( iI (i2 fT 2 Conventional origin

2) 2)

2) 2)

I

-, I

I

I

, ,

-I -I

-i -i

I I

I

-i I I

-1

Point origin

-I -I

I

-1

I

0 0

-I

-1

I 0 0

--I

-i -i

I

I

0 0

-I

-1

0 0 I

Conventional origin

'The superscript o n the left of the irreducible representations indicates the dimension. For the irreducible representation ofdimension greater than 1, only one of the combination is given. The combination is the one corresponding t o the last of the partner functions a s listed in Table V. In order t o obtain real matrix elements some of the SCPW have b a n multiplied by a n imaginary factor, and for the irreducible representations X, and X , the point origin has been used (from Ref. 24).

8 $

% b

3

2.

39

METHODS OF BAND CALCULATIONS

A typical matrix element between symmetrized combinations of Bloch functions described in Section 6 is ($? '@,r), H$&.7k741,

(8)

where H is the crystal Hamiltonian and $;,'(k7 r) =

1

C e x ~ ( i kRv) 14; un, *

~

J

N

1, m(r -

Rv) *

(9)

m

V

The matrix element (8) becomes a. i*.o Cm' am, 1 2 . 2 tun, i

1,

m(rh H

C e x ~ ( i kRv) un,1, m,(r - Rv)I.

m,

*

(10)

V

This matrix element can be separated into two parts; one is the product of the atomic eigenvalue enIm times the overlap matrix element Ol,2(k), the other is an appropriate sum of two-center and three-center integrals involving the crystal potential

Vc'

=

c' VA(r

-

R,) +

V,(r - Rv - d),

(11)

V

V

where V, and V, are appropriate atomic-like potentials, and the atom at the origin is left out of the sum. The prime on the V's indicates that the contribution to the crystal potential from one atom is left out of the sum; in fact, the potential on one atom has been used to bring the atomic eigenvalue out. The overlap matrix elements, as well as the contribution from the two-center integrals, can be expressed in terms of a few independent integrals by considering only the contributions from atoms which are not too far in the lattice. The mathematical techniques for obtaining explicit expressions in terms of the smallest number of independent integrals are described in detail by Slater and Koster." We give in Table VII the expressions for all matrix elements of interest in the zinc blende lattice involving s and p atomic functions. The table can easily be extended to include the d functions by using the technique of Ref. 10. By using the general expressions of Table VII with the symmetrized combinations of Bloch functions given in Table V one can readily obtain all matrix elements at the symmetry points. The matrix elements of the crystal Hamiltonian between plane waves and between the SCPW are very simple in form; they are appropriate combinations of expressions of the following type : 1 --((exp[i(k h,) r], H exp[i(k h2) r])} NR

+

lo

-

+

-

J. C . Slater and G . F. Koster, Phys. Rev. 94, 1498 (1954).

TABLE VII MATRIXELEMENTS BETWEEN BLOCHFUNCTIONS FOR A GENERAL k VALUE"'b

5

D

i

1

, . 5 . v . l

~(ssu),, cos - cos - cos - - I sin - sin - sin [ 2 2 2 2 2 2 4

-(spo),,

4

[-

4

6 . v . l

cos - sin - sin 2 2 2

3 bpa)AB + 2@pn)A, 4 3

- [(.pa)AB-

. . 5

v

+ z sin -2 cos 2- cos 2-l l

I[222 5

v

5

v . i

i

. . 5 . D . i

cos - cos - cos - - i sin - sin - sin 2 2 2

5 . v

1

(ppn)AB i cos cos - sin - - sin - sin - cos ] [ 2 2 2 2 2 q -

"The formulas apply to both overlap and energy matrix elements. Standard notations for the two center integrals are used; for instance , l ,d) ~ (dr, r for the overlap matrix elements, and (ppn),, = ~~.~,~,~(r)H~..,,~,,(rr - d) dr, for the energy matrix elements, the (ppn),, = ~ u ~ l , , , A ( r ) u " . ,axis of quantization being along A-B. The first term in the diagonal elements is 1 for the overlap matrix elements and is the crystal field integral IIuA(r)lZVC(r)drr with Vc(r) as given by formula (11) for the energy matrix elements.

b5

= tak,, q = )ak,,

5

= iak,.

2.

41

METHODS OF BAND CALCULATIONS

If the crystal potential is written as the sum of local potentials at the lattice sites, the expression becomes even simpler. In the case of the zinc blende lattice, when the origin is at a lattice point we have &(k

+ hl, + hZ)

= VA(lhl

- h21)

+ exp[i(hZ - hl) ' d1 v B ( b l

- hZ1) ; (13)

when the origin is at the midpoint between atoms A and B, we have

where VA(r) and V,(r) are the atomic-like potentials at the atoms A and B, and their Fourier transforms are defined as V((h()=

'S

exp( - ih * r) V(r) dr.

Explicit expressions for the matrix elements involving the SCPW reported in Table VI are given in Table VIII. The overlap matrix elements between Bloch functions and SCPW are also important. Any matrix element

v qi

@Y

r), s; ' i(kr)l

(16)

can be expressed simply in terms of an orthogonality coefficient which depends only on the atomic function and on the absolute value Jk+ 4.'' By expanding each plane wave in spherical harmonics and using the orthogonality of the associated Legendre functions one obtains

The coefficients u ; , ~of the plane waves and c:,,, of the spherical harmonics define the symmetry-adapted functions, the angles 0, and qj refer to the vector k + 4 s gives the position of the atom in the unit cell. The expression for the orthogonality coefficients is

where PJr) is the radial part of the atomic wave function multiplied by "

R . S . Knox and F. Bassani, Phys. Rev. 124, 652 Appendix (1961).

I

TABLE VIII MATRIX ELEMENTS BETWEEN SYMMETRIZED COMBINATIONS OF PLANEWAVESAT THE POINTS r. L, AND x.THEKINETIC PART IS INDICATED IN UNITS OF h 2 / 2 ~

r1

H,, = V ~ O )

+ Vs(0)+ 3Vs(8) H44 = I41 + Vs(0) + 4Vs(8) + Vs(16) H s s = [8l + Vs(0) + 4Vs(8) + 2Vs(16) + 4Vs(24) + Vs(32)

H z z = H33 = 131

HI2

= -2Vs(3)

HI3 =

-2VA(3)

H I , = -&VA(4)

rls

H , , = H~~ = [31 + VS(O) - V S ( ~

[4] + Vs(0) - Vs(16) H44 = [8] P(0) - 2Vs(16) + V'(32) H 5 5 = 181 + Vs(0) 2Vs(8) - V'(32) - 2Vs(24) H I , = -VA(4) + V"(12) = $[Vs(3) - Vs(ll)] Hi4 = Vs(3) - 2Vs(II) + V'(19) H33 =

r12

Hi, =

H n

=

+

+

[41 + Vs(0)- 2Vs(8) 181 + Vs(0) - 2Vs(8)

+ Vs(16) + 2Vs(16) - 2Vs(24) + Vs(32)

+ +

H , , = &[- VA(3) VA(19)] H23 = [ - VA(3) VA(1 l)] H , , = VA(3) + VA(l9) - 2VA(ll) H Z s = Vs(3) - Vs(19) H3, = 0 H 3 s = -2VA(4) + 2VA(20) H,, = 0

4

Jz + 2Vs(3)] + Vs(0) + 2Vs(8) + --[Vs(ll) 2

]:[

+ Vs(0) + 2Vs(8) - Jz -[Vs(ll) 2

H44 =

H55

HI,

=

=

[y]+

H 2 , = $VA(3)

+ 2Vs(3)]

3 - --[Vs(19) 2

+ 2Vs(11)]

H3,

= +VA(ll)

3 + Vs(o)+ 2Vs(8) + -[Vs(19) 2

+ 2Vs(11)]

H35

= -T[V’(~)

+

Vs(0) 2Vs(8)

3

--VA(3) 2

HI,

=

4Vs(3)

3

&

H3, =

3

+ 2VA(11)]

- 2VA(4) + -[VA(3) 3

+ 2VA(ll)]

H46 = -$[Vs(3) H,,

L3

Hi1

=

“a1 + -

fi

= +VA(19)

2

+ 2Vs(11)]

-+ 2VA(11)]

fi + -[Vs(ll) 2

- Vs(3)]

H,, = -$[VA(3)

4 + Vs(0)- Vs(8) - --[Vs(ll) 2

- Vs(3)1

H Z 3=

+ Vs(0)- VS@) + $[Vs(ll)

- Vs(19)]

H Z 5 = - g [2V A ( 3 ) - VA(ll)]

+ Vs(0) - Vs(S)- $[Vs(ll)

- Vs(19)]

H2,

Vs(0) - Vs(8)

=

N

+ 2Vs(11)]

-VA(12) - 2VA(4) - -[VA(3) 2

Ha5 = -VA(12)

H , , = - $VA(3)

+ 2VA(3)]

-

VA(ll)]

- VA(12) + VA(4) - $[VA(ll)

+Vs(8) - Vs(16)

+

9

- VA(3)]

[ V s ( 3 )- Vs(ll)]

TABLE VIII (continued)

H34 = - s [ V A ( l l ) - VA(19)] 2

+ V’(0) - V’(8) - <[Vs(27)

- Vs(19)]

Jz

H35 = y [ V s ( l l ) - Vs(3)]

-

-

HI, = - yJ [2 V A ( l I ) - VA(3)]

Jz

HI3 = +H24 = --[Vs(3) 2

H3,

XI

HI5

=

HI,

=

Vs(S) - V’(16)

Jz - -[Vs(3) 2

-

H22

+ V’(0) - VA(4) = [2] + V‘(0) + Vs(8) - 2VA(4)

H33

=

+ 2Vs(8) + V’(16)

- VA(3)]

H46

=

Vs(ll)]

H,,

= $[VA(27)

-

=

[61

HI2 = f i [ V s ( 3 )- VA(3)] HI3

=

-2[VA(4) - Vs(8)]

- VA(3)]

H35

-

VA(19)]

Hi4 = Vs(3)

+ V s ( l l ) - VA(3)- V A ( l l )

Hi5 = Vs(3)

+ VA(3) - V’(I1)

- VA(ll)

VA(4) - 2VA(12) - VA(20) H,, = J5[Vs(3) - VA(3) + ~ ’ ( 1 1 ) - ~ A ( 1 1 ) ]

+ Vs(0) + V’(8) + V’(16) + V’(24) H55 = [61 + V s ( o ) + V’(8) - Vs(16) - V’(24) H44

+ yJ [ 2V A ( 3 ) - VA(I1)l

H45 = VA(12) - VA(20) $[VA(ll)

[l]

[51 + V’(0)

VA(12) - VA(20)

+

- Vs(ll)]

H14 = VA(4) - VA(12) + $[VA(ll)

=

-

2VA(4) - 2VA(20) H,, = -,/5[VA(4)

- 2VA(20) + 2VA(4) H2, = H4, H34 = V’(3)

=

- 2Vs(8) +

vA( 12)l

0

- VA(3) + ZV’(11)

H35 = -Vs(3) - VA(3)

- 2 V A ( l l ) + V’(19) - VA(19)

+ Vs(19) + VA(19)

X3

H , , , H,,, H,,, H,,, H,,,H , , , H , , , H 3 4 may be obtained from X I with a change of sign of antisymmetric part H , , , H,,, H , , , H,,, H,, may be obtained from X , with a change of sign of symmetric part

X,

HI1

=

[2] + Vs(0) - V s ( 8 )

H,,

=

[5] + Vs(0)- Vs(16)

H,,

=

H,,

P

gEi P v1 U

a

H , , = [6] + Vs(0) - V s ( S ) - Vs(16) + Vs(24) H , , = [6] + Vs(0) - Vs(S) + Vs(16)

-

Vs(24)

0

H , , = f i [ V s ( 3 )- Vs(ll)] H , , = -&[VA(3)

- VA(ll)] ~

In this table the symbols Vs(g’) and VA(g2)indicate VB(g2)5 VA(gZ),gz being the square of a reciprocal lattice vector in units of 4n2/a2.

Y

m

46

FRANC0 BASSANI

and J , is the Bessel function. For the zinc blende structure, for every SCPW we have two terms of the type (17), one for each sublattice, which differ by the values of the phase factor exp(ihj.s). Since the coefficients a;j and cy, have been determined in the previous sections and are given in Tables V and VI for the symmetry points [C, ~ , 9 n ~ ~cp)( 8is, given directly in Table v], it is very simple to determine the explicit expressions for the matrix elements (17) at r, X , L. Such expressions are reported in Table IX. IV. Approaches to the Calculation of the Electronic States

9. INTRODUCTORY REMARKS In principle all the methods developed for calculating the electronic states in solids can be applied to the 111-V compounds.’2 However, methods of band calculations like the Wigner-Seitz cellular methodI3 and the Kohn-Rostoker variational method, l4 which have been successful with metals, have not yet been generally applied to semiconductors. The Slater method of augmented plane waves” has also mostly been applied to metals and only recently to strong insulators.’6 Though it is to be expected that an attempt will be made to compute the energy bands in the 111-V compounds with the above methods, it appears at present that the method of band calculation which has shown itself particularly suitable for covalent semiconductors is the orthogonalized plane wave (OPW) method of Herring.” HermanI8 first applied the OPW method to covalent semiconductors showing that one can obtain an energy gap in diamond between valence and conduction states and the right structure for the valence band. Similar results have been obtained by Herman and Callaway” and by Herman” on Ge, by Woodruff” and by BassaniZ2 on Si; results for the lowest conduction bands have also been obtained and compared with experimental For an extensive analysis of the methods of band theory and their applications see J. Callaway, “Energy Band Theory.” Academic Press, New York, 1964, and J. Reitz, Solid Srate Phys. 1, l(1955). l 3 E. Wigner and F. Seitz, Phys. Rev. 43, 804 (1933); 46,509 (1934). l4 W. Kohn and N. Rostoker, Phys. Rev. 94, 1411 (1954); F. S. Ham, ibid. 128, 82 (1962). ’*J. C. Slater, Phys. Rev. 51, 846 (1937). l 6 L. F. Mattheiss, Phys. Rev. 133, A1399 (1964). l 7 C. Herring, Phys. Rev. 57, 1169 (1940). See also T. 0. Woodruff, Solid State Phys. 4, 367 (1957). F. Herman, Phys. Rev. 88, 1210 (1952). l9 F. Herman and J. Callaway, Phys. Rev. 89, 518 (1953). 2 o F. Herman, Physica 20, 801 (1954). 2 1 T. 0. Woodruff, Phys. Rev. 103, 1159 (1956). 2 2 F. Bassani, Phys. Rev. 108, 263 (1957); Nuovo Cimeizto 13, 244 (1959). l2

OVERLAP MATRIXELEMENTS BETWEEN SYMMETRIZED COMBINATIONS OF BLOCHFUNCTIONS AND SYMMETRIZED COMBINATIONS OF PLANE WAVES AS GIVEN IN TABLE v AND TABLEVI, RESPECTIVELY *

1

s2

Ll

Sl

SA

B

C.I.

=,

1 --bx

8

+ P y + PA*

B

C.I.

TABLE IX (continued)

+ I

3

I I

3

I I

3

METHODS OF BAND CALCULATIONS

I

3

2.

+

+

I

3

I I

r3

49

50 FRANC0 BASSANI

SI

0

s4

ss

0

B

0

C.I.

s2

C.I.

C.R.

C.R.

0

C.I.

% 2 0

b tA

0

6 fi

- i)Ap,A(6)

C.R.

s5

0

0

O1 The matrix elements ($,, S,) are listed as functions of the orthogonality coefficients, the Sp being indicated in the same order as given in Table VI. The orthogonality coefficients A,,,A(f2) have been defined in formula (18), f’ being Jk + hI2 in units of 4xz/u2. The symbols C.I. and C.R. indicate that the coefficients of An,,B(f*)can be obtained from those of A,I,A(.fZ)by changing the sign of the imaginary and the real part, respectively.

Y

52

FRANC0 BASSANI

data. More recently the OPW method has been applied to gray tinz3 and to a number of 111-V compounds24 with reasonable success. It had previously been recognized that the gross results of the OPW method depend essentially on the symmetry of the lattice and not on details of the crystal potential. A diamond lattice made with sodium atoms, for instance, would have the same basic band structure as a lattice made of silicon atoms.25 The reason has been investigated and has led to a perturbation approximation to the OPW method which has been used to simplify the calculations and to gain physical insight.z6 Another simplification of the OPW method, the pseudopotential approximation, has been introduced by PhillipsZ7 and by Phillips and Kleinmanz8 and has been particularly useful as an interpolation scheme since in this approximation the energies can be computed at all points k with a relatively small number of parameters. The tight-binding approximation has been used for covalent semiconductors mainly as an interpolation procedure. l o Also in this approximation it turns out that the basic features of the band structure are mostly determined by symmetry. Recently, numerical calculations of the valence bands of 111-V compounds have been carried out with reasonable success by a modified tight-binding method which has been called the method of linear combination of bond orbital^.'^ We shall describe in the next two sections the approach of orthogonalized plane waves and the tight-binding approach, including the modifications and simplifications of the original methods. The two approaches complement one another: in genera1 the tight-binding approach is best suited to the low-energy electronic states and the OPW approach to the higher electronic states. 10. TIGHT-BINDING APPROACH a. Linear Combination of Atomic Orbitals

The wave function for the electronic states can be expanded in Bloch sums of atomic orbitals: Ya*'(k, r) =

C a,". iQ)t,b,"

'(k, r) .

(19)

C

F. Bassani and L. Liu, Phys. Rev. 132, 2047 (1963). F. Bassani and M. Yoshimine, Phys. Rev. 130, 20 (1963). 2 5 F. Bassani, J . Phys. Chem. Solids 8, 375 (1959). 26 F. Bassani and V. Celli, Nuovo Cimento 11, 805 (1959); Studiu Ghisleriana 11, 157 (1959). " J . C. Phillips, Phys. Rev. 112,685 (1958). J. C. Phillips and L. Kleinman, Phys. Rev. 116, 287 (1959). 29 A. 1. Gubanov and A. A. Nran'yan, Fiz. Tverd. Tela 1, 1044 (1959) [English Transl.: Soviet Phys.-Solid State 1, 956 (1959)l; C. A. Coulson, L. B. Rkdei, and D. Stocker, Proc. Roy. SOC. (London) A270,357 (1962). 23

24

2.

METHODS OF BAND CALCULATIONS

53

The &?@,r) are the symmetry-adapted Bloch functions described in Section 6 ; the subscript c stands for the appropriate atomic quantum numbers. The coefficients of the expansion in (19) and the corresponding eigenvalues can be obtained from the energy matrix Hc,c and the overlap matrix Oc,cby solving the secular equations

1afb(HCrc- E“O,,) = 0 .

(20)

C

The eigenvalues are the solutions of the secular determinant

llHc,c - EUOc,,)I= 0 .

(21) The expansion (19) will in general contain all the functions of the complete set J/,(k,r), and consequently the secular equations (20) are infinite in number. Furthermore every matrix element Hc,cand Ocjccontains in general an infinite number of terms, as indicated in Section (8). This method can be adopted and is of physical significance when the secular determinant (21) is of small order, i.e., when a small number of basic functions is sufficient to obtain a good approximation to the exact solutions in the expansion (19). Furthermore, in computing the matrix elements, one should be able to consider only a small number of significant terms, namely, overlap end energy integrals between functions located on the nearest atoms and on the neighboring atoms up to a certain order. There is no precise mathematical test to decide when the above conditions are satisfied, and we must rely on physical intuition. The tight-binding method can be applied for the states of low energy, because the crystal wave functions will be concentrated near the nuclei where they are not too different from the corresponding atomic functions. At very low energy one has the extreme case that only one atomic function is needed in the expansion (19). The overlap matrix element is 1, and the energy matrix elements are the atomic eigenvalues Ec plus an additional constant due to the crystal field ; those states are called “core states.” The “valence states” cannot always be conveniently expanded in linear combinatioqs of atomic functions. In solid rare gases” and in ionic crystals3’ such an expansion is justified, but in covalent substances, where there is a large overlap between the atomic orbitals centered on different lattice sites, the accuracy of the procedure is questionable. For the “conduction states” the method described is expected to fail. In the case of the 111-V compounds there are two sets of “core states,” one on sublattice A and the other on sublattice B, their eigenvalues do not depend on k and are equal to the atomic eigenvalues plus a constant shift, opposite on the two sublattices due to the ionicity. If the uncompensated 30

L. P Howland, Phys. Rev. 109, 1927 (1958).

54

FRANC0 BASSANI

charge on atoms A and B is i-Q, the lattice will produce near the nuclei a nearly constant electrostatic potential fctm(Qe2/a)which must be added to the atomic eigenvalueS 01, being the Madelung constant for the zinc blende lattice equal to 3.7829.j’ The “valence states” originate from the highest occupied s and p states of atoms A and B; there are eight such functions to consider, and at a general k point one must diagonalize a secular matrix of order 8. At a symmetry point the size of the matrix is reduced by using the symmetrized combinations of Table V. The general expressions for the matrix elements are given in Table VII. At every k point the lowest four eigenvalues are the “valence states” and will be occupied by the eight valence electrons per unit cell.

b. Linear Combination of Bond Orbitals For the valence states of covalent semiconductors a method essentially equivalent to the previous one29 has been put forward along the general lines suggested by Hall32 for diamond. It consists in expanding the valence wave functions in linear combinations of molecular orbitals. The molecular orbitals in turn are expressed as combinations of atomic orbitals s and p on the two nearest atoms A and B oriented in the direction joining the two atoms. This is a “bond orbital”

$t)= $)!

+

l$p,

(22)

and will be occupied by two valence electrons. In a unit cell there are four such bond orbitals oriented in the directions of the four nearest neighboring atoms. For the bond in the direction [ 11I ] for instance we will have

Ve= [s + (P, + P y + P A * + 4 s -

(Px

+ P y + Pz)lB,

(23)

and similarly for three other bond orbitals. Four antibonding orbitals of the type I@ = - p+$) can also be formed from the eight atomic orbitals. They correspond to empty states and are neglected in most calculations. From every “bond orbital” one forms a Bloch sum : 1 r) = __ exp(ik. R,) $g)(r - R,). J

N

V

The same procedure as in Section 10a can then be applied, and four linear equations result for every value of k. The overlap matrix elements and the matrix elements of the Hamiltonian involve integrals between bond orbitals. Clearly, if Iz is treated in the same way as the expansion coefficients and the antibonding orbitals are included in the expansion, the method is 31

32

V. Takahashi and Y. Sakamoto, J . Sci. Hiroshima Uniu. 24, 117 (1960). G. G. Hall, Phil. M a g . 43, 338 (1952); 3, 429 (1958).

2.

METHODS OF BAND CALCULATIONS

55

formally equivalent to the more conventional tight-binding approximation described in Section 1Oa. In fact this would correspond to choosing different combinations of the same atomic functions on both atoms and will lead to equivalent equations. A difference exists when 2 is determined independently from the band calculation by minimizing the energy per bond in the Heitler-London approximation as suggested by Coulson et al.29 One can hereby properly take into account the exchange integrals between the nearest atoms, although in practice one uses experimental atomic values for the integrals on the same atom and simplifying assumptions for the resonance integral (uA, Hu,), as is done in semiempirical molecular calculations. Another advantage is that one can define the fraction of valence electrons associated with atom A and atom B from the value of the parameter 2, and this gives the polarity of the compound before the band calculation is performed. In the 111-V compounds it is expected that the atom with five valence electrons carries more than five electrons and the atom with three valence electrons carries less than three electrons because of the difference in electron affinities. The above procedure may be criticized on fundamental grounds because the total energy to be minimized should really be computed from the valence wave functions of the crystal. The bond energy is a concept which cannot be given a basic physical definition. Nevertheless the bond orbitals point of view can provide a semiquantitative picture of the 111-V compounds which can be related to experiments on the width of the valence band and on vibrational absorption. The major shortcoming of the tight-binding approach is that it must be limited to the valence states and cannot give precise information on the energy gap and the conduction states. 1 1 . ORTHOGONALIZED PLANEWAVEAPPROACH a. OPW Method

The method of orthogonalized plane waves was introduced by HerringI7 as a modification of the simple method of expanding the valence and conduction states in plane waves. Herring recognized that if one imposes the condition that every state be orthogonal to the inner states, the convergence is greatly improved. This condition can be satisfied by orthogonalizing every plane wave to all the inner states by the Schmidt procedure. Let a plane wave be W@i h, r) = N-1/2exp[i(k + h). r]; an orthogonalized plane wave will be

+

X(k

+ h, r) = W(k + h, r) - C ($c, W)$c(k.

(25)

C

where $&r)

is an inner state represented by Bloch functions of the type

56

FRANC0 BASSANI

(19). The valence and conduction wave functions can be expanded in the X(k + h, r), and they will be automatically orthogonal to the inner states. At symmetry points of the Brillouin zone one can use the SCPW S"d(Jk+ hJ,r)as explained in Section 7 and for simplicity indicate them as S:'@) where p is used as an index, but p = k + h. We can always use this notation to indicate symmetrized combinations or single plane waves depending on the small group of k. The valence and conduction states will then be expressed in general as

Ya*'(k,r)= zc;.[S>' - ~ ( $ c , S ; i ) $ c J . Pt

(26)

C

A wave function of this type is smooth far from the nuclei and is similar to the atomic wave functions near the nuclei, because of the second term to the right of Eq. (26). The coefficients cp, of the expansion as well as the eigenvalues E" of the Schrodinger equation are determined by solving the secular equations (S;'IH - E"1Ya.i) = 0.

(27)

The eigenvalues E" correspond to the zeroes of a secular determinant whose matrix elements are Mp,,, = (S>i, (H - E ) S 2 i ) - C ( E - Ec)($c, S>')(S",.',

$c).

(28)

C

The above expressions can be made explicit at every point k of the reduced zone as functions of parameters which depend on the matrix elements of the crystal Hamiltonian H between plane waves and on the orthogonality coefficients of the atomic states of the core Anl(lpl) defined in (18). If the crystal potential is written as a sum of local atomic-like potentiais as indicated in Part 111 and one considers the core states on the two sublattices, the resulting expressions for the zinc blende lattice are very simple. At the symmetry points the results of Table VIII and Table IX can be directly substituted into Eq. (28) to obtain the most important matrix elements for the states of interest. In general one can write the matrix elements (28) as functions of the symmetric and antisymmetric parts of the crystal potential Vs(Ip - p'l) and VA((p- p'l) and of the symmetric and antisymmetric expressions : R,Sl(lpl, IP'I) = ( E - En*,A)A.~~,A(IPI)A,~,A(IP'I) + ( E - Enl,B)A~~,B(~P~)Anl,B(~P'~),

R$(lpl? IP'I) = ( E -

-( E -

AlAil,

A(IPoAnl, A(lP'l)

BlA:l,

B(lPl)Anl,B ( b ' I ) .

(29)

2.

METHODS OF BAND CALCULATIONS

57

When the origin is chosen at a lattice point one obtains the expression

where PI is the Legendre polynomial, o is the angle between p, and p,’, p, = k h,, p, = k + h,, and the coefficients u;? and u;: are determined by symmetry. With the conventional origin one obtains a similar expression except that every term in braces is multiplied by the phase factol; exp[ - i(p, - pi) * d/2]. At the special points r, L, and X the coefficients uiVand are given in Table V1, and one can obtain the matrix elements directly by applying the above formulas. The expressions of the type (I)~, Sp) given in Table IX are necessary to obtain the valence and conduction wave functions as indicated in Eq. (26). One of the nice features of the OPW method is the rapid convergence. This is due to the fact that in the secular determinant the off-diagonal elements are quite small and decrease very fast as (p’ - pI increases; in fact the terms in the matrix elements originating from the orthogonalization requirement cancel to a large extent the Fourier coefficients of the crystal potential. Such a cancellation is a general property of the orthogonality requirement and is of importance also in atoms and molecules, as was pointed out by Cohen and Heine.33 This fundamental property justifies the pseudopotential schemez7*z8 and the perturbation approximation to the method introduced by Bassani and Celli.34

+

b. Pseudopotential Scheme

The essential approximation of the pseudopotential scheme consists in assuming that in the matrix elements of the final equation not only the Fourier coefficients of the crystal potential but also the terms originating from the orthogonality condition depend only on the reciprocal lattice vectors h = p’ - p and are eigenvalue independent. Then the repulsive terms and the Fourier coefficients of the crystal potential appear in the matrix elements as the Fourier coefficients of a total pseudopotential 33

M. H. Cohen and V. Heine, Phys. Rev. 122, 1821 (1961).

34F.Bassani and V. Celli, J . Phys. Chem. Solids 20, 64 (1961).

58

FRANC0 BASSANI

VT(Jp- p’l) = Vdlp - p’l) + VR(lp - p’l). These Fourier coefficients are very small and decrease very rapidly as [p - p’l increases. Phillips” suggested that they can be taken as disposable parameters to obtain an interpolation scheme which is more accurate and convenient than the one based on the tight-binding approximation. Bassani and Celli34 noted that a very small number of parameters are needed because VT((p- p’() can be assumed to be 0 for values of the reciprocal lattice vector greater than a certain (hi. The idea of a repulsive potential due to the core electrons had already been adopted in connection with the “nearly free electron model” by Hellmann, by Gombas, and by AntonEik3’; they also introduced a crystal potential and a repulsive potential but their repulsive potential was derived from statistical arguments by using the Thomas-Fermi approximation on the atoms. For a priori numerical calculations the approximation is not as consistent as the full orthogonalized plane wave method. To elucidate further the idea of the pseudopotential, we can make some considerations on the meaning of the OPW method along lines discussed by many Let us consider the valence or conduction wave functions expanded as indicated in formula (26). They consist of two parts, a smooth function given by the expansion of the plane waves and an attached function rapidly oscillating near the core states. Following Phillips and Kleinman” we can write Eq. (26) as Ya.’&, r) = cpa. ’&, r) -

1WC,(pa*W C ,

( 3 1)

C

where cp is the smooth function. Substituting (31) into the Schrodinger equation with a model potential, one obtains

The Schrodinger equation on the crystal state has been transformed into a Schrodinger equation on the smooth function with an effective potential Vc + VR, where &(pa’

=

1( E -

Ec)($c

9

(Pa’

‘)$c

*

C

In general the effective potential is nonlocal and eigenvalue dependent. Equation (32) leads to the OPW secular equations (27) and (28) when p is H. Hellmann, “Einfiihrung in die Quantenchemie,” $9. Leipzig, 1937; P. Gombas, “Statistische Theorie des Atoms,” $19, $35. Springer-Verlag, Wien, 1949 ; E. AntonEik, Czech. J . Phys. 4, 439 (1954). 36 B. J. Austin, V. Heine, and L. J. Sham, Phys. Rev. 127, 276 (1962); V. A. Harrison, ibid 136, A1107 (1964).

35

2.

METHODS OF BAND CALCULATIONS

59

expanded in plane waves. The approximation discussed at the beginning of this section can be inserted in Eq. (32) by taking Vc + VR to be a local and smooth pseudopotential. Since V, depends on the cp and on E and on the core states it is not uniquely determined. As was first shown by Bassani and Celli,34 the same crystal wave functions and eigenvalues will result when VRcp =

1

($c

9

Aq)$c

7

(33)

C

where A is any arbitrary operator. In fact, Eq. (32) can be written as

with A arbitrary, since, by multiplying (34) to the left by any core function t,bc* and integrating, it can be shown that the exact solutions cp and E for any A operator will always satisfy the condition ($0

Acp) = ( E - Ec)($c, cp).

(35)

Various choices of the arbitrary operator A have been suggested.36 They all must lead to the same results for the quantities of physical significance at every degree of approximation because condition (35) must be satisfied to the same order of approximation. c. Perturbation Approach

The reasons discussed in Section 1l a for the fast convergence of the OPW method make it possible to develop a perturbation approach. It was first introduced to explain the dominant role that the lattice symmetry plays in the sequence of the energy statesz6 but can also be used in many cases as a technique for numerical calculations.". 34 A simple way to obtain the perturbation solutions up to the second order is to solve Eq. (34) and satisfy condition (35) to the same order. The unperturbed solutions are the plane waves S;i(r) which correspond to the unperturbed Hamiltonian pz/2m + .V(O),where V(0) is the space average of the crystal potential ; the unperturbed eigenvalues are

The perturbation term is

60

FRANC0 BASSANI

The perturbation corrections up to second order are

One has also to satisfy to the first order the condition (35) with

This can be done consistently if we separate first- and second-order terms in the energy and assume that l($c,Sp)12 is at most of first order. One finally obtains

When some unperturbed levels are very close in energy and the term

Ep"," - E;.,. would become too small, one can solve a small order secular equation for them and use the perturbation approximation to take into account the second-order contributions arising from the other unperturbed states which are well separated. This perturbation approach generally gives results very close to those of the full OPW method. Its main advantage, besides the simplicity in the calculations, is that one can derive qualitative arguments on the sequence of the energy states from the unperturbed solutions of the "empty lattice" and from the type of core states of the lattice. In fact, Eqs. (39) indicate how the levels of a given symmetry are changed by interaction with unperturbed

2.

METHODS OF BAND CALCULATIONS

61

levels of the same symmetry and by interactions with the “core states” of the same symmetry. The perturbation approach can be further simplified ; it reduces to ordinary perturbation theory if one uses the pseudopotential approximation described at the beginning of Section Ilb. Another perturbation approach which can be conveniently used in detailed calculations, particularly in connection with the pseudopotential approximation, is the one suggested by L o ~ d i n . ~The ’ contributions to the energy due to the interaction of well-separated states are put in the eigenvalue equations of the strongly interacting states, by correcting every matrix element H,,,, with the additional term

where the sum is over the distant states and E is the exact eigenvalue of the state of interest. V. Qualitative Features of the Band Structure and Preliminary Calculations

12. TIGHT-BINDING APPROACH We can first discuss the energy states at some symmetry points of the reduced zone by considering the general structure of the eigenvalue equation (21) for the zinc blende lattice. The crystal eigenvalues will depend first of all on the atomic eigenstates of the constituent atoms. At every k point a correction will be obtained which depends on the value of k and on the basic integral parameters described in the previous sections. The twocenter overlap integrals and the energy integrals can be estimated from the atomic functions and from a model crystal potential. Only small changes in their values will occur for different 111-V compounds because the basic functions are of the same type. Consequently the corrections on the atomic eigenvalues are expected to be very similar for all the 111-V compounds. At the point T(k = 0) the s-functions do not mix with the p-functions, as indicated in Table V. The s-functions on the two sublattices mix to give covalent Bloch functions of the type One then has two states rl corresponding to the two values of As obtained from the secular equation (20). The two states are widely separated in energy because of the large interaction between atoms A and B; it is to be expected that the lower state will be occupied by two electrons while the 37

P.Lowdin, J . Chem. Phys. 19, 1396 (1951).

62

FRANC0 BASSANI

higher will be an empty conduction state. The p-functions of the representation rI5give covalent Bloch orbitals of the type Pr,A

+ nppx,B

(41)

and their partners. Also in this case one has two states widely separated in energy. The following sequence results :

rlu)< r l s u ) < rl(2) and

l-lS(219

which is consistent with an insulator, since there are eight electrons per unit cell and the state F I 5 ( l )is occupied by six electrons. At the points L and X a similar analysis can be made simply by using Bloch functions of the appropriate symmetry and reasonable values for the parameters. The situation is schematically illustrated in Fig. 2 for the points r and L.

FREE A T O M S

> c

POINT

L

-. . ..

0 Y 1 LL

r,

(2)

\ \

FIG.2. Correspondence between atomic energies and crystal energies at r and L in a 111-V compound. Atom A indicates the group V element. The specific drawing refers to GaAs.

2.

METHODS OF BAND CALCULATIONS

63

At the point X the lowest states are: a bonding covalent function from sA and P ~ , a~ bonding , state formed with sB and p x S A , and a bonding state formed with the two remaining p functions at atoms A and B. The expected sequence of levels is

Xl(l) <= X3(l) < XJ1) 4 X1(2), etc. In the particular case of group IV elements atoms A and B are equal; the above considerations retain their validity except for the fact that the coefficients As, A, of formulas (40) and (41) and the like at L and X are determined by symmetry to be 1 or - 1. The functions which belong to X, and X, in the zinc blende structure have the same energy and become partners of the irreducible representation X I of the diamond structure. Calculations can be easily performed at a large number of k points by solving Eq. (21) with the same parameters used at the symmetry points; at every point one obtains for the valence states a bonding mixture of functions on sublattice A and on sublattice B. The fraction of electrons associated with atom A and with atom B could be computed from such a complete band calculation. We may say qualitatively that the further apart the energy states of the two atoms the smaller the mixing and the larger the fraction of electrons associated with the atom of lower energies. In general more than five of the eight valence electrons are on atom A, which then becomes electronegative. The above rule suggests that 11-VI compounds will be more ionic than 111-V compounds of the same row, and group IV elements will have ionicity 0; a compound like cubic Sic will be ionic, the C atom being electronegative. Arguments of this type can be pursued further and can lead to semiempirical rules connecting the energy gap to the cohesive energy and to the ionicity of the compound.38 Detailed calculations for diamond have been carried out by Hall3’ and by M ~ r i t a The . ~ ~latter has also used the method in a semiempirical way to obtain the shape of the energy bands in Si and Ge near the Fermi level. For the 111-V compounds detailed calculations have been carried out on the valence bands by a number of author^'^.^^ with the method of linear combinations of bond orbitals described in Section lob. The parameter A is determined semiempirically by using experimental atomic values and simplifying assumptions on the resonance integrals (uA,HuB); simple approximations developed for molecules have been adopted for the other integrals. The results obtained for the valence states are in agreement

+

E. Mooser and W. B. Pearson, Progr. Semicond. 5, 103 (1960); G. Leman and J. Friedel, J . Appl. Phys. 33, 281 (1962). 39 A. Morita, Sci. Rept. Tohoku trniu. 33,92 (1949); J. Phys. Chem. Solids 8, 363 (1959). 40 A. A. Nran’yan, Fiz. Tverd. Tela 2, 414 (1960) [English Transl.: Soviet Phys.-Solid State 2, 439 (1960)l; D. Stocker, Proc. Roy. SOC.(London) A270, 397 (1962).

38

64

FRANC0 BASSANI

with the qualitative analysis outlined above. Furthermore, the results indicate that the top of the valence band is the state rI5, in agreement with experimental results, and that the width of the valence band is of the order of 10 eV. The results on the ionicity of the 111-V compounds are also discussed; it turns out that the atom with more valence electrons is electronegative and the values obtained for Q for most 111-V compounds range between 0.4 and 0.5, which is the right order of magnitude. To fit the experimental values of Q quantitatively and to extend the method to the 11-VI and I-VII compounds would probably require a much more elaborate calculation than has been done heretofore. The main difficulty with a priori numerical calculations is that the parameters cannot be computed with sufficient accuracy because some of the two-center integrals depend strongly on the behavior of the atomic potentials far from the nuclei, where they are poorly defined. Furthermore one cannot check the convergence of the procedure by adding more atomic functions. We may conclude that the tight-binding approach gives a good general account of the situation in 111-V compounds but is not well suited for a priori numerical calculations. Slater and Koster” suggested that the method can be best used as an interpolation scheme, the values of the two-center integrals being disposable parameters to be determined from fitting the energies at a few symmetry points. The only difficulty with this procedure is that the number of parameters is very large as can be seen from Table VII, and is probably redundant. We think one can efficiently use the tight-binding approach if one constructs a simple model potential as a function of a few critical parameters to be determined from experimental data or from reasonable assumptions.

13. OPW APPROACH Qualitative rules for the sequence of the energy states at the symmetry points can be obtained from the perturbation approach on the “empty lattice,” as discussed in Section 11. The lowest states of the “empty lattice” will remain the lowest states also when the perturbation is present. The first order correction is positive and its magnitude can be related to the symmetry of the state; s-like states will be raised more than p-like and d-like states because their orthogonality coefficients are larger. In second order, the energy of a state is changed by interaction with unperturbed states of the same symmetry, and the change is smaller as the states are further apart. In Fig. 3 we present the eigenvalues of the “empty lattice” with the symmetry classification for the zinc blende lattice. The sequence of the energy states can be obtained at the symmetry points by applying the rules given above to the unperturbed states of Fig. 3. The same sequence is obtained

2.

65

METHODS OF BAND CALCULATIONS

as in Section 12 from the tight-binding approach. At the point r the lowest eigenvalue will be rl because it is the lowest in the “empty lattice.” The first-order correction to the energy is positive and mainly due to the repulsion of the core states. The second-order correction is rather small because the other rl unperturbed states are quite distant. The next unperturbed state of the empty lattice contains the irreducible representations rI5,r15,rl,and Fl. They are split by the perturbation and we expect the following order: r15(l) < r1(2), r&), r1(3).The state rI5is the lowest for two reasons: it is p-like and is raised less than the rl states by the

k = p (1.1.11 1 2 2

k = (0.0.0)

h*$

(1.0.01

FIG.3: Energy bands in a constant potential to which the symmetry of the lattice is incorporated (“empty lattice”). The eigenvalues are plotted as functions of k in arbitrary units. The states below the horizontal dashed line become the valence states and the lowest conduction states in the crystal. (From Bassani and Yoshimine, Ref. 24.)

66

FRANC0 BASSANI

core states, and it has a strong second-order interaction with the next unperturbed state of the same symmetry belonging to the set (200), as can be seen from the matrix elements of Table VIII. The order of the conr15(2), r,(3)is not so certain and is established on the duction states r,(2), basis of the magnitude of the matrix elements of Table VIII and Table IX. A similar analysis at the points L and X gives the sequences L,(1) < L,(2) < L3(2),and X l ( l ) < X3U) < XA1) < XI@) < X 3 (2) < X&). L3U) < POINT

POINT

L

r

POINT X

0

-0,:

)<111>

-1.0

- 1.5

-2.0

l -2,:

FIG.4. Lowest states of the "empty lattice" and crystal states in cubic S i c at r, L, and X . The first are indicated by black dots and the second by circles. The crystal states have been joined by lines to visualize the energy bands. The data on S i c are taken from Bassani and Yoshimine, Ref. 24.

2.

METHODS OF BAND CALCULATIONS

67

The situation is visualized in Fig. 4, where it is shown how the “empty lattice” eigenvalues will split because of the crystal potential and the core states. An energy gap exists and the material is an insulator when r15, L,, and X , are all lower than the next levels in the sequence; this cannot be decided on a qualitative basis alone; numerical calculations must be performed to prove this point. Similar qualitative remarks can be made concerning the relation between the zinc blende lattice and the diamond lattice. It can be noted that, in the expression (30) for the matrix elements, there are two terms arising from the crystal potential and the core states: a symmetric term which is the sum of the contributions from the two sublattices and an antisymmetric one which is the difference of the contributions from the two sublattices. The second term is much smaller than the first; it is zero in the group IV elements, it increases as atoms A and B are more different and it will be larger for 11-VI compounds of the same row in the periodic table. Some states interact only via the antisymmetric term, and this is the basis for the difference between the group IV elements, the 111-V compounds, and the 11-VI compounds of the same row. Herman and Callaway41 first made this point which has been efficiently adopted by Cardona and G r e e n a ~ a yto ~~ correlate a large number of optical data on different compounds. One can see from the symmetrized functions of Table VI and from the matrix elements of Table VIII that the states near the Fermi levels which interact only via the antisymmetric term are r15(l) - rI5(2), r,(2)- r1(3), L,(1) - L,(2); the states X I and X 3 differ by the sign of the antisymmetric term and are degenerate in the diamond structure. The effect of the antisymmetric interactions is to separate these states further. Consequently one expects that the valence band will be lowered going from the group IV element to the 111-V compound and to the 11-VI compound of the same row, and the energy gap will be increased provided the separation between the conduction states X, and X , does not increase greatly and the state r,(2)is not too much lowered. The detailed calculations performed to date have generally confirmed all the above expectation^.'^. 34*43 They have further shown that the splitting X, - X , in the conduction state is quite small for elements of the same row and decreases as the size of the atoms increases. The splitting 41

43

F. Herman, J. Electron. 1, 103 (1955); J. Callaway, ibid 2, 230 (1957). M . Cardona and D. Greenaway, Phys. Rev. 131,98 (1963); see also M. Cardona, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 3. Academic Press, New York, to be published. L. Kleinman and J. C. Phillips, Phys. ReL>.118. 1153 (1960); F. Herman and S. Skillman, Proc. Intern. Con/: Semicond. Phys., Prague, 1960 p. 20. Czech. Acad. Sci., Prague, 1961 ; F. Quelle, MIT Progr. Rept. No. 43, 77 (1962)(unpublished).

68

FRANC0 BASSANI

between the valence states X , and X 3 is generally quite large and causes a separation of the valence band into two subbands with an energy gap of the order of 1 eV. As an example of the results one obtains for III-V compounds, we show in Fig. 5 the band structure of AlP and for comparison the band structure of Si as computed by Bassani and Y ~ s h i m i n e . ~ ~ The detailed calculations have also revealed a difficulty first pointed out by Herman and Skillman.43 The energies of some of the electronic states are more strongly affected than the others by small changes in the crystal potential or by a shift in the core states. Two such sensitive states are r,(2)and L1(3), r2’and L1(2)in the diamond structure, as can be seen from the structure of their secular matrix. From Table VIII, for instance, we see that the state r1(2),having an unperturbed energy [31, has a diagonal matrix element strongly dependent on the potential through the term 3 Vs(S)and interacts strongly with a near state of unperturbed energy [4] through the matrix element J6 [ Vs(3) + Vs(1l)]. This observation has been useful for the interpretation of changes produced in the band structure by applying pressure and forming alloys44; on the other hand it explains the reason why precise numerical results for comparing with accurate experimental data have not yet been obtained in an a priori calculation. The crystal potential which has been used in connection with OPW calculations has been constructed as the sum of atomic potentials at the lattice sites. Every atomic potential consists of two parts : (a) A Coulomb term due to the electrostatic potential of the nucleus and the atomic electrons,

where Z e is the nuclear charge and p r is the radial density of electrons around the nucleus, p, = 47cr2p(r). (b) An exchange term, which is approximated by the Slater4’ formula :

where p(r) is the density of electrons and e the electron charge. The “core” eigenstates have been taken from Hartree-Fock atomic calculations. The general validity of the model potential described has been discussed by many author^,^^,^^, 45 with particular concern for the exchange 44 45

F. Bassani and D. Brust, Phys. Rev. 131, 1524 (1963); J. C. Phillips, ibid 125, 1931 (1962). J. C. Slater, Phys. Rev. 81, 385 (1951); “Quantum Theory of Atomic Structure,” Vol. 11, p. 14, Appendix 22. McGraw-Hill, New York, 1960; J. C. Phillips and L. Kleinman, Phys. Rev. 128, 2098 (1962).

-0.2

-0p2/ >0.4

\

a - 0.8

-1.6

-1.8

FIG.5. Band structure of AIP (right) and of Si for comparison (left). The values computed are indicated by circles. Note the lowering of the valence band and its splitting into two subbands going from the group IV element to the 111-V compound. Note the small separation X I - X , for the conduction state and the large separation for the corresponding valence state. Energies are in rydbergs. (From Bassani and Yoshimine, Ref. 24.)

70

FRANC0 BASSANI

contribution. From an operational point of view it turns out that the main source of error lies in the fact that the Hartree-Fock atomic states are not eigenstates of the model crystal potential. As pointed out by H e i r ~ eand ~~ Q ~ e l l e ,the ~ ~“core states” should be computed again with the model potential (42) and (43) to reach internal consistency. 14. SPIN-ORBIT SPLITTING It is of interest to consider the effect of the spin-orbit interaction :

where ax,a,,,a, are the Pauli matrices. Some of the electronic states split under the perturbation (44) as indicated in Table IV, and the values of the splittings are of great physical interest. The method used by Kane4’ and Dresselhaus3 for the 111-V compounds consists in diagonalizing the matrix of the operator (44) between the crystal wave functions with their spin functions

a=

(a

and

D=

(:)

-

This procedure is of particular value in connection with the k p method for obtaining the shape of the energy bands near the symmetry points as functions of the splittings at the symmetry points.47 For an a priori calculation of the splittings at the symmetry points it is convenient to evaluate the matrix element of H,, between combinations of crystal functions and spin functions which belong to the irreducible representations of the double group G(k). This can be done with the general procedure described in Section 5 from the character tables of the double group once the transformation properties of the space functions and of the spin functions are known. The transformation properties of the space functions are the same as those of the functions of any basic set and can be obtained from Table I and from the basic functions already used in Part 111. The transformation properties of the spin functions can be expressed in terms of the Eulerian angles associated with every symmetry operation.48 For the state rI5, for instance, including the spin functions but neglecting spin-orbit interactions, one has six degenerate functions f i a , f2a, f3a, fib, 46 47

48

V. Heine, Proc. Roy. Soc. (London)A240, 340 (1957). E. 0. Kane, J. Phys. Chen7. Solids 1. 249 (1957); see also the chapter by E. 0. Kane on the k . p method in this same volume.

M. Tinkham, “Group Theory and Quantum Mechanics,’’ Sec. 5. McGraw-Hill, New York, 1964.

2.

71

METHODS OF BAND CALCULATIONS

fZ/?,f38. By applying to these functions formulas (3) and (2) we obtain the irreducible representation of the double group functions 1

r,

for the two partner

(45)

1

F - -%8

2-Jz

+ $2. + f 3 8 ) ;

for the representation Fs we obtain the four partner functions

1

6

->

- -{flu -

$2.

+ 2f38)

*

As explained in Section 5, the off-diagonal matrix elements of the spinorbit operator (44) with the functions (45) and (46) are zero by symmetry; the diagonal matrix elements give the perturbation correction to first order. From the functions (45) of r7we obtain -2A/3; from the functions (45) of Ts we obtain A/3. The parameter A, essentially positive, is the splitting in first-order approximation, defined as

Similar calculations can be made at the other symmetry points. Higher order contributions must be included if the splitting is not small compared with the energy separation between the simple states. The values of the splitting can be computed in one of the approaches discussed in Part IV. The calculations are greatly simplified by the fact that the contributions to integrals of the type (47) are significantly large only in the region near the nuclei where the wave functions and the crystal potential change rapidly with distance; the expression (47) reduces to the more familiar form

where L, is the z component of the angular momentum operator.

72

FRANC0 BASSANI

Detailed a priori calculations have not yet been made for the Ill-V compounds, but an accurate study of the spin-orbit splitting on group IV semiconductors has been done by Liu49,23within the approach of the orthogonalized plane-waves method. He has shown that the main contribution to (48) is due to the parts of the wave functions arising from the “core states” in formula (26). Consequently the computational problem is reduced to that for the spin-orbit splitting of the “core states” and can be dealt with. It is found that the splittings of the valence states at the point r are slightly larger than the corresponding atomic values, while at the point L they are slightly smaller than the atomic values, in agreement with the experimental results. Similar results are expected for the 111-V compounds. The splitting at r,, will be slightly larger than the weighted average of the corresponding splittings of atom V and atom 111. A qualitative difference between the diamond and the zinc blende structures will occur at the point X where the uppermost valence state X , in the diamond structure is not split by spin-orbit interaction, while the corresponding state X , of the zinc blende lattice is split by spin-orbit interaction into X , and X , of the double group. If g, and g, are the partner functions of X , , the partner functions of X , and X , become 1 -k*U

4

+

k2B)

and

1 -{gd

Jz

k @,U}?

(49)

the second choice of signs referring to the representation X,. The firstorder corrections to X , and to X , are +Ax and -A,, respectively, with

In the tight-binding approximation the splitting 2 4 can be. related to the corresponding atomic splittings on the atoms A and B by using the covalent p functions of the type given in Table V. The coupling coefficients between lattice A and lattice B result in + A for g, and -A for g, from the secular equation (20). We obtain . 2 A A - 111, A B 26, = 3 (1 + A 2 ) 0 , ’

0, being an appropriate normalization factor. In the diamond structure AA = AB and A2 = 1 so that this splitting vanishes. It will become appreciable when the atoms are very different in atomic number and A is small. 49

L. Liu, Phys. Rev. 126, 1317 (1962).

2.

METHODS OF BAND CALCULATIONS

73

VI. Conclusions and Lines of Future Progress From the last sections it appears that, while the general structure of the energy bands of 111-V compounds can be understood from its quantum mechanical foundations, the calculations of precise physical quantities have not yet reached sufficient accuracy to be compared with the experimental data. Depending on the philosophy adopted with respect to energy band calculations in general there are two possible attitudes. The first is to use the band theory approaches as a framework with a number of adjustable parameters from which the quantities of physical interest can be computed and related to experimental data. This attitude is of particular advantage for the interpretation of complicated experiments, such as the structure in absorption and reflectivity and the data of cyclotron resonance. For instance, the semiempirical pseudopotential approach described in Section 11 has been used to derive optical constants in Ge and Si in good agreement with experiments.’’ This method and the k p perturbation method will certainly be used to a great extent in the 111-V compounds, and they seem to be capable of explaining most of the experimental data. A second attitude is to pursue the aim of accurate Q priori calculation, taking into account all effects which have been so far neglected. This leads to very difficult computational problems, and every small progress requires great effort. We feel nevertheless that the basic knowledge of the electronic properties of solids will be greatly increased in this way, and even the cohesive energy of crystal types will be eventually computed. For covalent semiconductors and the 111-V compounds the OPW approach lends itself to further improvements. As already mentioned, one can reach a better internal consistency by using “core states” computed with the potential of Section 13 ; calculations to this effect are under way and the atomic states in this approximation have already been obtained by Herman and Skillman.’l On the other hand, the model crystal potential described in Section 13 is only a rough approximation to the true potential; the Slater formula for exchange4’ is valid for a nearly constant electron density and will not represent accurately the contribution from the “core electrons.” Brinkm a d 2 has proposed a different approximation to the exchange potential on valence and conduction electrons. He separates the contribution from D. Brust, J. C. Phillips, and F. Bassani, Phys. Rev. Lerters 9,94 (1962); D. Brust, Phys. Reo. 134, A1337 (1964). 5 1 F. Herman and S. Skillman, “Atomic Structure Calculations.” Prentice-Hall, Englewood Cliffs, New Jersey, 1963. 5 2 W. Brinkman, Ph.D. thesis, University of Missouri (1964), to be published.

74

FRANC0 BASSANI

the core electrons from the contribution from the valence electrons. The first contribution can be included exactly, and the matrix elements of the resulting nonlocal potential can be computed in a way similar to the terms originating from the orthogonality requirement. The exchange potential due to the valence electrons can then be approximated with the Slater formula using the crystal density of valence electrons and not the atomic electron densities. This procedure can be made nearly self-consistent by recomputing the valence electron density from the band calculation as suggested by Kleinman and Phillips.43 Brinkman’s crystal potential leads very close to the solution of the one-electron problem. Correlation effects between valence electrons can also be taken into account along the lines adopted for the free electron gas. A general theory’ has been obtained by translating the OPW method in the second quantization formalism and using the random phase approximation for the interaction between valence electrons. This results in screening the exchange interaction between valence electrons by an effective dielectric constant which depends on the momentum transfer and on energy. By using the uniform electron gas and allowing the density of electrons to be a function of position, as in the Slater approximation, one can obtain a simple formula for taking into account correlation ~creening.’~ Relativistic effects other than the spin-orbit splitting seem to be of greater importance than previously anticipated. They can be obtained by reducing the Dirac equation to an ordinary Schrodinger equation with correction terms in the Hamiltonian operator. To second order in B = u/c the correction terms are5’ : the spin-orbit coupling (44),the mass-velocity correction -(h4/8m3c2)V4, the Darwin term (h2/8m2c2)V2VC.Only the spinorbit coupling term changes the degeneracy of some states and produces splittings; the other terms give corrections which are equivalent to small . made ~ ~ preliminary changes in the crystal potential. Herman et ~ 1 have estimates of the effect on the energy states produced by such relativistic terms on group IV semiconductors. They found that at the point r the net effect may be a lowering of the conduction state r2’by a quantity which is of the order of 1 eV in Ge and is larger when the atomic number increases. For this reason it appears that accurate a priori calculations must also include relativistic effects. F. Bassani, J . E. Robinson, R. S. Schrieffer, and B. Goodman, Phys. Rev. 127, 1969 (1962). J . E. Robinson, F. Bassani, R . S. Knox. and J. R. Schrieffer, Phys. Rev. Lerrers9.215 (1962). 5 5 A. Messiah, ”Mecanique Quantique,” Vol. 11, Ch. XX. Dunrod, Paris, 1960. 5 6 F. Herman, C. D. Kuglin, K. F. Cuff, and R . L. Kortum, Phys. Rev. Lefters 11, 541 (1963). 53

54

CHAPTER 3

The k p Method E. 0.Kane I. INTRODUCTION

. . . . . . . . . . . . . . . . 75

11. SCOPE OF PRESENTPAPER .

.

111. THE k p REPRESENTATION 1 . Spin-Orbit Interaction 2. Use of Symmetry . .

. . . . . . . . . . . . 77

. . . . . . . . . . . . . 78 . . . . . . . . . . . . . 81

. . . . . . . . . . . . .

82

IV. SPECIFIC DIAMOND AND ZINCBLENDESYMMETRILS. . . . . . 3. The r p o i n t ; k = 0 . . . . . . . . . . . . . . 4. 100 and 11 1 Directions . . . . . . . . . . . . . APPENDIX.. . . . . . . . . . . . . . . . .

83 83 96 99

I. Introduction The k p method was originally a device for exploring the properties of the energy bands and wave functions in the vicinity of some important point in k space with the aid of perturbation theory. As such, the method follows very straightforwardly from the Bloch form of the wave function and was employed for special purposes very early in the development of band theory without being dignified as a “method.” For example, the approach is used by Seitz’ to derive an expression for the effective mass. The mass formulas were extended to the more complicated case of degenerate bands by Shockley.’ Dresselhaus et al.3 added the important ingredient of spin-orbit interaction in their classic paper on cyclotron resonance. The paper of Dresselhaus et aL3 established the importance of the k p approach as a rigorous basis for the empirical determination of band structure. The k * p method, coupled with the use of symmetry, shows that the band structure in the vicinity of a point in k space depends on a small

-

’F. Seitz, “Modem Theory of Solids,” p. 352. McGraw-Hill, New York, 1940. l W . Shockley, Phys. Rev. 78, 173 (1950). 3G. Dresselhaus, A. F. Kip, and C. Kittel, Phys. Reo. 98, 368 (1955).

75

76

E. 0.KANE

number of parameters (band gaps and masses) which may be accurately determined by experiment. It is well known that the cell periodic part of the Bloch function, unk(r), forms a complete set of cell periodic functions for any fixed k.4 From this it follows that, if the energy and momentum matrix elements are known for all bands for any given value of k, the energies for all k are completely determined. We call this the k p representation. The limitations of the empirical k * p method are, thus, not prescribed by perturbation theory but rather depend on how much of the k - p matrix can be empirically determined. In practice, however, it appears that too many parameters are involved for it to be practical to represent the band structure throughout the zone empirically by the k p method. The pseudopotential method developed by Phillips5 appears to have the virtue of describing the band structure throughout the zone with many fewer parameters and hence will always be preferable from the standpoint of simplicity and ease of computation. Perhaps a judicious combination of the k - p and pseudopotential methods will nevertheless provide the ultimate in accuracy. To the extent that the k * p method is restricted to apply only to the masses in a given band, the results obtained depend entirely on symmetry considerations and are completely independent of the character of the Hamiltonian from which they were derived. However, when relations between masses in different bands are obtained or when nonparabolic effects are considered, all possible Hamiltonians are not equivalenL6 The k * p results are customarily derived on the basis of a one-electron Hartree Hamiltonian. Characteristic differences occur if a Hartree-Fock Hamiltonian is assumed.6 In the more general treatment, p is replaced by (- im,/h) [r, HI to which it is equivalent in the one-electron Hartree Hamiltonian without spin-orbit interaction. In the general case one must also consider [r, [r, H I ] which is equal to - h2/mo for the Hartree Hamiltonian, but is not a constant for other Hamiltonians. For nonparabolic effects, higher commutators of r with H also enter. Such commutators are zero for the Hartree Hamiltonian. Conduction band and light hole masses have been compared quantitatively for germanium’ and indium antimonide.8 Any possible effects due to [r, [r, HI] + h2/mo have been within the experimental error. The

-

’

J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955). J. C. Phillips, Phys. Rev. 112, 685 (1958). E. 0. Kane, J . Phys. Chem Solids 6, 236 (1958); E. I. Blount, Solid State Phys. 13, 305 (1 962). L. M. Roth, B. Lax, and S. Zwerdling, Phys. Rev. 114, 90 (1959). D. M. S. Bagguley, M. L. A. Robinson, and R. A. Stradling, Phys. Letters 6, 143 (1963); E. D. Palik, S. Teitler, and R. F. Wallis, J. Appf. Phys. 32,2132 (1961).

3.

THE

-

k p

METHOD

77

nonparabolic effects in InSb have been extensively studied.’ No effects due to [r, [ I , H I ] + hZ/rno or higher commutators of r with H have been detected. None of these experiments provide any information about whether (- irno/h)[r, HI can be replaced by p.

11. Scope of Present Paper In Part 111 we develop the theory of the k p representation for a Hartree Hamiltonian and describe the Lliwdin’O perturbation method which we use to treat it. The Hartree Hamiltonian with and without spin-orbit interaction is described explicitly. The use of symmetry is briefly mentioned. In Part IV the k - p method is applied to the diamond and zinc blende lattices. The zinc blende structure differs from the diamond only in the lack of inversion symmetry. Although theoretically the band structures in the two lattices differ because of the different symmetry, these differences are so small that they have not yet been reliably detected experimentally. This is partly because time reversal symmetry duplicates some of the important features of inversion symmetry. For example, &(k)and a(-k) are always degenerate due to time reversal.’ The symmetry properties and general band structure of zinc blende lattices have been discussed very thoroughly by Dresselhaus and also by Parmenter.12 We will generalize some of this work, particularly as regards nonparabolic effects for bands at k = 0, but we will restrict our discussion to only those symmetry types which are thought to represent conduction or valence extrema as presently identified. We follow Pannenter’s symmetry notation since that is the one most generally used. In the Appendix we treat the “k-p” approach from a more general point of view to demonstrate which features are dependent on the choice of Hamiltonian and which are independent of it. In the course of our presentation we will mention representative values of band parameters but will not attempt to give a thorough list. A complete tabulation has been given by Ehrenreich and more recent data are given by Cardona.

’

9R.Bowers and Y. Yafet, Phys. Rev. 115, 1165 (1959); S. D. Smith, T. S. Moss, and K. W. Taylor, J . Phys. Chem. Solids 11, 131 (1959); E. D. Palik, S. Teitler, and R. F. Wallis, J . Appl. Phys. 32, 2132 (1961); G. B. Wright and B. Lax, ibid 32, 2113 (1961). l o P. Lowdin, J . Chem. Phys. 19, 1396 (1951). L. P. Bouckaert, R. Smoluchowski, and E. Wigner, Phys. Rev. 50, 58 (1936). G. Dresselhaus, Phys. Rev. 100, 580 (1955); R. H. Parmenter, ibid 100, 573 (1955). l 3 H. Ehrenreich, J . Appl. Phys. 32, 2155 (1961); M. Cardona, J . Phys. Chem. Solids 24, 1543 (1963). See also M. Cardona, “Optical Absorption above the Fundamental Edge,” in Volume 3, this series.

78

E. 0.KANE

-

III. The k p Representation We treat the one-electron problem of an electron moving in a periodic potential V(r). The eigenvalue equation for the electron energy E is

Bloch showed that +h may be written $ = eik.runk(r),

(2)

where unk(r)has the periodicity of V(r), and where k lies in the first Brillouin zone and n is a band index running over a complete set of bands. Substituting Eq. (2)in Eq. ( 1 ) gives

For any given k, the set of all unk(r)is complete for functions having the periodicity of V(r). Hence if we choose k = k,, the wave function for any k may be expressed in terms of the wave function for k,, unk(r)

=

1

cnfno(

- kO)un’ko(r).

n’

(4)

We call this the ko representation. We define HkOto be

H

P2 h - - + -k,.p ko-2m m

h2kO2 ++ V(r). 2m

(5)

Then, by the above discussion, HkOUnko

=

En@O)UnIco

7

We can easily convert Eq. (7) to a matrix eigenvalue equation by substituting Eq. (4) in Eq. (7), multiplying both sides of Eq. (7) by unko(r),and integrating over the unit cell in which the u’s are normalized:

3 . THE

-

k p

79

METHOD

Equation (8) is the eigenvalue equation for the point k written in the k, representation. Although Eq. (8) as it stands is correct for any k, it is most useful when k is near ko so that the nondiagonal part of the Hamiltonian, h -Q m - ko) ’,.P can be treated as a perturbation. We use a type of perturbation theory described by Liiwdin.” In ordinary perturbation theory an attempt is made to diagonalize the Hamiltonian h completely by an iterative process which works when all off-diagonal matrix elements h, are small compared to the unperturbed energy separations (q - E ~ )of the interacting levels. In Liiwdin’s method one assumes that all states can be divided into two classes A and B. States in category A may interact strongly with one another but any state in category A interacts weakly with any state in category B. The interactions connecting states in A with states in B are then removed iteratively just as in ordinary perturbation theory, but no attempt is made in this first step to remove matrix elements connecting states in A. After removal of the interactions connecting A and B the states in A are left with “renormalized” interactions with one another. This “renormalized” interaction matrix must then be diagonalized exactly. Let h, be the initial interaction matrix and let hij be the “renormalized” matrix. Then Lijwdin shows 7

In Eq. (10) i,j are in A and fl is in B. Interactions between A and B have been removed in lowest order only. Ei is the eigenvalue of state i which may be determined self-consistently by the eigenvalue equation A

x ( h f j - E3ij)cji= 0 ;

i,j

in A .

(11)

i

The coefficients cji refer to the expansion of the perturbed wave function $; in terms of the unperturbed functions t,bj: A ,B

*i‘

=

1 cji*j. i

Equation (11) gives the coefficients cji for unperturbed states i , j in A . When i is in A but j is in B the coefficients are A

c.. = .It

1 Ei

h,

-

hjj

cki,

j in B,

i, k in A .

80

E. 0. KANE

The eigenvalue Ei in Eqs. (10) and (13) may usually be replaced by the unperturbed value hii. The Lijwdin perturbation method will converge rapidly as long as

(h,l

+

- EB;

(14)

i in A, j in B .

The set A is selected in order to satisfy (14). In particular, if a group of states is degenerate due to symmetry, such as the threefold degenerate p-bands, all degenerate bands would be included in the set A. The degenerate band problem was first treated in this way (prior to Liiwdin) by Shockley.2 Different energy bands for the same k, are separated on the average by 3 to 5 eV. Whenever any group of bands are separated by energies much less than 3 eV, it is appropriate to include them all in category A and study the bands in the vicinity of k,, treating (h/m)& - k,)*p in Eq. (8) as a perturbation by the Lijwdin method. In the case of a single band treated by ordinary perturbation theory, Eq. (8) gives, for the energy in the neighborhood of k,, to second order,

+

The expansion point k, is frequently an extremum so that pnn hk, = 0. Axes can always be chosen so that the quadratic terms in Eq. (15) have no cross product terms (principal form). Assuming k, is an extremum and using principal axes, Eq. (15) becomes

where i is a unit vector in the direction of the ith principal axis. If k, is a general point, the principal axes are a function of the crystal potential, but if k, lies on the (loo), (lll), or (110) axes in the diamond or zinc blende structure a principal axis must lie along the symmetry axis. For the (100)and (111)axes the energy surfaces must be ellipsoids of revolution about the symmetry axis, so that only two mass constants appear, m1 and m,,the masses along and transverse to the symmetry axis, respectively.

3.

THE

-

k p

METHOD

81

1. SPIN-ORBIT INTERACTION

Spin-orbit interaction plays an important role in band structure for many materials. The spin-orbit energy H , should then be added to the Hamiltonian of Eq. (1): h H,, = m [ V V X P ] * O 4m c

Equation (18) then generates two added terms in Eq. (5), namely, Hkso =

2 1

h

-I- *2,

s = -[VVX 4m c

p],

For some purposes it may be convenient to add Hk, to the Hamiltonian H,, of Eq. (6) which generates the k, representation. However, it is frequently preferable to take the k, basis without spin-orbit interaction. Hku, will then be nondiagonal and is usually diagonalized by perturbation theory. The advantage of the latter approach is due to the fact that spinorbit interaction lowers the symmetry of the Hamiltonian, so that states which were degenerate are split. The number of fundamental mass parameters is then increased because of the lowered symmetry. For example, if a trio of degenerate p-states is split by spin-orbit interaction the masses in the split bands are related to each other. If the initial triple degeneracy were not considered, there would be no relations between the masses of different bands. Spin-orbit interaction is very small for the light elements 6ut increases rapidly as one goes down or to the right in the periodic table of the elements. For example, the valence bands of silicon and germanium are split by 0.04 eV and 0.29 eV, respectively. A tabulation of atomic spin-orbit splittings has been given by Braunstein and Kane14 for columns 111, IV, and V. A more extensive tabulation has been given by Herman et aE.” The latter authors and also Johnson et have stressed the importance of relativistic corrections to the Hamiltonian which do not involve the spin. They find these terms are actually larger than the spin-orbit terms but are less noticeable because they do not lower the symmetry. Such terms have not been l4

R. Braunstein and E. 0. Kane, J . Phys. Chem. Solids 23, 1423 (1962).

’’ F. Herman, C. D. Kuglin, K. F. Cuff, and R. L. Kortum, Phys. Rev. Lerrers 11, 541 (1963). L. E. Johnson, J. 3.Conklin, and G. W. Pratt, Jr., Phys. Rev. Letters 11, 538 (1963).

82

E. 0. KANE

explicitly included or estimated in the k - pmethod to date. As we discuss further in the Appendix, to the extent that these or any other terms influence only the magnitude of [ r , H ] they will be automatically included in the empirical k p treatment where the parameters are determined from experiment. Contributions to [r, [r, HI] k2/mo or to higher commutators would not be included in the usual empirical k p representation and might show up in nonparabolic effects or in the comparison of masses in different bands. Such effects have not yet been seen. The smallness of the spin-orbit energy compared to an average band gap of the order of 3 eV usually justifies a perturbation treatment of spin-orbit interactions for the lighter elements. The maximum value of interest for lf2in Eq. (19)is for k = K/2, where K is a principal lattice vector. For Ga, Ge, and As this maximum value is less than 0.1 times the spin-orbit splitting as calculated from Hartree wave functions for the free atom. Since X I weights the core region more heavily than Z 2 ,the ratio of X 2 to lfl will decrease for atoms heavier than Ge and increase for lighter atoms. The absolute values of both Z l and S 2 increase on going down and to the right in the periodic table. we are led to consider Because of the smallness of X 2 compared to Z1 k dependent terms which come from second-order perturbation theory from the product of the XI and k p perturbations. Near k = 0, selection rules to be discussed in Section 3c make these second-order terms very small, probably smaller than X 2 ,but at other points in the zone the secondorder terms are probably more important than X 2 .

-

+

-

-

2. USEOF SYMMETRY

-

The use of symmetry is of great importance in the k p method since the number of undetermined constants is usually reduced by symmetry considerations. An elementary example of the role of symmetry is afforded by the simple case of ellipsoidal energy surfaces given by Eqs. (16) and (17). If the ellipsoid is centered on a 111 or 100 axis in diamond or zinc blende the surface must be an ellipsoid of revolution with the symmetry axis as axis of revolution. In the absence of symmetry five mass constants appear, the three principal masses and two constants to fix the orientation of the ellipsoid. With the above symmetry, the number of constants is reduced to two, the masses along and transverse to the symmetry axis, rn, and rn,. Symmetry also forces degeneracy in many cases of interest. The energy surfaces are frequently nonanalytic in the vicinity of points or lines of degeneracy. In this case the use of group theory to account properly for the symmetry is most important. The use of symmetry in the k - p method for the zinc blende lattice has been discussed most thoroughly by Dresselhaus and also by Parmenter. l 2

3.

THE

k * p METHOD

83

We use the results of symmetry in our discussion of the band structure in the vicinity of the symmetry points which have been empirically found to be important, namely, the zone center, and the (100) and (111) axes.

IV. Specific Diamond and Zinc Blende Symmetries 3. THEr POINT; k =0

The valence band maximum in all known column IV and 111-V band structures occurs at k = 0,17if we ignore the small linear k terms treated in Section 3c. The symmetry type is that given by bonding p-functions in the tight-binding picture, namely, rI5in zinc blende and rz5,in diamond structures. In the small gap materials InSb, InAs, GaSb, GaAs, and InP the lowest conduction band corresponds to antibonding s-states with symmetry type rl.In germanium the r;, minimum is only 0.898 eV above the valence band maximum.’* In all the above cases it is appropriate to lump the conduction and valence bands together in Liiwdin’s class A and put all other states in class B (see Part 111). We then remove all interactions between class A and class B to lowest order in perturbation theory. a. k p without Spin-Orbit Terms

.

If we neglect spin-orbit interaction, we have only the k - p interaction nondiagonal. The “renormalized” interaction matrix for class A may then be written

” In

Ref. 13 Cardona has suggested that this may not always be the case. In fact, using Hensel’s results for silicon [J. C. Hensel and G. Feher, Phys. Rev. 129, 1041 (1963)lthe mass in the 110 direction with spin-orbit splitting neglected is infinite within experimental error (m,/m* = +0.04 0.05).Hence, even in this well-known case, the valence band maximum is just “barely” at k = 0. Experimental evidence for the heavy (110) mass has been obtained by H. Miyazawa, K.Suzuki, and H. Maeda, Phys. Rev. 131, 2442 (1963). S. Zwerdling B. Lax, L. M.Roth, and K. J. Button, Phys. Rev. 114, 80 (1959).

84

E. 0. KANE

0;

B

=

P

= -

diamond structures,

h

i-(slpJx). m

The sums in Eqs. (24) through (29) are over all states uj transforming like Ti as indicated above the summation sign. The r on the left is for zinC blende, on the right, for diamond. The prime on the summation in Eqs. (24), (28), and (29) means that states in our four-dimensional subspace are not to be summed over. Due to time reversal, all ui at k = 0 can be taken to be real. All the constants in Eqs. (24) through (29) are then real. The phase factor (-i) is introduced in Eq. (30) in order to make P real also. The constant B in Eq. (29) is zero in diamond structures because s and z have opposite parity. The parity of p acting twice is positive, of course. All the other constants are closely related in the two structures, although this is not apparent from the r notation. We emphasize the similarity by listing the lowest spherical harmonics which would lead to a given symmetry type using tight-binding wave functions. The f signs give the parity of the (Jlt)symmetry operator for diamond. Either sign gives the same symmetry in zinc blende. Plus signs all give bonding functions and minus signs give

3.

THE

k p

METHOD

85

antibonding functions :

The symmetries on the right of the semicolon are diamond and on the left are zinc blende. The symmetries that are paired in Eqs. (24) through (28) are seen to have the same tight-binding origin. We would expect these constants to have the same magnitude in zinc blende and diamond structures. The constants L, M, N , F', G, H1, H, are nearly identical to those of Dresselhaus, Kip, and Kittel. The primed quantities differ because the interaction leading to the constant P in Eq. (30) is treated explicitly by us, whereas Dresselhaus et al. treated it by perturbation theory which then gives a large contribution to the constant F. F' is thus much smaller than F. L' and N' will also be much smaller than L and N. If we treat the interaction "P" by perturbation theory we obtain the Dresselhaus results : A'

+ P2/(E, - E J ,

F = F'

+ PZ/(E, - E J ,

A

=

L=F+2G,

N

=

F -G

+ HI

-

H2.

In this approximation, which is exact for masses at k = 0, we would replace A', L', and N' in Eq. (20) by A, L, and N , and the interaction P would have been removed. B would also be ignored as leading to energy corrections of higher order than k2. In the work of Roth et ~ 1 on. germanium ~ the constants A and F were found to be equal except for sign, suggesting very little contribution from A' or F'.

86

E . 0. KANE

,

b. Nonrenormalized .X Spin-Orbit Terms In this section we treat the spin-orbit effects” which make the principal contributions in diamond and zinc blende lattices, namely, the XI interaction of Eq. (19) within the s-p subspace. We neglect the interaction .X2 as well as any “renormalized” X , interactions which would arise by removing interactions with higher bands in the Lijwdin manner. In Section 3c we treat the linear k terms which are one feature of the interactions which we are neglecting here. When we consider spin, our four-dimensional s-p subspace becomes eight-dimensional in principle. In diamond, the existence of inversion symmetry guarantees a double band degeneracy of Kramers type,” so that only four distinct bands occur. In zinc blende the Kramers doublets split, but only by very small amounts. We use an energy ordering notation for the four Kramers doublets and affix a -t sign when we wish to indicate a splitting of the doublet. The subscripts increase with energy. In the conventional terminology our notation reads: El = “split-off’ band; E , = “low-mass’’ band; E , = “high-mass’’ band; E , E conduction band. If we transform to the IJ,mJ) representation, the band energies at k = 0 are diagonal H(k= 0)

*

l9

L. 1. Schiff, “Quantum Mechanics,” p. 321. McGraw-Hill, New York, 1949. E. 0.Kane, J. Phys. Chem. Solids 1, 83 (1956).

3.

THE

-

k p

87

METHOD

The choice of phase is such as to give the raising and lowering operators positive matrix elements

The constant E, is the valence band energy at k spin-orbit energy. In what follows we use

E,

=

E,.

+ A/3.

-

=0

in the absence of

(34)

The combined action of spin-orbit and k p interactions leads to an 8 x 8 interaction matrix or an eighth-order secular equation. Such an object is easily handled numerically by the large machines, but for analytic purposes one must make further simplifications. (1) “Small gap” or Three-Band Approximation for Finite A. A good approximation for the small direct gap materials is to first ignore all k p interactions except the constant P in Eq. (20). The interaction P and the spin-orbit coupling of Eq. (32) are then treated exactly. After this has been done, the neglected k p interactions of Eq. (20) are restored by Jirst-order perturbation theory. This approximation gives the masses at k = 0 for all bands exactly in the (100) and (111) directions for zero or finite spin-orbit coupling. The approximation was also tested for the (110) direction in germanium and found to be better than 5 % for the heavy mass band E,, which should be the least accurate case. We will refer to the above approximation as the “small gap” approximation. It could also be called a threeband approximation since it involves a cubic secular equation. The P interaction between a conduction state s and a valence state

-

-

I),=

-

a,x

+ ayy + a,z

The k p interaction polarizes the valence state entirely along the k vector. The spin-orbit interaction is also completely isotropic. The k vector determines the orbital wave function and may also be used as the axis of spin quantization. Because of this isotropy we can use the results we have already written down for k in the z direction. Using Eqs. (20) and (32) it is then easy to see that the states are coupled as follows:

88

E. 0. KANE

If we take the sequence of states

(x’- iy’)t

is1 ;

Jz

;z‘l

(37)

7

we obtain exactly the same matrix. We put primes on x,y, z in (36) and (37) to indicate that they are not the crystallographic directions but any orthogonal set of p functions with z‘ oriented along the k-vector. The secular equation corresponding to (36) is easily written down as

(E’- E,)(E‘

- E,)(E’

- E,

+ A) - P2kZ(E’- E, + +A) = 0 ,

(38)

E’ is the eigenvalue. Let uB b , ci be the real coefficients of the states in the order listed in (36) or (37) corresponding to a given eigenvalue E / of Eq. (38). By our energy ordering definition i will range over 1,2, and 4 since the heavy mass band 3 does not enter in (36) or (38). Then, first-order perturbation theory applied to the neglected portion of Eq. (20) gives the following approximate energies :

+ b i Z ( M k 2+ ( L - M N)(kxzkyz+ kX2kz2+ kY2kz2)/kZ) + Ci2(L’k2- 2(L - M - N)(kx2ky2+ kx2kz2+ kY2kz2)/k2} (kZ(kX2ky2 + kx2kz2+ ky2kzz)- 9kx2kY2k,2)1/2, k fiaibiB -

k

i = 1,2,4.

By Eqs. (21), (23). and (31a). L‘ expressed in terms of the E; as

-

(39)

N‘ = L - N . The coefficients are easily

ai = kP(E/ - E,

+ 2A/3)/N,,

bi = (&/3)(Ei’

-

E,)/Ni,

ci = (Ei’ - E,)(E,’ - E,

+ 2A/3)/N1;

(40) i = 1,2,4,

where N is a normalizing coefficient. The splittings of the Kramers doublets are given by the term in Eq. (39). They are proportional to the constant B which is the only term in Eq. (20) which is incompatible with inversion symmetry. B is, of course, zero in diamond structures. The product uibigoes like k for small k, hence the splitting is proportional to k3, as was first shown by Dresselhaus.” The splitting can never exceed the largest spin-orbit interaction in the problem,

3.

THE

k*p

89

METHOD

of course, hence in the present case the splitting cannot exceed A. At general k-values, splittings of the order of A are to be expected. To lowest order in k, the splitting, E4+ - E4-,for the conduction band is easily found to be E4’ - E4- =

+

4APB{k2(kx2ky2+ kX2kz2 kY2kZ2) - 9k,2k,,2k,2}’~2

3(Ec - E”)(EC - E ,

+ A)

9

(404

which is of the same form as Dresselhaus’ Eq. (20).” The heavy mass states ( x + i y ) t / f i and ( x - i y ) l / a have no interactions with each other or the states in (36) and (37) in the small gap approximation. Treating the neglected portion of (20) to first order, we get

E,

=

E,

h2k2 + kY2kz2) + __ + Mk2 + ( L - M - N )(kx2ky2+ kX2kz2 . k2

(41)

2% A useful feature of Eq. (41) is that it shows that the heavy mass anisotropy depends only on a single parameter y, namely, y = - ( L - M - N ) / ( M + h2/2m,). (4W

This fact has been used by Gobeli and Fanz1 to deduce y from an analysis of the p-type “Burstein shift” in InSb. Gobeli and Fan find a value of y = 2 which is probably within the experimental error of the value 1.6 found by Bagguley et a1.8 (2) “Small Gap” Approximation for A = 0. In the case that A = 0 the “small gap” approximation proceeds very much as in Section 3b(l). Equation (39) is correct for the conduction band and low mass bands, E , and E , , respectively, with bi = 0, and we can write

hZkZ 2% + Ci2{L’k2- 2(L - M - N )

Ei = E [

+ -+ ai2A‘k2

+ k Y Z k Z Z ) / k Z } , i = 1,4 ;

x (kx2kY2+ kx2kz2

a, = kP/N2, ci =

(Ei’ - E J N , ,

N 2 is a normalizing constant. 21

G. W. Gobeli and H. Y. Fan, Phys. Rev. 119, 613 (1960).

(42)

90

E . 0.KANE

The complication that arises in this case is that we now have two degenerate heavy mass bands after the kP interaction has been introduced. We must treat the remainder of the k p terms in Eq. (20) exclusive of kP by first-order perturbation theory as before, but we must use degenerate first-order perturbation theory between the two heavy mass bands. The results are h2k2 ky2kz2) E, = E , + M k 2 + ( L - M - N )(kX2ky2 kX2kz2 k2 2% (44)

-

+

ED

=

+

+

~

. >’”

IL - A4 - Nl{kX4ky4+ kx4kz4 + kY4kz4- k2kx ’k y 2k

ik2;

(3) Effective Mass Test of the “Small Gap” Approximation. We can use the results of Sections 3/44and 36(2) to calculate the masses at k = 0 in the small gap approximation. They are: A finite ; “small gap” P2 ti2 h2 L’ + 2M 2m1 - 2m,+ 3 3(E, - E , AY h2 - h2 2m2 2m, -h2 -

2m3

-

+ + 2L + A4 - ( L - M - N ) (kx2ky2+ kx2kz2+ kY2kz2) k4

3

(kX2ky2+ kX2kz2+ kY2kz2) + M + (L- M - N ) , 2m, k4 h2

-

h2 h2 +A’+____ 2m, 2m, p32 ( E , - E ,

A finite; h2 a’=-+ 2mo

);({ p’

h2

2%

+ exact

L+2M 3 ’

L - M 2

=

__ = a‘

>

(46)

(kx2ky2+ kX2kz2+ kyzkz2) k4

+ +[NZ - ( L - w21

+ p’,

The exact results are transcribed from Dresselhaus et aL3 The masses m, and rn4 are isotropic and agree with the “small gap” results as expected. The

3.

THE

-

k p

91

METHOD

m, and m3 “small gap” results are exact for the (100) and (111) directions. For the (110) direction the “small gap” result differs by less than 5 % from the exact mass in the case of germanium:

A

=

0;

h2

h2

P2

2m4

2m,

( E , - EJ’

-- --+A’+

h2

a =-

2m0

+ M + (L - M

small gap

(kX2ky2+ kX2kz2+ k;kz2) k4

- N)

9

(kX2ky2+ kX2kz2+ kY2kz2) k4 2m1 For a general k point the above expressions are only approximate, but they are exact for the 100, 111, and 110 directions and therefore should be quite accurate in general. The results of Eq. (48) have already been obtained by Shtivel’man.22Shtivel’man noted that the differences in the m3 mass for the A = 0 and A finite cases implied strong heavy mass nonparabolic effects in p-type silicon, as we have discussed in footnote 17. The high degree of accuracy of the “small gap” approximations for the masses implies that it is quite a good approximation for the direct gap III-V and column TV materials. For some purposes the simpler analytic expressions for the masses may prove useful. If the ultimate in accuracy is desired, the 8 x 8 secular equation is readily solved by numerical methods. (4) Two Band Model; A + E, - E,. When the band gap is small compared to the spin-orbit interaction, the analysis of the preceding section can be further simplified by including only two states in the interaction matrix of Eq. (36), namely, the conduction band and the upper valence band. The matrix then becomes ~h2 h2 --

+ L - 2(L - M 2m0

- N)

1

is 7

,/$IT 22

-

1 -(XI

3

+ iy‘u

K. Ya. Shtivel’man, Fiz. Tuerd. Tela 5, 348, 350 (1963) [English Transl.: Soviet Phys.-Solid State 5, 252, 254 (1963)l.

92

E . 0. KANE

The states

have the same interaction matrix. Analytic expressions for the eigenvalues of Eq. (49) are easily written as

El’

=

E, - A - P2k2/3(E, - E,

+ A),

-

The expression for El’ was obtained by perturbation theory. For InSb where A 0.9 eV, E, - E, = 0.23 eV, the above expression is fairly accurate. For small k, Eq. (50) becomes

E,‘

=

E,

+

E 2’ = E -

2P2k2 30% - E,)’ 2P2k2 3(E, - E,)‘

The parabolic approximation of Eq. (51) goes bad when 2P2k2/3(E, - E,) becomes of the order of ( E , - E,)/4 as shown by Eq. (50). Nonparabolic effects in indium antimonide have been observed in many experiments,’ a number of them prior to the derivation of Eq. (50).23 The accuracy of h.(50) can be improved by the use of first-order perturbation theory for the neglected interactions, as in the previous section :

E4’

=

E‘,

+

~

,,had +-B{k2(kX2k,’

$

23

p2 )k’ 3(E4’ - El’)

+ kX2kz2+ kY2kz2)

-

9k,2k,2kz2}”2 ;

(52)

H. J . Hrostowski, G. H. Wheatley, and W .F. Flood, Phys. Rev. 95, 1683 (1954): R. P. Chasmar and R. Stratton, ibid 102, 1686 (1956); R. J . Keyes, S. Zwerdling, S. Foner, H. H . Kolm, and B. Lax, ibid 104, 1804 (1956).

3.

E,

=

E,

THE

h2k02 +_ _ + Mk2 + (L - M 2m0

E,=E,-A+k’

-

k p

tkx2ky2+ kX2kZ2+ ky2kz2); (54) k2

- N)

-

3

3(E, - E ,

a2 =

{ ( E , - E,)’ + 8k2P2/3}”2 + E, - E, 2{(E, - E,)’ + 8kZP2/3)“’

d’

(1 - a’),

=

93

METHOD

+ A)

(55)

3

where a and d are the coefficients of the two states of Eq. (49) in the final wave function. For the case of InSb the corrections to the simple result for E‘, can be evaluated using Bagguley’s cyclotron masses and taking A‘ = B = 0. Bagguley et a1.* find G = -1.2(h2/2mo), H I = -4.7(h2/2rn0). Using these values in Eqs. (22),(31a), and (52) we find that for E,’ = E, + (E, - E,) the correction due to the split-off band (El) is about 11% of ( E , - E J , and the correction due to the free electron term is about +3% and the contribution from G and H, is about - 3 % of ( E , - E,) in the 100 direction.

+

c. Linear k Terms; Zinc Blende Spin-Orbit Effects at

r

Terms of the form (xlpyIz) are allowed by spatial symmetry in the zinc blende but not the diamond lattice. Dresselhaus” showed that such terms are zero with the use ofa combination of time reversal and spatial symmetry operators. Hence in the absence of spin-orbit splitting, linear k terms are absent at k = 0. In the presence of spin-orbit effects, linear k terms may appear. The interaction matrix Hlinfor these terms was shown by Dresselhaus to be of the form

94

E. 0.KANE

k,

0 H lin

c

=--

k, - ik,

2 -2k, &k,

+ ik,)

+ ik, 0

-2k,

+ ik,)

-fi(k,

- f i ( k , - ik,)

0

k, - ik,

2kz

$(k,

-

ik,)

2kz k,

+ ik, 0

.

Our constant C , is identical with Dresselhaus’ constant C. C, results from the interaction iWz of EQ. (19) taken in first-order perturbation theory. C , and C, result from Slof Eq. (19) together with (hjm) k - p in secondorder perturbation theory. Dresselhaus12 has estimated that C , K / 2 0.02 eV for InSb where K / 2 is a zone edge k vector. The other constants may be even smaller since they or higher spherical harmonics [see Eq. (31)]. The valence originate from “8’ band of Sb is 5 p but will contain some 5d admixture which can be coupled by the spin-orbit interaction to a 5d excited state as C b requires. The 5d spin-orbit splitting will be very small so C b and C , may be even smaller than C,. For lighter elements the constants C will be even smaller. No reliable observation of the constant C has yet been reported in zinc blende, but a similar type of term has been measured in CdS.24 Dresselhaus12 showed that the linear k energy had the form

-

E

= E, +_

C{k2

[3(kX2ky2+ kyZkz2

+ k~kz2)]1/z)1/2.

(59)

He also gave the exact fourth-order secular equation for the energy bands 24

G. D. Mahan and J. J. Hopfield, Phys. Rev. 135, A428 (1964).

3.

THE

k*p

95

METHOD

-

including linear k terms and quadratic k terms from the k p interaction. He wrote down exact solutions of the secular equation for the (100)and ( 1 1 1 ) directions. The combination of linear k terms and quadratic k terms results in a valence band maximum at k, away from k = 0. At this point the linear k terms are just twice the heavy mass quadratic terms in absolute value. In the “small gap” approximation, the light mass quadratic terms are then much larger than the linear k terms at k,. Hence we neglect the linear k interactions connecting light and heavy mass bands and obtain the following approximation for the bands which is good for IkJ2 lk,l but not for Ikl 4 lkml.

3fic 2k2

E,’

E2

= E , f ___ {(kx2

+ k?)(k,2 + kZ2)(ky2+ kz2)}112,

’= E , f @2k2 { ( k X 2 + k;)(kx2 + k:)(k;

(60) + k?) - 8kx2ky2k2}1/2,

where E , and E , represent quadratic k terms with mass coefficients as given by m2 and m3 in Eq. (46). If the extremum k, occurs in the (111) direction, as the mass values of Bagguley et al.* show that it will for InSb, the energy surfaces are ellipsoids of revolution with the axis of revolution in the (111) direction. The following expressions for the masses can be derived using Eq. (60):

1 1

2 +2M+-(L-M-N), 3

1 +3M--(L-M-N). 3

A necessary criterion for a maximum in the ( 1 1 1 ) direction is that m, of Eq. (62) be negative. Use of the definitions (22) and (31a) show that this criterion is well satisfied for InSb. The position of the extremum is then given by k, = IJzm,CP21

9

(63)

and the energy of the extremum Em is

Em = lm,C2/h21+ E , .

-

-

(64)

Taking Dresselhaus’ estimate of C K / 2 = 0.02 eV, we compute that lo-’ eV and k, 1 0 - 3 ( K / 2 )for indium antimonide; hence the effect of the linear terms will be extremely small. We should emphasize

Em - E,

96

E. 0. KANE

that the smallness of the constants C , and C , is due to the fact that they have no contribution from the p-like spin-orbit energy. At k points of lower symmetry, terms analogous to Cb and C , will have p-like contributions and should be much larger. 4. 100 and 1 1 1 DIRECTIONS

By analogy with silicon and germanium it is generally believed that there are low lying conduction band extrema in the 100 and 111 directions in the 111-V compound^.'^ Since symmetry requires an extremum at the L and X points in the absence of spin-orbit effects in the zinc blende lattice, it is assumed that the extrema occur at these points. In GaP and AlSb the lowest conduction band is thought to occur at X.’ In InSb, InAs, GaSb, GaAs, and InP the conduction minimum is at k = 0, but higher lying extrema at L or X are close enough in energy to be important in transport processes. As will soon be shown, the extrema cannot be exactly at L , or XI because the spin degeneracy at these points is lifted by energy terms linear in k. The situation is analogous to the valence band maximum at k = 0. However, the linear k terms at L , and X , are expected to be much larger than the terms at k = 0. The linear terms along symmetry lines were first determined group theoretically by Dresselhaus” using the double group. We derive the symmetry relations for the single group and add spin-orbit interaction as a perturbation. Our results agree with those of Dresselhaus, but we get an expression for the interaction constant which allows a better estimate of its magnitude. The energy bands in the vicinity of an L , or X , point or at a A, or A 1 extremum may be shown to have the form

where E , is the energy at the expansion point, k,, and k, is assumed to be an extremum for a,= 0. The quantity k, is the component of k - k, along the symmetry line, and k, is the component transverse to the symmetry line. The k signs indicate a splitting of the bands which are “spin degenerate” on the symmetry axis. The quantity a, is proportional to the spin-orbit interaction as we will indicate with explicit formulas. For finite a,, the actual extremum occurs away from the symmetry point k, for the values

3.

THE

-

k p

METHOD

97

The extremal energy is

(assuming is positive). To within the accuracy of Eq. (65), the extremal energy occurs over the circle k, = constant rather than at an isolated point or set of points. Since we find Em, to be very small, the extremal circle should be a very accurate approximation. The occurrence of such “toroidal energy surfaces” was noted for wurtzite structures near k = 0 by C a ~ e l l a who , ~ ~ also described some of the transport effects to be expected from such surfaces. A striking decrease in the Hall effect at low temperature for a magnetic field along the toroidal (c) axis was one important property noted. Such effects cannot occur here because the Hall effect must be isotropic in zinc blende structures. However, the magnetoresistance may show anisotropy. The possible existence of “extremal ~OOPS” for zinc blende structures has been noted by Sheka.26 RashbaZ7has studied the influence of these structures on spin resonance and cyclotron resonance. We have shown that “extremal loops” or circular rather than point extrema occur for A,, L,, A,, X , extrema, namely, for all the cases which are likely to occur. (The possibility of minima with other symmetry types is not definitely ruled out.) We now list explicit expressions for the coefficients in Eq. (65) for the A,, L,, A,, X , symmetry types : A, or L,. The double group corresponding to A, is A,, which is doubly degenerate :

where Re means “real part of.” S is defined in Eq. (19). A& denotes the ith wave function of the degenerate set j which transforms as A3. x and y are cubic directions, x, is in the 111 direction. 25

26

‘’

R. C. Casella, Phys. Reo. Letters 5 , 371 (1960). V. I. Sheka, Fiz. Tuerd. Tela 2, 1211 (1960) [English Trarrsl.: Soviet Phys.-Solid State 2, 1096 (1960)]. E. I. Rashba, Fiz. Tuerd. Tela 2, 1224 (1960) [English Trans/.: Soviet Phys.-Solid State 2, 1I09 (1960)l.

98

E. 0.JUNE

A l . When spin is considered, A1 goes into the double group symmetry As, which is twofold degenerate. We take the x-axis as the symmetry axis :

(AiOIPy(Ad> (A,jlSy(Aio) -k (AiolPdAdi) (Adlsyl 81') (Eo - E j ) A, and A4 are degenerate by time reversal. X, point. The symmetry type X I goes into the double group symmetry X , which is doubly degenerate : u: = -2h Rex m

j

In estimating the parameter a, we approximate the spin-orbit matrix element GlSlO) by the value it would have near k = 0 in the valence band, namely, I(jlSl0)l

-

A/3.

(71)

This is partly justified since we expect some valence band admixture in the conduction band (state 0), and we may take j as a valence band state. Equation (71) is probably an overestimate. Using (71) and considering just one virtual state j we calculate

where we have used Eq. (67) and ignored the operator Re. Our estimate has not reflected the fact that Eminshould be zero in diamond crystals. We feel that (72) is a considerable overestimate of the effect, probably by a factor of 10 or more. The effect is therefore quite small. AlSb may have an Emin between 0.001 and 0.01 eV.

3.

THE

k ’p

99

METHOD

Appendix

We derive the k, representation in a more general way to illustrate which features of the results depend on the choice of Hamiltonian. We take a general one-electron Hamiltonian H having the periodicity of the crystal.28 The wave function can still be written in the Bloch form: = eik.runk(r).

I)

(All

The Hamiltonian operating on u may be written

Z

1 k,k,k,[r,,

+-

3!jl,v,a

[ry,[YA, HI11 + . . . .

For the Hartree Hamiltonian of Eq. (1) the commutators have the form

h m

- i[r, HI = - p ,

For the Hartree Hamiltonian all higher commutators are zero. In the empirical approach the matrix elements of p are determined from experiment subject only to symmetry relations. The symmetry properties of --i[r, HI are no different from p, hence there is no empirical distinction between Hamiltonians for this term. The second-order commutator is a universal constant for the Hartree Hamiltonian but not for the Hartree-Fock Hamiltonian6 or the approximate Dirac Hamiltonian.”

H

P2

=-

2m

P“ - ___ ih ( V V . p) + ____(V - E ) P 2 + . + V ( r )+ s - 0 - 8m3c2 4m2c2 2mc2 2m ~

* *

.

(A6)

The second-order commutator contribution from the exchange operator has been estimated to be smalL6 Blount6 has noted that the commutator

’*

Blount has shown how the k, representation can be used for a many-electron Hamiltonian. See Ref. 6.

100

E. 0.KANE

relations for the exact Dirac Hamiltonian are extremely simple. It may be simplest to treat relativistic effects from this point of view. However, it is then necessary to think in terms of negative energy states. Relativistic effects have not yet been included in k p formulations though they have been found to be important for band structure calculations.’ All commutators higher than the second vanish for the Hartree Hamiltonian but not for the exchange or approximate Dirac Hamiltonians. For the exact many-electron Hamiltonian the commutator relations are also disarmingly simple. As in the case of the Dirac equation, the formal simplicity of the many-electron Hamiltonian as compared, say, to HartreeFock is more or less compensated by the much larger function space which is involved. For a given energy interval many more bands would exist in a many-electron vs. a one-electron picture. This may not be true at very low energies however. As we have noted, the empirical k - p results depend on the assumed Hamiltonians for commutator terms of second order or higher. The Hartree Hamiltonian predicts definite relations between masses in different bands with small energy separations. It also predicts definite nonparabolic effects. Even in the Hartree case higher bands make some indeterminate contribution to mass relations between bands and to nonparabolic effects, so that the Hartree results are not completely quantitative. Mass comparisons have been made by Roth et al.’ for germanium. Nonparabolic effects have been extensively studied in InSb.’ No effects from higher order commutators have been observed. The success of the empirical k p method based on the Hartree Hamiltonian undoubtedly rests on the empirical determinations of the matrix elements of p. The apparent unimportance of higher order commutators may be due to the simplicity of the true many-electron Hamiltonian where higher order commutators are also absent. If the additional bands due to the many electron degrees of freedom are at energies high compared to the conduction and valence bands, the “small gap” approximation may work well in the many-electron case also.

-

’9

-

CHAPTER 4

Effect of Heavy Doping on the Semiconductor Band Structure V. L . Bonch-Bruevich I . INTRODUCTION .

. . . . . . . . . . . . . . .

101

I1 . MAINFEATURES OF THE HEAVILY DOPED SEMICONDUCTORS . . . 1. Impurity Levels and Impuritv Bands . . . . . 2. Averaging over the Configurations. Crystal Momentum 3. Screening . . . . . . . . . . . . .

. 102 . . . . 102 . . . . 104 . . . . 106

111. CHARACTERISTICS OF THE MANY-ELECTRON SYSTEM. 4 . Charge Carriers . . . . . . . . . . 5 . Density of States . . . . . . . . . 6 . Complex Conductivit-v . . . . . . . .

. . . . . 108 . . . . . 116 . . . . . 118

IV . PERTURBATIONTHEORY. . . I . Electron-Electron Interaction 8 . Electron-Impurity Interaction 9 . Electron-Phonon Interaction

. . . .

. . . .

. . . .

. . . .

V . SEMICLASSICAL APPROXIMATION . . . . . 10. Green Functions . . . . . . . . 11. Density of States . . . . . . . 12. Remarks an the Density of States Tail . 13 . Fermi Level . . . . . . . . . 14. Momentum Distribution Function at Zero Screened Potential . . . . . . . 15 . RPsumP . . . . . . . . . .

. . . . .

108

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

119 119 123 126

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

128 128 131 137 138

Temperature and the

. . . . . . . 139 . . . . . . . 141

.

I Introduction This chapter is devoted to the problem of the influence of doping upon the band structure of semiconductors. In the second part. the qualitative features of the system in question are considered and the necessity of taking the many-body effects into account is stressed. In the third part. the necessary notions are introduced and the mathematical tools needed to handle the problem are reviewed briefly. The fourth part treats the different interactions from the perturbation theory point of view. This approach 101

102

V . L. BONCH-BRUEVICH

becomes invalid in certain cases of interest and then the other way of handling the problem must be used. This is the semiclassical approximation used in the fifth part of the paper to consider the density of states tail in the forbidden band.

11. Main Features of the Heavily Doped Semiconductors The study of the heavy doping effect on the band structure is the first necessary step in the theoretical investigation of the electronic properties of the semiconductors containing a high enough impurity concentration. The interest in these materials has been stimulated recently by the tunnel diodes physics as well as by a study of the optical properties of the A"' BV compounds. However, substances of this kind seem to be of interest from a somewhat broader point of view. Besides being widely used in thermoelectric devices, semiconductor catalysts, etc., the heavily doped materials present a part of the general problem of the energy spectra of the disordered structures. This is probably one of the main theoretical problems of condensed state physics. Important results have been obtained in this field concerning the phonon spectra.'-5 Our problem is similar to this in some respects. On the other hand the electronic spectra have some specific features which contribute to the basic concept of the heavily doped semiconductor. From the purely theoretical point of view the study of the heavily doped materials has a serious methodical interest. We shall see that this is essentially a many-electron problem, and the new methods developed recently in this field are of much use (if not completely necessary) here. 1 . IMPURITY LEVELS AND IMPURITY BANDS

It is well known that the impurity atoms when treated independently from one another may form discrete energy levels in the forbidden band. The wave function of an electron occupying such a level is concentrated in the vicinity of the atom in question. Clearly the notion of independent impurity atoms is justified only when their wave functions do not overlap appreciably (light doping). Once such an overlap takes place the impurity levels begin broadening-similar, to some extent, to what happens with the F. J. Dyson, Phys. Rev. 92, 1331 (1963). Cimento 3, Suppl. 4,716 (1956). I. M. Lifshitz and G. 1. Stepanova, Zh. Eksperim. i Teor. Fiz. 30,938 (1956) [English Trunsl.: Soviet Phys. J E T P 3, 656 (1956)l. A. A. Maradudin, P. Mazur, E. W. Montroll, and G. H. Weiss, Rev. Mod. Phys. 30, 175 (1958). S. Takeno, P r o p . Theoret. Phys. ( K y o t o ) 25, 102 (1961); 28, 33 and 632 (1962).

* I. M. Lifshitz, Nuovo

4. EFFECT OF HEAVY DOPING

103

regular lattice atoms when the crystal is being formed from the gas phase. Then the allowed energy region arises in the forbidden band. This region, separated by a gap from the normal conduction band,sa is usually called the impurity band. Yet the term is not so well designated: there is no periodic dependence of the energy on the crystal momentum in this case. Moreover, the very notion of the crystal momentum in the “impurity band” needs a justification since the impurity atoms are not, generally, periodically spaced. One should rather think of them as randomly distributed in the crystal. Once the sample is macroscopically homogeneous the impurity concentration averaged over the big enough part of the crystal is constant, of course. Yet there may be appreciable fluctuations of the impurity concentration in small volumes (of the order, say, of the range of the interatomic forces). This leads to the second mechanism of level broadening due, classically, to the fluctuations of the electron potential energy. The case when there is an impurity band, separated both from the conduction and valence bands of the host crystal, is naturally referred to as intermediate doping. The relevant impurity concentration n may be obtained experimentally by studying the electrical properties of the material at low enough temperatures6 When n increases the impurity band becomes broader, which fact reflects itself, for instance, in the corresponding decrease of the activation energy determining the electron concentration in the conduction band.’q8 Finally when the impurity concentration becomes high enough the activation energy is zero.8aThis means that the top of the impurity band merges into the conduction band. In such a situation the notions of the conduction and impurity bands taken literally lose their strict meaning. There is a sole region of the continuous spectrum. The material where the impurity band has merged into one of the “host” bands of the crystal will be called heavily doped. The first feature of the heavily (and intermediately) doped material follows directly from the definition : the impurity is responsible in this case both for the spectrum and for the scattering of the charge carriers. We recall that in the opposite case of light doping the electrons trapped in the SaHereand in what follows we consider the n-type sample. Nothing changes in principle in the p-type case (though there are of course quantitative differences due to the different spectra of electrons and holes in a perfect crystal). N. F. Mott and W. D. Twose, Advan. Phys. 10, 107 (1961). ’ G. L. Pearson and J. Bardeen, Phys. Rev. 75, 865 (1949). * H. Fritzsche and M. Cuevas, Pi-or. Intern. Conf. Semicond. Phys., Prague, 1960 p. 222. Czech. Acad. Sci., Prague, 1961. saThe critical concentration is, respectively, 3 x 10” C I T - ~ and 3 x 10l8cnC3 in Ge and Si doped with the group V donors; in the InSb case it is substantially lower.

104

V . L. BONCH-BRUEVICH

impurity levels take practically no part in the transport phenomena, and the structure of the conduction band is not practically influenced by the impurity. There follows as well the second feature of the heavily doped samples: the gas of the charge carriers is expected to be degenerate, at least partly (excluding the case of almost exact compensation). Because of this such materials are often called degenerate.

2. AVERAGING OVER

THE

CONFIGURATIONS. CRYSTAL MOMENTUM

To see the third feature of the heavily (and intermediately) doped materials recall that the solution of any purely mechanical problem of electron motion depends upon the configuration of impurities. This of course is never known precisely (even if it were known, the solution containing lo'* parameters would be of no interest); moreover, it may be different in the different parts of the crystal. On the other hand, the result of the macroscopic experiment refers usually to the whole crystal and not to its small parts. In other words, the macroscopic experiment automatically includes averaging over the impurity distribution. The necessity of such an averaging is the third feature of the materials in question. Let N be the total number of the impurity atoms in the sample and let their position vectors be R , , R,, . . . , R w Let P ( R l , . .. , RN) d R , . . . dR, be the probability of the first atom being in the volume element d R , near the point R , , the second atom-in the element dR, near the point R,, etc. Then the averagesb value of any physical quantity A(R,, .. . , RN) is

where the integral is over the volume of the sample, normalized to one :

a. The function P is

In what follows we shall assume the impurities to be distributed uniformly on the average. This means that an integral of the type (1) taken not over the whole sample but over a suficiently large domain of it does not depend upon the domain's position in the sample. The sample of such a type is naturally referred to as macroscopically homogeneous. Clearly it possesses the translational symmetry of the perfect lattice (once we are interested in the average quantities only). This makes it possible to introduce the notion of the crystal momentum of the charge carrier. Namely, due to the translational symmetry of the lattice the one-particle density matrix 8bNotto be confused with the thermodynamic average.

4.

EFFECT OF HEAVY DOPING

105

pI(r, r‘) averaged in the sense (1) has the properties

pl(r + a,r‘) = pl(r,r’

+ a) = pl(r + a , r ’ + a) = p l ( r , r ’ ) ,

(3)

where a is the lattice vector. Thus the eigenfunctions of the operator p 1 are the Bloch functions characterized in particular by the crystal momentum p and the band index L9 The eigenfunctions of the operator p1 are convenient when calculating the expectation values of the quantities described by additive operators (i.e., the current density). To see this one has to recall that the matrix elements of p1 may be expressed in the second quantized form as1’*” ( P h J P ’ ~ ’= ) (a,lap,,,) .

(4)

Here a,, and aplare the operators creating and annihilating the particle in a state (p, I), the symbol (. . means taking the quantum-statistical expectation value with respect to the full density matrix p a)

(...) = Tr(p-..),

(5)

and the trace is taken with respect to the second quantization variables. In what follows we shall always use the operators in the Heisenberg picture: in Eq. (4) they are taken at equal times. When there are no external fields present the matrix (4) is of course diagonal. Then the expectation value of the quantity described by some additive operator A is given by

where (pIIAIpl) are the corresponding matrix elements and the sum over p’ extends over, say, the first Brillouin zone. To get the nonzero value of the current density j one has to consider the system in presence of the external field (superconductors excluded). Then the matrix p1 becomes nondiagonal and, therefore,

The subscript f means that account is to be taken of the external fields when calculating the expectation value. In the case where no interband transitions are possible (low enough fields and frequencies), Eq. (7) may be lo

’I

S. V. Vonsovski, Usp. Fiz. Nauk 48,289 (1952). N. N. Bogoliubov, “Lectures on Quantum Statistics” [in Ukrainian]. Rad. Shkola, Kiev, 1948. V. L. Bonch-Bruevich and S. V. Tyablikov, “The Green Function Method in Statistical Mechanics,” Chaps. 1 and 2. North-Holland Publ., Amsterdam, 1962.

106

V. L. BONCH-BRUEVICH

transformed approximately to the intuitively clear expression (j> =

C @lljlph@,9.

(7a)

P, 1

Here

I ) =
3. SCREENING On raising the impurity concentration the free carrier concentration n, is usually increased as well. Thus the carrier-carrier interaction may become important. This leads, in particular, to the screening of the impurity fields by the free charges. What it may result in is best understood by considering first a rather special (though experimentally realizable) example. This is the case of the lightly doped sample under the conditions of strong injection, In other words, let us consider the semiconductor containing a small number of the donor atoms and many free electrons and holes. To simplify matters we treat the case of the usual “shallow” donors in the rigid lattice (no polaron effects). The standard effective mass method ‘‘*We shall see that in the limiting case of the very heavy doping the damping is small in a certain sense in the energy range that is of importance for a number of problems. In this case one may use the concept of the dispersion law but not of the impurity corrections to it.

4. EFFECT OF HEAVY DOPING

107

may be used” for the simple parabolic band. (This is by no means an essential limitation in the present context.) Then the electron energy spectrum is determined from the Schrodinger equation : h2

--V2+ 2m

+ V(r)rC/= E$.

Here m is the effective mass, V(r)the electron-donor interaction energy, and E the eigenvalue referred to the band edge. The region E > 0 corresponds to the conduction band, the negative eigenvalues, if there are any, form a discrete set and describe the impurity levels in the forbidden band. The point is that the interaction V ( r ) is screened by the other free charges.”” Were they absent, the function V ( r )could be approximated by the Coulomb law, - e2/&r,E being the lattice dielectric constant. The screening transforms this into a short-range potential. Its particular form has been calculated by various authors for different cases. For instance, in case of the nondegenerate electron gas the well-known Debye formula I/ = - e 2 / uexp( - r/ro) (9) was shownI3 to give a reasonable approximation under practically all experimental conditions. For our purpose just two points are important : (1) For any form of the screened potential JlVl dr < co, and, therefore, the number of discrete eigenvalues of Eq. (8) is finite.14 This number depends upon the value of the screening radius r o ; in particular it may be equal to zero. This fact has a very simple meaning: the screening weakens the electron-donor interaction and this results in the discrete levels coming closer to the band edge and finally disappearing when ro becomes small enough. (2) The value of the screening radius and therefore the spectrum of Eq. (8) depends upon the electron concentration (and temperature). Thus l6 it may be, to some extent, varied e~perimentally.’~. I z W. Kohn, Solid State Phys. 5, 258 (1957). IZaThe use of the screened potential in a one-electron equation (8) needs justification. This may be given, i.e., by the Green function method”; we shall return to this question in the next section. V. B. Glasko and A. G . Mironov, Fiz. Tverd. Tela 4, 336 (1962) [English Transl.: Soviet Phys.-Solid State 4, 241 (1962)J. l4 L. D. Landau and E. M. Lifshitz, “Quantum Mechanics-Non-Relativistic Theory,” Chap. VII. Moscow, 1948 [Translation by J. B. Sykes and J. S. Bell, Pergamon, London and Addison-Wesley, Reading, Massachusetts, 19581. V. L. Bonch-Bruevich, Fiz. Tverd. Tela, Akad. Nauk SSSR, Otd. Fiz.-Mat. Nauk, Sb. Statei 2, 177 (1959). l 6 V. L. Bonch-Bruevich, Fit. Tverd. Tela 3, 768 (1961) (English Transl.: Soviet Phys.-Solid State 3, 558 (1961)J.

’’

108

V. L. BONCH-BRUEVICH

Obviously the deepest discrete level merges into the band when the screening radius becomes of the order of the Bohr radius in the crystal, a, = &*/me2. In the case of the Debye potential (9), machine calculation was carried out." When E = 16, m = 0.2m0 (mo is the true electron mass), the critical free carrier concentration above which there are no discrete levels is about 10'' cmP3. This value is typical for heavily doped semiconductors (at any rate when there is no appreciable compensation). Thus the screening effect is of primary importance in these substances. Evidently the usual treatment of the impurity band as a result of the level broadening becomes irrelevant under level disappearance conditions. However, this does not mean as yet that there are no impurity bands. Really the case treated above is somewhat different from that of the heavily doped material since in the last case the free charge carriers come not from the exterior sources but from the impurity itself. This means that the problem is a self-consistent oneI6: the answer to the question of whether there are impurity levels (or bands) depends upon the electron distribution while the latter is itself determined by the energy spectrum of the crystal. In other words no a priori assumptions about the energy spectrum should be made; the answer must come from the solution of the appropriate dynamical problem, both the electron-donor and electron-electron interactions being taken account of. The only thing that may be conjectured beforehand in the heavy doping case is the absence of the usual discrete shallow impurity levels ; they certainly do not correspond to any self-consistent solutions, since the overlap of the corresponding localized wave functions would not be small at n l O I 7 to 10l8 cmP3. The necessity of the self-consistent and, therefore, fully many-electron approach to the problem of the energy spectrum is the fifth specific feature of the heavily doped materials.

-

111. Characteristics of the Many-Electron System

4. CHARGE CARRIERS

We have seen in the previous section that the study of the electron energy spectrum of the heavily doped semiconductor is, in principle, the problem of the many-electron theory of solids. Then the natural question arises as to what extent is it possible to use in such a theory the standard band theory concepts. For instance, what is the many-electron meaning (if any) of such notions as the band edge, the density of states, etc? The answer was given by various authors in the last decade. We give here a short review of some results that are needed to understand what follows. The proofs of the " V.

L. Bonch-Bruevich and V. B. Glasko, Opt. i Spectroseopia 14, 495 (1963).

4. EFFECT

OF HEAVY DOPING

109

statements made below may be found in the papers cited; in particular all the declarations of the type “it may be shown” with no reference given refer to Ref. 11. First of all it is necessary to define the notion of “the charge carrier.” This has been done by various authors 11,18-zz (see also the newly published textbook by Pinesz3)from the equivalent (though formally somewhat different) points of view. The idea of the solution is suggested by Eqs. (6H7a) as well as by the fact that there are no charge carriers in the pure semiconductor at very low temperatures ; the charge carriers are to be considered as the elementary excitations characterized by a definite crystal momentum and, possibly, by some discrete quantum numbers. In other words, the problem is to find the energy change of the whole many-electron system when one of the occupation numbers n@, I ) is changed by one. It seems natural to treat this energy change, W @ ,I ) as “the energy of the charge carrier,” yet the name needs a justification : the term “energy” is used to characterize the stationary state. Therefore one has, assuming the state @,l) to be prepared at the moment t’, to consider the probability of observing it at the later moment t > t’. Clearly the relevant probability amplitude is

It is clear from general principles that A,@, t’) may either oscillate (with the constant modulus) or decay when the difference t - t’ increases. In the first case the state is stationary, and the concept of the charge carrier energy has an exact meaning. In the second case the state @,l) is nonstationary, and the concept of the charge carrier energy cannot be strictly introduced. It may have but an approximate sense if the damping constant is small enough: the “level width” must be small compared to the energy W @ ,r) measured from its minimum value. In what follows we shall always have in view both the energy and the damping constant when speaking about the energy spectrum of the charge carriers. The procedure described above may be divided into two stages. Consider first the pure semiconducting material. At the first stage the elementary excitations of the many-electron system are separated out. They are the conductivity electrons (called simply “electrons” in what follows) and S. I. Pekar, Zh. Eksperim. i Tear. Fiz. 18, 525 (1948). S. V. Vonsovski, Izv. Akad. Nauk SSSR, Ser. Fiz. 12, 337 (1948). * O W. Kohn, Phys. Reu. 105, 509 (1957); 110, 857 (1958). “ A. E. Glauberman, V. V. Vladimirov, and J. V. Stasyuk, Fiz. Tuerd. Tela 2, 133 (1960) [English Trans/.: Souiet Phys.-Solid State 2, 123 (1960)l. 2 2 V. L. Bonch-Bruevich, Uch. Zap. L‘uousk. Gas. Uniu. 33, No. 1 (6), 59 (1955). 23 D. Pines, “Elementary Excitations in Solids.” Benjamin, New York, 1963. l9

110

V . L. BONCH-BRUEVICH

holes.23a They appear in pairs; the minimum energy A needed to create such a pair is just what is usually called "the forbidden band gap." Since the recombination time is very large compared to all the other characteristic electronic times, the total number of electrons and holes is practically constant. Thus we come to the (by now standard) problem of the mixture of the two imperfect gases. The forces between the quasiparticles that they consist of--electrons and holes-are mainly Coulombic. The influence of the crystalline lattice shows up in (a) the periodic field acting upon the quasiparticles and (b) the reducing of the Coulomb interaction due to the polarization of the medium. In a homopolar crystal the latter effect may be approximately taken account of just by introducing the static lattice dielectric constant E in the force law. This is valid provided the frequencies W(p,l)/h [W(p,r) being referred to its minimum value] are far from the dielectric dispersion regions. This means that the excitation energies must be small enough, a condition that is usually met for most of the carriers in such materials as Ge or Si. In polar or partly polar crystals (in particular of the A"' BV type) only the electronic part should be included in E, the ionic part of the dielectric permeability is to be treated separately, using some kind of the polaron concept (see Section 9). The situation is essentially the same in case of an imperfect crystal as well. The only trivial difference comes from the fact that in this case the pair of excitations may consist, for example, of the conductivity electron and the "hole bound to the donor." Again we come to the problem of imperfect gases of quasiparticles interacting both with one another and with the lattice imperfections. The study of an imperfect electron (or hole, or electron-hole) gas constitutes the second stage of the problem of the semiconductor energy spectrum. To be speclfic we consider the n-type material neglecting holes. The field of the perfect lattice may be excluded from explicit consideration by using some kind of the effective mass According to Bonch-Bruevich and Tyablikov" the problem reduces to studying the time dependence of the correlation function A,, . Consider the state of the thermodynamic equilibrium. Then the expectation value in Eq. (10) is to be evaluated with the help of the density matrix 23"Thereare still excitations of the exciton type. They are of no interest to us in the present context since they carry no electrical current. l4 S. 1. Pekar, Zh. Eksperim. i Teor. Fiz. 16, 335 (1946). 2 5 J. C. Slater, Phys. Rev. 76, 1592 (1949). 26 S. I . Pekar and M. F. Deigen, Tr. Inst. Fiz. Akad. Nauk Ukr. SSR 7 , 108 (1956). 2 7 C. Kittel and M. Mitchell, Phys. Rev. %, 1488 (1954). J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955).

4.

111

EFFECT OF HEAVY DOPING

p = C exp B(pN

- H ),

(11)

where fi is the inverse temperature (in energy units), C is the normalization factor determined from the condition T r p = 1, H and N are the total Hamiltonian and the total particle number, respectively, and p is the Fermi level. Evidently in this case the amplitude A,,(t,t’) may depend only on the difference t - t‘ and one may Fourier analyze it :

It may be shown that the spectralfunction I @ , 1 ; v ) is real and nonnegative. Clearly the time dependence of A,, and, therefore, the spectrum of the charge carriers is determined by the singularities of the spectral function in the complex v-plane. To calculate the spectral function it is convenient to use the retarded and advanced Green functions, Gr and G,. In the (p, 1) representation they are defined by the relations

r

G,@, 1 ; pi, ;t - t’) = ie(t - t‘)([a,,(t),a,.,.(t’)l+> (13)

r, C,(p, Z;p’, Z’; t - t’) = -iO(t’ - t)([a,,(t),
iiP,,4’)]+> (14)

Here O(t - t’) is the well-known step function: O(x) = 1, x > 0 and 6(x) = 0, x < 0, [. . . ,. . .I+ is the anticommutator. It follows from Eqs. (12)-(14) that the spectral function is expressed directly in terms of the diagonal elements of the Green-functions Fourier transforms :

E = hv. To obtain the equations for the Green functions we differentiate with respect to t and use the relation dU

ih - = (uH - H a ) . at

(16)

The Hamiltonian H contains in our case the effective kinetic energy W,@, I ) determined by the dispersion law in the perfect crystal, the electronelectron and electron-impurity interaction energy and, if needed, electronphonon interaction. (This will be taken account of only in Section 9.) The

112

V . L. BONCH-BRUEVICH

form is

H =

I~ dx’(xlW0 x + U,(x’)G(x)u(x’) + t l d x dx‘ V,(x

- x’)~(x)~~(x’)u(x’)Q(x)

+ electron-phonon interaction energy. It is convenient to use the coordinate representation and introduce the second Green function GJx”, x”, X, x’; t - t’) = Z(x”)u(x”)a(x) 1a(x ’) ) ,

(17)

x, x’, x” being the coordinate vectors. Then we obtain

{ E - Wo(r)- U(x)}G(x,x’; v) -/dy M(x, y ; v)G(y,x’; v)

= - h6(x

- x’). (18)

Here G is G, or G, (these functions differ only through the boundary conditions, see below), W,(f) is the kinetic energy (differential) operator, U(x) the screened potential of electron interaction with all the impurity centers, M is the mass operator defined by [G,(x”, x”, x, x’; v)I/,(x - x”) dx“ = { U(x)

- U,(x)}G(x, x’; v) + [dy M(x, Y ; ~ ) G ( Yx’; , v),

.

(19)

V , is the electron-electron interaction energy, U , the unscreened electronimpurity interaction. The U and U , Fourier transforms are related by

where

U(r) = /dk U(k) exp(2nikr),

(21)

e is the electron charge in Gaussian units, K(k, v) the Fourier transform of

the two-particle function K(x, x‘; t - t‘) = iO(t

[...,...I-

- t’)<[a(x, t)a(x, t),ii(x’, t’)a(x’, t ’ ) J - ) ,

(22)

is the commutator. It may be shown that K(k,O) < 0. The

quantity ro

1

=-

271

[-

nch ~

e2

lim K(k,

k+o

(23)

is just the screening radius (i.e., Debye radius if one evaluates K(k, 0) in the corresponding approximation). The functions U(x) and K(x) satisfy the neutrality condition. In case of the homogeneous system (which will interest us in what follows) this may

113

4. EFFECT OF HEAVY DOPING

be written in the form

Here N is the total number of impurity atoms, V,,,,, the screened Coulomb potential related to V , by an equation of the type (21). It was assumed above that the bands were nondegenerate at the extremum. If there is a degeneracy (such is the case in most of the hole bands of A”’ BV compounds) the operator Wo(0 is replaced by an operator matrix Wo”’. Likewise the Green functions and the mass operator acquire the matrix indices i, I‘, and the unit matrix Jll. appears in the right-hand side of Eq. (18). The rules of approximate calculation of the functions K and G , (and, therefore, M ) may be found in any textbook on the theory of the Green function^."*^^-^^ In this section it is enough to consider just the formal equation (18). The boundary conditions for Eq. (18) are not of much interest once the sample is large enough (for instance, the usual periodicity conditions in the fundamental cube may be used). The initial conditions follow from the definitions (14) and (13). Referred to the Fourier transforms they are formulated as the analyticity conditions of Gr (G,) in the upper (lower) v-halfplane.33a It follows from Eq. (18) that the poles of the spectral function are the eigenvalues of the auxiliary equation ( E - WON -

w4lYm - p f ( x , Y)$(Y)

dY

= 07

(25)

being the auxiliary function. Formally Eq. (25) resembles the usual “one-electron’’ eigenvalue problem ; it is called the effective Schrodinger equation (a detailed discussion of it may be found in Ref. 11). Consider in particular the perfect lattice, when U(x) = 0 and the system is homogeneous. Then the mass operator depends only upon the difference, x - y. Thus we may Fourier analyze it using an equation of the type (21), and then Eq. (25) leads to the “dispersion relation” :

)I

E - W&, 1) - M @ , E ) = 0 .

(26)

S. V. Tyablikov and V. L. Bonch-Bruevich, Advan. Phys. 11, 317 (1962). D. N. Zubarev, Usp. Fiz. Nauk 71, 71 (1960) [English Transl.: Soviet Phys.-Usp. 3, 320 (1 960)]. 3 1 A. I. Alekseev, Usp. Fiz. Nauk 73, 41 (1961) [English Transl.; Soviet Phys.-Usp. 4,23 (1961)l. 32 D. A. Kirzhnitz, “The Field Theory Methods in the Many-Body Theory” (in Russian). Gosatomizdat, Moscow, 1963. 3 3 P. Nozikres, “Le problkme a N corps.” Dunod, Paris, 1963. 33”Thephysical sheet is meant, see below. 29 30

114

V . L. BONCH-BRUEVICH

The solutions of Eq. (26) define the spectrum of the charge carriers in the perfect lattice with account taken of the electron-electron interaction. Since the mass operator is generally non-Hermitian, the function M(p, E ) may have both real and imaginary parts. Therefore the solutions of Eq. (26) have the form E = W@,1) - i ( P , 0 (27) 3

with Wand y real. The sign of y is easily established once the analytical properties of G , and G, are taken into account. It follows from Eqs. (15) that the set G, and G , may be considered as a multivalued function G of the complex variable E possessing a cut on the real axis. Fourier integrals of the type (12) define them on the “physical sheet,” where, according to what has been said previously, there may be only the branch points (or, in case y = 0, poles on the real axis). The complex poles of Eq. (27) (at y # 0) lie on the unphysical sheet. They may be determined formally as, say, the poles of the retarded Green function in the lower half-plane. Then, evidently, y >, 0, i.e., the amplitude A,, decays in time, as it should. It is clear from the preceding discussion that, at small y, all the essential concepts of the one-electron theory may be transferred to the manyelectron theory as well. By “essential” we mean the concepts that are really needed to interpret the experimental data, such as, for instance, the band edge, the effective mass, the Fermi surface, etc. This type of situation is realized in one of the two cases. First, the electron-electron interaction may actually be small, in the sense that the average potential energy is small compared to the average kinetic energy. It is clear from the dimensional considerations (and may be shown formally) that the corresponding parameter which is small in case of the small interaction is, in case of the degenerate gas,

A = (n;’3ao)-’.

(28)

In the nondegenerate case the relevant parameter is /z = Be’n,“’//~.

The mass operator calculated in the first nonvanishing approximation in ;1 turns out to be Hermitian and y = 0. Thus there is a certain range of the n, and fi values where it is possible to speak literally about the corrections to the charge carriers’ energy spectrum due to the electron-electron interaction. For instance, one can define a concentration-dependent effective mass, band gap, etc. Second, the non-Hermitian part of the mass operator may be small, not “uniformly” in the whole momentum space but just near the Fermi

4. EFFECT OF

HEAVY DOPING

115

surface. This is due to the Pauli principle and the conservation l a ~ s . ~ ~ - ~ ~ Assuming a priori that there is a Fermi surface this statement may be proven exactly, without any assumptions concerning the smallness of the i n t e r a ~ t i o n Thus . ~ ~ the assumption is confirmed a posteriori, i.e., the discussion is self-consistent. It should be noted, however, that the Green functions obey the nonlinear equations which may have other solutions as The problem of finding unambiguously the true state realized under given experimental conditions has not yet been solved in full generality. The main concepts of the band scheme may be transferred as well into the many-electron theory of imperfect crystals provided the impurity concentration is small enough. Indeed, we have by definition N

U ( X )=

C

V(X - Ri).

i=l

Let the screening radius be small compared to the average distance between the impurity atoms ro

< n-'I3.

Then Eq. (25) may be considered separately for each impurity atom. Once the parameter 2, Eq. (28), is small, we can neglect the mass operator and then (18) reduces to Eq. (8).36bThe only noticeable effect of the electronelectron interaction consists, in this case, in the screening which makes the number and spacing of the discrete levels depend upon the temperature and the electron concentration. This, however, does not change the principal features of the imperfect crystal band scheme. The situation is different when the impurity concentration becomes large enough. It was pointed out in Section 2 that the usual description in terms of the dispersion law may become inapplicable in this case, even if we use the n,-dependent effective masses. Then the question arises of finding the concepts and quantities with which it is appropriate to describe the many-particle system in the general case, when there is no dispersion law. A. B. Migdal, Zh. Eksperim. i Teor.'Fiz. 32, 399 (1957) [English Trans/.:Soviet Phys. J E T P 5, 333 (1957)l. 35 J. M. Luttinger and J. C. Ward, Phys. Rev. 118, 1417 (1960). 36 A. B. Migdal, Nucl. Phys. 30, 239 (1962). 36"The fact that they do have them in some cases is demonstrated by the phenomenon of superconductivity. 36bTheneglect of the mass operator is, in fact, not essential. Really the mass operator might be taken account of by introducing the concentration-dependent effective mass or, more precisely, by changing the dispersion law in accordance with Eq. (27).

34

116

V. L. BONCH-BRUEVICH

5. DENSITY OF STATES The answer to the question just raised may be found when one considers the general expressions for the macroscopic quantities of interest. Consider first the additive quantities that have nonzero equilibrium expectation values, such as, for example, the particle number and the kinetic energy. According to (6) to calculate such expectation values one has to evaluate the quantity (ZPppJ. From Eq. (11) it may be shown that this reduces to the same spectral function I that appears in (12):

In particular, if the mass operator is Hermitian one obtains, from (15) and (26),

I@,l;v)

=

[

1-

3-’ ~

6 [ E - W(p,l)].

The spectral function is the first of the exact quantities we are looking for. However, it is somewhat inconvenient since, even if the band index is omitted (one-band approximation), it depends upon the four variables, v, k. Related to the spectral function is the density of states, p(E), which turns out to be more convenient in a number of thermodynamic problems. To introduce it let us calculate the total number N of electrons in the system considered. One may use Eq. (6),taking account of the fact that in this case (pl(Alp1) = 1. Replacing the sum over p by an integral and making use of Eqs. (15) and (31) we obtain

where

p(E)=(2/h)CSmcImG,(k,1;v); E = h v ,

p = hk.

(34)

1

Equation (33) resembles formally the well-known equation of the usual perfect gas statistics (and this is the reason for the term “density of states”). When applied to the perfect gas, Eq. (34) does indeed give the usual density of states po (the number of quantum states per unit volume per unit energy interval). Indeed, in this case M = 0 while the sum over 1 reduces to that over the s p h components and gives the factor 2. Thus, according to (32) and (34L Po@) = 2

j& a E - Wo(k)l.

(35)

It is convenient to introduce the new integration variables, Wo,a, b, where

4.

117

EFFECT OF HEAVY DOPING

a and b are some suitable functions of

k, for instance, the polar angles.

Then po(E) = 2 J da dbZ(a, b, E ) ,

(354

where Z(a, b, W,) is the relevant Jacobian. Equation (35a) is the standard one in the usual one-electron theory of solids. When, in particular, W, = h2k2/2mthe usual expression is obtained from (35a): p , , ( ~ )=

.

[(2m)3/2/(2.)2~31~1’2~(~

(35W

In the case of an imperfect gas possessing a dispersion law (ImM = 0) the density of states, according to (27), is given by p(E) = 2 / d k 6 [ E - W,(k) - M I .

(36)

Finally, if there is a discrete level E l , the density of states has a deltafunction-like peak : in the vicinity of El, p(E) = N d E

- El),

(37)

where N , is the concentration of imperfections responsible for the level in question. Thus the density of states may serve as a characteristic of the charge carrier energy spectrum if there is any. In the general case, when damping is present, the variable E in Eq. (34) should not be considered as the energy of an electron, a quasiparticle, or anything else. It is just the variable to be integrated over in J3q.(33). The term “density of states” is not very fortunate in this case since there may be no stationary states with energy E. However, the function p(E) plays the role of the density of states in Eq. (33), and this is all that is needed for thermodynamic applications. Once p(E) is calculated, using Eq. (34), one may then relate the electron concentration to the Fermi level and to the temperature, with the help of the standard equation (33). It is seen that the density of states as defined by Eq. (34) is an exact property of the system. It may always be introduced regardless of whether there is or there is no dispersion law and, more generally, regardless of any assumptions whatsoever about the electron interactions. It is straightforward to generalize Eq. (34) to include the spatially inhomogeneous case. Then the Green function is nondiagonal in the (p, 1 ) representation; it is easily seen that Eq. (34) remains valid provided G,@, I ; v) is replaced by the diagonal element Q- ‘G,@, 1 ; p, I ; v). In other words, Eq. (34) defines the density of states as the trace of the Green function considered as the continuous matrix. Therefore any convenient basis may be used to calculate p(E).

118

V.

L. BONCH-BRUEVICH

For instance, in the coordinate representation one obtains p ( ~ =) ( 2 / h ~Jdx ) Im G J ~ x, ;V) .

(344

Sometimes it is convenient to introduce the density of states functions separately for different bands. Clearly they are given just by Eq. (34), with the summation over 1 omitted : p(E, 0 = (2/4

j

m(

Im G,(k 1;v) .

(34b)

[The spin sum is not carried out in (34b).] The concept of the density of states may be employed to give an exact many-electron definition of the energy band, namely, an I-band is the energy domain when the function (34b) is nonzero and nonsingular. Of course when a dispersion law exists this definition coincides with the usual one; yet if damping is present the band so defined has not the verbal meaning of a domain of permitted energies since the states @, 1) are nonstationary. 6 . COMPLEX CONDUCTIVITY

The density of states, being an exact quantity, does not, however, exhaust all the properties of the system. Indeed, it follows, i.e., from Eq. (7a), that to calculate the expectation value of the current density (in the nonequilibrium state when it may be nonzero) one has to know the occupation numbers n,@, r) in the presence of an external field. On the other hand, the density of states is defined by Eqs. (34) and (15) as an equilibrium property. The exact characteristics of the kinetic properties may be found by calculating the density matrix in presence of some external perturbation. In this way it is possible to introduce the conductivity tensor describing an exact reaction of the system to the weak external electric Let the field intensity be given by E , = Em,aexp( - iot) ,

(38)

where is the constant amplitude, and a, p are the vector indices. Then in the approximation linear in E 37 38

39 40

41

42

R. Kubo, J . Phys. SOC. Japan 12, 570 (1957). V. M. Galitskii and A. B. Migdal, Zh. Eksperim. i Teor. Fiz. 34, 139 (1958) [English Transl.: Sooiet Phys. J E T P 34, 96 (1958)l. M. Lax,Phys. Rev. 109, 1921 (1958). D. N. Zubarev, Usp. Fiz. Nauk 71, 71 (1960) [English Transl.: Soviet Phys.-Usp. 3, 320 (1 960)]. S . V. Tyablikov, Fiz. Tverd. Tela 2, 361 (1960) [English Transl.: Sooiet Phys.-Solid State 2, 332 (1960)l. V. L. Bonch-Bruevich, Zh. Eksperim. i Teor. Fiz. 36,924 (1959) [English Transl.: Soviet Phys. J E T P 36, 653 (1959)l.

4.

EFFECT OF HEAVY DOPING

119

Ga> = jm,aexp( - iwt) ,

(39)

the amplitude jm,abeing given by

im,,

=

am~(w)Ern,p 7

(40)

where the summation over repeated indices is implied and aas(w)is the complex conductivity tensor : oap(o)= in,(e2/or5)h,

J

-m

The expectation value in Eq. (41) is taken with respect to the equilibrium density matrix (11); r5 is the quantity with the dimension of a mass (called sometimes an “optical” effective mass; the general formula for it may be found in Ref. 11); vp(0)is the Heisenberg velocity operator at t = 0. According to (41) and (7a) the complex conductivity is expressed in terms of the two-particle Green function (22) [as should be expected in view of the fact that the function (22) describes a polarization of the medium; see (20)]. The Fourier transform of the function (22) is generally not reducible to the density of states and is thus an independent exact characteristic of the many-particle system. This leads to considerable difficulties when an attempt is made, for instance, to deduce the form of the density of states function from the optical data. Yet in some cases a direct, though approximate, relation may be found between K and p.43

IV. Perturbation Theory 7. ELECTRON-ELECTRON INTERACTION Our aim in what follows is to calculate the density of states of the heavily doped semiconductor. Due to the extreme complexity of the problem the calculation can be carried out only approximately. It is convenient to c o n s i d e r a t the simplified problem treating both the electron-electron and the electron-impurity interactions as small perturbations. Strictly speaking such a treatment may be inadequate since, as we know, the problem should be solved in a self-consistent way; on the other hand it is well known that the bound states, if there are any, cannot be obtained in any finite order of the perturbation theory if the unperturbed spectrum is continuous. In fact we shall see that some peculiarities of the heavily doped material do not appear in any finite order of the expansion in powers 43

V. L. Bonch-Bruevich and R. Rozman, Fiz. Tuerd. T eb 5, 2890 (1963) [English Trunsl.: Soviet Phys.-Solid State 5, 2117 (1964)J.

120

V. L. BONCH-BRUEVICH

of the impurity concentration. Therefore the perturbation theory treatment may refer rather to the case of an intermediate doping if we are interested not in the impurity band as such but in the influence of the impurities upon the conduction band structure. Nevertheless this problem is of some interest for the theory of the heavily doped materials as well. First of all we shall see that the electron-electron interaction may in fact be treated in such a way. Second, the perturbation theory calculation will enable us to justify, to some extent, the assertion made in Section 2 on the possible absence of the dispersion law in the doped material. Third, it will be possible to use the formula obtained below to calculate the Fermi level in the heavily doped sample (Section 13). The problem at hand has been treated by various author^^“-^' using formally different but essentially equivalent methods-with similar results. We shall follow the treatment given in Refs. 48,49, and 51, where practically the same methods were used. The results of the calculation depend upon the degree of degeneracy of the electron gas. We shall consider only the case of complete or almost complete degeneracy, setting the Fermi level p being referred to the bottom of the conduction band.51a The calculation carried out for the nondegenerate case4* shows that all the principal qualitative results to be obtained below remain valid in that case too. When the temperature T = 0 the parameter determining the strength of the electron-electron coupling is given by Eq. (28). It will be considered as small:

A < < 1.

(43)

R. H. Parmenter, Phys. Rev. 97, 587 (1955). H. Schmidt, Phys. Reu. 105, 425 (1957). 46 W. Kohn and J. M. Luttinger, Phys. Rev. 108, 590 (1957). 47 J. R. Klauder, Ann. Phys. (N.Y.) 14, 43 (1961). 48 V. L. Bonch-Bruevich and A. G. Mironov, Fiz. Tuerd. Tela 3, 3009 (1961) [English Transl.: Soviet Phys.-Solid State 3, 2194 (1962)l. 49 P. A. Wolff, Proc. Intern. Con$ Phys. Semicond., Exeter, 1962 p. 220. Inst. of Phys. and Phys. SOC.,London, 1962. E. M. Conwell and B. W. Levinger, Proc. Intern. ConJ Phys. Semicond., Exeter, 1962 p. 227. Inst. of Phys. and Phys. SOC.,London, 1962. I. P. Zvyagin, Fiz. Tverd. Tela 5, 581 (1963) [English Transl.: Soviet Phys.-Solid State 5, 422 (1963)l. 51*Hereand in what follows phrases such as “the bottom of the conduction band,” “the top of the valence band,” etc., refer to the pure material. 44 45

4.

121

EFFECT OF HEAVY DOPING

The weakness of the electron-impurity interaction is characterized, first, by the applicability condition of the Born approximation and, second, by the condition

nalk 4 1 .

(44)

Here 0 is the typical electron scattering cross section due to a single impurity atom and k the typical wave number of the scattered charge carriers. The relation (44) guarantees the smallness of the multiple-scattering effects. It is clear from dimensional considerations that, in case of complete degeneracy and ionized impurity scattering, both conditions will reduce after all to one and the same inequality (43). Numerically condition (43) holds rather poorly for most of the real materials (InSb excluded). Therefore the calculation based on it may have significance only as a model. Still more important is the following: Even if Eq. (43) does hold, the criterion (44), being true for most of the electrons, breaks down for the slowest of them. Thus the perturbation results may not be used to study the density of states near the band edge. When the degeneracy is not complete, criterion (43) still makes sense if the temperature is low enough [see Eq. (42)]. The new small parameter that appears, T / p , does not change the structure of the perturbation series if it is kept smaller than A (which will be assumed in what follows). Within the limits of the problem thus set up it is sufficient to use a oneband effective mass approximation. Note that the conditions of the smallness of both perturbations, described, respectively, by M and U,are the same. Therefore in the first approximation these perturbations may be treated separately, i.e., neglecting the electron-impurity interaction when calculating the mass operator M . Thus we come to the well-known problem of the high-density electron gas. The first approximation to the mass operator is given here by the dec~upling.’~

G~(x”,x”,X, X’;V) = neG(x, X’;V) - n(x”, x)G(x”, X’; V) .

(45)

Here n(x”,x) is the element of the first (one-particle) density matrix :

-

n(x“,x) = (a(x”)a(x)) = Sdk n(k) exp 2nik (x“ - x) .

(46)

The first term on the right-hand side of Eq. (19) may be neglected in this approximation. Using the neutrality condition (24) one obtains for the Fourier transform of M : M ( k , v ) = -jdk’K(k’)n(k - k).

(47)

The unperturbed value of the function n(k) should be used in (47). This is

122

V. L. BONCH-BRUEVICH

the usual Fermi function : no(k)

=

[exp P(P0 - Wo) + 11-

’

(48)

7

where p o is the Fermi level of a perfect electron gas. Equation (47) is just the usual exchange term. It is known that at zero temperature it contains a logarithmic singularity at k = kF, where k, is the Fermi wave number. The singularity is due to the long-range nature of the Coulomb forces and is removed by taking account of the ~creening.’~ Thus the mass This is done by the replacement of the type of (21).11,29,31-33 operator takes the form

M(k, v)

=

-(e2/nE) [dk’n,(k’ - k)[k” - (e2/.rrsh)K(k’,0)]-’.

(49)

Since the main trouble comes from the small values of k , a reasonable approximation may be obtained by replacing K(k’,0) by the constant value lim K(k’, O).” Then, in view of (23), k ‘- 0

M(k,v) = -(e2/xr)~&’O[po - Wo(k’ - k)][k’’

+ ( 2 ~ r ~ ) - ~ ] - ‘(50) ,

In an isotropic approximation,

Wo(k)= h2k2/2m,

(51)

the integral is easily evaluated, and we obtain

M(k) = -(2e2k&)

{+ 1

kF2

-

k2 + K 2

4kkF

In

(kF + k)’ - k)’

(kF

+ K2 + lc2

where K = (27110)-1, Note that the quantities (47), (49), and (50)-(52) are real (and do not depend upon v). As we know, this means that there is no damping in the approximation adopted, and the notion makes sense of the carriers’ energy spectrum calculated with the Coulomb corrections.’ Using field theory language, the “bare particles” dispersion law, W,(k), is replaced by a “renormalized” one, describing the “dressed particles” :

W,(k) = Wo(k)

+ M(k).

(53)

Once the renormalization (53) has been carried out, we may no longer take explicit account of the electron-electron interaction, except in using the screened potential and the neutrality condition. M. Cell-Mann and K. A. Brueckner, Phys. Rev. 106, 364 (1957). 52”ltmay be shown that such is always the case in the first nonvanishing approximation, once the instant forces are considered (no retardation).

4.

EFFECT OF HEAVY DOPING

123

8. ELECTRON-IMPURITY INTERACTION

Once the renormalized dispersion law is introduced, Eq. (18) may be rewritten in the following symbolic form :

(Gi’

where

G,’

+ U)G = 1 ,

=

h-’(W, - E ) .

(54) (55)

The problem is to study the zeros of the function (G)a;l. It goes without saying that (G)a;l # (G-l),,,; therefore one has either to calculate the Green function itself and to average it according to Eq. (1) or to average just Eq. (54). To use the perturbation theory the second way seems to be more convenient. Let us first separate out the average potential, putting

u = < u>,,+ U’, (Go’)-’ = G,’

( UO,, = 07

+ (U),,

Then Eq. (54) takes the form

(Go’)-’G

+ U’G = 1 .

(57)

The result of averaging Eq. (57) may be written as

(G-’),,

= (Go’)-’

+ M,,

(58)

where M , , the “mass operator” describing the electron-impurity interaction, may be found by iteration. In the first approximation we obtain

M,

= -

(59)

To perform the averaging explicitly one has to know the probability density P appearing in Eq. (1). Its calculation is by itself a very complicated problem and may (in principle!) be solved only in the limits of some special model. It is clear, however, that in a macroscopically homogeneous sample the function P must b e approximately constant, provided all the interatomic distances IRi - Rjl (i # j = 1,. . . ,N) are large enough. At small distances the correlation between the impurities’ coordinates comes into play. In particular it is impossible for the two impurity atoms to occupy one and the same lattice point: P = 0 if any two of its arguments coincide. Note, however, that the impurity concentration is small compared to the concentration of the regular lattice sites, even at the highest doping levels now attainableszb(n 5 10’’ to 10” ~ m - ~Thus ) . the probability of the two 5ZbThismay be considered just as the definition of the impurity doped sample as contrasted to the mixed crystal.

124

V . L. BONCH-BRUEVICH

impurity atoms occupying one and the same lattice point is small even if the correlation is neglected, and we may ignore the contribution of these regions of the coordinate space in the integrals (1) (except when the function being averaged is singular or very large just at the coinciding arguments). This means that a reasonable approximation may be expected to result if the Then, according to the norfunction P is replaced just by a malization condition (2), P(R,, . . . ,RN)= W N , (60) and we obtain, using (24), (29), and (56) and considering the limit N + 00, R + co, N/R = n < co,

M,(x, x’; t, t’) = -n[&

IV(k)12(G(x,x’; t, t’)),, exp 27cik. (x” - x’). (61)

Here use has been made of the relation V (-k) = V*(k). Evidently in the macroscopically homogeneous sample the averaged Green function depends only on the difference x - x’. Then the usual fourdimensional Fourier transform may be used to obtain

M , @ , v ) = -nJdk JV(k‘)12G(k - k’; v ) .

(62)

The averaging symbol has been omitted since only the averaged Green function will appear henceforth in this section. Inserting the right-hand side of Eq. (62) in Eq. (58) and taking account of Eq. (55) one obtains the equation for the mass operator:

Two approximations of essentially different nature were used when obtaining Eq. (63). First the interaction of an electron with every single impurity atom was considered small. Second, the impurity concentration was considered small in the sense of relation (44). The first of these assumptions may be easily released. Indeed, if one considers the higher order corrections to the mass operator one sees that they fall into two classes: those proportional to n and those proportional to higher powers of n. Corrections of the first type describe the electron scattering by a single impurity center and may be formally summed if one introduces the exact two-body scattering amplitude instead of IV(k)12.54This may be of value when the short-range forces are taken account of and, in addition to i ”‘The functional methods were used to obtain the mass operator at the arbitrary type of impurity-impurity ~orrelation.’~ 53 R. A. Suris, Fiz. Tverd. Tela 5, 458 (1963) [English Transl.: Soviet Phyx-Solid State 5, 332 (1963)J. 5 4 S. F. Edwards, Phil. M a g . 33, 1030 (1958).

4. EFFECT

OF HEAVY DOPING

125

[Eq. (28)], the other parameters may come into play. If, on the other hand, V(k)is just the screened potential of the charged donor (the case considered below) then the replacement of the Born amplitude by the exact one makes no sense: the difference between them is appreciable when the condition (43) fails and then the condition (44) fails as well. (Remember that the electron and impurity concentrations are practically equal in the case considered.) In th'e approximation adopted, only the first iteration of Eq. (63) is to be used. In this way we obtain, taking account of Eq. (27),

+nJ

W&) =

dk' I~(k')12[Wo(k)- WO& - k11-l

= W, -k AW, y

(64)

nnjdk' IV(k')12S[W(k) - W(k - k ) ] .

(65) To estimate the screened potential of the single donor the usual Debye formula54"may be employed. Then, in the normalization adopted above, N

V(k) = (e2/m)(k2 + xZ)-l.

(66) The integrals appearing in (64) and (65) are easily evaluated in the isotropic approximation (51). [In Ref. 50 the integral (64) was computed for some directions of the k-vector in the anisotropic case of the n-type Ge.] In particular, the energy correction and the damping constant at E > 0 come out as follows : A W ( k )= - (ne'/~~')[ W(K) 4 W0(k)]-' , (67)

+ y = (2nne"/~~~)[W(rc) + 4Wo(k)]-'k/ic.

(68)

Here W(K)= h2lc2/2m. The energy corrections may be taken literally only provided y $ (Awl. In our case this reduces to the relation (69) 2nkllc 4 1 , which may be fulfilled only for very small values of k (where, by the way, the perturbation calculation is at best doubtful). When k is of the order of the Fermi wave number k , n1I3, condition (69) certainly does not hold. In fact, in view of Eq. (43) the inverse relation is true. This means that the concept of the dispersion law m y become inapplicable once the corrections are taken account of due to the electron-impurity interaction. In such conditions the only "effective masses" that make sense are the ones that enter into the formulas for the directly observable quantities, such as the optical effective mass, density of states mass, etc. Being defined by different experiments they must not, generally, coincide, even if we start from the

-

s4aIf the degenerate system is considered, this is not obvious.5s Yet it turns to be true for the doped material (see Section 14). s 5 J. S. Langer and S. H. Vosko, J . Phys. Chem. Solids 12, 196 (1959).

126

V . L. BONCH-BRUEVICH

isotropic model (51). Note, however, that when the doping is heavy enough the damping constant y(kF) is small compared to the Fermi energy itself: ?%)/Po

= (son1131- 1

.

Therefore the impurity corrections are not very essential in this region, thus justifying the perturbation theory approach. 9. ELECTRON-PHONON INTERACTION Up to now we have completely neglected the existence of the lattice vibrations when treating the energy spectrum of the semiconductor. Such an approach has some raison d’&tre in case of homopolar crystals, such as germanium. However, it is subject to some doubts when there is a polar coupling present. It is well known, for example, that the polaron concept has rather radically changed the standard band picture of the purely polar ~ r y s t a l s . ~ ~The - ~ *A”’ BV compounds, though not completely polar, do possess some portion of the polar bond as is clearly seen, for example, from the electron scattering properties in comparatively pure materials.59 Thus some “polaron” effects may well be expected. The energy spectrum of the heavily doped semiconductor with a partly polar lattice was considered by Keldysh and Kopaev.60 Only one important feature of the heavily doped material was taken account of, i.e., the degeneracy of the electron gas. The electron-impurity interaction was ignored completely, so that the system under consideration was really a degenerate electron gas plus the quantum field of the polar phonons. The Green function for the particles interacting with some quantum field may as well be written down in form (58) if the appropriate form of the mass operator is employed.60a It was assumed in Ref. 60 that the electron-phonon interaction was small enough (not too large a part of the polar binding). Thus the standard perturbation theory could be used to calculate the mass operator. For the simplest case (51) the result was [W(k)]“’ - pol/* [W(k)]”’ pol/’

+

S. I. Pekar, “Studies in the Electron Theory of the Polar Crystals” (in Russian). Gostechisdat, Moscow, 1952. N. N. Bogoliubov, Ukr. Mat. J . 2, 4 (1950). 5 8 S. V. Tyablikov, Zh. Eksperim. i Teor. Fiz. 21, 397 (1951). 5 9 H. Ehrenreich, 1. Appl. Phys. 32, 2155 (1961). 6o L. V. Keldysh and Yu. V. Kopaev, Fiz. Tuerd. Tela 5, 1411 (1963) [English Transl.: Soviet Phys.-Solid State 5, 1026 (1963). 60aIn Ref. 60 the causal Green functions were used instead of the retarded ones. Of course this does not make any difference in the final results. 56

’’

4.

EFFECT OF HEAVY DOPING

127

Here A is a constant, depending upon the electron-phonon coupling constant, B is a function possessing no singularities at the Fermi surface, and wo is the polar phonon frequency (in energy units), assumed independent of the wave vector. Clearly only the term that is written down explicitly in l?q. (60) is of importance near the Fermi surface; the function B may be neglected in this energy region. Strictly speaking it is just this fact that justifies the use of the Fermi-surface concept: the mass operator is real in this energy range, and one may speak of the dispersion law. The density of states of the system considered is given by Eq. (36). Putting (70) in the right-hand side of (36) we obtain, near the Fermi level, p

=

p o { l - aIn[(~/p,)'/~- 1]>-',

(71)

u is the well-known electron-polar phonon coupling constant

= e2m1/2h-

1/ZW,

yE-m 1 - €0- I ) ,

(73)

and E, and E~ are, respectively, the high- and low-frequency dielectric constants. It is seen that the density of states is zero at the Fermi level. This result is analogous to some extent to that of the electron-electron Coulomb interaction when treated in the first approximation (47). However, the screening smears out the logarithmic singularity of the Coulomb mass operator as seen explicitly from Eq. (52). In the case now considered the situation is somewhat different since the phonon field is the high-frequency one. This leads the authors of the Ref. 60 to assert that there is no screening effect in the first nonvanishing approximation of the perturbation theory. This justifies to some extent the approach employed in Ref. 60; the electronphonon interaction induces radical changes of the density of states near the Fermi level while the electron-impurity interaction was seen to be insignificant in that energy range. The reduction of the density of states near the Fermi surface seems to explain the phenomenon observed by Hall et aL6' The differential conductivity of the tunnel diodes made of the A"' BV compounds has a minimum at low voltages (to the left of the maximum current), the value of the voltage corresponding to the minimum, being proportional to u. In fact, the tunneling in this voltage range is due to those electrons that have energies near pLo. 61

R. N. Hall, I. H. Racette, and H. Ehrenreich, Phys. Rev. Letters 4, 456 (1960); R. N. Hall, Proc. Intern. Conj Semicond. Phys., Prague, 1960 p. 193. Czech. Acad. Sci., Prague, 1961.

128

V. L. BONCH-BRUEVICH

Further, the form of the density of states (71) leads to some peculiarities in light absorption. Consider, for example, a p-type sample and let 0- 6 &, where ph is the hole Fermi level referred to the top of the valence band (the hole energy increasing downward). In the approximation adopted above, the electrons in the conduction band are practically free. Therefore the interband absorption constant ( is proportional just to the density of states near the Fermi level. In this way one obtains

Here A is the band gap, me and mh the electron and hole effective masses, 8 the usual step function, and a the hole-phonon coupling constant. It is seen that, due to the polaron effect, the absorption constant does not rise abruptly from zero to its full value at

v

=

A

+ ph(m, + me)m,

but changes gradually in the frequency range proportional to a(w0/ph)'/'. Something like this seems to have been observed by Roberts and Quarrington.62

V. Semiclassical Approximation 10. GREENFUNCTIONS

The perturbation theory while giving some information on the density of states near the Fermi level becomes inapplicable at lower energies. Thus, condition (44)may not hold for the slow electrons. The essence of this condition is the possibility of neglecting the simultaneous electron interaction with several donors. The last effect furnishes probably most of the dficulties encountered when treating the problem by some form of the perturbation theory. An attempt was made by Lax and Phillips6j and by Frisch and Lloyd64 to overcome this limitation by solving numerically the one-dimensional Schrodinger equation in the field of randomly distributed force The true donor potential was replaced by a delta function. However in the heavily doped material this approximation seems to be V. Roberts and J. Quarrington, J . Electron. 1, 162 (1955). M. Lax and J. C. Phillips, Phys. Rev. 110, 41 (1958). 64 H. L. Frisch and S. P. Lloyd, Phys. Rev. 120, 1175 (1960). b4"Seealso the qualitative discussion of A i g r a i ~ ~ ~ 6 5 P. Aigrain, Physica 20, 978 (1954). 62

63

4.

EFFECT OF HEAVY DOPING

129

inadequate. Indeed, it makes sense if the screening radius ro is small compared to the average distance between the impurity atoms n-1/3.When the electron concentration increases the screening radius is of course reduced, yet in case of no injection the distance, n-1’3, decreases as well. The last factor turns out to be dominant in the degenerate case. To estimate ro one may use the well-known Debye formula for the degenerate gas ro = &(n/3) 116 a.112n,- 1 / 6

.

(75)

If n, = n, it follows from (75) that

- &nao3)1/6, = (76) - L1/2-1/2 2 where 1 is the parameter (28). The right-hand side of Eq. (76) increases with ron

1/3 N

the impurity concentration. Thus in the degenerate case the delta-function approximation of the impurity potential may be justified only for sufficiently small doping We shall limit ourselves to the asymptotic case of the very heavy doping in the degenerate case where condition (43) and the estimate (76) do hold. (It will be shown below that Eq. (76) remains essentially unchanged when the corrections to ro are taken into account.) Under such conditions the electron-electron interaction may be treated by the simple renormalization method of Section 7. For simplicity we take the parabolic and isotropic renormalized dispersion law :

W , = p2/2m,.

(77)

The energy is now referred to the renormalized conduction band edge. The generalization to the anisotropic case of the n-type Ge causes no serious diffculty but is not needed for our purposes. Consider first We shall follow the treatment given by the 65’In the compensated sample, where n, < n, the product is still greater than the righthand side of %. (76) if the degeneracy is preserved. On the other hand, in the nondegenerate ’ / ~ .may be considerably greater than 1. case (at n, = n), r,91’/~= ( ~ / 4 @ e ~ n ” ~ )This 66 V. L. Bonch-Bruevich, Fiz. Tverd. Tela 4, 2660 (1962) [English Transl.: Soviet Phys.-Solid State 4, 1953 (1963)l. 6 7 V. L. Bonch-Bruevich, Fiz. Tverd. Tela 5, 1852 (1963) [English Transl.: Soviet Phys.-Solid State 5, 1353 (1964)l. 68 V. L. Bonch-Bruevich, Proc. Intern. Conf: Phys. Semicond., Exeter, 1962 p. 216. Inst. of Phys. and Phys. SOC.,London, 1962. 68“Anapproximation in the same spirit was adopted by Kane6’-’l and by Keldy~h.~’ 6 9 E. 0. Kane, Proc. Intern. Conf: Phys. Semicond., Exeter, 1962 p. 252. Inst. of Phys. and Phys. SOC.,London, 1962. ’O E. 0. Kane, Phys. Reu. 131, 79 (1963). 7 1 E. 0. Kane, Phys. Rev. 131, 1532 (1963). 7 2 L. V. Keldysh, Abstr. Intern. Conf: Phys. Semicond., Exeter, 1962 p. 25. Inst. of Phys. and Phys. SOC.,London, 1962.

130

V. L. BONCH-BRUEVICH

the “one-band” approximation. In the approximations adopted above, Eq. (18) takes the form

{E

+ i~ + C + (h2/2m,)VX2 + U(x)}G(x, x’ ;E ) = -hh(x

-

x‘). (78a)

The adjoint equation is

{E

+ k + C + (h2/2mr)V;, + U(X’)}G(X,X’;E) = -Frh(x

- x’). (78b)

Here C = nJdx V(x), E + + 0. The term iE was added to take explicit account of the analyticity condition of G, at Im E > 0. According to (76) it is impossible in this case to separate out the “sphere of action” of any individual impurity atom : the electron potential energy at any point of the crystal is due to many donors. Therefore the function U(x) is slowly varying on the distance of the order of n-1’3, and the solution to Eqs. (78a,b) may be obtained semiclassically, i.e., in the form of an expansion in powers of the derivatives of U(X).’~” It is convenient to use the new variables

2R=x+x’,

r=x-x’

(79)

and to choose units such that h = 2m, = 1 (the energy being measured in units of W, = m,e4/L2h2). Then the symbolic solution to (78a,b) reads (the higher order derivatives being neglected) :

ij

W

G(r, R ; E ) =

0

ds exp( - E S

+ is(E + C)}L,L,G(r),

(80)

where

L,

=

expis{$V,* - U(R)},

The method of “disentangling” such operators is well known,32 and an explicit formula may be obtained for the Green function containing an expansion in commutators in the exponent. In the equilibrium problems (when calculating the density of states, etc.) just the zero-order approximation suffices when all the derivatives of U(R) are neglected. When calculating the Fermi level this leads to a relative error of the order of /z as compared to the terms that are kept in the final result. Evidently this ’’”Strictly speaking, U(x) is a slowly varying random function. Therefore all the assertions concerning the smallness of the derivatives, etc., make literal sense only after the averaging over the impurity configurations has been carried out. This will be done in due time; for the time being it is sufficient to bear in mind that the quantities neglected in Eqs. (80)-(82) may not be at all small by themselves; they just give small contributions to the final results.

4.

131

EFFECT OF HEAVY DOPING

approximation is completely equivalent to the standard method of the bent bands.72b In the approximation just described, we obtain G(r, R;E ) = i

Im d s j d k exp{

--ES

0

+ is[E + C - U(R) - (2nk)’I + 2nikr). (82)

Taking the average over the impurity configurations, with the help of Eqs. (1) and (60) one finds easily :

s

(G(r, R ;~5))~“ = dk exp(2nikr) G(k, E) , G(k E ) = i$

0

ds exp{ --ES

(83)

+ is[E - (2nk)’I + a l ( s ) ) ,

(84) (85)

Note that Real -al”(0)

< 0,

al(0) = al‘(0) = 0,

= -2a2

=

-nldr P ( r ) < 0 .

(86)

If, in particular, the function V(r) is the Debye potential, V(r) = (Ze’/Er) exp(-r/ro), 2 being the charge on the donor in units of e, then

a = (Ze2/-E)(nnro)”2. (87) If there are several types of impurity atoms (such is, for instance, the case of appreciable compensation) a1 should be replaced by

The index j enumerates the types of impurity atoms present in the concentrations nj. 1 1 . DENSITY OF STATES

To calculate the density of states in the “one-band” approximation adopted above one may use Eq. (34), 1 being the spin index. Since the Green function (83)is spin-independent the sum over 1 gives just a factor 2, and we obtain (in dimensionless units): p(E) = (2/7c) / d k Im

i l mds exp{ -a + is[E 0

+

- (27~k)~l a l } .

(89)

72bIn kinetic problems it is necessary to take account of the corrections to the semiclassical approximation.

132

V. L. BONCH-BRUEVICH

It is convenient to interchange the order of k- and s-integrations, yet some care must be taken since the k-integral diverges at s = 0. To circumvent this formal difficulty note the identity Im i

I&Im exp{ ds

- ES

0

+ ( 2 ~ k ) ~ i=s )0 .

Subtracting this from Q. (89) gives an expression in which the order of integrations may be interchanged. In this way one obtains m

p ( E ) = ( 2 7 ~ ) - ~Im ~ exp(--xi/4)pfJ -~/~

s-3/2exp(a, SO

+ iEs - ES) ds,

(90)

where the symbol "pf" means the finite part of the integral when so -+ 0. An equation of the type (90) is practically all that is needed to calculate a number of directly measurable quantities, since most of them contain the density of states p(E) under the integral sign. It is interesting, however, to determine an explicit form of the density of states as well. There are three regions of interest : near the Fermi level, near the band edge, and deeply in the forbidden band. For positive values of E (the conduction band) the integral appearing in Eq. (90) may be evaluated by the stationary phase method. Really there is a big parameter in the exponent since, according to (87), a/W, = O(A-'/"). We shall assume, further on, that p %- a. (This will be justified by a subsequent calculation.) Then the most important extremal point is s = 0, and the simple calculation yields (in the usual units) :

where x = E2/8a2,

and I,/, , . . . are Bessel functions of the imaginary argument. When the energy E (referred, as usual, to the band edge) is near p, Eq. (91) takes the form p(E) = po(1 - a2/4E2).

(93)

Thus the density of states is but weakly perturbed in this energy range.72c The reason for this is clear. The increase in the impurity concentration is accompanied by a corresponding increase in the concentration of electrons, which leads to the more pronounced screening effects. As a result, the average potential energy of the electrons in the impurity field grows more slowly than the Fermi energy. "'The result (93) is, of course, consistent with the perturbation theory results cited in Section 8.

4.

EFFECT OF HEAVY DOPING

133

On the other hand, near the band edge the density of states changes radically. According to (91) we obtain, at small E,

where r is the Eulerian gamma function. The formula (94) turns out to be true for E < 0 if the absolute value IE1 is small enough, 1E( 4 WB1-0.3. Thus the density of states does not tend to zero at the band edge but is finite and regular there. This is what should be expected since the notion of the band edge has but a “genetic” meaning in our case. At greater values of (El the integration in Eq. (90) should be carried out more carefully if the electron-impurity forces are attractive. It is convenient to consider at first the model approximating the function V ( r )by a spherical well or “hump”:

V ( r ) = )gO(r, - r ) .

(95)

Here g and r, are positive parameters, O is the usual step function. Strictly speaking the semiclassical treatment used above does not apply to the sum of potentials of the form (95). It is clear, however, that the integral defining the function aI(s)will suffer but a small change if the step function in (95) is smeared out a little to make the total impurity potential a smooth function. The error induced by using approximation (95) may be estimated to some extent, noting that, in this case, a, = nu,{exp(

figs) - 1 T igs} ,

(96)

where u, = (4n/3)rC3.On the other hand, if one uses the Debye potential the function aI(s),while oscillating, increases in amplitude with increasing s like In’s. This increase is due to the singular behavior of the Debye potential at small distances which should, generally, be cut off in the semiclassical treatment. Therefore the true form of a,(s) should not be very different from that given by Eq. (96). In the approximation (95) there is no damping, and the electron energy spectrum is given by

W(k,m) = (2nk)’ k nv,g T mg, where in is an integral number. This result should not be taken too seriously : it is due to the approximation adopted and will disappear when it is corrected. The other point is essential. Putting (96) into Eq. (90) one obtains p(E)=

exp(-nu,) 2n2

1 Km !( a,

m=O

E k nuog T mg)’/28(E L- nuog f mg).

(97)

134

V. L. BONCH-BRUEVICH

Let the electron-impurity interaction be attractive [the “minus” sign on the right-hand side of Eq. (95)]. Then, as seen from (97), the density of states is nonzero at any energy range, the region E < 0 included. Indeed, given any value of E one can always find terms in the sum over m such that E - nu,g mg > 0. In other words, an infinitely deep density of states tail is formed in the “forbidden band.”72d The series appearing on the right-hand side of Eq. (97) is in fact nothing else but the expansion of the density of states in powers of the impurity concentration. It is seen that (in accordance with Ref. 47) the tail cannot be obtained in any finite order of the perturbation theory using such an expansion. It is easy to determine the asymptotic behavior of the density of states at E < 0 and (El -+ co. Then only the terms with large enough values of m contribute to the sum in (97), and the Stirling approximation may be used. In this way one obtains72e:

+

P(E)

-

Y112 exp{lEl

+y

- (mo

+ t )In [(mo + +)/nqJ},

(98)

where rn, is the least integral number exceeding JEl + nuog and y = rn, - ]El - nu,g;

0 ,< y d 1

Abandoning now the model of Eq. (95) and turning to the more realistic potential we do not have Eq. (97) any more. Yet, as before, the density of states is always nonzero if there are attractive electron-impurity forces. Deeply in the tail region ( E < 0, [El -+ 00) we obtain (in usual units)

where E , = 4WB2-’’2

= 4WB(na,3)’1”.

(100) Equation (99) is formally valid at (El & E,. The reason for the tail formation is easily understood. The point is that, though the impurity distribution is homogeneous on the average, there are always some local fluctuations of the number of impurity atoms in any particular volume.72fThis results in a local change of the electron energy as compared to its average value. In particular there are regions where the 7ZdRemember,however, that in the one-band approximation used above one cannot go further than, say, half of the forbidden band. ’*‘According to (98), the function p ( E ) oscillates with E passing through zeros at y = 0. This is a specific feature of approximation (95) and thus should not be taken too seriously. ”‘To avoid misunderstanding, note that the fluctuations considered have nothing to do with the usual thermal fluctuations in a gas. They are just clusters originating in the process of crystal growth, after which the impurities are assumed immobile.

4.

135

EFFECT OF HEAVY DOPING

energy is reduced. After averaging over the impurity configurations, this manifests itself as the tail. The fluctuation picture allows one to estimate the density of states in the tail by a very simple though not rigorous method (we resume in what follows the simplified version of the discussion given in Ref. 70). Consider some volume u and assume, for simplicity, that the addition of every atom above the average value nu changes the electron energy by a constant w. Then the problem is reduced to calculating the probability P of the rn atoms appearing in the volume u.72g Evidently, this is given by the Poisson formula : P(rn) = [(nu)”/rn!] exp( -nu). (101) The density of states is p(E)

- C P(rn)(E + rnw)”2. m

In the tail region the large values of rn are of importance. Then, using the Stirling formula we obtain, from (101),

P(m)= ( 2 7 ~ r n ) - ’exp{rn /~

-

nu - rn ln(rn/nu)}.

-

(102)

Let the volume u be of the order of the Debye sphere (u ro3). Clearly, then, w N E , = e2/Ero and nu % 1. In the energy region not very far from the band edge, the srnaZZ fluctuations are most important : Irn - nu1 << nu.

(103)

Then the exponent in Eq. (102) may be expanded in powers of rn - nu, and we obtain

By definition,

(rn - nu)2 = ( E / E o ) 2 , and the density of states has a Gaussian form: p(E)

where E,

=

- exp( - E 2 / E C 2 ) ,

-

E , ( 2 n ~ ) ” ~ WB(na03)5112.

(105) (106)

7zgTheconcentration of the free carriers is assumed constant. In reality the fluctuation in impurity number induces the fluctuation in the local electron density as well, this reducing somewhat the density of states in the tail. This effect is of especial importance if, in its absence, the bound states may appear in the potential well due to the fluctuation. We shall take account of it, but in a rough way, assuming that such bound states never form.

136

V . L. BONCH-BRUEVICH

Equations (105) and (106) are valid both for attractive and repulsive forces since the potential comes in squared. In other words, they describe both the electron and hole tails formed, respectively, near the conduction and valence bands irrespective of the type of the doping However, the domain of applicability of (105) is limited by the condition (103); fluctuations should be small. More deeply in the forbidden band the larger fluctuations are dominant when condition (103) does not hold: either the volume u is considerably smaller than the Debye one, or the number of atoms n greatly exceeds the average number nu. The last possibility is not improbable since .-‘I3 < r,, [heavy doping, cf. (76)]. When n 9 nu and the electron-impurity interaction is attractive, formula (102) leads directly to (99). On the other hand, in case of the repulsive forces Eq. (99) does not come out and there is no tail in the depth of the forbidden band. Indeed, in this case the electron energy is reduced on decreasing the number of impurity atoms, and this decrease cannot exceed - nu,. It should be kept in mind, however, that all the former results were obtained for the “one-band” approximation. This becomes invalid near the middle of the conduction band, and the simultaneous treatment both of the conduction and valence bands is needed67 (see also K e l d y ~ h ’where ~ the same considerations were used in another problem). Yet it turns out that the inclusion of this “band-to-band interaction” effect leads but to some corrections that are quantitatively unimportant. One of them, however, is essential in principle: it is nonzero only if there is some compensation and leads to the formation of the tail in the density of states of the minority carriers. The “intensity” of this tail is however rather small, so that its presence should manifest itself just as a certain decrease of the intensity of the majority carriers’ tail. The influence of compensation on the density of states of the majority carriers may be estimated with the help of Eq. (88). In the energy range where Eqs. (103) and (105) are valid, the compensation increases the density of states somewhat since it leads just to replacing n by the sum Z j n j and weakens the screening effect. More deeply in the forbidden band only the first of these factors remains, while the electron interaction with the compensating impurity, being repulsive, tends to reduce the density of states in the tail. The net effect seems to depend upon the particular model and the particular numbers used. 7ZhEquation(105) may be obtained from the general formula (90)by expanding a, in powers of s with only the first nonvanishing term left. 7 3 L. V. Keldysh, Zh. Eksperim. i Teor. Fiz. 45,364 (1963) [English Transl. ; Soviet Phys. J E T P 18, 253 (1964)l.

4. 12. REMARKSON

THE

137

EFFECT OF HEAVY DOPING

DENSITY OF

STATES

TAIL

Thus the self-consistent treatment gives a result that is, in a sense, intermediate between the two limiting cases considered in Section 3. The screening does not sweep away all the impurity states in the forbidden band but reduces their number considerably, as compared to what would be observed if there were no screening at all. The function (101) decreases very rapidly when one goes deeper into the forbidden band. The total electron concentration below the band edge, n- , is given by the relation

n- =IOmdEp(E)[exp/?(E-p)+ 1 1 - l .

(107)

At T = 0 this coincides with the total number of impurity states (per unit volume) in the forbidden band. Using Eq. (90) for the density of states and putting T = 0 it is easy to integrate over E . The remaining integral over s may be calculated approximately neglecting the factorially small number of the deep tail states. To this end it is sufficient to use just the first term of the a; expansion in s ~ ~ ~ : u1

Then one obtains easily

n-

= -a2s2.

-n(n~,~)-~'~.

Note, further on, that the density of states (90) has no singularity at the point E = - W,, where the discrete impurity level is situated in the case of light doping. Two remarks should be made in connection with the above results. First, it follows from what has been said that the tail formation is due not to the strong electron attraction to one of the donors but rather to the collective action of many impurity atoms situated close enough to one another. Therefore it is just this point where the assumption of the uncorrelated probability distribution (60) may become invalid ; the correlation between the impurity atoms will certainly exclude the formation of the too large donor clusters. This will leave but a finite number of terms in the series (97), and an effective cutting off of the tail will result. However in the heavy doping conditions this seems to be not very important. Indeed, the correlation of the type just discussed seems to show up at distances of the order of the lattice parameter d. Therefore the number of atoms in the cluster may be as large as, say, (ro/d)3.This corresponds to the tail states at the energy of about (ro/d)3E,. Formula (101) will still be valid in 4 x cm, d 5 x lop8cm; so this case if r,/d %- 1 (practically, a, that at n lo2' ~ m - r,/d ~ , 4). It follows from (101) that, in this energy

-

-

-

-

73aThisprocedure is clearly equivalent to the perturbation theory approach of Section 8.

138

V . L. BONCH-BRUEVICH

range, the density of states in the tail becomes very small, comparable in fact with that due to some casual imperfections like dislocations, small angle grain boundaries, etc. This means that under such conditions the tad is strictly cut off when the density of states in it practically becomes zero. Second, although no doubts exist concerning the existence of the tail and its rapid decrease, the problem of its exact asymptotic form is not completely clear as yet. The point is that, in the region of large negative values of E , the semiclassical approximation used above is not very convincing (as well as the approach used in Ref. 71). Therefore, the experimental study of the density of states in the forbidden band is desirable. Some possibilities in this respect will be mentioned in Section 15. 13. FERMI LEVEL

As shown in Section 5, to calculate the Fermi level, one has to set ne = n in Eq. (33) (no compensation, extrinsic range): +m

n

=

J

d~ p(E)[exp P(E - p) -03

+ 11-1 .

(110)

In view of the results obtained in the preceding section one might expect but a small difference between p and p,,, the unperturbed Fermi energy of the electron gas with the same concentration n and the dispersion law Wo(k).In fact, once the parameter 2 is small the electron-electron interaction causes just a small renormalizing term to appear, while the electronimpurity interaction changes the density of states appreciably only in the regions where it is small. Thus, we may put

P

= PO

+ b e e + APei

9

(111)

where Apee and Apei are the corrections due to the electron-electron and the electron-impurity interactions, respectively. In the first approximations that we limit ourselves to, these quantities may be calculated independently. For the Coulomb correction we obtain, using Eqs. (36) and (53),35 Af&e

= M(kF)

(112)

9

’ ~the Fermi wave number. [This differs from the usual where kF = ( 3 n / 8 ~ ) ’ is definition by the factor 27t due to our normalization of the Fourier transform; cf., e.g., Eq. (21).] Since K << kF, Eqs. (112) and (52) result in

A&,

=

-2e2k&

=

o(2).

(113)

To calculate the impurity correction it is convenient to insert Eq. (90)

4.

EFFECT OF HEAVY DOPING

139

into (110). Neglecting the factorially small terms one obtains, at T = 0,

where p' = p o

+ Apei. It follows from (114) that Apei < 0 and - - 0(19/8). APei

(115)

PO

Thus asymptotically at A 4 0 the electron-electron interaction gives the dominant correction to the Fermi level. The temperature corrections may be computed in an analogous manner. Retaining only the first nonvanishing term in 1, one obtainss1: p = po{ 1 - 0.23(nao3)-' I 3

-(n2/12)[1

+ l2(naO3)-l137t- 3 ( 3 ~ 2 )1- / 3 ] T / p o+ . . -} .

(1 16)

Thus, at T < ,uo the thermodynamic properties of the unipolar heavily doped semiconductor are very similar to those of the perfect degenerate Fermi gas. However, one essential difference between these materials and the degenerate samples doped but slightly (or intermediately) should be noted. It is known74 that in the latter case there are two degeneracy temperatures, the upper and the lower ones. The second of them is due to freezing out of the carriers passing from the conduction band to the impurity levels. In the heavily doped sample there is clearly no lower degeneracy temperature. The upper degeneracy temperature T* may be roughly estimated, with the help of Eq. (1 16), as the temperature at which the Fermi level coincides with the band edge in the pure material. It goes without saying that, strictly speaking, the approximations used above are invalid in this case, so Eq. (1 16) may give no more than an order of magnitude estimate. One obtains in this way :

T* To*(l - 0.21), (117) where To* = (12/n2)pOis the degeneracy temperature calculated for a perfect electron gas in the conduction band.74

14. MOMENTUM DISTRIBUTION FUNCTION AT SCREENED POTENTIAL

ZERO

TEMPERATURE AND THE

The statistical properties of the charge carrier gas do not reduce to the Fermi level only. Of considerable interest may be, for example, the 74

A. G. Sarnoilovich and L. L. Korenblit, Lisp. Fiz. Nauk 57, 577 (1955).

140

V. L. BONCH-BRUEVICH

momentum distribution n(k). This is given by

n(k) = (1/7c)

a,

-m

11,74a

dE Im G,(k, E ) [exp B(E - p ) + 11- ,

(1 18)

where, as above, G,(k,E) is the Fourier transform of the anticommutator retarded Green function averaged over the impurity configurations. If there is no damping and the mass operator does not depend upon the energy variable E, then, according to Eqs. (15), (32), and (118), n(k) is given by the usual Fermi formula, with renormalized energy, (53). If damping is present, the situation alters, and’ there is no universal form for the function n(k) (at equilibrium!). In particular at zero temperature the Fermi step function may be smeared off somewhat. Inserting the Green function (84) into (118), putting T = 0 and integrating over E, one obtains

The integral in (119) is regular in view of (86). To calculate n(k) explicitly note that the quantum corrections [present in (80) but omitted in (84)Jwould effectively cut off the integral (119) at large values of s. Therefore, we may approximate the function aI(s)by the Then simple calculation gives first expansion term (

n(k) = *{ 1

+ erf[p - W,(k)/2a]},

(120)

where erf x is the standard error function. It is seen that due to the electron-impurity interaction the Fermi distribution is smeared off, even at absolute zero. That is, the concept of the Fermi surface has no unique sense in the heavily doped materials.74c This is no wonder since, as we know, the very notion of the dispersion law is not exact in this case. The “smearing off” effect described by Eq. (120) is not large. In view of Eq. (87) the second term in the curly brackets tends rather rapidly to the ratio P - WWIP - Wk)I

when W,(k) recedes from the Fermi level. However this effect is of principal 74PTheright-hand side of (118) contains an extra numerical factor as compared to the formula given in Ref. 11. This is due to the different definitions of the Fourier transforms in Ref. 1 1 and in this paper. 74bThisis nothing but the use of the stationary phase method, allowed since the integral converges and a/W, is large, where W, is defined following Eq. (79). 74cThisstatement seems to contradict to the so-called Migdal theorem.j4 Yet the conditions under which this theorem was derived do not hold in our case.

4. EFFECT

OF HEAVY DOPING

141

importance when one calculates the screened potential. It is known55 that, for a perfect degenerate Fermi gas, the function K(k,O) appearing in Eq. (20)has a logarithmic branch point at k = 2k,. This leads to a peculiar behavior of the screened potential at large r : it oscillates and decreases considerably slower than exponentially : u(r)

-

I-

.

C O S ( ~ T ~ / ~ )

(121)

Were we to use Eq. (121) instead of the Debye formula, all the previous estimates would be altered somewhat (not to say that it would be doubtful to employ the semiclassical approximation in case of such a rapidly oscillating potential). However the form of the function (121) is due to the strictly stepwise character of the Fermi distribution at T = 0. It may be shown6’ that once there is some smearing off of the Fermi sphere the discussed branch point of the function K(k, 0) disappears and, consequently, the asymptotic behavior of the potential at large distances is given by the exponential formula (9) with the screening radius defined by (23). In the semiclassical approximation the function K(k, v) is easily evaluated, and the corrections to the screening radius coming from the electronimpurity and the electron-electron interactions may be found.66 They turn out to be rather small, of the order of A3/’, thus justifying all the previous estimates. In particular we obtain the result already used in the previous calculations : a/p

-

(nao3)-114

=

/23/4.

15. RESUME

We have seen that some features of the semiconductor energy spectrum are changed rather radically by the effect of heavy doping. The result of the electron-impurity interaction is not just the shift of the valence and the conduction band edges but the effective smearing off of them (the “tail” formation). The very notion of the band edge acquires a somewhat conditional sense in such a system. The impurity states in their usual meaning disappear too. The tail formation is in fact nothing but a heritage of the impurity bands left after the screening is taken account of. On the other hand, doping produces but a small effect on the density of states deeply in the allowed band and, in particular, near the Fermi level (in uncompressed samples). The only serious effect in this energy range seems to be due to the electron-polar phonon interaction. The fact that the density of states changes but little near the Fermi level while the total number of the tail states is relatively small leads to the result that the thermodynamic properties of the charge carrier gas in the heavily doped uncompensated samples do not differ appreciably from those of the

142

V. L. BONCH-BRUEVICH

perfect degenerate Fermi gas. Such seems to be the case in a number of standard kinetic problems as well. (Those of the static conductivity, thermoelectromotive force, etc.) The influence of the tail states seems to manifest itself in some subtle problems, among which the excess current in tunnel diodes first comes to attention. As is well-known the excess current may be caused by various factors; in particular the deep traps in the junction region seem to be of importance.” Yet when these are excluded the excess current does not tend to zero. The remaining part of it is probably just the usual tunnel current flowing between the tails formed at the opposite sides of the junction.76 To our knowledge the quantitative theory of this effect has not been developed in detail as yet. Qualitatively this point of view does not contradict the experimental data of Belova and K ~ v a l e v according ,~~ to which the position of the minimum tunnel current is shifted toward the higher voltages when the concentration of the doping impurity is increased (at constant temperature). Further on, the semiquantitative estimate7’ based upon essentially the same ideas was confirmed experimentally by Logan and Chynoweth.” The second phenomenon which might be sensitive to the presence of the tail in the forbidden band is the recombination radiation. Under certain conditions the form of the relevant spectral curves may directly reveal the density of states function.43 Last (but probably not least), interesting possibilities seem to be opened by the study of the piezoelectric effects in materials with anisotropic energy surfaces.79 A. G. Chynoweth, W. L. Feldman, and R. A. Logan, Phys. Rev. 121, 684 (1961). V. L. Bonch-Bruevich, Radiotekhn. i Elektron. 5, 2033 (1960) [English Transl.: Radio Eng. Electron. ( U S S R ) 5, 238 (1960)]. 7 7 N. A. Belova and A. N. Kovalev, Radiotekhn. i Elektron. 6, 160 (1961) [English Transl.: Radio Eng. Electron. (USSR) 6, 140 (1961)l. 7 8 R. A. Logan and A. G. Chynoweth, Phys. Rev. 131, 89 (1963). 79 H. Fritzsche and M. Cuevas, Proc. Intern. Conf Phys. Semicond., Exeter, 1962 p. 29. Inst of Phys. and Phys. SOC., London, 1962. 7s

76

CHAPTER 5

Energy Band Structures of Mixed Crystals of 111-V Compounds Donald Long I. INTRODUCTION . . . . . . . . 1. Importance of Mixed Crystals . . . 2. and Structures of I I I - vCompounds .

11. ENERGY GAPVERSUS COMPOSITION. . . . . . . . . 3. In(As,-,P,) 4. (In,_,Ga,)As . . . . . . . 5. Ga(As,-,P,) . . . . . . . 6. (In,-,Ga,)Sb . . . . . . . 7. (Ga,-,Al,)Sb . . . . . . .

. . . . . .

. . . . . . . . 143 . . . . . . . . 144 . . . . . . . . 145 . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . . 111. OTHERBAND PARAMETERS VERSUS COMPOSITION 8. EfSective Masses . . . . . . . . . . . . . . 9. Spin-Orbit Splitting . . . . . . . . . . . . .

. . . . . .

147 149 150 150 153 154

. 155 . 155

.

156

I. Introduction For the purposes of this article we define a mixed crystal as an alloy of two 111-V compound semiconductors. It now appears that most of the 111-V compounds are mutually soluble in all proportions, so that it will generally be possible to make a mixture of a pair of them in any composition.’ However, only five particular combinations have been studied in any detail to date. This article will therefore emphasize what is known about the energy bands in those five, which are all ternary compounds and can be represented by the formulas: In(As, -,PX), (In, -,Ga,)As, Ga(As, -,PX), (In, -,Ga,)Sb, and (Gal -,Al,)Sb, where x represents the mole fraction of the second group I11 or V element in the alloy. These mixed crystals can be thought of simply as an additional group of 111-V compound, zinc blende crystal structure semiconductors, in which the various energy band parameters are continuously variable with composition rather than discrete as in the ordinary binary 111-V compounds. This point of view is See J. Woolley, in “Compound Semiconductors” (R. K. Willardson and H. L. Goering, eds.), Vol. 1, pp. 3-20. Reinhold, New York, 1962, for detailed information on the metallurgical aspects of mixed 111-V compounds. See also E. K. Muller and J. L. Richards, J . Appl. Phys. 35, 1233 (1964).

143

144

DONALD LONG

valid because the transport and optical experiments done on mixed crystals have been interpretable in terms of well-defined band parameters having values which are simply intermediate to those in the binary components.'" Also, the band parameters can be expected a priori to vary continuously with composition, provided that the alloy remains random throughout and ordering effects do not occur.2*2a We deal here only with single-phase alloys. There are several useful general references which provide background information for the subject of the present article. They include the early review of semiconductor alloys by Herman et al.,' the review articles by Ehrenreich3 and by Long4 describing the energy bands in the 111-V compounds, and the books on 111-V compounds by Hilsum and Rose-Innes5 and by Madelung.'" 1. IMPORTANCE OF MIXEDCRYSTALS

Mixed crystals have been studied since almost the beginning of research on the 111-V semiconductors There are several reasons for the interest in them. First of all, by studying the variation of the energy gap and effective masses with composition one can often infer information about the band structures of the two components of a mixed crystal. If the band structures of the components are identical in form, the energy gap and the masses will exhibit a smooth and perhaps even linear dependence on composition. If the conduction or valence bands are of different forms, however, a kink or abrupt change of slope in the dependence may be seen at some intermediate composition. For example, such a kink is clearly evident in the gap vs. composition plot for Ge-Si alloy^,^^^ which is reproduced in Fig. 1. The explanation of the discontinuity at about 15 mol % Si in Ge is as follows.2 As Si atoms replace Ge atoms in the lattice, the three kinds of conduction band minima of Ge move away from the valence band in energy toward the positions they must ultimately have in pure Si? The [ I l l ] minima are lowest in Ge and the [lo01 minima are lowest in Si, so that the initial steeper slope at the Ge-rich end in Fig. 1 is due to a relatively rapid rise of the [ill] minima, while the flatter slope at higher Si concentrations is due to the slower movement of the [loo] minima, which have become the '"It appears that lattice-vibration data are not completely consistent with this interpretation. Gross features of the spectra suggest that results may be characteristic of pure germanium and silicon aggregates rather than a true disordered alloy. See Volume 3, Chapter on Multiphonon Lattice Absorption by W. G . Spitzer, Fig. 9 and discussion. * F. Herman, M. Glicksman, and R . H. Parmenter, Progr. Semicond. 2. I (1957). *"In regard to evidence for possible cases of aggregation, see footnote la. H. Ehrenreich, J. Appl. Phys. 32, 2155 (1961). D. Long, J. Appl. Phys. 33, 1682 (1962). C. Hilsum and A. C . Rose-Innes, "Semiconducting 111-V Compounds." Pergamon, London, 1961. "0. Madelung, "Physics of 111-V Compounds," Wiley, New York, 1964.

5.

BAND STRUCTURES OF MIXED CRYSTALS

1.21.1

0.7

I

I

I

I

145

I

-

60 80 100 MOL PERCENT SILICON IN GERMANIUM

0

20

40

FIG. 1. Energy gap vs. composition in Ge-Si alloys.

lowest-lying in the alloys containing more than 15% Si. The kink therefore results from the different positions in k-space of the lowest conduction band points in the two components of the alloy system. Thus one can hope to obtain information about the form of the bands in a particular semiconductor by making a series of alloys of it with a second semiconductor for which the band structure is already known, and then measuring the energy gaps in the mixed crystals. There is also considerable interest in mixed 111-V compounds for device applications. The continuous variability of parameters makes possible the design of a material with virtually any gap width within the range covered by the 111-V compounds (-0.17 to -2.3 eV at room temperature). This designability is especially useful for infrared or optical devices in which operation over particular wavelength ranges is required, such as in infrared sensors and injection lasers.

2. BANDSTRUCTURES OF 111-V COMPOUNDS Before beginning the discussion of the band structures of mixed crystals, we must first review the most important features and parameters near the edges of the conduction and valence bands in the binary 111-V compound~.~-”In all the semiconductors of this class studied to date the D. M.S. Bagguley, M. L. A. Robinson, and R. A. Stradling, Phys. Letters 6, 143 (1963). H. Piller and V. A. Patton, Phys. Rev. 129, 1169 (1963). 7PR.A. Stradling, Phys. Letters 20,217 (1966). * M. D. Sturge, Phys. Rev. 127,768 (1962). 8aH.Piller, in “Physics of Semiconductors” (Proc. 7th Intern. Conf.),p. 297. Dunod, Paris and Academic Press, New York, 1964. T. S. Moss, A. K. Walton, and B. Ellis, Proc. Intern. Con$ Phys. Semicond., Exeter, 1962 p. 295. Inst. of Phys. and Phys. SOC., London, 1962. l o T. S. Moss and B. Ellis, Proc. P hys. SOC. (London)83,2 17 (1 964). J. W. Hodby, Proc. Phys. SOC.(London)82,324 (1963).

146

DONALD LONG

valence bands are of the same form. The valence band maximum is, for practical purposes,11a at k = 0 and is doubly degenerate (excluding spin), with two E vs. k branches meeting at the center of the Brillouin zone; E is carrier energy and k represents the wave number. These two branches give different effective masses, corresponding to the so-called “light” and “heavy” holes. A third valence band branch is split from the above two by spin-orbit interaction. I

’

\

0.36eV

Gads

1.53eV

HEAVY HOLE BRANCH LIGHT HOLE

-0.5

0

0.5

1.0

k , in 10’ cm-’

FIG.2. Energy vs wave number in the [ l l l ] and [lo01 directions for GaAs (at 0°K). The position of the subsidiary [lo01 minima betwen k = 0 and the zone boundary, their exact energy separation from the central minimum, and the effective mass of these minima are not established, so that the drawing of them here is only suggestive Note that the very small splitting of the heavy hole branch due to lack of inversion symmetry is not shown: it has never been clearly resolved experimentally?

Of the seven most intensively studied 111-V compounds, five have their conduction band minima also at k = 0 (viz., InSb, InAs, GaSb, InP, and “”Because of the fact that in the 111-V compounds the unit cell contains two different atoms and the inversion symmetry of the diamond lattice is no longer present, the possibility exists of contributions to energy from terms linear in k. Any such terms must, however, be very small since they have not been detected e~perimentally,~ except perhaps in InAs.

5.

BAND STRUCTURES OF MIXED CRYSTALS

147

GaAs), whereas the other two (Gap and AlSb) have multiple minima elsewhere in k-space. Mixed-crystal studies have helped to establish these band structures, in cooperation with various other types of experiments. The k = 0 minimum in each of the five semiconductors listed first above is nondegenerate, so that the surfaces of constant energy are spheres, but it is a nonparabolic minimum in the sense that E deviates from a quadratic dependence on k at energies away from the band edge. Some of these five materials also have subsidiary conduction minima at energies of only a few tenths of an electron volt above the central minimum, notably GaSb ([lll] valleys at -0.1 eV above the k = 0 valley) and GaAs ([loo] valleys at -0.36 eV up). The lowest conduction band points in both GaP and AlSb appear to lie on the [loo] axes in k-space, so that symmetry requires them to be six in number (unless they should happen to occur at the zone boundary, in which case their number would be three). Each of these two semiconductors also has a k = 0 minimum, which lies about 0.35 eV above the [loo] valleys in GaP and 0.3 eV above in AlSb. The energy gap, effective masses, and valence band spin-orbit splitting energies of the above seven 111-V compounds are listed in Table I. An E vs. k diagram for the [lo01 and [lll] directions in GaAs is shown in Fig. 2 to illustrate the general nature of the band structure in all these semiconductors. Let us now review what is known about the energy bands in the five 111-V compound mixed-crystal systems for which sufficient reliable information is available. The main item of information for all these materials is the dependence of the energy gap on alloy composition, and this is presented for each alloy system immediately below. Additional lesser items are discussed later. Note that in each of the alloy systems considered here the lattice parameter has been found to be a linear function of composition,’ so that one can easily convert the composition scales in the following figures to lattice parameter scales if desired. 11. Energy Gap versus Composition

In the cases to be considered, the energy gap was measured either by optical (infrared) absorption techniques or from the slope of a logarithmic plot of resistivity or Hall coefficient vs. 1/T in the range of intrinsic conduction, where T is the absolute temperature. An optically measured gap is characteristic of the temperature of measurement, but a gap width determined from the resistivity or Hall effect vs. temperature behavior represents a linear extrapolation to absolute zero according to the rate of variation of the gap with temperature over the intrinsic range.’ Since the primary interest here is in the gap variation with composition, we will not

TABLE I ENERGYBANDPARAMETERS OF THE 1II-V

Average effective massesh

Energy gaps Compound" InSb [OOO] lnAs [OOO] GaSb [OW] InP [OOO] GaAs [OOO] AlSb [OOO] GaP [IOO]

-

0°K

0.235eV 0.43 0.81 1.41 1.53 1.60J 2.4

77°K

0.225 eV 0.41 0.77-0.82' 1.38 1.49-1.52' 1.58l -

6

COMPOUNDS

300°K 0.17 eV 0.35 0.70-0.74' 1.27 1.37-1.43' 1.52l 2.25

Electrons

Light holes

0.0155' 0.025 0.047' 0.073 0.07 0.39 0.35

0.02 0.025 0.052'

Heavy holes

-0.40.41 0.35' 0.68 -

-

0.12 -

-

Spin-orbi t splitting energy 0.9 eV 0.43 0.8 0.24 0.35 0.75 0.13

Referencesd 6 7,7a -

8,8a 9,lO 9,ll

U 0

z

The bracketed digits indicate the locations of the lowest conduction band minima in k-space. The values given are ratios of the effective masses to the free electron mass. Also, these are band edge values, and they are averages in the cases of the nonspherical energy surfaces. Value at 4°K ; other mass values are representative of room temperature. Most of the tabulated values were taken from Refs. 3 and 4. The recent references listed here were used to update certain values and fill in missing ones. Values are uncertain. Most optical data (including mixed-crystal results) agree with the first values shown, but the careful analyses of the recent references give the second values. J These gap values represent the consensus of early m e a s ~ r e m e n t sand ~ ' ~ agree well with our Fig. 7 ; however, a careful study of optical absorption by Turner and Reese [ P h y s Rcu. 117, 1003 (1960)], and recent evidence that the lowest conduction band minima in AlSb are o n [LOO] axes, indicate that better values may be the following: Indirect gap 77°K 300°K

1.40eV 1.33

Direct gap at k

=

0

1.70 eV 1.62

The band structure of AlSb is still not well understood and needs more study to resolve these discrepancies.

!i

s

3

5.

149

BAND STRUCTURES OF MIXED CRYSTALS

be overly concerned with accurate absolute values, but we will specify in each case whether the gap was measured by optical absorption or by the intrinsic electrical properties, so that one can know how to correct for the small differences. Note that in all the III-V compounds the energy gap decreases with increasing temperat~re.~

3. In(As, -xPx) This III-V mixed-crystal system was one of the first in~estigated.'*-'~ The energy gap vs. composition curve has been measured both by the optical and electrical methods, and the results are plotted in Fig. 3. The I

2

1.4

-

1.2

-

I

I

I

t

a- 1.0 4

a

z

0.4

-

-

/ I

I

I

I

0.4 0.6 0.0 1.0 0 0.2 X I nP Inns FIG. 3. Energy gap vs composition for In(As,-,P,) Solid line is from electrical measurements, and dashed lines are from optical absorption. The 0°K line actually represents a linear extrapolation from gaps measured at various temperatures between 77" and -470°K. The x's indicate the positions of the radiation emission lines of diodes made of these alloys.

optical results are taken from the experiments of O ~ w a l d , 'who ~ measured the absorption edges of a range of alloys at several temperatures between 90" and about 470°K; thus the 0°K plot involves an extrapolation of his data. The electrical results were obtained by Weiss.13 The smooth, linear variation of the gap with composition is consistent with the fact that InAs and InP have conduction and valence band structures of identical forms.

'' 0. G. Folberth, Z. Naturforsch. IOa, 502 (1955). I' l4

H. Weiss, Z. Naturforsch. l l a , 430 (1956). H. Welker and H. Weiss, Solid State Phys. 3, 1 (1956). F. Oswald, 2. Naturforsch. 14a, 374 (1959).

150

DONALD LONG

Laser action at 77°K has recently been reported in p-n junction diodes made of In(As, ,Po.,,) alloys16;actually both spontaneous and stimulated emissions were observed in this alloy, and only spontaneous emission alone was observed in an In(Aso.s2Po.,s) diode, presumably due to technical limitations. The wavelengths of the emission peaks observed in the two alloys agree fairly well with the energy gaps expected for these compositions at 77°K according to Fig. 3. 4. (In

-

.Ga,)As

This is an example of a mixed crystal in which the cation concentration varies in the alloys instead of the anion concentration. The energy gap dependence on composition has been measured by Abrahams et al." and by Woolley et al.," using room temperature optical absorption experiments; the results are reproduced in Fig. 4. The dashed curve in Fig. 4 has been drawn favoring the Woolley et al. data, because their samples are believed to have been more nearly homogeneous than those of Abrahams et al. and also because they found consistent results on samples prepared in several different ways. Still, the disagreement of the two sets of data is disturbing. Here again the gap variation appears smooth, although the data are perhaps not quite good enough to eliminate the possibility of a kink. However, it is known from other experiments that the band structures of InAs and GaAs are identical in form, so that a smooth curve would be expected in Fig. 4. 5. Ga(As - .PX) Thus far, this mixed 111-V compound system has received more attention than any other. Folberth" and Welker and WeissI4 reported early measurements of the optical gap at room temperature, and Ehrenreich3 has recently reported additional results of this type obtained by Weisberg et al. ; these three sets of data are plotted in Fig. 5. Although the data points show a fair amount of scatter, they do seem to establish the existence of an abrupt change of slope between two linear regions of the gap vs. composition dependence in the vicinity of a 50% alloy, as indicated by the dashed curve in Fig. 5. The explanation of the kink in Fig. 5 is undoubtedly analogous to that given earlier for Ge-Si alloys; i.e., it occurs because and when the k = 0 conduction band minimum (lowest in GaAs) and the [lo01 minima (lowest in Gap) lie at the same energy, assuming that the valence band remains unchanged in form over the entire composition range. The F. B. Alexander et al., Appl. Phys. Letters 4, 13 (1964). M. S. Abrahams, R. Braunstein, and F. D. Rosi, J . Phys. Chem. Solids 10, 204 (1959). J. C. Woolley, C. M. Gillett, and J. A. Evans, Proc. Phys. SOC. (London) 77, 700 (1961) l 9 0. G. Folberth, Z . Naturforsch. lOa, 502 (1955).

l6

5.

BAND STRUCTURES OF MIXED CRYSTALS

151

FIG. 4. Energy gap v s composition for (In,-,Ga,)As determined from room temperature optical absorption measurements by Abrahams et a1.” (0)and Woolley et a1.” (XI The x’s represent only the data taken by Woolley et al. on bulk single-phase samples.

FIG. 5. Energy gap vs. composition for Ga(As, -zPz).The dashed lines represent the results of room temperature optical absorption measurements by Folberth’’ (x), Welker and Weiss14 (o),and Weisberg ef aL3 (0). The two solid lines represent Spitzer and Mead’s determinations of the direct and indirect gaps v s composition, using surface barrier photovoltage data

152

DONALD LONG

optical absorption data leading to the dashed curve of Fig. 5 provided one of the original pieces of evidence for the conclusion that the lowest conduction band energy in GaP is not at k = 0. Following the publication of the optical absorption results discussed in the preceding paragraph, Spitzer and Mead” made a study of the energy gap vs. composition in the Ga(As, -,Px) system using measurements of the photovoltaic response of surface barriers at metal-semiconductor interfaces. It is found from the spectral dependence that the photoresponse increases sharply when the photon energy becomes greater than the energy gap, so that this effect can be used to measure the gap. Furthermore, when excitation can occur to two different sets of conduction band minima, as in the Ga(As, -,PJ alloys, the photoresponse spectrum is clearly resolved into distinct regions, corresponding to the two different types of electron transitions. This experimental approach therefore provides a sensitive means of studying separately the dependences of the gaps between the valence band and both sets of conduction band minima on composition. The Spitzer-Mead results are plotted as solid straight lines in Fig 5. The direct transition line (to the k = 0 minimum) is coincident below x = 0.53 with the dashed line determined earlier from optical absorption experiments, but the indirect transition line (to the [loo] minima) falls under the dashed line above x = 0.53 in such a way as to give a value of x = 0.38 instead of x = 0.53 for the composition at which the two sets of conduction band minima lie at the same energy. The photovoltaic technique appears to be more sensitive and less subject to possible uncertainties than optical absorption, so that its results may be more nearly correct, but the complete cause of the discrepancy in Fig. 5 remains to be determined. Fenner21 has found from an analysis of the effect of high pressure on electrical resistivity that the “crossover” point is at x z 0.44 at room temperature. Ga(As,_,P,) was the first mixed 111-V compound in which injection laser action was observed.22,23 Other studies have been made of both spontaneous and stimulated emission in such diode^.^^-^^" All of the p-n junction diodes which exhibited stimulated emission had a composition ’OW. G. Spitzer and C. A. Mead, Phys. Rev. 133, A872 (1964). ” G. E. Fenner, Phys. Reo. 134, A1113 (1964). N. Holonyak, Jr., and S. F. Bevacqua, Appl. Phys. Letters 1, 82 (1962). 2 3 N. Holonyak, Jr, S. F. Bevacqua, C. V. Bielan, and S. J. Lubowski, Appl. Phys. Letters 3,

’’

47 (1963).

S.-M. Ku and J. F. Black, Solid-State Electron. 6, 505 (1963). A. Fulton, D. B. Fitchen, and G . E. Fenner, Appl, Phys. Letters 4, 9 (1964). 2 6 N. Ainslie, M. Pilkuhn, and H. Rupprecht, J . Appl. Phys. 35, 105 (1964). 26aM.Pilkuhn and H. Rupprecht, J . Appl. Phys. 36, 684 (1965). 24

l 5T.

153

5 . BAND STRUCTURES OF MIXED CRYSTALS

parameter value of x 2 0.45. According to Fig. 5 this corresponds to GaAsrich alloys in which the GaAs band structure dominates. This result is consistent with the band structure vs. composition pattern in Ga(As, -px), since a direct gap was required for stimulated emission. Note also that the observation of stimulated emission only below x x 0.45 is consistent with the Spitzer-Mead conclusion about the gap vs. composition kink in Fig. 5 and with Fenner’s pressure result. Studies of fluorescent emission due to electron beam excitationz6b are in general agreement with all the above experiments; they give “cross-over” points of x = 0.43 at room temperature and x = 0.40 at 77°K.

6. (In, -,Ga,)Sb Woolley and co-workers have measured the energy gap vs. composition in this system by both the optical absorptionz7 and intrinsic electrical properties methods,28 and the resulting curves are plotted in Fig. 6. No 0.7

> Q

aa

/

W

/

/

/

/

a

0.2

-

/

0 IoSb

, ,/ 0.2

0.6

0.4

X

0.8

1.0

GoSb

FIG. 6. Energy gap vs. composition for (In,-xGa,)Sb. Solid curve is from electrical measurements, and dashed curve is from room temperature optical absorption (after Woolley et 4Lz8J7).

data points are shown, since there were many of them and their scatter was small enough that the shapes of the curves in Fig. 6 can be considered well established. The curve obtained from the room temperature optical 26bD.A. Cusano, G . E. Fenner, and R. 0.Carlson, Appl. Phys. Letters 5, 1 4 4 (1964). *’ J. C. Woolley, J. A. Evans, and C. M. Gillett, Proc. Phys. Soc. (London) 74, 244 (1959). J. C. Woolley and C. M. Gillett, J . Phys. Chem. Solids 17, 34 (1960).

154

DONALD LONG

absorption measurements is relatively smooth, although it almost exhibits possible breaks near x = 0.2 and x = 0.7. The band edges in InSb and GaSb are of analogous forms, however, so that a smooth curve is expected. The solid line in Fig. 6, representing the electrical results and giving the gap extrapolated linearly to OOK, extends from the InSb end only to x x 0.7; the data at the GaSb-rich end show complicated behavior, presumably due to the small separation of the [ l l l ] conduction band minima from the k = 0 valley in GaSb, and to a possible ordering effect in the GaSb-rich alloys.28 7. (Gal -,Al,)Sb The composition dependence of the energy gap has been measured for (Gal -,Al,)Sb by both the optical and electrical methods. The results are shown in Fig. 7, and the positions of the data points from the room temper-

07 0 GaSb

02

0.6

0.4

X

0.0

1.0 ALSb

FIG. 7. Energy gap vs. composition for (Ga,-,AI,)Sb. Solid curve is from electrical measurements, and dashed curve is from room temperature optical absorption. The data points from Burdiyan’s’’ optical experiments are indicated as open circles, and the data of Miller et aL3’ are represented by x’s.

I. I. Burdiyan, Fiz. Tuerd. Tela 1, 1360 (1959) [English Transl.: Soviet Phys.-Solid State 1, 1246 ( 1960)]. 30 J . F. Miller, H. L. Goering, and R. C. Himes, J . Electrochem. Soc. 107, 527 (1960). z9

5.

BAND STRUCTURES OF MIXED CRYSTALS

155

ature optical absorption experiments of B ~ r d i y a nare ~ ~indicated. Similar optical measurements by Miller et aL3’ gave data with even more scatter. There is probably a break in the optical curve at x x 0.7, although the data points are too scattered to permit establishing its existence and position definitely. The electrical curve3’ cannot be expected to show a very sharp change of slope, since the relatively large kT spread inherent in the electrical measurements would tend to smear out any abrupt kink. Thus the gap vs. composition curve is probably consistent with the fact that GaSb and AlSb have different conduction band structures.

111. Other Band Parameters versus Composition Energy band parameters other than the gap are also expected to vary with composition, but very few measurements of such quantities have yet been made. Let us briefly review what is known about composition dependences of effective masses and of the spin-orbit splitting energy.

8. EFFECTIVE MASSES Oswald determined the electron effective mass m,* as a function of composition from measurements of free carrier absorption during his study of the optical properties of the In(As,P,-,) system.” The absorption coefficient at wavelengths longer than the fundamental edge is inversely proportional to rn*2. The results are plotted in Fig. 8, and the curve is

w

: : : mfl -

m0

0.w

0.0

01

0

1

0.2

InAs

I

I

0.6

0.4 X

I

0.8

I

1.0

InP

FIG.8. Electron effective mass mn* vs. composition for In(As,-,Px) as determined from free carrier absorption data (after Oswald”).

smooth as expected because of the similar conduction bands of the two components. The disagreement of the curve end points in Fig. 8 with the masses for pure InAs and InP of Table I is presumably due to difficulties 31

I. I. Burdiyan and B. T. Kolomiets, Fiz. Tuerd. Tela 1, 1165 (1959) [English Transl.: Sooiet Phys.-Solid State 1, 1067 (196011.

156

DONALD LONG

and inaccuracies of absolute measurement in the free carrier absorption experiments, but the curve shape is still probably representative of the alloy system. Woolley and GillettZ8 have inferred values of m,* and the hole mass m,* in (In, -,Ga,)Sb alloys from their resistivity and Hall effect vs. temperature data. A rather smooth trend is found for each mass, but the curves are not presented here since masses determined in this manner are only roughly representative of the true values.

9. SPIN-ORBIT SPLITTING Hodby' has studied the infrared absorption corresponding to transitions among the three valence band branches in a series of Ga(As,-,P,) alloys and has deduced spin-orbit splitting energies from the results. His data points show very little scatter and indicate a linear dependence on composition between 0.33 eV in pure GaAs and 0.13 eV in Gap. It is worth noting that a linear composition dependence of the valence band spinorbit splitting energy has also been observed in Ge-Si alloys.32 These interesting results suggest that one would find a linear or nearly linear dependence in any mixed crystal of the 111-V compounds because of the similar valence bands in all these semiconductors. 32

R. Braunstein, Phys. Rev.

130, 869 (1963).

Magnetic Field Effects

This Page Intentionally Left Blank

CHAPTER 6

Magnetic Quantum Effects Laura M . Roth and Petros N . Argyres I. INTRODUCTION .

. . . . . . . . . . . . . . . 159

1 . General Discussion 2. Density of States .

. . . . . . . . . . . . . . I59 . . . . . . . . . . . . . . 162

11. MAGNETIC SUSCEPTIBILITY . . 3. De Haas-van Alphen Effect . 4. Steadv Effects. . . . .

. . . . . . . . . . . 169 . . . . . . . . . . . 169 . . . . . . . . . . . 172

111. QUANTUM GALVANOMAGNETIC EFFECTS . 5. Basic Theory . . . . . . . . 6. Shubnikowde Haas Eject . . . . 7 . QuantumLimit . . . . . . . 8 . Optical Phonon Scattering . . . . 9. Conclusion. . . . . . . . .

. . . . . . . . 174 . . . . . . . . 174 . . . . . . . . . . . . .

. . . . . 183 . . . . . 192 . . . . . 199 . . . . . 202

I. Introduction 1. GENERAL DISCUSSION Some of the most interesting magnetic effects occurring in solids are due to the profound changes in the density of states of the conduction electrons induced by the magnetic quantization of levels. Thus the level structure consists of a set of magnetic subbands' separated by the cyclotron frequency, and the density of states has a singularity at the bottom of each magnetic subband. In degenerate materials with a well-defined Fermi energy, if the magnetic field is swept, the density of states has a singularity each time a new subband crosses the Fermi level, which results in quasiperiodicity in 1/B. This quasiperiodicity is reflected in a number of properties. Perhaps the best known of these oscillatory effects is the de Haas-van Alphen effect,2 i.e., oscillations in the magnetic susceptibility. Actually, oscillations in the magnetoresistance of bismuth were discovered by Shubnikov and de Haas3 slightly earlier than the susceptibility oscillations. The

' L. D. Landau, 2. Physik 64, 629 (1930). W. J. de Haas and P. M. van Alphen, Leiden Commun. 208d, 212a (1930) and 220d (1933). L. Shubnikov and W. J . de Haas, Leiden Commun. 207a, 207c, 207d, 210a (1930).

159

160

LAURA M . ROTH AND PETROS N. ARGYRES

oscillatory effects, and in fact primarily the de Haas-van Alphen effect, have been important tools in probing the Fermi surface of metals and semiIn 111-V compounds the experimental results have been entirely of the Shubnikovde Haas type, i.e., oscillatory magnetoresistance. Previous reviews of the oscillatory effects have been given by Kahn and Frederikse6 and by Adams and Keyes.’ In addition to the susceptibility and magnetoresistance oscillations, analogous oscillations have been found in other properties, primarily in semimetals. These include specific heat,* thermomagnetic properties,’ and optical properties. O The classical galvanomagnetic effects, i.e., those effects not dependent upon the quantization of magnetic orbits, have been extremely important in determining band structure of semiconductors, and are discussed by Becker and by Weiss in this volume. Galvanomagnetic effects in semiconductors are also reviewed extensively by Beer.’ In metals, the classical magnetoresistance is related to the shape and connectivity of the Fermi surface. Thus the magnetoresistance due to closed magnetic orbits saturates, while that for open orbits does not saturate. Hence, for multiply connected Fermi surfaces there is rapid angular variation of the magnetoresistance.’2, l 3 In addition, in some metals magnetic b r e a k d ~ w n ’(“orbit-hopping”) ~ influences both de Haas-van Alphen’ and magnetoresistive effects.I6 We shall not discuss these interesting effects but refer the reader to the literature. The observation of metallic-type oscillations in semiconductors is due to the existence of certain 111-V compounds, for example, InSb and InAs, which are degenerate at low temperatures. These compounds have electrons with small effective masses, and the carriers do not freeze out. The reason for this has been given in terms of impurity banding,I7 i.e., the impurity D. Shoenberg, Progr. Low Temp. Phys. & 226 (1957). D. Shoenberg, “The Fermi Surface” (W. H. Harrison and M. B. Webb, eds.), p. 74. Wiley, New York, 1961. ’ A. H . Kahn and H. P. R . Frederikse, Solid State Phys. 9, 257 (1959). E. N. Adams and R. W. Keyes, Progr. Semicond. 6, 87 (1962). * J. E. Kunzler, F. S. L. Hsu, and W. S. Boyle, Phys. Rev. 128, 1084 (1962). M. C. Steele and J. Babiskin, Phys. Rev. 98, 359 (1955). l o M. S. Dresselhaus and J. G. Mavroides, Solid State Commun. 2, 297 (1964). A. C. Beer, Solid State Phys. Suppl. No.4 (1963). l 2 I . M. Lifshitz and V. G. Peschanskii, Zh. Eksperim. i Teor. Fiz. 35,1251 (1958); 38,188 (1960) [English Transl.: Soviet Physics J E T P 8, 875 (1959); 11, 137 (196O)J. 1 3 R . G. Chamber$ “The Fermi Surface” (W. H. Harrison and M. B. Webb, eds.), p. 100. Wiley, New York, 1961. l4 M. Cohen and L. Falicov, Phys. Rev. Letters 5, 544 (1960). M. G. Priestley, L. M. Falicov, and G. Weisz. Phys. Rev. 131, 617 (1963). l 6 R. W. Stark, Phys. Rev. 135, A1698 (1964). ’’ R . W. Keyes and R . Sladek, J . Phys. Cheni. SoIids 1, 143 (1956).

’

6.

MAGNETIC QUANTUM EFFECTS

161

wave functions are spread out due to the small effective mass, and overlap so much that the impurity band overlaps the conduction band. An alternative explanation due to Adams and Keyes’ is that conduction electron screening reduces the interaction between an electron and a donor to such an extent that the binding energy vanishes. Actually, a combination of the two effects probably occurs. In any case, in a sufficiently high magnetic field the carriers do freeze due to the contraction of the donor wave f ~ n c t i o n s . ’ ~ These * ’ ~ magnetic freeze-out effects are discussed by Putley in this volume. In this chapter we shall first discuss the magnetic levels and obtain the density of states for simple and complex bands, including the effect of level broadening. This will be applied to the de Haas-van Alphen effect, and results will also be given for the steady magnetic susceptibility, which is actually a quantum effect since classically the steady susceptibility vanishes.” The theory of quantum galvanomagnetic effects will be presented in some detail. Much of the interest in this subject is in the theoretical problems involved in generalizing transport theory to apply to quantum galvanomagnetic phenomena. Results will be given for the oscillatory magnetoresistance for simple bands and compared with experimental results. The generalization to complex bands will not be discussed explicitly, but by comparison with the de Haas-van Alphen effect the general form of the results should be obvious. In addition to oscillatory effects, profound changes in various properties are to be expected in the quantum limit, i.e., when the magnetic field is so large that all the electrons are in the lowest magnetic level. This can occur in semiconductors and semimetals when the effective mass is small. We shall discuss the quantum limit for the susceptibility as well as the magnetoresistance. Finally we note that the existence of oscillations in various phenomena does not necessarily require a sharply defined Fermi level, if an external energy is available to probe the density of states. One example of this is oscillatory magneto-optical effects, discussed by Lax and Mavroides in Volume 3 of this work, in which the energy is that of a photon. As shown by Gurevich and Firsov,” the energy can also be that of an optical phonon, and the effect results in a new kind of oscillatory magnetoresistance which we shall discuss in the final section of this chapter. Y. Yafet, R. Keyes, and E. Adarns, J . Phys. Chem. Solids 1, 137 (1956). R. Sladek, J . Phys. Chem. Solids 5, 157 (1958). 2o R. E. Peierls, “Quantum Theory of Solids,” p. 145. Oxford Univ. Press (Clarendon), London and New York, 1956. V. L. Gurevich and Yu. A. Firsov, Zh. Eksperim. i Teor. Fiz. 40, 198 (1961) [English Transl.: Soviet Phys. JETP 13, 137 (1961)]. l9

162

LAURA M. ROTH AND PETROS N . ARGYRES

2. DENSITY OF STATES a. Simple Band

In this section we review briefly the effects of a magnetic field on the motion of the electrons in a crystal. We shall be primarily interested in the density of one-electron states. We consider first a simple model which is in fact quite useful for 111-V compounds, namely, the case of a “free” electron with spin s having mass m*, g-factor g, and charge (-e) in a uniform magnetic field B in the zdirection. Classically, the motion of the electron transverse to the magnetic field becomes periodic with angular frequency o,= eB/m*c,

(1)

the cyclotron frequency, while the longitudinal motion remains the same as that of a free particle. Quantum mechanically the transverse motion becomes quantized, while the longitudinal remains unaffected. In this case the problem can be solved exactly.’ The Hamiltonian is

where A is the vector potential describing the magnetic field, B = V x A, and pB = et2/2mc is the Bohr magneton, with m the free electron mass. For the Landau gauge, A = (0, B x , 0), which proves convenient for the later applications, p,, and p, commute with the Hamiltonian and thus the energy eigenstates of the orbital part of H , are easily found to be $,&)

Here k

=

=

(L,L,)-1’2q,(x - Xk)eikyVk=z.

(3)

(k,,k,) is the two-dimensional wave vector associated with the

y- and z-components of the motion of the electron, and L,, L, are the

corresponding normalization lengths. The oscillator quantum number n takes on the values 0, 1, 2 , . . . . The q , ( x - X,) are the normalized wave functions of a simple harmonic oscillator of angular frequency o,centered at the point X , = -i2k,,

(4)

where 1 is the classical cyclotron radius of the lowest oscillator orbit. It is given by

Note that 1 is independent of the mass of the electron. The energy eigenvalues, now including spin, are

6. =

€nkf

( + :-1)+ n

163

MAGNETIC QUANTUM EFFECTS

- Am,

-

+

E,;

E,

=

h2kZ2/2m*,

the first term being the quantized transverse energy and the second the longitudinal kinetic energy. Here v = rn*g/2m; for free electrons with m* = m and g = 2, we have v = 1.

1

0

I 3/2

I

0

1/2

I

5/2

1 7/2

I

J

912

E/h wc FIG. 1. Density of states in the presence of a magnetic field. The zero field case is also shown.

We note that these states are degenerate since the energy is independent of x k . In fact, the density of orbital states per unit range of the wave number k,, with a given oscillator quantum number n, is dN -(n, dkz

k,)

=

e B / ( 2 7 ~ ) ~= h c1 / ( 2 7 ~ 4 ~ ,

(7)

i.e., proportional to B. We can now evaluate the density of states per unit energy interval at energy E, which is defined by

where R is the volume and evaluate Eq. (8) to give

E,kf

are given by Eq. (6). Using Eq. (7) we can

where the sum is understood to extend over all non-negative integers n for which the summand is real. Note that this makes dN/dE = 0 for E less than the lowest eigenvalue (1 - v)hwc/2. The density of states (neglecting

164

LAURA M. ROTH AND PETROS N. ARGYRES

spin) is shown graphically in Fig. 1, which displays the well-known quasiperiodic character with period ha,.We see that there is a singularity in the density of states at the bottom of each magnetic subband. This is due essentially to the one-dimensional character of the motion in the direction of the magnetic field. The density of states in the presence of a magnetic field should be contrasted with the density of states in the absence of the magnetic field, namely,

2

dN, dE -

*1

6(Ek

- E)

(9)

=

k

where the factor of two is for spin. This is also plotted in Fig. 1. The analytic relationship between the two densities of states is obtained through the use of the Poisson summation formula,22 m

+m

n=O

-m

m

1 @(n + 3)= 1 (- lrjo@(x)e2xirxd x .

f 10)

This shows that dN/dE is equal to dN0/& plus oscillatory terms, whose amplitude vanishes as B 0. More explicitly we have, for E / ~ o , % 1, --f

For arbitrary values of ha, the corresponding expression is obtained by replacing cos(x - in) in Eq. (11) by

aJ

(2x/n)”l

cos(x - &y2) d y ,

0

which involves Fresnel integrals. The value of the effective mass m* for a band edge such as found in most 111-V compounds is obtained by means of the k - p method, which is discussed by Kane in a previous chapter of this volume. The corresponding A recent tabulation of result for the g-factor is obtained by Roth et g-factors for 111-V compounds has been made by C a r d ~ n a ,and ~ ~ is reproduced in Table I.24a ” R . Courant and D. Hilberf “Methods of Mathematical Physics,” Vol. 1, p. 77. Wiley (Interscience), New York, 1953. 2 3 L. M. Roth, S. Zwerdling, and B. Lax, Phys. Rev. 114, 90 (1959). 24 M. Cardona, J . Phys. Chem. Solids 24, 1543 (1963). 24s The information is also given in Table IV of the article by Cardona “Optical Absorption above the Fundamental Edge” in Volume 3 of this series.

6.

TABLE I EFFECTIVE MASSESAND g-FACTORS FOR ELECTRONS AT k 111-V COMPOUNDS‘ Material

m/m*

InSb InAs InP GaSb GaAs GaP AlSb a

165

MAGNETIC QUANTUM EFFECTS

-44 - 12 0.60 -6.1 0.32 1.16 0.4

1.7

9.1

0 IN

lvl = lgm*/2ml

‘9

66 36 14 22 12

=

0.33 0.17 0.021 0.15 0.013 0.11 0.022

Based on calculation of C a r d ~ n a . ’ ~

b. Complex Band

Much of the character of the above results is preserved for the motion of the electrons in a crystal under the influence of a periodic electrostatic potential. These electrons behave as quasiparticles with an energy-momentum relation &@) appropriate to the particular energy band in which they move ; p is the crystal momentum hk, k being the Bloch wave vector. Their motion in the presence of a magnetic field has been considered by a number of authors including B l o ~ n tWannier ,~~ and Fredkin,26 and R ~ t h . ~For ’ bands which are nondegenerate except for spin, it is possible to obtain an effective one-band Hamiltonian of the form H o = S@

+ eA/c) + B W,@ + eA/c) + B2W,@ + eA/c) + ..

The first term of Eq. (12) is the zero-field band energy &@), with p replaced by the operator p + eA/c in a completely symmetric manner. The second and third terms (likewise symmetrized) are corrections coming essentially from the noncommutativity of the components of kinetic momentum p + eA/c:

[

p x + c , eAx

[l”t

&+-I ,A] = C

The second term is the Zeeman interaction and depends on spin. The third term is a complicated correction necessary (unfortunately) to obtain the correct steady magnetic susceptibility.

’’E. I. Blount, Phys. Reu. 126, 1636 (1962). 26

G. H. Wannier and D. R. Fredkin, Phys. Rev. 125, 1910 (1962). M. Roth, J . Phys. Chem. Solids 23, 433 (1962).

” L.

166

LAURA M. ROTH AND PETROS N. ARGYRES

The first term of Eq. (12) was postulated by Onsager2*who showed that, since the transverse kinetic momentum components behave, from Eq. (13), like a canonically conjugate coordinate and momentum, the Bohr-Sommerfeld quantization condition can be invoked for closed orbits. The result A(E,k,,

k,) = (n + YPR/12

(14)

gives the quantization of the cross-sectional area A(€,k,) of the energy surface €@) = E in k-space intercepted by the plane k, = constant perpendicular to the direction of the magnetic field. Here n is an integer and y a constant (but see below). The result can also be obtained in the WKB approximation, and gives implicitly the energy eigenvalues of the quasiparticle in the magnetic field in terms of the quantum numbers n and k,. The energy separation between two successive energy levels is now

for a fixed k,. This defines a cyclotron frequency m,. It has been found possiblez9 to generalize Eq. (14) to include the second and third terms of Eq. (12). The result is the same as Eq. (14X but with y replaced by

Here v = g p a / h o , gives the spin splitting in a generalization of gm*/2m. The last term in Eq. (16) is a field-dependent correction depending also on the energy and which we shall not need explicitly here. Finally, the spinindependent part of y for low fields is precisely 112, a result which is valid if none of the orbits inside the given one intersects itself. Although quasiclassical expressions are in general correct only for large quantum numbers n, we expect Eq. (14) to give fairly good results for low n if the energy surface is not too complicated. For example, for a parabolic band A ( E , ~ , ) = n(2rn*~- h2kZ2)h-', and Eq. (14) with y = 1/2 gives exactly the energy levels we obtained before for all quantum numbers n. In this quasiclassical approximation it is clear from Eq. (14) that the eigenvalues are degenerate, as in the previous simple case of parabolic bands, since they depend only on n and k,. In fact, the density of orbital states per unit range of the wave number k, for fixed n is the same as before, and it is given by Eq. (7). The form of the density of states per unit range of

'* 29

L. Onsager, Phil. Mag. [7] 43, 1006 (1952). L. M. Roth, Phys. Rev. 145, 434 (1966).

6.

MAGNETIC QUANTUM EFFECTS

167

energy dN/dE is determined by the behavior of

as a function of E . We see that when aA/ak, = 0 there is an infinite discontinuity in the density of states provided that aA/& # 0. This condition corresponds to there being an extremal cross-sectional area of a surface of constant energy. Thus in the vicinity of a given energy, such as the Fermi energy, there will occur sets of infinite discontinuities in the density of states separated by the cyclotron frequencies corresponding to extremal areas of the surface. c. Level Broadening Over and above the periodic potential we must consider the effects of imperfections such as impurities and phonons in the solid. The imperfections affect the motion of the electrons in a variety of ways, which we can classify as scattering, broadening, and energy shifting. The scattering is, of course, all important for the transport phenomena, and we shall discuss it later. The energy shifting and broadening of the unperturbed energy levels are usually small effects for most equilibrium properties. However, as we shall see below, the broadening of the unperturbed energy levels is quite important for the amplitudes of the oscillations for the magnetic susceptibility for metals and semimetals. They also prove to be of importance for the oscillations and the magnetic field dependence in the quantum limit of the transport properties. We shall, therefore, review here briefly the effects of collisions with imperfections on the density of states curve. In the presence of impurities, the infinite discontinuities in d N / k that the magnetic field brings about are rounded over. A simple way to see heuristically the effect of the broadening and energy shift on the density of states curve is to replace the &function in Eq. (8) by a Lorentzian function with a constant level width and a constant energy shift 4 i.e.,

dN d8

1

1

R

B(E

- = -Tr-

r - H,

-

A)2

+ r2'

where Tr stands for trace. This can be seen simply from the representation of - c) as (1/n)lims+o+Im(~nk - E - is)-' by the substitution enk-+ E , + A - iT. Carrying out the summation over k and neglecting the uniform level shift A, we obtain for the case of parabolic bands with effective mass m* and neglecting spin

~

168

LAURA M. ROTH AND PETROS N. ARGYRES

ir - (n

+ +)ho,3-

1/2

(19)

where Re stands for the real part. In particular, for the ground oscillator level n = 0, E,- ' I 2 in the unperturbed level density is now replaced by

which coincides with the unperturbed one for E, S r, has the finite value (2r)-'l2 at E, = 0, and goes to zero as E , - ~ / ~ for E, + - co.The quasiperiodic character of dN/& is now modified but not, however, completely destroyed. The oscillations can again be obtained by use of the Poisson summation formula. The result for E / ~ o , 9 1 is the same as the oscillatory part of Eq. (11) except that the amplitude of the rth oscillatory term is diminished by the factor exp{ - 2nTr/ho,}, as can be guessed by comparing Eq. (19) with Eq. (8) which suggests the substitution E + E k iT in the oscillatory part of Eq. (11). This factor describes the physical fact that the individual energy levels can be distinguished only if their separation ho, is greater than their broadening r. If r S b,, the oscillations are smeared out. This result was first obtained by Dingle.30 Dingle took r to be equal to h/22, where l/.c is the average transition rate for scattering by the impurities and was taken to be a constant independent of the energy E. While the constant r result has been most useful in interpreting oscillatory phenomena, the approximation gives divergent results in some situations such as the quantum limit, due to the long tail of the density of states curve for energies below the bottom of the magnetic subband. Therefore in recent years a more rigorous approach has been used. A rigorous theory of the effects of imperfections on the density of states can be given on the basis of the formula 1

_ dN - -Trd(E dE

0

- H);

H

=

H,

+ I/,

in which H includes the scattering potential V . For a collection of fixed, randomly distributed scattering centers, this has been considered by 3o

R. B. Dingle, Proc. Roy. SOC.(London) A211, 517 (1952).

6.

MAGNETIC QUANTUM EFFECTS

169

Bychkov3' and Kubo et d3' in connection with the de Haas-van Alphen effect and galvanomagnetic effects, respectively. It can be shown, from the general theory of damping, that Eq. (21) can be written in the form of Eq. (18) where, however, r and A are now operators depending on E and H,, so that the width and shift are different for different levels. The exact determination of r ( E ) and A(€) is quite difficult, and the reader is referred to the ' above references for details. The work of Bychkov31 and Kubo et ~ 1 . ~was concerned primarily with infinitely localized impurity potentials. Kubo et d3* found that the oscillatory part of the density states is given as above with r = h/2T(E), where z is the lifetime for an electron in the magnetic field, and 7 - l is proportional to the density of states as broadened by the collisions. Near the bottom of a magnetic subband the result becomes rather complicated and has been studied in some detail for the n = 0 level by Kubo et d3'for a short range potential. The density of states curve drops to zero at a certain point below the unperturbed ground state and remains zero for lower values of the energy, as it should on physical grounds, in contradistinction to the constant broadening case that gives a long tail for small E. This is of some importance for the conductivity in the quantum limit. For the oscillatory behavior the Dingle result is obtained if z is approximated by its zero field value. The effects of the interactions among the electrons on the single-particle excitation energies in a magnetic field, and, therefore, on the density of quasiparticle states, have been studied by L ~ t t i n g e r .Under ~~ the usual assumption of applicability of perturbation theory of the electron-electron interaction, it is found that for E 9 hac, kT the period and amplitude of the oscillations of the density of states are given correctly by the quasiparticle picture even in the presence of interactions.

11. Magnetic Susceptibility

3. DE HAAS-VANALPHEN EFFECT

According to Fermi statistics the occupation of levels is governed by the Fermi distribution function

where the Ferrni level [ is determined from the electron concentration N 31

32 33

Yu. A. Bychkov, Zh. Eksperim. i Teor. Fiz. 39, 1401 (1960) [English Transl.: Soviet Phys. J E T P 12, 977 (1961)l. R . Kubo, N. Hashitsurne, and S. J. Miyake, Solid State Phys. 17, 269 (1966). J. M. Luttinger, Phys. Reu. 121, 1251 (1961).

170

LAURA M. ROTH AND PETROS N. ARGYRES

by the relation

with dN/de the density of states, Eq. (8) or (9). For zero temperature, i.e., complete degeneracy, we have N = k F 3 / 3 z 2where kF is the Fermi wave vector in zero magnetic field. The thermodynamic properties of the system are determined by the free energy density of the system, which is given by

We shall be interested primarily in the magnetic moment, M

=

-dF/dB,

(25)

where the derivative is at constant volume. The free energy for electrons in a parabolic band with effective mass m* can be obtained from Eqs. (22) and (8). This calculation was first carried out by Landau with the use of the Poisson summation formula and was reported in the work of S h ~ e n b e r gand ~ ~ Peierl~,~’ while others36 have derived the same results by different mathematical methods. Since these authors were interested primarily in metals, they considered only the case of degenerate statistics, i.e., [ k T and ( % ho,.If we include the spin the result can be written as the sum of two terms,

+

F(L B) = F ,

+ F,,,

>

(26)

the first of which varies monotonically with the magnetic field, and the second of which gives the oscillations. F , gives the steady susceptibility x = - a2Fl/aB2,which we shall consider in the next section. The oscillatory part of F gives rise through Eq. (25) to an oscillatory magnetic moment

In Eq. (27) we have incorporated the effect of the level broadening we D. Shoenberg, Proc. Roy. SOC.(London) A170, 341 (1939). Appendix. R. E. Peierls, “Quantum Theory of Solids,” Chap. 7. Oxford Univ. Press (Clarendon), London and New York, 1956. 36 A. H. Wilson, “The Theory of Metals,” Chap. 6. Cambridge Univ. Press, London and New York, 1958. 34

35

6.

MAGNETIC QUANTUM EFFECTS

171

discussed earlier in Section 2c. We thus see that MoSc((,B)is a quasiperiodic function of B-’ with period A(l/B) = eA/m*cl. Since, as we shall see below, in this approximation it is consistent to ignore the magnetic field dependence of the Fermi energy, this period can also be written as A(l/B) = (2ne/hc)A,’, where A , = nk,’ is the maximum cross-sectional area of the Fermi sphere. Note that this is independent of the effective mass of the electrons ; it depends only on their concentration, lower concentrations giving larger periods. The amplitude of the oscillations is largest at T = 0°K and remains large as long as the thermal spread of the Fermi distribution is much smaller than the magnetic quantum of energy, i.e., for kT < ha,; in this case it varies as (B1/’/m*) exp( -2n1-/h0,) for the lowest harmonic. For higher temperatures the amplitude diminishes due to the spread of the Fermi distribution, and for 2n2kT 9 h a , it varies like B- l j 2 T exp[ - 2n2k(T + T’)/ho,] for the lowest harmonic, where we have written r in terms of the “Dingle temperature” T‘ = I-/&. In this case the higher harmonics (r > 1) contribute insignificantly. We note that a small effective mass enhances the amplitude of the oscillations. Finally the factor cos(nvr) in (27) results from the spin of the electron. In the case v = 1 it is only a phase factor (-ly. This is believed to account for the phase of the de Haas-van Alphen effect in Bi.37 In some cases g can affect the amplitude; for example, v = 1/2 results in the vanishing of the lowest harmonic. The effect of the magnetic field on the Fermi energy [ is obtained from Eq. (23). For a single band we assume that the concentration of electrons is For degenerate statistics (5 % k T ) and for [ % h0, the result from our simple band can be obtained by again using the Poisson summation formula :

5 = 5 0 + 5lB’ +

50,,,

(28)

where lois the Fermi energy at B = 0. (,,, is a sum of quasiperiodic func~ 0 is small tions in B-’,whose amplitude is proportional to ( h 0 J ~ 0 ) 3 / 2and compared to to.Due to their small amplitude these oscillations are not generally important. In addition, the smooth term goes as B2, so is not important in the steady susceptibility. Thus in the above result we can use

5 = 50. The result (27) has been generalized to the case of an arbitrary dispersion through the use of the semirelation F(k) by Lifshitz and Ko~evich,~’ classical quantization of the orbits, Eq. (14). For degenerate statistics 3 7 E. 1. Blount and M. Cohen, Phil. Mag. 5, 115 (1960) 3 7 a F ~the r case of more than one band there can be a transfer of carriers as a function of the magnetic field. 38 I . M. Lifshitz and A. M. Kosevich, Zh. Eksperini. i Teor. Fiz. 29,730 (1955) [English Trans/.: Soviet Phys. J E T P 2, 636 (1956)].

172

LAURA M. ROTH AND PETROS N . ARGYRES

+

+

(5 k T ) and many occupied states [[ ho, where o, is given by Eq. (15)], they obtained the following result for M,,, : m

M,,,

=

1 a, sin r= 1

where we have used y* = (1 f v)/2 from Eq. (16), and I = (hc/eB)’”. In Eq. (29) we find that the period is related to the extremal value A,([) of the cross-sectional area of the Fermi surface : A(l/B) = (271e/hc)& ‘(5). The phase fz/4 depends on the type of extremum, the upper sign corresponding to a maximum and the lower to a minimum. The amplitude now depends on the curvature ldZA/dkZ21,at the extremal area. The remaining factors are quite analogous to the simple model, except that the cyclotron frequency is now given by w, = 2x/h12(dA/d[)from Eq. (15). For complex Fermi surfaces in metals and semimetals, there can be a number of extremal areas, each contributing its own period. From the temperature dependence of the amplitude the cyclotron frequency w , can be determined, and [d’A([, k z ) / ~ k z 2 ]can , be obtained from the amplitude itself. These three quantities are sufficient in principle to determine the shape of the Fermi surface and the velocity of electrons at the Fermi surface.39 In practice, Fermi surface determinations are based on models with de Haas-van Alphen periods determining parameters in the models. 4. STEADY EFFECTS a . Steady Magnetic Susceptibility For our simple band model the calculations discussed yield a steady susceptibility from the smooth part of the free energy of Eq. (24). Unfortunately, the result [the first term of Eq. (30) below] is deficient in that a term is omitted which depends on the third term of Eq. (12) evaluated at the band edge [W,(O)].The need for such a term was first pointed out by Kjeldaas and Kohn?’ who showed that it was necessary to obtain the correct tight binding result for the susceptibility. The correct result for simple bands is thus

x =N 39 40

(X)’ds __

-(3v2

-

1) + 2NWz(O),

I. Lifshitz and A. Pogorelov, Dokl. Akad. Nauk SSSR 96, 1143 (1954). T. Kjeldaas and W. Kohn, Phys. Rev. 105, 806 (1957).

6.

173

MAGNETIC QUANTUM EFFECTS

where we reiterate that I' = gnz*/2m. The first term gives the Pauli paramagnetism (depending on v) and the Landau-Peierls diamagnetism. For band edge evaluation of the last term we refer the reader to Kjeldaas and K~hn.~' The calculation of the steady susceptibility for a general band has been the subject of a good deal of literature because of its complexity. Perhaps the simplest appearing result is that obtained by Roth2' and by Wannier and U~adhyaya,~'"valid for doubly (spin) degenerate bands :

Here the first term is the Landau-Peierls result. The second term is essentially the paramagnetic susceptibility, and the trace is to include spin coordinates. The third term is discouragingly complicated, and we refer the reader to the above authors for discussion. The above calculations were based on expansions of the free energy which omitted the oscillations. A way of obtaining both the oscillations and the steady susceptibility together has been found by one of us29 and is based on the generalized quantization condition, Eqs. (14) and (16). The term y , l F 2 in Eq. (16) contains a part dependent on W, and also a diamagnetic correction. Inclusion of y l results in the correct steady susceptibility. A different approach to the problem, based on an evaluation of the energy levels and useful for III-V compounds, has been developed by Yafet4' and has been applied to InSb suc~essfully.~~

b. Quantum Limit For large enough magnetic fields (a practicality in III-V compounds), the electrons are all in the lowest quantum state, and the Fermi level is changed drastically. This case of the quantum limit occurs for < 3hoc/2 for a band without spin, and C < (1 + v)ha,/2 for our model of a band with spin. Considering the latter case we have for degenerate statistics

c

The latter relation follows from Eq. (231 with use of (8) and then (9), assuming no change in total carrier concentration in the band as a result of the magnetic field. 40sG.H. Wannier and U. N. Upadhyaya, Phys. Rev. 136, A803 (1964). 4' 42

Y. Yafet, Phys. Rev. 115, 1172 (1959). R. Bowers and Y. Yafet, Phys. Rev. 115, 1165 (1959).

174

LAURA M. ROTH AND PETROS N. ARGYRES

The free energy in this case is given by

while the magnetization becomes, in the limit of large B,

he

M = N -(1 (Zmc

- V) +

(34)

if we neglect the second term in F which goes to zero as l/BZ.The Landau level contribution to the susceptibility thus saturates. c. Classical Statistics For classical statistics we have eilkT 6 1 and the distribution function becomes

f(4

-

exp(i - E)/kT,

(35)

where from Eq. (23) the Fermi level is given by ellkT =

sinh(hoJ2kT) N ( 2 7 ~ h ~ / r n * ) ~ ' ~"2(hoc)(kT)cosh(vho,/2k T )'

(36)

The free energy is obtained from Eqs. (24) and (36):

F = N ( < - k T + WZB'),

(37)

where is given by Eq. (36), and these results are valid for arbitrary h o c / k T . Clearly the oscillations of the susceptibility have now been entirely masked by the broad tail of the Boltzmann distribution, and the free energy is a monotonic function of the magnetic field. The quantum limit result for the magnetization is again given by Eq. (34).

111. Quantum Galvanomagnetic Effects 5 . BASICTHEORY a. Introduction In this section we review the quantum effects of a magnetic field B (which we take in the z-direction) on the transport properties of the conduction electrons in a uniform static electric field E. These are pronounced for strong magnetic fields in the sense w,z $ 1, where T is a measure of the mean free time (or, relaxation time), and tzo, 2 kT. Under these conditions the quantization of the electronic energy levels is not blurred either by the collisions with the impurities or by the thermal spread of the electron

6.

175

MAGNETIC QUANTUM EFFECTS

<

distribution. For degenerate statistics with Fermi energy > hot, we find oscillations in the transport coefficients, i.e., the Shubnikov-de Haas effect, whereas in the quantum limit we find smooth field variations characteristic of the scattering mechanism. For later convenience let us recall here the relationship between the measured transport coefficients and the calculated components of the conductivity tensor oij.The latter is defined by

J i = C aijEj, j

where J i and E j (i, j = x, y, z ) are the components of the current density and the electric field, respectively, and they satisfy the Onsager relation ai,(B) = ojd-B). The Hall coefficient (R ) and the transverse (pl) and longitudinal (p resistivities are then, for a magnetic field in the z-direction and for isotropic bands,

For the transverse case the quantum galvanomagnetic effects have been observed so far at large Hall angles, i.e., when oYx$- D ~ , ,in which case

R

E

- I/BO,,.

(43)

In calculating the conductivity tensor for electrons in a high magnetic field, the longitudinal and transverse effects present somewhat different theoretical problems. The longitudinal magnetoconductivity was calculated by Argyres and and by Bar~-ie,~’ using a rather straightforward generalization of Boltzmann transport theory. The principal effect is due to the change in relaxation rate because of the altered density of states in a magnetic field. For the transverse effects, on the other hand, the Boitzmann equation approach is inadequate for describing the effects of quantization. The transverse effects were first considered by T i t e i ~ a who , ~ ~ assumed that the transverse current is produced by the drift of the centers of the 43

P.N. Argyres and E. N. Adams, Phys. Rev. 104, 900 (1956).

P. N. Argyres, J . Phys. Chem. Solids 4, 19 (1958). R. Barrie, Proc. Phys. SOC.(London) B70, 1008 (1957). 46 S. Titeica, Ann. Physik [5J 22, I29 (1 935).

44

4s

176

LAURA M. ROTH AND PETROS N. ARGYRES

cyclotron orbits of the electrons due to the electric field, and the scattering by the imperfections in the crystal. Essentially the same method of calculation was followed by a number of different w0rkers,4~,~* and in particular by Davydov and P o m e r a n c h ~ k . ~ ~ The quantum mechanical approach for the transverse current has been based upon the calculation of the density matrix which, for high enough magnetic fields, can be expanded in powers of the scattering potential (e.g., impurities or phonons). This has been carried through by Kubo et al.,50,51Adams and Holstein,52 Argyres and R ~ t h , ’ Kosevich ~ and A n d r e e ~and ~ ~K~l i ~ ~ g eThese r . ~ ~ calculations provide a rigorous justification of the semiclassical procedure of Titeica More recently a number of refinements of the theory have been made; in this connection we shall refer extensively to the work of MiyakeS4”and Kubo et who have considered in detail the effects of level broadening, among other things. We shall be primarily concerned with the large Hall angle calculations, for which quantum effects are pronounced. For arbitrary Hall angles, i.e., for low as well as high magnetic fields, quantum transport equations have been developed for the transverse current by Argyre~,’~ Gurevich and N e d l i ~Horing ~ , ~ ~ and Argyres5’, 5 8 and Kli~~ger.’~” The earlier quantum transport equations of L i f ~ h i t z Lifshitz ,~~ and Kosevich,60and Argyres61 were in error in the following respects : The G. E. Zil’berman, Zh. Eksperim. i Teor. Fiz. 29, 762 (1955) [English Transl.: Soviet Phys. JETP 2,650 (1956)J. 48 J. Appel, Z . Naturforsch. 11A, 892 (1956). 49 B. Davydov and I. Pomeranchuk, J . Phys. USSR 2, 147 (1940). 5 0 R. Kubo, H. Hasegawa, and N . Hashitsume, Phys. Rev. Letters 1, 279 (1958). 5 1 R. Kubo, H. Hasegawa, and N. Hashitsume, J . Phys. SOC. Japan 14, 56 (1959). 5 2 E. N. Adams and T. D. Holstein, J . Phys. Chem. Solids 10, 254 (1959). 53 P. N. Argyres and L. M. Roth, J . Phys. Chem. Solids 12, 89 (1959). 53aA.M. Kosevich and V. V. Andreev, Z k . Eksperim. i Teor. Fiz. 38,882 (1960)[English Transl.: Soviet Phys. J E T P 11, 637 (1960)l. 5 4 M. I. Klinger, Fiz. Tverd. Teh 3, 1342, 2507 (1961) [English Transl.: Soviet Phys.-Solid State 3, 974 (1961), 1824 (1962)l. 54aS.J. Miyake, Doctoral Thesis, Tokyo University (1962); J . Phys. SOC. Japan 20,412 (1965). 5 5 P. N. Argyres, Phys. Rev. 117, 315 (1960). 5 6 L. E. Gurevich and G . M. Nedlin, Zh. Eksperim. i Teor. Fiz. 40,809 (1961) [English Transl. : Soviet Phys. JETP 13, 568 (1961)l. 5 7 N . J. Horing and P. N . Argyres, Proc. Intern. Con$ Phys. Semicond., Exeter, 2962 p. 58. Inst. of Phys. and Phys. SOC.,London, 1962. 5 8 P. N. Argyres, Phys. Rev. 132, 1527 (1963). 58aM. I. Klinger, Fiz. Tverd. Tela 3, 1354 (1961) [English Transl.: Soviet Phys.-Solid State 3,983 (1961)l. 5 9 1. M. Lifshitz [E. (sic) M. Lifshitz], J . Phys. Chem. Solids 4, 11 (1958). 6 o I. M. Lifshitz [E. (sic) M. Lifshitz] and A. M. Kosevich, J . Phys. Chem. Solids 4, 1 (1958). P. N. Argyres, Phys. Rev. 109, 1115 (1958). 47

6.

MAGNETIC QUANTUM EFFECTS

177

former two t h e ~ r i e s ~ ~ did . ~not ’ take into account the effect of either the magnetic field or the electric field on the scattering. In the latter,61although the effect of the magnetic field on the scattering was considered, the influence of the electric field on the collisions was not. We review below the theory of the transport coefficients for longitudinal and transverse cases, presenting first the results of the Born approximation for the scattering. We consider scattering either by fixed, randomly distributed scattering centers that scatter the electrons elastically, or by phonons that can change the energy of the electrons as well as their momentum. The theoretical calculations, as mentioned above, are based on the density matrix p. For elastic scattering we can consider p to be a one-electron density matrix which satisfies the equation of motion, up to first order in the electric field,

where H , is given by Eq. (2), Vis the scattering potential, and f is the Fermi distribution function. For electron-phonon scattering, a corresponding equation can be writtens3.” for the coupled system from which the oneelectron density matrix p is to be extracted. We shall indicate the derivations for the elastic scattering case but quote also the results for inelastic scattering. The current density is given by e J = --Trpv, R

(45)

with v = (m*)-‘(p + eA/c) the velocity operator. In the representation given by Eq. (3) we have

b. Longitudinal Case

For the longitudinal case and treating the scattering in the Born approximation, we obtain the result in terms of a transport equation for the diagonal matrix elements pr of the density matrix in the nk+ representation introduced in Section 2. If we let p represent n, k, & collectively, we have55,58

178

LAURA M. ROTH AND PETROS N. ARGYRES

where the W,, are transition probabilities in the Born approximation between the Landau states ,u and v. If we assume a standard form of solution

where df/& = - (l/kT)f( 1 - f ) , we have to first order in cp

c w,vfp(l

- fv)(cpp - cp,)

= eE,~,'fp(l - f,).

(49)

V

For scattering by the impurity potential V(r) = X iu(r - ri) the transition probability is

and this gives elastic scattering. We are assuming that V(r) is independent of spin. The matrix element IVpv12in Eq. (50) should, of course, be averaged over the random positions of the scattering centers, which are assumed to be distributed homogeneously in the crystal. It is then found that (for p and v having the same spin) /Kk,,,ylZ =

(Ni/a)

lW(q)I2

I(nkleiq"ln'k')lz,

(51)

9

where N i is the concentration of scattering centers and where

-

w(q) = Jv(r) exp( - iq r) dr

(52)

is the Fourier transform of the scattering potential for each center. A model much used in calculations corresponds to a b-function potential

v(r - ri) = a6(r - Ti),

(53)

where a is the strength of the b-function. The quantity m*a/2nh2 is often referred to as the scattering length. For such elastic collisions the Fermi factors cancel in the transport equation (49), which simplifies to

For some scattering interactions [Eq. (53)is an example] the collision operator simplifies further. For a solution of Eq. (54) of the form $ p = k, times a function of the energy E, the scattering operator can be simply described by a momentum relaxation time, namely,

6.

179

MAGNETIC QUANTUM EFFECTS

where

For a solution of this form to exist, it is sufficient that the relaxation times znk+as given by Eq. (56) turn out to depend on n and k only through the energy Erik. The solution of the transport equation is trivial in this case, and from Eqs. (54) and (45)we obtain

For phonon scattering the transition probability in Eq. (49) is ?-

x [(N,

+ I)&,

-

E,

-

ho,)+ Nq&@ - E,

+ ho,)],

(58)

where q = (j,q) denotes collectively the branch and wave vector for the phonon mode with energy ha,, N , = [exp(ho,/kT) - 11-l is the Planck distribution function for the phonons (assumed in thermal equilibrium), and C(4)denotes the strength of the electron-phonon interaction. The two terms in Eq. (58) give the contributions to the scattering rate of the phonon emission and absorption processes, respectively. For acoustic phonon scattering in semiconductors and semimetals, the deformation potential theorem gives for our spherical band edge and longitudinal phonons lC(q)l2 = hElZq/2PQvs,

(584

where El is the deformation potential parameter,62 p is the mass density, and v, is the sound velocity. For sufficiently high temperature the energies of the phonons involved in the scattering processes can be neglected. In such a case, since N , + 1 E N , E kT/ho, = kT/hv,lql, the scattering rate, Eq. (58), is equivalent, according to Eq. (51),to that of an infinitely localized scattering potential, as in Eq. (53) with a Z N i replaced by ElZkT/pvs2.At lower temperatures, inelasticity can however play a role as we shall see later, e.g., in the quantum limit. c. Transverse Case For the transverse case, since the quantum galvanomagnetic effects under study here are most pronounced for magnetic fields so large that the Hall 62

J. M. Ziman, “Electrons and Phonons,” p. 205. Oxford Univ. Press (Clarendon), London and New York, 1960.

180

LAURA M. ROTH AND PETROS N. ARGYRES

angle is quite large, the simplest procedure is to expand the steady-state density matrix p in Eq. (44) in powers of the scattering interaction. This is possible in this case, because the magnetic field limits the mean acceleration of the electrons by the electric field, and thus a finite steady-state current is present even in the absence of any scattering mechanism. This is in clear contrast to the case of B = 0. The current is in a direction perpendicular to both B and E, and it is easily found to be equal to its classical value J , = NecE,/B. By iteration of Eq. (44) it can be showns3 that to second order in Vthis value of J , does not change. Furthermore the current density in the direction of the electric field is now found to be different from zero, and the following result is obtained for both elastic and inelastic scattering : ox, =

(e2/2kTQ)Cf,(l - fV)W,,(X, - XVl2 2

(59)

PV

o,, = N e c / B .

(60)

Here X , is the center of the electron motion in the x-direction when the electron is in the state p, and it is given by Eq. (4). WPyis again the transition probability due to scattering in the Born approximation. Thus, in this approximation the Hall coefficient becomes equal to the classical value R = l/Nec, and the transverse resistivity pL is, from Eq. (42), p I = (B/Nec)20xx.

A more general solution of Eq. (44),correct to all orders in V, has been ' the case of elastic scattering: obtained by Kubo et ~ 1 . ~for

where H = H , + r! and where X is the operator for the x-component of the orbit center, whose eigenvalues X , are given by Eq. (4). This equation is useful, as we shall see below, for situations in which the perturbative solution, Eq. (59), breaks down. A corresponding equation has been derived for inelastic scattering by the same authors. For phonon scattering Eq. (59) can be written in the general form (omitting spin for the moment):

where G,,,(E) is a smooth function of E. The denominators in Eq. (62) arise from the density of initial and final states [see Eq. (S)]. We note that because of the presence of the phonon energy in the second factor in the denominator, we have a convergent expression for ox..

6.

181

MAGNETIC QUANTUM EFFECTS

For elastic scattering the conductivity, Eq. (59), can be written in the simple form cTxx =

--

wnk f ,n'k' f(xk

-

xk')2

9

(63)

which can be expressed in analogy to Eq. (62) as

A problem now arises in that Eq. (64) does not give convergent results, as was first noticed by Davydov and P o m e r a n c h ~ k .The ~ ~ divergence is logarithmic and arises from the two density of states factors coinciding in elastic scattering when n = n'. We note that inelasticity in the collisions is sufficient to avoid the divergence, but for elastic scattering improvements in the theory are necessary to remove this divergence. This is particularly important for classical statistics in the quantum limit, for which n = n' = 0. We discuss this problem below.

d. Elimination of the Divergence There are several mechanisms which can be responsible for cutting off the divergence, which have been discussed by Adams and Holstein" in terms of a cutoff energy E, for the kinetic energy in the z-direction. While more sophisticated treatments of each mechanism have been carried through and will be discussed, the cutoff energy will be a useful guide as to which mechanism is important in a given case, since the mechanism giving the largest cutoff energy will dominate. We now discuss in turn the various mechanisms which have been considered. ( 1 ) Collision Broadening. We have seen in Section 2 that the effect of collision broadening on the density of states is to remove the infinite discontinuities for E, = 0. The cutoff occurs when E, is of the order of the width r of the levels. This argument, due originaily to Davydov and Pomera n c h ~ k has , ~ ~been applied by Kubo et ~ l . , ~ who ' find for the model of a short range potential with scattering length f [see Eq. (53) and below]

where 1

is the mean free time for an electron in zero magnetic field.

182

LAURA M. ROTH AND PETROS N. ARGYRES

A more rigorous approach to the collision broadening cutoff using damping theory can be based on Kubo’s expression, Eq. (61)’and is discussed also by Kubo et aL3’ This gives essentially the same result as above. As we shall see below, collision broadening is the dominant cutoff mechanism for phonon scattering at high temperatures and for impurity scattering at relatively high impurity concentrations. (2) Non-Born Scattering. Another way to avoid the divergence of Eq. (64) for impurities is to calculate the scattering cross section of each impurity center more exactly than in the Born approximation. Certainly for very slow electrons the Born approximation is not valid, because if an electron moves slowly along the direction of the magnetic field it interacts with the same impurity repeatedly on account of its cyclotron motion. The problem has been treated by S k ~ b o v Kahn,64 ,~~ and Bychkov,6s and discussed by Kubo et al.32The discussion has been based primarily on the case of a short range potential. S k ~ b o has v ~ treated ~ the scattering by a short range potential in a magnetic field with the use of a scattering length f . This theory gives a cutoff energy

h2f 2/2m*14, (67) where l2 = hc/eB. The results of Kahn and Bychkov are in essential agreement. MiyakeS4”has applied the results to the case of the quantum oscillations as well as the quantum limit. Kubo et aL3’ have compared this expression with that for the collision broadening and have found that non-Born scattering can become more important than impurity collision broadening at low impurity concentrations and high magnetic fields. (3) Inelasticity. For inelastic collisions the transverse conductivity is given by Eq. (62), for which there is no longer a divergence, since the two density of states peaks are displaced by a phonon energy ho,.We thus expect the cutoff energy E, to be a typical phonon energy. The case of acoustic phonon scattering has been considered in detail by Kubo et aL3’ and Gurevich and Firsov2’ for the quantum limit, and they have found that the cutoff corresponds to the energy of a phonon whose wavelength is equal to a Landau orbit radius, i.e., E,

E,

-

hvJl.

(68)

V. G. Skobov, Zh. Eksperim. i Teor. Fiz. 37, 1467 (1959), 38, 1304 (1960). [English Transl.: Soviet Phys. J E T P 10, 1039 (1960), 11, 941 (1960)]. 64 A. H. Kahn, Phys. Rev. 119, 1189 (1960). b 5 Yu. A. Bychkov, Zh. Eksperim. i Teor. Fiz. 39,689 (1960) [English Transl. : Souiet Phys. J E T P 12,483 (1961)l. b3

6.

MAGNETIC QUANTUM EFFECTS

183

This is in disagreement with an earlier incorrect estimate of Adams and H~lstein.’~ Kubo et have compared this cutoff, Eq. (68), with that for collision broadening, and have found that inelasticity of acoustical phonon scattering can become more important than collision broadening at sufficiently low temperatures. For optical phonons we might expect E, to be the optical phonon energy. However, this is often large compared to ho,,and if we recall that the density of states is actually a series of peaks, we notice that if two of these differ by an optical phonon energy, the divergence can reappear. This is the basis of Gurevich-Firsov2’ oscillations, which will be discussed in the final section of this chapter. ( 4 ) Classical C u t o f This mechanism arises from the fact that, if the scattering rate becomes large near k, = 0, the condition o,z % 1 is no longer satisfied and the perturbation treatment of the collisions is not applicable. The approach through a quantum transport equation5’ is valid for arbitrary o,z and so avoids this divergence. However, in practice this mechanism does not seem to be important according to Adams and Holstein.”

6. SHUBNIKOV-DE HAASEFFECT a. Theory As in thermodynamic properties, oscillatory behavior in the magnetic field dependence of the resistivity can occur primarily (but not exclusively) in the case of degenerate statistics, i.e., [ B kT, and when a number of oscillator states are occupied, i.e., > ho,. In view of the complexity of the general expressions, calculations have been performed only for the simplest energy bands, and in particular for a parabolic band with effective mass m*. Furthermore, we shall restrict ourselves to the easily tractable model of isotropic and elastic scattering. This is the case of scattering by impurity centers with a force range much smaller than the de Broglie wavelength of the electrons, for then we may use the &function potential given by Eq. (53). Another isotropic and elastic scattering mechanism is the interaction with acoustic phonons in semimetals and semiconductors at sufficiently high temperatures, as discussed in Section 5b. For the longitudinal case it can be seen from the work of A r g y r e ~that ~~ a relaxation time as given by Eq. (56) exists. In fact, since the backward and forward scattering probabilities are equal, this relaxation time is equal, as in the B = 0 case, to the lifetime, i.e.,

184

LAURA M. ROTH AND PETROS N. ARGYRES

Here dN/dEnk* is the density of states for each spin in the presence of the magnetic field as given by Eq. (8), with the constant C = (2n/h)a2Ni for scattering by impurity centers of density N i and C = (2n/h)(E12kT/pu,2)for acoustical phonon scattering. In the absence of the magnetic field the relaxation time is Z;~(E) = *CdN,/&, with dNo/dz given by Eq. (9), the factor giving us the one spin density of states. Thus, in this simple case the effect of the magnetic field on the scattering is given directly through its effect on the density of states, whose oscillatory behavior we have discussed. The longitudinal resistivity, as given by Eq. (57b is then found for kT bc to be

4

+

where po = m*/Ne2z0([,) is the resistivity when B = 0, and N , is the density of electrons with spin f.This expression is a little complicated because the resistivities for the two spin cases must be combined in parallel. hw, we may approximate by simply averaging the relaxation rates For for the two spins. For arbitrary kT/ho, the oscillations can then be exhibited explicitly, for it is found that

where

( - l y ho, b,=- r1I2 __ 2c)

(

'I2

2n2rkT/hoc sinh(2n2rkT/ho,) COS(XVI) e - 2nrrlhwc

>

(72)

where we have included collision broadening as discussed in Section 2c. Thus we see that the oscillatory behavior of the longitudinal magnetoresistance in 1/B is quite similar to that of the diamagnetic susceptibility; both have the same period and temperature dependence of the amplitude. For the transverse case and again for isotropic scattering, we can express cxx,Eq. (63X in terms of the relaxation time of Eq. (69). We find52.55 gxx.=

e212

-~

1

df

+ +)T'(Enk)-l

--(?I nk+ dEnk+

(73)

If we evaluate Eq. (73) at OOK, we find, using Eq. (69), PL = Po-

1

3(hJ3 16L2 ..,+[[

*

n++ - (n

+

f ~ ~ ) h o , ][[" ~ (n' +

1

+ * +~)ho,]'/~' (74)

For any finite temperature, we must integrate over the energy, which is

6.

MAGNETIC QUANTUM EFFECTS

185

5 here, and we obtain the logarithmic singularity discussed earlier. Furthermore, in putting this in a form analogous to Eq. (71), we must include one or another of the broadening mechanisms considered above. This has been done by MiyakeS4”for both collision broadening32 and non-Born scattering, and the result for collision broadening (75)

where b, is given by Eq. (72) and m

1

(77) r- 1

1

(78) The second term in Eq. (75) is the contribution to the oscillatory part of the transverse resistivity due to transitions in which the quantum number n changes. The quantity R represents the contribution of the transitions which do not change n, and it clearly diverges if the level width r vanishes. Its relative importance depends on the ratio r/[.In practice R would appear to be unimportant when collisions are frequent enough to damp out r > 1 harmonics in the oscillations. This conclusion was reached by Adams and Holsteins2 on the basis that the divergent term is quadratic in the oscillatory part of p .

b. Experiment The oscillatory magnetoresistance effects in III-V compounds have been studied in InSb by Frederikse and Hosler66 and in InAs by Sladek67 and by Frederikse and Hosler.68 More recently results have been obtained for GaSb by Becker and Fan69 and by Becker and Yep.” In addition we shall 65”Thisresult can be obtained phenomenologically by expanding each summation in Eq. (74), including the Dingle factor, and then multiplying. bb H. P. R . Frederikse and W . R . Hosler, Phys. Rev. 108, 1136 (1957). 67 R. J. Sladek, Phys. Rev. 110. 817 (1958). “ H . P. R. Frederikse and W. R. Hosler, Phys. Rev. 110,880 (1958). 6 9 W. M. Becker and H. Y. Fan, in “Physics of Semiconductors” (Proc. 7th Intern. Conf.), p. 663. Dunod, Pans and Academic Press, New York, 1964. 7 0 W. M. Becker and T. 0. Yep, Bull. Am. Phys. SOC.10, 106 (1965).

186

LAURA M. ROTH AND PETROS N . ARGYRES

B (kG) FIG.2. Longitudinal and transverse magnetoresistance of n-type InSb at 1.7"K. Sample and results described in Table 111. [After Frederikse and H0sler.6~1

include in our discussion the experiments of Hinkley and Ewald7' on grey tin which are comparable to the other cases. In all the cases cited, the oscillations have been due to electrons at the zone center, for which the results of our simple model should apply. In GaSb, oscillations due to the subsidiary maxima were not observed, but effects of their occupation occur as discussed below. As discussed in the introduction, the existence of degenerate free carriers at low temperatures can be attributed to screening out of the shallow impurity potential due to the high concentration of carriers. We shall be concerned here only with situations in which the carrier concentration remains constant, i.e., freezeout effects are not important, and we refer the reader to the chapter by Putley in this volume for a discussion of these effects. Typical magnetoresistance curves are shown in Figs. 2, 3, 4, and 5 for InSb, InAs, GaSb, and grey tin. The InAs and GaSb results are plotted to show the periodicity in 1/B.The results have been found to be well described by an oscillatory term of the general form of the second term in Eq. (71) or (75),with r = 1, and with -z/4 replaced by a phase 9.Experimental data on the various cases are summarized in Table 11. We discuss separately various features of the results. (I) Periods. The period and phase of the oscillations can be obtained by plotting nodal or extremal values of l/B versus integers. A typical such plot

'' E. D. Hinkley and A. W. Ewald, Phys. Rev. 134, A1261 (1964)

6.

187

MAGNETIC QUANTUM EFFECTS

0.15

\

/

L'

I

-0.10

I

I

I

I

I

I

I

I

I

I

I

I

0.10 -

- 0.10

3

5

7

9

1076 (gauss') FIG.3. Longitudinal and transverse magnetoresistance of n-type InAs. Sample and results described in Table 111. [After Sladek.67]

=

FIG. 4. Longitudinal and transverse magnetoresistance of n-type GaSb (carrier density 1.3 x lo'* ~ m - ~Results ). described in Table 111. [After Becker and Fan.69]

188

LAURA M. ROTH AND PETROS N. ARGYRFS

0

4

8

1

2

1

FIG.5. Longitudinal and transverse magnetoresistance of n-type grey tin at 1.2"K. Sample and results described in Table 111. [After Hinkley and Ewald."]

is shown in Fig. 6 for InAs. The periods of the oscillations should be directly related to the electron density for spherical energy surfaces independent of the effective mass or deviations from parabolic behavior. The results bear this out. For GaSb the agreement was with the low temperature Hall coefficient, which reflects the high mobility carrier concentration. (2) Phase. The phase cp should theoretically be given by -n/4 for our simple model, for both the longitudinal and transverse cases. A phase TABLE I1 EXPERIMENTAL DATAON THE SHUBNlKOV-DE HAASEFFECT ~

~

Material Ref.

~~

Carrier Observed Calculated density period period ( ~ r n - ~ ) (G-' x lo5) (G-' x lo5)

m*/m (exptl)

T'

(exptl)

Tm' (exptl)

(OK)

(OK)

InAs (S-1)

67 7.6 x 10l6

1.85(ll) 1.84(1)

1.77

0.02

16.2(jj) 16.8(1)

6.6

GaSb

69 1.3 x 10"

0.275(/1) 0.275(1)

0.267

0.05

6.60)) 6.6(1) 19.6(ll,l

4.9

1.0 x 10'8

InSb (S-4)

66 2.30

Greytin (B-1)

71

x 1015

1.14 x 10l6

0.05

19W) 19.6(1) 5.72(11) 5.84(1)

18.4 6.30

0.010

0.024

-

3.2

6.0 __

0.36

6.

189

MAGNETIC QUANTUM EFFECTS

Integers

FIG.6. Positions of the nodes of the longitudinal magnetoresistanceplotted versus integers. [After Sladek.67]

reversal due to the g-factor is not expected in any of our cases, although a reduction in amplitude should occur. Unfortunately, the phase is difficult to obtain accurately due to the extrapolation. The best agreement is found in Sladek’s lowest temperature InAs results, while the worst is in Hinkley and Ewald’s grey tin results in which the longitudinal phase was found to be - 4 2 .

0.0011 0

I

I 2

I

_I

4

I

6

105/B (gauss-’)

Fie. 7. Dependence of amplitudes of oscillations of longitudinal magnetoresistance upon magnetic field of GaSb. [After Becker and Fan6’]

190

LAURA M. ROTH A N D PETROS N. ARGYRES

(3) Amplitude. The amplitude of the oscillatory magnetoresistance is seen from Eq. (72) to be dependent on temperature and magnetic field. The temperature dependence yields a value of the cyclotron frequency and hence the effective mass. It is found67,69that the x/sinhx dependence (where x = 2n2kT/timc)is followed for the lowest temperature. The magnetic field dependence of the amplitude involves both the temperature and the collision broadening. The effects can be separated by dividing the amplitude by (hoc/[o)"2x/sinh x and plotting the logarithm of the result versus the reciprocal of the magnetic field. The slope of the resulting straight line yields the Dingle temperature given in Table 11. Such a plot is given in Fig. 7 for GaSb, which shows the excellent fit with the form of the theoretical expression. If we extrapolate the plot of Fig. 7 to B-' = 0 the intercept should yield (cos 71v)/,/2 for the longitudinal magnetoresistance from Eq. (72). For the transverse case, if R, Eq. (76),can be neglected as it apparently can here,54a the amplitude should theoretically be 2.5 times the longitudinal amplitude. Experimentally the transverse amplitude is greater than the longitudinal amplitude, but by a somewhat smaller factor. The 2.5 prediction is dependent on the isotropic scattering mechanism assumed. A calculation based on ionized impurity scattering, which is thought to be the mechanism involved at least in GaSb69.7 2 would be desirable. The order of magnitude of the extrapolated amplitudes is in agreement with the theoretical One aspect of the experimental result which has not been calculated theoretically is the steady component of the magnetoresistance, which is particularly noticeable in the transverse case. (4) Efectiue Mass. The results for effective masses in the III-V compounds in Table I1 agree well with other experiments. For GaSb, Becker and Yep7' report a concentration dependence of the effective mass, which is expected from the nonparabolicity of the bands. For grey tin the magnetoresistance oscillations establish that the band edge is at the zone center, but the effective mass is an order of magnitude larger than a theoretical prediction73 of 0.0036q based on a band structure similar to InSb, with the k = 0 conduction band being the s-like r7- band and based on a thermal gap74 of 0.085eV. On the other hand, Groves and Paul7' have "J. E. Robinson and S. Rodriguez, Phys. Rev. 135, A779 (1964). ""Hinkley and Ewald state that their amplitudes were an order of magnitude smaller than the theoretical result for the transverse magnetoresistance. This can be partly accounted for by a factor of two due to an error in Adams and Holstein's result, Eq. (6.24) in Ref. 52. There is probably an additional reduction due to the g-factor. 7 3 F. Bassani and L. Lin, Phys. Rev. 132, 2047 (1963). 74G.A. Busch and R. Kern, Solid State Phys. 11, 1 (1960). l 5 S. Groves and W. Paul, Phys. Rev. Letters 11, 194 (1963).

6.

MAGNETIC QUANTUM EFFECTS

191

proposed a model in which the r7-band is below the Ts+ band which corresponds to the InSb valence band. The former “light hole” band now turns up and becomes the conduction band, and the k = 0 energy gap is zero, the thermal gap corresponding to subsidiary minima such as we found in GaSb. This model apparently accounts for the results and may also account for the very large steady part of the magnetoresistance. (5) Level Broadening. The Dingle temperature T’ given in Table I1 can be compared with a “mobility temperature” T,’ = h/2nkzm obtained from the mobility relaxation time75a.,z For the 111-V compounds T‘ correlates which can be expected well with T,‘. In fact T‘ is somewhat larger than Tm‘, for ionized impurity scattering. The former depends on the carrier lifetime, while the latter depends on the momentum relaxation time, and the two are equal only for isotropic scattering. The near equality of the two times is in contrast with the case of de Haas-van Alphen measurements in metals in which the oscillations are, in general, due to small groups of electrons while the conductivity is due to the bulk of the electrons in the crystal. For GaSb,69 as we discuss below, more than one band is occupied, but T’ and T’, are both due to the same band. For grey tin T’ was found to be considerably larger than T i , a fact that Hinkley and Ewald7’ attribute to inhomogeneities, which can also produce broadening by introducing a spatial variation of the period. The results of Becker and F a d 9 on tellurium-doped GaSb are especially interesting in that the collision broadening effect decreases as carrier concentration is increased, resulting in a striking change in amplitude with carrier concentration, as is evident from Fig. 7. GaSb has interesting galvanomagnetic properties75b because of the existence of subsidiary (1 11) minima with a high density of states located 0.08 eV above the low mass k = 0 conduction band minimum. The transport properties thus change considerably when, due to high carrier concentration or high temperature, the subsidiary minima are occupied. The decrease in Dingle temperature occurs in just the region of concentration in which the subsidiary minima are becoming populated. A calculation by Robinson and Rodriguez7’ has shown that the change in collision rate can be explained on the basis of ionized impurity scattering within the k = 0 minimum. The ionized impurity potential is modified by electronic screening. The Fermi-Thomas screening wave number q, for degenerate electrons is given by

75“T,,,’ is related to T,,’, introduced by SIadek,6’ by T,‘ = T,,‘/2.

7 5 b W .M. Becker, A. K. Ramdas, and H. Y. Fan, J . Appl. Phys. Suppl. 32, 2094 (1961).

192

LAURA M. ROTH AND PETROS N. ARGYRES

where d N / d [ is the density of states at the Fermi level and K is the dielectric constant. When the carrier density becomes large enough for the subsidiary bands to be populated, the increased density of states results in a greatly enhanced screening, so that the impurity scattering decreases. The screening effect is dominant despite the onset of intervalley scattering. The theory predicts a greater decrease in T’ than in T,’, a fact that agrees with experiment. Becker and Yep7’ found that by diffusion of lithium into telluriumdoped GaSb, the mobility could be increased enough to observe oscillations in less heavily doped material for which only the zone center states are occupied. 7. QUANTUM LIMIT a. Theory In the quantum limit (see Section 4b) the results for the conductivity become qualitatively different from low field results, and the magnetic field dependence of the longitudinal and transverse resistivities is strongly dependent upon the scattering mechanism and the degree of degeneracy. We will consider here in particular the “extreme quantum limit” cases2 for which [, or kT < ho,. The impurity scattering mechanisms which we shall consider here are the &function potential of Eq.(53)and in addition ionized impurity scattering for which

where we have included screening. qs is given by Eq. (79) for degenerate statistics, while for classical statistics 4ne2N 4s2 =

For phonon scattering the transition probability is given by Eq. (58). For acoustic phonons and deformation potential scattering C(q) is given by Eq. (58a). For polar crystals that lack an inversion center there is also a piezoelectric coupling to acoustic p h ~ n o n s ,which ~ ~ results in a C(q) for longitudinal phonons given by

Here we have included screening as introduced by Adams and Holstein,52 and P 2 is a coupling constant. Consideration of optical phonon scattering will be postponed until Section 8. 7b

H. J. G. Meijer and D. Polder, Physica 19, 255 (1953).

6.

MAGNETIC QUANTUM EFFECTS

193

Considering first elastic scattering for the longitudinal case a relaxation time exists and is given, from Eqs. (56), (50),and (51), by

where

is now a one-dimensional density of states, and where the averaged matrix element is given by

with w(q) given by Eq. (52). The longitudinal resistivity is now given by

For the transverse resistivity and elastic scattering we can obtain from Eqs. (63) and (42), using Eq. (50) for the transition probability,

where zI is defined by Eq. (83) with ( w ~ ) ~replaced , by the transverse averaged matrix element

x [Iw(4,, q y , 0)l2 + Mq,,

2k)121.

(87) In Eq. (86) we have included the appropriate cutoff energy E, from Section 5d. In Tables I11 and IV we give results for the longitudinal and transverse resistivities for the various scattering mechanisms and for degenerate and classical statistics, respectively. The elastic scattering results are essentially those of Adams and Ho1stei1-1.~~ We should note that the high temperature acoustic phonon results (both deformation potential and piezoelectric) correspond to elastic scattering and that, as discussed in Section 5b, the high temperature deformation potential results are identical to the 6function potential model. In Tables 111 and IV the entries are normalized to the appropriate zero field resistivities (which are also given in the table) under the assumption that neither carrier density nor statistics changes. It should, however, be noted that in the quantum limit the degeneracy qy,

TABLE 111 LONGITUDINAL ANI) TRANSVERSE MAGNETORESISTANCE I N THE EXTREME QUANTUMLIMIT.^ DEGENERATE STATISTICS Scattering mechanism &function impurity potential

poNe2/m*

( ) ( ) ( )

PII/PO

E,’kT _ (o”z 2m* ~ ” _ _ p . 2 2sh h’

Low temp. acoustical phonon [(m*v,2(o)’i26 kT 6 hu,/l]

EI2kT[o”2 2m* _____ pv,2 27th h’

+

3 (!!)’

5 0 ” 2 2m* 3 i 2 N,a 2 __ 2nh h2

High temp. acoustical phonon (kT ha,//)

3’2

P J P O

8

lo

_ 27 64

-J ,

( h o ) 3 (kT)4 2__ (rn*u,2)2(2i0)’

” Compiled from calculations of Adams and H ~ l s t e i n , ~Kubo ’ er and present authors. We have assumed only one spin state occupied, and for the low temperature results we have taken T < Debye temperature. Definitions as in text and as follows:

6.

MAGNETIC QUANTUM EFFECTS

195

temperature [i.e., cz, Eq. (32)] decreases with increasing field, so that the statistics can change as a function of B. If several scattering mechanisms are important we would need a weighted average of the quantities in the table. In the transverse resistivity for classical statistics the appearance of the factor In(kT/.z,) is due to the cutoff introduced in Section 5d and involves the assumption that E, G kT. More precise calculations for various cutoff mechanisms32differ in the same limit only in that E, is multiplied by a factor of order one. . ~ Turning now to inelastic acoustic phonon scattering, Kubo et ~ 1 have considered the effect of inelasticity upon the scattering in some detail, and have it to be important for low temperatures. Similar results have been found by Klinger.77 In the calculation of Adams and Holstein,52 the inelasticity of the scattering was not included correctly, so their low temperature phonon results are in error. In calculating the zero field resistivity for semiconductors and semi metal^,^^ it has been found that inelasticity is ’ / ~ , E is equal to for degenerate unimportant for kT S ( ~ * u ~ ~ E ) where statistics and kT for classical statistics. For cases of interest this is quite a low temperature. However, in the quantum limit the relevant parameter becomes hvs/i, which is the same as the aforementioned cutoff energy and which can be larger than m*u: for sufficiently large fields. Thus, for the quantum limit case, inelasticity can be important in a range of temperature where it is not so for zero field. In evaluating the transverse resistivity for inelastic scattering we must now use Eq. (59) for n = 0 with the transition probability given by Eq. (58). This has been carried out by Kubo et and the results are included in Tables I11 and IV. The approximation involved is k T $ [ ~ * u ~ ~ The E ] ~high / ~ . temperature region corresponds to kT % hu,/l, in which case the inelasticity introduces the cutoff for classical statistics, as discussed earlier. The low temperature region corresponds to [ m * ~ , ~ z 4 ] ’kT ~ ~4 hu,/l. For the longitudinal case the resistivity is obtained from the Boltzmannlike equation, Eq. (49), where now, due to the inelasticity, a relaxation time does not necessarily exist. For kT S=- [ r n * ~ ~ ~ e however, ]~’~, it turns out that one can define a relaxation time, and we have included results for this case in Tables 111 and IV. It is interesting to note the effect of the range of the scattering potential upon the elastic scattering results. If ro denotes this range (for example, ro = 0 for the b-function potential, and r,, qS-l for the ionized impurity potential), we can compare it with the Landau orbit radius 1. In evaluating

-

M. 1. Klinger, Z k . Eksperim. i Teor. Fiz. 31, 1055 (1956) [English Trans/.: Soviet P h y ~ J E T P 4, 831 (1957)l. 78 A. H. Wilson, “The Theory of Metals,” p. 264. Cambridge Univ. Press, London and New York, 1958. 77

~

LONGITUDINAL AND

TABLE IV TRANSVERSE MAGNETORESISTANCE IN THE EXTREME QUANTUM LIMIT.‘CLASSICAL

Scattering mechanism

poNeZ/m*

STATISTICS

PLIPO

PII/PO

-(-) -(-)

&function impurity potential

hw, 371 k T 1

3 (%%) kT ! High temp. acoustical phonon ( k T 9 hu,/l)

P’kT 3G(2kT) pv: 32h(xkT)“’

Low temp. piezoelectric (m*u,’ < k T < hvJI), (4;’ S I )

P’kT

z 3

4

s>

E,

’

kT

In-E~

> z

c2 ___

High temp. piezoelectric ( k T >> huJI), (4;’ 9 I )

(4,’

>

’ In-k T

kT 4m*v,’

Low temp. acoustical phonon (m*uS2< k T 6 hu$)

Screened ionized impurity

1 hw, 371 k T

! 3 (%) kT

r

(F)

3G(2kT)

2m*

4nZe2

~

”’

4 1 (kT)’/m*v: 371 G(2kT) 4kT + E,

1/2

NiF(3kT)

,/,_4_ _I_

kT 371’~’ G(2kT) m*u,’

- _ _ In

(F)

pv,’ 32h(akT)“’

tl

4 -__ 1 hw, k T -In 371 C(2kT) k T E,

4 1 ha, -~ In 371 G(2kT) 4kT + E~

2m*

(7 1 2 8 h) n”’(kT)’~’(~)

8 _

8

k T_

F(3kT) 4kT

1

_ _~ _

+ E,

71

F(3kT)

In (4kT

kT + E,)E, In Ec

Kubo et al.,” and present authors. We have assumed only one spin state occupied, a Compiled from calculations of A d a m and and for the low temperature results we have taken T 4 Debye temperature. Definitions as in text and Table Ill and as follows:

’’

dy

-x(x

+ y)”’le’

- 11

I-’

z of order 1 ;

x”dx

6.

MAGNETIC QUANTUM EFFECTS

197

the averaged matrix elements of Eqs. (84) and (87X we see that if ro 4 I, the short range case, then w z is constant over the region in which the exponential factor is large, so that the ( w ’ ) , ~and (w’), are equal and independent of the magnetic field. The longitudinal scattering rate then just depends on the density of states, which is proportional to B. A comparison of the two resistivities shows that pI and pII differ essentially by a factor ho,(l/~,), which is of the order of hOJ, for degenerate statistics and h J k T classically, if we ignore the divergence in our estimate. This indicates, incidentally, that the transverse magnetoresistance should be large compared to the longitudinal magnetoresistance, a fact which is borne out experimentally. This short range case corresponds to the &function potential in Tables 111 and IV. A different situation arises for a long range potential for which io9 1. In principle, since the Landau orbit radius decreases with field, this case should occur for sufficiently high magnetic fields for any scattering potential of finite range. In the long range limit, we can ignore the exponential field dependence in Eqs. (84) and (87), so that ( w ’ ) ~ ,goes as B-’ and (w’), as B - (This result depends, unfortunately, on the convergence of the integrals and so, in special cases, may not apply exactly). If we ignore the k, dependence of (w’), we find that both pII and pI saturate for classical statistics, and for degenerate statistics they increase less rapidly with field than for the short range case. The long range case is exemplified by ionized impurity scattering in the extreme quantum limit. For acoustic phonon scattering the effective range of the interaction corresponds to hvJk7: which is the wavelength of a phonon of energy kT The short range case then corresponds to high temperature and the long range case to low temperatures, as can be seen in Table 111.

’.

b. Experiment Experimental results in the quantum limit range of magnetoresistance are not nearly so complete and well correlated with theory as for the oscillatory effects. Since the high field effects are sensitive to the scattering mechanism it was hoped that the effects could be used to identify scattering mechanisms. However, the experimental results have not borne this out. The case of InSb has received the most attention in the literature. Results of Sladek79and others”, 8 1 for the transverse magnetoresistance are shown in Fig. 8. Rather similar appearing results were obtained by Amirkhanov 79

R. J. Sladek, J . Phys. Chem. Solids 16. 1 (1960). Haslett and W. F. Love, J . Phys. Chem. Solids 8, 518 (1959). R. Bate, R. Willardson, and A. Beer, J . Phys. Chem. Solids 9, 119 (1959).

so J. C.

198

LAURA M. ROTH AND PETROS N. ARGYRES

/

hwc kT I

10

5

P I I l l

4

10

I

I

20

I

I

l

l

50

B (kG) FIG. 8. Transverse magnetoresistance of several samples of InSb at 77°K: D Sladek79 (1014 C I Y - ~ , 6.2 ( N = 2.8 x 1014 ~ m - mobility ~ , = 5.3 x lo5 cm2/V-sec). 0 Bate et x 10’ cm2/V-sec). 0 Frederikse and HosleP‘ (10” cm-’, 3.3 x 10’ cm2/V-sec). A Haslett and Loveso (1.2 x 10l6 c ~ r - lo5 ~ , cm*/V-sec). [After A d a m and

et in pulsed magnetic fields up to 800 kG. It is seen that p J p 0 is very nearly proportional to B for all cases given. The longitudinal magnetoresistance, on the other hand, has been observed to saturate at the value 1 . 3 ~for~ pll, in the case of Sladek’s results. The ratio of longitudinal to transverse magnetoresistance was found by Sladek to be proportional to hoJk?: in agreement with the theoretical short range result for classical statistics, which apply in all the cases quoted. Sladek has found that the best fit to his high field magnetoresistance results is obtained from the high temperature piezoelectric scattering mechanism. It is not clear, however, whether this mechanism should dominate at zero field. Experimental results have also been obtained on n-type InAs by ~ found for a typical sample that the transverse Amirkhanov et u E . ~ who magnetoresistance varied as B’.’ at 77°K and the longitudinal magnetoresistance saturated at pIl/po= 1.2. These authors conclude that the field and temperature dependence of their results is in qualitative agreement with the theory for scattering by both acoustical phonons and impurities, and for classical statistics. Kh. I. Amirkhanov, R. I. Bashirov, and Yu. E. Zakiev, Dokl. Akad. Nauk S S S R 132, 793 (1960) [English Transl.: Soviet Phys. “Doklady” 5, 556 (196O)l. 8 3 Kh. I. Amirkhanov, R. I. Bashirov, and Yu. E. Zakiev, Zh. Eksperim. i Teor. Fiz. 41, 1699 (1961) [English Transl.: Sooiet Phys. JETP 14, 1209 (196211. 82

6.

MAGNETIC QUANTUM EFFECTS

199

Considerable doubt has been cast on the meaningfulness of the high field experimental results by the work of who showed that a linear magnetic field dependence for the transverse magnetoresistance is expected even classically, due to inhomogeneities in the distribution of the impurities. The resulting fluctuations in carrier concentration can cause large variations in the Hall field leading to transverse currents in the sample, which increase the dissipation and produce an enhanced magnetoresistance. Puri and Geballe85 have pointed out, however, that the effects of the inhomogeneities are not so pronounced for thermomagnetic effects. They obtain results for the magneto-Seebeck effect on n-Ge which are consistent with a B2 dependence of the electronic scattering rate, as expected from Table IV, for acoustic phonon scattering. The thermomagnetic effects are discussed by Puri and Geballe in this volume. For additional experimental results in the quantum limit for semiconductors in general the reader is referred to Beer's review article." 8. OPTICAL PHONON SCATTERING

In the case of scattering by optical phonons with frequency w o (assumed independent of q) there is a new type of magnetoresistance oscillation with B, as was pointed out by Gurevich and Firsov.21 The transition probability in this case is given by Eq. (58) with

(88) where A is a coupling constant defined by Gurevich and Firsov [Eq. (7) in Ref. 211, and wq = coo. The origin of these oscillations in the case of the transverse resistivity can be seen from Eq. (62). If wo < w c , the situation is similar to that of acoustic phonon scattering. If, however, wo > w, the singularities of the two factors can coincide when wo = mu, with m = 1,2,3,. . . . Thus the transverse resistivity pI becomes infinite whenever the magnetic field is such that w; = m*,/eB = m/wo. This gives pL a quasiperiodic behavior as a function of B-' with period A(l/B) = e/m*co,. It is clear that the presence of this quasiperiodic behavior of pI is independent of the statistics of the electrons. The expression Eq. (62) has been evaluated for Boltzmann statistics by Gurevich and Firsov,21 under the assumption hw, % kT, which would ordinarily correspond to the quantum limit. When w, % wo the result is lC(dI2 =

PI =

84

85

Po-,

2kT

C. Herring, J . Appl. Phys. 31, 1939 (1960). S. M. Puri and T. H. Geballe, Phys. Rev. Letters 9, 378 (1962).

200

LAURA M. ROTH AND PETROS N . ARGYRES

where p o is the zero field resistivity:

When w, < w o ,the result is obtained in terms of the parameter 6 = o o / w c - M , where M is the largest integer smaller than wo/o,. For hoc6/kT < 1, Gurevich and Firsov find

The factor in front of Eq. (91) is the classical high field magnetoresistance which differs from the zero field value as discussed by Gurevich and Firsov." The second term is singular when 6 goes to zero, and so gives the quantum oscillations. For degenerate statistics the evaluation of Eq. (62) has been carried out by Efros,86 who found that, for [ % hao 2 hw,,

where p1 is a constant. The effects of an anisotropic effective mass on these oscillations have been considered qualitatively by Gurevich et ~ 1 . ' ~ The logarithmic singularities of p1 for w, = wo/m are, of course, smoothed out by any of the mechanisms we mentioned before in the case of elastic scattering. This was examined by Firsov and Gurevich," who arrived at the conclusion that the dominant mechanisms are the level broadening and the electron-electron interaction and gave expressions for the appropriate cutoffs for 6. Oscillations of a similar nature may also be expected to occur in the longitudinal magnetoresistance. Firsov et al. 89 refer to investigations which indicate that the sign of the oscillations in the longitudinal magnetoresistance depends on the relative importance of acoustical and optical phonon scattering, a maximum occurring if optical phonon scattering predominates, and a minimum if acoustical phonon scattering dominates. A. L. Efros, Fiz. Tuerd. Tela. 3, 2848 (1961) [English Transl.: Soviet Phys.-Solid State 3, 2079 (1962)l. " V. L. Gurevich, Yu. A. Firsov, and A. L. Efros, Fiz. Tuerd. Tela. 4, 1813 (1962) [English Trunsl.;Soviet Phys.-Solid State 4, 1331 (1963)l. Yu. A. Firsov and V. L. Gurevich, Zh. Eksperim. i Teor. Fiz. 41,512 (1961) [English Transl.: Soviet Phys. JETP 14, 363 (1961)l. 8 9 Yu. A. Firsov, V. L. Gurevich, R. V. Parfeniev, and S. S. Shalyt, Phys. Rev. Letters 12, 660 (1964).

6.

MAGNETIC QUANTUM EFFECTS

201

B(kG)

FIG. 9. Longitudinal and transverse magnetoresistance for n-type lnSb samples at 90”K, demonstrating Gurevich-Firsov oscillations. 11: N = 4.1 x 1013~ m - ~mobility , = 5.5 x lo5 cm2/V-sec. I: N = 6 x 1013 ~ m - mobility ~ , = 6.7 x lo5 cm2/V-sec. [After Firsov et aLE9]

It should be pointed out that these oscillations due to scattering by optical phonons differ from the corresponding oscillations due to other scattering mechanisms in the following respects: (a) they can occur either with Boltzmann or Fermi statistics; (b) they persist even in the quantum limit kT or [ < ho,< hw,; (c) their period A(l/B) = e/m*co, is independent of the concentration of the carriers and dependent on their effective mass, whereas the period of the previous oscillations, A(l/B) = Ae/rn*c[, depends on the carrier concentration and is independent of their effective mass; (d) their amplitude is proportional to exp( - ho,/kT) when ho, > kT, i.e., it increases with increasing temperature. Observation of the Gurevich-Firsov oscillations has been reported in InSb by Puri and Geballe” and more extensively by Firsov et af.*’ who have investigated experimentally all the points we mentioned above. Their experimental results are shown in Fig. 9, for both transverse and longitudinal effects. The transverse effect shows maxima at the expected field values and the period of 3.0 f 0.2 x lo-’ G - ’ IS . consistent with the theory (0, = 3.7 x 1013 sec-’, m* = 0.016, B 34 kG).In the range of carrier concentrations 5.2 x 10’3cm-3 to 1.3 x 1015cm-3 the period of the oscillations was indeed found to be independent of concentration, although no such osciliations were observed with samples of n-InSb with concentrations larger than 5 x l O I 5 ~ m - The ~ . amplitude of the oscillations increased

-

S . M. Puri and T. H. Geballe, Bull. Am. Phys. SOC.8, 309 (1963); “Thermomagnetic Effects in the Quantum Region,” Chapter 7, this volume.

202

LAURA M. ROTH AND PETROS N . ARGYRES

with increasing temperature up to liquid nitrogen temperatures, in conformity with the theory. Furthermore, all their samples were nondegenerate. Their experimental results on the oscillations of the longitudinal magnetoresistance are also in agreement with their, as yet unpublished, calculations for the change of phase with temperature, as well as for the period.

9. CONCLUSION In concluding this review of the quantum galvanomagnetic effects on the III-V semiconductors, we may say that, although the study of these effects has not yielded as yet the detailed information about the scattering mechanisms that was hoped for originally, it has provided us with useful information for checking the band structures and effective masses of the carriers and for at least indicating the dominant scattering mechanisms. Furthermore, they have given us a quantitative measure of such fine aspects of the scattering interaction as level broadening. More detailed experimental and theoretical work seems, however, necessary in order to clarify the problems that have arisen and to enhance the effectiveness of these experiments as sensitive probes of the detailed nature of scattering interactions.

ACKNOWLEDGMENT It is a pleasure to express our indebtedness to Mrs. Helen F. Reilly for her patience and expertness in typing this review.

CHAPTER 7

Thermomagnetic Effects in the Quantum Region* S. M. Puri and T. H . Geballe I. INTRODUCTION. . . . . . 1. Survey . . . . . . . . 2. Definitions and General Relations

. . . . . . . . . . 203 . . . . . . . . . . 203 . . . . . . . . . . 208

11. THERMOELECTRIC POWER IN HIGH MAGNETIC FIELDS. . . . . 209 3. Electronic Part of Thermoelectric Power . . . . . . . . 2 10 4. Phonon-Drag Thermoelectric Power. . . . . . . . . . 214 5. Experimental Results on n-Type Germanium . . . . . . . 223 111. COMPOUND SEMICONDUCTORS OF 1II-V GROUP. 6. Experiments on n-Tvpe InSb. . . . . . 7. Quantitative Formula ,for Phonon Drag . . . 8. Analysis of the Experimental Data . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. 232 . 232 . 238 . 245

Iv. OSCILLATIONS INDUCED BY OPTICAL PHONONS .

. . . . . . 255

I. Introduction 1.

SURVEY

There are two general fields of interest in the study of transport properties of solids. First, these studies give information about the energy bands of the solids, the orientation and the shape of the band edges, and the position of the Fermi level. Secondly, one can learn about the scattering mechanisms of the charge carriers and of the phonons. An external magnetic field provides a convenient parameter for unscrambling the different effects. The program of determining the band structure from these studies was carried out extensively’ in the 1950’s and is the subject of other chapters. The emphasis in this chapter will be mainly on what can be, and the little that has been, learned about the scattering of electrons and of phonons *The work at Stanford was supported in part by the U.S. Army Research Office, the U.S. Air Force Office of Scientific Research, and the US. Office of Naval Research under the joint Services Electronics Program by Contract Nonr 225(48), and in part by the Air Force Office of Scientific Research of the Office of Aerospace Research under Contract No. AF 49(637)1429. (a) 9. Lax, Reu. Mod. Phys. 30, 122 (1958). (b) B. Lax and J. G. Mavroides, Solid State Phys. 11, 261, (1960). (c) 9. Lax, Proc. Intern. School Phys. “Enrico Fermi” XXII, 240 (1963).

’

203

204

S. M. PURI AND T. H. GEBALLE

by studying the effect of a magnetic field on the thermoelectrjc properties. For this it is essential to know the energy band structure of the solid. It is difficult to develop a rigorous theory of transport phenomena which is also tractable enough for explicit calculations to be compared with experimental data for arbitrary strength of the magnetic field. Different approximations and different mathematical approaches are necessary under different conditions. Throughout the following it is assumed that the problem can be formulated in terms of the energy states of an ideally pure crystal which has been stripped of its lattice excitations; the presence of impurities and of the phonons in a real crystal causes scattering of electrons between these energy states and can be treated by the perturbation theory. Furthermore, it is convenient to classify the regions of the magnetic field in the following manner. (1) Classical region, where hw << E. The classical field can be subdivided into (i) a low field region (wz 6 1) and (ii) a high field region (oz 1). (2) Quantum region, where ho 2 E. We shall assume that oz 9 1 in the quantum region. The situation of a low field quantum region where o‘s << 1 does not arise in high mobility semiconductors, such as are under discussion. Here w is a cyclotron frequency of the crystal electrons, and consequently hw is the magnetic part of the energy of the electron; we also have o = eB/mc where e is the electron charge, B is the magnetic field, m is the cyclotron effective mass,’ and c is the velocity of light; E is the energy of a typical electron which contributes to the transport properties. If the electrons are described by degenerate statistics, E is of the order of . the nondegenerate statistics to which we shall the Fermi energy E ~ For confine ourselves, we have i sz kT. The relaxation time ‘s is of the order of mean free time between scattering collisions. The classical region can be studied adequately by treating the magnetic part of the energy as a perturbation which adds the Lorentz-force term to the usual classical Boltzmann e q ~ a t i o n As . ~ a consequence of this assumption, the transition probability between different electron states, caused by the interaction of the electrons with phonons and impurities, is independent of the magnetic field. In the quantum region, the magnetic field cannot be treated as a perturbation, and therefore enters into the unperturbed states of the electron. This has the consequence that the transition probability for scattering depends explicitly upon the magnetic field and gives rise to the so-called quantum effects in transport properties. The effect of the magnetic field on scattering due to various mechanisms will depend upon the range and the shape of the scattering potential. The transport properties in the quantum region therefore show a different

+

W. Shockley, Phys. Reo. 79, 191 (1950). See, for example, A. C . Beer, Solid State Phys. Suppl. No. 4 (1963).

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

205

functional dependence on field and temperature for different scattering mechanisms. The study of transport properties in this region thus has some advantages over similar studies in the classical region, because the functional dependence in the quantum region provides a direct method for sorting out the various scattering mechanisms and their relative importance at any temperature. In certain instances a particular scattering mechanism can be more or less isolated from other mechanisms. For example, scattering of electrons between different magnetic energy levels caused by optical phonons gives rise to oscillations in the transport properties4 The study of the oscillatory part can give information about the electron coupling to optical phonons and the frequency-momentum dispersion of the latter. As a second example, we shall see that the phonon-drag thermoelectric power in transverse magnetic fields in the quantum region is not sensitive to the scattering of electrons by impurities. It is, in fact, determined by the electron scattering on low energy acoustic phonons and the mean free path of these phonons. At sufficiently low temperatures, the mean free path of the phonons is determined by the size of the crystal specimen and is known. The phonon drag at such temperatures, therefore, provides information on electron-phonon interactions. This information, combined with similar data at higher temperatures, yields information on the relaxation mechanisms of the low energy acoustic phonons. The Nernst effect, on the other hand, depends on electron scattering from all mechanisms and can be combined with other data to learn about electron-impurity scattering. However, there are certain difficulties which must be resolved before the full potential of the above methods can be used. The quantum transport theory in the presence of high magnetic field is complicated and involves some subtle point^.^ The correct approach, which can lead to tractable results in a situation which is experimentally realizable, is not always obvious. This has been demonstrated time and again by the conflicting results obtained by different author^.^ As an example, it has not been possible so far to incorporate into the quantum transport theories the effect (a) V. L. Gurevich and Yu. A. Firsov, Zh. Eksperim. i Teor. Fiz.40,199 (1961) [English Transl.: Soviet Phys. JETP 13, 137 (1961)l (b) M. I. Klinger, Fiz. Tuerd. Tela 3, 1342 (1961) [English Transl.: Soviet Phys.-Solid State 3, 974 (1961)l (c) S. M. Pun and T. H. Geballe, Bull. Am. Phys. SOC. 8, 309 (1963).(d) Yu. A. Firsov, V. L. Gurevich, R. V. Parfeniev, and S. S. Shalyt, Phys. Reu. Letters 12, 660 (1964). (a) E. N. Adams and T. D. Holstein, J . Phys. Chem Solids 10, 254 (1959). (b) P. N. Argyres, Phys. Rev. 117, 315 (1960). (c) YLLN. Obraztsov, Fiz. Tuerd. Tela 4 414 (1964) [English Transl.: Soviet Phys.-Solid State 6,331 (1964)l.(d) P. S. Zyryanov and V. P. Silin, Zh. Eksperim. i Teor. Fiz. 46,537 (1964) [English Transl. ;Soviet Phys. JETP 19,366 (1964)].(e) A. I. Anselm and B. M. Askerov, Fiz. Tuerd. Tela 3,3668 (1961) [English Transl. ;Soviet Phys.-Solid State 3, 2665 (1962)l. (f) L. M. Roth and P. N. Argyres, “Magnetic Quantum Effects,” see Chapter 6 of this volume.

206

S. M. PURI AND T. H. GEBALLE

of random fluctuations in the density of charge carriers. Some idealized cases have been discussed' which only emphasize the extreme importance of the spatial variation of the charge carrier density in, at least, some of the experimental situations. The use of Born approximation to calculate scattering from impurities' where the potential can be long range has not been clarified. In the above situation, a logical procedure of development could be to check the present theories experimentally. The experiments can be performed on samples in which the scattering mechanisms are known with some certainty from previous knowledge, and the results compared with the predictions of the theory. The choice of experiments must be such that the experimental results are not affected significantly by fluctuating density of charge carriers, surface conduction, etc., which have not been accounted for by the theory. There is some theoretical evidence6(")that the thermomagnetic effects, especially the variation of thermoelectric power with the magnetic field, are most suited for this purpose. Such experiments have been done with some success on the phonon-drag thermoelectric power of n-type Ge.8 It was possible in these experiments to establish8'b) that the model of a homogeneous medium used in the theory is valid in the case of thermomagnetic effects, even when it is not valid in the case of galvanomagnetic effects. The spurious effects due to surface conduction are also less serious in the case of thermomagnetic effects. In brief, it is reasonable to assume that the present theories of quantum transport are essentially correct in describing the field and temperature dependence of thermomagnetic effect. The thermomagnetic data on 111-V compound semiconductors can, therefore, be analyzed with some confidence to study electron-scattering mechanisms. The available information, especially at low temperatures, is at present neither very extensive nor very precise. It will be worthwhile to present some discussion of the results for n-type Ge which not only is illustrative, but also gives some confidence in the quantum-region studies. We shall discuss the Seebeck coefficient Q in detail because more experimental data are available for it, and because a great deal of information about scattering mechanisms can be obtained by simple modifications of the available theories for galvanomagnetic effects. The Nernst effect is not treated in any detail because of the lack of sufficient and precise experimental data. Some of the theoretical treatment has been given by (a) C. Herring, J . Appl. Phys. 31, 1939 (1960). (b) R. T. Bate and A. C. Beer, J . Appl. Phys. 32, 800, 806 (1961). ( c ) J. A. Morrison and H. L. Frisch, J . Math. Phys. 5, 1681, 1672 (1964). ' A. H. Kahn, Phys. Reo. 119, 1189 (1960). * (a) S. M. Puri and T. H. Geballe, Phys. Rev. Letters 9, 378 (1962).(b) S. M. Puri, unpublished Ph.D. dissertation, Columbia University, New York, 1964.

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

207

Obraztsov.' The other thermomagnetic effects will not be considered at all. We refer to the galvanomagnetic effects, magnetoresistance and Hall effect only, in so far as the results have a bearing on the transport theory in general. Because of the requirements of higher magnetic fields, there are very few measurements of the thermoelectric power on p-type material extending into the quantum region. The degeneracy of the valence bands presents additional problems for theoretical analysis which have not been worked out. This limits our discussion to the case of n-type material in the extrinsic range. We have made no attempt to present the quantum transport theory from a rigorous basis. The emphasis here is on experimental results. The theoretical discussion which we have given is, we believe, necessary in order to analyze the experimental results and to bring out the physics. Even though it is confined to its elementary form, it occupies a large fraction of this chapter. In discussing the theory, we have found the n-approach advocated by Herring to be very convenient, i.e., we calculate the Peltier coefficient IT and then obtain the thermoelectric power Q using Kelvin's relation. This procedure avoids the difficulties and pitfalls of incorporating spatially varying temperatures in the quantum region theory. In the remainder of this part we recall briefly the definitions of the various quantities and some thermodynamic relations that exist between them. The qualitative features of the thermoelectric power in the quantum region are discussed in Part 11. We avoid carrying out any detailed calculations in this part and make use of only physical arguments. Section 3 is concerned with the electronic part of Q. In nondegenerate semiconductors, a major fraction of Q,, the electronic part of Q, is determined just by the position of the band edge and the Fermi energy, the transport term due to the electron motion within the band being relatively small. Therefore, Q, can be conveniently calculated by studying the effect of magnetic field on the position of the Fermi energy with respect to the band edge. This approach that we have adopted leads to a logarithmic increase of Q, with B. There had been some controversy about a linear but now there seems to be general agreement that the above approach yields the correct field d e p e n d e n ~ e . ~ " ' ~lo~ ' ~ ) ~ The phonon-drag part of Q is treated in Section 4. We first argue that the relation between Qp, the phonon-drag part of Q, and the electrical resistivity, derived by Herring is valid with some slight modifications even in the quantum region. Borrowing the results of the theory of magnetoresistance in the quantum region and using physical arguments, we obtain the functional dependence of Q, on the magnetic field and the temperature. In Yu.N. Obraztsov, Fiz. Tverd. Tela. 7 , 5 7 3 (1965) [English Transl.: Soviet Phys.-Solid State 7 , lo

455 (1965)]. S. M. Puri and T. H. Geballe, Phys. Rev. 136, A1767 (1964).

208

S . M . PURI AND T . H. GEBALLE

Part I11 we have tried to make the arguments mathematical, and complete expressions are obtained for Q,. We have included the discussion of Section 4, even at the cost of being repetitious, because of the insight that such a discussion lends to the physical phenomena. In Section 5 we describe some experimental results in the case of n-type Ge. These experiments study the effects of slight inhomogeneities in the charge carrier distribution and the spurious surface conduction on the magnetoresistance and the thermoelectric power measurements. The results confirm the theoretical expectation that these spurious effects distort the high field magnetoresistance results but do not significantly affect the thermoelectric power results. The functional dependence of Q obtained in Part I1 is verified. The results of measurements of Q on n-type InSb are described in Section 6. These results show that a large fraction of Q in the high field region is due to the phonon-drag. Next we have listed the type of information that one can obtain from these experiments. In Section 7 we calculate the phonon drag Q in transverse magnetic fields in the quantum region. The phonon distribution is treated classically. The effects of quantization are taken into account by recalculating the electron-phonon scattering term, which constitutes the driving term for the phonon transport. The assumption is made that phonon scattering on other phonons and on the walls of the specimen is not affected by the magnetic field. The effects of the piezoelectric mode of electron coupling to acoustic phonons are considered. The effects due to the nonparabolicity of the bands are also discussed. A comparison of theory and experiment is made in Section 8. In Part IV we present the available data on the oscillatory behavior of the transport properties of a nondegenerate semiconductor caused by scattering of electrons on optical phonons. 2. DEFINITIONS AND GENERAL RELATIONS

The Seebeck coeficient" or the thermoelectric power denoted by Q is defined in terms of the electric field E, established by a temperature gradient

VT E =Q.VT, (14 with the auxiliary condition that the current J equals zero. The Peltier coeficient, denoted by T, is defined in terms of the heat flux density F that accompanies an electric current density J flowing under isothermal conditions. The defining equation is F 'I

=T-J,

For a detailed discussion of definitions see J. P. Jan, Solid State Phys. 5, 3 (1957).

(1b)

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

209

with the auxiliary condition that the temperature gradient V T equals zero. We have represented Q and TT in their most general forms as secondorder tensors. In cubic crystals in the absence of a magnetic field, the symmetry of the crystal requires that the tensors Q and TT be diagonal and, furthermore, all the elements are equal. However, in the presence of a magnetic field, the nondiagonal elements are, in general, different from zero. In this case the diagonal elements represent the thermoelectric power or the Peltier coefficient, while the nondiagonal elements are related to the Nernst effect. The Nernst coefficient /3, defined analogous to the Hall constant, is given by Q , = -BB. Sometimes a distinction is made between the isothermal and the adiabatic coefficients. We shall not make this distinction in what follows, because the difference between the two definitions in our case is very small. The coefficients Q and TT are not independent; they are connected through the Kelvin relation which, in its most general form, reads zi,(B) = TQj( - B). (2) Kelvin’s relation is one of a set of thermodynamic relations called Onsager’s reciprocal relations. Theoretically, it is simpler to calculate TT directly, because in this case one of the driving forces, VT, is zero. On the other hand, it is much more convenient experimentally to measure voltages and temperature gradients which determine 0 than to measure heat fluxes required for determining TT. Kelvin’s relation can then be used to compare the experiment with theory. However, care must be shown in this because, as Callen” has shown, theoretical definitions of Q and TT have some flexibility, and we must use the definitions that are appropriate to the experimental measurements.

11. Thermoelectric Power in High Magnetic Fields In this part we discuss the qualitative behavior of the thermoelectric power in the quantum region, i.e., in magnetic fields so high that the spacing of the Landau levels is greater than or comparable to the energy of a typical conduction electron. We shall start the discussion by giving a simple physical picture of the phenomena, making use of only qualitative arguments. This gives some insight into the problem and is also useful in estimating the order of magnitudes of the various quantities. In simpler situations, it is possible to predict the functional dependence of various quantities on the magnetic field and the temperature.g Later we shall try to make these arguments more quantitative.

’’ H. B. Callen, Phys. Reu. 85, 16 (1952); 73, 1349 (1948).

210

S. M. PURl AND T. H. GEBALLE

Because of the electron-phonon interaction, any disturbance in the electron distribution is communicated to the phonon system and vice versa. For example, an electric field, although it does not affect the phonon system directly, because of the absence of an electric charge on the phonons, does disturb the phonon spectrum indirectly through its effect on the electron di~tribution.’~ We shall assume that the departure of the phonon spectrum from its thermal equilibrium is so small that it does not have an appreciable effect on electron-phonon scattering. This is the same assumption as is generally made in the classical transport theories to decouple the two transport equations for the electrons and the p h ~ n o n s . ’ Under ~ these conditions, the thermoelectric power can be considered as composed of two independent terms : the electronic part Q, and the phonon-drag part Q , . I 5 These terms are explicitly given by Eqs. (5) and (8). In the limit of low charge carrier density n, Q, is independent of n. For increasing n, a correction term to Q, arises, which is negative and depends upon n. This decrease in Q , with increasing n has been called the “saturation effect” by Herring.” The electronic part Q, is also altered slightly, but, unlike the correction to Q,, it is not possible in this case to differentiate experimentally between the correction term and the zero-order term. The correction to Q , is therefore not treated separately. For large departure of the phonon spectrum from equilibrium, it is no longer possible to discuss the electron and the phonon transport independently, and one therefore cannot speak of the electronic part and the phonon-drag part of Q. In the quantum theories this implies that the total density matrix of the electron-phonon system cannot be approximated by a product of two terms, describing the electron and the phonon distributions. We shall not discuss this situation at all. In the experiments to be described, the correction due to the “saturation effect” on Q, is, at most, small and can be approximated by a first-order perturbation calculation. For carrier densities finite but smaller than a certain critical value the correction is negligible. In the quantum region, the critical value of the density depends upon the magnetic field because it affects the scattering rate. 3. ELECTRONIC PARTOF THERMOELECTRIC POWER

By definition,12 the electric current J and the heat flux carried by electrons F , are given by J = ne
l4

Is

R. Peierls, Ann. Phys. ( N . Y . ) 3, 1055 (1929); “Quantum Theory of Solids,” Oxford Univ. Press (Clarendon), London and New York, 1955. See, for example, M. Dresden, Rev. Mod. Phys. 33, 265 (1961). C. Herring, in “Halbleiter und Phosphore” (M. Schon and H. Welker, eds.), p. 184. Friedr. Vieweg und Sohn, Braunschweig, 1958.

7.

211

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION Fe

= U((E -

>

(3b)

where v and E are, respectively, the velocity and the energy of the electron, and EF is the electrochemical potential. The sign ( ) denotes averaging over the steady state electron distribution. If E~ is the energy of the ground state in the band, we can write F,

=

~ ( € 0- EF)(v)

+ U < O AE) .

(3c)

The electronic part of the Peltier coefficient, n,, is given by using (lb): 1

71, = -(Eo

- EF)

e

+ -1 -(v, AE) e

(0)

(4)

where we have written E = E, + AE. Using (2), the electronic contribution to the thermoelectric power is given by

The first term of (5) is determined uniquely by the band structure of the solid and the number of free charge carriers. The second term depends, in addition, upon the electron-scattering mechanism. For a system obeying nondegenerate statistics, in the absence of a magnetic field, the second term has a value between two and four in the relaxation time approximation.15 The lower value holds for pure acoustic scattering; the value increases for increasing amounts of ionized impurity scattering. In many situations of interest to us, both scattering processes are present, and the second term is relatively uncertain. However, this uncertainty is always less than 100 pV/deg. The above arguments hold even in the presence of a magnetic field of arbitrary strength whether in the quantum region or the classical region. Equations (4) and (5) remain valid since they state nothing more than the definition of the respective quantities. In quantum theories, the angular brackets ( > must be interpreted appropriately as the trace of the product of the density matrix with the operator corresponding to the quantity inside the bracket. However, we must make the provision that E~ and EF can depend upon the magnetic field. Keeping this in mind, we can discuss the behavior of Qe in the classical and the quantum region. In the classical region, the magnetic field leaves the first term unaltered, and the total change in Q, arises from the variation of the second term. In the limit of high fields, the electron distribution approaches a limiting formI6 ; consequently, the second term attains a limiting value, and Q, is said to saturate classically. In the transverse fields, the limiting value is ,:

-

“.I. A. Swanson, Phys. Rev. 99, 1799 (1955).

212

S . M. PURI AND T. H. GEBALLE

irrespective of the electron scattering mechanism. l7 In longitudinal fields, the limiting value is the same as if the magnetic field were absent.” We may note that in the classical high field region, Q, in the transverse field case can be calculated precisely, if the band structure and the electron density are known, even when scattering mechanisms for the electron are not known. In the quantum region both terms in Eq. (5) are altered by the magnetic field, and Q, no longer saturates. If .z0 is raised by an amount i h o , which is the energy of the lowest Landau level, and if nothing else is changed, EF would also be raised by exactly *hm; I.z0 will then remain constant. But due to the degeneracy of the quantized levels, the density of states near the band edge is increased. If all or most of the electrons are in the - E ~ I , as is lowest Landau level, the altered density of states increases necessary to keep the total number of occupied levels constant. It can be shown” that, for a fixed value of carrier concentration, - E ~ I increases under the above conditions as kT In(ho/kT), provided the spin splitting of the Landau levels is small compared to kT. If the spin splitting is large, as in the case of InSb, it must be taken into account explicitly. If the charge carrier concentration also changes with the field, we must add a term kT ln(n,/n,), where nB and no are, respectively, the carrier concentrations in field B and zero field. Since the first term of EQ. ( 5 ) depends only on - +I, the effect of magnetic field on this term is the same regardless of whether the fields are transverse or longitudinal. The effect of the magnetic field on the second term depends on its orientation with respect to the electric field, as we discuss now. In transverse fields, the Hall velocity u,, which is perpendicular to the external fields, is independent of electron scattering and much greater than the velocity u, parallel to the electric field. To a first approximation, the effect of collisions can be neglected, in which case u, = 0 and u, = cE/B. The second term of Eq. (4), is reduced to”

(

:$)

ho hw coth - kT In 2 sinh + I k T + kT. (6) 2 2kT The first and second terms of (6) together represent the contribution due to electron motion in the x-y plane, which is quantized. The second term, which is -ho/2 for ho + kT, is subtracted because of a shift in the zero of energy. The third term is due to electron motion parallel to the magnetic field. The origin of the last term which equals k T is discussed in Ref. 10. It is the difference between the flow of energy density in a thermodynamically “closed system” as compared to an “open system.” A “closed system” cannot exchange particles (electrons here) with the outside world while an (AE)

” C.

=

-

~

~

Herring, T. H. Geballe, and J. E. Kunzler, Bell System Tech. J . 38, 657 (1959).

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

213

BAND EDGE

FERMl ENERGY

-

4 !!T

ZERO FIELD

CLASSICAL

EXTREME QUANTUM

LIMIT

FIG. 1. Effect of transverse magnetic field on the different contributions to the Peltier coefficient. On the extreme left is the case corresponding to zero field and on the extreme right, the situation corresponding to the extreme quantum limit. The center portion represents the picture for high field in the classical region.

“open system” can. The experiment corresponds to the “closed system,” while it is convenient to carry out the calculations for an “open system,” since, in this case, no extra boundary conditions on the wave functions need be imposed. It was pointed out in Ref. 10 (Appendix) that to calculate energy flux in an “open system” we must calculate the enthalpy density of the system in the steady state. This gives rise to the last kT term of Eq. (6). The arguments given therein to derive the extra term are rather empirical. ObraztsovI8 has explicitly calculated the energy transport in a “closed system” by imposing boundary conditions on electron wave functions which vanish beyond the walls of the specimen. These boundary conditions give rise to extra surface fluxes. The results of his rigorous calculations agree”” with the results of Ref. 10. In the extreme quantum limit, when only the lowest Landau level is occupied, the first two terms cancel out; the motion in the x-y plane is completely frozen and does not contribute to (AE). In the transverse magnetic field, the increase in Q, above its classical limiting value becomes Q,(B) - Q,(saturation)

=

e

,

hw B k?:

(7)

Figure 1 demonstrates qualitatively the effect of magnetic field on the Peltier coefficient. The effect of any kind of collision is to produce a finite u, and to add terms of the order of (vx/u$’ to (7). This correction is very small in practical cases. In longitudinal magnetic fields, an expression similar to Eq. (7) holds ; but in this case the one in the brackets should be replaced by a number r, Yu. N. Obraztsov, Fiz. Tverd. Tela 7, 573 (1965) [English Transl.: Sooiet Phys.-Solid State 7, 455 (1965)l. Isa C. Herring, private communication. We are indebted to C. Herring for discussions on this point.

214

S.

M.

PURI AND T. H. GEBALLE

which is obtained only through an explicit calculation of the second term of Eq. (5). The value of r depends upon the dominant mode of electron scattering and is of the order of unity. Ordinarily, r can be neglected as compared to the other terms of Eq. (7). However, in case of scattering by optical phonons, r varies periodically with the magnetic field and contributes an oscillatory part to the thermoelectric power in longitudinal fields4 These oscillations are due to a marked change in the transport properties whenever the spacing of the Landau levels is such that Zhw = hQ,, hQo is the energy of the optical phonons and 1 is an integer. Under these conditions, it is possible for an electron to be scattered without changing its momentum parallel to the magnetic field. The optical phonon exchanges energy only with the magnetic part of the energy of an electron. The well-known discontinuity in the density of electron states at k, = 0 causes the periodic variations of transport properties. These oscillations have an analog in the oscillatory magnetooptic effectslg which are due to similar transitions”” caused by photons. The oscillations in Q , are superimposed over a logarithmically varying monotonic part. We also note that these oscillations will not be observed in the thermoelectric power in transverse magnetic fields since it does not depend on electron scattering. We discuss these oscillations in Part IV.

4. PHONON-DRAG THERMOELECTRIC POWER As was pointed out in the beginning of Part 11, even under isothermal conditions the phonon system in a solid will not be in its equilibrium distribution if an electric current exists in the solid. This deviation of the phonon distribution results in an energy flow which, in the case of semiconductors,20 is in the same direction as the energy flow due to electron motion.”” The heat transported by the directed flow of phonons per unit current under isothermal conditions is defined as the phonon-drag component of the Peltier coefficient, np. The net departure of the phonon distribution from equilibrium depends upon two opposing factors, namely, (i) the rate at which the flow of electrons feeds momentum into the phonon system, and (ii) the rate at which the phonons can lose the excess momentum l9

S. Zwerdling, B. Lax, L. M. Roth, and K. J. Button, Phys. Rev. 114, 80 (1959); L. M. Roth, B. Lax, and S. Zwerdling, ibid. 114,90 (1959); E. Burstein, G. Picus, H. Gebbie, and F. Blatt, ibid. 103, 826 (1956).

“”In magnetooptic experiments, the transitions are between the Landau levels of the valence band and the conduction band. The oscillations that we are discussing occur due to transitions between Landau levels of the same band. See, for example, D. K. C. MacDonald, “Thermoelectricity,” Wiley, New York, 1962. ’‘‘In some metals, the net flux of phonons is in a direction opposed to that of electrons. This can happen only if Umklapp collisions between electrons and phonons are possible (see Ref. 20).

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

215

and come back to the equilibrium conditions. The steady state is obtained when a balance between the two rates is established. The first factor is intimately connected with electrical resistivity; to be precise, with that part of resistivity which is due to electron-phonon collisions.21 If, by some means, the electron-phonon interaction were turned off, the resistivity due to phonons would go to zero and so would the first factor. On the other hand, if the resistance is increased by, say, introducing a magnetic field, the first factor, and therefore np, would follow this increase. Regarding the second factor, it is generally assumed that all processes tending to restore phonon equilibrium can be described by a relaxation time for phonons, zp. The various processes which contribute to zp are phonon-phonon collision, phonon scattering from the boundaries of the specimen, and scattering from defects, impurities, etc. In general, zp is a function of the phonon wave vector q and the lattice temperature T. From the above arguments, it is clear that we can write the Peltier coefficient, and therefore the thermoelectric power, due to phonon drag in terms of the electrical resistivity p and the phonon relaxation time zp. Herring has derived the required relation” :

=

1 fneS2?,p T

-

Here f is a dimensionless number lying between 0 and 1, which measures the fraction of all electron collisions in which a phonon is involved; p is the total resistivity (fp is roughly the resistivity due to electron-phonon collisions); and S is the sound velocity. The bar on zp and S denotes that we take zp and S corresponding to a certain average wave vector of all phonons that take part in phonon drag. The last line of Eq. (8) is obtained by putting p = l/o = m/ne2fe.If a relaxation time approximation can be made for electron scattering, f is given by’

where z, is the total relaxation time of electrons, zep is the electron relaxation time due to collisions with acoustic phonons, and hK is the crystal momentum of the electron. The average is taken over the steady state distribution for electrons.

’’ C. Herring, Phys. Rev. 96, 1163 (1954).

216

S. M. PURI AND T.

H. GEBALLE

Herring's arguments were originally given explicitly for the case where there is no magnetic field present. We shall follow his arguments assuming that a magnetic field of arbitrary strength is present, and show that the above relation is valid with only slight modifications necessary for interpretation. The advantage of the above approach is that, if the above relation holds as it stands or with slight modifications, we can use the results of the quantum theory of magnetoresistance to study the phonon drag Q,. Consider the transport phenomena in a solid in which the electric field E is applied in the x-direction and a strong magnetic field B in the z-direction, so that the quantum region conditions hold. The current is along a direction in the x-y plane. The y-component of the current is entirely determined by the electric and the magnetic fields; to a first approximation, it is independent of scattering.22 This component is non-ohmic in the sense that there is no energy dissipation accompanying it. In the absence of scattering, there is no current flow in the x-directionz2; the current flow in the x-direction is caused by scattering. To the lowest order in scattering, the different scattering processes do not interfere with each other, and J , is simply the sum of the currents caused by each type of scattering acting independently, i.e., J,

=

+

JXp J,'

+ -..,

(10)

where JXpand J,' are the currents caused by scattering on phonons and impurities, etc. The net current has a magnitude (5,' + J:)'I2 and makes with the y-axis. To get the Peltier an angle of cos-'[Jy(J,2 + Jy2)-1/z] coefficient, we want to calculate the component of heat flux in the direction of the net current. For this we use the momentum balance arguments of Herring. The rate of change of electron momentum is equal to the Lorentz force : dt

+

= .(E

q.

In the geometry we have considered, uy is of just the right magnitude and direction to make h dK,/dt = 0. This is what we said earlier: The electron motion in the y-direction does not dissipate momentum. The electric current in the x-direction is accompanied by dissipation of momentum in the y-direction given per electron by dKY - vxB h---e-dt C 22

1 J,B n c

A. H. Kahn and H. P. R. Frederikse, Solid State Phys. 9, 257 (1959).

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

217

The dissipated momentum is delivered to the scattered phonon, impurities, etc. Equation (10) tells us that the total momentum dissipation due to scattering on phonons alone equals JxPB/C. If the crystal momentum is conserved in electron-phonon collisions, as it is in the case of a nondegenerate electron assembly, the whole of the dissipated momentum is fed into the phonon system. The phonon distribution reaches a steady state by losing momentum due to collisions with other phonons on the boundaries of the sample and on impurities. Assuming that these later processes can be described by a relaxation time Zp,we have the steady state equation for phonons : relaxation

which gives

or

- JXPE = neTp-, JY

where P, is the net momentum of phonons in the y-direction, and we have set J, = neEc/B. This directed flow of phonons causes an energy flux F, in y-direction given by F, =

1N,$w,U,,

(1 5 )

4

where U, is the y-component of the group velocity of the phonons. Also, for long wavelength phonons we can make the acoustic approximation 0,=

sq,

(16)

Equation (17) is not quite correct in elastically anisotropic crystals, but it is a good approximation if a suitable average sound velocity is used. Therefore, we have F, = C S2Nqhq, (18) 4

=

SZP,

=

EJxp neS2Tp-. JY

218

S. M . PURI AND T. H. GEBALLE

The component of the energy flux in the direction of net current F,, is given by

The Peltier coefficient n is simply Fil/JJI.By substituting for J X p E / ( J Y Z+ J x 2 ) = p:, where p z x is the electrical resistivity due to phonons in the transverse field, we get np = neS2fpp:, .

(20)

Using the Kelvin relation, we obtain Eq. (8) for Qp, except that fp is replaced by p:,. If the magnetic field is in the classical region, pEx of Eq. (20) is replaced byfp,,. Here p x x is the resistivity due to all scattering mechanisms [which in the classical high field region cannot be separated as in Eq. (lo)], and the factorfgiven by Eq. (9) is introduced to take account of the fact that we are concerned with only those collisions in which a phonon is involved. The effect of the magnetic field is contained in factorsf, Zp, and pxx, but the major effect is through the change of resistivity. The case of longitudinal fields is simpler in principle because when the fields are parallel the transport problem, even in the quantum region, can be set up in terms of a Boltzmann-type e q ~ a t i o n , ' and ~ the formal structure of Eq. (8) remains the same with p replaced by p z z . The only difference from the classical calculation is that both the effect of the magnetic field on the electron transition probability and the density of states must be taken into account in calculating 7,. The arguments leading to Eq. (20) are not rigorous, because in the quantum region the effect of the magnetic field is not completely given by the addition of the Lorentz force term [Eq. (ll)]. However, one can argue intuitively that, for a given electric field, the effect of the magnetic field on electron-phonon collision is contained entirely in its effect on the electrical resistivity. Therefore, the factor pxx (or pzz)is kept intact in the final results. We remedy this objection in the next part, where we derive the more quantitative formulas that we need. For the moment, we assume that the final results are valid and discuss separately the case of transverse and longitudinal fields in the extreme quantum region, i.e., when all electrons occupy the lowest Landau level. a . Transverse Fields

If JYz s J x 2 , as is true for the weak scattering case, then p$ is independent of any scattering other than of that on acoustic phonons, in particular, 23

P. N. Argyres and E. N. Adam, Phys. Rev. 104, 900 (1956).

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

219

of the ever present impurity scattering. If, in addition, phonon scattering on impurities is also negligible, Q, is totally unaffected by the presence of impurities. The independence of Q , of impurity scattering is obtained only in the transverse fields in the quantum region; in the classical region or even in the quantum region in longitudinal fields, this simplification is not realized. The independence is of considerable practical importance in analyzing the experimental results because it is difficult to calculate f precisely. In some crystals, the value off may be so small that Q, cannot be measured easily in the zero field or classical region. A sufficiently strong magnetic field applied in the transverse direction may then make available the information contained in Q,. We obtain next the field and temperature dependence of Q,, which is contained in pBx and Z., The field dependence of transverse magnetoresistance in the quantum limit has been calculated by several authors. Of interest to us are the results for electron scattering on acoustic phonons by deformation potential in the high temperature limit, i.e., kT >> ha,, where oq corresponds to a typical phonon taking part in the scattering of the electron. In this case pBX is given by5(”) pBx

N

pT-”2.

(21)

It may be remarked that the “high” temperature limit approximation is valid for all temperatures higher than about 1°K. To determine the field and the temperature dependence of Q, completely, we must know how Zp varies with the magnetic field. Phonons that contribute most to electron-phonon scattering are those that have wavelengths of the order of a certain characteristic length of the electron. In the classical region (hw < kT), this characteristic length is the wavelength of the interacting electron. But in the quantum region (hw >> kT), the wavelength of an electron is not a defined quantity for motion perpendicular to the magnetic field, and the characteristic length is, instead, the size of the cyclotron orbit, 2r. This size can be determined from the following argument : r

=

vJw,

where u I is the component of velocity normal to B and in general is given by =

(Z + $)hw,

where I is an average of the magnetic quantum number 1 (see Part 111) of the electron levels that are occupied. In the extreme quantum region, all electrons are at the lowest Landau level 1 = 0, therefore, we have

41 = 1/r w (eB/hc)’”.

(22)

220

S. M. PURI AND T. H. GEBALLE

From this we find that, in the extreme quantum limit, q1 of the interacting phonons increases as B'12. To translate the q dependence of zp into the magnetic field dependence, we replace q by ql. This is reasonable in the extreme quantum limit because, 4

=

4 1(1 + qz2/4L 2 P 2 .

(23a)

Using Eq. (22) and taking q, for an average thermal energy electron in the z-direction, this reduces to

which means that in this case the wave vectors of phonons interacting with electrons lie in a short circular cylinder of height -(mk7'/h2)'/2 and radius -(eB/hc)112. For phonons of small q in the acoustic branch, the single mode relaxation time due to phonon-phonon interaction can be written asz4 zp(q) cc

[q2-T+s+V

I-',

which, combined with Eq. (22), gives 7,

-

I-'.

[ ~ ( 2 - ~ ) / 2 ~ 3 + s + ~

(24a)

According to Herring,24 for elastically anisotropic crystals of class Oh, s = 0 for longitudinal phonons and s = 1 for transverse phonons; v] = 0 provided the temperature is much less than the Debye temperature and v] approaches a value of -2.0 as the temperature approaches the Debye temperature. For the high purity semiconductors under discussion, there will be a region of temperature where tp is determined by phonon-phonon scattering, and the above formula holds. At lower temperatures, the relaxation time due to phonon-phonon scattering becomes very long, and the effective mean free path of phonon-drag phonons in samples of size ordinarily used in the experiment is limited by scattering from the boundaries of the specimen. At sufficiently low temperatures boundary scattering of phonons dominates, and the relaxation time is independent of q or 7: In the boundary scattering region of temperatures, we have zp = constant,

(24b)

where the constant is related to the size and the shape of the sample cross section. The expected field and temperature dependence of Q , can be obtained by using Eqs. (21) and (24) in Eq. (20). The results are listed in Table I. 24

C. Herring, Phys. Rev. 95, 954 (1954).

TABLE I. THE MAGNETIC FIELDAND

Zero field

THE

TEMPERATURE DEPENDENCE OF AQ,

I N THE

EXTREME QUANTUM REGION"

Extreme quantum limit Transverse fields Longitudinal fields

Phonon branch Assumptions

Remarks

Q,(o) Acoustic, any

Longitudinal acoustic

Transverse acoustic In general

T,(T)= const. T-"' independent of q and T

A T,(T) = 2 T-'"

q2T3

T,(T) =

A

T-4

AQpb ~ 2 ~ - 3 / 2

AQ,lQ,(o) B ~ T - ~

AQP BT-

112

BT-'

- A 4" T'

~-(r+m/2-1/2)

2 r,(T) = const. is valid for boundary scattering of phonons. For samples with crosssectional dimensions 1 mm, the assumption is valid for T < 15°K.

-

BT-Q/Z

BT-I

g 0 ~ - 7 / 2

B"To

~ 3 / 2 ~ - 1 1 / 2

83/2~-3/2

~ 1 / 2 ~ - 9 / 2

~ 1 ' 2 1~ 0 -

4T4 T,(~)

?

AQdQ,(o)

~(2-"/2)

B(2 - m / 2 )

g(l-"/z'

B"

T -(r + 3/21

T-(z-m/Zl

T-('+l/2)

T-"-~'~)

-n1/2)

(1) The form of T,(T)chosen is that predicted by Herring for the case of infinitesimal q and T + Debye temperature. The form is expected to be valid in nondegenerate semiconductors of high purity for T > 30°K. (2) Transverse phonons can be coupled to electrons by deformation potential only if electron energy surfaces are nonspherical.

a For two orientations of the fields: transverse fields (column 4) and longitudinal fields (column 6). Also given are the functional dependence of the quantity AQ,/Q,(O) in the two cases (columns 5 and 7, respectively); the temperature dependence of Q,(O) (column 3) is the ideal value given in Ref. 15. It is assumed that in each case the electron coupling to the acoustic phonon is described by the deformation potential theory of Bardeen and Shockley. The contributions of longitudinal and transverse branches of the phonon spectrum are listed separately. In a real solid, the two contributions are added in proportion to their coupling constant to the electrons and the factors A, and A, of column 2. Note. There will be no effect of impurity scattering in the extreme quantum limit in the case of transverse fields. In the case of longitudinal fields, slight amounts of impurity scattering decrease the magnitude of Q, but the field dependence is not changed significantly. For large amounts of impurity scattering, both the magnitude as well as B and Tdependences are altered. See text in Section 5 for the reasons why AQ,, the increase in the phonon-drag thermoelectric power above its zero field value, is used rather than Q,(B) in columns 4 and 6.

H

p

% 0

3

$

5=!

0

5

'

3

J

8

3 C

3 z

8

E; Z

h)

222

S. M. PURI AND T. H . GEBALLE

b. Longitudinal Fields In Ref. 23 it is shown that a relaxation time approximation can be made for elastic scattering. Consider the simple case, also the most relevant in practical situations, in which the electrons are scattered by acoustic phonons (deformation-potential type) and by ionized impurities. The respective relaxation times ,z, and ze, have been calculated for a spherical energy surface case,23

where A , and A , are constants independent of the magnetic field pr the energy of the electron and E, is the energy due to motion parallel to the magnetic field. The total relaxation time z, is given by

The steady state distribution function is given by23

where fo is the Fermi distribution function. Using Eq. (9), expressions for f and pzzcan be obtained in a straightforward manner. The average relaxation time of phonons contributing to the phonon drag 7, is obtained as in the case of transverse fields, from Eq. (24). The discussion leading to Eq. (24) is valid for the longitudinal fields also. The complete magnetic field and temperature dependence of Q , in longitudinal fields is obtained from Eq. (8) where pzz replaces p. Rather unwieldly expressions are obtained, which cannot be simplified and must be handled by numerical integration. If the scattering by ionized impurities can be neglected,24af approaches unity as expected, and the field dependence of pz, is given by23

-

p,, BT”2. (28) The field and temperature dependence of Q , in longitudinal fields can be found by the use of Eqs. (8), (24), and (28). The results are listed in Table I. 24“The scattering rate due to phonons increases more rapidly with the magnetic field due to the factor hw in Eq. (25a) as compared to scattering on ionized impurities. So if impurity scattering is small in zero field, one can neglect it in sufficiently high magnetic fields.

7.

223

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

We notice that the phonon drag Q in longitudinal fields is quite sensitive to the presence of electron scattering other than by acoustic phonons through the factor f whereas, in transverse fields, Q, depends only on scattering by acoustic phonons. The assumption has been made that the presence of impurities does not alter the phonon relaxation times. 5. EXPERIMENTAL RESULTSON TYPE GERMANIUM In this section, we shall describe some results of measurements on magnetoresistance and the thermoelectric power of n-type Ge in the quantum region. The purpose is to verify the functional dependence predicted by the theory and to track down experimentally the reasons for discrepancy in the case of transverse magnetoresistance. Before we discuss the experimental results, we shall briefly summarize the predictions of the theory regarding the qualitative behavior of the magnetoresistance and the change in thermoelectric power. In the classical region, the magnetoresistance, as well as the electronic and the phonon-drag part of thermoelectric power, becomes independent of the magnetic field. The saturation value depends slightly on the scattering mechanism in the case of p and Q , in transverse as well as longitudinal fields, and on Q , in longitudinal fields only. In transverse fields, Q, does not depend on electron scattering. In the quantum region, the magnetoresistance, in general, depends on the magnetic field, the functional dependence being determined by the orientation of the field and by the dominant scattering r n e c h a n i ~ m . ~ ' " ) ~ ~ ' ' ' ~ ~ ~ Again the electronic part of Q is independent of electron scattering in transverse fields and depends slightly on electron scattering in longitudinal fields. The increase of Q, above the classical saturation value is determined by Eq. (7). The phonon-drag component, on the other hand, is approximately related to magnetoresistance by Eq. (8). In the quantum region, theory obtains expressions for the total resistivity p ( B ) and the total thermoelectric power Q,(B) in the limit hw/kT+ 1. When extrapolated to zero field, these expressions give zero values for p and Q,, obviously because of the approximations made in the calculations. It is not quite obvious how the experimental results in the practical situation where hw/kT 10, should be analyzed. However, the experimental results for n-type Ge suggest that it is more appropriate to regard the theoretical expressions as if they were obtained for the change in the respective quantities ( A p and AQ,) rather than their total value. It is clear that since p and Q, increase very rapidly with the field in the quantum region, this difference will become very small in the limiting case. The analysis of the results for n-type InSb

-

25

E. N. A d a m and R. W. Keyes, Progr. Semicond. 6, 85 (1962).

224

S . M . PURI AND T. H . GEBALLE

will not be significantly affected by these uncertainties since Q,(O) is negligible in this case, anyway. The functional dependence of AQ, is given in Table I. In particular, at low temperatures, transverse AQ, varies as B 2 T - 3 / 2and, , at higher temperatures, the exponent of B lies between 1 and $, while that of ( 1 / T ) is between 4.5 and 5.5. In longitudinal fields, the variation of AQ, is more complicated because the presence of impurity scattering directly affects it. However, if impurity scattering can be neglected. the longitudinal AQ, varies as B T - ” 2 at low temperatures. At higher temperatures, the exponent of B is zero or 1/2, and that of (l/T)lies between 3.5 and 4.5. The basis for the functional dependence of AQ, is that the magnetoresistance is correctly given by the Adams-Holstein formula^,^'") namely, B 2 T - ’ / 2for the transverse field case and BT”2 for longitudinal fields, and that the phonon relaxation times are given by Herring’s formula, Eq. (23). The experimental measurements on the longitudinal magnetoresistance of n-type InSb26-28and n-type Ge8+29demonstrated qualitatively that the saturation predicted by the classical transport theories is violated in the quantum region. However, the observed functional dependence of p on B and T does not agree with the predictions of the theory in the case of InSb. Similar difficulties are encountered in the transverse magnetoresistance of n-type Ge, in addition to the fact that transverse magnetoresistance does not saturate even when the quantum effects are expected to be unimportant.30s31 It is now believed that the experimental results on high field magnetoresistance are very sensitive to gross inhomogeneities, small fluctuations in charge carrier density, and surface conduction in a specimen of finite size.6(a1,6(b),6(c)*2s These results, therefore, cannot be compared to the theory developed for an ideally homogeneous medium. Herring6(”’has also predicted, on a theoretical basis, that the thermoelectric power is not sensitive to the spurious effects cited above. It is our purpose in this section to demonstrate experimentally that his prediction is correct and that the thermoelectric measurements are not affected to any significant degree by the inhomogeneities normally present in good material. This is done by R. J. Sladek, J . Phys. C h e m Solids 16, l(1960.) 27

’*

J. C. Haslett and W. F. Love, J . Phys. Chem. Solids 8, 518 (1959).

R. T. Bate, R. K. Willardson, and A. C. Beer, J . Phys. Chem. Solids 9, 119 (1959). W. F. Love and W. F. Wei, Phys. Rev. 123, 67 (1961). 30 T. J. Diesel and W. F. Love, Phys. Rev. 124, 666 (1961). 3 1 (a) I. G. Fakidov and E. A. Zavadskii, Zh. Eksperim. i Teor. Fiz. 34, 1036 (1958) [English Transl.: Soviet Phys. J E T P 7 , 716 (1958)l; E. A. Zavadskii and 1. G. Fakidov, Fiz. Metal. i

29

Metalloved. 10, 495 (1960); 11, 145 (1961) [English Transl.: Phys. Metals Metallog. ( U S S R ) 10, No. 3, 180 (1960); 11, No. 1, 141 (1961)l. (b) V. R. Karasik, Dokl. Akad. Nauk SSSR 130, 521 (1960) [English Transl.: Soviet Phys. Doklady 5, 100 (1960)l. (c) Y. Kanai, J . Phys. Soc. Japan 10, 718 (1955).

7. THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

225

studying the effect on the magnetoresistance and the thermomagnetic measurements of changing the relative amount of surface conduction or by comparing the measurements on one sample which is more inhomogeneous than another. The great sensitivity of magnetoresistance to inhomogeneities is due to the distortion of current lines produced by a magnetic field in an inhomogeneous sample. The distortion of current lines is necessary to produce a uniform transverse Hall field in adjoining local regions of different carrier As the magnetic field increases, the Hall field increases linearly, and the distortion increases too. The reason that Q is not as sensitive to inhomogeneities is that, first, the transverse field in the presence of a temperature gradient is not as sensitive to electron density as the Hall field, and, second and more important, the transverse field is small, in fact approaches zero, in the classical high field region. These conjectures about the respective sensitivity of the two types of measurements to inhomogenieties are confirmed by the experimental results8@’of Figs. 2 and 3. 4.5

40

35

30

25

A f -

Po

2.0

15

10

05

0 IN KG

FIG. 2. Effect of gradients in electron density on transverse Ap. The solid curves show measurements on one side of the bridge shaped sample of n-type Ge, and the dotted curves show measurements on the other side. The magnetic field pushes the current lines to one side of the sample (which measures larger Ap). Reversing the field pushes the current lines to the other side.

226

S. M. PURI AND T. H. GEBALLE

which show the magnetoresistance and the thermoelectric power, respectively, of a sample of n-type Ge. Measurements of the Hall effect at two different points of this sample 0.56 cm apart showed the electron density to differ by about 20% at these two points, presumably varying continuously along the length of the sample. In the figures are plotted the relative change in the potential drops,

versus the magnetic field B. The terms V ( B ) and V(0) are the potential drops across two probes in a magnetic field 3 and in zero field, respectively. In Fig. 2, the potential drops are due to a constant current in the sample (magnetoresistance), and in Fig. 3 they are due to a constant temperature gradient (thermoelectric power). Measurements on two different sets of potential probes, ab and a'b', situated on the opposite sides of the brjdgeshaped sample, have been plotted separately for two directions of the

TRANSVERSE TEMP 35'K

AQ I

I

60

FIG.3. Effect of gradients in electron density on transverse AQ. Measurements are made on the same n-type Ge sample as in Fig. 2. In contrast to Ap case, no distortion of the equipotential lines is produced in this case because of zero current conditions of measurements (Curve 1 has been displaced for clarity.)

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

227

magnetic field transverse to the primary gradient. Referring to magnetoresistance results in Fig. 2, it is seen that for one direction of magnetic field, e.g., + B , voltage drop on one side of the sample is much greater than the voltage drop on the other side. We ascribe this to the concentration of current flow lines on the side of the sample registering the higher potential drop, caused by the magnetic field in an inhomogeneous sample. Reversing the direction of B moves the current lines to the other side of the sample, as is inferred from the figure. An analysis of this kind of distortion of current lines in an inhomogeneous sample in the presence of a magnetic field is given by Bate and Beer.6(b)Turning our attention to the measurements of Q on the same sample, shown in Fig. 3, we find that the thermoelectric power measurements are not affected to any significant extent by inhomogeneities even as large as in this sample. (For clarity, we have shifted the two curves vertically.) The two sets of probes measure the same voltage. The signal changes slightly on reversing the magnetic field, but the effect is not serious. Experiments have also been carried out to investigate the effect of surface conduction on the magnetoresistance and thermoelectric power measurements. In a sample of finite size, the conductivity of the surface layer may not be the same as that of the bulk crystal. The mechanism of conduction in the surface layer is different from that in the crystal; it is probably a lowmobility conduction process in the surface layer where the lattice structure is damaged during cutting and other processes of sample preparation. If the conduction mechanisms are different, the magnetoresistance properties

1

B

IN KG

FIG. 4. Effect of surface conduction on transverse magnetoresistance. Measurements on the original sample of n-type Ge are repeated after sandblasting its surface. which introduces appreciable surface conduction, thereby increasing the magnetoresistance.

228

S. M. PURI AND T. H. GEBALLE

are not expected to be the same. Because of this, the proportions of the current carried by the surface and by the bulk crystal change with the magnetic field, which introduces serious distortion of current lines and hence a spuriously large magnetoresistance. Since there is no current flow in thermoelectric measurements, the effects are not expected to be as serious in this case. Experimental measurements shown in Figs. 4 and 5 again confirm this. The surface conduction of a sample is enhanced by sand blasting the surface of a chemically etched crystal. A comparison of measurements on a sand-blasted sample with those on an etched sample

0 {kG) FIG.5. Effect of surface conduction on transverse A Q of n-type Ge. Sandblasting the sample surface does not alter appreciably the AQ effect. The absolute magnitude of AQ differs in the two cases because of difference in temperature.

show that surface conduction indeed increases magnetoresistance considerably. When the difference in the mean temperature of the sample for measurements shown in Fig. 5 is taken into account, it is found that the thermoelectric power is not changed by sand blasting the sample. While the inhomogeneities and surface conduction do not explain entirely the discrepancy between theory and experimental results on magnetoresistance,8@)the serious effect that they produce on such measurements is obvious. On the other hand, it is gratifying that AQ measurements are free of such spurious effects. One is then prompted to look at the functional dependence of Q and compare it with the theoretical results. In n-type Ge, phonon drag makes a large contribution to Q below liquid nitrogen temperature^.^^ The change*@)in Q of a high purity sample of

’’H. P. R. Frederikse, Phys. Rev. 91, 491 (1953);

T. H. Geballe, ibid 92, 857 (1953)

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

229

n-type Ge is shown in Fig. 6 for transverse magnetic fields up to 88 kG. The magnetic field in this case is directed along the [loo]-type axis. Due to the-many-valley structure of the conduction band of Ge, the cyclotron effective mass has four values, corresponding to four valleys for a general field direction, but for B along [loo], all the four values coincide because all the valleys (major axes of the energy ellipsoids are along [llll-type axis) are symmetrically placed with respect to the field direction. For this reason, the theory applicable to the isotropic electron mass can be used at least

FIG. 6. Dependence of transverse AQ on magnetic field for n-type Ge sample. Magnetic field is directed along [lo01 axis. The mean temperature of the sample (in degrees Kelvin) is marked on each curve. The sample has about 8 x 101Z/cm3excess donors.

230

S. M. PURI AND T. H . GEBALLE

qualitatively, and the results of Table I are valid for comparison. As seen in Fig. 6, which is typical of results on high purity samples, the thermoelectric power attains complete saturation and starts increasing again for hw/kT > 0.6 as the quantum limit is approached. The calculated contribution of the electronic part to AQ is less than 0.1 % of the observed value in the quantum region and can be neglected. The observed AQ is then almost entirely due to the change in the phonon-drag contribution. At 15"K, AQ/Q,(O) varies approximately as B2. The exponent of B decreases with increasing temperature, having the value 1.4 at 85°K. The value two for the exponent at 15°K is what one expects theoretically for the boundary scattering of phonons (Table I). At 85°K the phonon relaxation is mostly due to phonon-phonon scattering. It has been generally assumed that the long wavelength phonons in the transverse branches have a much shorter relaxation time than the longitudinal phonons of the same frequency. Therefore, one might have expected that the exponent of B at 85°K would be closer to one which is the value for contribution of longitudinal phonons than to 1.5 which holds for transverse phonons. The experimental value of 1.4 seems to indicate that the contribution of transverse phonons to Q , dominates over the contribution of longitudinal phonons, implying thereby a much longer relaxation time for the former. The unexpected importance of transverse mode contribution is also evident in the classical high field thermomagnetic data.33 This point should be investigated further, preferably in a case where only one branch of the phonon spectrum contributes to Q,. In Fig. 7 we have plotted the value of AQ/Q,(O) vs. temperature for fixed magnetic field in the above mentioned case of n-type Ge. The experimental data for 88 kG approximately follow T 3 I 2 dependence. The dependence for AQ,/Q,(O) is given in Table I for various ideal situations and is seen to vary between T - ' and T - 2 .The agreement is as good as one can expect considering that the observed Q,(O) itself does not follow the ideal theoretical dependence." The lower curve of Fig. 7 shows AQ/Q,(O) for a field of 10 kG. which is well within the classical field region. The transport properties in a longitudinal field show quantum effects only when ho 2 1SkT. In the temperature region where phonon-phonon scattering is the important mechanism for the relaxation of phonon-drag phonons, the increase in electron-phonon scattering is counterbalanced to a large extent by the decrease in the average relaxation time of phonons. The increase in Q, is therefore much slower than the increase in the magnetoresistance ratio. The existing measurements of Q, in longitudinal fields for n-type Ge do not establish accurately the field dependence. At lower temperatures, where the phonons are scattered by the boundaries, the 33

C. Herring, T. H. Geballe, and J. E. Kunzler, Phys. Rev. 111, 36 (1958).

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

231

5 4

3

2

1.5

AQ

Qp

1.0

0.9 0.8 0.7

0.6 0.5 0.4

0.3

0.2 0.15

10

TEMPERATURE IN DEGREES KELVIN

FIG. 7. Temperature dependence of transverse AQ/Q,(O) at constant magnetic field for n-type Ge sample. The curve at 10 kG is representative of the classical saturation region and the curve at 88 kG shows temperature dependence in the quantum region. Magnetic field is along [lo01 axis.

increase in Q , is sharper and varies linearly with the magnetic field as expected (Table I). Some typical curves for this case are shown in Fig. 8. The data are consistent with theoretically predicted behavior, although not sufficiently extensive to determine the field dependence at higher temperatures. These results on n-type Ge presented above can be considered to imply the following. First, that the effects of inhomogeneities, surface conduction, etc., are not significant in the case of thermoelectric power, and a quantitative comparison with the theory worked for infinite, homogeneous media is

232

S. M . PURI AND T. H. GEBALLE

LONGITUDINAL B 100 V T 100

-

AQ QP

15

20

30 40 [KGI

50 60 70 809( 00

FIG.8. Longitudinal AQ for n-type Ge sample. The magnetic field and VTare directed along the same [ 1001 axis. The mean temperature of the sample is marked on each curve.

justified. Second, that, at least for electron scattering on acoustic phonons via deformation potential, calculations based on the theory of Adams and Holstein give correct functional dependence on the fields.

III. Compound Semiconductors of In-V Group 6. EXPERIMENTS ON +TYPE InSb The results on Ge point to the success and the usefulness of the study of thermomagnetic effects in the quantum region. If the experiments are performed to measure the phonon-drag effect, one can get information not only about the electron-phonon scattering but also about the relaxation mechanisms of the low energy acoustic p h o n o n ~ .On ~ ~ the other hand, if the phonon-drag component of Q is very small, an investigation of the change in the thermoelectric power with magnetic field is not very interesting from the point of view of studying scattering mechanisms, because the electronic part of Q in high fields is rather insensitive to electron scattering. Our first task, therefore, is to estimate from experiments as precise a value of Q, as possible. This is usually done by subtracting from the measured value of Q the theoretically calculated electronic component. For this reason, it is important to know the behavior of Q, in the quantum region. There has been some controversy, as discussed in Section 1, about the expected variation of Q, in transverse fields. We shall not enter into a detailed discussion of this point, but refer the reader to the literat ~ r e . ~ ( ~ ) * ’The ( ~ )controversy ,~* now seems to be resolved. On our part, we

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

233

have adopted for Q, the formula

![ );(

Q,(B) - Q,(Class. Sat) = - In -

no + flu + In coth nB 2kT

- I n k sinh(&)]

-

;].

Equation (29) holds for a parabolic band with no spin splitting. In the case of high purity n-type InSb at low temperatures, the correction due to the nonparabolicity of the band is small while the spin splitting amounts to about 60 pV/deg. These effects have been considered explicitly in the

T

( O K )

FIG.9. Thermoelectric power of an n-type InSb sample (584) with carrier density 4 x IOl4/cm3.The solid line represents a smooth curve through the experimental points. The other two curves represent the theoretical values of the electronic contribution for different assumptions about the electron scattering: (i) dashed curve for scattering on ionized impurities, (ii) dotdash curve for scattering on acoustic phonons.

analy~is.~’ The information contained in Q, can be unscrambled piece by piece by changing the experimental conditions. It will be seen that this can be done with a fair degree of success. The magnitude of Q, in n-type InSb is very small in zero field; the maximum value is less than 50 pV/deg at 20°K. Figure 9 shows the results of measurements on a sample with an electron density of about 4.0 x lOI4/cm3. In this figure we have also shown the value of Q, calculated using Eq. (5). The contribution from the second term of Eq. (3, which is

234

S. M . PURI AND T. H . GEBALLE

sensitive to electron scattering, is relatively large in the case of n-type InSb because the small effective mass of the electrons makes the first term small. In the figure we have plotted the computed values, using the relaxation time approximation for solving transport equation (i) scattering on ionized impurities, z cc E~~~ ; (ii) scattering by acoustic phonons, z cc E - 'Iz.A value of 0.013 x 9.1 _+ gm is used for the electron mass. The experimental values agree fairly well with the first curve. At the upper end of the temperature scale, a small admixture of acoustic scattering improves the agreement between the calculated and the measured values. At temperatures below 10"K, as the sample becomes degenerate, electron screening of the charged impurity potential, which has been neglected, would, if included, decrease the calculated value. The difference in the observed and calculated values between 10" and 3WK, which amounts to less than 50pV/deg, is larger than the estimated uncertainty of the measurements. This implies that a phonon-drag contribution of less than 50,uV/deg exists between lo" and 30°K and is negligible outside this temperature range. This result is not totally unexpected, since Qp can be estimated by comparison with Ge by a correspondence argument. The observed electron mobility in InSb has a value of -2 x lo5 cm2/V-sec at 20°K. Extrapolating Ehrenreich's results,34 the mobility due to scattering on acoustic phonons alone is expected to have a value of -lo8 units at 20°K. This gives, as a crude estimate, the value for to be used in Eq. (8). At the same temperature, Q, for 2 x 10,000pV/deg, with f sz 1 and an electron n-type Ge has a value of mobility of -4 x lo5 units.j3 Upon using these data and Eq. (8), one obtains, for the ratio of Q, in InSb and in n-type Ge

-

or a value of 40 ,uV/deg for Q, in InSb. The estimate is much too crude to take the exact agreement with experiment seriously ; nevertheless, the order of magnitude should be correct. The value of Q is observed to decrease with the magnetic field in the classical region. This further confirms the above observation, because Q, cannot decrease with magnetic field while Q, can, depending upon the predominant mode of electron scattering. One reason for the small value of Q, in n-type InSb is the predominance at low temperatures of ionized impurity scattering, even in the purest specimens, which reduces the value off: In transverse magnetic fields in the quantum region, Q , does not depend onJ; and we may expect to see a sizable amount of Q,. It is more convenient to measure the change in Q from the saturation value in the classical high 34

H. Ehrenreich, J. Phys. Chem. Solids 2, 131 (1957).

7 . THERMOMAGNETIC EFFECTS IN THE QUANTUM

REGION

235

FIG. 10. Some typical values of A p , the increase in the thermoelectric power in the quantum region over its limiting value in classical high fields. The calculated values of the electronic part of AQec are shown by the dashed curves. The solid curve shows the phonon-drag part of AQpc obtained by subtracting the electronic component from the total measured values.

fields. We define AQc = Q(B) - Qsaluration, where Q(B) is the measured value of Q in field B and Qutlturation is the value of Q in the plateau in the classical region when Q is plotted against B. This plateau is found to be quite well defined. The electronic component of AQc, which we denote by AQec, is given directly by Eq. (29) modified for nonparabolicity and the spin splitting following Ref. 42. The experimental values of AQ,"are plotted against the magnetic field B in Fig. 10. In this figure we have also plotted the calculated value of AQ,". The difference between the two curves is the phonon-drag component of AQeS which increases with the magnetic field. Below about 20"K, this difference does increase quadratically with the

236

S. M. PURI AND T . H . GEBALLE

magnetic field for not too large values of the field. At still higher fields the behavior is complicated due to effect of the band structure. The presence of a large phonon-drag component can be confirmed by making measurements at low temperatures on samples of different crosssectional areas.j5 At low enough temperatures, around 20"K, phononphonon scattering is almost completely quenched, and for low-energy phonons, TP of Eq. (20) is determined effectively by the size of the specimen.

T (OK) FIG. 1 1 . Ratio of the thermoelectric power of two samples of n-InSb of different crosssectional area for a fixed field of 80 kG plotted as a function of temperature.

In fact, around 10°K or so, Zp, and therefore Qp, is directly proportional to the size of the specimen; Q, is not sensitive to the size of the sample. We have made measurements1o on two samples cut from the adjoining parts of a crystal, but of different cross-sectional area. The AQc ratio for the two samples in a field of 80 kG is plotted as a function of temperature in Fig. 11. The great sensitivity of Q to the size of specimen at liquid hydrogen temperatures indicates the presence of a large contribution from phonon drag, although the AQc ratio for the two samples is not as large as is to be expected from sample sizes for complete boundary scattering of phonons. The discrepancy may be due to the experimental difficulties, as discussed in Ref. 10, or, as has been pointed out by HerringI5 for the case of Ge, there may be an additional scattering mechanism present for phonons which gives a relaxation length comparable to the size of the specimen.

'' T. H. Geballe and G. W. Hull, Conf. phys. basses temp. p. 460. Inst. Intern. du Froid, Paris, 1955. See also Ref. 15.

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

237

More careful experiments than the one described should be done to clarify the uncertainties and obtain quantitative data. However, the size dependence experiment in its present state is a convincing demonstration of the existence of a large amount of phonon drag. Before we discuss the experimental results in detail we list the problems and the type of information that we are looking for in these experiments. The first thing that we want to establish is the mode through which the electrons in InSb interact with acoustic phonons, i.e., whether the scattering is of the deformation potential type or of the piezoelectric type. Because of a lack of inversion symmetry of the zinc blende structure of the crystal, InSb is potentially piezoelectric, and therefore the polarization charge produced by phonon-induced strains can scatter electrons. Although the piezoelectric coefficient has not been determined, the experiments on the mobility of hot electrons36 have been interpreted to show that the piezoelectric scattering is much stronger than the deformation potential type scattering. There is also no direct experimental measurement of the deformation potential constant qD. E h r e n r e i ~ hestimated ~~ the value of qD from the experimentally measured37 change in the energy gap with uniaxial strain. He quotes a value of 7.2 eV on the assumption that the valence band maxima in InSb are not as sensitive to pressure as the conduction band minima. More recently, Haga and Kimura3* have analyzed the experiments on free carrier infrared absorption39 and found a value of 30 eV for qD. Phonon-drag experiments in the quantum limit give a more direct answer to these questions. In summary, we shall find that the piezoelectric scattering is negligible from 40" down to about 7"K, and qD has a value of 8.2 eV, which is closer to Ehrenreich's initial estimate. The second kind of information that we would like to get from these experiments concerns the relaxation time of low energy acoustic phonons. The behavior of these phonons is not very well understood. Early theories gave a relaxation time that was too large and led to the prediction of infinite thermal conduction at low temperatures unless some undefined cutoff was introduced. Herringz4 was the first to consider the effects of small elastic anisotropy and showed that the divergence could be removed in crystals of certain symmetry groups. He predicted the following expression for zp(q, T ) : zp(q, T )

-

At/q2T3

for longitudinal phonons

(304

(a) R. J. Sladek, Phys. Rev. 120, 1589 (1960). (b) G. D. Peskett and B. V. Rollin, Proc. Phys. SOC. (London) 82,467 (1963). 37 R. W. Keyes, Phys. Rev. 99, 490 (1955). 3 8 E. Haga and H. Kimura, J . Phys. SOC. Japan 18, 777 (1963). 39 W. G. Spitzer and H. Y. Fan, Phys. Rev. 106, 882 (1957). 36

238

S. M. PURI AND T . H. GEBALLE

-

A,/qT4

for transverse phonons.

(30b)

These expressions hold for small values of q and for temperatures much less than the Debye temperature. The coefficients A are related to the thirdorder anharmonic term in the lattice vibrational energy. The order of magnitude of the above expressions for t, have been verified experimentally.'7,21,33Th e most detailed information to date has been obtained from the phonon-drag experiments on n-type Ge in the classical region. However, in the case of Ge, the transverse as well as the longitudinal phonons contribute to phonon drag, and the information obtained from the experiments relates to a certain average of the combined behavior. It has not been possible to extract from the experiments separate information about zp for different branches of the phonon spectrum. This introduces a certain element of uncertainty in comparing the experimental results with the theory, because the relaxation times of phonons in the two branches are expected to be very different both in magnitude and in their dependence on frequency and temperature. In this respect InSb is especially interesting because it provides a system in which only the longitudinal phonons contribute to phonon drag. This is because, with a spherical conduction band in InSb, transverse phonons can interact with electrons only through the piezoelectric mode and not via deformation potential. We already mentioned that the piezoelectric scattering in InSb is negligible at low temperatures, and therefore transverse phonons are not disturbed from the thermal equilibrium. Thus the information about t, that we obtain from phonon drag in InSb relates only to longitudinal phonons. In the following we discuss how the above information about electronphonon and phonon-phonon scattering is extracted from the experimental results. To achieve our objective we have to refine the expression for Q , obtained in Section 4. To be precise, we shall (i) obtain the various constants of proportionality for the results of Table I, (ii) show how these formulas are changed if the electrons interact with phonons through the additional piezoelectric mode, and (iii) show how the average of zp used in the formulas is obtained. Also, it is not very satisfactory to derive the results in the quantum region using the idea of a Lorentz force on electrons. We shall avoid this difficulty but otherwise retain Herring's model of phonon drag.15 We confine ourselves to the case of transverse fields. 7. QUANTITATIVE FORMULA FOR PHONONDRAG The foregoing expressions for phonon-drag thermoelectric power can be made more precise using the correct electron wave functions in the presence of a magnetic field. For this, we calculate the phonon distribution and the energy flux carried by phonons in a semiconductor under the

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

239

influence of crossed electric and magnetic fields. The quantum region condition ho % kTis supposed to be satisfied, and the temperature gradient is zero. The phonon distribution is driven by the phonon collisions with electrons that are drifting under the influence of external fields, while collisions with other phonons or boundaries of the specimens, which we call relaxation processes, tend to restore thermal equilibrium. In the steady state, we have, assuming spatial homogeneity,

which depends The right-hand side is described by a relaxation time T~(T), only on the frequency of the phonon mode q under consideration and the temperature of the lattice. In particular, it is assumed that T, is independent of the distribution of other phonons. The justification for using this single mode relaxation time has been given by Herring” and car rut her^.^' The essence of their argument is that in semiconductors a very small fraction of the phonon modes, namely, those having wavelengths much longer than the wavelength of a thermal phonon, are significantly shifted from their equilibrium distribution. The major part of the phonon spectrum around thermal energies which is most effective in scattering the phonon-drag phonons retains the equilibrium distribution. These arguments remain valid even in the presence of a magnetic field, but impose an additional condition in the quantum r e g i ~ n ~that l , ~(eB/hc)’’2 ~ 4 (kT/AS). Assuming that this condition is satisfied, we have

Here Nqo is the number of phonons of mode q in thermal equilibrium. We first consider the case of a free electron gas, neglecting all band structure effects. This model will approximate the crystal electrons in a nondegenerate and parabolic band. The case of nonparabolic band will be discussed later. The electron Hamiltonian can be written as

= H, 40

+ v,

P.Carruthers, Rev. Mod. Phys. 33, 92 (1961).

L. E. Gurevich and G. M. Nedlin, Fiz. Tverd. T e h . 3, 2779 (1961) [English Transl.: Soviet Phys.-Solid State 3, 2029 (1962)J 42 S. M.Puri, Phys. Rev. 139, A995 (1965).

41

240

S. M . PURI AND T. H . GEBALLE

where m-’ is the effective mass tensor, A is the vector potential describing the magnetic field, and V is the perturbation due to scattering by phonons, etc. We have neglected the Zeeman interaction term between the electron spin and the magnetic field. The eigenfunctions and the eigenvalues of H o are given by (32b) $l,ky,kz = exp[i(kyy + k z z ) l q , ( x - xO) >

+

+

+

(324 ( I +)ha ykz2 eEx,, (pi's are wave functions of a simple harmonic oscillator centered at xo = -(h/mo)[k, + (eE/hi%)] and y = h2/2m. We calculate the transition rate between the eigenstates of H,, which we specify by the single quantum number p for brevity: El,ky,k,

=

Here W,f‘-’ are the probabilities for an electron to be scattered from state p to p’ by emitting a phonon of mode q (+ sign) or absorbing a phonon of mode q (- sign), andf, is the steady state electron distribution. The transitions are caused by the electron-phonon interaction. Using the Born approximation, we get

where V, is the q component of the Fourier transform of that part of I/ which is due to electron-phonon interactions. In the extreme quantum limit (ho% k T ) , the expression obtained for dN,/dt is4’

where B, is a dimensionless constant given by

n is the total number of electrons and kB2 = eB/hc. In writing ( 3 3 , we have neglected terms of the order of E2 and higher. Equation (31c) can be solved by iteration using Eq. (35). The energy flux F, transported by phonons of mode q is given by

F,

=

ho,(N, - N,O)U,,

(37)

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

241

where U , is the group velocity of the phonons given by

u, = ((l/Iql)S-

(38)

In elastically anisotropic crystals, S will depend on the direction of q but not on its magnitude; variations as large as about 30% are observed. We assume S to be isotropic and correct the final results by using a suitable average value for the sound velocity. The total energy flux transported by scattered phonons is obtained by summing F, over all q and all branches of the phonon spectrum. Because the value of S for optical phonons is vanishingly small, and also because their relaxation time is very small, we can neglect their contribution and confine our attention to acoustic phonons. The only nonzero component of F is along the y-direction; other components vanish on integration over q. Considering this we can write the Peltier coefficient as4' nyy

=

(39)

(Fy/E)P,,

where p x y = - B / n e ~ . ~ 'The thermoelectric power is obtained by making use of the Kelvin relation42

k

Qy y = -e

[(J-) akT

'I2

h

q2

( k ~ ) - ' ~ SIqzl~ I h l ~ ~ ~ ~ (40)

where we have neglected terms - ( N , - N,')/N: compared to unity. These neglected terms give rise to the saturation effect, to be discussed later in this section. The form of IV,12 depends upon the mode of electron coupling to acoustic phonons, and we consider this next. a. Scattering Potential; Efect of Piezoelectric Coupling

For electron scattering by long wavelength phonons, the deformation potential theory of Bardeen and S h ~ c k l e yhas ~ ~been very successful. For isotropic energy bands, scattering is defined in terms of a single constant, called the deformation potential constant qD. In terms of qD, the Fourier transforms V, are given by

where 6 is the density of the crystal. In this approximation only the longitudinal phonons interact with the electrons. 43

J. Bardeen and W. Shockley, Phys. Rev. 80, 72 (1950).

242

S . M. PURI AND T. H . GEBALLE

For anisotropic energy surfaces, a single deformation potential constant does not suffice. Herring and V ~ g have t ~ shown ~ that if the energy bands consist of several surfaces of revolution placed symmetrically in the Brillouin zone, scattering within each band can be described by two constants. We shall not concern ourselves with this complication and consider here only the isotropic energy bands described by Eq. (41a). If the crystal is piezoelectric, a propagating acoustic phonon produces a polarization charge because of the induced strain. The electrons get scattered from this piezoelectric charge. This mode of electron-phonon coupling is in addition to the usual deformation potential type of coupling. The Fourier components for this piezoelectric mode of coupling are given by45

where qp is a constant of the dimensions of force and depends on the piezoelectric constants of the crystal. If the crystal lacks a center of symmetry, the piezoelectric constants do not, in general, vanish. Such is the case for 111-V compounds that have the zinc-blende structure. The two scattering modes given by (41a) and (41b) arise from the same source, and therefore may have a phase correlation between them. Therefore, we have, in general,

where v is the phase difference between the two scattering modes. This is the expression for V, that we shall use in analyzing the experimental results on 111-V compounds. b. Band Structure Eflects

The free electron model that we have considered can be used for electrons in a realistic crystal provided the energy band in which the electrons move is removed far enough on the energy scale from other bands. One has to, in this case, simply replace the free electron mass by the effective mass of the crystal electrons, or, in the more general case of anisotropic energy surfaces, by an effective mass tensor. This approximation is sufficiently accurate for the conduction bands of Ge and Si. But for the valence bands of these materials and for the valence as well as the conduction bands of 44 45

C. Herring and E. Vogt, Phys. Rev. 101, 944 (1956). H. J. G. Meijer and D. Polder, Physica 19, 255 (1953).

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

243

111-V compound semiconductors, the free electron model is not appropriate. This is because, due to the close proximity of several energy bands, there is a strong interaction between them, and the effect of this interaction changes with changing magnetic field. Consequently, the energy levels and the wave functions are somewhat different from those obtained for the free electrons. For example, the spacing of the Landau levels is not uniform, which gives rise to the well-known quantum effects in the cyclotron r e s o n a n ~ e . ~ ~ , ~ ’ In the extreme quantum region that we are considering, all electrons are in the lowest Landau level; we need to know the energy and the wave function of only this level. The spacing between the Landau levels affected most by the band structure effects is of no concern to us. First of all, we note that the spin-orbit interaction splits the spin degenerate Landau levels by an amount that depends upon the magnitude of the g-factor. The values of g-factor can be quite large; in n-type InSb, the spin splitting of the levels is comparable to the Landau level spacing.48 We assume that there is no scattering of electrons between the spin-split levels. We can then carry out the calculation for each state separately and add the final results, taking into account the difference in the Boltzmann occupation probability of each state. The Boltzmann factor, in the case of n-type InSb, makes the contribution of the higher of the spin states vanishingly small, and we neglect it completely. Also the curvature of each quantized level, y , which relates energy t o crystal momentum parallel to the magnetic field, is not constant as in the free electron case but changes with the energy of the quantized level and the magnitude of k,. In the quantum region most of the electrons occupy states with small k, because of the higher density of states2’ near k, = 0. This makes our calculation much less sensitive to the change in curvature with k,. Therefore, we can consider y to be a function of B but independent of k,. This is a “parabolic approximation” to a nonparabolic band. The effects considered so far are small and taken into account in a trivial manner. In addition to these, we must also consider the modification of the electron wave functions Eq. (34b), which affects the electron scattering rate. This effect arises from the interaction between the various bands which are close to each other in energy. In the case of 111-V compounds, as Kane4’ has shown, it is sufficient to consider only the interaction between the J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955); J. M. Luttinger, ibid. 102, 1030 (1956). 4’ J. C. Hensel, Proc. Intern. Conf. Phys. Semicond., Exeter, 1962 p. 281. Inst. of Phys. and Phys. SOC., London, 1962. 48 G. Bemski, Phys. Rev. Letters 4, 62 (1960). 49 E. 0. Kane, J . Phys. Chem. Solids 1, 249 (1957); also see Chapter 3 of this volume, “The k * p Method”. 46

244

S.

M. PURI AND T. H. GEBALLE

-

conduction band and the valence bands due to the k p term of the Hamiltonian. In the absence of a magnetic field, the wave functions with angular momentum $, which to a first approximation comprise the heavy hole band, do not mix with the conduction band wave functions which have angular momentum 3. In the presence of a uniform magnetic field in the z-direction, the momentum operator p is replaced by p + eB x x/c. In this case the wave functions of heavy hole bands also enter the conduction band wave function because of the nonvanishing matrix elements of x between the two wave functions. The admixture of heavy hole bands to the electron wave functions increases with the magnetic field and is greater than the contribution of the other two valence bands in the extreme quantum region. The problem of finding the electron energy states in the presence of a magnetic field has been solved independently by Yafet” and by Roth et al.,” following the formulation of Luttinger and K ~ h nUsing . ~ ~their results and making the further assumption whereby the contributions of the valence bands other than the heavy hole bands are neglected, it can be shown42 that the phonon drag thermoelectric power is given by an expression similar to Eq. (40), except that Bq of Eq. (36) is now multiplied by a factor

where ap is a small coefficient which determines the amount of admixture of the valence band wave functions to the conduction band wave functions. Also, y is now a slowly varying function of B given by

where E, is the energy gap and P is the interband matrix element defined by Kane?’ and y s and yo are the values of y in the magnetic field B and the zero field, respectively. The net effect of the corrections due to the band structure effects is that Q, varies somewhat less rapidly with the magnetic field than in the case of free electrons. The corrections are negligible for small values of the field but increase rapidly as the field is increased. c. Saturation Eflect

The results leading to Eq. (40) are obtained by replacing N , in Eq. (35) by its thermal equilibrium distribution N,’, assuming thereby that the electron-phonon scattering rate is not appreciably changed by the shift in 50 5’

Y . Yafet, Phys. Rev. 115, 1172 (1959). L. Roth, B. Lax, and S. Zwerdling, Phys. Reu. 114, 90 (1959).

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

245

the phonon distribution. This is true as long as the shift N , - N: is small compared to N , . In effect, this approximation neglects the contribution of electron-phonon scattering to the relaxation time of phonons. The magnitude of this correction, of course, depends on the density of electrons; also, in the quantum region, it depends upon the magnetic field which affects the electron-phonon scattering rate. Finally, the correction is significant only if the relaxation time of phonons due to principal processes considered in the theory, i.e., phonon-phonon scattering and phonon scattering from the walls, is sufficiently long. A first-order correction can be easily obtained by solving Eqs. (31c) and (35) by an iteration procedure. The correction to the thermoelectric power due to the finite density of electrons is given by42

8. ANALYSIS OF

THE

EXPERIMENTAL DATA

The parameters of the band structure are known in the case of n-type InSb. The only unknowns in the theory are the phonon relaxation time ~~(7') and the electron-phonon coupling constant lV,12. At sufficiently low temperatures the phonon-relaxation time is determined by the boundary scattering of phonons, the other relaxation mechanism, viz., phononphonon scattering, being relatively ineffective. In the case of boundary scattering of phonons, the relaxation time is independent of the temperature as well as the frequency of phonons. It is limited by the size of the specimen and is given by T,, = b/S, (46) where b depends upon the cross-sectional size and the length-to-width ratio of the sample. For a sample of square cross section, T,, is given correctly to within 10-20% if b is chosen to be the width of the sample. The changeover from a phonon relaxation time dominated by phonon-phonon scattering to that determined by boundary scattering is marked in the experiments by a sharp change in the temperature dependence of QP.In the case of InSb the experimental data show" that, below 12"K, Eq. (46) is a very good approximation for samples with sides 1 mm. Assuming that the phonon relaxation time is given by the known quantity of Eq. (46), the low temperature data can be analyzed to obtain information about V,.

-

--

a. Piezoelectric Versus Deformation Potential The coupling constant V, involves three unknowns: qD, q,, and v. Our first task is to obtain a rough estimate of the relative magnitudes of qD

246

S . M. PURI AND T. H . GEBALLE

and q,. This can be easily done: We know that, in the extreme quantum region, q is scaled as (eB/hc)'/2. Therefore, when the summation over q is carried out for Eq. (40), the q dependence of any factor changes into the magnetic field dependence of Q,. Now since the exponent of q in (V4l2 is different for the piezoelectric scattering and the deformation potential scattering, we expect th-at an inspection of the magnetic field dependence of Q, will reveal the relative importance of the two types of scattering. Neglecting, for the moment, the corrections due to the band structure effects and the saturation effect, and using Eqs. (42) and (46) in Eq. (40), we get,

where

const =

(k)

(L)'(T) nkT hb

',

e 271

and Z's are numbers42 of the order of unity which have a weak logarithmic dependence on B. It is seen from Eq. (47) that the magnetic field dependence of Q, is determined by the relative magnitudes of constants qp and qD, which determine the piezoelectric and the deformation potential modes of scattering. If the electrons are scattered mostly by the piezoelectric mode, Q, varies linearly with the magnetic field; and if scattering by deformation potential dominates, the variation is quadratic. On the other hand, if the two types of scattering are comparable, the Q, vs. B curve is bent upwards, changing its curvature from linear to quadratic. This is consistent with the results of Adams and Holsteins(a) for magnetoresistance. The experimental values of Q, for a sample of n-type lnSb are shown in Fig. 12. Since Q,(O) is negligibly small, the dependence of AQ,' and Q,(B) are the same. The magnetic field dependence of AQpcchanges with the field; starting with a quadratic dependence, the curve bends down. The observed change in curvature has the opposite sign from what may be expected as due to intermixing of the piezoelectric mode and deformation potential scattering. The observed change in slope at higher fields may be due to collision of electron energy levels and will be discussed later. The initial slope of the 7.7"K curve is even greater than two. This can be explained as due to the influence of ionized impurity scattering when the condition ho/kT >> 1 is not fully satisfied; this effect becomes more prominent as the temperature decreases. The initial quadratic rise with the magnetic field B of the experimental data indicates that the piezoelectric scattering is not important. This is further confirmed by detailed data fittings4' and the internal consistency of the analysis. 52

R . Kubo and H. Hasegawa, J . Phys. SOC. J a p a n 14. 56 (1959).

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

241

lo4 1

5

2

==--=

10.1O-

/11.90

---17.7' $0

3

-0 0

3

a.

Y

000

Q

5

2

lo2 10

20

40

60

80

100

B(kG1

FIG. 12. Measured values of the phonon-drag component of the thermoelectric power of n-type InSb in the quantum region plotted as a function of magnetic field for various temperatures. The range of temperatures is where the mean free path of phonon-drag phonons is determined by the size of the experimental specimen. The quadratic variation of AQpc with B indicates that the piezoelectric mode of electron scattering is unimportant as compared to scattering via deformation potential.

b. Size of Deformation Potential Once we accept that the piezoelectric mode of scattering is negligible and the phonon relaxation time is given by Eq. (46), the only unknown of the

248

S . M. PURI AND T. H . GEBALLE

theory is the deformation potential constant qD. The value of qD can therefore be determined by matching the experimental values of AQ; with the calculated values. The effect due to nonparabolic band and the saturation effect are also taken into account, as discussed earlier [Eq. (43)and Eq. (45)J.

FIG. 13. The comparison of the calculated values of phonon-drag thermoelectric power for boundary scattered phonons with the measured values for n-type InSb. Curve I shows the calculated values neglecting the admixture of valence band wave functions and the saturation effect Curve I1 is the correction to be subtracted due to the admixture of valence band wave functions. The correction due to the saturation effect, also to be subtracted, is given by curve 111. The solid line gives the sum of the three contributions. The deformation potential constant qD which is the only unknown of the theory is adjusted to match the experimental values shown by the solid dots.

A typical example of such a match is shown in Fig. 13. The curve marked I is the value calculated from Eq. (40) using the appropriate effective mass of electrons given by Eq.(44);the curve marked I1 is the negative correction

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

249

due to the valence band admixture; and the curve marked I11 gives the saturation effect, also to be subtracted. It is seen that both of these corrections increase with the magnetic field. The solid curve is the sum of the three contributions. If the constant b of Eq. (46) is taken simply as the sample width, the best fit is obtained by taking qD = 8.31 eV. The fit is seen to be quite good except for fields higher than 60 kG. This deviation at very high fields is discussed later in the section. The uncertainty in determining q,, is quite small. Due to the fact that AQ," varies quadratically with qD, the percentage error in qD is only half that of AQ,". The major experimental error in determining AQ," arises from measuring the temperature gradient in the sample. An error of a few tens of microvolts per degree in AQ," due to a small error in the calculated values of AQeCis not too serious since AQ," is several thousand microvolts per degree. The uncertainty of qD from the above effects should be 5 5 %. This is borne out by the consistency of the values of qD determined at a number of temperatures as given in Table I1 of Ref. 42. An additional uncertainty arises from the choice of the value of b. Again the percentage error in qD is only half that in b. If b is taken to be equal to the width of the sample, an average value of 8.25eV is obtained for q,, using the data at several other temperatures. These values are obtained from measurements on a high purity sample (sample 8081°), electron density 3.0 x 1013/cm3 and maximum mobility near 60°K of greater than 7.0 x 105cm2/v-sec. Less complete data on another sample having 1.0 x loi4 electrons/cm3 (sample 801lo) gave consistent results for temperatures above 16°K. The value of the deformation potential constant obtained from these experiments is close to E h r e n r e i c h ' ~estimate ~~ of 7.2 eV for the same constant. If we were to choose a boundary scattering length b [Eq. (46)] of 20% greater than the width of the sample, which has been shown from the thermal conductivity datas3 to be more nearly correct, it would reduce the value of qD obtained from phonon drag to 7.5 eV. The effects of this magnitude can best be determined directly by making phonon-drag measurements on samples of different cross-sectional sizes. The only other estimate of qD is that of Haga and K i m ~ r a , ~who ' find a value of 30 eV, from the analysis of free-electron infrared-absorption data. Such a large value of qD is not only in wide disagreement with thermomagnetic data but is also irreconcilable with the zero-field thermoelectric data It would predict a phonon-drag contribution in zero field which is greater than the total measured thermoelectric power.

-

53

(a) H. B. G. Casimir, Physica 5, 595 (1938). (b) R. Berman, F. E. Simon, and J. M. Ziman, Proc. Roy. Soc. (London) A220, 171 (1953). (c) R. Berman, E. L. Foster, and J. M. Ziman, Proc. Roy. SOC.(London) A231, 130 (1955).

250

S. M. PURI A N D T. H. GEBALLE

c. Phonon Relaxation Time

To obtain information about the phonon-phonon scattering we turn to the data at higher temperatures. Guided by Herring's expressions for zq (Table I), we assume a phonon relaxation time of the form z,(T)

A q" T'

= __

(49)

and analyze the data in order to obtain values for m, r, and A. As explained earlier, any q dependence when integrated over q is changed into the

FIG.14. The comparison of the measured values ( T = 33.4"K) of the phonon-drag thermoelectric power with the calculated values for n-type InSb. Curve I is calculated assuming that T I/q. Curve 11 is calculated assuming s(q) l/qz.

-

-

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

251

magnetic field dependence. Therefore, the value of m can be obtained from the magnetic field dependence of Q, in the phonon-phonon scattering region. Figure 14 shows a comparison of the calculated values and the experimental data at T = 33.4"K. Curve I is calculated assuming m = 1,

5 33.4O

37.5O

$ ,/

2 (

u

10 10

30

40

B(kG) FIG. 15. Measured values of the phonon drag component of the thermoelectric power in quantum region plotted as a function of magnetic field. The data are shown for temperatures of 33.4", 37.5", 43", and 54°K where phonon-phonon scattering is the major mechanism for phonon relaxation.

252

S.

M. PURI AND T. H. GEBALLE

and curve I1 is for rn = 2. The experimental points shown by dots follow more nearly curve I indicating a value of rn = 1. Deviations are observed at fields above 60kG, which arise from reasons to be discussed later. Deviations of the order of ten microvolts per degree are also observed for fields less than 20 kG. This is probably due to small uncertainties in calculating the electronic part, which are appreciable with respect to the relatively small phonon drag component. We try to compare theory and experiment at a temperature high enough to minimize the effects of boundary scattering of phonons, and yet where the observed values of phonon drag are large enough so that the uncertainty of a few microvolts per degree in calculating Q, does not produce too serious an effect. Figure 15 shows similar data at a few other temperatures; the slopes of the curves again indicate rn = 1. The value of the exponent r of Eq. (49) is obtained by plotting AQ," for fixed magnetic fields against temperature. Such a plot is shown in Fig. 16. The smaller slope for T 5 20°K is due to boundary scattered phonons. The slope of the curve on its steepest part, between 25" and 45"K,is 4.5 for each of the three curves, which implies a value of 3 for I when the temperature dependence of other factors in Eq. (41) is taken into account. The value of A is obtained by putting y~ = 8.25 eV, as obtained earlier. and matching the calculated values to the observed data. From such a procedure, we find A x 4.4 x lo3 sec c ~ - ' ( " K ) - ~ . The change in the slope of curves of Fig. 16 for T > 45°K is probably due to the approach of Debye temperature (-205°K for InSb) where the value of r is expected to decrease. Also, small remaining uncertainties in calculating AQec have a relatively large effect on AQ," at these temperatures. In concluding, we shall say a few words about the discrepancy between the observed and the calculated values of AQ," for fields higher than 60 kG. There may be two explanations for this: .First, we see that in a high magnetic field the wave vector qP of the phonon-drag phonon becomes comparable to the q, of thermal phonons. For example, at 10°K we have 4# % kT/hS = 3.7 x lo6 cm-', which is only about three times qp z (eB/hc)"' at 100 kG. Our calculations of AQpc are in error inasmuch as they violate our basic assumption of 4t 9 qp. This assumption permits the use of a "singlemode relaxation time" and the treatment of each phonon mode independently from others. The theory of phonon drag in extremely high magnetic fields when B (Ac/e)(kT/hS)' has been discussed by Gurevich and Nedli~~ They . ~ ~predict that in this situation the thermoelectric power becomes independent of the magnetic field once again. Instead of the saturation predicted by them we have observed that, at 6.5"K, Q, goes through its maximum value at 90 kG and starts decreasing as the field is increased further. Second, we observe that in the theory we have neglected the

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

253

t IV

6

8

10

40

20 TEMPERATURE

I

(OK)

FIG.16. The phonon-drag thermoelectric power of n-type InSb for fixed magnetic field in the quantum region plotted as a function of temperature. The straight line drawn through the data points between 25" and 45'K has a slope of 4.5 indicating that the relaxation time of phonons varies as T-j.'.

broadening of the electron energy levels due to collisions. Collision broadening is one of the mechanisms invoked to remove divergence in the theory of magnetoresistan~e~'~)*~(")~~~ in the quantum region. The inelasticity of electron-phonon collisions is another. We have considered only the latter mechanism. Since the scattering rate increases with the field, so does the collision broadening of the electron energy, and it is therefore possible that the collision broadening becomes dominant in high fields while the inelasticity of collisions dominates at low fields. These cutoff mechanisms operate as follows.42 The finite energy of the interacting phonons gives rise to the last exponential factor in Eq. (36) with the consequence that Eq. (35)

254

S. M. PURI AND T. H . GEBALLE

remains finite for q, = 0. The inelasticity in effect prevents phonons with qz < h ~ , / ( 4 y k T ) ' / ~from scattering electrons and contributing to Q,. Due to collision broadening, on the other hand, the electron energy states have a finite width -h/z and a corresponding spread Akz in momentum along the magnetic field where z is approximate collision time for an electron. A collision is not significant unless it changes k, by an amount greater than Akz, which means only phonons with qz 2 Ak, can take part in electron scattering. Collision broadening thus changes the lower limit of the integral over 1q,1 from zero to a finite value -Ak=. If the value of Ak, is such that the last exponential factor of Eq. (36), with q, replaced by Ak, which arises due to inelasticity, is already small, then the effect of collision broadening will be negligible. In the opposite case a finite Akz reduces the value of integrals I, of Eq. (47) and hence the calculated values of Qp. The value of Ak, depends on the magnetic field because of its dependence on z. At present, no quantitative estimates have been made. It is of interest to compare how the information about zq( T ) is obtained from the phonon-drag experiments in the classical and in the extreme quantum region. Herring et al.' 7,33 divided the conduction electrons into groups of different energies and assumed that each group contributes to the transport properties independently. From the analysis of the several transport coefficients, they determined the variation of n, of each electron group with its energy E. In the classical region, the q of phonons which interact with the electrons scales as the square root of the electron energy and, [Eq. (49)J. therefore, ?, of phonons interacting with each group varies as The relaxation time of electrons due to electron-phonon scattering varies as E - '/', which gives [Eq. (8)] : R P x & ' / Z - 1'2m

.

The value of m can be determined from the experimentally observed energy variation of n,. In the experiments in the quantum region, on the other hand, the q of interacting phonons scales with radius of the lowest Landau orbit and therefore actually changes with the magnetic field. The magnetic field, in this case, brings phonons of different q into interaction with the electrons. In discussing phonon drag we have limited ourselves to transverse fields because in this case the electronic component is independent of electron scattering; it can be calculated precisely and subtracted from the total measured Q to get the phonon-drag component. Apart from the difficulty of theoretically determining precisely the magnitude of Q, in longitudinal fields, the analysis is difficult. From Table I, it can be readily seen that the phonon-drag effect changes less rapidly with the magnetic field.

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

255

Furthermore, the presence of electron scattering other than that on acoustic phonons seriously affects the magnitude and is difficult to treat.

IV. Oscillations Induced by Optical Phonons The part of Q, which is sensitive to electron scattering in longitudinal fields is too small to be of any practical interest. Not much reliable information about electron scattering can be obtained from such data. There is one exception, however; an interesting case occurs when electron scattering on optical phonons is significant. It is easy to see that, in this case, the electron scattering has singularities whenever the spacing between any two Landau levels is equal to the energy of the optical phonon. The transport properties show an oscillatory behavior with respect to the magnetic field.53aThis was pointed out independently by Klinger 4b and Gurevich and F i r s o ~ . ~The ' origin of these oscillations can be understood as follows: In the presence of a magnetic field the density of states of an electron is given by

where =.,I

is the largest integer for which the condition &

2 (I

+ +)ho

is satisfied, and kBZ = eB/hc as before. It is seen from this expression that the density of states N ( E ) goes to infinity for E = (I + #ho,i.e., when k, of the electron state is equal to zero. It is this infinity in the density of states which is at the root of these oscillations. The density of states, as given by Eq. (50), is shown in Fig. 17 by the dotted line. The finite lifetime of an electron state due to scattering broadens the electron energy level. The net result of the collision broadening of the levels is to replace the infinity in N ( E )by maxima of finite amplitude and width. This is shown qualitatively by the solid line of Fig. 17. For comparison we have also shown the density of states in the absence of a magnetic field by the dot-dash curve which shows a monotonic behavior. Of course, we are discussing a simple isotropic and parabolic band. In Fig. 18 we show a set of Landau levels, and we consider scattering due to optical phonons. Neglecting dispersion in the optical branch, 53These oscillations are quite distinct from the better known Shubnikov-de Haas type oscillations which can be observed only in degenerate material. We shall discuss the distinguishing features of the two types later.

256

S.

M. PURI AND T. H. GEBALLE

12.5

ELECTRON DENSITY OF STATES

I!

10.0

7.5 N(€1

5. 0

2.5

0

I 0.5

I

I

1.5

2.5

1

3.5

($1 FIG. 17. The density of states of an electron gas. The short and long dash curve shows the density of states in the absence of the magnetic field. The dashed curve shows the density of states in the presence of a magnetic field for a single electron undergoing no collisions This curve has singularities for half-integer values of E / ~ o . The solid curve is a qualitative representation of the density of states of conduction electrons in a solid in the presence of magnetic field. I t is assumed that the collision determined lifetime of each state gives it a width equal to 5 % of the Landau level spacing. The solid curve has been normalized to have the same area under the curve as the dotted line. Collisions smooth out the singularities.

phonons have a fixed energy shown by the length of the arrow alongside. We distinguish between two types of scattering: (i) first, that in which the magnetic quantum number I changes by maximum amount and the change in z-component of electron momentum is minimum; such an event is shown by arrow marked 1, (ii) arrows 2, 3, and 4 show events where the change in 1 is less than the maximum possible. The transition rate W due to any scattering event is given by W = number of electrons in initial state x lmatrix elementlZ x density of final states.

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

257

Now if we increase the magnetic field, the scattering rate for each event varies smoothly until Ak, for the first type of scattering becomes zero, i.e.,

hsz = A1 h ~ ,

(51)

where A1 is an integer and h 0 is the energy of the optical phonon. In this case the density of final states for first event passes suddenly through infinity (neglecting, of course, collision broadening) to a small value, and the transition rate goes through a singularity. For each case that satisfies

1

I

I kz

-

I

1

FIG. 18. A set of Landau levels and the possible transitions caused by an optical phonon of energy shown by the arrow alongside.

Eq. (51), cXxgoes to infinity (because transverse current is caused by scatteringz2).In the limit of weak scattering, where the Hall angle remains nearly go”, the magnetoresistance (in the absence of the “real” effects to be considered below) also goes to infinity in each case. Thus the transport properties show an oscillatory behavior each time condition (51)is satisfied for A1 = 1,2, 3, etc. The period of these oscillations is given by

Furthermore, no oscillations are observed when

ho>hn.

(53) In real solids the amplitude of these oscillations is not infinite, as may be inferred from above, but finite due to the following reasons :

258

S. M. PURI AND T. H. GEBALLE

(1) Collision broadening. As is indicated in Fig. 17, collision broadening of the electron levels makes the density of states finite at the point of singularities. Furthermore, due to broadening, the energy conservation condition is somewhat relaxed, which further limits the amplitude of the oscillations. It is clear that the oscillations will be observed only when oz P 1, where z is the collision time. (2) Dispersion ofphonons. Phonons of different wavelengths have slightly different energies, even in the optical branch. This will give a finite width 6B e 6 Q / m (where 6 Q is the spread in the energy of optical phonons) and hence a finite amplitude to these oscillations. (3) Nonparabolic band. The nonuniform spacing of the Landau levels for a nonparabolic band reduces the amplitude of at least the higher order oscillations. Since electron scattering on optical phonons does not contribute to phonon drag, it is clear that these oscillations will not be observed in Q,. Furthermore, those transport properties which are independent of electron scattering in strong magnetic fields (for example, Hall effect and Q, in transverse fields) will also not show these oscillations. The theory of oscillatory magnetoresistance in both t r a n ~ v e r s e ~and (~) longit~dinal~~ field has 'been worked out. The present theory is elementary in the sense that it does not take into account in a quantitative manner any of the factors which make the amplitude of the oscillations finite. Such a theory is straightforward, although it involves tedious algebra, especially in the case of longitudinal fields54 with several scattering mechanisms interfering with each other. Gurevich and fir so^^^ find that if scattering other than that by an optical phonon exceeds a certain limit, one observes minima instead of maxima at resonance. This complication is, of course, not present in transverse fields because various scattering mechanisms operate independently. The oscillations were first observed in the measurement of thermoelectric power in longitudinal field^.^(")*^^ The change in Seebeck voltage of n-type InSb with magnetic field is shown in Fig. 19. The mean temperature of the sample is 120"K, and it has an electron density of 6.2 x lOI3/cm3. At least five maxima can be easily identified for values of magnetic field at 42.5, 21.0, 13.8, 10.0, and 7.7 kG. A close observation of the curve shows kinks at other values of the field which are too close to be identified separately. The first resonance at 42.5 kG can be interpreted as due to scattering of electrons from the

-

V. L. Gurevich and Yu. A. Firsov, Z h . Eksperim. i Teor. Fiz. 47, 734 (1964) [English Transl.: Soviet Phys. J E T P 20, 489 (1965)l. 5 5 V. M. Muzhdaba, R. V. Parfen'ev, and S. S. Shalyt, Fiz. Tverd. Tela 6, 3194 (1964) [English Transl.: Soviet Phys.-Solid State 6, 2554 (1965)l

54

7.

259

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

lowest Landau level, 1 = 0, to the next higher Landau level, 1 = 1, so that the spacing between these two levels at 42.5 kG equals the energy of the optical phonon. The spacing between Landau levels can be calculated from the known band structure constants. Using the spectrum of energy levels

900 840 780 720 -

660 600 540 480 -

A0

420 -

360 -

300 .240 -

t

l

0 2

0

10

20

30

50

40

:

:

5

7

':

10 60

! .

: '

I3 I5 17 70

:

20

'

1

' ! '

25 80

H IkGl

FIG. 19. The change in thermoelectric power (arbitrary units) of n-type InSb (sample 804) in longitudinal magnetic field^.^'') The mean temperature of the sample is 120°K. the temperature difference across the potential probes is undetermined but of the order of 20". The peaks in AQ at 42.5, 21.0, 13.8, 10.0, and 7.7 kG are caused by resonances in electron scattering on optical phonons. The inset shows the x-y recorder plot of the raw data.

in high magnetic fields as given by Eq. (2.18) of Ref. 42, we can calculate the spacing of the levels, which is 0.0288 eV for spin up electrons and 0.0262 eV for spin down electrons. This is in fair agreement with the published values for the energy, 0.025 eV, of the long wavelength optical phonons. The small discrepancy may be due to the possibility that the resonances in thermoelectric power are slightly shifted from the resonance condition [Eq. (51)] predicted from the simple considerations given above.

260

B (XG)

FIG. 20. The relative change in the thermoelectric power of n-type InSb (sample 804) in longitudinal magnetic fields. The mean temperature of the sample is marked on each curve. There is no observable shift in the position of the peaks as the temperature of the sample is changed.

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

261

Also, we observe that the period of the resonance is not strictly constant. This, of course, is due to the nonparabolic nature of the conduction band of InSb. Figure 20 shows similar data for other temperatures. Similar oscillations are also observed in l~ngitudinal~'~) and trans~ e r ~ e ~ magnetoresistance. ( ~ ) ~ ~ ( ~ ) The ~ ~representative ~ data are shown in Figs. 21 and 22. Firsov et al. believe that they observe minima in longitudinal magnetoresistance at the points of resonances. Measurements (Fig. 21) which extend to 100 kG,4(c)are similar to their observations in the range of overlap regarding the positions of maxima and minima. The data in Fig. 21 as well as more complete considerations4") fail to support the conclusion drawn by Firsov et ~ 1 . ~ The ' ~ ' resonances are marked by maxima even in longitudinal magnetoresistance. The positions of resonances in longitudinal Ap are shifted slightly, just as in the case of longitudinal AQ, from the simple resonance condition given by Eq. (51). So far no attempt has been made to analyze the amplitude of the oscillations in terms of collision broadening or the phonon dispersion. The reason is lack of quantitative calculation and the fact that it probably involves too many unknown parameters. The experimental information about the oscillations can be summarized as follows : (1) The oscillations have been observed in nondegenerate material where the Ferrni level is below the lowest Landau level. Although theory predicts that the oscillations should be observed even in degenerate material, this has not yet been confirmed experimentally. Probably the reduced mobility of electrons in degenerate material makes the amplitude of the oscillations very small. (2) The position of resonances does not depend upon the density of charge carriers or the temperature of measurements. The density of charge carriers can be altered by as much as a factor of 10 without causing any observable change in the resonances. The temperature has also been varied from about 60" to 200°K without any observable change in period. (3) The oscillations are observed at temperatures around liquid nitrogen and higher. The temperatures are much higher than -4"K, where Shubninkovae Haas type oscillations are observed. The reason obviously is that at low enough temperatures there are not enough optical phonons to excite electrons. (4) The amplitude of the oscillations increases with temperature. Ultimately the amplitude should attain a maximum value and decrease with further increase of temperature because of a reduction in electron mobility. 56

D. Y. Mashovets, R. V. Parfen'ev, and S. S. Shalyt, Zh. Eksperim. i Teor. Fiz. 47,2007 (1964) [English Transl.: Sooiet Phys. J E T P 20, 1348 (1965)]; S. S. Shalyt, R. Y. Parfen'ev, and M. Y. Aleksandrova, Zh. Eksperim. i Teor. Fiz. 47, 1683 (1964) [English Transl.: Soviet Phys. J E T P 20, 1131 (1965)l.

262

B (kG)

FIG. 21. Longitudinal magnetoresistance of an n-type InSb sample (81 1). The peaks mark the resonance condition for electron scattering on optical phonons. The position of the peaks is markedly different from the corresponding resonance in thermoelectric power measurements.

7.

THERMOMAGNETIC EFFECTS IN THE QUANTUM REGION

263

The temperature at which the maximum amplitude is obtained probably depends upon the purity of the material. Mashovets et al.56 have observed the maximum amplitude at 104°K. In some of our samples, at 120°K the amplitude was still increasing with the increase of temperature. The reason these oscillations were observed first in thermoelectric measurements is

/

2

I

4

0

M

I

5

10

15

20

25

30

FIG. 22. The curves of the transverse (Ap J p 0 ) and longitudinal (Apll/po)magnetoresistance for n-type InSb samples with carrier concentrations 6 x l O I 3 ~ 1 1 and 3 ~ 4.1 ~ x l O I 3 ~ r n and - ~ mobilities 6.7 x lo5 cm2/V-sec and 5.5 x lo5 cm2/V-sec, respectively, at 90°K. The broken lines correspond to the monotonic background on these curves. The oscillatory part of the transverse and longitudinal magnetoresistance as a function of the inverse magnetic field is given in the upper part of the figure. The vertical lines correspond to the resonance condition, given by Eq. (51). [After Firsov et al., Ref. 4(d)].

that the monotonic part, in this case, is small, varying logarithmically with the magnetic field. The fixed oscillatory part therefore becomes a sizable fraction of the total thermoelectric power. In the case of magnetoresistance the monotonic part is large and rapidly varying with the field. It may also have contributions from spurious nonbulk effects like nonuniformity of electron density and surface effects. It therefore required close and careful measurements to detect the oscillatory part over a large

264

S. M . PURI AND T. H. GEBALLE

monotonic part, especially in transverse magnetoresistance. It would be advantageous to do the quantitative analysis on AQ measurements where more reliance can be put on the separation of the monotonic part from the measured values.

CHAPTER 8

Band Characteristics near Principal Minima from Magnetoresistance W. M . Becker . . . . . . . . I . INTRODUCTION 1. Band Structure of the Compounh . . . . . . . . 2. Theory . . . . . . . . . . . . . . . . 3. Complicating Effects . . . . . . . . . . . . I1. SURVEY . . . . . . . . . 4 . Conduction Band . . . . . 5. Valence Band . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

111. EXPERIMENTAL MEASUREMENTS

6. n-Type InSb I . p-Type InSb 8. n-TypeInAs 9. p-Type InAs 10. n-Type InP . 11. n-Type GaSb 12. p-Type GaSb 13 . n-Type GaAs

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . IV . NEGATIVE MAGNETORESISTANCE . . 14. InSb . . . . . . . . . 15. n-Type InAs . . . . . . . 16. GaAs

. . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . . . . . . . . . . . . .

265 266 267 270 212 212 213 213 273 215 277 217 278 278 281 283 284 284 285 286

.

I Introduction The study of magnetoresistance has long been an important tool in the investigation of the band parameters of semiconducting materials. The early work by Pearson and Suhl' on germanium and by Pearson and Herring' on silicon clearly anticipated the results of cyclotron resonance with regard to the tensor properties of the effective mass. However. it was not until the full development of a transport theory for anisotropic energy surfaces by Abeles and Meiboom3 and by Shibuya4 that magnetoresistance G. L. Pearson and H. Suhl. Phys. Rev . 83. 768 (1951). * G. L. Pearson and C. Herring. Physica 20. 915 (1954). B. Abeles and S. Meiboom. Phys. Rev . 95. 31 (1954). M. Shibuya. Phys. Rev. 95. 1385 (1954).

265

266

W. M. BECKER

results were successfully applied in establishing energy surface symmetries in cubic materials. With the initiation of galvanomagnetic studies on the 111-V compounds it has become clear that the quality of the single-crystal samples is of primary importance in magnetoresistance studies for the determination of band parameters. The required degrees of purity and homogeneity have been achieved by present crystal growth technology for only a few of this family of compounds. In some of these materials, the usefulness of magnetoresistance as an investigative tool is further limited by contact and surface effects which tend to mask the behavior characteristic of the bulk. For a number of the compounds the carrier mobilities may be extremely low, even for the best crystals obtainable, in terms of purity and homogeneity. As a consequence the bulk magnetoresistance might be quite small and perhaps easily obscured by secondary effects. A considerable amount of work on magnetoresistance has been carried out on a number of 111-V compounds. Those pieces of evidence which were obtained mainly at low magnetic field strengths and which relate to the band structure of the compound are discussed below. Although a number of the investigations yield band parameters consistent with those obtained from other types of measurements (optical absorption, piezoresistance, etc.) and theory, others do not. Where discrepancies arise, they will be commented on with due consideration of the aforementioned complicating effects. A special aspect of magnetoresistance to be considered is that of negative magnetoresistance at low temperatures. Recent theoretical treatments5. indicate that study of this effect may be fruitful in elucidating the conduction mechanism in impurity bands. A number of investigations of negative magnetoresistance in 111-V compounds have been carried out and are reviewed below. 1. BANDSTRUCTURE OF THE COMPOUNDS

Both theoretical investigations and a wide variety of experimental studies7 indicate that the lowest lying conduction bands in the 111-V compounds may be expected to be of the following types: (1) a spherical energy band centered at k = 0, (2) multivalley spheroidal, energy surfaces lying along either the [ 1001 or [ 1111 directions, or (3) combinations of these two cases wherein the band edges are separated only slightly in energy and ' Y . Toyozawa, Proc. Intern. Conf. Semicond. Phys., Prague, I960 p. 215. Czech. Acad. Sci., Prague, 1961. Y.Toyozawa, J . Phys. SOC.Japan 17,986 (1962). ' See H. Ehrenreich, J . Appl. Phys. 32, 2155 (1961) for a recent summary of theoretical and experimental studies on the 111-V compounds.

8.

BAND CHARACTERISTICS

267

may both be populated at low carrier concentrations or moderate temperatures. investigations indicate that the valence band structures of the 111-V compounds are similar to those of the semiconductors Ge and Si, with differences arising due to the lack of inversion symmetry in the zinc-blende structure. In particular, a number of energy maxima lying at points away from k = 0 are predicted for the 111-V compounds whereas the maxima occur at the center of the Brillouin zone for materials with the diamond structure. The maxima appear to be important only for a heavy hole band that is twofold degenerate at k = 0, the degeneracy being lifted along [l111 directions for small k by energy terms proportional to k. Occurrence of maxima for a light hole band, also twofold degenerate at k = 0, does not appear to be of any experimental significance. Both the light and heavy hole bands are degenerate at k = 0. As in the case of the elemental semiconductors, the heavy hole band is characterized by a warped energy surface, a large effective mass, and low mobility, while the light hole band is nearly spherical with a small effective mass and high mobility. A third band with the edge centered at k = 0 is separated in energy from the other bands by spin-orbit splitting. A recent comparison ' of experimental data with theoretical calculations appears to verify the essential similarity between the valence band structures for all the 111-V compounds. Spheroidal energy surfaces are usually characterized by the anisotropy parameter K = (ml,*/m,*) x (zl/zl,) where the m*'s are the effective masses and the z's the relaxation times referred to the principal axis system of the energy ellipsoid. The subscripts 11 and I denote the values of the parameters parallel and perpendicular to the unique axis of the ellipsoidal energy surface. The theory of magnetoresistance effects has been extensively reviewed by a number of authors.I2*l 3 We here confine our discussion to very qualitative considerations, so that the main features of the experimental results listed may be readily related to salient features of the band structure.

'

2. THEORY If a current carrying sample is placed in a magnetic field, changes in resistance occur which are a function of the band parameters of the material R. H. Parmenter, Phys. Rev. 100, 573 (1955). G. Dresselhaus, Phys. Rev. 100, 580 (1955). l o E. 0. Kane, J . Phys. Chem. Solids 1, 249 (1957). I ' R. Braunstein and E. 0. Kane, J . Phys. Chem. Solids 23, 1423 (1962). " A . C. Beer. Solid State Phys. Suppl. No. 4 (1963). '' A. H. Wilson, "The Theory of Metals," 2nd ed. Cambridge Univ. Press, London and New York. 1953.

268

W . M. BECKER

and the relative orientations of the current and field. These changes take place due to the lateral deflection of the carriers by the Lorentz force. The simplest case is that of a degenerate semiconductor with spherical energy surfaces and isotropic relaxation time. Since the drift mobility depends only on a single relaxation time z, all carriers will have the same velocity under the action of the applied electric field. The Hall field will thus be equal to the Lorentz force for each carrier, no lateral deflection of the carriers will result, and the magnetoresistance will vanish. Departure of the relaxation time from isotropy in 111-V semiconductors with spherical energy surfaces does not appear to be experimentally important, and will not be further discussed. In a nondegenerate semiconductor with spherical energy surfaces, the magnitude of the magnetoresistance will depend on the energy dependence of the relaxation time. A constant relaxation time implies the same velocity for all the carriers and, as before, the magnetoresistance will vanish. However, if the relaxation time is energy dependent, the carriers will have a distribution of velocities, and the Hall field will cancel out the Lorentz force for carriers of only one mobility. Thus transverse currents associated with individual electrons will exist. Their further interaction with the magnetic field results in a change in the current in the longitudinal direction causing an increase in resistance. The initial deflection is proportional to v,H or p H , the second-order interaction would then be expected to vary as p 2 H 2 . Competing relaxation processes may have a profound effect on the magnitude of the transverse magnetoresistance, which are not brought out by the previous simple considerations. In the case of the 111-V compounds with spherical energy surfaces, both theory and experiment show that a small amount of impurity scattering, although decreasing the carrier mobilities only slightly, may lower the magnetoresistance to values far below those expected in pure material^.'^^'^ Although conduction by degenerate carriers in a spherical energy band with isotropic scattering does not produce transverse magnetoresistance, conduction by carriers in two such bands does give an effect, provided the mobilities of the carriers are different from each other. This case is analogous to the situation described previously for nondegenerate material, since again the Hall field can cancel out the Lorentz force for only one carrier velocity. In the case of longitudinal magnetoresistance (field applied parallel to the current direction), no effect should be observed in materials with scalar effective mass and isotropic scattering since the carrier motion is always parallel to the magnetic field. l4 l5

R. T. Bate, R. K. Willardson, and A. C. Beer, J . Phys. Chem. Solids 9, 119 (1959). R. K. Willardson and J. J. Duga, Proc. Phys. SOC. (London)75,290 (1960).

8.

BAND CHARACTERISTICS

269

Both the longitudinal and transverse magnetoresistance are sensitive to the tensor properties of z/rn*. In anisotropic media, application of a longitudinal magnetic field gives rise to some current components which are not colinear with the magnetic field. As a result groups of carriers are deflected by the magnetic field producing an increase in resistance. Seitz16 has given a phenomenological equation for the current density in cubic materials which is linear in the electric field E and contains terms up to second order in the magnetic field H.For experimental investigations, this equation is most useful in its inverted form’

E = poi

+ R,H

X

J

+ po[bH2j + cH(j - H) + dTj] ,

(1)

where po, R,, b, c, and d are constants of the material which depend on the band structure and carrier relaxation processes, and

the subscripts on H being referred to a coordinate system coincident with the crystal axes. The change in resistivity Ap from the zero field value p o can be obtained from the above equation and has the form

where t j and qj are the direction cosines of the current and field, respectively. In following discussions, reference to “orientational effects” will denote experimental investigations of the dependence of A p / p o H 2 on the orientation of E and H with respect to the crystal axes. The Seitz formulation implies a quadratic dependence of the magnetoresistance on magnetic field strength. Evaluation of the transport integrals for the determination of the b, c, and d are made under the approximation pH 4 10’ where p is the mobility in cm2/V-sec and H is the field in gauss. Reference below to “weak field” behavior indicates that both conditions have been experimentally realized for the fields used in the investigations. The magnetoresistance coefficients b, c, and d have been shown to obey certain relations which depend on the symmetry of the energy surfaces. These are given in the accompanying tabulation. l6

F. Seitz, Phys. Rev. 19,312 (1950).

270

W. M . BECKER

Energy surface symmetry Spherical ( K = 1) [ 11 11 Ellipsoids [lo01 Ellipsoids [ 1101 Ellipsoids

Relationship”

b+c=O, b+c=O, b+c=-d, b f c = d,

d=O d>O d0

EFFECTS 3. COMPLICATING A number of theoretical and experimental studies have shown that the galvanomagnetic effects which depend on the band parameters may be obscured by perturbing influences such as inhomogeneities, sample geometry, contacts, and surface conduction. Weak field magnetoresistance appears to be particularly sensitive to such influences, particularly in the high mobility semiconductors, and indeed it has been proposed” that departures of sample behavior from that expected on the basis of the known band structure and scattering mechanisms be used as indicators of perturbing effects. When the band structure of a material is not known, however, recourse must be made to the comparison of orientational measurements to the energy surface symmetry relationships appropriate to the crystal system for a guide as the validity of the data. The influence of sample dimensions and potential probe positions on galvanomagnetic measurements has been a subject of continued interest. Early work on Hall effect” showed that, in the usual sample geometry, the influence of the current electrodes on the Hall field is minimized by using large length-to-width ratios (I/w 3 3). More recently the effects of sample geometry on magnetoresistance have been calculated by several authors.’’. 2 1 Their treatments showed that the presence of the current contacts perturbs the Hall field in the vicinity of the contacts giving rise to transverse currents and consequent spurious magnetoresistance. Broom” has demonstrated that shorting due to the potential contacts may give anomalous contributions to the angular variation of the transverse magnetoresistance. This latter effect appears to be quite pronounced in the 111-V compounds and has made difficult the obtaining of reliable orientational data for determination of weak-field magnetoresistance coefficient^.^^ M. Glicksman, Progr. Semicond. 3. 3 (1958). R. T. Bate, J. C. Bell, and A. C. Beer, J . Appl. Phys. 32, 806 (1961). l 9 I. Isenberg, B. R. Russell, and R. F. Greene, Reu. Sci. Instr. 19, 685 (1948). ’O R. F. Wick, J . Appl. Phys. 25, 741 (1954). * I J. R. Drabble and R. Wolfe, J . Electron. Control 3, 259 (1957). ” R. F. Broom, Proc. Phys. SOC. (London) 71, 500 (1958). 2 3 M. Glicksman. J . Phys. Chem. Solids 8, 511 (1959).

8.

BAND CHARACTERISTICS

271

Herring et aLZ4pointed out that both surface conduction and inhomogeneities in bulk conduction could produce a spurious field dependence of the high field transverse magnetoresistance. A theoretical treatmentz5 was carried out for the model of small fractional variations in the local conductivity, the spatial fluctuations occurring over lengths small compared to the size of the crystal, but large compared to such quantities as the Debye length or some suitably defined mean free path. At weak fields, the magnetoconductivity tensor components calculated from the model were found to differ from the values appropriate to homogeneous material. In particular, the theory predicted that nonzero contributions to the longitudinal magnetoresistance could occur even though the band structure parameters indicated very small or zero longitudinal magnetoresistance. That inhomogeneities on a microscopic scale occur in 111-V compounds has been amply demonstrated by experimental investigations of InSb crystals grown by the Czochralski technique. Ingots grown from Te-doped melts preferential segregation of Te impurity along the [ 1111 direction indicating that the distribution coefficient k is much larger for (1 11) facets than for other growth planes. Similar results were obtained for Se-doped ingots of InSb.28*29 Etch studies2' on Czochralski grown Se-doped ingots revealed structures transverse to the growth axis, which were taken as indicators of periodic spatial fluctuations of impurity concentration. The sensitivity of galvanomagnetic effects to conductivity gradients was analyzed by Bate and Beer.30 They examined a model of an exponential variation of carrier concentration in the current direction. Solutions of the potential problem indicated that distortion of the current lines took place under the action of a magnetic field, and spurious contribution to the weak field transverse magnetoresistance arose as a result. Anomalous transverse magnetoresistance effects have also been predicted from a model of a sharp magnetoconductivity discontinuity in the current direction when each contact of the measuring pair is in a different region of the sample." It has been shown by a number of authors3', 3z that spurious magnetoresistance properties of InSb may depend on the growth direction of the ingots and on the angle between the current direction and the growth axis. C. Herring, T. H. Geballe, and J. E. Kunzler, Bell System Tech. J: 38, 659 (1959). C. Herring, J . Appl. Phys. 31, 1939 (1960). 26 K. F. Hulme and J. B. Mullin, Phil. Mag. 4, 1286 (1959). " J. B. Mullin and K. F. Hulrne, J . Phys. Chem. Solids 17, 1 (1960). W. P. Allred and R. T. Bate, J . Electrochem. SOC.108, 259 (1961). z9 H. C. Gatos, A. J. Strauss, M. C. Lavine, and T. C. Harman, J . Appl. Phys. 32, 2057 (1961). 3 0 R. T. Bate and A. C. Beer, J . Appl. Phys. 32, 800 (1961). 3 1 H. Rupprecht, R. Weber, and H. Weiss, 2. Naturforsch. 15a, 783 (1960). 3 2 H. Rupprecht, Z . Naturforsch. 16a, 395 (1961). 24

25

272

W. M . BECKER

we is^^^ has discussed a model of a layer structure in which the electron concentration changes periodically as a function of position. The results of his analysis indicate that, in such a case, a longitudinal magnetoresistance may be observed if the current direction and the perpendicular to the layers are at an angle to each other different from zero. Experimental measurements confirmed this behavior in n-type InSb where one normally expects zero longitudinal magnetoresistance because of the isotropy of the conduction band. From careful consideration of various perturbing effects, Weiss concluded that the transverse magnetoresistance relating to the band parameters of n-type InSb with electron concentration > 1OI6 cm-3 accounts for a fractional change of less than 1 % in the resistance at 10,000G, and disappears completely for n > 10'7cm-3, a result to be expected on the basis of the statistical degeneracy of the carriers. 11. Survey 4. CONDUCTION BAND

Magnetoresistance results bearing on the band structure of various 111-V compounds are listed below. In the cases of the conduction band structure of four of the materials discussed, InSb, InAs, InP, and GaAs, the weight of experimental evidence from a variety of measurements obtained to date indicates that the lowest lying conduction band is spherical and is centered at k = 0, and other band edges are far away in energy. Where magnetoresistance results on these materials appear to give evidences of more complicated band structure, such as added conduction in nearby anisotropic bands, further work on the influence of perturbing contributions to the magnetoresistance in these materials would seem to be necessary to clarify the situation. In GaSb the evidence for two band conduction based on magnetoresistance behavior is in excellent agreement with other experimental results bearing on the band structure of this material. Further, the measurements show that at least one of the bands is germanium-like with valleys lying along [111] directions. Reliable values of the anisotropy are not however derivable from existing data. To the author's knowledge, no magnetoresistance results have been reported for the compounds AlSb and GaP up to the present date (August, 1964),33aalthough single crystals of these materials have been available for H. Weiss, J . Appl. Phys. 32, 2064 (1961). Since the completion of this article, magnetoresistance measurements on n-A1Sb have been reported by Stirn and B e ~ k e rTheir . ~ ~ ~results confirm the picture that the lowest conduction band of AlSb is silicon-like, with valleys lying along [lo01 directions. Details of the measurements and analysis of the results will be incorporated in an article to appear in a future volume of this series. 3 3 b R. J. Stirn and W. M. Becker, Phys. Rev.. 141, 621 (1966).

33

33a

8.

BAND CHARACTERISTICS

273

several years. In both AlSb and Gap, the conduction band structures are presumed to be silicon-like. The study of their magnetoresistance properties would prove quite interesting because of the possibility of deriving the tensor properties of the carriers in the El001 minima. However, the extremely low mobilities of the carriers in n-type AlSb and GaP (- 100 cm2/V-sec) produced to date presents an extremely difficult measurements problem, and magnetoresistance studies probably cannot be fruitfully pursued unless higher mobility materials become available. 5. VALENCEBAND

Magnetoresistance measurements relating to the valence band structure have been confined so far to InSb and GaSb. In the case of both of the compounds, the results appear to confirm the prediction of conduction by carriers in both light and heavy hole bands. For InSb, the evidence indicates that the valence band is nearly isotropic or has a number of maxima with nearly spherical surfaces of constant energy. Theory" predicts that the energy difference between light and heavy hole band edges in InSb should be of the order of kT at liquid helium temperatures Accordingly, low temperature measurements on very pure material should in principle give evidence for details of the anisotropy of predicted off-center maxima. However, impurity band behavior appears to dominate the conduction processes at low temperatures preventing the observance of such effects. Magnetoresistance results on GaSb give a clear indication of anisotropic effects in the valence band When combined with recent piezoresistance data, the behavior appears to favor a warped energy surface structure such as has been shown for Si and Ge. The inability to experimentally separate the magnetoresistance contributions from the several hole bands appears to be a limiting factor in the study of the valence band structure of this compound, particularly since cyclotron resonance measurements yielding effective mass data have not as yet appeared in the literature. In. Experimental Measurements 6. TYPE InSb

Pearson and T a n e n b a ~ mcarried ~ ~ out measurements on polycrystalline material at room temperature. A large transverse magnetoresistance was observed, but the longitudinal magnetoresistance was small. This behavior was taken to be consistent with the presence of spherical energy surfaces Tanenbaum and collaborator^^^ extended the measurements to single-crystal material with similar results. 34

G. L. Pearson and M. Tanenbaum, Phys. Rev. 90, 153 (1953).

'' M. Tanenbaum, G. L. Pearson, and W. L. Feldmann, Phys. Rev. 93, 912 (1954).

274

W.

M. BECKER

Frederikse and H ~ s l e rstudied ~ ~ orientational effects in single-crystal samples at 78°K. Indication of weak field behavior was deduced from the H Z dependence of the transverse magnetoresistance between 100-200 G. The sign, magnitude, and field dependence of the longitudinal magnetoresistance was found to vary from sample to sample, as seen in Fig. 1.

H (gourd

lcl

-

FIG. 1. Magnetoresistance of several n-type InSb samples at 78°K illustrating different behavior in the longitudinal effects in samples with the same concentration (a) and (b), n 2.4 x lOI5 cm-3, and similarity in behavior of the transvers effects to that of a purer sample (c), n 1 x 10” cm-3. (After Frederikse and H o ~ l e r . ~ ~ )

-

However, the longitudinal magnetoresistance was always at least an order of magnitude smaller than the transverse magnetoresistance, consistent with an isotropic one-carrier model for the conduction band At the temperature of measurement, the samples were nondegenerate, and the magnitude of the transverse effect could be accounted for on the basis of lattice and impurity scattering. The difficulties in obtaining consistent results on the longitudinal effect were ascribed to geometry and inhomogeneity effects. Champness3’ carried out orientational studies of the magnetoresistance on a-type single crystals at room temperature and liquid air temperatures. The samples investigated were nondegenerate in these temperature ranges. The results showed that the longitudinal magnetoresistance depends on the crystallographic direction ; the largest value occurs for current and field 36

”

H. P. R. Frederikse and W. R. Hosler, Phys. Rev. 108, 1136 (1957). C. H. Champness, Can. J . Phys. 39, 452 (1961).

8.

275

BAND CHARACTERISTICS

along the [ 1001direction. This result suggested that conduction was taking place not only in the k = 0 band, but also in a nearby multivalley band with valley edges lying along [ 11 11 directions. It is to be noted that the samples investigated were all cut from the same single crystal. The work of we is^^^ and others has since shown that the above quoted results might be expected on the basis of periodic changes in electron concentration as a function of position in the as-grown ingot. Champness had already established the presence of inhomogeneity in his samples from an analysis of Hall effect measurements, but the nature of the inhomogeneities was not clear from the data. The strong possibility therefore exists that the longitudinal effects observed by Champness are due to inhomogeneities. All weak field magnetoresistance results obtained to date on n-type InSb still appear therefore to substantiate the picture of conduction taking place predominantly by carriers in a spherical band centered at k = 0.

7. p-TYPEInSb Early magnetoresistance m e a s u r e m e n t ~ ~on ~ p-type single-crystal material indicated spherical energy surfaces in the valence band. Extensive measurements were later carried out by Frederikse and H ~ s l e on r ~single~ crystal samples showing a high degree of homogeneity in carrier concentration. Characteristic sample behavior is shown in Fig. 2. As is indicated by the graphs, the longitudinal magnetoresistance is more than an order of magnitude smaller than the transverse effect in the weak field region. These results imply conduction predominantly by carriers in an isotropic energy

FIG.2. Magnetoresistanceof several p-type InSb samples at 78°K (a) and (b), p 2.3 x 1015 ~ r n and - ~ (c) p 4 x l O I 4 c m - 3 showing large ratios of transverse to longitudinal effects in the weak field region (H < lo00 G). (After Frederikse and H o ~ l e r . ~ ~ )

-

38

H. P. R. Frederikse and W. R. Hosler, Phys. Rev. 108, 1146 (1957).

N

276

W. M. BECKER

band However, much evidence7 indicates that the InSb valence band structure is similar to that of germanium. Analyses of 78°K data on transv e r s magnetoresistance and Hall effect data have been carried out by several author^.^^' They obtained reasonably good agreement with the data by assuming that both a light and a heavy hole band, degenerate at k = 0, are present. In their analyses, the light hole band was considered to be spherical. Confirming evidence of two band conduction is seen (Fig. 3) in the decrease of the Hall effect with increasing magnetic field. However, the smallness of the longitudinal magnetoresistance, indicated in Fig. 2, appears to preclude any evaluation of the anisotropy of the heavy hole band from the data

H (klogaussl

-

FIG. 3. Hall coefficient as a function of magnetic field at 78°K; sample R, p ~ m - sample ~ ; U, p 4 x 1014 cm-’. (After Frederikse and Hosler.”)

-

2.3 x 10’’

Charnpness4O has analyzed data on transverse magnetoresistance and Hall effect to obtain mobility and carrier concentration ratios for the twohole bands at 77°K. By assuming (1) negligible intrinsic electron concentration at 77”K, (2) spherical energy surfaces for both holes, and ( 3 ) energy independent relaxation times for both hole bands, a light-to-heavy hole mobility ratio of 5 -7 and a light-to-heavy hole concentration ratio of 1 % is obtained for carrier concentrations from 3 x 1014 to 6 x 1015cm-3. Additional information on the light and heavy hole bands, as obtained from transport studies, is presented by Weiss in this volume. In particular, it was noted that both the density ratio and the mobility ratio of fast and slow holes increased with temperature above 22°C. The former effect was attributed to the nonparabolicity of the light-mass band.

-

39

40

H. J. Hrostowski, F. J. Morin, T. H. Geballe, and G. H. Wheatley, Phys. Rev. 100, 1672 (1955). C. H. Champness, Phys. Rev. Letters 1, 439 (1958).

8.

BAND CHARACTERISTICS

277

8. TYPE InAs Weiss41 and Champness and Chasmar4’ found slight ( 2 % at 10,OOOG) transverse magnetoresistance effects at room temperature in partially degenerate material, but did not report orientational studies, presumably due to the smallness of the effects measured Frederikse and H ~ s l e r ~ ~ measured single-crystal material at 77°K and found that the longitudinal magnetoresistance was very much smaller than the transverse magnetoresistance (see Fig. 4), consistent with the picture of conduction principally by carriers in a spherical energy band centered at k = 0. Additional experimental data on the magnetoresistance in n-type InAs are given by Weiss in Chapter 10.

I@

5x10’ 10‘

5 x 10.

ti (gauss)

FIG.4. Magnetoresistance of n-type lnAs at 77°K. (After Frederikse and

9. p-TYPEInAs

Magnetoresistance data relating to the valence band structure of I d s have not yet appeared in the literature. The study of this material appears to be complicated by conductivity contributions from an n-type surface layer.44 41

42 43

44

H. Weiss, 2. Nnturforsch. lZa,80 (1957). C. H. Champness and R. P. Chasrnar, J . Electron. Control 3, 494 (1957). H. P. R. Frederikse and W. R. Hosler, Phys. Rev. 110,880 (1958). H. Rupprecht and H. Weiss, Z . Nnturforsch. 14a,531 (1959).

278

W . M. BECKER

10. TYPE InP Glicksman's measurements of the orientational effectsz3indicate that in weakly doped material with electron concentrations of the order of 10l6~ r n or - ~less, the longitudinal magnetoresistance is small compared to the transverse effect. At higher concentrations a small anisotropy of the [loo] type is indicated. The first result was taken as consistent with the lowest lying conduction band being spherical. The second result suggested either the onset of anisotropic scattering, or conductivity contributions from carriers in nearby silicon-like subsidiary minima.

0

0.001L

4.2"K

I 10.

10' H

in

10'

Oersteds

FIG. 5. Transverse magnetoresistance as a function of magnetic field in n-type samples of GaSb. The 300" and 77°K data were obtained on a sample having R(300"K) = - 4 cm3/coul, and the 4.2"K data were given by a sample having R(300"K) = - 3.2 cm3/coul. (After Becker, Ramdas, and Fan.45)

11. TYPE GaSb Magnetoresistance measurements have been carried out by Becker Their results clearly showed an H 2 dependence of the transverse magnetoresistance in magnetic fields up to 14 kG, as seen in Fig. 5. For samples with IRI > 5 cm3/coul, the magnetoresistance of the samples decreases with decreasing temperature. Figure 6 shows this behavior. The result is expected for carriers in a spherical conduction band which become more degenerate with decreasing temperature. In the higher concentration

et

W. M. Becker, A. K. Ramdas, and H. Y . Fan, J . Appl. Phys. 32,2094 (1961).

8.

279

BAND CHARACTERISTICS

samples, however, the effect increases with decreasing temperature. This behavior was taken as a clear indication of the existence of a second conduction band, consistent with the conduction band structure derived from a wide variety of other measurements.' The sharp rise assumed to be due to onset of conduction in a higher band occurs at R - 5 cm3/coul corresponding to an electron concentration of n = 1.25 x lo'* ~ m - ~ . The variation of the transverse magnetoresistance at 4.2"K is represented by the dotted curve in Fig. 6 and is based on calculations using the expression (1 - YHY --b (4) (Roo)2 - xy(lxy&'

-

+

where b is the Seitz coefficient important for transverse magnetoresistance, R is the Hall coefficient, oo is the total conductivity, x = nz/nl the ratio of \

\

\ A*

I [IlO] H I [IIO] A 3&K x 77% 4.2%

*\

-~(rm'/cculonb) at 4.2-K

FIG. 6. Transverse magnetoresistance of n-type GaSb at three different temperatures plotted against Hall coefficient at 4.2"K. (After Becker, Ramdas, and Fan4')

the carrier concentrations in the two bands, yH = pHZ/pH1 the ratio of the Hall mobilities, and y = p2/p, the ratio of the conductivity mobilities for the carriers in the bands. By combining the results of optical absorption, magnetoresistance, and Hall effect measurements, the authors deduced : The density-of-states mass of the lower band mdl= 0.052m0,and the density-of-states mass ratio md,/mdl= 17.3. Using the density-of-statesmass value of the lower band, the position of the minimum of the second band was calculated to be at an energy of 0.08 eV above the central minimum at 4.2"K. Clear indication of anisotropic conductivity contributions by carriers in the higher-lying band was obtained from the room temperature measurement of the angular variation of the transverse magnetoresistance (Fig. 7).

280

W. M. BECKER

Following are the relations between the magnetoresistance coefficients at room temperature :

b = 2.21 x 10-’oG-2,

d = 10.8 x 10-12G-’ .

These results satisfy the symmetry conditions for [ l l l ] multivalley conduction.” At 4.2”K the longitudinal magnetoresistance was found to be less than 1/30 of the transverse magnetoresistance for samples where the

I

iii

.

I

iio

001

I iii

I

ooi

HI

FIG. 7. Variation of transverse magnetoresistance with field orientation for n-type sample of GaSb at 300°K. (After Becker, Ramdas, and Fan.45)

Fermi level was well inside the upper conduction band. Furthermore, no angular variation in the transverse magnetoresistance was detectable. This result appeared to be inconsistent with the room temperature magnetoresistance data. The authors showed however that the absence of anisotropic effects at low temperature could be ascribed to the low mobility of the carriers in the upper band, although the anisotropy factor K of the carriers in the upper band might be very large. The recent extension46 of magnetoresistance measurements in n-type GaSb to strong magnetic fields (pH %- lo8) has led to the observation of Shubnikov-de Haas oscillations. (See other chapters for discussion of transport effects at large magnetic fields.) Analysis of the data indicated that the behavior could be understood on the basis of the two-band model 46

W. M. Becker and H. Y. Fan, in “Physics of Semiconductors” (Proc. 7th Intern. Conf.), p. 663. Dunod, Paris and Academic Press, New York, 1964.

8.

BAND CHARACTERISTICS

281

applicable to the weak field magnetoresistance results. The oscillatory behavior was ascribed to modulation of the conductivity principally by carriers in this central minimum An effective mass value for carriers in this band was derived which agrees closely with the weak field results. The effect appears however only when the Fermi level is close to the [1111 valley edges, suggesting that carriers in the higher lying band influence the effe~t.~~,~’ 13

I

I

. I d-J

E

0

H

in Oarsteds

FIG. 8. Transverse and longitudinal magnetoresistances and Hall coefficient as functions of magnetic field for a p-type sample of GaSb at 77°K (After Becker, Ramdas, and Fan.45)

12. P-TYPEGaSb

Early measurements on single crystals were carried out by Tanenbaum and c o - ~ o r k e r s who , ~ ~ found the magnetoresistance behavior to be consistent with spherical energy surfaces. Becker et aL4’ examined the dependence of the Hall coefficient and magnetoresistance on magnetic field strength in single-crystal samples at room temperature and 77°K. As shown in Fig. 8, the transverse magnetoresistance and Hall coefficient decrease with increasing field, whereas the longitudinal magnetoresistance remains constant. This result was assumed to indicate clearly the presence of two types of holes, similar to the case of germanium and silicon. The 47 48

J . E. Robinson and S. Rodriguez, Phys. Rev. 135 A779 (1964). J. E. Robinson and S. Rodriguez, Phys. Rev. 137,A663 (1965).

282

W. M. BECKER

decrease in R and AplpJI’ was ascribed to the violation of the condition m c% ~ 1 for the light holes. Lack of field dependence of the longitudinal

magnetoresistance suggested that the longitudinal effect results only from the heavy hole contribution. The continued presence of the field dependence of R and Ap/pHZ at 77°K led to the conclusion that the two-hole bands are nearly degenerate at the minimum Similar results on the 77°K field dependence of the transverse magnetoresistance have been observed by MatyaS and S k a ~ h a who , ~ ~also interpreted their data as an indication of conduction by two kinds of holes. The angular variation of transverse magnetoresistance obtained by Becker et aL4’ is shown in Fig. 9. From these R ( 3 0 0 O K b + 55crn’/coulornb H =13,400 Oersteds

lo] ,

I [I

0.0701

I

I

I

HI[IIO]

I

1

data and the longitudinal effect, the following values for the Seitz coefficients are obtained :

(b + C) = 9.1 x 10-l’ G - 2 , d

=

b = 45.1 x 10-’2G-2

-8.9 x 10-’’G-’,

+

These results satisfy the relations b c = - d , d < 0 and were taken as an indication of a band symmetry of the [loo] type. It was noted that the same relations had been observed in magnetoresistance measurements on p-type Ge and p-type Si wherein the effects were ascribable to warped energy surfaces5’ rather than to a multivalley structure. From orientation data for the same directions of current and field shown in Fig. 9, MatyaS and Skacha 49 50

M. MatyiS and J. Skacha, Czech. J . Phys. 12, 566 (1962). J. G. Mavroides and B. Lax, Phys. Rev. 107, 1530 (1957).

8.

BAND CHARACTERISTICS

283

also derived the [ 1001 symmetry relation. Recent piezoresistance measurementssl indicate that the band structure is either that of p-type Ge or is a many-valley band structure with energy minima lying along [ 1113 directions. The combination of the piezoresistance and magnetoresistance results thus appear to favor a germanium-like valence band structure. 13. n-TYPE GaAs Early magnetoresistance measurements on single-crystal samples were made by Broom and his collaborators. 52 They investigated orientational effects for samples cut in various crystallographic directions. The results were found to be inconsistent from sample to sample, and furthermore did not agree with the requirements for cubic symmetry. It was conjectured that these discrepancies arose from contact effects. G l i c k ~ m a nexplored ~~ the inhence of contact size on magnetoresistance behavior at room temperature for a number of samples in the concentration range 4 x 10I6 to 4.7 x 1017cm-3. He found that when welded side contacts of 0.002-in. diameter gold wires are used, the resulting sample behavior appears to agree with the picture of conduction in a spherical energy band, whereas larger diameter contacts give inconsistent results. Kravchenko and Fans3 studied single-crystal samples with carrier concentrations ranging from 3 x 1014 to 10l8 ~ m - They ~ . found anisotropy of the [lo01 multivalley type in all except the highest concentration sample. An analysis was made in terms of a model consisting of a minimum at k = 0 and a set of [lo01 minima. The data indicated that the two band edges were separated in energy by not more than a few hundredths of an electron volt. Although experimental evidence for higher lying minima in the conduction band of GaAs has been well established by a variety of experimental methods including infrared ab~orption,’~Hall coefficient measurem e n t ~ , ~ ~pressure -’~ effects on optical a b ~ o r p t i o n ,and ~ ~ electrical resista n ~ e , ~ the ’ . ~results ~ indicate band separations of the order of 0.2-0.4 eV. As yet, no corroborative evidence for the presence of closer bands such as the one indicated by Kravchenko and Fan’s work exists in the literature. 0. N. Tufte and E. L. Stelzer, Phys. Reu. 133, A1450 (1964). R. Broom, R. Barrie, and I. M. Ross, Proc. Intern. Coiioq. Semicond. and Phosphors, Garmisch-Purtenkirchen, 19S6 p. 453. Wiley (Interscience), New York, 1958. 53 A. F. Kravchenko and H. Y. Fan, Fiz. Tuerd. Telu 5, 660 (1963) [English Transl.: Soviet Phys.-Solid State 5, 480 (1963)l. 5 4 W. G. Spitzer and J. M. Whelan, Phys. Rev. 114, 59 (1959). 5 5 J. T. Edmond, R. F. Broom, and F. A. Cunnell, Rept. Meeting Semicond., Rugby, 1956 p. 109. Phys. SOC.,London, 1956. 5 6 L. W. Aukerman and R. K. Willardson, J . Appl. Phys. 31, 939 (1960). H. Ehrenreich, Phys. Rev. 120, 1951 (1960). A. L. Edwards, T. E. Slykhouse, and H. G. Drickamer, J . Phys. Chem. Solids 11, 140 (1959). 5 9 W. Paul, J . Appl. Phys. 32, 2140 (1961). 51

52

’’ ’*

284

W . M . BECKER

IV. Negative Magnetoresistance Negative magnetoresistance at low temperatures was first observed in p-type InSb," and has since been reported for a number of other semiconductors. Until recently, a plausible theory for the phenomenon has been lacking. As a consequence, experimental investigations have emphasized the need to establish whether the effect is due to bulk conduction processes or may result from the accidental influences of surface conduction, defects, etc. The fact that regularities in sample behavior seem to depend generally only on carrier concentration and temperature now seem to indicate that negative magnetoresistance is indeed a bulk phenomenon. Furthermore, the effect appears to be associated always with impurity band conduction, as seen in the results listed below.

14. InSb Negative magnetoresistance in InSb was first reported by Fritzsche and Lark-Horovitz,60 who observed a change in sign of the transverse magnetoresistance from positive to negative values in p-type InSb, with decreasing temperature. As seen in Fig. 10 the change occurs at a temperature somewhat below that at which a maximum in the Hall coefficient is reached. Frederikse and Hosler have observed negative magnetoresistance both in the longitudinal and transverse magnetoresistance for p-type InSb at 4.2" and 1.7°K.38These authors found that the effect depends somewhat on the surface treatment. Broom6' showed that the effect in n-type material at 4.2"K was not a bulk property but was largely due to introduction of defects into the lattice during sample preparation. Sasaki and co-workers62 investigated the effect at 4.2"K in both n- and p-type InSb and found a systematic dependence on impurity concentration, indicating that the effect is characteristic of the buIk material and the conductivity type. The result was qualitatively explained on the basis of a theory by Toyozawa5 which involves electron population changes between several impurity bands with different mobilities. According to the theory, the magnetic field increases the energy between the bands, and the electron population increases in the band of higher mobility leading to a decreasing resistance. A second, more recent theory of negative magnetoresistance has been proposed by Toyozawa,6 which is based on scattering by localized spins. According to the model, scattering is reduced by the alignment of the spins in the magnetic field, resulting in an increased carrier mobility. As yet, the 6o 6' 62

H. Fritzsche and K. Lark-Horovitz, Phys. Rev. 99, 400 (1955). R. F. Broom, Proc. Phys. Sue. (London) 71, 470 (1958). W. Sasaki, C. Yamanouchi, and G. M. Hatoyama, Proc. Intern. Conf Semicond. Phvs., Prague, 1960 p. 159. Czech. Acad. Sci., Prague, 1961.

8.

BAND CHARACTERISTICS

285

Absolute temperature ('Kelvn)

I /Absolute ternperature('Kelwn)

FIG. 10. Hall coefficient, resistivity, and transverse magnetoresistance plotted as a function of IIT for a sample of InSb with RH(78"K)= 1200cm3/coul. (After Fritzsche and LarkHorovitz.60)

experimental evidence has not given a clear choice between these two models. 15. TZ-TYPE InAs

Shalyt6j has observed negative magnetoresistance at 4.2"K in H < 3 kO in a sample containing n = 3 x 10'6cm-3 at 77°K. Zotova and cow o r k e r ~found ~ ~ negative magnetoresistance below liquid nitrogen tempera~ .seen in Fig. ll(a), ture in various samples containing 1.2-4 x 10l6~ m - As the magnetoresistance behavior varies in a systematic way with the carrier concentration. Furthermore, the onset of negative magnetoresistance is clearly correlated with Hall effect behavior indicating impurity band conduction [Fig. 1 l(b)]. The magnetoresistance behavior observed was interpreted as due to scattering by localized spins of electrons associated with partially isolated impurity atoms.6 b3

S. S. Shalyt, Fiz. Tverd. Tefu4, 1915 (1962) [English Transl.: Soviet Phys.-Solid State 4, 1403 (1 963)].

64

N. V. Zotova, T. S. Lagunova, and D. N. Nasledov, Fiz. Tuerd. Tela 5, 3329 (1963) [English Transl.: Soviet Phys.-Solid State 5, 2439 (1964)].

286

W . M. BECKER

I000

600

0, .OOF E

Ir

100 I

10

100

500

T, OK (a)

(b)

FIG. 11. (a) Transverse magnetoresistance in n-type InAs at 4.2%. At 300”K, R,(N7) = R,(Nl) = -206 cm3/coul. R,(N5) = - 336 cm’/coul, and R,(N8) = -450 cm3/couI.(b) Temperature dependence of the Hall constant and of the negative magnetoresistance. (After Zotova, Lagunova, and N a ~ l e d o v . ~ ~ ) - 174 cm3/coul,

16. GaAs Broom et al.52 observed negative magnetoresistance for both the longitudinal and transverse effects in an n-type sample with n = 7.6 x loi6~ r n - ~ . The behavior was thought to relate to impurity band conduction, as indicated by the temperature behavior of the Hall effect and resistivity. Nasledov and Emel’yanenk~~’ observed negative magnetoresistance at temperatures 65

D. N. Nasledov and 0. V. Emel’yanenko, Proc. Intern. Conf. Phys. Semicond., Exeter, 1962 p. 163. Inst. of Phys. and Phys. SOC.,London, 1962.

8.

BAND CHARACTERISTICS

287

below 30"-60"K in samples with n = 0.55-6 x 1016cm-3 (measured at 300°K). It was concluded that the effects were determined by the properties of the crystal impurity band Recently the observations of Nasledov and Emel'yanenko have been confirmed by Woods and Chen66 on crystals grown by a variety of techniques, indicating that the effect is due to bulk conduction. Also, small but measureable negative magnetoresistance was found in Cd-doped p-type GaAs samples by Woods and Chen. bb

J. F. Woods and C. Y. Chen, Phys. Rev. 135, A1462 (1964).

This Page Intentionally Left Blank

CHAPTER 9

Freeze-Out Effects, Hot Electron Effects, and Submillimeter Photoconductivity in InSb E . H . Putley I.

FREEZE-OUT EFFECTS .

. . . . . . . . . . . . . 289 . . . . . . . 289

1. Hydrogen Atom in a Strong Magnetic Field 2. Freeze-Out Effect in Ins6 . . . . 3 . Comparison of Results with Y I< A Theory .

. . . . 11. HOT ELECTRON EFFECTSAT LOW TEMPERATURES . 111. SUBMILLIMETER PHOTOCONDUCTIVITY . . . . . 4. Behavior in Zero or Small Magnetic Fields . . 5 . Behavior in Larger Magnetic Fields . . . . 6 . Other Photoconductive Effects . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. 292 . 297 . 299 . 305 . 305 . 3 11 . 31 3

I. Freeze-Out Effects 1 . HYDROGEN ATOMIN

A

STRONGMAGNETIC FIELD

This effect was discovered in the course of experiments suggested by the theoretical work of Yafet, Keyes, and Adams (YKA),' which was concerned with the behavior of the ionization energy of a hydrogen atom in a magnetic field. It was shown that when a hydrogen atom is placed in a magnetic field so large that the energy difference between the Landau levels of a free electron becomes comparable with the Rydberg ionization energy for a hydrogen atom in zero magnetic field, then this ionization energy is increased. The behavior can be described in terms of a parameter y = hwJ2RY,

(1)

where R, is the Rydberg energy and w , is the angular frequency for cyclotron resonance of a free carrier in the magnetic field. The quantity h o J 2 is therefore the zero point energy of the free carrier. When y 5 1, the wave function of the ground state of the hydrogen atom becomes considerably distorted. In the plane perpendicular to the magnetic field the wave function is compressed considerably, but along the direction Y. Yafet. R. W. Keyes, and E. N. Adams, J . Phys. Chem. Solids 1, 137 (1956).

289

290

E. H. PUTLEY

40

30

20

*

0 x

*> 10

I

/El84

5

3

2

, Y

,

14 B 16 KG 18

4 ,-I0

15

22 24 26 28 30 32 FIG.I . Variation of ionization energy of hydrogen atom in magnetic field based on Ref. I for n-type InSb. Observed values calculated from data of SIadek3 and Putley.’ 4

6

8

10

12

20

of the field there is a much smaller compression. So that, as y increases, the originally spherical wave function becomes progressively barrel shaped and finally cigar shaped. Because there is some compression in all directions the average distance of the electrons from the nucleus is reduced, and the ionization energy increases. The result of this calculation is shown in Fig. 1, which shows that when y = 2 the ionization energy is doubled, but at greater values of y the increase is not so rapid. To observe this effect, values of y 1 are required. From Eq. (1) an expression for the value of B, the magnetic induction making y = 1, can be obtained: B

=

2.35 x 109(m*/mK)’

G,

(2)

where m is the free electron mass, m* the effective mass, and K the high frequency dielectric constant. For a real hydrogen atom the field required to satisfy (2) is 2 x lo9 G, which is quite unattainable. For hydrogenic impurity centers in semiconductors, the combination of large dielectric constants and small effective masses can bring the field required into the experimentally accessible region. As Table I shows, the value for Ge is 3.7 x lo5 G, which is too large to permit y to be raised to really large values. For InSb it is about 2 kG, which means that with a 100-kG installation a value of y of about 50 should be reached. There are other semiconductors for which Eq. (2) indicates an even smaller field.

9.

291

FREEZE-OUT EFFECTS, HOT ELECTRON EFFECTS

TABLE I PROPERTIES OF HYDROGENIC IMPURITY Magnetic induction at which y

=

CENTERS

1 in various media

Medium

m*/m

K

B (gauss)

Free space Ge n-type InSb PbTe n-type Cd0.14Hg0.86Te

1 -0.2 1.3 x 10-2 -0.1 3 x 10-3

1 16 14 200 100

2 x loy 3.1 x 10’ 2.03 103 580 2.1

-

Ionization energy, Bohr radius, and maximum concentration for weak interactions

Medium Free space Ge n-type InSb PbTe n-type Cdo.l4Hg0.,6Te

13.6 0.01 9.0 x 3.5 x 1 0 - 5 4 x 10-l

5.29 x to-’ 4.2 x lo-’ 5.7 x 1.0 x 1.8 x

-

3.4 1 0 1 5 1.3 x 10’’ 2.5 x 10” 4.3 x lo7

It would thus appear easy to study the YKA theory by observing the behavior of one of the semiconductors mentioned in Table I in a strong magnetic field. There is one difficulty, however. The YKA theory applies to an isolated hydrogen atom. For the impurities in a semiconductor to behave like isolated atoms the spacing between nearest neighbors must be about ten times the Bohr radius. The factors that reduce the ionization energy increase the Bohr radius : E

=

13.6(m*/mK2) eV,

uB = 5.29 x 10-9(m/m*)K cm.

(3) (4)

Table I also gives the ionization energy and Bohr radius for hydrogenic impurities in various semiconductors. The impurity concentration at which the average spacing between neighboring impurities is about lOu, is also given. The value obtained for G e (-3.4 x 10”) corresponds roughly to the maximum concentration at which impurities behave as isolated centers. The value obtained for InSb (1.3 x 10”) is considerably smaller than the concentration in the best available material, which is probably greater than In both PbTe and HgTe-CdTe alloys the minimum obtainable impurity concentration is about 10l6 to 1017~ r n - ~ .

292

E. H. PUTLEY

Table I shows that there is no material which can be used for a direct test of the YKA theory of the behavior of a hydrogen atom in a very strong magnetic field. Of the possible semiconductors, Ge is the only one which can be prepared with adequate purity. In the purest n-type InSb the interaction between the impurities is so strong that it behaves in a metallic fashion down to at least 1°K. However the behavior of p-type InSb suggests that the impurities should show hydrogenic behavior if the concentration could be reduced sufficiently. Some evidence of discrete behavior might be expected for concentrations about one order of magnitude greater than the values given in Table I, so that InSb approaches closest to the requirements for this experiment. It should also be pointed out that, apart from the question of purity, in both PbTe and in HgTe alloys the free carriers are mainly produced by deviations from stoichiometric composition. It is not certain that the behavior of an electron trapped on a stoichiometric defect can be described by the hydrogenic model.

2. FREEZE-OUT EFFECTIN InSb Although InSb is not suitable for a direct test of the theory, it would be expected that application of a large magnetic field could produce effects that would provide indirect evidence. If the Bohr radius is reduced sufficiently, the interaction between impurities would be reduced. If a sufficient reduction were obtained, then localized impurity levels would be formed. This possibility led Keyes and Sladek2*3and Frederikse and Hosler4 to measure the Hall coefficient and conductivity at low temperatures and as a function of magnetic field. Later, when somewhat purer material was available, these measurements were repeated by Putley.' Typical results are shown in Figs. 2 and 3. Material sufficiently pure to test the YKA theory is still not available, which is shown by the fact that in small magnetic fields (how small depends on the purity of the sample) the behavior of all samples studied is metalli~.~" The most striking feature of the results shown in Figs. 2 and 3 is that in high enough magnetic fields the Hall coefficient increases roughly exponentially as the temperature is reduced to about 13°K. This behavior is to be expected if the carriers are falling out of the conduction band into a R. W. Keyes and R. J. Sladek, J . Phys. Chem. Solids 1, 143 (1956).

R. J. Sladek, J . Phys. Chem. Solids 5, 157 (1958); 8, 515 (1959). H. P. R. Frederikse and W. R. Hosler, Phys. Rev. 108, 1136 (1957). E. H. Putley, Proc. Phys. Sor. (London) 76, 802 (1960); J . Phys. Chem. Solids, 22, 241 (1961). '"Vinogradova et aL6 have described apparently purer InSb, but study of this material (supplied by courtesy of the Academy of Sciences, U.S.S.R.) has shown that its behavior is not fundamentally different from that described here. K. 1. Vinogradova, V. V. Galavanov, and D. N. Nasledov, Fiz. Merallou i Metallouendenie 16, 385 (1963).

9.

293

FREEZE-OUT EFFECTS, HOT ELECTRON EFFECTS Temperature (OK)

77 6 x105p

10

5

3

I

I

I

2

I.! I

4

3x1041

0

I

I

0.1

I

1

0.2 0.3 0.4 I / T ("K)

0.5

0.6

(

7

FIG.2. Dependence of Hall effect upon magnetic induction. (After Sladek.')

I

8230 X

I

1.3 FIG.

1.4

1.5 1.6 1.7 1.8

2

2.5 T" K

3

4

5

6

4

I , ,

8 10

I

I

20 50

3. Dependence of Hall effect upon magnetic induction. (After P ~ t l e y . ~ )

294

E. H. PUTLEY

shallow impurity level. Below about 13°K the Hall coefficient becomes constant or falls slightly as the temperature is lowered to about 1°K. This behavior may be the result of impurity band conduction or other types of interaction between the centers, or possibly of surface conduction. Considering the behavior above 1.8”K, the variation of electron concentration with temperature may be calculated in a way similar to that for a normal impurity level. In this case, the magnetic field alters the density of states in the conduction band, which must be allowed for. The conduction band is split into a series of sub-bands (Landau levels), which represent the fact that in the plane normal to the magnetic field the motion of the electron is represented by a quantized harmonic oscillator. The energy difference between states is ho,where o,is the cyclotron resonance frequency. Along the direction of the magnetic field the motion of the electron is the same as in zero field. The density of states in the conduction band, neglecting spin, is now7

The energy of the lowest state is raised by ) h a , above the bottom of the conduction band in zero field. Because the motion along the field direction is not quantized by the field at energies greater than $ho,more than one term appears in the sum. But if a sufficiently large magnetic field is applied so that firstly ha, 9 k T and secondly the total number of states in the lowest Landau level is greater than the number of available electrons, then only the first term in the sum need be considered. This situation is sometimes known as the “extreme quantum limit.” We will assume that these conditions are satisfied and later show that this is justified. The Fermi energy is calculated by writing

n is the total number of electrons in the conduction band and the factor 2 is introduced to include both directions of spin. In InSb the large g-factor splits the Landau levels further, into two bands with opposite spins, and So,. that in the extreme quantum limit the splitting is comparable with +$h the lowest level will contain electrons of only one direction of spin, and the factor 2 must be omitted from Eq. (6). This equation can now be rewritten n

=

eB (2m*kT)li2y F h

I/~(Y*),

’ A. H. Kahn and H. P.R. Frederikse, Solid State Phys. 9, 257 (1959).

(7)

9. FREEZE-OUT EFFECTS,

HOT ELECTRON EFFECTS

295

where q* is the reduced Fermi energy and is measured from the bottom of the lowest Landau sub-band. If classical statistics can be used, F-1,2(q*)-+

or n

=

d/’ exp q*

eB ( 2 ~ m * k T )’ ~exp ~ q* h2

=

NB exp q*.

(8) (9)

Using this expression for the Fermi energy and supposing that with the applied field B the concentration NDof donors has a ground state energy E below the lowest Landau level and that there is also a concentration NA of acceptors, the following expression is found for the dependence of n upon temperature :

This equation differs from that usually derived for a compensated impurity level in a semiconductor’ only in the expression for NB, which now represents the effective density of states in the Landau level. By fitting Eq. (10) to the experimental data between 4” and 2”K, the values of E and of N , and N, can be found. Measurements at 77°K are used to obtain (ND - NA). The results obtained from experimental data, such as that shown in Figs. 2 and 3, are shown in Fig. 1 in which values of E are plotted and in Table I1 which also gives values for ND and NA.While E depends on B, the values for ND and NA are, within limits of about 20%, independent of B. The values of the impurity concentrations were also calculated for the purer specimens by applying the Brooks-Herring expression for ionized impurity scattering to the mobility at 20°K. This temperature is sufficiently low that lattice scattering can be neglected but is sufficientlyhigh that the electron gas is nondegenerate, and the assumptions used in deriving the Brooks-Herring expression are valid : these conditions are not satisfied at 4°K. The impurity concentrations calculated in this way are also given in Table 11, and on the whole are seen to agree well with the calculation based on Eq. (10). The values for both the ionization energies and the impurity concentrations for Sladek’s specimens differ from those given in his paper.3 This is because his method of calculation implies uncompensated impurities, which is not consistent with other evidence, such as the behavior of the mobility, on the impurity concentrations in the material. The values obtained for N D and NAshow that all specimens appear to be heavily compensated, ND 5(ND- NA). Possibly this implies some

-

E. H. Putley, “Hall Effect and Related Phenomena.” Butterworths, London, 1960.

TABLE 11 MEASURED IONIZATION ENERGIES AND DONORCONCENTRATIONS IN n-TYPE InSb Sample No.: ND-N, ( ~ m - ~ ) :

(ev), N , (cm-3) for following values of B (gauss)

Cl58/84 4.9 1013

4.2

M1 1013

HC27/149 4.16 1013

C259/19 3.19 10'3

E

8230 7125 6142 4417 2936 Value" of ND from mobility measurements at 20°K

ND

f:

7.5 x 10-4 6.2 5.1 2.7

4.3 x 1014 4.4 4.5 3.9

ND

E

7.1

10-4 5.8 4.8 2.5

3.3

28 24 20 a

7.6 x 6.6 5.4 3.3

3.1 x 10'4 3.1 2.7 2.8

2.9

Sample NO.^: N,-N, cm : ~ ( e v )N, , ( c ~ - ~ ) for following B (kG)

2.8 x 10'4 3.1 2.9 2.6

ND

E

1.4 E

6.0 x 10-4 4.7 1.8

N,

9.0 x 10-4 7.8 6.8 4.7 2.4

2.4 x 1014 2.0 1.8 1.7 1.3

3.0

B1 1014

3.3 N,

1.4 x 1 0 1 5 1.2 1.3

8

E

5.2

10-4 3.3 1.1

N3 1014

ND

3.4 x 1015 4.2 2.5

These values are calculated from the Hall mobility measured with a weak magnetic field ( - 100 G ) and taking for the mobility p, = 315n/512Ru. These results were calculated from Sladek's experimental results (Ref. 3).

P

?

9.

FREEZE-OUT EFFECTS, HOT ELECTRON EFFECTS

297

aggregation process between donor and acceptor centers since if both concentrations were varying independently the samples might be expected to show greater inhomogeneity in their behavior. 3. COMPARISON OF RESULTS WITH YKA THEORY

Comparison of the values of the ionization energies with the YKA curve shows that although the existence of magnetically dependent energy levels is confirmed, the behavior differs markedly from that predicted by the theory. This is not surprising since the theory applies to an isolated impurity. While comparison of the impurity concentration given in Table I1 with the limit for interactions estimated in Table I shows that in all samples studied metallic behavior should be expected. This is indeed found in low magnetic fields, but above a certain critical field (depending on the purity of the sample) discrete levels appear. The limited results available suggest that above the critical field the ionization energy increases linearly. Measurements have not been extended to sufficiently high fields to show whether at high enough fields the behavior tends to the YKA result. The effect of screening has been referred to in general terms by Keyes and ad am^.^ A variational calculation of bound states in a screened Coulomb field has been carried out by March," which suggests strongly that in the high-field limit detailed consideration of screening is essential to understand the concentration dependence of E. With a screening radius chosen to fit Putley's data,s it turns out that all other discrete states except the ground state are excluded over the range studied, although the first excited state will appear as a discrete state in higher magnetic fields (- 50 kG). Unfortunately the only attempt to consider the Brooks-Herring theory generalized to high fields is that of Argyres and Adams," which only included the isotropic effect of carrier freeze-out. More recently, ~ shown that the screening is anisotropic, and in the Durkan et ~ 1 . ' have plane perpendicular to the magnetic field the "magnetic radius" enters into the theory in addition to the carrier density n(B). Further calculations are in progress. Screening effects cannot account for the complete disappearance of the ionization energy in small magnetic fields. Here the interaction effects between impurity atoms are more important. Although the simple model of an impurity center with a magnetic field dependent ionization energy accounts for the behavior in high fields and at temperatures above about 1.8-2"K, the behavior at lower temperatures R. W. Keyes and E. N. Adams, Progr. Semicond. 6, 85 (1962). N. H. March, private communication. " R. N. Argyres and E. N. Adams, Phys. Rev. 104, 900 (1956). J. Durkan, J. E. Hebborn, and N. H. March, Proe. Intern. Con$ Magnetism, Nottingham, 1964 p. 26. Inst. of Phys. and Phys. SOC., London, 1965. lo

298

E. H. PUTLEY

does not accord with this model. As Fig. 3 shows, the Hall coefficient does not continue to increase as the temperature falls but passes through a broad maximum and then decreases slowly. This behavior might be explained by the following possible mechanisms : (a) conduction in an impurity band3 produced by overlap of the wave function of either the

0 529.5

4417

6142 InSb C158/84

7125

/

8230 I0

1.2

1.5

2 T

3

4

O K

FIG.4. Dependence of conductivity upon magnetic induction. (After P ~ t l e y . ~ )

ground states of the impurity centers or of their excited states; (b) interactions between neighboring pairs of i m p ~ r i t i e s;' ~(c) at sufficiently low temperatures the electron gas becoming degenerate and the screening effect then leading to a temperature-independent concentration ; (d) surface conduction shorting out the bulk conduction, as can occur in p-type InSb at low enough temperature^.'^ Figure 4, which illustrates the behavior of the conductivity of one of the purer specimens, shows that, in contrast to the behavior of the Hall coefficient, there is no discontinuity in its temperature dependence. This suggests that the same mechanism could be responsible for the conduction above 2°K as below it. Thus, of the four explanations l3

l4

S. C. Miller, Phys. Rev. 133, A1138 (1964). E. H. Putley, Proc. Phys. SOC. (London) 73, 128 (1959).

9.

299

FREEZE-OUT EFFECTS, HOT ELECTRON EFFECTS

suggested for the low temperature behavior, the multiple interactions or degeneracy effects are more probable than impurity band conductions or surface effects. 11. Hot Electron Effects at Low Temperatures At 4°K and at lower temperatures, marked deviations from Ohm's law are o b ~ e r v e d ~ . ' ~in. ' ~the samples of n-type InSb in which the freeze-out effects discussed in the first section occur. This part will not be concerned

0' INDUCTION KG

InSb Cl58/84

T - 4 23OK

03-

0 2-

In

0

I 77

0 0

0 5

ImA

lo

l5

20

25

30

FIG.5. Voltage-currentcharacteristics for sample at 4.2"K.

with a general account of hot electron effects but will discuss them in relation to the freeze-out effect and the photoconductive effects to be considered in the last section. When the temperature is too high or the magnetic induction too low for freeze-out to occur, behavior of the type shown in Fig. 5 is observed. This shows the current-voltage relations for a specimen at 4°K and in magnetic fields up to 6 kG (for this sample a field of about 20 k G was required to produce observable freeze-out at 4°K). In zero magnetic field, Ohm's law is obeyed for electric fields less than 0.04 V/cm, and for fields up to 0.1 V/cm the behavior follows the expression CT

Is

=

n0(l

+ PE').

E. H. Putley, Proc. Phys. SOC.(London) 73, 280 (1959).

'' E. H. Putley, Phys. Stat. Solidi 6, 571 (1964).

(11)

300

E. H. PUTLEY

FIG.6. Voltage-current characteristics for sample at 1.3"K

Since in this temperature range ionized impurity scattering is dominant, increases with increasing temperature or mean electron energy so that /I will be positive. For electric fields greater than 0.1 V/cm, 0 at first increases more rapidly than (11) would predict, but the effective value of p[ =(l/o) do/d(E2)] then falls, and for fields greater than 0.2 V/cm the relation between current and voltage is substantially linear. The behavior in magnetic fields too weak to produce the freeze-out effect is similar to 0

TABLE 111 VALUESOF uo and /3 OBTAINED BY FITTING EQ. (1 1 )

B

B=O

(kG)

T ( O K )

4.2 3 2 1.3

GO

0.15

0.13

RESULTS FOR SAMPLE ClS8j8.1

B

4.44

B 22 19 17 15

UO

B

(ohm-' cm-') (cmZV - z ) 4.8 x lo-' 2.2 x lo-' 8.0 1 0 - 3 3.2 x lo-'

= 6.22

(kG)

(kG)

(ohm-' cm-') (cmZV-') 0.3 1 0.23

=

TO

3.0 2.5 1.7 1.3

00

B

(ohm-' cm-')

(cm' V-')

2.7 x lo-' 1.0 x lo-' 2.0 1 0 - 3 5.0 10-4

1.5 1.2 0.7 0.35

9.

FREEZE-OUT EFFECTS, HOT ELECTRON EFFECTS

301

FIG.7. Voltage-current characteristics in weak magnetic fields at 4.23" and 1.22"K.

that in zero field, as the results in Fig. 5 show. The behavior at lower temperatures is also very similar providing freeze-out does not occur. This is illustrated by Fig. 6 which shows results at 1.35"K. For fields up to 3.55 kG the general form is similar to that at 4°K. Typical values for oo and B are given in Table 111. When freeze-out occurs the resistance in the ohmic region will be higher, so that the curves shown in Fig. 6 measured for the larger magnetic fields rise very steeply. When the electric field reaches 0.5 to 1.0 V/cm impact ionization occurs. The electric field at breakdown is proportional to the magnetic field, a typical result being E = 0.13B for B 3 4 kG. In this region the electric field is practically independent of current, although discontinuities occur in it. When all available carriers have been ionized the current-voltage relation again becomes approximately linear. At sufficiently large current densities all carriers are ionized and their mean energy is large compared with that corresponding to the lattice temperature. Hence the behavior should be independent of the lattice temperature. This is illustrated by Figs. 7 and 8, in which results obtained at different temperatures are plotted. The results shown in Figs. 5 to 8 were obtained using a slowly varying direct current. The maximum current is limited by thermal effects to about 1.5 A/sq cm.

302

E. H. PUTLEY

lo

ImA

25

20

FIG.8. Voltage-current characteristics at 6.22 kG, 4.23"K and 1.22"K,and for large currents.

These measurements were all made using a transverse magnetic field. When a longitudinal field is applied, the freeze-out effect is not affected by the change in orientation, but the conductivity is larger. This corresponds to the behavior of the magnetoresistance effect at higher temperatures, which is very small in longitudinal fields. Table IV compares typical results for transverse and longitudinal magnetic fields. TABLE IV HOT ELECTRON EFFECTS I N TRANSVERSE AND LONGITUDINAL MAGNETIC FIELDS AT 1.5"K" Transverse

B (kG)

"0

(ohm-' cm-')

0 1.8 3.55 5.35 7.1

0.102 0.049 0.0084 0.0021 0.00068 ~~~

Sample C143/14/3m. Breakdown field.

B (cm' V-') 18.6 8.54 5.78 1.69 0.73

Longitudinal Eb

(Vjcm) -

0.40 0.56 0.77

"0

(ohm-' cm-') 0.094 0.057 0.016 0.0043 0.0014

B (cm2 V-') 22.7 13.2 5.0 1 2.71 1.31

Eb

(Vjcm) -

0.23 0.29 0.39

9.

FREEZE-OUT EFFECTS, HOT ELECTRON EFFECTS

303

A closer look at the region where impact ionization occurs shows that a number of discontinuities 0ccur.l Figure 9 illustrates the main features. At the point A a rapid fall AB in electric field occurs as the current is increased, and a further fall CD occurs at higher currents. On reducing the current it follows the path DEFBA. When the observed voltage lies on the plateau BFC observations with an oscilloscope across the sample show that a relaxation oscillation occurs. A typical wave form is shown in Fig. lqa). This type of behavior is similar to that of the discharge tube relaxation oscillator. The wave form and frequency are determined mainly by the resistance and capacity in the circuit supplying current to the sample, A

Current

FIG. 9. Diagram showing region in voltage-current characteristics where oscillations occur.

while the sample itself acts as a bistable device. This oscillation probably corresponds with the one first reported by Haslett and Love” and more recently described by Phelan and Love.” At higher current densities, near the point where the high field linear region starts, there is another discontinuity in the current voltage curve (G in Fig. 9). This denotes the occurrence of a second oscillatory effect which unlike the relaxation oscillation is practically independent of the external circuit. Typical wave forms are shown in Figs. 10(b) and (c). There appears to be a fundamental frequency in the region of 15-150 kc/sec with large higher harmonic components. These results were observed in the presence of a transverse magnetic field which had to be greater than 3.5 kG.So far no experiments have been made to see if a similar effect occurs in a longitudinal magnetic field. Although the presence of a magnetic field is essential, the frequency was found to be independent of the size of the magnetic field but to vary with the current as approximately Z3/2.

l9

E. H. Putley, in “Phyics of Semiconductors” (Proc. 7th Intern. Conf.), p. 443. Dunod, Paris and Academic Press, New York, 1964. J. C. Haslett and W. F. Love, J . Phys. Chem. Sofids 8, 518 (1959). R. J. Phelan, Jr., and W. F. Love, Phys. Rev. 133, A1134 (1964).

304

E. H . PUTLEY

FIG. 10. Oscilloscope photograph of oscillations. (a) Relaxation oscillation. Current supplied through 100-kohm resistor; 0.02 pF condenser across sample; magnetic field 7.1 kG; sample current 120 pA; time base 10 psec/major division. (b) Magnetic field 5.33 kG;sample current 8.25 mA; time base 20 p.ec/major division. (c) Magnetic field 6.22 kG; sample current 21 mA; time base 5 pseclmajor division.

9.

FREEZE-OUT EFFECTS, HOT ELECTRON EFFECTS

305

If the transit time for electrons is calculated by assuming that all available electrons are ionized, this is found to correspond to roughly half the period of oscillation, implying that the sample behaves as a half-wave resonator. The value found for the drift velocity is about lo4 cm/sec. As this is small compared with the velocity of sound (-2 x lo5 cm/sec) this oscillation does not appear to depend on phonon interactions. Another type of oscillation which is well understood is the helicon resonance2' This requires a longitudinal magnetic field. The frequency is proportional to field but independent of current. For a helicon oscillation at 100kc/sec in the sample used, a longitudinal magnetic field of only 0.2 G would be required. Since the main transverse field is several kilogauss, a stray longitudinal component of this value could easily be present, but as its value should be proportional to the main field a frequency dependence on magnetic field rather than on sample current should be observed. So far a more detailed analysis of these results has not been made. For this to be possible Hall effect measurements are required to show how the carrier concentration changes with electric field.20a If the behavior is similar to Ge there will probably be an appreciable change before the avalanche ionization field is reached, so that this information is required before an analysis at low electric fields can be made.

III. Submillimeter Photoconductivity 4. BEHAVIORIN ZERO OR SMALL MAGNETIC FIELDS

The ionization energy of the impurity levels observed in the magnetic freeze-out experiment described in Part I corresponds to a wavelength of a few millimeters. This suggested that photoconductive effects should be observable at submillimeter wavelengths. Since the longest wavelength at which extrinsic photoconductivity can occur in Ge is about 0.1 mm, this possibility was of both scientific and practical interest. When photoconductive effects were first observed,' it was found that there was a small effect at zero magnetic field which was enhanced considerably by applying a magnetic field of 6-7 kG. There was no long wave cutoff, the effect remaining large at wavelengths of at least 8 mm. Although a magnetic field enhanced the effect it did not, at long wavelengths, alter the wavelength dependence of the effect. Some typical results are shown in Figs. ll(a) and ll(b). A. Libchaber and R. Veilex, Phys. Rev. 127, 774 (1962). '''Using a differential method, Mitchell, Putley, and Shaw [Phys. Stat. M i d i Oct. (1966)l have measured the Hall effect in large electric and magnetic fields. The results show that the behavior of the photoconductive effect in magnetic fields below 8 kG can be accounted for qualitatively by the same model as used with zero magnetic field.

306

E. H . PUTLEY 1000 .

(a)

I

0.01

Current d e n s i t y m A / c m z

1000

I

I1

0.I

(bl

,

1.0 h.M M

FIG.11. (a) Dependence of responsivity upon current density and magnetic induction: wavelength, 0.2 mm; temperature, 1.65"K.(b) Dependence of responsivity upon wavelength and magnetic induction at optimum current density.

9.

FREEZE-OUT EFFECTS, HOT ELECTRON EFFECTS

307

These results showed that the behavior was quite different from that expected for a photoionization process. Studies of microwave cyclotron resonance absorption and hot electron effects in Ge had shown that together these could produce free carrier photoconductivity.” If the free carriers absorb energy from the radiation, their mean energy will be increased, so that the nonlinearity in the current-voltage characteristic will cause a change in resistance. This process is analogous to a bolometric effect, but because only electronic transitions are involved, the response time is less than sec. Since, as Part I1 shows, marked hot-electron effects are observed in InSb, the possibility of these being responsible for the submillimeter photo~ K~gan,~~ conductivity was suggested by Rollin,” and by Lifshits et ~ l . , ’by and by others. Consider first the free carrier absorption. The classical theory of conductivity is adequate for this case. The absorption coefficient a is given by a = 4r~na/cK‘~~,

(12)

where K is the dielectric constant and o the high frequency conductivity: 0=

o,(l

+ w%,2)-1.

Here oQ= ne2z,/m*

is the low frequency conductivity and z, is the relaxation time. Thus at low frequencies (wz, < 1) the absorption is largest, but at high frequencies (wz, 9 1) it falls as m-’ (or as A’). For the high purity n-type InSb showing the freeze-out effect, typical values of the parameters at 4°K are n = 5 x 1013cm3 and z = 8.5 x sec, so that oz = 1 at a wavelength of 1.6mm and the absorption coefficient is 22cm-’ at 1 mm and 0.30 cm-’ at 0.1 mm wavelength. Thus a sample of practicable dimensions should be an efficient absorber of millimeter wavelength radiation, but the absorption will fall off as the wavelength is reduced below 1 mm. Since the photoconductive signal will vary as the absorption coefficient when this is small,I6 the behavior shown in Fig. 11 is thus accounted for. Measurement of the transmission through InSbI6 (Fig. 12) shows that, at 1.2mm wavelength, the absorption of the free electrons with zero

*’ D. W. Goodwin and R. H. Jones, J . Appl. Phys. 32,2056 (1961).

’’ B. V. Rollin, Proc. Phys. Soc. (London) 77, 1102 (1961); M. A. Kinch and B. V. Rollin, 23

24

Brit. J . Appl. Phys. 14, 672 (1963). T. M. Lifshits, Sh. M. Kogan, A. N. Vystavkin, and P. G. Mel’nik, Zh. Eksperim. i Teor. Fiz. 42, 959 (1962) [English Transl.: Souiet Phys. J E T P 15, 661 (1962)l. Sh. M. Kogan, Fiz. Tuerd. Tela. 4, 1891 (1962) [English Transl.: Soviet Phys.-Solid State 4, 1386 (1963)l.

308

E. H. PUTLEY

magnetic field is high, the value for the absorption cross section being 1.2 x lo-’’ cm2. On applying a magnetic field of about 7 kG the majority of the free electrons fall into the impurity level, which has a much smaller cross section, being estimated at 1.0 x 10-’3cm2. On reducing the wavelength to 0.197 mm the free electron cross section in zero field falls to 8 x cm’, while that of the impurity level rises to 4 x cm’. Thus at wavelengths longer than 0.2mm absorption by free carriers is the dominant

lnsb c143/14 0.8 rnm thick

40

‘0

I

2

3

4

5

6

7

8

Magnetic field KG

FIG.12. Transmission through a sample of InSb at 1.4 and 0.197 mm.

absorption process. This implies that at long wavelengths free carrier photoconductivity is the dominant process. Simple expressions for the responsivity and the response time for the free carrier photoconductive process under open circuit conditions can be obtained from Kogan’s paper.24 The responsivity R and the time constant z are R = PV/uo V/watt, (15 ) z = $p(k/e)d T / d p

sec,

(16)

where j3 is given by Eq. (ll), or, more generally, j3 = c- do/d(E2),

Vis the voltage applied to the sample and u its volume. These equations

9.

FREEZE-OUT EFFECTS, HOT ELECTRON EFFECTS

309

show that the photoconductive properties can be estimated from measurements of the electrical characteristics of the sample. The equation for R implies that all the radiation is absorbed so that it represents the long wavelength limit. Typical values for the parameters at 4°K are = 20 V-’ cm2 14

12

10

8

0.06

v)

c ._

c 2

4 LL

6

0.6

0.06 do -

dT

r

4

2

O(

psec 0.4 0.04

0.02

0.I

0.2

0.3

0.4

E vcm

FIG.13. Responsivity R and time constant 7 of hot electron detector calculated from du/dT determined from current-voltage characteristics.

and

and d T / d p = 6.7 x OK cmpz Vsec. These values give R = 130 V/watt sec, which are of the right order. A more detailed and z = 1.7 x example is shown in Fig. 13. Here the values of B and of d a / d T ( = n e d p / d T in zero magnetic field) are plotted against E, and the values of R and z

310

E. H. PUTLEY

calculated from Eqs. (15) and (16) are also given. These results show that R should pass through a well-defined maximum and that z at first increases up to values of E somewhat higher than that for maximum R. z then also passes through a maximum. These calculations may be compared with Fig. 14, which shows the dependence of R on E, found by Danilychev and O ~ i p o v and , ~ ~Fig. 15, which shows values for z measured at electric fields up to the value giving maximum responsivity. The experimental results for z confirm that initially it increases with the electric field, but they were not extended sufficiently far to show the maximum.

E v/cm FIG.14. Variation of responsivity with electric field. (After Danilychev and O s i p ~ v . ~ ' )

If free carriers ire responsible for the photoconductive effect, then the means by which the magnetic field enhances the effect has to be explained. From the current-voltage characteristics measured with an applied magnetic field, values of p and d T / d p can be obtained, which can then be used to calculate the behavior from Eqs. (15) and (16). Typical values for a magnetic field of 6 k G and at 1.3"K are G = 3.5 x 10-40hm-'cm-', a = 0.3 V - 2 cm2, V = 0.2 V, and dTJdp = 2.1 x lop4. These values give

R

=

3 x lo4 V/watt,

z = 1.0 x lo-'

sec.

Thus the effect of the magnetic field, by reducing o and increasing Vand despite the reduction in p, leads to a marked increase in responsivity and also to a reduction in time constant. Both conclusions are confirmed by the experimental data shown in Figs. 11 and 15. So far this calculation has not been extended to the dependence on electric field. Until Hall effect measurements to determine the variation of carrier concentration have been completed2'" a detailed analysis of the current-voltage curves cannot 25

V. A. Danilychev and B. D. Osipov, Fiz. Tverd. Tela. 5,2369 (1963) [English Transl.: Soviet Phys.-Solid State 5, 1724 (1964)l.

9.

FREEZE-OUT

EFFECTS,

HOT ELECTRON EFFECTS

311

be made. As with the zero field case the calculation of responsivity implies that all radiation is absorbed. At sufficiently high magnetic fields the number of free carriers will be too small, so that a falling off in performance will eventually occur. This is also shown by the results in Fig. 11.

,/:/--/. _-+-' ,====,/

,//C

s.5-

0

Specimen No. C143/14/9 Temperature 1.24 O K Wavelength 0.9 mm

0' Potential difference across detector mW

FIG. 15. Variation of time constant with electric field. (After Putley.16)

Measurements to wavelengths beyond 8 mm with the same sample show that the photoconductive effect is practically constant beyond 2 mm and that its dependence on magnetic field is the same as at the shorter wavelengths. Danilychev and 0sipovz5 find that the photoconductive effect extends to several meters, but the behavior of their material differs in some ways from that produced at the Royal Radar Establishment. Since all this work requires material at the limit of the best available purity, differences in the behavior of material from different sources are perhaps to be expected. 5 . BEHAVIOR IN LARGERMAGNETIC FIELDS

So far, we have seen that the photoconductive effects in zero or small magnetic fields can be accounted for by free carrier effects. With magnetic fields of 10 kG or greater a marked photoconductive effect is observed near the wavelength at which cyclotron resonance occurs, while at longer wavelengths the photoconductive effect is much smaller.z6 By varying 26

M. A. C. S. Brown and M. F. Kimmitt, Brit. Commun.& Electron. 10, 608 (1963).

312

E. H. PUTLEY

the magnetic field the wavelength for this resonant effect changes, as shown in Fig. 16. This figure shows that the size of the peak falls as the wavelength approaches 60pm.This is due to the presence of a restrahlen band. When a sufficiently large magnetic field is applied the photoconductive effect is observed again on the short wave side of the restrahlen band. With a field of 76 kG the resonance occurs at 26 pm. Magnetic induction K G 25

15

10

7 / //

/

/

//

Responsivity at peak

T = 4.2

/'

"K

Detector at 1.8 OK. 8.5.5 KG

Detector responsivity B=14KG. T.4.2 K \

\ I

50

100

I50

200

Wavelength - microns FIG. 16. Tunable photoconductive effect. (D* is the detectivity defined as the square root of the sensitive area divided by the noise equivalent power.)

When fields of more than 10 kG are applied practically all the electrons

will be in the impurity level. Then the most likely transition is not an interLandau level one but rather the excitation to a higher impurity state in which the electron rotates about the magnetic field axis, but still remains attached to the impurity center.27Assisted by phonon emission, the electron can now fall into the lowest Landau level. This process produces an increase in conductivity. An exact comparison between this process and cyclotron resonance has not yet proved possible, as an experiment is required in which both processes can be studied under similar conditions.

*'

H. Hasegawa and P. E. Howard, J. Phys. Chern. Solids 21, 179 (1961).

9.

FREEZE-OUT EFFECTS, HOT ELECTRON EFFECTS

313

6. OTHERPHOTOCONDUCTIVE EFFECTS In addition to bulk photoconductivity, Vystavkin et aLZ8have considered the situation in which the radiation is absorbed near the surface of a thick sample, setting up a gradient in the electron temperature. In zero magnetic field a photothermal emf is observed, but when a moderate magnetic field is applied a transverse voltage which is the photoconductive analog of the Nernst effect appears. The photothermal effect is very small, but the photo-Nernst effect is comparable with the photoconductive effe~t.’~,~~

’*

A. N. Vystavkin. Sh. M. Kogan, T. M. Lifshits, and P. G. Mel’nik, Radiotekhn. i Elektron. 8, 994 (1963) [English Transl.: Radio Eng. Electron. ( U S S R ) 8, 995 (1963)l. 2 9 Recently, Jonscher (private communication) has shown that the photo-Nernst effect is less important than the calculation of Vystavkin et al. indicated. 30 Footnote added in proof. The following papers should be referred to for more recent work: M. A. C. S. Brown and M. F. Kimmitt, Infrared Phys. 5,93 (1965); E. H. Putley, Appl. Opt. 4, 649 (1965);A. N. Vystavkin, V. N. Gubankov, and V. N. Listvin, Fiz. Tuerd. Tela 8, 443 (1966 [English Transl. : Soviet Phys. Solid State 8, 350 (1966)l.

This Page Intentionally Left Blank

CHAPTER 10

Magnetoresistance H . Weiss I. INTRODUCTION..

. . . . . . . . . . . . . . . 315

11. SHAPEOF SPECIMEN AND ELECTRODES . 111.

. . . . . . . . . 317 INHOMOGENEITIES . . . . . . . . . . . . . . . .324

IV. THE HOMOGENEOUS RODLIKESEMICONDUCTOR . . . . . . ,340 1 . Indium Antimonide. . . . . . . . . . . . . . . 341 2. Indium Arsenide . . . . . . . . . . . . . . .364 3. Indium Phosphide . . . . . . . . . . . . . . . 310 4. Gallium Antimonide . . . . . . . . . . . . . .310 5. Gallium Arsenide . . . . . . . . . . . . . . . 312 V. INFLUENCE OF PRESSURE .

. . . . . . . . . . . . . 315

I. Introduction The trajectories of the electrons in a solid body are changed in a magnetic field by the deflecting Lorentz force. Thus, even in an isotropic body, the current lines and direction of the electrical field form an angle different from zero. This angle is identical with the Hall angle 9 if the magnetic field is perpendicular to the direction of the electric current. Generally, the specific resistance of an electronic semiconductor in a magnetic field is different from that with no magnetic field. The reason for this behavior can be better understood after referring to Fig. 1. Figure l(a) shows the path of an electron under the influence of both an electrical dc field E applied in the negative x-direction and of a perpendicular magnetic field with the magnetic induction B in the z-direction. At the origin of the coordinate system, the particle has zero velocity. It moves along a cycloid in the x-y plane with the mean velocity E/B in the ydirection. In contrast to this motion in a vacuum, the electron in a semiconductor does not move in the direction of the arrow along the cycloid in Fig. l(a), because it soon undergoes a collision with a lattice atom. In this collision it loses some of its velocity [point I in Fig. lfb)]. Then it starts again on a curve like that shown in Fig. l(a). The electron is accelerated in

315

316

H. WEISS

FIG.1. Motion of an electron in crossed electric and magnetic fields: (a) in a vacuum, (b) in a solid.

the x-direction and is more and more deflected with increasing velocity in the y-direction by the influence of the magnetic field until again it loses its velocity by a collision with the lattice at point 11. This is repeated during the following collision at point 111. On the average, the electron moves at an angle 9 (the so-called Hall angle) from the direction of motion in the absence of a magnetic field. The relation between 9 and the Hall mobility p B of the electrons is given by Eq. (1): tan 9 = p B B ,

(1)

where p B is the Hall mobility which is defined as the ratio of the Hall coefficient R , and the specific resistance pB in a field with magnetic induction B :

(2) In the above considerations, the thermal velocity of the electrons was neglected. It is usually large compared to the additional velocity which the electron obtains from the external applied field. Following the linear equations of motion, this additional velocity is superimposed on the thermal velocity. Since the thermal velocity does not contribute to the current, it may be neglected. If one knows the effective mass of the electrons in the crystal and the collision probability, i.e., the correlation of the energy of the colliding electron with the time between two collisions with the lattice, as well as the distribution of the charge carriers in the allowed energy states in k-space, then the Hall angle 9 and resistance pB in the magnetic field, i.e., the ratio of electrical field strength in direction of the current to the current density, can be calculated. On the other hand, the information concerning the PB =

RJPB.

10. MAGNETORESISTANCE

317

mechanism of charge transport given by magnetoresistance is somewhat more complicated. For example, if two or more bands contribute to the conductivity, a magnetoresistance may exist even though each band by itself has none. The shape and inhomogeneities of the specimen being measured can also cause a change in resistance in a magnetic field that is different from what would be observed in a homogeneous semiconductor. For materials with a high electron mobility, these effects are very large and depend only on the Hall angle 9, but not on other parameters of the semiconductor. If one has measured an increasing resistance in a magnetic field for material with a high electron mobility, there remains the problem of finding which of the different mechanisms that result in a magnetoresistance effect is predominant. To find the simple magnetoresistance, as exhibited by a homogeneous rodlike semiconductor, can be difficult. In the following sections, how the shape of the solid body and inhomogeneities of the conductivity give rise to a change of the resistance in a magnetic field will be discussed first. After this, experimental magnetoresistance results for the different 111-V compounds and a discussion of band structure and transport mechanisms will be given.

II. Shape of Specimen and Electrodes First we must distinguish between the transverse and longitudinal magnetoresistance effects. For the former effect, the magnetic induction is perpendicular to the current direction ; for the latter, the directions of current and induction are parallel. Generally, one assumes a rodlike specimen for the theoretical as well as experimental treatment of the problem. Boltzmann found in 1886 that the geometrical shape of the specimen had an influence on the magnitude of the measured magnetoresistance. This shape influence is highly pronounced for large Hall angles, i.e., for materials with high electron mobilities. With n-type indium antimonide at room temperature, the Hall angle 9 in a magnetic field of 10,000 G amounts to about 80". From the above description concerning the reason for the Hall angle 9 between the directions of current density and electric field, one concludes that this angle is independent of the shape of the specimen for an isotropic semiconductor as long as the current is perpendicular on the magnetic induction B. To understand more precisely the influence of the shape on the measured magnetoresistance values, let us look at Fig. 2. It shows the current lines and equipotential lines in a rectangular plate made of isotropic material with current electrodes on the short edges. The magnetic induction B is perpendicular to the plane of the drawing. The mobility of the electrons in

318

H. WEISS

the electrode metal is less than that within the semiconductor by orders of magnitude. Because of this, the Hall angle in the metallic electrodes may be neglected. The electrodes are equipotential lines in a magnetic field as well as in the absence of a magnetic field. Therefore, the electrical field in the semiconductor near the electrodes must always be perpendicular to the electrodes. In the absence of a magnetic field, the current lines are parallel to the electrical field, while in the magnetic field, they are rotated by the Hall angle 9 in comparison with the vector of the electrical field. Therefore, the current lines going into the semiconductor in a magnetic field are no

I-

FIG.2. Equipotential lines (--) and current paths (-) for a semiconductor in the form of a rectangular plate with the magnetic field normal to the plane of the drawing.Current electrodes are at the ends of the plate, and contacts 1 and 2 are potential probes.

longer perpendicular to the electrodes but are at an angle 9 with the direction of the current lines obtained with no magnetic field. The boundary condition along the edges of the semiconductor plate is just opposite, since the charge carriers cannot leave the plate. The currents there do not have a transverse component either with or without the magnetic field. Thus, they must always move along the edge of the plate. At the center of the semiconductor plate, the electrical field is rotated by the Hall angle 9 compared with its direction in the absence of a magnetic field. As discussed above, the specific resistance both with and withoyt a magnetic field is always measured in such a way that the electric field is measured along the current lines. It follows from Fig. 2 that the potential leads must be arranged so as not to include the area near the electrodes where the current lines are not parallel to the length direction of the plate. The influence of the current electrodes on the magnetoresistance effect is the smaller the larger the distance between them and the measuring probes.

319

10. MAGNETORESISTANCE

If the distance of the probes from the current electrodes is more than twice the width of the specimen-and the length of the specimen is at least 10 times its width-the shorting effect of the current electrodes on the measured magnetoresistance is negligible. This experimental result was confirmed by the calculations of Drabble and Wolfe.’ Indium antimonide, which possesses the highest known electron mobility and which, on the other hand, does not exhibit a significant increase of the specific resistance in a magnetic field, is an ideal model for studies of the

20

15 RB/RO

10

5 1

O:;

i d

t B; i ; i

;16IG

FIG.3. Relative resistance of four specimens of InSb of equal doping but of different shape as a function of magnetic induction at room temperature. Specimen shapes are indicated on the curves. (After Weiss and Welker.’)

influences of the geometric shape on galvanomagnetic effects. A large magnetoresistance effect is observed in specimens with a high ratio of electrode width to electrode separation. Figure 3 shows the ratio of the resistances obtained with (Rs) and without a magnetic field (R,) for specimens of indium antimonide with the same carrier concentration but different length-to-width ratios as a function of B. The magnetoresistance increases with decreasing electrode distance and approaches a limiting value which is ideally obtained with a disk-shaped specimen (Corbino disk).’ For the lowest curve, the ratio of specimen length to electrode width is 10. For the next curve this ratio is 1 : 1 and for the J. R. Drabble and R. Wolfe, .I Electron. . 3, 259 (1957). H. Weiss and H. Welker, Z . Physik 138, 322 (1957).

320

H. WEISS

second from the top 1:3. In a disk (upper curve) the equipotential lines remain radial in a magnetic field, and the current lines are logarithmic spirals. The circular disk with concentric inner and outer electrodes is not the only shape that gives the higher limiting value for the magnetoresistance by geometry. The condition for obtaining this limiting value is only that the equipotential lines both with and without a magnetic field must have 15 14 13 12

11 10

9

I

I

1

I

I

2

I

fan fl

I

I

3

I

4

I

5

FIG.4. Relative resistance as a function of the tangent of the Hall angle for different lengthwidth ratios of specimen. (After Lippmann and K ~ h r t . ~ )

the same position. Therefore, the current lines are rotated by the full Hall angle. This is always the case if the semiconductor is a single coherent area and if one current electrode is surrounding the other. The circular disk is a shape which satisfies the desired boundary conditions and which, at the same time, is the simplest and is easy to manufacture. The calculations of the influences of the shape of the electrodes on the measured magnetoresistance were made by Lippmann and K ~ h r t With .~ a method of conformal mapping they found an integral which can be H. J. Lippmann and F. Kuhrt, Z. Naturforsch. 13a, 462 (1958).

10. MAGNETORESISTANCE

321

developed for small and large Hall angles. The transition region between small and large angles was calculated by numerical evaluation of the integrals. The results of these calculations, shown in Fig. 4, correspond quantitatively with the experimental results of Fig. 3. The values R$Ro, shown in Fig. 3, are not the highest possible with indium antimonide ; for an appropriate doping still higher values can be obtained (compare Fig. 5). The highest magnetoresistance effect is given by a circular disk, the socalled Corbino disk. The resistance R E of the Corbino disk in a magnetic

FIG. 5. Calculated (dashed curves) and measured (small circles) values of the relative resistance R,/Ro at room temperature for field disks (Corbino disks) as a function of the conductivity of the InSb for two values of magnetic field. (After Weiss.’)

field B related to the resistance R , with no magnetic field can be approximated by the following formula :

where p E / p o is the ratio of the specific resistance in field B to the resistance with no field; it is measured on a long rod. Knowing p E / p o and Hall mobility p B , one may determine the magnetoresistance effect of the Corbino disk. The electron mobility and pB/po depend on the doping of the semicond~ctor.~ H. Rupprecht, R. Weber, and H. Weiss, Z. Naturforsch. 15a, 783 (1960).

322

H. WEISS

Figure 5 shows the calculated and measured values of RJR0 for Corbino disks made of the indium antimonide as a function of the specific cond~ctivity.~ It can be seen that disks of large and small diameters exhibit exceptionally large values which correspond to the theoretical expectations.

FIG.6. Relative resistance Rs/Ro of field disks for two values of conductivity of InSb as a experimental data, u = 240(ohm-cm)-’; Function of the magnetic induction B : experimental data, u = 800 (ohm-cm)-’ ; -, calculated curves for each of the above. (Unpublished measurements, H. Weiss.)

-O-o-o-,

+++,

The highest factor measured at 10,000G is 38. The disks of Fig. 5 were fabricated from single crystals which were pulled in the [ 1111 direction. They were cut in a plane perpendicular to the pulling direction. The test rods for measuring the ratio p B / p o and pB lay perpendicular to the [ill] axis. The measured values of test rods as well as of Corbino disks were not influenced by inhomogeneities such as those discussed by Bate.6 The two H. Weiss, J . Appl. Phys. Suppl. 32, 2064 (1961). R. T. Bate, “Electrical Properties of Nonuniform Crystals” (to appear in a subsequent volume of this series).

10. MAGNETORESISTANCE

323

dashed lines in Fig. 5 were calculated from the measured values of pB/po and pB for magnetic fields of 6OOO and 10,OOOG . In large magnetic fields, the contribution of the holes to the conductivity is not negligible. That is the reason why the maximum of both curves in Fig. 5 does not lie in the intrinsic range with 0 = 200 (ohm cm)-' at 300°K but near CT = 300 (ohm cm)-' even though the electron mobility is smaller than for intrinsic material. The maximum is shifted to higher values of the specific conductivity as the magnetic field increases. Thus, an increase in resistance to the 200-fold value is obtained for intrinsic material in a field of 100 kG, while a factor 260 for 0 = 800 (ohm cm)-' is obtained with only 40 kG in good agreement between calculation and experiment (Fig. 6). To obtain the magnetoresistance Ap/po, the potential probes have to be put, as mentioned above, inside the current electrodes. They should be

- / / m/ / / FIG.7. Principle of the field plate consisting of a semiconductor with short-circuit strips. Current paths are indicated by arrows. (After Welker.')

point-shaped as far as possible. Soldered probes as well as lateral arms on the specimens generally have an appreciable area and distort the current lines in a manner which depends on the magnitude of the magnetic field and may even lead to a longitudinal magnetoresistance. In a longitudinal field of 10,000G with indium phosphide, Glicksman' found a magnetoresistance with soldered probes which was 10 times higher than with pointshaped probes. The fact that a semiconductor with a high electron mobility, and with an appropriate shape, exhibits a large increase in resistance in the magnetic field can be used for fabricating devices, the resistance of which is changed in a magnetic field. For a circular disk, the Corbino disk, one gets the highest influence of the magnetic field on the resistance. For the application of the magnetoresistance in the usual electronic circuits, especially in connection with transistors, generally, one needs resistances of the order of 100 to 1OOOohms. With the Corbino disk, this requirement cannot be fulfilled because of the high specific conductivity of indium antimonide. Units with resistances of more than 1 or 2 ohms in the absence of a magnetic field are difficult to fabricate. Hence, another shape for the field-dependent

' M. Glicksman, J . Phys. Chem. Solids 8, 511 (1959).

324

H. WEISS

resistance must be chosen. An alternative consists of a thin, long semiconducting platelet which is mounted on an insulating support. The plate is covered with short-circuiting strips' as shown schematically in Fig. 7. One may regard the device made in this way as a series of single semiconducting platelets for which the ratio of electrode width to electrode distance is large. In this manner a magnetoresistance is obtained which is about half the value resulting from the circular disk configuration. In a field

1-

....... . .-. -. . ...... ........

FIG.8. Rotation of the current lines in a magnetic field at a discontinuity in the doping level for B normal to the plane of the paper. Upper half: No rotation of the current lines, no change in resistance. Lower half: Rotation of the current lines by the angle a,increase in resistance. (After Weiss.")

of 10,OOO G an increase in resistance to twenty times the zero-field value can be achieved. Large variations of the resistance are possible by varying the length, width, and thickness, as well as the number of platelets in the series. HI. Inhomogeneities There does not exist a semiconductor which is completely homogeneous. The inhomogeneity may be a concentration step, that is, a sudden change in carrier concentration but with no change in the type of conductivity. There may also be periodically changing concentrations or gradients of concentrations. In the following section it will be shown how these inhomogeneities influence the specific resistance of a semiconductor H. Welker, Elektrotech. Z. 76, 513 (1955).

10. MAGNETORESISTANCE

325

in a magnetic field. To demonstrate these influences, again indium antimonide is a very appropriate model. Figure 8 helps one understand how the changing resistance near a concentration step arises. The perpendicular line in the middle separates two areas of different electron concentrations; on the right side, the concentration is larger. It is known that the electron mobility in indium antimonide depends only slightly on doping. For simplicity, let us assume the mobility in both areas is identical and independent of magnetic field. Then one gets the same Hall angle in both areas. Since the current densities on both sides of this boundary must be identical, the electrical field strengths left and right of the border are inversely proportional to the electron concentrations and in the same directions as the current in the semiconductor with no magnetic field. By adding a magnetic field perpendicular to the plane of the drawing, one gets a transverse electric field. If the current lines on both sides of the concentration step maintain their direction also in a magnetic field, the result is that the Hall field strengths E , and E,,,, indicated in the upper part of Fig. 8, are not identical. This is not in agreement with the condition curl E = 0. The current lines must be rotated by the same angle o! in order to get identical projections of the electrical field strength on the border line from the right and left regions. In this way, the electrical field strength in the right area, which has a higher concentration, is rotated by more than the Hall angle of the homogeneous semiconductor. Thus, the component of the resulting field strength perpendicular to the border line is diminished, while in the left area with the lower concentration it is increased. Figure 9 gives results of measurements which were made by Schonwald on a single-crystal n-type specimen of indium antimonide. The electron concentration increased from 3.7 x 10”/cm3 to 2.5 x 101*/cm3 over a very short di~tance.~ The upper part of Fig. 9 shows the measured electrical field strength along a line in the middle of the specimen with and without a magnetic field. The measurement was made with small point probes. In the left part, with lower concentration, the electrical field strength in a magnetic field increases as the border line is approached. It is always appreciably greater than for the case with no magnetic field. In the right part, with higher concentration, the field strength in the magnetic field is always smaller than it is with no magnetic field, and near the border it even becomes negative. The results of these measurements are identical for positive and negative magnetic field. The negative potential difference is easily explained. A small rotation of the current lines is sufficient to produce a negative potential difference in the direction of the current lines in the absence of a magnetic field because there is only a small angle between current and equipotential lines.

326

H. WEISS

Bate et al.’ have calculated the current densities and potentials near a concentration step and given a solution which consists of a complicated power series for the general case. By measuring the electrical field strength, not in the middle of the plate, but as usual along the edge with potential probes in the plane of the -

cm

n=3.7.1017crn-3

n=2.5-1018crn-3 cm2 p=16,000V-sec

8=’5 kG 1=100 mA

E

B=O

2-

0

1.200

1.400 uV

1.325 1.365

I

I

1

I

,

1.5

1

0.5

0

.

I

0.5

I

1

I

1.5 rnrn

FIG.9. Top: Electric field intensity E along the center of a specimen containing a doping discontinuity, for B = 0 and B = 15,000 G (B normal t o the paper) and a current of 100 mA. Bottom: Measured equipotential lines (- - --) and current lines (-) constructed from a knowledge of the Hall angle. (After Weiss.’)

drawing, one gets different, and also negative, values of the potential variation for both directions of the magnetic induction. By averaging the measured values for positive and negative magnetic induction, one always gets a positive magnetoresistance. Figure 10 demonstrates how careful one must be in measuring magnetoresistance on materials with a high electron R. T. Bate, J. C. Bell, and A. C. Beer, J . Appl. Phys. 32,806 (1961),

10. MAGNETORESISTANCE

327

mobility.'O In this case, the specimen has an electron concentration of 5.5 x lOl6/cm3. It was cut from a single crystal perpendicular to the pulling direction. The Hall coefficient exhibits a variation of 15% along the rod. In the figure, the relative potential is given as the function of a transverse magnetic field up to 30,000G. The symbols 1-3 and 2-4, respectively, on the measured curves indicate that the voltage was measured between points 1 and 3, and between points 2 and 4,as given in the upper part of the figure. The data of the full lines were measured in the usual position for transverse magnetoresistance with measuring probes perpendicular to the direction of the magnetic field. The data of the dashed curves were measured with the arrangement of measuring probes in the plane of the magnetic field. With the latter arrangement of the probes, both positive and negative values of the transverse magnetoresistance were achieved. With the usual probe arrangement only positive values were obtained. This figure exhibits two interesting features : First, the curves 1-3 and 2-4 exhibit a great difference in magnetoresistance. Also, the magnetoresistance of the homogeneous semiconductor is certainly smaller than is shown by curve 2 4 . Second, the dashed curves exhibit partly negative, partly positive values of the changing resistance. This suggests an oscillatory behavior of the resistance. Steps and gradients in the carrier concentrations can be analyzed in terms of gradients of the Hall coefficient. This is not the case with periodic variations of the concentration, for a period length smaller than 0.1 mm. They may be recognized by etching"*12 but not quantitatively. Measurement of electrical conductivity in single layers is not possible. Some information can be achieved by measuring the infrared absorption with a mi~roprobe.'~The basic difficulties in studying inhomogeneities is that they generally are not reproducible. Muller and WilhelmI4 were able to detect temperature oscillations in the liquid semiconductor, which produce periodic variations in impurity concentrations in indium antimonide. They were also able to indicate a method for producing a homogeneous crystal. Such variations in impurity concentrations give rise to a strong anisotropy in the magnetoresistance for polycrystalline as well as for single-crystal samples of indium antimonide. For example, values of the transverse magnetoresistance vary as much as two orders of magnitude for specimens cut in different directions from a single crystal of indium antimonide. However, there cannot be an anisotropy in the homogeneous semiconductor, because of the spherical energy surfaces of the conduction band. H. Hieronymus and H. Weiss, Solid-state Electron. 5, 71 (1962). W. P. Allred and R. T. Bate, J . Electrochem. SOC.108, 258 (1961). H. C. Gatos, A. J. Strauss, M. C. Lavine, and T. C. Harman, J . Appl. Phys. 32, 2912 (1962). " F. R. Kessler and H. J. Metzger, Phys. Kondensierten Materie 1, 263 (1963). l4 A. Miiller and M. Wilhelm, Z . Naturforsch. 1%, 254 (1964). lo

328

H. WEISS

Figure 11 shows typical results for the dependence of the transverse magnetoresistance on the orientation of the specimen^.^ The measurements were made on rods cut in different directions from a single crystal which was pulled by Czochralski method in the [ l l l ] direction with a pulling

% 100

50

A VIVo 0

-50

-100

lo

B

20

IkG

FIG.10. Relative potential change AV/Vo as a function of the magnetic induction B. The specimen w a s cut normal to the direction of pull of the crystal, and potential probes were perpendicular t o the plane of B (-), and in the plane of B (- -- -). The two figures on each curve indicate the positions of the measurement probes on the specimen according to the sketch. (After Hieronymus and Weiss. lo)

rate of about 1 mm/min. The highest value of Ap/po is obtained for rods parallel to the growing direction of the crystal. One gets the smallest values if the specimens are cut perpendicular to the direction of growth. For the other directions the values lie between these limiting values. The crystallographic directions [ 1111 and [ 1111, which are equivalent for

10. MAGNETORESISTANCE

329

ordinary magnetoresistance, are seen to exhibit different values for Ap/p, in these nonhomogeneous specimens. The same is true for the equivalent directions [ 1121 and [ 1121. Rupprecht” has found that, if the pulling rate used for the growth of the crystal is increased, the magnetoresistance is smaller and the ellipse, shown in Fig. 11, is reduced to a single point. Similar behavior was also

FIG.11. Transverse magnetoresistancein single crystals of InSb for different angles between the direction of cut of the sample and direction of growth of the crystal. The direction of growth is parallel to the [I 111 direction, and R, = - 15 cm3/C, B = 10 kG. (AEter Rupprecht et ~ 1 . 4 )

found in crystals which were pulled in the [113] and [lo01 directions. Doping with selenium instead of tellurium results in the same behavior. Apparently, this type of magnetoresistance is an effect which is caused by the manner of pulling the crystal and which is not a characteristic property of indium antimonide. The reason for this high anisotropy of magnetoresistance is a periodic structure of the donors in layers of varying concentration. These lie perpendicular to the growing direction. They appear H. Rupprecht, 2. Naturforsch. M a , 395 (1961)

330

H . WEISS

in polycrystalline as well as in single-crystal material and are practically independent of the crystallographic orientation of the growth direction. In contrast to the behavior described above, in intrinsic indium antimonide the same values for the increase of resistance in the magnetic field are always obtained, independent of the orientation of the specimens and the arrangement of the probes.4Jo These results are reasonable because the

(b) FIG.12. (a) Current lines in a magnetic field in a sample with the length axis normal to the layers of equal doping. (b) Current lines with and without magnetic field in a sample which was cut at an angle 6 to the normal t o the doping layers.

effect of donors is negligible in an intrinsic specimen and thus it would not be expected to exhibit any effects related to inhomogeneities in donor concentrations. This model was confirmed by comparing the calculations of Herring16 and etching experiments made by Muller and Wilhelm.I4 Herring calculated the galvanomagnetic effects for a stratified medium with periodically varying doping levels. The results are valid for a material in which the electron concentration varies periodically and the electron mobility remains constant. The latter assumption is reasonable since the mobility depends only slightly on the electron concentration? For a stratified medium one has a series of steps of the type demonstrated by Figs. 8 and 9. In a rod with striations along the length, the current lines in the absence l6

C. Herring, J . Appl. Phys. 31, 395 (1961).

10.

331

MAGNETORESISTANCE

of a magnetic field lie parallel to the edges of the specimen. With a magnetic field perpendicular to the plane of drawing, current lines with the longer zigzag path behavior demonstrated in Fig. 12(a) are obtained. If there is no longitudinal as well as no transverse magnetoresistance in a homogeneous semiconductor, one can deduce from Herring's calculations the following resistance tensor for a stratified medium' :

[+

A = ( ( n ) (l/n) - 1) 1

pZ(B2

I.

- B,2)

1 + (PBJ2

The x-axis is perpendicular to the layers. The quantities (n) and (l/n) are the spatial averages of the electron concentration n and of l/n. The tensor in (4) is composed of the tensor of the isotropic material corrected by the factor n / ( n ) and an additional tensor with only one term different from 0, which describes the strong anisotropy of the resistance in a magnetic field. At the same time, Eq. (4)contains the dependence of the resistance on the angle cp between current and magnetic induction. The magnetoresistance is not proportional to sin2rp. This has been confirmed by experiments. For example, Fig. 13 shows a comparison of calculated and measured magnetoresistance as a function of the angle cp for a specimen cut perpendicular to the layers in the x-direction ( [ l l l ] direction). A longitudinal magnetoresistance was not found in this specimen, which has an electron concentration of 5.4 x 1016/cm3. It is important to note that for small magnetic fields the dependence is almost proportional to sin2cp, but in high magnetic fields it becomes singular. The relation between magnetoresistance Ap/p, and angle between magnetic induction and current is given by Eq. (5), when the current is in the x-direction:

Po

If the magnetic field has no longitudinal component (B, = 0), the magnetoresistance is proportional to B2. The addition of a longitudinal component of the magnetic field decreases the rotation of the current lines for the same

332

H. WEISS

0

30

60

90

120

150

1800

P FIG.14. Magnetoresistance as a function of angle 9 between current and magnetic field of two specimens (-and --) of InSb with u = lOOO(ohm-cm)-', cut at two different angles (20" and 45") with respect to the direction of pull of the crystal. Maxima of comparable size belong to the same specimen, and the respective data were taken for B moving in two different planes, perpendicular t o each other, as 9 traversed the range 0 I 9 I n. (After Weiss.')

10. MAGNETORESISTANCE

333

transverse component, and the current obtains a component in the direction of B. For higher magnetic fields, therefore, one obtains an almost singular dependence on cp. Comparison of calculated and measured curves shows that the striation model is valid for indium antimonide. For B = l0,OOO G and p = 50,000cm2/V-sec (i.e., p B = 5 ) and a concentration difference of 20% between two layers of equal thickness, an increase in resistance of 31% is obtained from Eq. (5). This is the observed order of magnitude. This indicates that in indium antimonide the usual variations in the doping level are of the order of 20%. The validity of the model of stratified structure is evident if one cuts a specimen at an angle to the growth direction. By rotating the coordinate system, the tensor in Eq. (4) can be calculated for any direction of magnetic field B and a current density j. In rotating the coordinate system, the first tensor on the right side of (4) retains its appearance with the transformed components of B, but each term of the second tensor generally depends upon B2.If we cut a specimen with an angle 8 to the normal on the layers, as shown in Fig. 12(b), the current lines exhibit a zigzag behavior, without magnetic field. This gives a higher resistance than is obtained in the case of Fig. 12(a), and the magnetoresistance is given by Eq. (6): Ap Po

-=y

p2(B2 - B;) 1 @BJ2 '

+

For all values of 8, the maximum of the magnetoresistance is reached when the direction of B lies in the layers. Its value is independent of the particular direction of B within the layers. For 0 not equal to 0 or 90°, there is always a longitudinal magnetoresistance effect. The resistance does not increase if the magnetic induction is perpendicular to the layers. By measuring the magnetoresistance effect as a function of the angle between the current and the magnetic field in a specimen cut according to Fig. 12(b) with the normal to the layers lying in the plane of the drawing, one can represent A p / p o for the case when B is also in the plane of the drawing by Eq. (8) below :

The shape of the curve for the magnetoresistance as a function of the angle cp is identical with that for 8 = 0. The maximum is no larger at 90" but is shifted by the angle 6. If B is not restricted to the plane of j and the normal to the layers, there is no angle cp, for a nonzero 8, for which the

334

H. WEISS

magnetoresistance vanishes completely-as is apparent from Eq. (6) inasmuch as B > B,. Furthermore the shape of the curve depends on the angle 8. A reasonably simple case to consider is when B lies in the plane formed by j and the normal to the plane of Fig. 12, so that Eq. (6) becomes AP _ Po

+

sin3cp cos'cp sin' 8 y(pB)Z1 + (pB)2 cos2cp cos28'

(84

In Fig. 14 is shown the dependence of the magnetoresistance upon the angle cp for 2 single-crystal specimens. They were cut from the same single crystal at an angle of 20" and 45" to the pulling direction ([ 1111-direction) and had a Hall coefficient R , = - 55 cm3/C. The Hall coefficient along these rods is practically constant. The dependence of the magnetoresistance upon the angle between current and induction B was measured in two planes perpendicular to each other for both specimens. In agreement with the calculations the amplitudes of both maxima were almost identical. The difference was only 10%. The half-width of the maximum positioned nearer to 90" is larger than the half-width of the maximum positioned farther away. The minima do not lie at 0" and 180" but between 150" and 170". If cpl and cp2 are the angles corresponding to the two maxima of one specimen, the angle 8 between specimen axis and normal to the layers of varying concentration is given by Eq. (9) below :

The preceding relationship is simply established by recalling from Eq. (6) that the maxima obtain when B lies in the layers. Thus unit vectors along the current and along the two directions with cp, = 90" and cp2 = 90" in the two planes of measurement may form an l,m, n Cartesian base system. Then the unit vectors MI, M, along B in the maximum positions may be written as follows:

M,

M,

=

=

coscp,l

+ sincp,

m,

coscp,l+ sincp,n.

The product [MI x M,] gives a vector N parallel to the normal to the layers. The components are : N,

=

sin cpl sin cp,,

N,

=

-coscp, sincp2,

N,

= -

sincp,coscp,.

(9b)

10.

335

MAGNETORESISTANCE

The angle 8 is given by cos 9

=

+

(sin'q, sin'q,

sin q1sin q z cos'q, sin'q, sin'q, cos2q,)""

+

which leads at once to (9). It follows from Eq. (9) that respective values of 8 for the two specimens are 51" and 68", indicating an angle of about 20" between the pulling direction and the normal to the layers. This is also in good agreement with the results obtained on a third specimen cut parallel to the pulling direction-not shown in the figure-and for which both maxima were at about 70". This crystal was pulled without rotation. The above considerations demonstrate how, with a material of high electron mobility, e.g., indium antimonide, one can measure a magnetoresistance effect even if p B = p o . In addition, it is possible to separate the magnetoresistance of the homogeneous isotropic material from the

............

.......

...........

..........

Y-L

.........

.........

E

500& 0.

m

400 1 n

"0'

I

90'

..........

..........

180"

0"

90'

180'

0'

90'

180"

FIG.15. Magnetoresistance of high-purity InSb at 77°K and 6 k G as a function of angle cp between current and magnetic field. Three samples were measured; the direction of cut, the electron concentration, and the electron mobility are given at the top. The upper series illustrate the observed data, measured in two different planes lying perpendicular to each other; the central series indicate the magnetoresistance due to the inhomogeneities ; and the lower series show the magnetoresistance of the homogeneous semiconductor. (After Weiss.17)

336

H . WEISS

magnetoresistance effect caused by inhomogeneities. To do this, at least two specimens must be cut in two different directions from a single crystal and measurements made of the dependence of magnetoresistance on the angle between magnetic induction and direction of the specimen. Figure 15 shows a measurement of pure indium antimonide for which it was possible to separate the different effects. The indium antimonide used was a single crystal with an electron concentration of about 7 x lOI3/cm3 in a field of 6000 G and at 77°K. The upper horizontal series demonstrate the measured relative increase in resistance as a function of the angle between magnetic induction and electric current for 3 different specimens from the same single crystal. The direction of cutting is given at the top of the figure where in each case the single crystal was grown from left to right. The left rod was cut parallel to the growing direction, the center one at an angle of 45", and the right one perpendicular to the growth direction. Below each rod, the electron concentration and mobility are indicated. They were calculated from the Hall coefficient at 6000G. Between lo00 and 6OOOG the Hall coefficient changed by less than 1%, but increased somewhat with decreasing magnetic induction below 1OOOG. The values of n and p given for the rods show that the single crystal was homogeneously doped. Two curves were measured for each specimen. The difference between these curves is that the angle between specimen and magnetic field was varied in two planes perpendicular to each other. The comparison of the three upper curves shows that one has to do with the superposition of two effects, one "true" magnetoresistance which is equal for all the specimens and a "nonhomogeneous" magnetoresistance effect which depends on the position of the specimen. The middle and the lower series attempt to separate the measured curves of the upper series into these two effects. The middle series shows the magnetoresistance caused by inhomogeneities, while the bottom series shows the magnetoresistance effect of the conduction band which is characteristic of homogeneous n-type indium antimonide. The middle series indicates that the normal to the layers was rotated by about 10" to the growth direction of the single crystal. Both maxima lie at about 90" for the left crystal, at 60" and 135" for the center crystal, and the right one does not exhibit a maximum. The magnetoresistance shown in the bottom series is the same for all three specimens and amounts to about 300%.'7 The experiments shown in Fig. 15 demonstrate the significant information a few measurements of the transverse magnetoresistance effect alone can provide. H. Weiss, Izv. Akad. Nauk SSSR, Ser. Fiz. 28, 969 (1964) [English Transl.: Bull. Acad. Sci. USSR, Phys. Ser. 28, 872 (1964)l.

10. MAGNETORESISTANCE

337

Bate and Beer18s'9 calculated the influence of a longitudinal gradient of electron concentration on the transverse magnetoresistance in a rod with infinite length in the x-direction. They assumed the body to be isotropic, the electron mobility independent of the electron concentration, and the magnetic field to be homogeneous. From the condition div j = 0, for a two-dimensional case with B directed parallel to the z-axis, the following equation for the potential I/ is obtained :

where K(x) = (l/n) dn/dx is the relative change of the electron concentration along the x-axis independent of y . If K is also independent of x and a constant, Eq. (10) may be separated, and one obtains the solution

Also in this case the equipotential lines remain straight lines which are rotated in a magnetic field by the Hall angle 9 as in the case of a homogeneous semiconductor. Their density decreases exponentially with x and y as well as with the electrical field strength. Figure 16 is a diagram of a specimen with equipotential lines corresponding to Eq. (11).

FIG.16. Equipotential lines (- -) and current lines (-) in the magnetic field in a small rod with a constant relative impurity gradient in the length direction; B is normal to the paper.

The current density indicated in Fig. 16 is independent of the x-direction, but is larger along the one edge of the specimen (lower part in the figure) l8 l9

R. T. Bate and A. C . Beer, J . Appl. Phys. 32,800 (1961). R. T. Bate and A. C. Beer, Proc. Intern. Conf. Semicond. Phys., Prague, 1960 p. 177. Czech. Acad. Sci., Prague, 1961.

338

H . WEISS

than along the opposite edge. From Eq. (11) one obtains, for the current density j , ,

Since, according to our assumptions, p is proportional to e-&, J , depends only on y . For large values of pB, the shift of the current lines is very large, and the differences in the current density along the two edges of the sample can amount to an order of magnitude. From Fig. 16, it can be seen that the probes on opposite edges measure different potential differences. With the upper pair (3-4) one obtains a decrease, while with the lower pair (1-2) an increase of potential difference in a magnetic field is obtained, corresponding to the decrease and increase, respectively, of the current density along the edges. By reversing the polarity of the magnetic field, the current lines are shifted to the opposite edges of the figure, and the potential differences measured with the probes are interchanged. Averaging both values of a pair of probes delivers a positive value of the measured variation in potential, as was discussed in the above case with a concentration step. However, by measuring with two probes in the middle of the specimen, as indicated in Fig. 16 by two crosses, a negative potential difference for both directions of B is obtained, since the current density in the middle is smaller for both directions of the magnetic field than with no field. The calculations, of Bate and Beer, show that one measures an increase in resistance of S%, with probes at the edge of a specimen which has a change in electron concentration of 20%/cm, a width of 1 cm, a mobility of 50,000 cm2/V-sec,and is in a field of 10,000 G ; and one obtains a decrease in resistance of 4% with probes in the middle of a specimen. These numbers demonstrate virtually that, in a body with a high electron mobility, only a small gradient in concentration may give rise to a distinct apparent magnetoresistance effect. The vanishing of the transverse current density j,, in the whole volume requires that the relative concentration gradient K ( x ) in Eq. (10) be a constant. Generally, this is not the case4specially near a concentration step. If one has not only a longitudinal but also a transverse gradient of electron concentration, the values of A p / p , measured with both pairs of probes (1-2) and (3-4) in Fig. 16 will not agree. Muller and WilhelmI4 have shown the reason for the variations in impurity concentration in indium antimonide as well as a possible means for avoiding them. They were able to correlate conditions of preparation, etch figures of concentration variations, and electrical measurements of magnetoresistance. They were able to establish that with occurrence of

10.

MAGNETORESISTANCE

339

etch figures there is also a magnetoresistance effect which is caused by striations. When they prepared homogeneous crystals that did not show etch patterns, the magnetoresistance effect did not indicate inhomogeneities.

FIG.17. Photomicrograph of an InSb-NiSb eutectic, with direction of growth parallel to the axis of the NiSb needles (enlargement 500x). Left: Micrograph parallel to the direction of growth. Right: Micrograph perpendicular to the direction of growth.

The most pronounced type of inhomogeneities was found in conductors with inclusions of metallically conducting phases which are not soluble in the basic material. Weiss and Wilhelm'' were able to produce welldefined anisotropic structures. Utilizing the indium antimonide-nickel antimonide eutectic which lies at 1.8 % by weight nickel antimonide, they were able to align the needle-shaped nickel antimonide inclusions parallel in the whole crystal. Micrographs of these two-phase semiconductors are shown in Fig. 17. On the left side is a cross section parallel to the nickel antimonide needles and on the right side a cross section perpendicular to the needles. The effect of this alignment on the magnetoresistance is analogous to the fieldplate structure of Fig. 7. Instead of short-circuiting strips on the semiconductor, we now have parallel short-circuiting needles H. Weiss and M. Wilhelm, 2. Physik 176, 399 (1963).

340

H. WEISS

in its interior. The structure is by far more subtle, since the average distance between the 50-,u long needles is only about 3 4 p . Two-phase intrinsic indium antimonide exhibits a 20 times higher resistance in a magnetic field of 10,000 G than with no magnetic field. Similar structures can be produced in indium antimonide with inclusions of FeSb or CrSb." The two-phase indium antimonide may be used for the production of magnetically controllable resistances, and results are similar to those obtained using the geometry shown in Fig. 7.21a

IV. The Homogeneous Rodlike Semiconductor

In the two preceding sections it was shown how shape and inhomogeneities give rise to a change of resistance in a magnetic field. Both effects increase with an increase in the Hall angle. However, one often desires to use the magnetoresistance effect of the homogeneous semiconductor in studying the transport properties in solid state physics. In semiconductors like germanium and silicon the increase in resistance is considerable because of the many-fold energy minima in the conduction band. In contrast, the magnetoresistance effect caused by shape and inhomogeneities can be neglected because the electron mobility is not very high. With 111-V compounds just the opposite is true. The compounds with a high electron mobility, like indium antimonide, indium arsenide, gallium arsenide, indium phosphide, have a simple spherical conduction band with a minimum at k = 0 and a very small magnetoresistance effect for n-type conductivity. There exists then first the problem of separating the different mechanisms for an increase in resistance. The influence of shape can be eliminated, as mentioned in the first section, by using a crystal with a high ratio of length to diameter and a large distance of the potential probes from the current electrodes, i.e., a rodlike semiconductor. As for the inhomogeneities, one has to distinguish between gradients in doping and quasiperiodical variations in doping. Concentration gradients can be found by measuring the Hall effect and by the fact that the values of ApIpo for the two directions of the magnetic field are different and depend very strongly on the position of the potential probes. It is possible to determine periodic inhomogeneities by e t ~ h i n g " , ' ~ . ' ~ or by measuring the inhomogeneous infrared absorption.' These methods are complicated and are usually not quantitative or reliable enough. For instance, the absence of striations in etching does not always prove the absence of variations in doping. The most reliable method was described 2 1 B. Paul, H. Weiss, and M. Wilhelm, Solid-state Electron. 7, 835 (1964). ""H. Weiss, Elektrotech. Z. 17, 289 (1965).

10. MAGNETORESISTANCE

341

with reference to Figs. 14 and 15. With this method, directions which are crystallographically equivalent and lie at different angles to the growth direction of the crystal, are examined by measuring the magnetoresistance effect as a function of the angle between axis and magnetic induction. In this way, one is able to separate the magnetoresistance of the homogeneous crystal from the influences of inhomogeneities. Also, it is possible to produce homogeneous crystals by maintaining certain conditions of temperature and pulling rate during production of the cry~ta1.l~ In the following section, the magnetoresistance effect of indium antimonide is discussed first, because it is the best known of the compounds, and because with this compound it was possible in many cases to separate the possible mechanisms for magnetoresistance by preparation or by special measurements. 1. INDIUM ANTIMONIDE u. Mixed and Intrinsic Conductivity

Figure 18 shows the transverse magnetoresistance of single crystals of indium antimonide for a magnetic field of 10,000G at 295°K as a function of the concentration of electrons and holes. The figure combines the results of Hilsum and Barrie” on p-type indium antimonide and of Rupprecht et a1.4 on n-type indium antimonide. The specimens with an electron concentration higher than the intrinsic concentration of n, = 1.6 x loi6/ cm3 were homogeneous single crystals doped with tellurium. The p-type specimens were prepared by doping with zinc. In the range of intrinsic conductivity, the relative magnetoresistance amounts to 55 % and decreases with increasing electron concentration to below 1% for n = 4 x 1017/cm3. In the transition range from intrinsic to p-type conductivity, at first, Ap/p, is still increasing, but decreases again, after a maximum of 160%, for higher concentrations. Longitudinal magnetoresistance was never observed at room temperature. Figures 19 and 20 give information concerning the dependence of transverse magnetoresistance on magnetic field. Figure 19 shows the dependence of magnetoresistance for one intrinsic and two slightly p-doped specimens up to fields of 150 kG. For the intrinsic specimen number 80, Ap/p, in small fields is proportional to BZ; for higher fields, it is less increasing than proportional to B2. With p-doped specimens 50 and 27 a deviation from the quadratic law is observed even at lower magnetic fields. This resuit is especially pronounced for the specimens with higher hole concentrations and is shown in Fig. 20, where the higher the concentration the sooner Ap/po begins to deviate from a quadratic dependence upon magnetic fields.

’*

C. Hilsurn and R. Barrie, Proc. Phys. SOC. (London) 71, 676 (1958).

342

H . WEISS

70 lo2

10’

dQ 40

loo

t

FIG. 18. Transverse magnetoresistance Ap/po of InSb as a function of the electron concentration n in a field of 10 kG at room temperature. The dashed and dashed-dotted lines indicate the theoretical dependence of Ap/po for n-type InSb. (After Rupprecht et aL4 and Hilsurn and Barrie.”)

In contrast to the behavior in the intrinsic range, the increasing resistance tends toward a saturation value in the range of mixed conductivity. In Fig. 20, the Hall coefficient is also given as a function of B. It decreases with increasing magnetic field, has a zero transition, and reaches a saturation value, which is given by the hole concentration p with R , = l/ep. Champne~s,’~ who has extended his measurements at room temperature on an intrinsic specimen of indium antimonide up to 300kG, achieved results which agree with those for the intrinsic specimen shown in Fig. 19. From the results, shown in Fig. 18, one can conclude that the electrons in the conduction band at room temperature do not produce magnetoresistance. The theoretical explanation of these experimental results will be 23

C. H. Champness, Can. J . Phys. 41, 890 (1963).

343

10. MAGNETORESISTANCE

I

2

B

5

10' kG

2

FIG.19. Transverse magnetoresistanceAp/p, at 27°C as a function of the magnetic induction B for four specimens of InSb having the following characteristics: 80 and 105, intrinsic; 50, acceptor concentration of 5.8 x 10'5/cm3; 27, acceptor concentration of 1.3 x 10i6/cm3. (After Hieronymus and Weiss.")

discussed later. The magnetoresistance effect in the range of mixed conductivity, i.e., when both holes and electrons contribute to the conduction, is caused by the superposition of electron and hole conductivity. To understand this, let us examine Fig. 21. In the left half, current density j and electric field strength E are shown for pure electron conduction; in the right half, for mixed conduction. In the upper part, the behavior is given for no magnetic field and in the lower part with a magnetic field perpendicular to the plane of the drawing. For pure n-type conductivity in the absence of a magnetic field, the current density and electric field are parallel. In a magnetic field, that component of the electric field parallel to the current density remains unchanged because of the absence of magnetoresistance. In addition, there is the transverse Hall field E,. This

344

H. WEISS

FIG.20. Hall coefficient ( x __ x )and transverse magnetoresistance (0-0) of six p-type specimens of InSb as a function of B at room temperature. Acceptor or hole concentrations are indicated at the top. (After Hilsum and Barrie.*’)

component plus the component along j form the total field strength E, which is rotated by the Hall angle 9. In the range of mixed conductivity, the total current density j is composed of the electronic part j, and the hole part j,., With no magnetic field both have the same direction as E. In a magnetlc field, the directions of j, and E must have the same angle 9 as in pure n-type conduction. The total current no longer has the direction of j, since the hole current j, still has the direction of E because of the very low mobility of the holes. It follows from the drawing that the projections of the electric field strength, parallel to j, are by far larger in a magnetic field than with no magnetic field. Thus a magnetoresistance effect can

FIG.21. Sketch illustrating the existence of transverse magnetoresistanceas a result of twoband conduction (at right); as compared to single-band, n-type conduction without magnetoresistance (at left). The magnetic field is normal to the paper.

occur in the range of mixed conductivity even with no magnetoresistance contribution from the conduction or the valence bands. Also, it can be seen, from Fig. 21, that the Hall angle 9' has a sign corresponding to n-type conductivity, but it is reduced in size by the additional hole current as compared to the angle 9 for pure n-type conduction. Chambersz4 has derived the following formula for the resistance in a magnetic field for two-band conductivity :

R, and R , are the Hall coefficients of the electrons and holes, and therefore are of opposite signs; 6,is the specific conductivity of the conduction band; and b, which is here a positive quantity, is the ratio of electron to hole mobility. All quantities on the right side of Eq. (13) may still depend on the magnetic field. It follows from Eq. (13) that for R, # - R p , i.e., for nonintrinsic conductivity, p B has a saturation value if the variables p,, R,/RP, p,, and b have saturation values. Using Eq. (13), S ~ h o n w a l dcalculated ~~ the relative magnetoresistance Ap/po as a function of p,,B with b = 100 for several values of R,,/Rp.The results of this calculation are given in Fig. 22. All variables of Eq. (13) were assumed independent of B. The largest magnetoresistance effect at small fields is found for - R,/R, = 100, that is, for equal partial currents j , and j , with no magnetic field. The saturation 24

25

R. G. Chambers, Proc. Phys. Soc. (London) A65, 903 (1952). H. Schonwald, 2. Naturforsch. 19a, 1276 (1964).

346

H. WEISS

values are identical for RJR, = x and R J R , = x-I. The same conditions are valid for the saturation values of the Hall coefficient except the signs of R, for p- and n-type conductivity are different. For the magnetoresistance effect of n-type specimens in magnetic fields below 10,000G, the following approximation is useful :

PJP,= b 9 1, n

(14) where pnois the Hall mobility of the electrons, measured with no magnetic field, and p, is the Hall mobility of the holes which may depend upon B. A d p o = pppnoB2p/n,

2 P,

lo2 10'

deleo loo

m-3

FIG.22. Calculated transverse magnetoresistance Ap/p, for n- and p-doped InSb as a function of the product p p B for pn/pp= 100 with R J R , as a parameter. (After S ~ h o n w a l d . ~ ~ )

Formula (14) is valid if Ap/po in the conduction band is much smaller than for intrinsic conductivity. From the increase of magnetoresistance with magnetic field in small fields, a value of 750cm2/V-sec is obtained for the hole mobility with no magnetic field. Using Eq. (14)it is possible to calculate the magnetoresistance effect for n-doped indium antimonide in the mixed conductivity range with n > p . The known values for the independence of the electron mobility upon doping4 are used, and a similar dependence is assumed for pP. The dashed curve in Fig. 18 was calculated in this manner under the assumption of a constant effective electron mass and classical statistics. If the degeneracy is taken into account, the lower part of the curve is shifted to lower values. However, the increase of the effective electron mass with increasing electron concentration must not be neglected. Therefore, the second dotted curve was calculated. For this curve, an increase of the

347

10. MAGNETORESISTANCE

effective electron mass of a factor of 2 was assumed for an increase of the electron concentration from 1.6 x 1OI6 to 1.6 x 1017/cm3.The measured points in Fig. 18 are still somewhat above the calculated curve, but it has essentially the right behavior. The scatter in the measured points is larger than expected from the accuracy of measurement, but very small inhomogeneities or influence of the electrodes may produce this difference as mentioned above. The reasonable agreement between measured and calculated values again indicates a vanishing magnetoresistance effect in the conduction band of indium antimonide. % 1

2 102

5 do/&

2 10’

5

2 2

3

I

I

I

I

I

I

I

4

5

6

7 1/T

8

9

10

I

I

11 12.10-ioK’

FIG.23. Transverse magnetoresistance of three n- and of two p-conducting single-crystal specimens of InSb as a function of 1/T at 10 kG. Impurity concentrationsare indicated on the curves. The n-type specimens are homogeneous. (After S c h o n ~ a l d . ~ ~ )

For p-doped specimens and magnetic fields below 10,000G, Eq. (13) should be used. Hilsum and Barrie22calculated the curves in Fig. 20, which agree very well with the measured values. Equation (14) may be used for all magnetic fields for intrinsic indium antimonide with n/p = 1 because of the small magnetoresistance effect in the conduction band. The deviation of transverse magnetoresistance of intrinsic indium antimonide from the quadratic law may then be explained as a decrease of the hole mobility with increasing induction. Using Eq. (14) at room temperature for the low magnetic field limit, one gets a hole

348

H. WEISS

mobility of 750cm2/V-sec and for 150kG a mobility of 620cm2/v-sec. That is, the hole mobility in a field of 150 kG is smaller by 13% than it is without a magnetic field. This may be the limiting value of magnetoresistance equal to 13.2 %, which one expects for acoustical scattering.2'j Using this method of calculating the hole mobility in pure indium antimonide, S ~ h o n w a l dattempted ~~ to determine the hole mobility at temperatures above room temperature. Figure 23 shows the transverse magnetoresistance as a function of 1/T for several p - and n-type specimens.

FIG.24. Mobilities of the heavy ( p , J and light ( p p 2 )holes and the electrons (p")in InSb as a function of the absolute temperature (-), and calculated results (---), using Eq. (14), for 0 and 10 kG. (After S ~ h o n w a l d . ~ ~ )

First, let us examine the intrinsic conductivity above room temperature. Evaluating the curve by use of Eq. (14) one does not obtain a monotonic decrease of ,up with increasing temperature but rather a minimum at about 200°C and an increase at higher temperatures, as shown by the dashed curve in Fig. 24. From this behavior of the hole mobility in indium antimonide, it is apparent that the explanation of transverse magnetoresistance in the intrinsic range using the simple two-band model is not valid. From the calculations of Kane,27 it is known that in indium antimonide the valence band is composed of a band with a large effective mass and a low 26

27

J. Appel, Z . Nnturforsch. %, 167 (1954). E. 0. Kane, J . Phys. Chem. Solids 1,249 (1957)

349

10. MAGNETORESISTANCE

hole mobility and a second band with a maximum at k = 0 with a small effective mass and a high hole mobility. Therefore, it appears necessary as well as reasonable to explain the magnetoresistance effect in the intrinsic range, especially at higher temperatures, with the three-band model. Schonwald derives the following equation for the specific resistance for three bands in a magnetic field : pB =

c

a + bB2 + cB4 a2 + hB2 + ZB4'

3

a=

0,;

r= 1

3

b

=

C Or2Rr2(Or+1 + 6 , + 2 ) ,

ok, k

6k+3

=

1,2,3 ;

r=l 3

3

E FIG.25. Transverse magnetoresistance Ap/p, of a specimen of intrinsically conducting InSb at different temperatures as a function of magnetic induction. Solid curves denote calculated values; crosses, measured values. (After S c h o n ~ a l d . ~ ~ )

350

H. WEISS

TABLE I CARRIER MOBILITIE~ AND CONCENTRATIONS I N INTRINSICINDIUM ANTIMONIDE AT HIGH TEMPERATURES'

a

"C

P1lP2

PP1

22 174 337 492

90 80 60 33

650 300 160 100

PP2

3 1.7 1.25 9.5

x 104 x lo4

x 104

x lo3

n

PI2

7.56 4.1 2.4 1.3

x

104

x

lo4

x 104

x 104

1.73 1016 1.56 x 10" 5.8 1017 1.25 1018

From Schonwald, Ref. 25.

The coefficients in (15) are functions of the Hall coefficients R, and conductivities CT,of the electrons in the conduction band as well as of the fast and slow holes in both valence bands. Since, at high temperatures, the magnetoresistance increases at a lower rate than one proportional to B 2 , it is possible to adjust the parameters in Eq. (15) so that one obtains information about the two hole bands. Figure 25 shows how good this agreement is between calculated curves and measured points if one uses the values given in Table I for the concentrations n, pl, p 2 and mobilities pn,ppl,pp2. The values for the mobilities of the slow and fast holes used for these curves are plotted in Fig. 24 as a function of 1/T. At room temperature, the mobility of the fast holes amounts to about 30,000cm2/V-sec. This is about half of the mobility of the electrons. The mobility of the slow holes decreases proportional to T-' with increasing temperature, the mobility of the fast holes with T - '.'. The number of fast holes related to the total number of holes increases from 1.1 % at room temperature to 3 % at 492°C.27aThis result is reasonable, since, according to Kane,27 the fast hole band deviates from parabolicity in the same manner as the conduction band, while the band of slow holes remains parabolic. With increasing energy, the density of states in the fast hole band increases faster than that in the parabolic slow hole band. Thus, the ratio of fast holes to slow holes increases. The relatively large magnetoresistance effect of intrinsic indium antimonide at higher temperatures may also be easily explained by the Kane model with two types of holes.

b. Indium Antimonide @-Type Conductivity) It can be seen from Fig. 23 that magnetoresistance of p-type specimens decreases to very low values following the transition into the extrinsic conductivity range. This behavior of magnetoresistance of p-type specimens 27"lnagreement with measurements made by Wagini.27b

27bH. Wagini, 2. Naturforsch. 20a, 239 (1965).

10. MAGNETORESISTANCE

351

was found by Weiss28and by Harman et as well as by Champne~s.~' The experimental data of S ~ h O n w a l dare ~ ~given in Fig. 23. Champness3' attempted to use magnetoresistance measurements at low temperatures to get information about the fast holes in p-type indium antimonide. Figure 26 shows the magnetoresistance of p-type specimens of indium antimonide as a function of the magnetic field B at 77°K according to S c h O n ~ a l d .The ~ ~ magnetoresistance of the specimen with an acceptor concentration of 7 x 10'5/cm3 is higher than that of the more heavily %

B FIG.26. Transverse magnetoresistance A p / p o as a function of magnetic induction in two p-type specimens of InSb at 77°K without pressure (0and A , of respective hole densities 7 x 10" and 2.6 x 1016/cm3), as well as during tension (+, 152 x 106dynes/cm2) and compression (A, 320 x lo6 dynes/cm2),as indicated on the curves. (After S c h o n ~ a l d . ~ ~ )

doped specimen with 2.6 x 1 O I 6 acceptors/cm3 and has a value of about 5.5 % at 10,000G. Ginster and S z y m a n ~ k afound ~ ~ a similar dependence of the specific resistance upon the magnetic field for two p-type specimens of indium antimonide at 77°K. It should be noted that even at relatively low magnetic fields the magnetoresistance is not proportional to B2. Schonwald explains the magnetic field dependence of the resistance by the existence of the two hole bands. Assuming the ratios of the concentrations of the fast and slow holes to be 1.1 x lo-' and 1.5 x lop2 in the two H. Weiss, Z. Naturforsch. 8a, 463 (1953). T. C. Harman, R. K. Willardson, and A. C. Beer, Phys. Rev. 95, 699 (1954). 30 C. H. Champness, J . Electron. Control 4, 201 (1958). " C. H. Champness, Phys. Rev. Letters 1, 439 (1958). 32 J. Ginster and W. Szymanska, Phys. Stat. Solidi 3, 1398 (1963). 28

29

352

H. WEISS

specimens, respectively, and the ratios of the mobilities to be 6.5 and 5.3, respectively, one gets the two solid curves in Fig. 26. They are in good agreement with the measured values. These results agree with the concentrations and mobilities of p-type indium antimonide at 77"K, determined by Champness under simplified assumptions. It was found from the experiments by Schonwald, in connection with measurements of the magnetic field dependence of the Hall coefficient, that the magnetoresistance of the purer of the two specimens in Fig. 26 cannot be explained using two valence bands which alone have no magnetoresistance effect. One has to assume a magnetoresistance effect in one of the simple valence bands. Figure 18 indicates that a p-type specimen at room temperature has a small magnetoresistance effect if it is highly doped, that A p / p o increases as the intrinsic range is approached, reaches a maximum, and decreases to the intrinsic curve. One obtains the same behavior by starting with p-type specimens at low temperatures and going to higher temperatures, as is shown in Fig. 23. At low temperatures, i.e., with a high p/n ratio, one has a small magnetoresistance, then an increase of A p / p o with increasing temperature in the mixed conductivity range and, after a maximum, an asymptotic approach to the intrinsic curve at higher temperatures. The dependence of the magnetoresistance on magnetic field is also similar to that shown in Fig. 20 for mixed conductivity on p-doping at room temperature. When cooled below the temperature of liquid nitrogen into the range of liquid helium, the magnetoresistance of p-type indium antimonide exhibits an anomalous behavior. Fritzsche and L a r k - H o r ~ v i t zfound ~ ~ a reversal in the sign of the magnetoresistance (shown in Fig. 27) near 10°K for specimens with hole concentrations of 1 x 10"/cm3 and 5 x 10'5/cm3. For the specimen with the higher impurity concentration the reversal of sign takes place below IWK, while for the purer specimen it is above 10°K. The dependence of magnetoresistance at 4.2"K in the range of negative magnetoresistance is quadratic in low magnetic fields and linear above 2000G. The specimens of Fritzsche and Lark-Horovitz had ground surfaces. Using specimens with a concentration of p = 3 x 10i5/cm3, Frederikse and H ~ s l e rfound ~ ~ longitudinal and transverse negative magnetoresistance of the same magnitude at 4.2"K. They found that, for a specimen with p = 4 x lOl4/cm3, the magnetoresistance effect is strongly dependent on surface treatment (etching or sandblasting). Also, at these same temperatures they found there may exist a negative magnetoresistance at low magnetic fields, while, in high fields, the magnetoresistance is positive. With different surface treatments, magnetoresistance goes through 33 34

H. Fritzsche and K. Lark-Horovitz, Phys. Rev. 99, 400 (1955). H. P. R. Frederikse and W. R. Hosler, Phys. Reu. 108, 1146 (1957).

10.

MAGNETORESISTANCE

353

zero at different magnetic fields. However, much of the negative magnetoresistance data are still questionable. Fritzsche and Lark-Horovitz showed that the transition from negative to positive magnetoresistance takes place at the temperature at which the Hall coefficient exhibits a maximum, that is, where the transition from extrinsic conductivity into impurity conductivity takes place. Lien ChihCh'ao and N a ~ l e d o v ~ found ~ , ~ the ~ same behavior as Fritzsche and L a r k - H o r ~ v i t zwith ~ ~ an impurity band conduction at helium temperatures for hole concentrations ranging from 1014 to 1015/cm3in the temperature range between 4.5" and 300°K. For etched p-type specimens with hole concentrations less than 1013/cm3, impurity band conduction was not

FIG.27. Transverse magnetoresistance Ap/p, of three p-type specimens of InSb as a function of 1/Tat 3.5 kG. For points A and 0, p = 1015/cm3;for points x, p = 5 x 10'5/cm3. (After Fritzsche and L a r k - H o r ~ v i t z . ~ ~ )

found at low temperatures. Putley3' did not observe impurity band conduction with a hole concentration of 5 x 1013/cm3. To explain negative magnetoresistance for impurity band conduction, the simple theory is not sufficient. In addition, the experimental values are not very reliable because of the dependence on surface treatment. Mackintosh38 and Stevens39 tried to explain the effect by assuming that 35

36

37 38 39

Lien Chih-Ch'ao and D. N. Nasledov, Fiz. Tuerd. Tela 1, 570 (1959) [English Transl.: Souiet Phys.-Solid State 1, 514 (1959)l. Lien Chih-Ch'ao and D. N. Nasledov, Fiz. Tuerd. Tela 3, 1458 (1961) [English Transl.: Souiet Phys.-Solid State 3, 1058 (1961)l. E. H. Putley, Proc. Phys. SOC.(London) 73, 128 (1959). I. M. Mackintosh, Proc. Phys. SOC. (London) B69,403 (1956). K. W. H. Stevens, Proc. Phys. SOC.(London) M9,406 (1956).

354

H . WEISS

an acceptor level is split into two levels in a magnetic field and that the occupation probability of the acceptor states as well as those of the conduction band is varied. Using these assumptions, they obtained a negative magnetoresistance, but it was smaller than the measured values by an order of magnitude and exhibited a different magnetic field dependence. Another explanation of negative magnetoresistance was given by Toyo~ a w a . ~According ' to him, the holes move like quasifree particles in an impurity band, and, at the same time, a small number of localized spins are formed. The electric current is determined by holes which are scattered on the one hand by impurities and on the other hand by the interaction with localized spins. The negative magnetoresistance effect is essentially caused by the fact that the spins are frozen in a magnetic field. The dynamical variations of the spin-dependent potential decrease with decreasing temperature. In this manner, the interaction with the spins is reduced and the electron scattering is decreased. c. Indium Antimonide (n-Type Conductivity)

Indium antimonide has the highest electron mobility of all known semiconductors. The condition pB = 1 is fulfilled at room temperature with p,,= 76,000cm2/V-sec in a field of 1300G and for the highest purity specimens at 77°K with p,, = 650,000 cm2/V-sec for only 160 G. Now, one has

where z is the mean collision time, m,, the effective electron mass, and o,the cyclotron frequency for an orbit in the magnetic field. From Eq. (16) it follows that, for relatively weak fields, p B = w,z & 1, i.e., an electron passes through several orbits before undergoing a collision with the lattice. Then the Hall angle is nearly 90" and the electrons move, on the average (see Fig. l), parallel to the y-axis with a mean velocity E/B. Landau41 was first to point out that such a circular periodic motion has to be quantized. With no magnetic field, the energy E of a particle in an isotropic semiconductor depends upon the wave vector k in the following manner :

40

41

Y. Toyozawa, Proc. Intern. Con$ Phys. Semicond., Exeter, 1962 p. 104. Inst. of Phys. and Phys. SOC.,London, 1962. L. Landau, Z . Physik 64, 629 (1930).

10. MAGNETORESISTANCE

355

In a magnetic field one obtains the relation

where k, is the component of k parallel to B. The energy of an electron is composed of the quantized energy for the motion perpendicular to the magnetic field and of the energy in the direction of the magnetic field. The latter is not influenced by the magnetic field. In this manner, one gets a sequence of one-dimensional bands in k-space separated by the energy hw, . The lowest band is shifted by the value $ha, (see Fig. 28).

n= FIG.28. Band model of an isotropic semiconductor in a magnetic field. The quantities o, and wcpare the cyclotron frequencies for electrons and holes.

The distribution of the possible states in k-space which vary with B and the change of z connected with this redistribution lead us to expect a magnetoresistance oscillating with B in a certain temperature range. Therefore, the following experimental results obtained with indium antimonide are presented before they are discussed. It was shown in Fig. 18 that no magnetoresistance effect could be found in the conduction band of indium antimonide at room temperature. For single crystals with impurity concentrations from 2 x lOi6/crn3 up to

356

H. WEISS

1 x 10'8/cm3 at the temperatures of liquid nitrogen and liquid helium, the transverse magnetoresistance is of the order of 1 to 2 % for homogeneous specimens measured in a field of 10,000G . With an electron concentration n = 1017/cm3, the mobility at 295°K is about 50,000cm2/v-sec, while at 77°K and 4.2"K it is 60,000~m~/V-sec.~ Since a magnetoresistance of a

'i

4

lo-'

10-2

z

B

I

I

103

1046

FIG.29. Transverse magnetoresistance of three n-type homogeneous, single crystals of InSb as a function of magnetic induction at 77°K. Impurity concentrations are indicated on the curves. (After Schonwald4z)

few percent can be produced by tiny inhomogeneities, the small value of measured magnetoresistance should be regarded as the upper limit of the magnetoresistance effect of the homogeneous semiconductor. Thus, we can conclude that indium antimonide, with electron concentrations higher than 2 x 1016/cm3 and for magnetic fields up to 10,000G, does not possess a magnetoresistance effect in the conduction band at all temperatures. But this picture is changed at lower impurity concentrations. The measurements,

10. MAGNETORESISTANCE

351

shown in Fig. 23, were obtained with homogeneous n-type single-crystal specimens produced by the method of Miiller and Wilhelm.I4 For small impurity concentrations the magnetoresistance in the extrinsic range is appreciable but does not strongly depend on temperature. Figure 29 represents the transverse magnetoresistance effect of two homogeneous n-conducting specimens of different doping as a function of the . ~should ~ be noted magnetic induction at 77°K according to S c h o n ~ a l d It that the transverse effect for the pure specimen with n = 5.2 x lOI3/cm3 depends quadratically on a magnetic field below 100 G and linearly upon B above 100 G. The Hall coefficient of such a specimen varies less than 1 % between lo00 and 10,000G, but at 10G it is higher by 10 to 15%. The dependence upon field, shown in Fig. 29, was found for homogeneous specimens as well as for inhomogeneous specimens after subtraction of the magnetoresistance effect caused by inhomogeneities. Bate et al.43 observed a similar dependence of the transverse magnetoresistance on magnetic field for a specimen with 1.7 x l O I 4 electrons/cm3. In these measurements, the influence of inhomogeneities was not considered, but it seems to be a specimen which was cut from a single crystal in the most favorable direction across the variations of concentration. Frederikse and H o ~ l e rstudied ~~ transverse and longitudinal magnetoresistance on n-type specimens with between 9 x lOI4 and 2.4 x 10’’ electrons/cm3 in the range between 200 and 8000 G at 77°K. Transverse magnetoresistance increased with decreasing electron concentration and pointed to a linear dependence on magnetic field above 1000 G. The longitudinal magnetoresistance was always lower by an order of magnitude and was, at times, negative. With both types of magnetoresistance, especially with the longitudinal, Frederikse and Hosler were very careful in discussing the results, because they realized the importance of inhomogeneities. Their specimens were cut from zonerefined crystals across the pulling direction. We now know that this reduced the influence of doping gradients as well as periodic inhomogeneities, but did not eliminate all inhomogeneities ; therefore, they did not discuss the results in detail, especially those regarding longitudinal magnetoresistance. The behavior of n-type indium antimonide down to liquid nitrogen temperatures is easily explained, but at liquid helium temperatures the situation is much more complicated. Figure 30 shows the conductivity of n-type indium antimonide with an electron concentration of 1.4 x 1014/cm3 as a function of 1/T as given by Sladek.45 The transverse magnetic field 42

43 44

45

H. Schonwald, (unpublished measurements). R. T. Bate, R. K. Willardson, and A. C. Beer, J . Phys. Chem. Solids 9, 119 (1959). H. P. R. Frederikse and W. R. Hosler, Phys. Reo. 108, 1136 (1957). R. J. Sladek, J . Phys. Chem. Solids 8, 515 (1958).

358

H. WEISS

in kilogauss is given for each curve. After cooling below 10"K, a distinct increase of magnetoresistance is observed. For higher concentrations, oscillations may be observed. Figure 3 1 shows the transverse and longitudinal magnetoresistance for fields up to T

"K

10

5

3'

2

1.5

Q-'cm-'

2.5 5

2 loo

loo

5

2

2

5 10'

lo-'

5

2

2

5

e

6

lo2 5

2

2

5 103

5

2

2

5 104 2 9 cm

5

3O

01

0.2

0.3

1IT

0.4

0.5

0.6

OK-

FIG.30. Conductivity u and resistivity p of an n-type single crystal of InSb with n = 1.4 x 1014/cm3as a function of the reciprocal temperature. The parameter is the magnetic induction B. (After Sladek4')

8000 G of an n-type specimen with n = 2.3 x 10'5/cm3 at 1.7"K according to measurements of Frederikse and H o ~ l e r .These ~ ~ authors observed such a behavior for electron concentrations of 1 x 2.3 x and 9 x lOl5/cm3. Also, Broom46 observed oscillations with specimens of n = 6 x l O I 5 and 7 x 10'5/cm3 but not with a concentration of 46

R. F. Broom, Proc. Phys. Soc. (London) 71,470 (1958)

10. MAGNETORESISTANCE

359

5 x 1016/cm3.This is in agreement with the results of Rupprecht et aL4 According to these authors neither a transverse nor a longitundinal magnetoresistance effect can be observed in the range of electron concentrations between 2 x 1OI6 and 101'/cm3 at 4.2"K for magnetic fields below 10,OOO G. 0.4

0.3

0.2

dQ

eo

0.1

0

0.1

nn

u.L

0

1

2

3 B

kG

FIG.31. Transverse (-O-O-) and longitudinal)-.-.-( magnetoresistance at 1.7"K of an n-type single crystal of InSb (n = 2.3 x 10'5/cm3) as a function of the magnetic induction. (After Frederikse and H o ~ l e r . ~ ~ )

Amirkhanov et aL47 were able to observe oscillations of the resistance at 20" and 77°K with heavily doped specimens in transverse and longitudinal magnetic fields above 50,000G. Figure 32 shows the longitudinal 4'

Kh. 1. Amirkhanov, R . I. Bashirov, and Yu. E. Zakiev, Dokl. Akad. Nauk S S S R 148, 1279 (1963) [English Transl.: Soviet Phys. Doklady 8, 182 (1963)l.

360

H. WEISS

magnetoresistance as a function of 1/B for a specimen with n = 6.6 x 1017/ cm3 at 77” and 20°K.47a Now, let us consider the vanishing magnetoresistance effect in the conduction band for the higher concentrations. It should be noted that the Hall coefficient does not exhibit a measurable dependence upon the magnetic field4 in the same range that the magnetoresistance vanishes. 1.2

1.0 0.8

0.4 0.2

FIG.32. Longitudinal magnetoresistance of an n-type single crystal of lnSb (n = 6.6 x 1017/cm3)as a function of reciprocal magnetic induction 1/B at (a) 77°K and (b) 20°K. (After Amirkhanov et ~ 1 . ~ ~ )

For calculations utilizing the Boltzmann equation and spherical energy surfaces, the longitudinal magnetoresistance effect vanishes for all types of isotropic scattering mechanisms. The specific conductivity ( T ~for an isotropic semiconductor with parabolic conduction band in a transverse magnetic field is given by Eq. (19):

+ N2/M,

oB = M

where

1;

M =

2 e2m, 3 n2h3

N =

2 e2mn 3 7 r 2 h 3 j r lE

1 1 + ( , Y B )dE’ ~

lE

%

”

1 + (pB)2

dE;

47aGurevichet al. observed magnetoresistance oscillations at 90°K caused by current-carrier scattering by optical ph0nons.4~~ 47bV.L. Gurevich, Yu. A. Firsov, R. V. Pareniev, and S. S. Shalyt, in “Physics of Semiconductors” (Proc. 7th Intern. Conf.), p. 653. Dunod, Paris and Academic Press, New York, 1964.

10. MAGNETORESISTANCE

361

1 is the mean free path and fo the undisturbed distribution function. If p does not depend upon the energy E, the terms with p B can be taken outside the integral, resulting in the same value for oBas for go, i.e., a vanishing transverse magnetoresistance. Since p = ez/m,, it follows that the mean collision time z is independent of the energy for the special case just cited. Since the shape of the conduction band in indium antimonide deviates strongly from the parabolic shape and m, depends on the energy of the electrons, Kotodziejczak and Sosnowski4* calculated the transport integrals using Kane's formulas for the conduction band in indium antimonide. The density of states as well as the wave vector are no longer connected with the energy by a mass which does not depend on energy. The following expression must be used:

In addition, different scattering mechanisms can result in a different dependence of the collision time z on the energy than is obtained in the case of constant effective masses. Guseva and Tsidil'k~vskii~~ calculated the transport integrals for the transverse magnetoresistance effect at 200" and 600°K for several electron concentrations in indium antimonide. These calculations were made for scattering by both acoustical and optical lattice vibrations. The smallest magnetoresistance was obtained for scattering by optical phonons, and it was the same order of magnitude as the small experimental va1~es.4~' Now, consider the magnetoresistance of high-purity indium antimonide at liquid nitrogen temperatures. The classical theory of transverse magnetoresistance indicates a B2 dependence at low magnetic fields and a saturation value for p B B 1. This saturation value depends strongly upon the type of scattering mechanism. It must also be assumed that quantum effects play a role since at 100OG the condition p B b 1 is valid and according to classical theory there should be saturation. According to Eq. (18), for electron concentrations smaller than lOl4/crn3 only the lowest Landau level with n = 0 is occupied in fields above 1000 G . This case is called the "quantum limit." Bate et have tried to explain the results at low magnetic fields with the classical theory and in high fields by quantization. However, they were not able to obtain quantitative agreement between 48

49

J . Kokdziejczak and L. Sosnowski, Acta Phys. Polon. 21, 399 (1962). G. I. Guseva and I. M. Tsidil'kovskii, Fiz. Tuerd. Telu 4, 2490 (1962) [English Transl. : Soviet Phys.-Solid State 4, 1824 (1963)].

49aThiswas confirmed for the temperature range between 20" and 450°C.49b

49bH. Wagini, Z. Naturforsch. 19a, 1541 (1964).

362

H. WEISS

theory and experiment. Also the assumption nf scattering by acoustical lattice vibrations is not fulfilled in high-purity indium antimonide. Sladek" calculated magnetoresistance in the quantum limit for several types of scattering, but was not able to explain conclusively the measured values in pure indium antimonide at 77°K. A substantially higher magnetoresistance effect is shown in Fig. 30 for pure indium antimonide below 10"K, although measurements of the Hall coefficient by Putley5' indicate that the electron mobility at 4.2"K is smaller by one order of magnitude than at 77°K. Yafet et al.52 have shown that, in a high magnetic field, the charges of the valence electrons surrounding a hydrogen atom are compressed in a plane perpendicular to B. Therefore the ionization energy should increase in large magnetic fields. However, in n-type indium antimonide all donors are ionized, even at the lowest temperatures, with no magnetic field. Hence, with increasing magnetic field the probability of the occupation of donor levels increases because of the increasing activation energy, and impurity band conduction exists. Sladek53showed that the increase in activation energy with magnetic field could be determined from the experimental Hall coefficient data, using the theory of two-band conduction. The experimental values are in agreement with the theoretical ones. If one knows the distribution of the electrons in the conduction band and impurity band from the measured Hall coefficient, one can calculate the electron mobility in the impurity band as a function of temperature and magnetic field using the curve in Fig. 30. The mobility decreases from 5 x lo4 cm2/V-sec with no magnetic field to about 5 cm2/vsec in a field of 20 kG at 1.56"K. This is reasonable, since, with increasing distance of the donor levels from the edge of the conduction band, the probability of occupation of donors increases and the probability decreases that a level remains empty or that an electron finds an unoccupied impurity level. In the same sense, for a given value of B a decrease in temperature leads to a decrease of electron mobility in the impurity band. For example, a decrease in mobility by one order of magnitude is observed in a field of 28 kG by cooling from 4" to 2°K. This model with a single donor level is not sufficient for a quantitative explanation. It appears that one must take into account an additional level into which the electrons must be lifted thermally in order to jump to a neighboring atom. The energy difference between this level and the basic level is then dependent upon the electron concentration but independent of B and is about 38% of the activation

'* 53

R. J. Sladek, J. Phys. Chem. Solids 16, l(1960). E. H. Putley, Proc. Phys. SOC. (London) 73, 280 (1959). Y. Yafet, R. W. Keyes, and E. N. Adams, J . Phys. Chem. Solids 1, 137 (1956). R. J. Sladek, J . Phys. Chem. Solids 5, 157 (1958).

10. MAGNETORESISTANCE

363

energy of the donors for the specimen with n = 1.4 x lOl4/cm3 shown in Fig. 30. For electron concentrations above 101’/cm3, much less influence of the impurity band conduction is seen, since impurity and conduction bands begin to separate at much higher magnetic fields than in the foregoing case. In this range, one observes oscillations of longitudinal and transverse magnetoresistance at liquid helium temperatures. For the existence of oscillations in addition to the condition p B 9 1, the electron concentration must have such a magnitude that the Fermi energy [ is equal to a multiple of ho,.Also since the distance hw, of the 10-’*1

n

s-’ 1 ‘\

no

B=l kG

”

0

\\\\

-

o

O

‘\ \

\

2kG

0

\

I - -

\\

\

-

h.

\

\ \\

A

10-l~

A

\

\ \\

-

\ \\

I

I

I 1 1 1 1 1

I

I

I

I 1 1 1 1 l

\ \\

‘\

FIG. 33. Average relaxation time 7 in indium antimonide as a function of the electron concentration n at helium temperatures. The A’s represent crystals exhibiting nonoscillatory magnetoresistance.(After Sasaki and Y a m a n o ~ c h i . ~ ~ )

single bands must be large in comparison with kT, one gets the further condition

t % f i h o , 9 kT. (21) The ratio ho,/kT is 1 at 77°K for a magnetic field of 7700G, and at 2°K only 200 G are required. According to (21) one can define a concentration range in which the conditions for oscillations are fulfilled. By substituting in both inequalities, w,z

5 > ha,,

> 1,

(22)

the sign of equality, one gets

5

=

h/z.

(23)

364

H. WEISS

The Fermi level i can be expressed by the electron concentration n if the effective mass is known. Sasaki and Y a m a n ~ u c h calculated i~~ the relation 7 = f(n) for indium antimonide with m, = 0.015 mo and obtained the dashed line in Fig. 33. The points for crystals with no oscillations lie left of the dashed line, whereas the points for specimens with oscillations lie on the right side. The mean collision time z of the specimens was calculated from the Hall mobility according to (16). For a large Hall angle, one observes a maximum of the specific resistance according to Shalyt and E f r ~ s if, ~the~ Fermi level just coincides with the lower edge of a single band. For the period of the oscillations, one has the following relation : gauss- . Thus, the length of the periods is independent of the effective mass and is determined solely by the electron concentration n. The authors’ demonstrated that there is good agreement between their theory and experimental results by using the absolute positions of the maxima of the specific resisr ~well ~ as those of tance for the measurements of Frederikse and H o ~ l e as ~ Broom.46 From the experiments of Amirkhanov et ~ 1 in . high~ fields up to 400 kG, this picture is not sufficient. Because of the high g-factor of about 50 of the electrons in indium antimonide, the splitting of the partial bands by the electron spin must be considered. Such calculations were made by Gurevich and EfrosS6 For the transverse magnetoresistance effect in highly doped indium antimonide, they got a good agreement with this theory, but for the longitudinal magnetoresistance the simple theory neglecting the spin gave better agreement with experiments.

2. INDIUM ARSENIDE Apart from indium antimonide, indium arsenide exhibits the highest electron mobility among the 111-V compounds. The highest value observed as yet at room temperature is 30,000 cm2/V-sec 5 7 for an electron concentration of 1.7 x 1016/cm3. For lower concentrations p, was smaller. The concentration quoted above corresponds to the lowest impurity content yet obtained in indium arsenide. The band structure of indium arsenide is similar to that of indium antimonide. Because of the higher 54 55

56

57

W. Sasaki and C. Yamanouchi, J . Phys. SOC.Japan 14, 849 (1959). S. S . Shalyt and A. L. Bfros, Fiz. Tverd. Tela 4, 1233 (1962) [English Transl.: Soviet Phys.Solid State 4, 903 (1962)l. L. g. Gurevich and A. L. Efros, Zh. Eksperiin. i Teor. Fiz. 43, 561 (1962) [English Transl.: Soviet Phys. J E T P 16, 402 (1963)l. T. C. Harman, H. L. Goering, and A. C. Beer, Phys. Rev. 104, 1562 (1956).

10. MAGNETORESISTANCE

365

effective mass of 0.02m0 and the larger band gap, the shape of the conduction band deviates less from the parabolic form than it does in indium antimonide. Apart from the oscillations at low temperatures, there are only a few experiments on the magnetoresistance effect in indium arsenide. There are several reasons for this. Mainly, it has not been possible to achieve a higher purity than 1016/cm3 by zone refining. In addition, the preparation of single crystals is much more complicated than with indium antimonide because the vapor pressure of arsenic at the melting point is not insignificant. %

FIG.34. Transverse magnetoresistance of n-type specimens of I d s as a function of magnetic n = 5.5 x 1016/cm3,single crystal, with current along induction at room temperature: -O-o-, n = 5.7 x 1016/cm3 (after Charnpnes~~~); -----, the [Ill]-direction (after WeissS8);-, n = 3.2 x lOI6/cm3 (after Amirkhanov et aL6').

Experiments concerning the influence of shape on the magnetoresistance58*59gave the expected results. The influence of inhomogeneities has not been studied in indium arsenide. The experimental results now in the literature may therefore have to be corrected. Measurements of transverse magnetoresistance at room temperature were first carried out by Weiss5' in magnetic fields up to 33 kG on a single crystal with n = 5.5 x 1016/cm3 58

H. Weiss, Z . Naturforsch. 12a, 80 (1957).

59

C. H. Champness and R. P. Chasmar, J . Electron. 3, 494 (1957).

366

H. WEISS

which was cut parallel to the [ 1111 direction. Figure 34 shows results, which were later confirmed by Champness and C h a ~ m a r . ’In ~ fields up to 30 kG, A p / p , is proportional to B1.65.ChampnessZ3 studied A p / p , on specimens with 5.7 x lOl6/cm3 in pulsed fields up to 260 kG. At low fields the data are identical with those in the two publications mentioned above. The %

5 4

3 2

1

0

-.-.-,

FIG.35. Transverse magnetoresistance of three n-type specimens of InAs as a function of temperature T : - x - x - - , single crystal, n = 101’/cm3, B = 6 kG (after Weber and Weiss.61); -0-0-, single crystal, n = 3 x 10i6/cm3, B = 4 k G (after Zotova et aL6’); single crystal, n = 1.3 x 10i6/cm3, B = 3 kG (after Zotova et aL6’).

measurements of Amirkhanov et aL6’ were made in pulsed fields between 50 and 400 kG. In cooling to liquid nitrogen temperatures, the transverse magnetoresistance increases. Figure 35 shows A p / p o measured on a single crystal with n = 10”/cm3 as a function of 1/T in a field of 6000G according to Weber and Weis6’ The increase of magnetoresistance during cooling 6o

61

Kh. I. Amirkhanov, R. I. Bashirov, and J. E. Zakiev, Zh. Eksperirn. i Teor. Fiz. 41, 1699 (1961) [English Transl.: Soviet Phys. J E T P 14, 1209 (1962)l. R. Weber and H. Weiss (unpublished measurements).

10. MAGNETORESISTANCE

367

agrees qualitatively with the increase of the electron mobility p,, but Ap/po increases stronger than proportional to p;. Two n-type specimens studied by Zotova et a1.62 exhibited negative values of A p / p o below 40" and 25"K, respectively. These values were

FIG.36. Transverse magnetoresistance of four n-type specimens of InAs as a function of the single magnetic induction B at 77°K: ---, single crystal (after Weber and Weiss6'); -, crystal (after ShalyP4); -- (after Frederikse and H ~ s l e r ~ ~ ) ;(after Amirkhanov et ~ 1 . ~ ' ) . The points lying close to one another with little scatter were omitted.

independent of the method of preparation, inhomogeneities, surface treatment, and orientation of the specimens. It should be noted that the negative magnetoresistance effect in small magnetic fields is proportional to B2,passes through a maximum in higher fields, and becomes again smaller at about l0,OOO G. 62

N. V. Zotova, T. S. Lagunova, and D. N. Nasledov, Fiz. Tuerd. Tela 5, 3329 (1963) [English Transl.: Sooiet Phys.-Solid State 5 , 2439 (1964)l.

368

H. WEISS

Figure 36 shows the dependence of magnetoresistance upon the magnetic field at 77°K for a polycrystalline specimen with n = 2 x 1016/cm3, according to Frederikse and H ~ s l e ;rfor ~ ~a single crystal with 3 x 10l6/ cm3 according to ShalyP4 and a single crystal with n = 1017/cm3according to Weber and Weiss.61 The curve, measured by Weber and Weiss, lies at lower values presumably because of the higher impurity concentration and the correspondingly smaller electron mobility. The two curves measured by Frederikse and Shalyt, respectively, on polycrystalline and on a singlecrystal specimen lie close to each other and possess an inflection point at about 8000 G. This seems to be a characteristic property of indium arsenide. The same is true for the field dependence in small magnetic fields, which is proportional to B1.65at room temperature as well as at 77°K. Amirkhanov et ~ 1 . ~ observed ' transverse magnetoresistance on a polycrystalline specimen with n = 3.2 x 1016/cm3 in fields up to 400kG. At room temperature in the highest fields, an increase in the specific resistance of 29 times was observed (Fig. 34). At liquid nitrogen temperatures the increase was a factor of 50 (Fig. 36). At 77°K and 200 kG, Ap/po increases more strongly than proportional to B and corresponds roughly to an extrapolation of the measurements of Frederikse and Shalyt. Weber and Weiss61 studied magnetoresistance as a function of the angle between specimen current and magnetic field on a single-crystal rod, which was cut in the [ 1111 direction. The measured curve has an exact sine shape, which indicates the absence of inhomogeneities (compare Fig. 13). The longitudinal magnetoresistance was about 10% of the transverse effect, but it was not reproducible. The 1.65 exponent in the field dependence of A p / p o for indium arsenide has not been explained. The negative magnetoresistance observed at low temperatures by Zotova et ~ 1 . ~ 'is related to a maximum in the Hall coefficient at temperatures above the zero transition of Aplp,. The authors deduce from these C U N ~ S an impurity band conduction which is related to the negative magnetoresistance similar to that for indium antimonide. The results of experiments with the oscillations of transverse and longitudinal magnetoresistance at lower temperatures are more easily explained. Such experiments have been made by Frederikse and H0sler,6~ Sladek,65 et ~ 1 Figure . ~ 37 ~shows the oscillaShalyt and E f ~ 0 . and s ~ ~Amirkhanov ~ tions for transverse and longitudinal magnetic fields at three different 64

65

66

H . P. R. Frederikse and W. R . Hosler, Phys. Rev. 110, 880 (1958). S. S. Shalyt, Fiz. Tuerd. Tela 4, 191 5 (1962) [English Transl.: Soviet Phys.-Solid State 4, 1403 (1 963)]. R. J. Sladek, Phys. Rev. 110, 817 (1958). Kh. I. Amirkhanov, R. I. Bashirov, and Yu. 8. Zakiev, Fiz. Tuerd. Tela 5,469 (1963) [English Transl.: Soviet Phys.-Solid State 5, 340 (1963)l.

369

10. MAGNETORESISTANCE

temperatures according to Sladek.6s This is the typical picture for indium arsenide measured on a single crystal with n = 7.6 x 1016/cm3.The periods are clearly distinguishable because a strong increase in the ordinary magnetoresistance effect is not superimposed, as is the case for indium 0.15 0.10 0.05

d

-020

-0.25

-0.30 G-'

3 0.20 0.15

b

-0.10

FIG.37. Transverse and longitudinal magnetoresistance Ap/pe of an n-type single crystal of lnAs with n = 7.6 x lOI6/cm3 as a function of the reciprocal magnetic induction 1/B at three different temperatures. (After Sladek.")

antimonide. They can be determined very precisely. The agreement between the periods, calculated from Eq. (24), and the measured values is very good. This is also true for the data in the other publications mentioned above. Amirkhanov et al. used specimens with electron concentrations up to

310

H. WEISS

2.7 x 101'/cm3 and in magnetic fields up to 400 kG in their experiments. Satisfactory agreement between their measurements and calculations was obtained when they utilized spin splitting with a g-value of 14. For the longitudinal magnetoresistance they had to neglect the spin splitting, as was done for indium antimonide. Sladek discussed the amplitude of the oscillations as a function of magnetic field and temperature. Also, he showed that the nature of the oscillations for monocrystalline and polycrystalline indium arsenide were practically identical. This was a confirmation of the isotropic spherical conduction band. Furthermore, it was observed that the oscillations in transverse and longitudinal fields possessed comparable amplitudes at nearly the same position, i.e., higher harmonics are not present in the oscillations. The oscillations depend exponentially upon 1/B. From this follows a collision broadening. The temperature dependence of the resistance oscillations fits the picture given by the general transport theory. An effective mass of the electrons which agrees with values determined by other methods can be derived. 3 . INDIUMPHOSPHIDE

Glicksman' studied a single crystal of indium phosphide (n = 6 x l o k 5 / cm3, p,, = 4600 cm2/V-sec) in a transverse magnetic field of 10,000 G at room temperature. If the specimens were cut parallel to the [ 1101-direction, magnetoresistance amounted to 1.1 % compared with 1.3% for cuts parallel to the [1001-direction. The longitudinal magnetoresistance was smaller by two orders of magnitude in both cases. With increasing impurity concentration, magnetoresistance decreases and is only 0.1% when n = 4.5 x cm3 and pn = 2750cm2/V-sec. Upon cooling to 77°K magnetoresistance increased. For a doping of n = 5 x 1016/cm3 the transverse magnetoresistance effect increased from 0.28 % at room temperature to 6.2 % at 77°K. In this case, the current was parallel to the [llO]-direction and the magnetic field parallel to the [001]-direction. It follows from the experiments of Glicksman that with an electron concentration below 10"/cm3, the conduction band is isotropic, i.e., the effective mass is a scalar. For higher concentrations, a small anisotropy may be observed. This can be explained by the second minimum in the conduction band, the energy of which is only slightly higher than that of the main minimum at k = 0. 4. GALLIUM ANTIMONIDE Indium antimonide and indium arsenide have only one minimum in the conduction band and it is near k = 0. Therefore, they are isotropic semiconductors. In contrast, gallium antimonide has a second minimum in the conduction band which has been successfully used to explain the galvano-

10. MAGNETORESISTANCE

371

magnetic6'* and optical6' as well as piezore~istance~~ effects. Magnetoresistance, which is a second-order effect in B, becomes anisotropic and is useful in providing information about the position of the second energy ' defines the minimum in k-space. According to Pearson and S ~ h l , ~one three coefficients b, c, and d of magnetoresistance in a cubic crystal in weak magnetic fields as follows :

where i, and B, are the components of the current i and of the magnetic induction B, respectively. In an isotropic crystal d vanishes, b describes the transverse, and b + c the longitudinal magnetoresistance. A nonvanishing d is characteristic of anisotropy. The simplest method for determining the three coefficients utilizes a specimen cut parallel to the [ 1101-direction. If one measures the magnetoresistance with B parallel to the specimen as well as with B parallel to the [ilOI-direction and to the [001]-direction, one gets the following relation between magnetoresistance and the coefficients : longitudinal

B1/[llO]:Ap/poBZ = b

transverse

BII [Ool] :Ap/poB2 = b, BII[T10]:Ap/poB2 = b

+ c + d/2, + d/2.

(26)

The relative transverse magnetoresistance effect for n-type gallium antimonide at 4.2", 77", and 300°K as a function of the Hall coefficient R was presented by Becker in Fig. 6 of Chapter 8. The length direction of the specimens was parallel to the [ 1101-direction,permitting data representative of relation (26) to be obtained. For all specimens studied, the magnetoresistance was proportional to B2 up to fields of 20,000 G. It should also be noted that the magnetoresistance vanished at helium temperatures for IRI > 5 cm3/A-sec. This behavior corresponds to the increasing degeneracy of the electrons with spherical energy surfaces in k-space. The steep increase of magnetoresistance at low temperatures with lRJ < 5 cm3/A-sec is caused by a mixed conductivity magnetoresistance [compare Eq. (13)], which is obtained when the second minimum starts to be occupied. The energy separation of the second minimum from the first is the order of 0.08eV. Information about anisotropy is obtained, as mentioned above, by measuring longitudinal as well as transverse magnetoresistance in different directions of B. The transverse magnetoresistance of a specimen with W. M. Becker, A. K. Ramdas, and H. Y. Fan, J . Appl. Phys. 32, 2094 (1961). A. .I. Straws, Phys. Rev. 121, 1087 (1961). 69 A. Sagar, Phys. Rev. 117, 93 (1960). 'O G. L. Pearson and H. Suhl, Phys. Rev. 83, 768 (1951).

'6

372

H. WEISS

n = 5.5 x 1016/cm3at 300°K as a function of the direction B was shown in Fig. 7 of Chapter 8. Although the variations with direction are small, Becker was able to determine the coefficients in Eq. (26). For example, b 2 x 10- l o gauss-', whiled 10- l1 gauss-'. Therefore, the anisotropy is very small. This is reasonable because it is caused by the higher positioned minimum. The data are consistent with the conclusion that these minima lie on the [ 11l ] - a x e ~ . ~ l At 4.2"K, Becker et al. did not discover an anisotropy of Ap/po in nconducting gallium antimonide. This does not prove the absence of anisotropy. Because of the degeneracy, the influence of the anisotropy on magnetoresistance is expected to be very small and below the accuracy of measurement of the apparatus used by Becker et al. On a p-type specimen with current parallel to the [IlOJ-direction and p = 1.1 x 1016/cm3,Becker et found a longitudinal magnetoresistance proportional to B' at 77"K, whereas the transverse effect increased less rapidly with B. The simultaneous decrease of the Hall coefficient with increasing B indicates conductivity by two types of holes. Since, in spite of the small hole concentration, there is a distinct two-band conduction, it follows that the minima of both hole bands lie near each other. The behavior of the transverse magnetoresistance of the p-type specimen for different directions of B was shown in Fig. 9 of Chapter 8. In contrast to Fig. 7, which applied to the n-type material, directions of the maxima and minima are interchanged. Thus the sign of d is negative, and b + c is about equal in magnitude to d, in agreement with studies of MatyaS and S k a ~ h a . ~ ~ This could suggest bands with minima on the [lo01 axes,'l but nonspherical valence bands such as those in germanium and silicon are also possibilities.

-

-

5 . GALLIUM ARSENIDE

Glicksman7 studied the anisotropy of the magnetoresistance effect of n-type single crystals of gallium arsenide at room temperature. The specimens were cut parallel to the [Oll] direction. The transverse magnetoresistance was 0.45% for an electron concentration of 4 x 1016/cm3with p,, = 3400cm2/v-sec and 0.27% at a concentration of 4 x 1017/cm3 with p,,= 3800cm2/v-sec. The field was 10,OOOG. An anisotropy could not be found; at least, it was below the accuracy of measurement. In contrast, on a specimen with n = 8.5 x 101"/cm3 and an electron mobility between The relationships between the magnetoresistance coefficients for various symmetries of the band minima were presented by Shibuya7l" and appear many places throughout the literature.' I b 71aM.Shibuya, Phys. Rev. 95, 1385 (1954). 71bSee,for example, A. C. Beer, Solid State Phys. Suppl. No. 4, 231 (1963). 72 M. MatyiS and J. Skicha, Czech. J . Phys. 12, 566 (1962). 7'

10. MAGNETORESISTANCE

373

4000 and 4700 cm2/V-sec at room temperature, Kravchenko and Fan73 found a remarkable longitudinal magnetoresistance as well as a distinct anisotropy. The measurements of Fritsch and we is^^^ shown in Fig. 38 did not confirm the results of Kravchenko and Fan. In this case, we have a much purer and very homogeneous specimen with an electron mobility of 6000 cm2/V-secat room temperature and 13,000cm2/V-secat 77°K. The electron concentration decreased only slightly on cooling and was 1.2 x 1016/cm3 at 77°K. Figure 38 shows Ap/po for a rod which was cut parallel to the [ O l l ] direction as a function of the orientation of the magnetic field. For B perpendicular to the current direction (upper curve), the relative magnetoresistance was independent of the special direction of B. The lower curve demonstrates that longitudinal magnetoresistance practically vanishes. It was not found on other specimens either. From this it follows that the conduction band is isotropic with no longitudinal magnetoresistance for a concentration of 1016 electrons/cm3. Willardson and Duga7' analyzed the magnetoresistance of gallium arsenide but did not consider the question of anisotropy. They assumed isotropy and studied magnetoresistance and Hall coefficient as a function of the magnetic field as well as temperature on single crystals with n between 1015 and lOI6/cm3. Their calculations were based on a spherical parabolic conduction band, charge carriers scattered by acoustical lattice vibrations, and ionized impurities, with the latter being predominate. The mobility of the electrons in pure gallium arsenide at room temperature was assumed to be 11,500 cm2/V-sec.Willardson and Duga obtained satisfactory agreement between their calculations and their measurements of Hall coefficient and magnetoresistance at 55", 77", 196", and 300°K. Willardson and Duga were able to explain magnetoresistance measured in small magnetic fields by assuming thermal scattering and scattering by ionized impurities. However, to simplify the calculations they used acoustical rather than polar lattice vibrations as the minority scattering mechanism. Above 400"K, as was pointed out by E h r e n r e i ~ hthe , ~ ~predominant scattering mechanism is via polar optical modes. a negative transverse Emel'yanenko and N a ~ l e d o v7 ~8 ~observed , A. F. Kravchenko and H. Y. Fan, Proc. Intern. Con$ Phys. Semicond., Exeter, 1962 p. 731. Inst. of Phys. p d Phys. SOC.,London, 1962. 74 D. Fritsch and H. Weiss (unpublished measurements). " R . K. Willardson and J. J. Duga, Proc. Phys. Soc. (London) 75,280 (1960). 7 6 H. Ehrenreich, Phys. Rev. 120, 1951 (1960). 77 0. V. Emel'yanenko and D. N. Nasledov, Zh. Tekhn. Fiz. 28, 1177 (1958) [English Transf.; Soviet Phys.-Tech. Phys. 3, 1094 (1958)l. 78 D. N. Nasledov and 0. V. Emel'yanenko, Proc. Intern. ConJ Phys. Semieond., Exeter, 1962 p. 163. Inst. of Phys. and Phys. S o c , London, 1962. 73

374

H . WEISS

4

oil]

YO

1001

[oiil

3

4 QO

2

1

90"

0

120"

150"

\ 1 10"

FIG.38. Transverse magnetoresistance at 6 kG and 77°K of an n-type specimen of GaAs with n = 1.2 x 1016/cm3as a function of the direction of the magnetic induction. Specimen axis is along the [Oll] direction. (After Fritsch and we is^.'^)

%

FIG.39. Transverse magnetoresistance at different temperatures as a function of the magnetic induction B, measured on an n-type single crystal of GaAs with n = 7 x 1015/cm3. (After Nasledov et a1.")

10. MAGNETORESISTANCE

375

magnetoresistance on n-type gallium arsenide at temperatures below 60°K. The change of the sign of magnetoresistance was combined with a maximum of the Hall coefficient. From these results Emel'yanenko and Nasledov derived an impurity band conduction that is related to a negative magnetoresistance effect like that of indium antimonide. Figure 39 illustrates the dependence of magnetoresistance upon magnetic field at six different temperatures. The measurements were made on an n-type single crystal with an electron concentration of 7 x 1015/cm3. V. Influence of Pressure

The influence of uniaxial pressure and tension upon magnetoresistance has been studied on p-type indium antimonide by Schonwald.25 Measurements were made under uniaxial pressure up to 142 kp/cm2 on an intrinsic specimen orientated in the [ 1111 direction at 20"C.79The Hall coefficient increased by 1.3% and the conductivity decreased by 1.6% in agreement with results of other authors. The influence of the pressure upon magnetoresistance was less than the accuracy of the measurements. This result is reasonable. According to Eq. (14) magnetoresistance in the intrinsic conductivity range is determined by the product of hole mobility and the electron mobility. Because uniaxial pressure gives rise to a change in energy gap only, but does not vary the mobility in first approximation, a change of Ap/po is not expected for an intrinsic specimen. This is also the case at higher temperatures where no change of Ap/po could be found. At 150°C TABLE I1 RATIOOF CONCENTRATION AND MOBILITY OF HEAVY(pl, p p l ) AND LIGHT(p2, pP2) HOLES FOR UNIAXIAL PRESSURE AND TENSION PARALLEL TO THE [ l l 11 DIRECTION AT 77°K IN InSb P=O P p L /I+

PJP2 PPllPP2 PJP2

P

=

0.153 91

0.19

64

320 kp/cm2"

P

=

- 152 kp/cmz

P

0.16 84

0.175 76

7

1 0 1 5 / ~ ~ 3

The unit kp is equal to lo6 dynes.

no variation of Hall coefficient with pressure could be established. Different results are obtained with p-type specimens at 77°K. With no magnetic field, uniaxial pressure decreases the specific resistance, but it increases

'' The unit kp is equal to lo6 dynes.

316

H. WEISS

with tension. The same is true for the magnetoresistance effect, it is smaller with pressure than without pressure but larger with tension. To explain his results, Schonwald assumes that under the influence of pressure and tension the two valence bands are shifted with respect to each other. An increase in resistance under tension means that the fast hole band is pushed downward from the slow hole band, so that the ratio of fast to slow holes is diminished. On the other hand, for compression of the specimen, the fast hole band is lifted above the slow hole band and the number of the fast holes increases. This picture must be found in magnetoresistance. As mentioned above in discussing p-type indium antimonide, magnetoresistance may be described quantitatively by assuming two-valence bands. It is not necessary to assume a magnetoresistance in either of the two bands by themselves. By variation of conductivity and Hall coefficient of the two bands, Schonwald obtained curves for the Hall coefficient and conductivity as a function of the magnetic field with and without pressure. The calculated curves for A p / p o as a function of B, which are shown in Fig. 25, confirm this model. The variation of the concentration of the slow holes under tension and pressure remains below the accuracy of measurement, since the Hall coefficient is not changed in a measurable manner. Yet the concentration of the fast holes is changed as is shown in Table 11. With P = -152 kp/cm2 it was diminished by 8%; with a pre~sure’~ of P = 320kp/cm2 it was increased by 16%. These effects are too large to be obscured by the accuracy of measurement. Although the tension was about half as large as the pressure, both values are in good agreement. In addition to the change in carrier concentration, caused by the relative shift of the two-valence bands, a change in mobility was also observed.

Plasma, Effects

This Page Intentionally Left Blank

CHAPTER 11

Plasmas in Semiconductors and Semimetals Betsy Ancker-Johnson I . DEFINITIONS . . . . . . . . . . . . . . . . . 319 1 . Plasma . . . . . . . . . . . . . . . . . 319 2 . Solid State Plasma . . . . . . . . . . . . . 381 3 . Nonequilibrium Plasma Production . . . . . . . . . 382 I1 . PLASMA OSCILLATIONS . .

. . . . . . . . . . . . 386

4 . Conduction Electron Plasmons 5 . Valence Electron Plasmons . . 6 . Plasmons in a Magnetic Field .

. . . . . . . . . . 386

. . . . . . . . . . 391

. . . . . . . . . . 394

111 . MAGNETOREFLECTION AND ABSORPTION. . . . . . . . . 395 I . Magnetoreflection and Absorption as a Function of Magnetic Field 395 8 . Magnetoreflection and Absorption as a Function of Frequency . . 402 IV . WAVEPROPAGATION. . . . . . 9 . AlfvPn and Helicon Waves . . . . 10. Wave Propagation and Doppler Effects 11. Helicon Interactions with Phonons . V . PINCHEFFECT . . . . . . 12. Pinch Effect in Semiconductors . 13. Pinch Effect in Semimetals VI . INSTABILITIES . . . . . 14. Helical Instabilities . . . 15 . Recombination Oscillations . 16. Relaxation Oscillations . 1I . Other Observed Instabilities 18 . Two-Stream Instab .

VII . CONCLUSIONS . NOTATION .

. . . . . .

. . . . . . . . 416 . . . . . . . . 416 .

.

.

.

.

.

.

. 421

.

.

.

.

.

.

.

. 431

. . . . . . . . . . 433

. . . . . . . . . . 433 . . . . . . . . . . 453

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. 455 . 455 . 469 . 412 . 414 . 418

. . . . . . . . . . . . . . . 480

. . . . . . . . . . . . . . . . 481

.

I Definitions 1 . PLASMA

A plasma is a large collection of electrically charged particles possessing opposite signs. The collection is electrically neutral. capable of being characterized by not too low a kinetic temperature.and large enough so that

379

380

BETSY ANCKER-JOHNSON

the particles shield each other from their Coulomb fields as well as impinging fields. A consequence of charge neutrality is the occurrence of a natural resonant frequency, the plasma frequency. It arises if electrons as a group are displaced a small distance [ ( t ) from the positive carriers (assumed stationary). Then the charge density which crosses a plane (x + [) is nec, where n and e are the electron density and charge, respectively. An electric field is produced at this plane, E = -4xne[/~,. The quantity E~ is the dielectric constant of the medium in which the plasma exists; it is essentially unity for ionized gaseous plasmas and may exceed several hundred for solid state plasmas. This electric field causes an acceleration of the displaced electrons whose effective mass is meff = m*m,:

d2[ e -=-E= dt2 meff

-

4xne2 Elmelf

r.

Thus, the plasma frequency is

(1) The shielding capability of a collection of charged particles is tested by the application of the Debye criteria: The Debye shielding distance, expressed as

(2) must be much smaller than the dimensions of the plasma and the Debye sphere, $ d D 3 , should contain many plasma particles. The temperature of the plasma is given by T, and k is Boltzmann’s constant. The Debye length iDis invariably much less than any dimension of the crystals used in plasma research (typically, AD = cm) because the densities of such to plasmas are relatively high and their temperatures are relatively low ; however, the Debye spheres of solid state plasmas often contain only a few plasma particles. This means the fields in the plasma no longer have simple spatial dependencies and, if the number of particles within the sphere is too small, determining the fields may even pose an intractable problem. If the plasma is degenerate, the appropriate average velocity for substitution into Eq. (2) is the average of the squared Fermi velocity uF2. This very broad definition of plasma is adequate, but in order to investigate the properties of plasmas, the fulfillment of a subsidiary condition is

1 1.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

381

helpful. If randomizing collisions dominate over long-range fields and forces, then the plasma lacks a collective response. For example, in the presence of an applied magnetic field the helical paths which charge carriers execute around the flux lines must not be interrupted by collisions so often that their motion approximates the straight line paths which they follow in the absence of the applied field. To ensure that an applied magnetic field does not have a negligible effect on the organized motion of a plasma, the product of the cyclotron angular frequency eB/m*c and the time between collisions must exceed unity, o,z > 1. This condition is barely fulfilled in some solid state plasma experiments, for example, those on the helical instability oscillations (Section 14), when small magnetic fields and nonequilibrium plasmas are employed. Some of the plasma effects to be described are very sensitive to the band structure of the semiconductor or semimetal in which they occur. This is particularly the case for magnetoreflection and absorption (Sections 7 and 8); indeed the investigation of these effects has contributed to the understanding of band structure in a number of materials. On the other hand, some plasma effects are independent of band structure, for example, helicon wave propagation (Section 8). The plasma properties studied by employing nonequilibrium plasmas (Sections 12, 14-18) yield information about the plasma state rather than about band structure. 2. SOLIDSTATEPLASMA Three types of plasmas occur in solids. Electrons (or holes) interacting with themselves in a uniform ion background constitute an “uncompensated plasma,” for example, the electrons in a metal, the electrons (or holes) of an extrinsic semiconductor, or those of a doped semimetal. Carrier motion in this type of plasma is always constrained by the necessity to maintain space charge neutrality with the fixed lattice or impurity ions. This does not prevent the occurrence of longitudinal oscillations, as discussed in Section 4. The second type, “compensated plasma,” possesses equal densities of oppositely charged particles, electrons and holes, both of which are highly mobile compared with the positively charged particles of an uncompensated plasma, namely, the lattice or impurity ions. Examples are the plasmas of intrinsic semiconductors and pure semimetals This type of plasma is also unable to engage in the gross motions that the third type of plasma can perpetrate. The third type is composed of nonequilibrium carriers in essentially equal densities, produced, for example, by injection from current contacts, by optical injection, and by impact ionization. These charge carriers are not required to neutralize fixed ionic charges, and therefore can readily engage

382

BETSY ANCKER-JOHNSON

in gross displacements such as occur during the onset of the pinch effect (Section 12). Uncompensated plasmas occur only in crystals. Compensated plasmas have some properties similar to thermally ionized gaseous plasmas (relatively high temperature plasmas). Nonequilibrium plasmas are completely analogous to partially ionized gaseous plasmas. Effects occurring in all three types of solid state plasmas are discussed below.’ 3. NONEQUILIBRIUM PLASMAPRODUCTION

The plasma properties of nonequilibrium carriers produced by impact ionization were chronologically the first to be investigated. As is well known, electrons under the influence of easily obtainable electric field strengths can strike atoms in some semiconductors with sufficient energy to excite electrons into a conduction band, leaving behind holes in a valence (or impurity) band. This production of electron-hole pairs by across-thegap ionization is an avalanche multiplication process yielding a plasma of relatively high density. This has been demonstrated, for example, by Glicksman and Steelela as shown in Fig. 1. These data were obtained by simultaneously observing the current passing through an n-type lnSb sample at 77°K and the voltage difference between two points along its length. Then the sample was immersed in a transverse magnetic field, the observations were repeated, and the Hall coefficient was measured as well. A pulsed power supply was used to avoid heating the lattice. The avalanche threshold is seen to be E = 200-300 V/cm. The sharp decrease in the Hall coefficient beginning at the same electric field for which the conductivity enhancement starts (both the R, and associated conductivity curves were recorded for a field intensity of 7 kOe) confirms that plasma has been produced. Its density can be calculated assuming a uniform carrier distribution ; how much greater than this the true density may be cannot be determined from these data because of the occurrence of the magnetic pinch effect, discussed in Section 12. Nonequilibrium plasmas have also been produced by injection of electronhole pairs from current-carrying contack2 Since soldered contacts that are genuinely “Ohmic” are extremely difficult to fabricate, almost any pair of soldered contacts will produce an injected plasma. Plasmas have been produced by this method in a number of semiconductors, whereas impact ionization as a method of plasma production is limited to semiconductors possessing a relatively small energy gap or to impact ionization of impurity

’ A discussion of the uncompensated plasma occurring in a metal is beyond the scope of this review. An exception is made in Section 11. IaM. Glicksman and M. C. Steele, Phys. Rev. 110, 1204 (1958). B. Ancker-Johnson, R. W. Cohen, and M. Glicksman, Phys. Reo. 124, 1745 (1961).

'

1 1. PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

383

'05

4

40

400

E (Volts/crn) FIG.1. Conductivitycharacteristics and Hall coefficient of a n-type InSb sample at 77°K illustrating plasma production by impact ionization. (After Glicksman and Steele.'')

states in semiconductors at very low temperatures. Current voltage characteristics, Fig. 2, show that a steep rise in conductivity occurs in p-type InSb at electric field strengths about an order of magnitude lower than the threshold for impact ionization in either n- or p-type InSb at the same ambient temperature. The three different magnitudes of conductivity in Fig. 2 produced by the same sample are controlled by the nature of the current contacts.2 Still more copious injection at even lower fields is obtained by using a pronounced p + p n structure with the p-section being very long compared to the p + and n sections. The mild increase of current at fields between 100 and 300 V/cm compared with the steep rises at larger and smaller fields is the result of the magnetic pinch effect. The conductivity changes from 2.6 mho/cm (Ohmic magnitude) to approximately 50 mho/cm at a power input just less than that required for the reduction in the slope of the conductivity curve; assuming a uniform distribution of plasma, its density at the highest injection level for sample C-14 of Fig. 2 is 3 x l O I 4 ~ m - The ~ . Debye length of this plasma is 1.5 x 10-5cm, and a Debye sphere thus contains about four charge carriers. Its coCz magnitude for B = 1 kG is somewhat greater than unity.

384

BETSY ANCKER-JOHNSON

The Hall coefficient of an injected plasma as a function of current is shown in Fig. 3. In the Ohmic range R, is positive and constant, as expected

m

FIG.2. Conductivity characteristics obtained using two samples of p-type InSb at 77°K illustrating plasma produced by injection from current-carrying contacts. (The x 's correspond to sample 2K-E, and the other symbols to three different sets of injecting contacts on sample C-1-4). The first steep rise in conductivity is caused by injection of carriers and the second by impact ionization. (Data from Ancker-Johnson et aI.Z*3).

for a p-type semiconductor. As the current increases the R , decreases to zero and then acquires a negative value characteristic of n-type material, or intrinsic material with electron mobility pe greater than hole mobility p,,. At large currents, when the density of carriers produced both by injection and impact ionization is much greater than the initial hole densities, B. Ancker-Johnson, Proc. Intern. Con$ Phys. Semicond., Exeter, 1962 p. 141. Inst. of Phys. and Phys. SOC.,London, 1962.

11.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

385

n-', hence the saturation values shown by the curves. For R H = 0 the expression for the injected plasma density assumes a simple form if the electron and hole relaxation times are independent of energy, namely, RH K

n = po(l

+ p e 2 ~ 2 ) [ b 2+( 1ph2B2)1-',

(3)

where b = p,/ph, po is the equilibrium hole density, and B is the applied magnetic field intensity. The preceding expression also neglects the contribution from a second hole band-specifically, the light hole band in InSb.

0

2

4

6

8

10

12

14

16

18

20 22 24

1 [Amps] FIG.3. The Hall coefficient as a function of total current for two levels of injection corresponding to the intermediate and lowest injection levels of Fig. 2, sample C-1-4. (After AnckerJohnson and Cohen, unpublished.)

The experimental results show that typically an injected density of only 1 % of the initial density p o is required to reverse the sign of R H in p-type InSb consistent with the large value of b in InSb. The formation of an injected occurs as follows: Electrons injected from a negative contact into a p-type crystal are constrained by the applied field to drift into the sample at their usual drift velocity peE. M. F. Berg, Bull. Am. Phys. SOC.9, 318 (1964).

386

BETSY ANCKER-JOHNSON

Approximately an equal number of holes become associated with these excess electrons to avoid large space charges. Their ultimate source may be either the plus or minus contact, and their immediate source is probably the free holes initially present in the semiconductor. A plasma thus forms at the negative contact and then fills the sample at a velocity that is a function of the applied field. The velocity of the front saturates at 6 x lo6 cm/sec beginning at approximately 200 V/cm in a typical p-type InSb crystal4 ; this velocity is an order of magnitude less than the electron drift velocity at the same electric field since it is an ambipolar-like velocity. In order to obtain reasonably uniform plasma distribution throughout a crystal, it is vitally important that the lifetime of the plasma exceed the time required for the plasma front to traverse the sample. The traverse time for a 5-mm long p-type InSb sample is about 80 nsec at the maximum velocity and twice that time with 100 V/cm applied. The m e a ~ u r e d lifetime ~.~ of the injected plasma is 1 psec through a plasma density range of 1 x 1 O I 2 to 2.5 x loL3~ m - This ~ . lifetime is four orders of magnitude larger than the electron lifetime reported6 for densities in the lo9 cm-3 range. If the lifetimes were as short at the higher density as at the lower, plasma experiments would be very difficult to perform. Each of these two methods of producing nonequilibrium plasmas has certain advantages. Impact ionization so far has provided the highest density plasmas. This density, however, cannot be controlled as can the injected plasma density. The higher electric field strength required for impact ionization than for injection means the plasma is considerably “hotter” than in the former case. Also, impact-ionization plasmas usually undergo pinching promptly after production, so that their density distributions are unknown and essentially uncontrollable. Because of the enumerated advantages of injected plasmas it is likely that optical injection utilizing lasers will become a popular way to produce high densities of nonequilibrium electrons and holes for plasma studies. 11. Plasma Oscillations

4. CONDUCTION ELECTRON PLASMONS A review of plasma effects may appropriately begin with a discussion of plasma oscillations, since their discovery in a partially ionized gas7 constituted the birth of plasma physics. The present discussion is very cursory because little work exists on plasma oscillations in semiconductors and B. Ancker-Johnson and M. F. Berg in “Physics of Semiconductors” (Proc. 7th Intern. Conf.), p. 513. Dunod, Paris and Academic Press, New York, 1965. A. R. Beattie and R. W. Cunningham, Phys. Reo. 125, 533 (1962). ’ L. Tonks and 1. Langmuir, Phys. Rev. 33, 195, 990 (1929).

11.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

387

semimetals. A plasma oscillation is an organized motion of a collection of charged particles resulting from the long-range correlations caused by Coulomb interactions. There are two branches in the longitudinal spectrum* : During the higher frequency mode the electrons and holes move out of phase, whereas they move in phase in the lower frequency mode. At the long wavelength limit the higher frequency mode oscillates with the plasma frequency, and the lower frequency mode has a linear dispersion relation. For these reasons the modes are called optical and acoustic, respectively. Such a longitudinal oscillation of a dense electron or hole plasmauncompensated usually, but in principle any type-ccurring in a solid is a quantized, elementary excitation called a plasmon.8 The plasmons occurring in metalssa possess such high frequencies that their energies are greater than the energies of electrons at the top of the Fermi distribution; these plasmons are thus not thermally excited. They can be generated when charged particles of energy greater than hw, pass through the electron gas producing inelastic scattering. In a free electron gas the resulting fundamental energy loss is

(4) where

In a semiconductor, plasmons can exist in thermal equilibrium since the average electron energy is usually much greater than hv,. For example, the plasmon energy in n-type Ge with n = 10'6cm-3 (degenerate) at temperatures for which impurity scattering dominates is about 2 x eV. At T = 100°K the plasmon energy for n = 2 x lo1' cm-3 is approximately equal to the average energy of the electrons. At 4°K the mean electron thermal energy exceeds the plasmon energy for n < 3 x 1014 ~ m - The ~ . purer the semiconductor, the greater the discrepancy in energy at a given temperature. The electron absorption energy in a solid state plasma may differ from (4) because of interband electron transitions; only if the latter have much smaller average frequencies than w, does the observed plasma frequency D. Bohm and D. Pines, Phys. Rev. St, 609 (1953); D. Pines, Can J . Phys. 34, 1379 (1956); D. Pines, Nuow Cimentu Suppl. 7, 329 (1958); P. Nozieres and D. Pines, Phys. Reu. 113, 1254 (1959); D. Pines and J. R. Schrieffer, Phys. Rev. 125, 804 (1962); also references cited in these papers. '"The extensive work on metals is omitted here to remain within the scope of this review.

388 equal tion

BETSY ANCKER-JOHNSON 0,. In

general the plasmon frequency is defined by a dispersion rela-

where IC is the wave vector and 1, is given by Eq. (2). This approximate dispersion relation may be derived by a variety of methods, such as the random phase approximation? the Green f u n ~ t i o n , ~ and, more simply, by using the Schrodinger equation.” The Green function method is the most general since the validity range of the dispersion relation

“I

FIG.4. A frequency-wave number diagram for two plasmons possessing different temperatures, TI z 7’’.

is a natural consequence of this theory rather than being imposed. This range is given by the first of these limits: for

IC

-e 1;

IC

9 2;’

= I C ~

collective behavior prevails;

whereas for

individual particle behavior prevails.

The frequency-wave number diagram of Fig 4 helps illuminate the meaning of Eq. (5). As the wave number approaches K~ and the phase velocity of plasmons approaches the mean thermal velocity of the plasma

lo

V. L. Bonch-Bruevich and S. V. Tyablikov, “The Green Function Method in Statistical Mechanics.” North-Holland Publ., Amsterdam, 1962. J. E. Drummond, “Plasma Physics,” p. 2Sff. McGraw-Hill, New York, 1961, and private communication.

1 1. PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

389

particles, Landau damping dominates. Thus, plasmons only exist at frequencies up to 10 to 15% greater than cop. More plasmons occur as the temperature is reduced because Landau damping is postponed to larger wave numbers since K , a T-’’’. As the density is decreased, Landau damping becomes more dominant, because xDcc n“’. The increased slope at c is the result of the third term in Eq. (5). It has little the tail of the a ~ i curves influence on plasma existence since the charge in slope occurs at wave numbers far larger than K , . Collisional damping is also significant in semiconductors because the period of a plasmon is of the same order of magnitude as the time between collisions, so that the absorption per cycle in semiconductors is large compared to that in pure metals. Tomchuk” has investigated the possible influence of thermal equilibrium plasmons in semiconductors on the conductivity via scattering. He has also included the effects of impurity scattering, since it is important at low temperatures, the temperature region in which plasmons are least damped (see above). His work is based on some results of Pines and Bohm.” They expressed the energy loss per unit distance of single electrons to the plasmon field as

+

L, = ( 7 ~ n e ~ / ~ ,In(1 ~ d , ) 2uOz/(v’)>,,)

(6) and the energy loss for random electron collisions within the Debye sphere as (7) where do and uo are the incident particle energy and velocity, respectively. If some thermal electrons are “incident” with mean kinetic energies of kT then the ratio of the plasmon loss to the random-collision loss is L, = (2nne4/~,2~0) 1n(;iDEfio/e2),

-

where b, is the impact parameter, b, = e2/3s,kT. Thus scattering on plasmons is relatively unimportant if ln(llD/bo)S 1. Tomchuk’ has sharpened this condition to conclude that plasma scattering is insignificant if h(AD/bO) 2 2. This result may be interpreted by noting that

AD/bo = 9(4/37cnlD3). (9) Thus, if a Debye sphere contains 0.82 or more plasma particles, plasmon l1

l2

P. M. Tomchuk, Fiz. Tuerd. Tela. 3, 1019 (1961) [English Transl.: Soviet Phys.-Solid State 3, 740 (1961)l. D. Pines and D. Bohm, Phys. Rev. 85, 338 (1952).

390

BETSY ANCKER-JOHNSON

scattering is negligible for energy exchange. (A similar argument holds for momentum transfer.) This boundary for E~ = 16 and T = 100°K occurs at n = 1.1 x 10l6 ~ m - and ~ , for T = 4°K it occurs at n % 7.3 x loll cmP3. These results are only valid for nondegenerate plasmas, however, this is not an important limitation since degenerate densities are much greater than the densities at which plasmons can be thermally excited. The density limit for thermal excitation was noted above. The excitation of plasmons in metals by high energy electron beams produces some absorptions at frequencies less than the fundamental plasmon frequency. Such measurements appear to be adequately explained’ by surface polarization causing a plasmon loss at

6, = k 0 d ( l

+

(10)

if the surface is a thin film with plane, parallel sides. R ~ m a n o v ’has ~ suggested that very thin p-n junctions can act as such a surface for a solid state plasma, and that “surface plasmons” may affect the conductivity. To show this he has derived a dispersion relation taking into account spatial dispersion arising from a system consisting of a semi-infinite electron plasma at z > 0 in contact with a semi-infinite hole plasma at z < 0. It is possible to show, he states, that an electron passing through a p-n junction can lose energy corresponding to four quasidiscrete excitations of plasmons, one each for the hole and electron plasma related to the usual plasmons in the volume, and two associated with the surface. Another approach to plasmons has been taken by Frohlich and Pelzer’ who noted a phenomenological connection between possible plasma frequencies and the optical constants; thus, in their view plasma oscillation theory is a special case of the theory of absorption and dispersion. The connection exists because, in the limit of small wave number (far from Landau damping and degeneracy) and for cubic crystals, the dielectric constant tensor is a scalar.I6 Therefore, the response of the electron plasma to either transverse or longitudinal excitations, for example, photons or fast electrons, can be described using the same quantity, namely, the longitudinal dielectric constant. l 7 The condition for the existence of a plasma at a given frequency and wave number is15 E ( 0 , K) = 0 . (11) R. H. Ritchie, Phys. Rev. 106, 874 (1957); E. A. Stern and R. A. Ferrell, ibid 120, 130 (1960). Yu. A. Romanov, Fiz. Tverd. Tela. 5, 2988 (1963) [English Transl.: Soviet Phys.-Solid State 5, 2187 (1964)l. H. Frohlich, and H. Pelzer, Proc. Phys. SOC. (London) A68, 525 (1955). l 6 S. L. Adler, Phys. Rev. 126, 413 (1962). l 7 H. R. Philipp and H. Ehrenreich, Phys. Rev. 129, 1550 (1963).

l3

l4

11. PLASMAS

391

IN SEMICONDUCTORS A N D SEMIMETALS

The real and imaginary parts of this expression yield two experimentally accessible criteria for determining the plasmon frequency : Re(c) = Re[c,

(1

-

w(O

+ i/z)

=

0

when oRz = up2- z-’

(12)

and -Im(c-’)

=

maximum

when

0; x 0; -

(2z)-’. (13)

+

The latter approximation obtains for o p z 1, where z is the time between collisions. The plasmon lifetime, as derived from the half-width of the absorption band, is given by (see Ref. 18 and references cited therein) where q is the index of refraction (see Section 7), k the extinction coefficient, and 27cvI is given by the w , in (13). Arai18 has applied this theory to heavily doped n-type Si. He measured the carrier scattering time as a function of density and found it to be nearly constant (-1 x 10-14sec) at the higher concentrations in his series of samples, whose carrier concentrations ranged from 1 x 1OI8 to 6 x 10’’ cm-3. Thus, he anticipated being able to separate the effects of density and scattering time on the plasmon lifetime. In the absence of sufficient data on the dielectric constant as a function of frequency, he calculated its value. The frequencies at which conditions (12)and (13) are fulfilled are plotted in Fig. 5, along with the plasmon frequency vp, calculated from (3) using 0.27~1,for the average electron effective mass and 11.7 for the dielectric constant of Si in the absence of plasma. The results obtained from the real part of Eq. (11) yield the lower plasmon frequencies of the figure, as expected from condition (12) in comparison with condition (13). The calculated plasmon lifetime to sec) is exactly proportional to the carrier (range 2 x scattering time, and hence Arai concludes that the plasmon lifetime is independent of density in his samples. 5. VALENCE ELECTRON PLASMONS

When the impinging particles possess energies much greater than the fundamental absorption loss, the valence electrons can be considered essentially unbound and are also able to oscillate collectively. The fast electrons in the usual characteristic energy loss experiment^,'^.'^ as well as the photons in some reflectivity measurements (e.g., “region 2” of Ref. 17), possess considerably more energy than the fundamental plasmons in semi” T. l9

Arai, Proc. Phys. SOC.(London)84, 25 (1964). 0. Klemperer and J. P. G . Shepherd, Aduan. Phys. 12, 355 (1963), and references cited therein.

392

BETSY ANCKER-JOHNSON

-

I

I

u

2

1

>. K

g

10”

L

LL

C

E c

“F B 0-

10l2 10’8

1019 Carrier Concentration (cm-3)

1020

FIG.5. Plasmon frequency as a function of carrier concentration for heavily doped n-type Si. The frequencies vR and vIr as obtained from conditions (12) and (13), are also plotted. (After Arai. 18) 1.2 1.8

1.0

1.6

1.4

0.8 1.2

-1

-Irnc-‘

rnE-1

1 .o

0.6

0.8 0.4

0.6

0.4

0.2 0.2

E (eV) FIG.6. A comparison of the energy loss function -Im(e-’) obtained from the results of optical (solid lines) and characteristic energy loss experiments20.21(dashed lines) for Ge and Si.

(After Philipp and Ehrenreich.”)

1 1. PLASMAS

393

IN SEMICONDUCTORS AND SEMIMETALS

TABLE I PLASMA OSCILLATION DATAFOR SEMICONDUCTORS AND SEMIMETALS” Calculated energy Material Ge Si InSb CdTe ZnTe GaP GaAs Sb As Bi

Nlatom 4 4 4 4 4 4 4 5 3 3

Measured energy

fro,

firnpsff

fro,

Beam

Optical

15.5 16.6 12.7 13.0 14.0 16.6 15.5 15.0 13.2 10.8

16.2 16.6 11.5

13.8 15.0 10.9

-

-

16.4b 17.W 13.v 16.9d 16.3d 15.3” 20.2’ 14.7b

16.0 16.4 12.0 16.9 14.7 -

-

-

16.3 12.3 -

13.3 9.7 -

-

-

-

-

“The plasmon energies rimpv are calculated for N/atom participating (“free”) valence electrons The effective plasmon energy (column four) is taken from Ref. 17. The energies h o , are given by the dispersion relation Re E(O) = 0 and are also from Ref. 17. The last two columns contain the maxima of the energy loss functions obtained from electron beam experiments (referenced individually) and optical energy loss experiments (Ref. 17). AU energies are in electron volts. C. J. Powell, Proc. Phys. Soc. (London) 76, 593 (1960). C. Kunz, 2. Physik. 167, 53 (1962). B. Gauthe, Phys. Rev. 114, 1265 (1959). J. Geiger, Z. Naturforsch. 17A,696 (1962). ’J. L. Robins, Proc. Phys. SOC.(London)79, 119 (1962). References to the calculations tabulated for ha, may be found in Ref. 19.

conductors. Philipp and Ehrenreich” have compared the results of their optical experiments with those of the characteristic energy loss experiments, Fig. 6. The dashed lines represent the loss when 1.5-keV electrons are incident on GeZ0and 47 keV are incident on Si.” The solid lines are the optical results normalized so that the peak heights coincide. The comparison shows good agreement in both the energy at which the peaks occur and the width of the absorption. The electron scattering curves show a low-lying loss attributable to surface plasmons, which is absent in the optical data. The Ge optical data exhibit a further rise at higher energies, whereas the Si data do not. Philipp and Ehrenreich” tentatively attribute this to the presence of d bands in Ge and their absence in Si. They have obtained an which is modified expression for an effective plasmon frequency opcff, 2o

C. J. Powell, Proc. Phys. Soc. (London) 76, 593 (1960).

*’H. Dimigen, Z. Physik 165, 53 (1961).

394

BETSY ANCKER-JOHNSON

compared with the “free” valence electron values because of coupling between the valence and d band electrons. Table I summarizes plasmon oscillation data for semiconductors and the semimetals Sb, As, and Bi. The second column lists the number of electrons per atom, namely, the usual number of s and p valence electrons, used to calculate the plasmon energy in column three. The data from fast electron experiments and from optical experiments are given in the last two columns. The agreement among the various methods for obtaining the plasmon energy is reasonably good for the semiconductors listed, but the discrepancies occurring for As and Bi suggest that the theories are not adequate to describe their plasmas. Low-lying plasmon losses have been measured in Si (10 eV)22 and Ge (10.7 eV).” The calculated surface plasmons, using = 1 in Eq. (lo), are 12 and 11.5 eV, respectively. The discrepancies may be because c l is greater than unity as a result of surface contamination or because the effective possesses some contribution from the crystal. The recent detectionz3 of monochromatic photons radiated by decaying plasmons in Al and Mg thin films as predicted by Ferre1Iz4 permits improved resolution and thus more detailed investigations of the plasmon spectrum than are possible by measuring electron energies. NO such results have yet been reported for plasmons in semiconductors or semimetals. 6.

PLASMONS IN A

MAGNETIC FIELD

Mermin and Cane125have generalized the Bohm-Pines theory’ to include the effects of a uniform magnetic field. In the absence of such a field and within the long wavelength limit noted above, the local Fermi sphere of electrons oscillates without distortion along the direction of propagation. When the plasma experiences a finite magnetic field the local Fermi sphere moves along an elliptical orbit in velocity space. These orbits have an easily visualized form for certain limits. If the field is parallel to the direction of propagation K, the plasmon is unaffected and the orbit follows the usual cyclotron motion. If the magnetic field is transverse, the plasmon follows an orbit whose projection in the plane orthogonal to the field is an ellipse with its major axis parallel to K. At the limit o,-+ 0 the motion reduces to a linear oscillation and at the limit o,9 op ordinary cyclotron motion results. The Landau damping of a plasmon is increased considerably by a magnetic field not exactly parallel or perpendicular to K. Mermin and ” W.

Hart1 and H. Raether, Z. Physik 161, 238 (1961); M. Creuzburg and H. Raether, ibid

171, 436 (1963). 23 24

E. T. Arakawa, R. J. Herickhoff, and R. D. Birkhoff, Phys. Rev. Letters 12, 319 (1964). R. A. Ferrell, Phys. Rev. 111, 1214 (1958). N. D. Mermin and E. Canel, Ann. Phys. (N.P) 26, 247 (1964).

11.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

395

Canel describe additional resonances which may be excited by plasmons; these, however, are heavily damped unless the plasma temperature is near zero. Stephen26 has obtained a dispersion relation for a degenerate plasma in a magnetic field also within the random phase approximation He discusses several special cases. III. Magnetoreflection and Absorption26a

7. MAGNETOREFLECTION AND ABSORPTION AS A FUNCTION OF MAGNETIC FIELD The reflection coefficient of orthogonally incident waves may be expressed as

if the extinction coefficient is negligible. In this limit the terms involving the collision frequency and random motion of the carriers in the expressions for the index of refraction are also negligible. Then the following relations are obtained from Maxwell's equations and the equation of motion for isotropic mass particles in a semi-infinite plasma exposed to electromagnetic radiation and a static magnetic field:

Right- and left-handed circularly polarized waves propagating with wave vector K along the magnetic field in the medium have indices of refraction q+ and q-, respectively. A linearly polarized wave propagating with both its wave vector and its electric vector transverse to the magnetic field (but whose electric vector may have components parallel and normal to K ) has an index q k The index qI1corresponds to propagation in the absence of a magnetic field. l6 M. J. Stephen, Phys. Reo. 129, 997 (1963). 26aAnexcellent review of the early work in this field has been provided by Lax and Mavroide~,~' pp. 343-355. 2' B. Lax and J. Mavroides, Solid State Phys. 11, 261 (1960).

3%

BETSY ANCKER-JOHNSON

Total reflection occurs in the limits q co, q = 0 (or for q imaginary), whereas zero reflection ensues if q = 1. In these cases the conditions for propagation parallel to the magnetic field are : -+

R = l ; ?-+a Reflection edge

(only one condition),

w = w,

plasma edge

[

0, R=O; q = 1 -= + I - - Reflection minimum

El:

10‘1-

The occurrence of these conditions as a function of magnetic trated in Fig. 7 (see Lax and Mavroides” for further illustrations). The

FIG.7. The reflection coefficient as a function of WJW for lossless plasmas of three different densities when K 11 B. The dielectric constant of the medium without a plasma is E , = 18.

position of the reflection edge, condition (19), is independent of plasma density. Condition (20) gives the width of the frequency range over which reflection occurs in units of wJw; the larger (w,/o)’, the broader the total reflection region. The cutoff at w, = o - w t / o is valid for any orientation of magnetic field with respect to propagation direction, since the propagation constant is zero at cutoff. It is very sharp because the dielectric constant of the medium containing the plasma is relatively large, for example, z 17 for InSb.

1 1.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

397

Some data which show the influence of the plasma as the experimental conditions deviate from classical cyclotron resonance absorption, for which wp 4 w, are reproduced in Fig. 8. The sample contained a compensated plasma of n x 2 x 1OI6 cmV3 corresponding to vp z 2.6 x 10” sec-’. For v z 1 . 8 the ~ ~ deviation of the reflected minimum from w J o = 1 is slight, but at v x 1 . 2 ~the~ deviation is about 45%, clearly showing the tendency of the plasma to shield out ordinary cyclotron resonance. The cyclotron mass has been obtained from such measurements by using the corrected o,values. The large collision frequency in the sample accounts for the absence of such sharp changes in the reflection coefficient as simple theory, Fig. 7, predicts. 1.30

-

0 .c

-0-C-Q-

p0 :

1.10 1.00

~=3.6 0=3.2

1

-

.90

.so

I

0

0.2

I 0.4

I

I

0.6

0.8

I I 1.0

I

I

1.2

1.4

1.6

*C/W

FIG.8. Cyclotron resonance data for InSb showing the influence of the plasma. [After H. G. Lipson, S. Zwerdling, and B. Lax, Bull. Am. Phys. SOC.3, 218 (1958),as reproduced in Ref. 27.1

Hebel and WolffZ8 have discovered quantum effects in the magnetoreflection and absorption of bismuth. These are observable in the reflection coefficient, as a function of frequency and magnetic field, of infrared radiation normally incident on slabs whose surfaces are either parallel or perpendicular to a magnetic field. The effects occur because : (a) the energy band of the conduction electrons is nonparabolic, hence the splitting of the cyclotron frequency is a function of momentum along the applied magnetic field, (b) the Fermi energy is small, E , = 27 meV, and (c) the cyclotron masses are very light. Thus the Landau level spacings in modest magnetic fields and the Fermi energy are comparable. 28

L. C. Hebel and P. A. WolfF, Phys. Reu. Letters 11, 368 (1963); L. C. Hebel, in “Plasma Effects in Solids” (Proc. 7th Intern. Conf.), p. 117. Dunod, Paris and Academic Press, New York, 1965.

398

BETSY ANCKER-JOHNSON

The relationship of the Landau levels to the Fermi energy &, is illustrated in Fig. 9a for a field parallel to the binary axis with magnitude 6.8 d B d 13.6 kG. There are two types of transitions that influence the reflectivity, the transitions between levels 0,l and those between 1,2. The density of states is largest at K , = 0. Thus, a reflection edge should occur at w z wqin the stated magnetic field range, as is also the case for parabolic

t

>.

P u c

Conduction Bond

Ot

I

I 0 xz

FIG.9a. The Landau level energy plotted for Bi as a function of the wave vector in the magnetic field direction. The three lowest levels are shown with the field magnitude chosen so that the n = 2 level is above the Fermi surface and the n = 1 and 0 levels intersect it. (After Hebel and W01ff.*~)

energy bands. Both types of transitions contribute to absorption in the frequency ranges wl’< w < to1 and w2’ < w2 These ranges are unique to nonparabolic bands. Three quantum effects emerge as a consequence of the energy level relationships of Fig. 9a. One occurs when the incident radiation has a frequency o > w1 (and, therefore, > w 2 ) when B is at the lower limit of the range given above. Since there are then no allowed transitions, the plasma is lossless except for collisional damping. As B is increased, the usual reflection minimum, Eq. (21), and plasma cutoff, Eq. (20),are observed

1 1. PLASMAS IN SEMICONDUCTORS A N D SEMIMETALS

399

and then, as B reaches B , , which puts the bottom of the n = 1 level at EF, w = w,, the plasma becomes dispersive, and a reflection dip occurs. The magnitude of B at which this dip occurs is independent of the orientation between K and B, whereas classical plasma effects are dependent on these orientations [cf. Eqs. (16)-(18)]. If B is made larger than B,, only 0 , l transitions are allowed (extreme < w < wl. quantum limit), so absorption may occur only in the range 0,' When B is held constant at a B > B , and w is varied, a reflection minimum

I

6

7

I 8

I 9

I

I

I

I

I

10

11

12

13

14

B

In

15

kG

FIG.9b. Experimental reflectivity at w = 205 cm-' as a function of magnetic field with this field normal to the surface of a Bi crystal at 2°K and parallel to a binary axis. (After Hebel and W01ff.~~)

occurs for r] = 1 at a frequency just <wl'. This new reflection minimum can only occur if the bands are nonparabolic, since otherwise, the stated absorption range does not exist. The third quantum effect is the result of unequal Landau level spacing, C ? ~ + ~ ( K ,) &"(K,), for the various principal quantum numbers n. This means that the frequency of allowed transitions at K, = 0 is a discontinuous function of B as the bottoms of the various Landau levels pass through the Fermi energy level. This is demonstrated in Fig. 9c for the two transitions of Fig. 9a. A reflection minimum occurs at w 2 when B is such that o = w2, as seen in Fig. 9b. A higher frequency of radiation calls for a larger B to produce this minimum until B , is reached. Then the o2transition can no

400

BETSY ANCKER-JOHNSON Field (kG) 0

4.5 I5

8 10 12 13

1 1Ill 190

230

210

I

250

I

16

173

I I1 I 1 I I I 270

290

Frequency (cm-’1

FIG. 9c. The magnetic field and corresponding frequency at which reflection minima are observed in Bi with the magnetic field along a binary axis. (After Hebe1.28)

longer occur, and there is no reflection minimum until B is large enough to make o = ol. Thus, Fig. 9c shows that a “Jump” in plasma frequency has occurred. Linearly polarized radiation traversing a plasma immersed in a longitudinal magnetic field experiences Faraday rotationz9 of the plane of polarization :

--

where d is the path length and q k are given by Eq. (16). In the limit of small

240

200

160

8

2 g a

120

80

40

0 0

4

12

8

H

16

20

Kilogauss

FIG. 1Oa. Faraday rotation angle as a function of magnetic field for two samples of n-type lnSb at 77°K. 29

L. M. Roth, Phys. Rev. 133, A542 (1964).

1 1. PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

H’

401

Kilogauss’

FIG. lob. Voigt phase shift as a function of magnetic field squared for the same plasmas as (a). (After Teitler, Palik, and W a l l i ~ . ’ ~ )

magnetic fields, o2% wC2and w 2 > 1/r2, the rotation 8 is linear in magnetic field H for a given plasma density:

2xe3dn e = c2qm*202 H This dependence is illustrated in Fig. 10a for two different plasma densities in InSb. A large literature exists on Faraday rotation in semiconductors which has yielded much information about band structure, including the relaxation time3’ when elliptical radiation is produced near cyclotron resonance. Very recently Furdyna31 has investigated nonlinearity in the Faraday rotation dependence on magnetic field. He has expanded the high frequency current in the presence of a magnetic field to terms which are cubic in magnetic field and also anisotropic for nonspherical energy bands. His measurements on n-type Si of BIB as a function of B2 yield a straight line,

3‘

S. D. Smith and C. R. Pidgeon, Proc. Intern. Conf. Phys. Semicond., Exeter, 1962 p. 342. Inst. of Phys. and Phys. SOC.,London, 1962. J. K. Furdyna, Phys. Rev. Letters 13,426 (1964).

402

BETSY ANCKER-JOHNSON

and he used its slope to determine the scattering average, (r4)/K~2>(r>Z). If the plasma, on the other hand, is immersed in a transverse magnetic field, plane polarized radiation oriented at 45" to the magnetic field is composed of equal components of electric field vectors parallel and normal to the magnetic field.32 The indices ql and qll are different, as shown in Eqs. (17) and (18), so a relative phase shift results with attendant elliptically polarized radiation :

If w w, and up,the phase shift resulting from this magnetic double refraction, also known as the Voigt effect, is 2ne4dn 6= H2. An example is shown in Fig. lob. Both the Faraday and Voigt effects are functions of frequency as well as wavelength and have been arbitrarily discussed in this section ; they could, of course, have been just as logically treated in the next section.

8. MAGNETOREFLECTION AND ABSORPTION AS A FUNCTION OF

FREQUENCY

a. wp > w , Another approach to observing the plasma effects on electromagnetic interactions in solids is to observe the reflections as a function of frequency less than the plasma frequency. If w, is also larger than w,, attention is concentrated on the plasma cutoff at q = 0 and the reflection minimum at q = 1 (cf. Fig. 7). The conditions for both longitudinal and transverse propagation (with E[ 1) are33:

+

Plasma edge w * = wl = f[(wc2+ 4wp2)'/2 k w,]

x wp k o J 2 + wC2(8wp)-'; Reflection minimum

32

33

w+

(26)

x op (27)

S. Teitler and E. D. Palik, Phys. Rev. Letters 5, 546 (1960); S. Teitler, E. D. Palik, and R. F. Wallis, Phys. Rev. 123, 1631 (1961); for anisotropy studies see E. D. Palik, J . Phys. Chem Solids 25, 767 (1964). B. Lax and G . B. Wright, Phys. Rev. Letters 4, 16 (1960); B. Lax, and S. Zwerdling, Progr. Semicond. 5, 221 (1960).

1 1. PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

403

If no magnetic field is applied, o = wp at the plasma edge for both longitudinal and transverse propagation, as Eq. (18) also shows (see Fig, 11). With a magnetic field applied the splitting of the edge is exactly equal to the cyclotron frequency. The theoretical curve for circularly polarized propagation is so nearly identical to that for linearly polarized transverse propagation that only the latter is plotted in Fig. 11. The splitting is still clearly observable when unpolarized radiation is used. -Zero ;-------; ---yllH r-------;

.t

80

-

0 0.85

II

I

I

!

_----

0.95

I I

1.05

!

Field

ElH

-.- y llH Unpolarizel

1.15

1.25

FIG.11. Theoretical curves of the plasma edge for isotropic carriers with WT % 1 and uP/oc = 5. The propagation factor is designated by y. The frequency splitting for both longitudinal and transverse propagation is equal to w,. (After Lax and Mavroides.”)

The plasma cutoff at o = opis an entirely different type of phenomenon than cyclotron resonance. In the absence of losses the motion associated with cyclotron resonance increases without limit, that is, the diameter of the cyclotron orbits grow without limit. In the plasma case there is an inherent restoring force. This is readily seen considering the magnetic interaction : The acceleration of charge carriers normal to the propagation direction causes an induced magnetic field, which by Lenz’ law induces an electric field opposing that of the incident radiation. The shift in the plasma cutoff, as a function of plasma density with no magnetic field applied, has been observed, for example, in indium antimonide,34 Fig. 12. Such measurements (in association with others) yield effective mass data. S~hmidt-Tiedernann~’ has pointed out that generalizing Eq. (18) to include mass anisotropy produces optical anisotropy, namely, double refraction or birefringence. The effective mass in a many-valley semiconductor can be made anisotropic by forcing carriers with an electric or strain field into a nonequilibrium distribution among the valleys. Then a plane-polarized incident beam splits into two waves which experience a 34

35

W. G . Spitzer and H. Y. Fan, Phys. Rev. 106, 882 (1957). K. J. Schmidt-Tiedemann, Phys. Rev. Letters 7 , 372 (1961); Proc. Intern. Con$ Phys. Semicond., Exeter, 1962 p. 191. Inst. of Phys. and Phys. SOC.,London, 1962.

404

BETSY ANCKER-JOHNSON

relative phase shift as they traverse the plasma. This birefringence has been made directly visible in silicon by applying a stress field to a slab containing a sharp boundary between undoped and highly doped silicon, Fig. 13(a). The boundary region was placed between two Nicol prisms in the path

h 100

5

10

15

6

3

2

(Microns) 20

25

30

1.5

1.2

1

3!

80

&

60

Y

LL

2.

c ._ > .c _

Yl

E (I,

OI

40

20

0

XI(

3

u (Set-') FIG. 12. The reflectivity as a function of wavelength (or frequency) for five n-type lnSb samples. (After Spitzer and Fan.34)

of approximately lp radiation. An image converter tube produced the photograph of the sample reproduced in Fig. 13(b). The fringe spacing, according to the theory of stress birefringence, is inversely proportional to the square root of the plasma density, hence the lower density plasma has a more widely spaced fringe pattern in Fig. 13(b). Such experiments have yielded the deformation potential ~ o n s t a n t . ~ ’

1 1.

PLASMAS I N SEMICONDUCTORS A ND SEMIMETALS

p

Undaped

405

Heavily Doped

(a)

FIG.13. (a) Four-point loading device for stress birefringence studies of Si. (b) Double refraction observed under the conditions in (a). (After S~hmidt-Tiedemann.~~)

The splitting of the plasma edge in a magnetic field was first recognized as a useful tool in solid state physics by Lax and Wright,33*36and demonstrated by measurements on indium antimonide and mercury selenide. Their zero magnetic field data for the former, reproduced in Fig. 14, are consistent with Spitzer and Fan’s results, Fig. 12. The splitting measured at isoreflection points (more accurately determinable than the cutoff) 36

B. Lax, IRE Trans. MTT-9, 83 (1961). This is an excellent succinct review of “Magnetoplasma Effects in Solids” on which this section of the present review is based to a large extent.

406

BETSY ANCKER-JOHNSON

indicated by the arrows, allows a direct calculation of the cyclotron effective mass. If the dielectric constant is known from measurements with o $ o,, the plasma density may be calculated. The shift of the plasma edge in Cd,Hg, -,Te with both magnetic field and temperature has yielded much information about its energy band structure.37

l o o i i i 90

, , , , , , o

I

38.5 kG

)

)

4

Photon Energy (ev)

n

FIG.14. The splitting of the plasma edge in a magnetic field as observed in n-type InSb with 1.8 x 10l8 ~ m - (After ~ . Lax and Wright.33)

=

Magnetoplasma reflection yields another constant of great importance, namely, the scattering time. Wright and Lax38 drew a theoretical curve which very closely fits their data for magnetoreflection in indium antimonide sec under the conditions listed on and thus determined that z = 2.9 x Fig. 15. They33 pointed out that this technique can be extended over a frequency range from millimeter waves to ultraviolet to determine many band structure constants not accessible otherwise. Magnetoplasma reflection in the limit up> o,may be used, furthermore, to study more complicated energy band structures, namely, those having degenerate bands and anisotropic energy s ~ r f a c e s .This ~ ~ .may ~ ~ be demonstrated by considering the close analogy Lax36 has shown to exist between classical galvanomagnetic phenomena and dispersive magnetoplasma effects: The current density component along an applied electric field may 37

38

T. C. Harman, A. J. Straws, D. H. Dickey, M. S. Dresselhaus, G. B. Wright, and J. G. Mavroides, Phys. Rev. Letters 7, 403 (1961). G. B. Wright and B. Lax, J . Appl. Phys. 32,2113 (1961).

1 1. PLASMAS

IN SEMICONDUCTORS A N D SEMIMETALS

407

be expressed as an expansion in the magnetic field:

Ji = oiEi + oijEiHj + o i j k E i H j H k + . . . ,

(28)

where the coefficient (ii is the usual conductivity along the component of the electric field Ei, (iij is associated with the Hall effect, and o i j k with the magnetoresistance. The equation can just as well represent the current density of a plasma in a solid exposed to electromagnetic radiation and a static magnetic field. Under the assumption w7 $ 1 the three coefficients in Eq. (28) are associated with the plasma frequency, the plasma-edge splitting equal to w, (described above) as well as plasma Faraday rotation

60

.=5

-

40

=0.041

U

-

0.002

al p:

c

g

20

v, al

n n

060

,065 ,070 Photon Energy .kw (eV)

,075

FIG.15. Transverse magnetoplasma reflection in InSb. (After Wright and Lax.38)

(discussed in Section 7), and the quadratic shift in the plasma edge, respectively. Anisotropy in solids is better studied utilizing magnetoplasma effects than galvanomagnetic effects because the effective mass is obtained directly, revealing anisotropy in the scattering time as well as in the effective mass, whereas galvanomagnetic measurements yield only the ratio of effective mass to scattering time. The intensity measurements which have been described above33.36* 37 are related to the plasma edge splitting which is linear in magnetic field. The rotation and ellipticity of linearly polarized radiation reflected from a plasma, which have been collectively known as the Kerr magnetooptic effect for many decades, are also related to this linear splitting. The electric vectors of the two counter-rotating waves reflected by a magnetoplasma whose surface is normal to the magnetic field are turned through angles given by

408

BETSY ANCKER-JOHNSON

where q and k are the index of refraction and coefficient of extinction, respectively. The ellipticity may be expressed as the ratio of the minor axis to the major axis of the resultant elliptically polarized wave,

P+l- b - I =

I E + , + lE-li

by using the amplitude relationships

The plane of polarization of the reflected elliptically polarized wave is rotated through an angle [compare Eq. (29)] $=

8, - 92

The ellipticity of a wave reflected from a collisionless, and therefore purely dispersive, plasma is illustrated in Fig. 16. The maxima for each curve (associated with a particular o,/op)occur with a separation equal to o,as q+ + 1, since then either E , or E - vanishes and 5 = 1. The intervening minimum is the result of both polarizations experiencing the same mismatch and so causing ( E + (= 1E-l. in The magnetooptic Kerr effect has been measured by Palik et n-type indium antimonide, Fig. 17. A quantity equal to the ratio of the minimum-to-maximum transmission of the optical system containing two imperfect polarizers, which approximates the square of the ellipticity [Eq. (30)], is shown as a function of frequency in Fig. 17(a). The ratio of w,/o, is approximately 0.4; the narrow minima observed in the theoretical curves for 0.1 and 0.6, Fig. 16, are, however, unresolved because of scattering effects. The Kerr rotation data in Fig. 17(b) also clearly exhibit the splitting of the plasma edge. As already indicated, the quadratic splitting of the plasma edge is not isotropic as is the linear splitting. Furthermore, in noncubic crystals even the zero-order effect exhibits the crystal a n i ~ o t r o p yas~ ~studies4' of Bi illustrate, Fig. 18. Linearly polarized radiation was reflected perpendicularly from the bisectrix plane. The two curves correspond to polarization parallel or orthogonal to the trigonal axis, respectively. The minima in the reflection curves correspond to condition (27), hence o can be calculated for both orientations, and some band parameters deduced. E. D. Palik, S. Teitler, B. W. Henvis, and R. F. Wallis, Proc. Intern. ConJ Phys. Semicond., Exeter, 1962 p. 288. Inst. of Phys. and Phys. SOC.,London, 1962. 40 W. S . Boyle, A. D. Brailsford, and J. K. Galt, Phys. Rev. 109, 1396 (1958).

39

1 1.

409

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

FIG.16. The ellipticity occurring in linearly polarized radiation reflected from a collisionless magneto-plasma in a crystal whose dielectric constant is 16.

antimonide

50

55

60

65

70

75

Wavelength in Microns

FIG.18. The reflection coefficient of Bi at 4°K. (After Boyle et 0 1 . ~ ~ )

410

BETSY ANCKER-JOHNSON

h. o,> op Magnetoreflection and absorption in this second limit of interest w, > ophave properties both similar to and distinctive from those described in the limit cop > 0,.The conditions for cutoff ( r ] & = 0) are, according to

Eq. (161, 0 1

+

= 0, op2/o,,

0 2 = op2/w,.

(33)

(34)

The first condition is equivalent to Eq. (20) and shows the correction term for determining cyclotron resonance when the plasma is significantly dense. Equation (34) is a condition unique to the limit o,% up> w and has been labeled “magnetoplasma resonance” [although it is not a resonwho first investigated the effect. ance as (34) shows] by Dresselhaus et Magnetoplasma resonance occurs when the dimensions of the plasma are small compared with the wavelength of the impinging radiation and its penetration depth into the crystal as limited by the plasma. In this case depolarization charges that accumulate on the boundaries cause appreciable modification of the applied electric field. A general solution of the equations of motion for the plasma electrons in the presence of a static magnetic field has been worked out by Lax and Ma~roides.~’ The components of their general conductivity tensor simplify to the following when the plasma has the configuration of a cylinder and is exposed to circularly polarized radiation :

where

When the plasma density is negligible o+ exhibits cyclotron resonance as usual, Fig. 19. If the positive and negative components were added, as in linear polarization, this resonance could not be observed unless w7 > 1. When 0,’is comparable to the radiation frequency, which defines the transition region, no absorption is observed independent of the magnitude of o7.Magnetoplasma resonance occurs when wp’ > o,and in contrast to the cyclotron resonance absorption, it is the negatively circularly polarized field which interacts with the plasma. The greater the plasma density, the larger the frequency separation between cyclotron resonance and magnetoplasma resonance. 41

G. Dresselhaus, A. F. Kip, and C. Kittel, Phys. Rev. 100, 618 (1955).

1 I . PLASMAS

411

IN SEMICONDUCTORS AND SEMIMETALS

O+

4%

0

2

1

3

5

4

6

wc/o

FIG.19. Magnetoplasma effect for a circular cylinder with the magnetic field along its axis. Here oz = 1, and w' has the values shown. The absorption for linear polarization is proportional to u+ and u - . (After B. Lax and L. M. Roth, as published in Ref. 27.)

Dresselhaus et aL41 observed magnetoplasma resonance in a small disk of indium antimonide. They modulated the carrier density (by modulating the incident light that produced the carriers) and so recorded derivatives of the absorption, Fig. 20. Condition (34) is determined approximately by the

' ' '

'

l

0

~

~

2

'

'24000M'CS

l

l

4

l

6

H

9600 k C s '

l

l

8

l

10

l

l

l

1

in Kilo-Oersfeds

FIG.20. Experimental plasma resonance absorption signals obtained with carrier modulation in a thin disk of n-type InSb at 77°K. The static magnetic field is directed normal or parallel to the plane of the disk in separate runs. The broken lines connect the curves below 9000 Oe with single terminal points determined at higher fields. (After Dresselhaus et ~ 1 . 4 ' )

412

BETSY ANCKER-JOHNSON

crossover points for the applied magnetic field parallel and perpendicular to the large face of the sample. The agreement between experiment and theory is only fair, probably because41 determining accurate depolarization factors for the small irregular specimen used was very difficult. The experiments, however, clearly show the existence of magnetoplasma resonance and its predicted dependence on magnetic field, o = wp2/oc, since large magnetic fields accompany small resonance frequencies. c. The Transition Region

In the transition region between the dominion of cyclotron resonance and that of magnetoplasma resonance, the dependence of carrier mass on the resonance magnetic field does not cancel as it does in the case of magnetoplasma resonance. Bemski4’ has calculated this effect for various plasma densities, Fig. 21. The resonance field is H , for an arbitrary mass nz,”. When upis only slightly larger than the applied frequency, the dependence of the resonance magnetic field on mass is strong. The magnitude of H decreases as m* increases, just the reverse dependence from that which occurs for cyclotron resonance. Bemski has verified this result by heating the electrons in n-type indium antimonide above the lattice temperature (held at 4°K) through application of pulsed, moderately high electric fields. “Warm” electrons have an enhanced energy and therefore increased effective mass because of the curvature of the conduction band in indium antimonide. Bemski’s measurements definitely show that the resonance field decreases as the carrier energy increases, but quantitative comparison with theory was not possible because the carrier concentration and the distribution of the heated carriers were not clear. The most impressive experiments on magnetoplasma resonance have been done in a multicomponent uncompensated plasma whose constituents were the light and heavy holes in p-type germanium.43 Michel and Rosenblum calculated the position of the resonance lines in this coupled system for circularly polarized radiation as a function of total carrier concentration, Fig. 22. The dashed lines indicate the resonant fields in the absence of coupling. The arrows point to the magnitude of the field for cyclotron resonance in the absence of plasma effects. The thickness of the lines suggests the relative peak intensity of the two resonances at a given plasma density. The absorption corresponding to light hole cyclotron resonance at low plasma densities shifts with increasing plasma density through zero magnetic field, increasing in intensity and eventually becoming the main magnetoplasma absorption. The absorption which was initially caused by 42 43

G. Bemski, J . Phys. Chem. Solids 23, 1433 (1962). R. E. Michel and B. Rosenblum, Phys. Rev. Letters 7, 243 (1961); Phys. Rev. 128, 1646 (1 962).

1 1.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

413

15

\0.7

I

-

0.5 -

1.o

1.2

1.4

m*/m b

1.6

1.8

2.0

FIG.21. Calculated magnetic fields, at resonance in the transition region between cyclotron resonance and magnetoplasma resonance, as a function of effective mass for various ratios of plasma to operating frequency. (After Bem~ki.~’)

FIG.22. Calculated and measured magnetic fields for magnetoplasma resonance as a function of plasma densities forp-type Ge. The thickness of the lines is approximately proportional to the relative intensity of the resonances. The dashed lines indicate what the resonance fields would be if there were no coupling between the light and heavy hole plasmas. (After Michel and R o s e n b l ~ r n . ~ ~ )

414

BETSY ANCKER-JOHNSON

B

FIG. 23a. Schematic plot of qz and the absorption coeficient A as a function of applied for U T 4 1. magnetic field in the presence of two hybrid resonances at w,,, and oh2

cyclotron resonance of the heavy holes never goes through zero magnetic field, but rather asymptotically approaches a field near the light hole cyclotron resonance line while decreasing in intensity. The resonances, once plasma effects become appreciable, are the result of coupling through the depolarization fields ; both resonances are the result of cooperative effects involving the whole collection of carriers. The open circles in Fig. 22 correspond to measured absorption maxima in small ellipsoids of p-type germanium in which the carrier concentration was controlled by temperature. The slight discrepancy in the heavy hole cyclotron resonance line and its shift with plasma density is probably the result of the assumption of isotropic masses, which is well fulfilled for the light holes but not for the heavy have studied such hybrid resonances in bismuth plasmas Smith et composed of two types of electrons and one type of hole. Instead of employing small ellipsoidal samples immersed in a microwave field, they used 44

G. E. Smith, L. C. Hebel, and S. J. Buchsbaum, Phys. Ren 129, 154 (1963)

1 1.

PLASMAS IN SEMICONDUCTORS A N D SEMIMETALS

415

x 103

0

1

2

3

4

5

6

B in Kilogauss FIG. 23b. Theoretical and experimental plots of power absorption coefficient as a function of magnetic field; the applied magnetic field is along the binary axis in the plane of the sample.

The incident microwave field at 70 G c is polarized such that E is parallel to the trigonal axis and H the binary axis. (After Smith et ~ 1 . ~ ~ )

“semi-infinite” plasmas with normally incident radiation. They have shown that x distinct types of carriers with different e/m ratios produce (x - 1) hybrid resonances and one other resonance at a frequency much nearer to op(inaccessible in their experiment). The two hybrid resonances in bismuth are: (1) an electron-electron hybrid caused by the oscillation of the two electronic charge clouds out of phase with one another (the holes affect this resonance insignificantly); and (2) an electron-hole hybrid which is the result of joint oscillations of the electrons and holes with a small slippage between them. The hybrid frequencies depend on the mass and relative concentrations of the carriers but not on their absolute densities. The schematic graph of the index of refraction squared and the absorption coefficient A = 1 - R as a function of applied magnetic field in the plane of the sample for normally incident radiation is shown in Fig. 23a. For

416

BETSY ANCKER-JOHNSON

B > 100G the analysis based on a local relationship between the incident microwave field and current is valid since the skin depth is large compared to the cyclotron radius. The onset of the electron-electron hybrid resonance is accompanied by an onset in absorption; the absorption drops very steeply at the “dielectric anomaly”39(q = 0) and is followed by the onset of absorption related to the electron-hole hybrid. Finally a linear relationship between magnetic field and absorption occurs at large fields which is associated with Alfven propagation (see Section 9). Figure 23b reproduces some of Smith et al.’s results. The electron-electron hybrid resonance is at 500 G, the dielectric anomaly occurs at 900 G in very good agreement with the calculated value of 950 G, and the electron-hole hybrid is at 3000 G. With the same geometry K I B and E IB, if carriers are distributed among several mass ellipsoids that have the same cyclotron mass but nonequivalent cyclotron orbits, the plasma cannot shield out completely the cyclotron resonance of the carriers. Thus ordinary cyclotron resonance occurs in addition to hybrid resonances. With the geometry K IB and E /I B “tilted-orbit” cyclotron resonance occurs associated with the particular band structure of bismuth. Since the electron mass ellipsoids are slightly tilted out of the trigonal plane, even though the microwave electric field is transverse in this geometry, cyclotron resonance can occur. Similar magnetoabsorption studies in lead telluride4’ have been reported recent 1y. IV. Wave Propagation

9. ALFVENAND HELICONWAVES Alfven and helicon wave propagation in solids occurs at frequencies far from those at which the effects discussed in Sections 4 through 8 can be observed. For these waves to propagate the impinging wave must have a frequency very much less than either the cyclotron frequency or the plasma frequency. Konstantinov and Pere1’46 first proposed the possible transmission of electromagnetic waves through a metal, and independently A i g r a i ~ ~ ~ ~ suggested their propagation in semiconductors. Aigrain named the excitation “helicon” because its propagation path is helical and its properties are particle-like. Buchsbaum and Galt48 interpreted some previously published 45

4h

47

48

R. Nii, A. Kobayashi, H. Numata, and Y. Uemura, in “Plasma Effects in Solids” (Proc. 7th Intern. Conf.), p. 65. Dunod, Paris and Academic Press, New York, 1965. 0. V. Konstantinov and V. I. Perel’, Zh. Eksperim. i Teor. Fiz 38, 161 (1960) [English Transl.: Sooiet Phys. J E T P 11, 117 (1960)l. P. Aigrain, Proc. Intern. ConJ Semicond. Phys., Prague, 1960 p. 224. Czech. Acad. Sci., Prague, 1961. S. J. Buchsbaum and J. K. Galt, Phys. Fluids 4, 1514 (1961).

11. PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

417

data of Galt et d4’ on the absorption of microwaves in bismuth as Alfven wave propagation, a phenomenon which has long been known to occur in gaseous plasmas.50 The dispersion relation as obtained from Maxwell’s equations [also used to derive Eqs. (16) to (18)] is o2 K X (K X E) - & . E = 0. (37)

+ C2

The tensor dielectric constant may be expressed ass1 El

E(O) = Ex

0

0

-Ex

E l

0

O

Ell

.

(38)

The dielectric constant component for the ordinary wave sIl is given by Eq. (18). The other two components E~ and E, may be obtained from Eq. (16) by using the coordinate transformation identitie?’

+

=

-

(39) Thus, assuming no interaction among the various different types of isotropic carriers, and letting v be the collision frequency €1 = (&+

E-)/2;

Ex

i(E+

E-)/2.

The summations are over the plasma constituents. The terms on the righthand side obtain for a collisionless plasma, that is, when losses are negligible. , ~ IE,~ 9 1 ~ unless ~ 1 the plasma is Within the stated limit, ocj9 o,I E ~ 9 composed of equal densities of positively and negatively charged carriers (compensated plasma or nonequilibrium plasma, Section 2). In this case AlfvCn waves propagate, E, = 0, and hence IE,,I lell % [&,I.

+

J. K. Galt, W. A. Yager, F. R. Merritf B. B. Cetlin, and A. D. Brailsford, Phys. Rev. 114, 1396 (1959). H. AlfvBn, “Cosmical Electrodynamics.” Oxford Univ. Press (Clarendon), London and New York. 1950. ” S. J. Buchsbaum in “Plasma Effects in Solids” (Proc. 7th Intern. Conf.), p. 3. Dunod, Paris and Academic Press, New York, 1965. ’* See, for example, W. P. Allis, S. J. Buchsbaum, and A. Bers, “Waves in Anisotropic Plasmas,” p. 20. M.I.T. Press, Cambridge, Massachusetts, 1963. 49

418

BETSY ANCKER-JOHNSON

For isotropic plasma constituents5' the dispersion relation yields (see Refs. 53 and 54 for approximate solutions when the plasma consists of anisotropic carriers) helicon waves n, 9 nh

K~

w2 cos c1 = f l I ~ x / , c w2

Alfven waves n, = nh

XI2 = c 2 E~

w2

K~~

cos2u = -E c2

(43)

(ordinary),

(extraordinary).

(44)

In these expressions c1 is the angle between the direction of propagation and the applied magnetic field. One mode of Alfven propagation is independent of a and is, therefore, an ordinary wave (also called fast wave). In the extraordinary (or slow) mode energy is propagated only along the magnetic lines of force although a may be arbitrary. No helicon wave propagation can occur in an orthogonal magnetic field. These equations also show that helicon waves have a quadratic spectrum, w K l c 2 , whereas AlfvCn waves have a linear spectrum, w oc K. The helicon waves are circularly polarized in the plane perpendicular to the applied magnetic field. Both types of AlfvCn waves are linearly polarized. The electric vector of the ordinary wave is orthogonal to both the magnetic field and the wave vector. The extraordinary wave's electric vector lies in the plane formed by the magnetic field and the wave vector while being transverse to the magnetic field.51 The phase velocities for K B neglecting damping are helicon waves

vH

=

5 W P

(7) '/2

=

(coB/4~en)'/~,

(45)

where p is the mass density, Xjnjmj. Thus, the helicon is dispersive and independent of carrier mass and band structure of the medium, in contrast 53

54

M. S. Khaikin, V. S. Edelman, and R. T. Mina, Zh. Eksperim. i T w r . Fiz. 44, 2190 (1963) [English Transl.: Souiet Phys. J E T P 17, 1470 (1963)l; M. S. Khaikin, L. A. Falkovskii, V. S. Edelman, and R. T. Mina, Zh. Eksperim. i Teor..Fiz. 45, 1704 (1963) [English Transl.: Souiet Phys. J E T P 18, 1167 (1964)l. E. A. Kaner and V. G. Skobov, Zh. Eksperim. i Teor. Fiz. 45, 610 (1963) [English Trans/.: Soviet Phys. J E T P 18, 419 (1964)].

1 1.

419

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

to the AlfvCn wave. For n = 1014 ~ m - vH ~ ,= 1.8 x lo7( f B ) ' ' 2 cmjsec wheref is the frequency in megacycles and the magnetic field is measured in kilogauss. In bismuth vA z 107Bcm/sec. The influence of collisional damping is found by solving55 for Im(E)/Re(E) using these expressions for the dielectric constant : helicon waves

ww, =

where 6

-0

when w 4 a,,o,z

Alfvenwaves

1

W p 2

E=-

EH

f1

+ w/w, +

i(wcT)-

f 1 + w/wc- i(w,z)1+6

(47)

5

>> 1 ;

E=E~[(.I+---)0

i

Och

wchzh

(il--+-)] 0

wce

I WceTe

The minus sign between the bracketed terms in (48) occurs because sH. A ~ t r o m first ~ ~noted the cancellation in the AlfvCn wave equation of the f terms which are responsible for the circularly polarized nature of helicon waves. From Eqs. (47) and (48) it is seen that all low frequencies may propagate as helicon waves with little damping because Im(&)/Re(c)= (mcz)- ' ; AlfvCn waves, however, are damped at frequencies below the microwaves range since their loss term is (wz)-' s6a Table I1 summarizes the properties of AlfvCn and helicon waves.s6b The Alfvkn wave 5 0 s 6 long known in gas plasmas is identical to the wave which propagates in a solid state plasma with n, = n h ; nevertheless, some Russian53 workers insist on different nomenclature for the solid state case. Helicon waves were first observed in sodium and have been extensively studied in metals.57 Their propagation has also been investigated in the cHh= -cHe

" D. 56

J. Bartelink, IEEE Solid State Device Research Conf., Boulder, 1964 (unpublished).

E. h o r n , Arkiu Fysik 2, 443 (1950).

'6aFor a discussion of Landau damping see Kaner and S k ~ b o v . ~ ~ 'bbAn alternative treatment of helicon wave propagation has been given by R. G. Chambers and B. K. Jones [Proc. Roy. SOC. (London) 270, 417 (1962)l who employ only the Hall coefficient, the sample resistivity, and its geometry. 5' R. Bowers, C. Legendy, and F. Rose, Phys. Rev. Letters 7, 339 (1961); see R. Bowers, in "Plasma Effects in Solids" (Proc. 7th Intern. Conf.), p. 19. Dunod, Paris and Academic Press, New York, 1965.

420

BETSY ANCKER-JOHNSON

A COMPARISON BETWEEN

THE

TABLE rr PROPERTIES OF HELICON A N D ALFVENWAVES Helicon waves

Density Propagation term : Re(&) Phase velocity Collisional damping term : Im(e)/Re(&) Polarization Modes of propagation

Conditions

f

ne

nh

cH = 4ncen/wB

vH = (coB/4sen)* 1 -

Ci r cu 1a r One for component of Bll~ 6

0,;OlflJ,

n, = nh eA = 4nc2p/B2 v,, = B(4np)-* 1 UT

wc5

0

AlfvCn waves

< Wp2

Linear One independent of B orientation and one for component of B ~ / K w 6 w, < wp

semiconductors InSbS8and PbTe59at the higher frequencies dictated by the lower plasma densities in these solids. Libchaber and Veilex'* qualitatively demonstrated the validity of the helicon dispersion relation (43) by showing that InSb is essentially transparent to the sense of circulatory polarized radiation which has the same direction as the rotation of electrons about the magnetic field, and opaque to the opposite sense of rotation. More recent highly quantitative results have been achieved by Furdyna." Figure 24(a) shows the transmission peaks of 35-Gc radiation which are Fabry-Perot dimensional resonances. These peaks occur when the thickness d of the sample satisfies their expression : ~

Y2

-

2

=

~

Y l o - Yxco,'/2

2412

- opw1/2E;/2

Fabry-Perot dimensional resonance (49a)

where &, and l are the wavelength in air and in the sample, respectively. The usual angular dependence on sin 6' in Eq. (49) is absent because in a medium of high dielectric constant the propagation is normal to the sample surface. The negative B curve in Fig. 24a corresponds to helicon wave propagation. The positive B circularly polarized mode is not transmitted until Eq. (34), w , = wp2/w, is satisfied. The carrier concentration obtained A. Libchaber and R. Veilex, Phys. Rev. 127, 774 (1962); Proc. Intern. Con$ Phys. Semicond., Exeter, 1962 p. 138. Inst. of Phys. and Phys. SOC.,London, 1962; see also J. Gremillet, Compt. Rend. 256, 2792 (1963). 5 9 Y. Kanai, Japan. J. Appl. Phys. 1, 132 (1962); "Plasma Effects in Solids" (Proc. 7th Intern. Conf.), p. 45. Dunod, Paris and Academic Press, New York, 1965. ''O J. K. Furdyna, Phys. Rev. Letters 14, 635 (1965).

58

11. PLASMAS

IN SEMICONDUCTORS AND SEMIMETALS

421

from these data agrees very well with the results of standard Hall measurements. These data also yield the dielectric constant of the solid in which the helicon is propagated. Furdyna6' has also observed Rayleigh interference patterns produced by circularly polarized waves transmitted through intrinsic InSb interfering with a reference signal, Fig. 24b. One arm of a 35-Gc microwave bridge transmitted the reference signal while the other arm contained the sample in which helicon wave propagation occurred. The corresponding interference condition is d = y,l Rayleigh interference. (49b) The electron concentrations calculated from the data of Fig. 24b are in excellent agreement with the known intrinsic values. KanaiS9 has measured the frequency of helicon waves transmitted in PbTe at 4°K and has observed the predicted linear dependence of frequency on applied magnetic field with a slope inversely proportional to the plasma

0

MAGNETIC FIELDlkG)

FIG. 24a. Transmission of circularly polarized microwaves as a function of longitudinal positive and negative magnetic fields. Fabry-Perot dimensional resonance peaks occur for both polarizations according to Eq. (49). The (-B) curve represents propagation of the helicon mode. The other circularly polarized mode is highly damped until the condition (34), wc2 = ww,, is satisfied, which occurs in the ( + B ) curve near 12.5kG. These data yield a carrier concentration n = 3.1 x l O I 4 cm-3 and a dielectric constant E, = 19.7. The concentration obtained from the Hall measurement is 3.3 x 1014cm-3 (after J. K. Furdyna, private communication).

422

BETSY ANCKER-JOHNSON

1

I

0

lnSb 35Gc

I

I 20

I

I

I

I

I

40 60 MAGNETIC F I E L D ‘ k G ’

I 80

I

1

FIG.24b. Recorder traces of Rayleigh interference patterns caused by helicon wave propagation in intrinsic lnSb and observed at the temperatures indicated. The sample thickness d was 4.32 mm. The electron concentrations shown were calculated from the observed period. (After Furdyna.60)

density, Eq. (43), o cc B/n. Recently59 he reported some discrepancies between the measurements of frequency as a function of magnetic field obtained by direct observation of the decaying oscillations and, alternatively, from observations of dimensional resonances. These discrepancies he tentatively attributes to depolarization associated with the sample dimensions. Fewer experiments have been performed on Alfven waves than on helicon waves because the frequency of the latter, in contrast to the former, may be arbitrarily low. Semimetals are better candidates for supporting Alfven wave propagation than semiconductors since high purity intrinsic semiconductors would require too high a frequency and degenerate semiconductors have higher collision frequencies than pure semimetals. As already noted the first indirect observation of Alfvkn waves48,49 occurred in Bi at 4°K. In these experiments the absorption coefficient A was measured.

423

11. PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

It results from the wave impedance interior of the plasma, and hence it wave velocities in the two regions, Alfvin velocity. Thus when Alfvtn magnetic fields

A

=

1- R

mismatch between free space and the determines the difference between the namely, the velocity of light and the waves propagate at sufficiently high

1 - [(c -

=

UA)/(C

+ UA)]*

%

4UA/c,

(50)

and the absorption is expected to be linearly proportional to the magnetic field and, at a given magnetic field strength, independent of both frequency and the polarization, hence of the direction of the magnetic field. Figure 25(a) reproduces the absorption in pure Bi, and Fig. 25(b) in Bi doped with Te, so that n, = 4n, and Alfvin waves cannot propagate. The difference in the absorption characteristic as a function of B and o for the two cases is a qualitative confirmation of AlfvCn wave propagation in the former case and helicon in the latter.

H in Oersteds (72000 Mclsec) B, x lo4 inWebers/M2 -18 -12 - 6

I

xln-21

6

12

18 XI03

A'

N =P

x 10-2 4.65

\\ \\

L

'r

0

72000 M cl rec

c

\\

L

0.00

W

3

2

0

n O.Oo14 -12 -10 - 8

H

- 6 -4

-2

0

in Oersteds (24000 M c l s e c )

2

4

6

8

B, x lo4 in

10 12

14x103

Webers/M2

FIG. 25. Plots of power absorption coefficient at two frequencies as a function of magnetic field for pure Bi with B normal to the sample surfae and along a trigonal axis. Vertical arrows indicate the cyclotron absorption fields for electrons and holes. The dashed lines show the experimental curves in the absence of the saturation which occurred in the experiments. The theoretical plots are based on a classical skin effect theory which agrees with the Alfven wave interpretations at large magnetic fields. (a) Pure Bi.

424

BETSY ANCKER-JOHNSON

(b) FIG.25. (b) Alloy of Bi and Te. (After Buchsbaum and Galt4* and Galt ri u / . ~ ' )

The first direct observation of Alfvtn wave propagation in a solid was reported by Williams6' and published by Williams and Smith.61 Bismuth (and also antimony) has been irradiated with 12 to 18 Gc so that COT > 1 but o,> o at reasonable magnetic fields. The transmission through a plane parallel sample was detected and produced the two different types of intensity-peak or fringe patterns also observed recently in connection with helicon wave propagation. One pattern occurred at high magnetic fields when the Alfven waves made several passes through the sample at various dimensional resonances before being significantly damped. An example of these Fabry-Perot fringes is shown in Fig. 26(a). The other type occurred at low magnetic fields when the damping prevented multiple reflections. Interference between microwave radiation that leaked around the sample and the AlfvCn waves passing through it produced a beat pattern, because the velocity of the latter changes with magnetic field whereas the leakage radiation velocity does not. A typical Rayleigh fringe pattern resulting from such interference is reproduced in Fig. 26(b); such patterns have twice the period of the Fabry-Perot fringes. The number of half-wavelengths G. A. Williams, Bull. Am. Phys. SOC.7, 409 (1962); 8, 205 (1963); G. A. Williams and G. E. Smith, IBM J . Res. Develop. 8, 276 (1964).

11.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

[cf. Eq. (49a)l

y

=

2d

--Ey cc [nf(m*)]'"/B

425

(51)

LO

plotted as a function of 1/B is linear, independent of the direction of B, and dependent on frequency only through ,lo,that is is independent of frequency. These are all hallmarks of AlfvCn propagation. The quantity nf(m*) is the mass density for anisotropic plasma constituents. Small deviations from linearity in (51), that is, in the phase velocity dependence

d = 3.54 miin T = 1.2'K

BI 70-02

Freq: lq.609 k M c l S e c 61 Surtace B I1 2 - Fold Axis Erf I1 Bisectrix Axis

-

,111

"-" 100

1

I

a0

60

I

I

40

20

l

o

I

l

20 40 Kilogauss

I

I

I

60

a0

101

H>O

H
4

6

8

10

12

14

16

Kilogauss

(b) FIG.26 (a) Fabry-Perot type interference pattern produced by dimensional resonances of Alfven waves. (b) Interference pattern produced by beating of incident radiation with Alfvtn waves. These occur at magnetic fields too low for the production of Fabry-Perot fringes. (After Williams and Smith.61)

426

BETSY ANCKER-JOHNSON

1.85

168

1.50 1.34 1.19 1.06

t

0

Ill

I,' 0.125

I

.

I

1

0.25

0.50

1 lo3H-',

OK'

FIG.27. The surface resonance of Bi as a function of reciprocal magnetic field demonstrating the simultaneous excitation of the ordinary and extraordinary Alfven waves. Part of the trace is shown with larger amplification on the right. The angle between the sample surface and the magnetic field was lo". C, gives the direction of the binary axis and C , the trigonal. (After Khaikin et d 5 j )

on magnetic field, have been interpreted as quantum oscillations in the mass density of the carriers and are being studied to yield further information about the band structure of Bi.61 Khaikin et were the first to publish examples of Fabry-Perot type interference patterns much like those of Fig. 26(a). Their experimental approach was the measurement of surface impedance rather than transmitted signals.61 By varying the orientation between the sample surface and the magnetic field Khaikin et aLS3have observed both the ordinary and extraordinary Alfvtn waves, Eq. (44). In a transverse magnetic field, oscillations TABLE 111

MEASUREMENTS OF THE

VELOCITIES OF

Ordinary wave

ALFVENW A V E S

__ cos a

u,H-'

(lo4 cm sec 'Oe- ')

Bi"

Extraordinary wave

OH-'

Orientation

IN

Experimental Theoretical ratio ratio

~

"After Khaikin et

(104cm sec-'Oe-') 5.2 & 0.1

1.4

1.2

11.1 & 0.2

2.8

3.0

427

11. PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

(Khaikin’s “s-oscillations”) of only one periodicity in H - corresponding to the ordinary wave were observed, whereas in fields inclined to the surface a second wave (Khaikin’s “p-wave”) of shorter period in H - ’ and lower velocity also occurred, Fig. 27. The AlfvCn velocity measurements corresponding to certain orientations are listed in Table 111. The ratio of the ordinary to extraordinary velocities can be compared with theory without reference to a particular energy band model ; the agreement between Khaikin et d ’ s measurements and theory is seen to be good. This type of data is also being used to study band structure of semimetals. Nonlinear effects related to helicon wave propagation have been reported by Maurer et a1.62They produced hot electrons in n-type InSb at 77°K by a high power 10-Gc signal and observed strong nonlinear effects in 500-G fields. A third harmonic and its dimensional resonances were also described. A theory is not yet available explaining their work.

-

10. WAVEPROPAGATION AND DOPPLEREFFECTS Random carrier motion in the presence of a static magnetic field and electromagnetic radiation causes the carriers to experience a Dopplershifted wave frequency. In the absence of other applied fields this motion is the result of the Fermi velocity distribution (or the thermal velocity distribution). A carrier, then, may engage in cyclotron resonance at a frequency other than w,; the maximum magnetic field at which this absorption can occur is given by the ~ o n d i t i o n ~ ~ , ~ ~ w = 0, - UOzKA (52) where I C ~is the AlfvCn wave and uoz is the maximum velocity component of a carrier at the Fermi surface in the direction of the magnetic field and in the direction opposite to the wave propagation. Thus the previously stated condition for AIfvCn wave propagation w 4 w, can be precisely restated : At magnetic fields just larger than that given by condition (52) essentially undamped AlfvCn waves propagate. Doppler-shifted cyclotron resonance was first observed by Kirsch and Miller63 in Bi at 2°K. They noted a large absorption of 9-Gc radiation at 1500G which is more than twice the cyclotron field for the heaviest cyclotron mass, Fig. 28(a). They observed excellent agreement with the calculated location of the absorption edge for holes, Fig. 28(b). J. Maurer, A. Libchaber, and J. Bok, in “Plasma Effects in Solids” (Proc. 7th Intern. Conf.), p. 49. Dunod, Paris and Academic Press, New York, 1965. J. Kirsch and P. B. Miller, Phys. Rev. Letters 9, 421 (1962). 64 J. Kirsch, Phys. Rev. 133, A1390 (1964). 64aOrhelicon wave vector. The effect has been observed in metals (but not semiconductors). Since helicons propagate at very low frequencies, the Doppler-shifted cyclotron resonance occurs at frequencies several orders of magnitude less than w, (see review article cited in Ref. 57).

428

BETSY ANCKER-JOHNSON

Doppler-Shifted Resonance

I

I 1000

I

500 Doppler-Shifted Heavy Electron

! (Arbitrary Units1

Yh Dielectric Anomaly

4000

2000

I

1500

0

DOPPl t

hifted

A

Undamped Alfven Wave Propagation

-2000

-4000

H FIG.28. (a) Experimental plot of surface resistance as a function of magnetic field parallel to the binary axis of Bi and normal to the surface. (b) Theoretical curve. (After Kirsch and

1 1. PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

429

Stern6' has pointed out that condition (52) can be used to measure the shape of the Fermi surface in metals, semimetals, and semiconductors, since the absorption yields information about the Fermi velocity in the direction of propagation. Doppler-shift effects may also be produced by applied fields. In this case an additional term is needed in Eqs. (47) and (48) as Bok and Nozieres66 pointed out:

co

where uD is the drift velocity (which is much less than the Fermi velocity in the experiment described below). Bartelink6' has observed the rotation of the plane of polarization of an Alfven wave as it traversed a Bi sample, a rotation caused by the Doppler effect. Since the velocity of one polarization direction is enhanced and the other direction's velocity is retarded by the drift field, a net rotation in the plane of polarization occurs. Bartelink employed damped Alfven waves (propagating at 100 kc) to avoid the detection of reflected signals which would have complicated the results. His experimental arrangement is shown in Fig. 29(a). A magnetic field was imposed along a current filament in the central square formed by two pairs of mutually perpendicular channels in the bismuth. The current filament was confined by the high magnetoresistance of bismuth. The source and detector coils for the low radio frequency waves were thus placed in quadrature in the channels, so that only a signal whose plane of polarization was rotated could be detected. Figure 29(b) shows a typical transmitted signal A , as a function of current. A nearly sinusoidal variation occurs as the AlfvCn wave plane of polarization is rotated through 180". The maximum signal corresponding to rotation through 90" has a linear dependence on applied field, Fig. 29(c), and is independent of frequency as expected from theory, since A , = A , sin(Jz/2H) where z is the distance traveled by the wave. At sufficiently high drift currents the damped waves are converted to

-

E. A. Stern, Phys. Rev. Letters 10, 91 (1963). J. Bok and P. Nozieres, J . Phys. Chem. Solids 24, 709 (1963). " D. J. Bartelink, Phys. Rev. Letters 12, 479 (1964); in "Plasma Effects in Solids" (Proc. 7th Intern. Conf.), p. 93. Dunod, Paris and Academic Press, New York, 1965.

65

66

'

430

BETSY ANCKER-JOHNSON

circularly polarized propagating waves ; they have, therefore, the polarization properties of helicon waves. The current path then becomes a wave guide. Bartelink6' has also observed this phenomenon in Bi. Nanney6* has observed traveling helicon waves in a solid state plasma wave guide in n-type PbTe.68a Bok and Nozieres66 predicted amplification of transverse waves in a drifted plasma of sufficient velocity : they also predicted the occurrence of

f 100

-160 -80I(A';;y) o

ao 1

6

7

Q

F

d 75' 8

-

50 . 25 .

FIG.29. (a) A 2 x 3 x 4cm Bi sample containing a small current channel (I), low rf source (S), and detector (D). (b) The amplitude in the detector coils varies sinusoidally with current as the plane of polarization of the signal arriving from the source rotates in and out of parallelism. (c) The position of the maximum in current (at 90" rotation) varies linearly with applied magnetic field. (After B a r t e l i ~ ~ k . ~ ~ )

instabilities. Misawa7' independently predicted instabilities in a drifted plasma but judged that amplification is not possible because he found the spatially growing wave to be evanescent. (See Section 18 for further discussion about these theories.)

'' C. Nanney, Phys. Rev. 138, A1484 (1965). 68aTodaand H i r ~ t have a ~ ~observed microwave transmission through a waveguide of n-type InSb in a transverse magnetic field, an effect not attributable to helicons but mentioned here in connection with plasma wave guides. 6 9 M. Toda and R. Hirota, IEEE Solid State Device Research Conf., Boulder, 1964 (unpublished). 7 0 T. Misawa, Japan. J . Appl. Phys. 2, 500 (1963).

11. PLASMAS IN SEMICONDUCTORS A N D SEMIMETALS

431

1 1. HELICONINTERACTIONS WITH PHONONS The fact that the helicon wave velocity can be as low as the sound velocity in a solid has led several workers54*7 1 - 74 to investigate helicon-phonon interactions. The dispersion relation for the two acoustic shear waves is (55)

0 = +SIC,

where s is the velocity of sound in the medium. The frequency at which the normal modes of the system could become admixtures of the helicon-like and phonon-like waves is found from this equation and (43): 0; 0,= E,-

();

.

0,

This is an accessible frequency for metals and semiconductors-for the latter assuming s = lo5cm/sec, n = 10” cmP3,and B = lo3 G, j , z 0.3 Mc. The wave vector at crossover assuming very long relaxation times is given by” IC,2

=

s2

20p4 El ~

0 :

c4

(1

+

Consequently, the wavelength of the coupled mode is inversely proportional to q(1 n)”’ and would be orders of magnitude larger in a semiconductor than in a metal. This means much larger semiconductor than metal specimens would be required to match both the frequency and the wavelength of the coupled mode. An interaction between transverse electromagnetic and acoustic waves should be observable if, within the limitations mentioned, a reasonably efficient mechanism exists. Grimes and B u c h ~ b a u m ’ have ~ written the dispersion relation in a form which explicitly expresses the coupling parameter. Their model is an elastically anisotropic, infinite medium composed of free electrons and an ion background (uncompensated plasma). In the absence of a magnetic field the anisotropy produces two independent shear waves with phase velocities u 1 and u2 when the propagation is along a twofold direction in the crystal. The simplifying assumptions employed are: a collisionless plasma (t + a), 0,9 tcuF (the Fermi velocity would be replaced by the average thermal velocity for a nondegenerate plasma),

+

G. Akramov, Fiz. Tverd. Tela 5, 1310 (1963) [English Transl.: Soviet Phys.-Solid State 5 , 955 (1963)J. l 2 D. N. Langenberg, and J. Bok, Phys. Rev. Letters 11, 549 (1963). ’3 J. J. Quinn and S. Rodrigueq Phys. Rev. Letters 11, 552 (1963); Phys. Rev. 133, A1589 (1 964). 7 4 C. C. Grimes and S. J. Buchsbaum. Phys. Rev. Letters 12, 357 (1964). ”

432

BETSY ANCKER-JOHNSON

VH/VSa d H FIG.30. Comparison of calculated dispersion curves for helicon-sound wave interactions with data points obtained from measurements on potassium at 4°K. (After Grimes and B~chsbaum.'~)

and w << w,. The resulting dispersion relation (tailored to potassium) can be written in the form74 x

[(UH2/U2

-

l)(Ul'/U2

- 1 - R/o)- R/ol

where u = o / u and R is the ion cyclotron resonance frequency. The term on the right-hand side couples the right and left circularly polarized components and so produces elliptically polarized waves. If u l = u2, the two circularly polarized components are independent and the three roots of Eq. (57) correspond to right- and left-hand circularly polarized sound waves and the right circularly polarized helicon wave. In the interaction region, uH z u l , Grimes and Buchsbaum have expressed this parameter in terms of the applied magnetic field and the elastic constant of the crystal. They found that the strongest coupling occurs for the largest magnetic fields applied to a medium with the smallest stiffness modulus. Potassium

1 1. PLASMAS

IN SEMICONDUCTORS AND SEMIMETALS

433

fulfills the conditions for coupling and is one of the few, if not the only, solid which does. Their results on a slab of potassium 2.5 x 2.5 x 0.3 cm using fields between 50 and 100 kOe are shown in Fig. 30. The quantity u/v, plotted along the ordinate is the phase velocity of an excitation normalized to the velocity of sound. The lines are solutions of Eq. (57) for the frequencies indicated. The lower branches of this dispersion relation represent excitations which are helicon waves for uJv, < 1 and become sound waves for uH/u, % 1. The upper branches change from sound waves to helicon waves These results are a clear confirmation of heliconwith increased u&. phonon interactions in potassium. V. Pinch Effect 12. PINCH EFFECTIN SEMICONDUCTORS

a. At Low and Intermediate Power Levels Thus far in this review the emphasis has been on equilibrium plasmasthe first two types described in Section 2. Henceforth most of the attention will be focused on nonequilibrium plasmas (except in Section 13) which have been produced by one of the methods described in Section 3. In 1934 Bennett7’ introduced the concept of current pinching in electronion plasmas. A discussion of the simple theory may be found in many places; for examples see Ref. 76. With the assumptions of an isotropic plasma possessing charge neutrality with more massive positive than negative carriers, these simplified expressions obtain for the pressure gradient appropriate Maxwell’s equation : Vp

1

= -J

c

x B

1

= L(-

c

J,BO

V x B = 4nJ/c.

+ JOB,), (594

Hence 4nJ0/c = - ZB,/Zr, 4n c

--J

lo“ = - - ( rBfJ) >

r Zr

where B, is the self-magnetic field and B, is an applied longitudinal magnetic field which inhibits pinching. Cylindrical coordinates are the natural choice

’’ W. H. Bennett, Phys. Rev. 45, 890 (1934). 76

L. Spitzer, Jr., “Physics of Fully Ionized Gases.” Wiley (Interscience), New York, 1956; D. J. Rose and M. Clark, Jr., “Plasmas and Controlled Fusion.” Wiley, New York, 1961.

434

BETSY ANCKER-JOHNSON

for this problem when the pinch is produced by an axial current. Integration of Eq. (59c) yields 2nr'J,(r') dr'

=

4n -Zz(r). C

Then

can be integrated by parts with the result

where a is the initial radius of the plasma column and I , is the critical or minimum current required for the onset of pinching. With the assumption that the temperature is independent of position, P(r) = 4MT, +

GI.

The pressure dependence on radius is such that p first term in (62) is zero when B, = 0 and =

(63) =

0 at r

2c2Nk(T, + Th).

= a,

hence the (64)

N is the total number of particles per unit length of the plasma column. When the applied current is given by (64) the plasma is in radial equilibrium; for I > I, pinching occurs. This result does not yield the pinch radius. Assuming a constant electron drift velocity u,, , the Bennett condition for pinching may be written

for temperature expressed in electron volts and ue, in centimeters per second. If an axial magnetic field B, is applied, Eq. (62) yields

I,

=

+

2cZk(T, TJ eve,

u2B~2 +-,c 241,

(66)

provided that the conductivity in the pinch is large enough to trap the field within the pinch, and that the conductivity in the pinch-free region remains high enough to cause a slow diffusion of B, into that region. The three existing theories for the pinch effect in semiconductors will be reviewed following a description of the experiments at low and intermediate power levels.

11. PLASMAS I N SEMICONDUCTORS A N D SEMIMETALS

435

30 28 26 24

22

460

o;,

2b0

2;o

2:o

2k0

Field In Volts

2180

3b0

10

cm-’

FIG. 31. Current-voltage characteristics for various axial magnetic fields in n-type InSb. (After Chynoweth and Murray.78)

The occurrence of pinching in a solid was first deduced by Glicksman and Steele” from conductivity measurements on a plasma produced by impact ionization in n-type InSb at 77°K. These results were verified by Chynoweth and Murray,” Fig. 31. Impact ionization apparently occurs at 180 V/cm in this sample, but the current does not rise as rapidly thereafter as avalanche multiplication dictates, unless a large axial magnetic field is applied. (B, is assumed to be uniform during pinching because the B, diffusion time is much less than the pinch time.78)This is precisely the type of behavior predicted by Eq. (66) : an axial magnetic field postpones pinching to higher currents. The pinch effect has also been extensively studied’ in plasmas produced by injection into p-type InSb, also at 77”K, Fig. 32. Again a “knee” in the conductivity curve, indicative of an increase in resistance, is

-

77

M. Glicksman and M. C. Steele, Phys. Rev. Letters 2, 461 (1959). A. G . Chynoweth and A. A. Murray, Phys. Rev. 123, 515 (1961).

436

BETSY ANCKER-JOHNSON

to be associated with pinching and, as shown in Fig. 31, increasingly larger currents are required for pinching in the presence of stronger and stronger applied magnetic fields. The I-E characteristics of the two types of plasma differ drastically because the injected plasma density decreases with increasing magnetic field before pinching occurs. Helical instabilities are responsible for this loss of plasma, as is discussed in Section 14. In the presence of magnetic fields large enough to make the B,-term in (66) dominant, the slope of I, as a function of B, should be 4 2 . The theoretical magnitudes of the average crystal diameters so obtained are: (a) for the Chynoweth and Murray7* data, 0.05 cm compared with the actual dimensions of 0.04 cm x 0.06 cm; and (b) for the Ancker-Johnson et aL2 data 0.045 compared with the actual dimensions of 0.042 x 0.051. Both sets of results represent good agreement between observations and theory. The increase in resistance associated with Figs. 31 and 32 has been observed under constant current 'conditions in a finite time after the

E (V/cm) FIG. 32. Current-voltage characteristics for various axial magnetic fields in p -type InSb. (After Ancker-Johnson et at.')

11.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

431

production of an impact ionization plasma7’ and an injected plasma.2 A typical oscilloscope trace of the average field strength in the sample is displayed in Fig. 37(b). Such a result was qualitatively interpreted as follows: After the plasma has filled the crystal, as deduced by the sharp drop in voltage, an equilibrium is established. Then, after a lapse of time (in the example of Fig. 37(b), equal to -0.2 psec), the voltage rises because of the increased magnetoresistance which results from the enhanced B, in the pinched condition, and possibly because of enhanced electron-hole scattering in the pinch as well.79 The reciprocal of the time at which the new equilibrium was established, called the pinch time, was plotted against plasma current (Fig. 36) :

where G,, is the conductivity of the sample without The data points fell on nearly straight lines which were extrapolated to infinite pinch time in order to deduce I,. The value obtained for impact ionization plasmas was 1 A,’9 and for injected plasmas’ it was an increasing function of plasma density. The former result is very plausible because, using (65), it yields a value for the sum of the electron and hole temperatures at the onset of pinching equal to 43OoK, just slightly less than twice the lowest optical phonon temperature (224°K”). The excitation of such phonons provides a very large heat sink for the plasma, provided there are opportunities for the transitions to occur (see the discussion of thermal pinching 0 in Section 12b). The magnitude of the resistance enhancement associated with pinching did not agree well with magnetoresistance calculation^.^^ An alternative explanation of the resistance increment, namely, a pinch “explosion,” has been given by Drummond and Ancker-Johnsons1 and is discussed below. Two experiments have been performed to determine the plasma distribution during pinching, both utilizing n-type InSb. Osipov and Khvoshchevs2 have measured the electron-hole recombination radiation in the wavelength range of 4 to 6 p . A scanning mirror was used to focus this emitted radiation on an InSb varactor. The resolving power of this system was one-fifth the width (2.6 mm) of their sample, The distribution of emitted 7 9 M. Glicksman and R. A. Powlus, Phvs. Rev. 121, 1659 (1961). 79aTheinitial carriers, electrons in the impact ionization plasmas and holes in the injected plasmas, remain distributed throughout the solid to prevent the establishment of space charges. S. J. Fray, F. A. Johnson, and R. H. Jones, Proc. Phys. Soc. (London)76,939 (1960). J. E. Drummond and B. Ancker-Johnson, in “Plasma Effects in Solids” (Proc. 7th Intern. Conf.), p. 173. Dunod, Paris and Academic Press, New York, 1965. 82 B. D. Osipov and A. N. Khvoshchev, Zh. Eksperim i Teor. Fiz. 43, 1179 (1962) [English Transl.: Soviet Phys. J E T P 16, 833 (1963)J.

438

BETSY ANCKER-JOHNSON

intensity across the sample undergoing pinching, both in the presence and in the absence of an applied field, at I , , = 50A is shown in Fig. 33. The authors estimate that the radius of the pinch is 0.1 mm, considerably less than the resolving power. Their results yield a mean concentration of -2 x 10l6~ m - assuming ~ , that longitudinal velocity saturation occurs at - 8 x lo7cm/sec. The pinch radius increases in the presence of the magnetic field and becomes unstable at 600 G. At 800 G the pinch appears to have a saddle, that is, to be somewhat hollow. The amplitude of detected radiation increases with B,, apparently because the volume of the surrounding, absorbing lattice becomes smaller as the pinch diameter increases. (The absorption coefficient for InSb is > lo3cm- ’ for wavelengths between 4 and 6 p . 8 3 )

-

-

FIG.33. The intensity of recombination radiation in n-type InSb at 77°K undergoing pinching with I,, = 50 A as a function of sample width equaling 2.6 mm. Curve 1 is for no applied magnetic field, 2 is for B, = 200 G, 3 is for B, = 400 G, and 4 is for B, = 2 kG.(After Osipov and Khvoshchev.”)

To estimate the temperature of the pinched plasma Osipov and KhvoshchevS2 used an orthogonal magnetic field to push the hot plasma to the sample surface, and thus they avoided the intervention of a cold absorbing region. By fitting a blackbody radiation spectrum to the observed emission, the total temperature of the electrons and holes was estimated at 500°K. This is a lower limit to the temperature of the plasma when it is pinched into a smaller diameter column in the center of the sample without a B , , at the same current. (The current was not given by the authors.) Todaa4 has scanned a sample of n-type InSb at 220°K undergoing pinch, both in the presence and in the absence of applied magnetic fields, with a microwave probe. He measured the reflected signal, which is a function of the impedance of the contact between the probe and the sample. This 83 84

E. 0. Kane, J . Phys. Chem. Solids 1, 259 (1957). M. Toda, J a p a n . J . A p p l . Phys. 2, 467 (1963).

1 1. PLASMAS IN SEMICONDUCTORS A N D SEMIMETALS

439

impedance is sensitive to the plasma density ; the carrier distribution could thus be measured. The resolving power of this method was not given by the author but appears to be also about 0.1 mm. His data show the pinch radius to be constant for plasma currents up to 1 7 A (the highest for which the distribution was measured) at a magnitude approximately equal to the resolving power of his method, a result that is inconsistent with pinch theory. It indicates that the pinch is probably smaller than can be measured by either of the techniques just described.

Volts FIG.34. Conduction characteristics and intensity of recombination radiation in n-type InSb as a function of current carried by the sample. The inset shows that the detector response (upper horizontal scale) is -0 until the plasma voltage reaches the magnitude required for impact ionization. (Data from Mare~hal.’~)

Marechal’’ has also reported measurements of recombination radiation emitted from n-type InSb immersed in liquid hydrogen. The conductivity characteristic and simultaneously obtained total emission are shown in Fig. 34. The former shows the usually observed Ohmic conductivity portion, followed by a decrease due to mobility reduction as the electrons acquire a temperature exceeding their equilibrium temperature. Breakdown occurs at a current of 8 A followed by pinching at a somewhat higher current as indicated by the “knee” in the conductivity curve. The recombination

-

85

Y. Marechal, J . Phys. Chem. Solids 25, 401 (1964).

440

BETSY ANCKER-JOHNSON

radiation is just detectable before impact ionization (see inset in Fig. 34) and then increases markedly ; the author attributed this to pinching. No explanation was offered for the great reduction in radiation output beginning at 15 A. An alternative explanation for the intensity pattern of emitted radiation is that copious recombination occurs after breakdown, but before pinching, and then, when pinching occurs, the observed emission is greatly reduced because of absorption by the intervening crystal, as discussed by Osipov and Khvoshchev.82

-

FIG.35. The full amplitude of oscillation at maximum and the frequency plotted as functions of the current in p-type InSb while undergoing pinching. The currents at which growing oscillations occur are indicated. (After Ancker-Johnson.86)

Reports on the first pinch experiments in impact i o n i ~ a t i o nplasmas, ~~ and injected plasmas’ included observations of small amplitude oscillations in the voltage during pinching while the current was held essentially constant. Oscillations related to pinching have also been observed in the presence of applied transverse7** and longitudinal’, 3* 79, 84 magnetic fields (see Sections 14 and 17). Ancker-Johnson3 distinguished in the latter situation between the oscillations associated with pinch and those related to helical instabilities (see Section 14). She also reported86 much larger, growing amplitude oscillations in p-type InSb which occur during pinching. A typical dependence of their amplitude and frequency on current is reproduced in Fig. 35. Steele and HattoriS7 have suggested that sound wave generation may accompany pinching in impact ionization plasmas. If the transverse plasma velocity at the surface of the sample exceeds the velocity

’’

86 87

B. Ancker-Johnson, Phys. Rev. Letters 9, 485 (1962). M. C. Steele and T. Hattori, J . Phys. SOC. Japan 17, 1661 (1962).

11.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

441

of sound, coherent acoustic waves may result, and these could instigate a coherent oscillating component in the transverse drift of a pinch. Then the interaction of the self-magnetic field and the oscillating transverse currents may produce oscillating longitudinal Hall currents. A rough calculation yielded approximate agreement with experiments on n-type InSb79and n-type InAs for the n = 2 mode. (The results of Drummond and Ancker-Johnson’s8 theory of pinch oscillations will be described below.) Three theories have been published describing the response of nonequilibrium electrons and holes when compressed by a self-magnetic field. Their assumptions, methods, results, and weaknesses are discussed below. Glicksman88 produced an adiabatic theory, that is, a theory in which the thermal contact between plasma and lattice is assumed to be negligible. The assumptions that the generation and recombination rates are equal and that the total number of carriers is constant during pinching are also unique to his treatment. He further required the longitudinal drift velocity u,, to be independent of electric field strength and Be. Common to at least one of the other theories are Glicksman’s assumptions that scattering can be described by a relaxation time, carrier energy can be described by a temperature, that the current produces forces much greater than diffusion forces, and that pinching begins when (T, + Th) equals twice the lowest optical phonon temperature. He begins with the moment, continuity, and Maxwell equations, and obtains the restoring forces primarily through the rising temperature of the plasma and also through the dependence of the mobilities on magnetic field. In Glicksman’s theory two time scales for pinching emerge, one related to the inertial response, in the range of lo-’’ sec which is comparable to the scattering time, and the other related to mass motion of the plasma in the tenths of microsecond range. The ambipolar radial velocity rises to this magnitude during the initiation of pinch :

where B, is given by (60). The second time scale solutions yield the following results : (a) A temperature of the plasma

T = To(4r,)4’3, which produces a temperature profile that is sharply peaked on the axis. M.Glicksman, Japan. J . Appl. Phys. 3, 354 (1964).

442

BETSY ANCKER-JOHNSON

(b) A condition for pinching which is identical to Bennett's equation (65)75 except for a small numerical factor. (c) A pinch radius which is weakly dependent on current,

(a is the sample radius and rp = 0.15~at 10 A, the maximum plasma current

for which this author88 thinks his theory is likely to be valid.) (d) Time-dependent graphical solutions for the pinch radius and the radial velocity; the latter becomes zero during the second time scale. (e) The dependence of the reciprocal pinch time on plasma current as deduced from (d) which can be compared with experiment, Fig. 36.

FIG.36. Reciprocal of the plasma pinch time as a function of plasma current for three values of hole mobility; experimental points are for n-type InSb with no = 2 x 10'4cm-3 for the sample corresponding to 0 , and no = I to 4 x IOl3 for the sample corresponding to x . (After Glicksman.88)

The adiabatic assumption of this8* theory implies that the lattice thermal conductivity is zero, hence the theory is limited in validity to small currents and short duration plasmas. A precise criterion for the limit of validity is lacking. The estimated maximum of -10 A implies a very high plasma temperature, 2700"K, and a radial velocity that goes to zero at only -40 nsec. Beyond that time such a hot plasma, confined to a small volume, begins to heat the lattice within its path according to the known8' thermal conductivity of InSb. The assumption of equal generation and recombination rates is inconsistent with the shorter time scale in the theory, because impact ionization requires a much longer time, lo-' sec." The plasma

-

89

N. H. Nachtrieb and N. Clement, J . Phys. Chem 62, 876 (1958). B. Ancker-Johnson, Bull. Am. Phys. SOC. 10, 345 (1965).

11.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

443

generation time is so long that it is comparable to the duration of a finite radial velocity when the plasma current is 10 A. A further difficulty with this theory is the need to know u,, in the pinch in order to compare the theory with experimental results, since the velocity distribution during pinching is not known. Marecha18' has presented a theory which assumes a linear recombination

dn/&

=

-n/z,

zp = plasma lifetime

(71)

and a generation which is independent on an unspecified function of electric field strength

dn/dt

=

(no

+ n)f(E).

(72)

He alone assumes a constant plasma temperature, and, therefore, his is a purely magnetic pinch theory applicable only at low currents. He also employs another limiting assumption, namely, that the nonequilibrium plasma density before pinching must be much less than the equilibrium density of the carriers, and thus his theory is surely not valid for impact ionization plasmas but does, perhaps, apply for injected plasmas in the initial stages of pinch. He employs the continuity equations with the diffusion terms and arrives at steady state solutions for the radial electric field strength and the plasma distribution. The latter, n(r), is expressed in a second-order inhomogeneous differential equation' ~

with

2 + (:+ ): -

Qx -

+ (rn- 1)y = - f ( E ) z ,

(73)

where p o = 471 x lo-' H/m and D, is the hole diffusion constant. The solution, which employs confluent hypergeometric functions, yields n(r) practically uniform for E, < 200 V/cm, assuming values in (73) appropriate for InSb, and n(r) sharply peaked for E , > 200 V/cm because there is no amplitude limitation on the axis as there is at lower E,. The pinch radius is given by

r p = ( ~ D , T / Q ) ' / ~= (0.4/E,) meter,

(74)

which is independent of sample radius and very much smaller (for 91

The right-hand side of this expression is incorrectly written in Ref. 85, as pointed out by D. A. Jerde (private communication).

444

BETSY ANCKER-JOHNSON

E , = 250 V/cm, r, = 16 p) than that given by the previously described theory.88 It is surely too small because this theory does not employ adequate restoring forces. Another major weakness of the theory is its extremity limited range of applicability. Distinctive to Drummond and Ancker-Johnson's' theory, and diametrically opposed to the others, is the assumption that the plasma is in good thermal contact with the lattice, hence theirs is a magnetothermal pinch effect. They therefore use temperature-dependent mobilities and plasma densities. They assume that the plasma is at the lowest optical phonon temperature at the onset of pinching, in agreement with Glicksman, and that pinching occurs in times comparable to those required for plasma production. Their two fundamental equations are those for the conservation of carriers and energy. In the former, n(r) is expressed in terms of the particle pressure and the Lorentz force. In the second equation, the product of the specific heat per unit volume of the crystal and the position-dependent, time differential of temperature is set equal to the electrical power input minus the divergence of the heat flux. They divided the pinch into two regions (to fit the calculation to a moderate-size analog computer) and solved for the density, temperature, and longitudinal electric field strength as a function of time, and also for the asymptotic solutions at long times. The time-dependent results are shown in Fig. 37(a). According to the assumptions of this theory,81 plasma production and pinching are both occurring at the beginning of the applied power pulse. The resulting small, conducting plasma column rapidly becomes hot. It swamps the ability of the crystal to conduct away all this heat to the surrounding bath at a low input power, -400 W/cm. The temperature on the axis therefore rises rapidly as shown in the Fig. 37(a), and then more plasma is produced as a result of thermal ionization. The attendant excess pressure on the pinch axis causes radial flux of carriers to the outer part of the pinch, and therefore the density in that region rises in time as shown in the figure. In the meantime the pinch column has shorted out the sample, so that the field strength drops very rapidly as the theoretical and experimental results both indicate, Fig. 37. Consequently the temperature on the axis begins to decrease and approaches an asymptotic value because the power input decreases with time. The density in both the inner and outer regions of the pinch thus decreases, and the field strength readjusts to a higher value. Hence the rise ~ .the ~~ in resistance (cf. Figs. 31 and 32) which was formerly a t t r i b ~ t e d to onset of pinching is attributed by Drummond and Ancker-Johnson' I to the explosion of the pinch into a somewhat hollow or tubular form. After the pinch expansion, the equations produce a steady state, as in Fig. 37 or, at slightly higher power levels, oscillatory behavior, Fig. 38, both in agreement with observations. Even the nonsinusoidal form of the observed

1 1.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS 7

I

-4

445

+ ._ c

<' -4

-e2.-e

Q C

-3

.g

-e C

8

5

-2

c Temperature

4

5 CD

e lz

-1

0 i i I

0

0

Time (Arbitrary Units) (0)

V/cm

98 49

0

FIG.37. (a) The computed concentration in the outer regions of the pinch, the temperature on the axis, and the axial electric field strength in a nonequilibrium InSb plasma as a function of time. (b) A typical oscilloscope trace at power levels close to the threshold for pinching in p-type InSb. (After Drurnrnond and Ancker-Johnson.*')

oscillations in Fig. 38 emerges from the equations describing magnetothermal pinch oscillations. The oscillations are observed to occur, Fig. 40, when the plasma is dissipating moderate power, namely, in InSb between 0.4 kW/cm (the upper limit for the magnetic pinch effect) and -2 kW/cm (the lower level at which thermal pinching begins; see the next section). The amplitudes of three observed modes of oscillation as a function of power

446

BETSY ANCKER-JOHNSON

Time (Arbitrary Units) (a)

FIG.38. (a) Computed electric field strength as a function of time. (b) Oscillogram showing a typical oscillatory response observed during pinching in p-type. InSb. (After Drummond and

Ancker-Johnson.’ I )

are also shown in this figure, in agreement with previous results,86 Fig. 35. The asymptotic results yield smooth temperature profiles of the pinch (in contrast to Glicksman’s peaked temperature profiles). As the input power increases, the temperature on the axis rises and ejects some of the carriers being produced by thermal ionization. In agreement with the timedependent results a somewhat hollow pinchg2 distribution is produced, Fig. 39.

’*

A similar hollow distribution has been derived theoretically for electron-ion pinches by G. Ecker and 0.Zoeller, Phys. Fluids 7, 1996 (1964).

11. PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

447

This theory" does not discuss the mechanism of energy transfer from the plasma to the lattice. The transfer apparently requires a time of at least tenths of a microsecond since very high-power, short-duration pinches have been produced without apparent damage to the ~ r y s t a l , ~ ' , whereas '~ pulse I0

I

I

I

I

I

I

1

1

.

1

Dimensionless Distance

FIG.39. The relative plasma concentration as a function of dimensionless distance from the axis. The associated powers dissipated by the plasma are indicated on the concentration profiles. (After Drummond and Ancker-Johnson.")

lengths > 1 psec at similar power levels melted the sample94 (see next section). b. At High Power Levels

As already suggested by Fig. 40 a qualitatively different kind of pinch effect occurs at high power levels, those exceeding -2 k W / m in InSb, and for durations exceeding 1 psec. Thermal pinching has no analog in electron-ion plasmas since it involves the melting of the crystal in the path of the pinched plasma. It was the visual observation of such melted channels that led to the discovery of the thermal pinch effect.94 Glicksman and W. A. Hicinbothem, Jr.. Phys. Rez,. 129,1572 (1963). B. Ancker-Johnsonand J. E. Drummond, Phys. Rev. 131, 1961 (1963); 132, 2372 (1963).

93 M. 94

448

BETSY ANCKER-JOHNSON

FIG.40. The types of pinch effect as a function of power into the plasma. The amplitudes of three observed modes of magnetothermal pinch oscillations are also shown.

Square cross-section, single-crystal samples of p - and n-type InSb at 77°K were pulsed with total currents up to 42 A for durations of 1.4 to 7.2 psec at a maximum repetition rate of 1.4 sec- '. The pinch channels so produced left permanent evidence of their existence by disturbing the lattice within the channels. At the highest power levels the crystals spontaneously cleaved while undergoing pinch and exposed sections of the column typically 2 to 6 mm long. At the lower power levels, from -2 to -6 kW/cm, the samples were cleaved after pinching under the stimulus of ultrasonic vibrations. An example is shown in Fig. 41. The channel diffuses incident light, whereas the crystal outside the column reflects it from flat cleavage surfaces. X-ray back reflection patterns indicated that the crystal within the channels had been melted and rapidly recrystallized. Such channels were found in the eight samples in which they were sought. Two attempts to find channels when the power into the plasma was 1.9 kW/cm were fruitless. Curved columns were produced by passing current through side contacts. The boundaries of the channels often look fluted, similar to boundary shapes visually observed in electron-ion plasma (magnetic) pinches. At the beginning of pinching the balance between magnetic and carrier pressure as given by Eq. (65) requires a temperature, assuming T, = Th, of kT

=

epP/4c2,

(75)

where P = ZE is the power input per unit length of plasma, and the sum of the electron and hole mobilities is p

=

7 x 108T-1.6 cm2/v-sec,

(76)

1 1.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

449

FIG.41. Photograph of a crystal after melting occurred in the path of the pinched plasma. The melted column rapidly recrystallized after the power was turned off. The power dissipated by the plasma was 6.2 kW/cm at the onset of pinching and dropped to 5.5 kW/cm after 1.5 psec. The radius of the melted channel shown is 2.8 x 1 O - j cm. (After Ancker-Johnson and Dr~rnrnond.~~)

450

BETSY ANCKER-JOHNSON

for T > 200°K in The radius of the pinch, however, is not determined by the simple theory described at the beginning of this section until

I

I

I

I

I

I

)

Using Bennetts‘ Temperature

FIG.42. The pinch radius divided by the plasma current as a function of power input per unit length of plasma: experimental data at the onset of pinching ( x ) , at the end of the pulses The upper curve is the result of magnetic theory and the lower, thermal pinch theory. The numbers associated with the arrows correspond to the net energy transferred to the pinch column in units of energy required to melt the column. (After Ancker-Johnson and Dr~mmond.~~) (0).

the pressure, 2nkT, is known on the pinch axis. This was calculated from the intrinsic electron concentration for InSb given byg6

(where E~ = 0.25 eV for InSb) provided this concentration is greater than the concentration in the pinch of electrons produced by injection and/or impact ionization. The pinch radius, defined by I = nrp2J, where J, is the maximum current density of the pinch, is given by

ZPI is the plasma current [Eq. (67)]. C. Hilsum and A. C. Rose-Innes, “Semiconducting 111-V Compounds,” p. 126. Pergamon, Oxford, 1961. 96 G. Bush and E. Steigmeier, Helu. Phys. Acta 34, 1 (1961). 95

11. PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

451

This quantity is graphed in Fig. 42 and marked “Bennett’s temperature.” The experimental points marked by the x’s correspond to momentarily stationary pinches very near the beginning of the current pulses. Thereafter the whole pinch column, lattice as well as carriers, rises to the melting temperature of the lattice because the pinch radii are so minute and the net power in the plasma column is so great. The net power is the input power P less the conduction loss of heat per unit length per unit time, W, x 0.4 kW/cm in InSb. As the lattice melts it produces an intrinsic carrier concentration, 1.6 x 10” ~ m - appropriate ~ , to this temperature, 800°K. The second curve in Fig. 42 is a plot of Eq. (78) for this temperature. The production of such a large density of carriers increases the conduction of the sample so much that the voltage across it, with nearly constant current supplied, drops by a large percentage during the pulse, as much as 45% in the experiments by the arrows, reported by Ancker-Johnson and D r ~ m m o n dAs . ~indicated ~ each of the experimental points on Fig. 42 moves away from the curve determined by the Bennett critical temperature and toward the curve determined by the thermal condition as thermal pinching occurs. The theory of the pinch effect proposed by Drummond and AnckerJohnson’’ (discussed in Section 12a) yields the temperature profiles shown in Fig. 39. The lattice reaches its melting temperature at 2.9 kW/cm, in good agreement with the experimental value of -2 kW/cm (compare Fig. 40). By contrast a purely magnetic theory requires P = 18 kW/cm for me1ting. The approximate net energy transferred to the pinch column, in units of the energy required to melt the column, is calculated as follows: Net power times the ratio of the pinch duration to the heat of fusion of the crystal within the column. For each experimental point this net energy is indicated in Fig. 42. In those cases for which the temperature reached values much greater than those necessary for lattice melting, evidence was seen of molten crystal “squirting” out the channel at the contacts, Fig. 43. This was shown to occur as the crystals cooled, consistent with the fact that InSb shrinks on melting. The plasma concentrations accompanying thermal pinching are much greater than the concentrations estimated by Osipov and Khvoshchev,’* who did not take into consideration thermal ionization. If their estimates were used, the theoretical curves in Fig. 42 would be moved up the ordinate by a factor of -6. The conclusions to be drawn from the good agreement between the theoretical curves and experimental points in Fig: 42 are that (1) thermal ionization occurs within the very short times required for Bennett pinching to take place (20 to 50nsec at high power levels) and (2) a unique kind of pinching or thermoelectric instability results at high

452

BETSY ANCKER-JOHNSON

power levels in electron-hole plasmas. The thermal pinch effect is initiated by the magnetic pinch effect but dominates the conduction at high power levels immediately after the onset of pinching and for the entire remainder of the pulse. The plasma obeys magnetic criteria [Eq. (65)] until the net

FIG.43. Photograph of the region near a current contact showing a conically shaped piece of InSb crystal which appeared on the surface of the sample after pinching. (After AnckerJohnson and D r ~ m r n o n d . ~ ~ )

power input to the plasma exceeds zero ;then the thermal condition controls its response and power balance prevails. There results an “absolute” instability, not an “over-stability,” because once the condition for this instability is reached there is no oscillation between the two conditions. An equilibrium model for thermal pinching in which thermal conduction balances joule heating has been proposed by Burgess.97 A conductor 97

R. E. Burgess, Proc. Intern. Con$ Semicond. Phys., Progue, 1960 p. 818. Czech. Acad. Sci., Prague, 1961.

1 1,

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

453

experiencing increasing power input while immersed in a coolant will heat up preferentially along the axis. This causes enhanced conduction along the axis and an attendant higher concentration of current there. An analog computer using parameters appropriate to the host crystals in which the experimental observation of thermal pinching occurred, has shown that an equilibrium thermal pinch at 800°K would have a radius considerably more than an order of.magnitude larger than the observed thermal pinch radii.

10

100

Volts/cm FIG.44. The conductivity of Bi according to Hattori and Steele9' and to Schillinger.'OO

13. PINCHEFFECTIN SEMIMETALS Recently experimental and theoretical work has been reported on the conductivity of bismuth at very high current level^.^^-'^^ Hattori and Steele98 passed currents up to 100 A along the binary axis in a very pure slab-shaped sample at both 77" and 243°K. They observed a sublinear conductivity beginning at 19 A when the sample was at 77"K, and linear behavior up to 96A at the other temperature, Fig. 44. These results were attributed to a large self-magnetoresistance, but the measured deviations from Ohmic conductivity were smaller than they expected for a pure magnetoresistance effect. Their calculation^^^ were based on the usual model of magnetoresistance applied to a simplified energy-band structure

-

T. Hattori and M. C. Steele, J . Phys. SOC.Japan 18, 1294 (1963). S. Tosima and R. Hirota, Jr Phys. Soc. Japan 19, 468 (1964); IBM J . Rex Develop. 8, 291 (1964). l o o W. Schillinger, IBM J . Rex Develop. 8, 295 (1964).

98

99

454

BETSY ANCKER-JOHNSON

appropriate for Bi, that is, inward particle flow crossing the self-magnetic field and so producing longitudinal Hall currents which oppose the primary current. The small observed enhancement in resistance compared to the magnetoresistance predicted by these rough calculations is ascribed to a distorted carrier distribution, namely, a pinched distribution. Such a distribution would produce enhanced diffusion of carriers toward the surfaces and so tend to cancel some of the self-magnetoresistance. A complete magnetoresistance theory for bismuth is formidable because of the large anisotropy of its magnetoconductivity tensor, therefore the conclusion that pinching has occurred is necessarily very tentative. Obtaining a density distribution that deviates from equilibrium in a semimetal involves different physics than the pinching of a nonequilibrium plasma in a semiconductor. In the latter case the nonequilibrium carriers do not depend on pinching for their existence (except at relatively high power levels), but they do depend on it in the semimetal case. Interellipsoid transitions of equilibrium carriers can produce a transverse, inward particle flow (but no net current) under the influence of a Lorentz force. This inward flow depletes the carrier density below the equilibrium value on the surface, and so electron-hole pairs are generated there and hence the conduction is enhanced. If the bulk recombination time is long compared with the time required to establish the pinch distribution, then one condition for the occurrence of pinch is satisfied. The diffusion time must also be long compared to the pinch time. Good agreement between theory and experiment on self-magnetoresistance is found if the scattering times between ellipsoids are taken to be of the order of lO-’sec and if diffusion effects are not dominated by the s ~ r f a c e . ’ ~ * ~ ’ Schillinger’’’ has also done experiments and theory on non-Ohmic conductivity in Bi. His conductivity measurements yielded similar results to those of Hattori and Steele,98 Fig. 44. The arrows indicate the critical currents for pinching, assuming the validity of the simple magnetic theory [compare Eq. (64)] with kT replaced by the Fermi energy (assumed equal for electrons and holes). I, z 32A for Schillinger 77°K data. Increasing the current beyond the last recorded point in the 300°K data destroyed the sample, perhaps because of the occurrence of thermal pinching. Schillinger has employed a similar approach to that of Tosima and H i r ~ t ato~express ~ the steady-state, radial particle flux and B, in differential form as functions of the electron and hole mobilities, the applied electric field strength and the carrier Fermi energies. These equations were solved for the current and field strength normalized to their pinch magnitudes in the absence of self-magnetoresistance effects, Fig. 45. The average diffusion length L is a parameter, as is the surface recombination velocity S. The L = 0 curve is the ordinary self-magnetoresistance effect with no carrier

1 1. PLASMAS

IN SEMICONDUCTORS AND SEMIMETALS

455

,

I

10

1

0.1

E/Ep FIG.45. Computed steady state conduction characteristics in Bi for some limiting values of the recombination parameters. (After Schillinger.lOO)

redistribution. The L = a curves correspond to a diffusion length equal to the radius with, respectively, an infinite surface velocity S and one approximating the sound velocity (lower curve). This diffusion length in a 0.1-mm radius sample corresponds to a bulk recombination time of 6 x lo-* sec. Schillinger's'OO results show that the conductivity characteristics are dominated by the self-magnetoresistance and vary only slightly for a wide range of generation-recombination parameters, even though the computed results exhibit large changes in the radial distribution and total number of carriers. The occurrence of pinching in Bi is, therefore, difficult to prove.

VI. Instabilities 14. HELICALINSTABILITIES a. Linear Properties

A brief communication by Ivanov and Ryvkin"' in 1958 reported current oscillations in n-type Ge at room temperature, immersed in parallel electric and magnetic fields. Apparently similar oscillations have been observed in n-type InSb at 77°K.''' Larrabee and Steele'03 investigated these lo'

1. L. Ivanov and S. M. Ryvkin, Zh. Tekhn. Fiz. 28,774 (1958) [EnglishTransl.: Soviet Phys.Tech. Phys. 3, 722 (1958)j. J. Bok and R. Veilex, Compt. Rend. 248, 2300 (1959). R . D. Larrabee and M. C. Steele, J . Appl. Phys. 31, 1519 (1960); Proc. Intern. Con$ Semic o d . Phys., Prague, 1960 p. 227. Czech. Acad. Sci., Prague, 1961 ; R. D. Larrabee, J . Appl. Phys. 34, 880 (1963).

456

BETSY ANCKER-JOHNSON

oscillations in detail and named the device resulting from their work an oscillistor. Using n- and p-type Ge and InSb at 300" and 77°K they noted oscillations at frequencies between a few kc and -10 Mc which were independent of both the load and their attempts to synchronize the oscillations to an external source. They demonstrated that a necessary condition for the occurrence of oscillation is the presence of an injected electron-hole plasma. G l i ~ k s m a n generalized '~~ a theory of Kadomtsev and N e d o s p a s ~ v ' ~ ~ (which had been successful in explaining some observed instabilities in the positive column gaseous plasmas) to apply to an electron-hole plasma. The oscillator effect is thus attributed to the temporal growth of a helical perturbation, superimposed on the unperturbed steady state plasma density and electric potential, in a plasma column subject to parallel electric and magnetic fields. The theory assumes that the plasma response is collision-dominated ; its sources are injection from the ends of the column or impact ionization in the bulk, and its sink is primarily surface recombination with some volume recombination. Holter106 took into account the influence of the initial charge carriers in the semiconductor, so that his theory applies to extrinsic or intrinsic semiconductors, while Glicksman's applies strictly to insulators. Holter also allowed for the heating of the plasma above the lattice temperature. A dispersion relation for the frequency is derived from the continuity equations and the equations of motion, assuming an essentially neutral plasma. The generation-recombination coefficient 2 used in the continuity equations has these meanings : Z > 0:

impact ionization plasma, recombination at boundaries

2 < 0:

injected plasma, volume recombination

Z z 0:

injected plasma, recombination at boundaries ; or impact ionization, volume recombination

The first and third conditions yield similar results, and the latter is used in the calculations. The boundary between a stable and unstable plasma, in which the helical density perturbation grows, occurs when Im(w) = 0. The growth conditions are expressed in terms of the electric field strength, the wavelength, and the frequency of the helix as a.function of the threshold magnetic field, with the ratios of injected plasma density to initial carrier lo4 lo' lob

M. Glicksman, Phys. Rev. 124, 1655 (1961). B. B. Kadomtsev and A. V. Nedospasov; J . Nuci. Energy: P t . C 1,230 (1960). 0.Holter, Phys. Rev. 129, 2548 (1963); Arbok Uniu. Bergen, Mat.-Nut. Ser. 8, 3 (1963).

1 1. PLASMAS

IN SEMICONDUCTORS AND SEMIMETALS

457

concentration (injection level) as a parameter. Quantitative comparison with the experimentally determined boundary is made by calculating the plasma temperature from the measured electric and magnetic threshold field strengths and using it to calculate the threshold frequency. Not only the threshold frequency but also the threshold electric field could be compared directly with the experimental values if a relation between temperature and electric field were known (as it is for the positive column). A physical interpretation of the helical instability was proposed by Hoh and Lehnert,'07 who studied a helical disturbance of the same form investigated previously by other workers: exp(iKz + imY - iot), where K is the wave vector along the z-axis, m is the mode number in the azimuthal direction (m = 1 for helix), and o is the frequency of the perturbation. They made the reasonable assumption that the ion helix (in a gas plasma) is fixed in space compared with the electron helix, which is shifted by the applied electric field at a velocity equal to the drift velocity of the electrons in the plasma. This axial displacement of the helices is equivalent to a rotation in the azimuthal direction. The attendant charge separation induces an azimuthal electric field strength E,', which contributes to a Ey'x B, drift that propels the electrons radially outward. Thus, the plasma is destabilized. Acting in opposition to this is the dissipation of the electron helix, caused mainly by diffusion along the magnetic field which results in a counterrotation of the electron helix as dictated by the unperturbed radial ambipolar electric field. Conduction caused by the perturbed axial electric field also tends to smooth out the helical distribution. The opposition of these stabilizing and destabilizing forces can produce oscillations and an enhanced diffusion of the plasma toward its boundaries. This explanation, however, is subject to an important criticism made by Ichikawa et ~ 1 . ' ~ ' They explored the effect of an internal electric field induced by charge separation, instead of first assuming the quasineutrality condition not only to describe the plasma, but also the perturbed state, as previous workers had. They found that the E,' x B, destabilizing force is not essential to the occurrence of helical instability oscillations. The validity of this criticism needs further evaluation. Gurevich and Ioffe"' have studied possible conditions for the existence F. C. Hoh and B. Lehnert, Phys. Rev. Letters 7, 75 (1961); F. C. Hoh, Phys. Fluids 5, 22 (1962). Y. H. Ichikawa, S. Misawa, and Y . Sasakura, Progr. Theoret. Phys. (Kyoto) 27, 1277, 1279 (1962), and private communication. "'(a) L. E. Gurevich and I. V. Ioffe, Fiz. Tuerd. Tela 4, 2641 (1962) (Engiish Transi.: Souiet Phys.-Solid State 4, 1938 (1963)l. (b) Fiz. Tuerd. Tela 4, 2964 (1962) [English Transl.: Soviet Phys.-Solid State 4, 2173 (1963)l. (c) Fiz. Tuerd. Tela 5, 2674 (1963) [English Transl.: Soviet Phys.-Soiid State 5, 1954 (1964)J.(d) Fiz. Tverd. Tela 6, 445 (1964) [English Transl.: Soviet Phys.-Solid State 6, 354 (1964)l.

lo'

458

BETSY ANCKER-JOHNSON

of current instability oscillations. Gurevich’ lo has suggested that every possible fluctuation of carrier concentration in a semiconductor has a wave form in an external electric field. He shows this by using the continuity conditions for positive and negative carriers : an‘ dt

-

+ V . {p+[n’E, + n,+E‘])

an‘ - + V.{pL-[n‘Eo + n o - E l } at

=

0,

(79) =

0,

where diffusion is neglected and the quasineutrality condition is employed. These equations have wave solutions of frequency w = KU where u is some average drift velocity. If the plasma possesses concentration gradients Vn,’ or Vn,-, the alternating electric field E’ does not coincide in phase with the alternating carrier concentration n’. There results an imaginary part of the frequency which produces an instability if the sign of Vn,’ is appropriate. Gurevich and Ioffe”’ have explored a number of possible ways to produce such a density gradient both parallel and normal to the applied electric field. These include (1) illuminating one side of a (2) producing Hall fields by using nonparallel electric and magnetic fields, ‘09(a),(b) (3) assuming the presence of a gradient in impurity concentration,’ 09(‘) (4) and assuming impact ionization. Assuming a plasma density with such a gradient in the x-direction for slab geometry and in the radial direction for cylindrical geometry, the occurrence of a time-dependent density perturbation d ( t ) = nO‘(t)eiKsr results in a field E ’ ( K , ~because ) of charge separation in E,,, the applied electric field. This treatment is in agreement with previous theories. The transverse component of E’ interacts with an applied magnetic field H,, to produce a time-dependent density variation, or “Hall flux” (&/at), cc V n , (Ex H), which is opposed by a “diffusion flux” (an/&), = D K ’ ~ ’ where D is the diffusion coefficient and K the wave vector of the perturbation. When the former is larger than the latter the instability can grow. This condition is shown to be valid when the surface recombination velocities and the mobilities are, respectively, unequal for the two types of charge carriers. The particle fluxes to the boundaries when bulk generation occurs result in a momentary excess of carriers of one sign there, and hence lead to a transverse field E,, equalizing these fluxes. Then

-

(g)H

= (-V-J),cc

‘lo

L. E. Gurevich, private communication.

Vn,.(E x H ) + Vn’aE,,.

1 1.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

459

TABLE IV THRESHOLD ELECTRIC FIELD THRESHOLD FREQUENCY IN THE STATEDLIMITSFOR SLABGEOMETRY

GUREVICH AND IOFFE'S1og'd' PREDICTED

STRENGTH AND

Limit

a,

Ec

The oscillatory nature of the instability is produced by a phase shift between the carrier flux caused by the variable electric field E and the flux caused by the effect of the static field Eol on the variable part of the concentration n'; that is, the first term on the right-hand side of (80) is proportional to K and the second to i K . The boundary conditions separating stable and unstable plasma are worked out for a few simplified cases employing slab and cylindrical geometry; some results for the slab geometry are recorded in Table IV. In case (1) Gurevich and Ioffe assumed an InSb sample with the thickness d = 0.3 cm and B = 200G and calculated w, = lo5 rad/sec and E , = 20 V/cm. TABLE V A COMPARISON BETWEEN THE THEORIES FOR TEMPORAL GROWTH OF A HELICAL DENSITY PERTURBATION IN THE B + 0 LIMIT

Author Gli~ksman'~~-Holter'~~ Gurevich-loffe

'

09.b

4,"

wth

B-IR-I

For extrinsic: B - ' R - 2 For intrinsic: BR-'

~ - 2 / 5 ~ - 6 / 5

B8'5R-16'5

R is the radius of the plasma and B is the applied magnetic field. For the generation-recombination coefficient proportional to E3l2.

The first three methods of producing a density gradient yield results in agreement with the insulator theory described above (Gurevich and Ioffe have not taken into account the influence of the equilibrium carriers nor the possibility of plasma heating.) In their most recent work treating impact ionization, the generation-recombination coefficient 2 is taken proportional to E3", following Keldysh.' Since the instability condition contains the

"' L. V. Keldysh, Zh. Eksperim. i J E T P 37,509 (1960)l.

Teor. Fiz. 37, 713 (1959) [English Transl.: Soviet Fhys.

460

BETSY ANCKER-JOHNSON TABLE VI A COMPARISON BETWEEN EXPERIMENTALLY AND THEORETICALLY FOR THE HELICAL DETERMINED THRESHOLD FREQUENCY INSTABILITYI N TYPE InSb AT 77°K Ancker-Johnson3Data B(G)

620 490 435 285 170

Holterio6 Theory kT(eV)

Eth(V/cm)

40 65 82 103 152

27.5 27.0 25.0 25.0 22.5

30.1 24.6 20.1 17.0 13.1

1.5 x 2.8 3.8 3.9 4.1

product ZE, which is proportional to E in the Glicksman-Holter theory and to E5/' in the Gurevich-Ioffe theory, different critical fields and frequencies emerge, as shown in Table V. It is clear that experiments should easily be able to determine whether impact ionization contributes to a density gradient, and therefore whether it influences the nature of the resulting instability. Gordeev"2 has proposed an explanation for the oscillistor which is similar in approach to Glicksman's, but he has calculated the threshold conditions only for a "zero frequency helix." Several experiments support the helical instability theory. Two teams of J a p a n e ~ e " ~ , have shown independently that the oscillations are caused by a rotating density wave. Phase differences were measured both by a microwave reflection t e ~ h n i q u e " and ~ by orthogonally located voltage probes. l4 Simultaneous measurements3 of the threshold electric field and frequency as a function of the magnetic field and the injection level have enabled quantitative comparisons with the theory for p-type InSb. In these experiments (0,~)N 1 and (o,z)+ N 5 x so comparison with the small magnetic field limit of the Glicksman-Holter theory is appropriate. Table VI shows good agreement between experiments and the helical instability theory. The table also shows the predicted temperature of the plasma. Typical x-y oscilloscope traces showing electric versus magnetic fields for different injection levels are reproduced in Fig. 46. The dashed line, passing through the E minimum, depicts the boundary between stable and unstable plasma. The oscillations do not appear regular in Fig. 46 because a sampling 'I2

'I3

G. V . Gordeev, Fiz. Tverd. Tela 4, 3144 (1962) [English Transl.: Soviet Phys.-Solid State 4, 2303 (1963)l. T. Misawa and T. Yamada, Japan. J . Appl. Phys. 2, 19 (1963). F. Okamoto, T. Koike, and S. Tosima, J . Phys. SOC.Japan 17. 804 (1962).

11.

PLASMAS IN SEMICONDUCTORS A N D SEMIMETALS

461

FIG.46. The dependence of electric field on magnetic field for eight different initial densities of nonequilibrium carriers as obtained using a sampling oscilloscope. The densities are derived from conductivity measurements on the same sample in the absence of magnetic field. The minimum in each curve denoted by the dashed line corresponds to the instability boundary. (After Ancker-Johnson.' ")

of thousands of pulses was used to produce each trace. The oscillations are actually very reproducible from pulse to pulse both in phase and in amplitude, as demonstrated by Fig. 47. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l l l l l l l ~ l l l l l l l l l l l l l l ,

I = 0.62A

I

B=375G

FIG.47. Helical instability oscillations as they typically appear well beyond threshold. Horizontal time scale is 0.2 p sec/large division.

Hurwitz and McWhorter"' have discovered (unidirectional) spatial growth of helical density waves in thermal equilibrium electron-hole plasmas. This is a convective instability unique to semiconductor plasmas in which unequal densities of positive and negative carriers can occur, in contrast to the instability just described, in which temporal growth of the helical density wave results for essentially equal densities of oppositely charged carriers. C. E. Hurwitz and A. L. McWhorter, Phys. Reo. Letters 10, 20 (1963); Phys. Rev. 134, A1033 (1964).

462

BETSY ANCKER-JOHNSON

Hurwitz and McWhorter's physical model for the growth mechanism is much like that proposed by Hoh and Lehnert ;lo7however, Hurwitz and McWhorter"' take into account not only an azimuthal, but also a radial electric field produced by charge separation. The interaction of these two fields with the applied axial magnetic field drives carriers not only radially as previously noted, but also azimuthally. In the work described above, the unperturbed distribution of nonequilibrium carriers has a radial gradient caused by a finite surface recombination velocity, hence the particle flux is not without divergence. Particles are thus fed from the main distribution into the helix with proper phase, causing growth of the perturbation. Hurwitz and McWhorter pointed out that growth can also be achieved in an equilibrium plasma in which no radial gradient is initially present, if the surface recombination velocity is sufficiently low so that radial and azimuthal flows can pile up or deplete carriers at the surface in the proper phase."'" The motion of the helix is composed of rotational and translational components, although the influence of these motions in producing a rotation of the perturbed carrier density pattern is indistinguishable to an observer at some point along the semiconductor. The translational component is simply the ambipolar drift velocity of the perturbation in the direction of the minority carrier drift. The rotation is caused by diffusion and conduction of carriers across the magnetic field in the thermal equilibrium plasma, and by the radial ambipolar electric field in the nonequilibrium plasma case, When n = p the ambipolar mobility is zero and the motion is purely rotational. If n # p translation dominates. The growth and motion of the helix produce either instabilities or amplification depending on the magnitude of the ambipolar drift velocity. If it is small (n w p), the perturbation grows temporally while essentially fixed in space, producing a nonconvective instability as discussed at the beginning of this section. If the drift is large (n # p), the perturbation is translated along the applied fields, growing as it drifts. This physical model is also presented in precise terms by Hurwitz and McWhorter.'" The threshold dependencies for spatial growth in the small '~~ The magnetic field limit, Table V, are the same as G l i ~ k s m a n obtained. experimentally observed thresholds in Ge are in excellent agreement with theory. Hurwitz and McWhorter clearly proved by phase and velocity measurements that the density perturbation is a helical wave moving in the direction of ambipolar drift, and the theoretically expected exponential "'"Depletion can occur only in a linear theory. Such a theory can adequately describe an amplifier over a considerable operating range (cf. Fig. 48), whereas an oscillator requires a nonlinear theory to describe its properties.

11.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

463

“I

7 - 4 1

,

,

Ij, A

47 67

,

; ;1

,

-6

0

2

4

6

Magnetic Field 6,

10

8

12

(kG)

FIG.48. Growth constant I C ~of the helical density perturbation as a function of magnetic field at several temperatures. The applied field was 50 V/cm, with its frequency adjusted to the threshold value at each temperature. The points are experimental, the solid curves are exact solutions in the low magnetic field limit (not fulfilled we11 at highest fields here), and dashed curves correspond to a simplified approximation (valid only at threshold). (After Hurwitz and McWhorter.’ Is)

increase [exp(x,z)where x i is the growth constant] of wave amplitude in the axial direction was confirmed by measurements of the relative signal strength along the semiconductor bar. Figure 48 shows the theoretical and experimental dependence of I C ~on magnetic field for a given electric field strength at various temperatures. The highest measured value of xi corresponds to an impressive gain of 35 db/cm.

-

b. Nonlinear Properties Several properties of the finite amplitude helix have been investigated although a theory is still lacking, Early experiments indicated that, with constant current, both frequency and amplitude are enhanced slightly beyond the threshold with increasing magnetic field.lo’ More thorough recent studies3* 1 6 , 11’ revealed that frequency is a nonmonotonic function

‘I7

B. Ancker-Johnson, Phys. Rev. 135, A1423 (1964). B. Ancker-Johnson, in “Plasma Effects in Solids” (Proc. 7th Intern. Conf.), p. 165. Dunod, Paris and Academic Press, New York, 1965.

464

BETSY ANCKER-JOHNSON

of B > B,. Different modes of oscillations occur, as illustrated in Fig. 49, with three types of transitions: very abrupt, frequency mixing, and noisy. The abrupt transition can produce, in successive applied current pulses, regular oscillations of two different, nonharmonic frequencies without any change in the operating conditions. Fig. 50 illustrates frequency mixing. In the course of noisy transitions the helical instability can be completely suppressed by a magnetic field several times larger than Bth.The amplitude as a function of magnetic field is complicated, as is also shown in Fig. 49. The response of the electric field strength beyond the threshold is displayed in Fig. 46.

Area =0.81 X

30 -

0

P'p=6700 cm2/V-sec

-

200 .

0.83mm2

= 0.43(aern).'

400

600

800

1000

1200

1400

B (GI FIG.49. The frequency of helical instability oscillations as a function of applied magnetic field for three different plasma densities (as determined by the current) in p-type InSb at 77°K. The numbers which are not underlined indicate the amplitude of oscillations in volts per centimeter. (After Ancker-Johnson.''6* 1

A startling property of the finite amplitude helix is the very large internal magnetic fields it produces. The effect was discovered118 by applying a triangular current wave form to p-type InSb at 77°K. Figure 51 shows a typical voltage response. In a given magnetic field the oscillations start at E,, here 74 V/cm, build up in amplitude as the current increases, and then cease at an E < E , as the current decreases. The hysteresis in electric field is, in this example, 46% of the threshold field. A series"' of such oscillograms yields the instability boundary, that is, the E,B, product at threshold, and the corresponding Ehysfields. An example of such data is reproduced in Fig. 52. The line connecting the observed threshold conditions (indicated by 'la

'I9

B. Ancker-Johnson, Appl. Phys. Letters 3, 104 (1963). B. Ancker-Johnson, Phys. Rev. 134, A1465 (1964).

11.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

465

I=2.OA

12 t / c m thAr-

16.26 MC

t:

b 0 . 4 4 psec

FIG.50. Example of a transfer from one mode of helical instability oscillation to another involving frequency mixing. Each exposure is for three or four successive pulses. (After AnckerJohnson.' ")

circles) defines the instability boundary. It has no characteristic locus since the third controlling parameter, injection level, depends on the properties of the particular contacts. With an applied field of 345 G, oscillations begin in this sample at 164 V/cm. The helix forms and carries current, thus E hys = 34.2v/crn.

2.5 V/lg. div.

1 1 1 I l l IIIIIII IIIIII Ill1 1 1 I 1 I IIIIIIII 1 11 IIIIIII

B=325G

t

= 0.5 psec/lg. div.

FIG.51. An oscillogram showing hysteresis in the threshold electric field strength while the plasma is immersed in a constant magnetic field intensity. (After Ancker-John~on."~)

4 6 6 ’

BETSY ANCKER-JOHNSON

producing additional magnetic field. As E is decreased after an excursion into the nonlinear region, oscillations cease at 124 V/cm, and the internal field vanishes. Thus a hysteresis loop is traced whose shape is not given by these experiments. The magnitude of the hysteresis magnetic field is deduced assuming that the boundary between stability and instability is the same,

3 2 0 k

110 I

I

130

1

150 I

170 I

I

190 ’

E (v/cm) FIG.52. A typical B-E threshold diagram showing the boundary between a stable and an unstable plasma. The circles which define this boundary correspond to the measured E,, at the onset of oscillations and the triangles indicate the measured electric field strengths at cessation (Elh-EhyS).Simplified hysteresis loops corresponding to these thresholds are drawn. (After Ancker-Johnson. ’ 19)

independent of the side from which the boundary is approached. An infinite number of loops can be generated between limits determined by the onset of pinching at high electric fields and the vanishingly small plasma densities at low electric fields. The internal magnetic field has also been approximately determined by direct measurements.”’ A plasma is normally diamagnetic,”’ but negligibly so compared with the huge paramagnetic effect just described. That the helical instability produces a paramagnetic field .is evident from the measurements,’ l 9 the theories,’0”’06’ lZ1 and simple physical reasoning: Since an applied field IZo

A. R. Moore and J. 0. Kessler, Phys. Rev. 132, 1494 (1963). T. Misawa, Japan. J . Appl. Phys. 1 , 67, 130 (1962).

11. PLASMAS

IN SEMICONDUCTORS AND SEMIMETALS

467

FIG.53. B-H loops resulting from the hysteresis in the threshold conditions for the helical instability. (After Ancker-Johnson.'19)

I=3.OA

Eo=128 (v/cm)

B

(GI 152

76

0 t: 4 0.1 I+

p sec

FIG.54. Oscillograms showing the simultaneous recording of B and E as a function of time in a sample of p-type InSb at 77°K. The two minima in E-traces No. 1 to 5 are labeled 1, 1' for E l , etc. (After Ancker-Johnson.'16)

468

BETSY ANCKER-JOHNSON

greater than Bth causes a shift in the equilibrium position of the plasma toward the walls, the magnetic pressure inside the helix must be greater than that outside, therefore, the field produced by the helix adds to the applied B. A set of deduced B-H loops for one sample is shown in Fig. 53. They do not encircle the origin as do familiar types of hysteresis loops. If the applied magnetic field is made a function of time while the applied electric field is held constant, a different effect occurs than that just described when the time dependencies of the fields are reversed. Oscillograms of the voltage response for a constant plasma density as a function of dB/dt are reproduced in Fig. 54. When the plasma is exposed to a small dB/dt, it is stabilized and then destabilized as usual (compare Fig. 46, which gives the field strength response as a function of B in the steady state). At relatively large dB/dt, the plasma exhibits two instability thresholds.

Terminal Current Oscillations Begin \

T =77°K E=, 32 V/cm 0

-0

2

4 6 8 Magnetic Field BO (kG)

10

12

FIG.55. Average sample resistance (n-type Ge) as a function of applied magnetic field. (After Hurwitz and McWhorter."')

Nonlinear effects have also been observed in helical wave arnplifi~ation."~ Since the growth is spatial the amplification saturates first at the detector probes farthest from the input signal. At relatively high temperatures (>47"C) and magnetic fields, the gain could be made large enough to saturate the farthest probes on amplified internal noise. Saturation was accompanied by a marked enhancement of the dc sample resistance and by the emergence of oscillations in the current familiar from the studies on temporally growing waves. In Fig. 55 this resistance is given as a function of B (whose dependence on the growth constant was illustrated in Fig. 48). The conditions at which the resistance begins to increase coincide with saturation at the farthest probe and the onset of oscillations. This behavior indicates that the density in the helix has become comparable to the equilibrium density. The strong bunching in the traveling helix produces a net increase in the resistance.

1 1. PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

469

Hurwitz and M ~ W h o r t e r " ~pointed out that linear theory does not predict the current oscillations observed by so many workers This is true because all the time-dependent quantities in the theory vary as exp(ircz imY - iot), and so the density integrated over a cross section is independent of time. Thus no oscillations in the total resistance can occur. The current oscillations, therefore, must depend on nonlinearity or sample nonuniformity.'2' Hurwitz and McWhorter's results show that oscillations may result from the saturation of stable amplification (spatial growth) in uniform samples, as well as from temporal growth. A recent research note'22 reports that the helical instability can be stabilized by application of a magnetic field emanating from quadrupolar wall mirrors. The intensity of this cusped field need be only a few tens of gauss to cause a marked decrease in bath the amplitude of the oscillations and the electric field strength which supports them. (Compare Fig. 46 which shows that the onset of the instability is accompanied by a large enhancement in electric field strength, as well as the appearance of oscillations.) The lifetimes of the nonequilibrium carriers can be appreciably enhanced in such a magnetic field configuration, which suggests that it is a magnetic trap.'

+

1 5. RECOMBINATION OSCILLATIONS Several authors have reported oscillations which they attribute to recombination at impurity levels. Nad123has described briefly the occurrence of oscillations in p-type Ge compensated with Au. These oscillations are associated with an increase in current and with hysteresis in the conductivity characteristic. They are quenched when nonequilibrium plasma is produced by optical injection. have studied Ge doped with Sb, and compensated with Ryvkin et Cu and radiation defects. The oscillations they observe require a dc voltage and steady illuminatiorr. The optical injection of plasma fills the shadow levels, and the applied voltage induces impurity breakdown. The resulting emptying of the traps causes a sharp rise in current When the optical injection refills the traps sufficiently, the process is repeated The frequency of these nonsinusoidal oscillations increases with illumination intensity and the magnitude of the applied voltage. Holonyak and B e v a c q ~ a observed '~~ oscillations both before and after a

"* Iz3

lZ4

lZ5

B. Ancker-Johnson, Phys. Fluids 7, 1553 (1964). F. J. Nad, Radiotekhn. i Elektron 6, 1775 (1961) [English Transl.: Radio Eng. Electron. (USSR) 6, 1582 (1961)l. S. M. Ryvkin, V. P. Dobrego, B. M. Konovalenko, and I. D. Yaroshetskii, Fiz. Tverd. Tela 4, 1911 (1962) [English Transl.: Soviet Phys.-Solid State 4, 1400 (1963)l. N. Holonyak, Jr., and S. F. Bevacqua, Appl. Phys. Letters 2, 71 (1963); N. Holonyak, Proc. IRE 50,2421 (1962).

470

BETSY ANCKER-JOHNSON

negative resistance region in the conductivity characteristics of n-type Si, Ge, and GaAs compensated with various impurities. The frequencies ranged from very low to - 5 Mc. They attribute the oscillations to “impact ionization (or hole recombination) in waves at electron-filled, deep-lying energy levels.” Some manifestations of the oscillations depend on the type of impurity. No theory has yet been proposed to account for these oscillations. A theory of injection in semiconductors with deep impurity levels has been worked out by Sondaevskii and Stafeev.lz6 The theory explains qualitatively the occurrence of negative resistance : Preferential trapping of either injected holes or electrons results in a variation of the conditions for recombination of these nonequilibrium carriers and, hence, a variation in their lifetimes. At a sufficiently high injection level a sharp increase in lifetime occurs because the density of trapped carriers is constant, equal to the impurity concentration. When this high injection level is reached in the part of the semiconductor nearest the injector of the predominantly trapped carrier species, the conductivity of the semiconductor suddenly increases, resulting in a negative resistance segment in the I-V characteristic. This persists until the entire sample is filled with a density corresponding to the high injection level; then a further increase in current again results in an enhancement in voltage across the sample. Using p-type Ge with Au impurities, Sondaevskii and Stafeev found good qualitative agreement with the theory and also observed oscillations as the negative resistance region was approached. The amplitude increased monotonically with current and, in the negative resistance region, sometimes markedly exceeded the magnitude of the dc supply voltage. It was impossible to plot the I-Vcharacteristic in this range because the voltage measurement included the averaged value of the amplitude of the oscillations. The frequency of oscillation was in the tens of kilocycles range and was unaffected by various load impedances. Current and/or voltage oscillations could be obtained depending only on the magnitude of the load resistance. Although aware of Holonyak and Bevacqua’s’*’ suggestion about the cause of oscillations in semiconductors with deep-level impurities, Sondaevskii and Stafeev prefer the proposal of Smith,lZ7 namely, that the oscillations may be associated with the interaction of carrier flux and acoustic phonons in a region of strong fields. Current oscillations in n-type Ge containing Au acceptors partially compensated with Sb at 0.04 eV have been studied (in the liquid hydrogen temperature range) by Kurova and Kalashnikov.lZ8 These oscillations are V. P. Sondaevskii and V. 1. Stafeev, Fiz. Tverd. Tela 6 , 80 (1964) [English Transl.: Soviet Php-Solid State 6. 63 (1964)l. R. W. Smith, Phys. Rev. Letters 9, 87 (1962). I. A. Kurova and S. G . Kalashnikov, Fiz. Tverd. Tela 5, 3224 (1963) [English Transl.: Soviet Phys.-Solid State 5, 2359 (19641.

1 1.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

471

unique compared to others described in this review in that their threshold is accompanied by a sharp drop in current and rise in voltage. The chemistry of the contacts influences the conductivity characteristics, the threshold current, and the threshold frequency of the instability. However, for all samples and contact alloys investigated (injecting and noninjecting), the oscillation

a

2

a 6 4

2

30

35

40

lo3/ T”K FIG. 56. The temperature dependence of the recombination oscillation periods in four Ge samples. The solders used to make the current contacts were composed of: (1) and (2), 95 % Sn and 5 % Sb; (3) and (4), In. The concentration of impurities x l O I 4 cm-’ was: (I), 32 Sb and 13 Au; (2) and (3), 3.6 Sb and 1.6 Au; (4), 7.2 Sb and 3.4 Au. (After Kurova and Kalashnikov.’’’)

period decreased exponentially as the temperature was enhanced, with an activation energy z 0.04 eV, Fig. 56. No oscillations could be found in n-type Ge containing Sb but no Au, nor in p-type Ge containing Au in which the lower donor level of Au was partially compensated by shallow acceptors. The authors suggest on the basis of this evidence that the oscillations are the result of. a recombination instability involving “fast” and “slow” traps. This model says that a negative resistance occurs because hot electrons are

472

BETSY ANCKER-JOHNSON

more readily trapped at local levels than cooler electrons (as first suggested by Ridley and W a t k i n ~ ” ~ )thus , decreasing the concentration. Since there was no external oscillatory circuit, and the internal reactances of the crystal did not match the observed low frequencies, the occurrence of negative resistance by itself is insufficient to explain the oscillations. However, oscillations of the required frequency may be generated if the crystal contains fast and slow capture centers, so that a periodic variation of electron concentration in the conduction band is associated with periodic fluctuations of electrons between the two types of traps. Bonch-Bruevich and KalashnikovI3O recently presented a theory in support of this model. The assumption that the Au impurity ions are the slow traps is compatible with the temperature dependence of the oscillation period, Fig, 56. The nature of the fast centers is not clear, but they may be surface states. Oscillations in n-type GaAs experiencing 800 to 3000 V/cm have been attributed to negatively charged centers which capture another, energetic e 1 e ~ t r o n . lSuch ~ ~ capture causes an increased electric field strength in a local region where trapping began. This high field region traverses the sample and is followed by another wave. The result is a very low frequency oscillation (< 1 cps). 16. RELAXATION OSCILLATIONS Koenig and Brown’32 have reported oscillations in n-type Ge at liquid helium temperatures. A critical quantity of joule heating which occurs preferentially on the axis of the sample (as far away from the bath as possible) causes a reduction in the impact ionization threshold, and, hence, a plasma is produced there. This results in a negative resistance and high frequency oscillations (typically 1 Mc). These are relaxation oscillations of the circuit including the sample. The heating causes a decrease in the cross section for single-electron capture and so destroys the balance between the latter and impact ionization. High purity samples generally do not oscillate, which is consistent with the dependence of the breakdown electric field strength on impurity concentration as well as temperature. Cardona and R ~ p p e l ’observed ~~ current oscillations in the 5-kc to 5-Mc range in p-type Ge with an injecting point contact at room temperature. The modulation of the current was as high as 80%. Kikuchi and

-

B. K. Ridley and T. B. Watkins, J . Phys. Chem. Solids 22, 155 (1961); Proc. Phys. SOC. (London) 78, 293 (1961). 1 3 0 V. L. Bonch-Bruevich and S. G. Kalashnikov, in “Plasma Effects in Solids” (Proc. 7th Intern. Conf.), p. 193. Dunod, Paris and Academic Press, New York, 1965. 1 3 ’ A. Barraud, Compt. Rend. 256, 3632 (1963). ‘32 S. H. Koenig and R. D. Brown, 111, J . Phys: Chem. Solids 10, 201 (1959). 1 3 3 M. Cardona and W. Ruppel, J . Appl. Phys. 31, 1826 (1960).

129

1 1. PLASMAS IN SEMICONDUCTORS A N D SEMIMETALS

413

Abe134 have studied oscillations in constricted semiconductor samples possessing two different diameters immediately adjacent along the sample length. They have reported many characteristics of this oscillator, some of which are inconsistent, and named it a sogicon (semiconductor oscillation generator by ionization and constriction). Most of these observations appear to be neatly explained by a theory'35 based on the work of Lampert and Rose.I3" According to Steele et a1.135impact ionization occurs at the constriction, producing a copious reservoir of electrons and holes for double injection into the larger portion of the semiconductor. This lowers the voltage necessary to sustain constant current, and therefore induces a negative resistance. Relaxation oscillations result if the transit time of the minority carrier is less than the time required for the establishment of the negative resistance. The voltage at which the transition occurs from Ohm's law to the square law conductivity characteristic is the voltage at which injection is first observed,'36

where L,, is the length of the large part of the sample. Thus, a plasma is generated in the bulk of the sample only when the minority carrier transit time is comparable to or less than the plasma lifetime.'36" This transit time determines the maximum frequency of the relaxation oscillations resulting from the negative resistance. The actual frequency also depends, of course, on the RC time constant of the oscillator circuit. This maximum frequency is approximately equal to the reciprocal of twice the transit time and is in good agreement with the measurements of Steele et Kosman and M ~ r a v s k i i ' have ~ ~ observed relaxation oscillations in Si and Ge diodes, which they attribute to fast surface states. Such states can form a depletion layer in which impact ionization occurs causing shunting of the p-n junction. This surface breakdown, which occurs at an electric field strength less than that required for bulk breakdown, produces an 134

M. Kikuchi and Y. Abe, J . Phys. SOC. Japan 17, 881, 1268 (1962), and earlier papers cited

13'

M. C. Steele, K. Ando, and M. A. Lampert, J . Phys. SOC.Japan 17, 1729 (1962); 18, 591

therein. (1963). M. A. Lampert and A. Rose, Phys. Rev. 121, 26 (1961). 136aThevoltage at which injection begins in the plasma studies on InSb (see Fig. 2 for an example) does not agree well with the predicted value, Eq. (81). See Refs. 4 and 5 for measurements of the injected plasma lifetime in InSb. 13' M. S. Kosman and B. S. Muravskii, Fiz. Tuerd. Telu 3, 2504 (1961) [English Trunsl.: Soviet Phys.-Solid State 3, 1822 (1961)l; B. S. Muravskii, Radio Tekhn. i Elektron 8, 162 (1963) [English Transl.: Radio Eng. Electron. (USSR) 8, 143 (1963)l; B. S. Muravskii, Fiz. Tverd. Tela 4, 2485 (1962) [English Trunsl.: Soviet Phys.-Solid State 4, 1820 (1%3)]. 136

474

BETSY ANCKER-JOHNSON

increase in the conductivity and a redistribution of the voltage. As the base voltage increases to its maximum, the depletion layer voltage approaches zero because of the sharp rise in conductivity. The fast states, therefore, return to equilibrium and begin to retrap the nonequilibrium carriers. As the latter disappear the relaxation becomes complete and the diode voltage returns to its initial magnitude; hence a new avalanche of carriers is produced. In agreement with this model Kosman and Muravskii observe a sharp increase in current and sawtooth oscillations associated with a critical voltage. At a higher voltage the oscillations appear to be essentially sinusoidal and undamped with a frequency 5 x lo4 cps.

-

17. OTHEROBSERVED INSTABILITIES

a. I n the Absence of Applied Magnetic Fields A plethora of oscillatory phenomena have been reported which as yet are not fully explained. They are briefly reviewed here. Ancker-Johnson86 presented experimental evidence for the existence of extremely large amplitude oscillations in p-type InSb at 77"K, Fig. 57. These oscillations exhibited growth and decay in time, Fig. 58. The peak amplitude often exceeded the applied voltage amplitude as the oscillogram shows. A characteristic of this phenomena is the extreme constancy of the frequency in a given sample (in the present example 30.8 k 0.2Mc) when other parameters, such as current, are varied over a wide range.'38 Sayer'40 noticed oscillations in n-type Ge with large contacts. They are associated with current saturation at 77°K and have a threshold at relatively low field strengths (-70 V/cm). Apparently they are associated with the nature of the contacts. These oscillations have some characteristics in common with Kurova and Kalashnikov's'28 oscillations described above ; their frequency range, however, is higher, namely 100 to 500 kc.

-

b. I n the Presence of Applied Magnetic Fields

Many instabilities have been observed in semiconductor plasmas subjected to large magnetic fields. The best known of these is the helical instability (Section 14). An important new observation by Larrabee and Glicksman recently reported13' that K. Ando [Japan. J . Appl. Phys. 3, 757 (1964)l has apparently reproduced these oscillations. By minimizing injection so as to produce avalanche breakdown followed by copious injection, he bas obtained as much as a 90% drop in voltage and therefore a very large negative resistance. Hence, this effect may be a relaxation oscillation. M. Glicksman, in "Plasma Effects in Solids" (Proc. 7th Intern. Conf.), p. 149. Dunod, Paris and Academic Press, New York, 1965. 140 M. Sayer, Solid State Electron. 5, 409 (1962).

13*

1 1.

PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

475

I (A) FIG.57. The full amplitude at maximum of “giant-constant frequency” oscillations as a function of current in p-type InSb at 77°K. (After Ancker-Johnson.86)

FIG.58. Oscillogram of the voltage response (above) and current (below) as a function of time (0.2 psec per large division). I = 5 A per large division and V = 50 V per large division.

H i ~ i n b o t h e m ’ ~involves ’ microwave emission from n- and p-type InSb at 77°K subjected to large electric and magnetic fields. This microwave energy emission requires the presence of an impact ionization p l a ~ m a ,l ’a ~Fig. 14’

R. D. Larrabee and W. A. Hicinbothem, Jr., in “Plasma Effects in Solids” (Proc. 7th

Intern. Conf.), p. 181. Dunod, Paris and Academic Press, New York, 1965. I4laA. G. Chynoweth and S. J. Buchsbaum have observed microwave emission at electric field strengths an order of magnitude below that required for breakdown (private communication).

476

BETSY ANCKER-JOHNSON

59, and is usually accompanied by oscillations in the resistance. (These oscillations may be related to T o d a ’ ~ ’ ~ work; ’ described below.) The threshold for emission is higher than the threshold for oscillations, and any orientation of the magnetic field with respect to the direction of current flow produces both effects (at different thresholds). A minimum of 3 kG is required for observation of either effect. The emission is broad band, covering the spectrum from - 3 to 44 Gc.

FIG.59. The average conductivity characteristics of a 10 x 10 mil conductive post of n-type InSb, mounted in a rectangular wave guide, with a magnetic field of 6 kG, parallel, at 45” and normal to its axis. The lower line intersecting each curve represents the threshold conditions for current oscillations and the upper one represents the threshold conditions for microwave emission at 15 Gc. (After Larrabee and Hi~inbothem.’~’)

Steele’43 has investigated the applicability of Bok and Nozieres’66 traveling-wave helicon theory (discussed in Section 10) to these observations. He assumed both a saturated drift velocity (consistent with the high fields shown by Fig. 59 to be required to produce the microwave emission) 14’ 143

M. Toda. Japan. J. Appl. Phys. 2, 461 (1963). M. C. Steele, in “Plasma Effects in Solids” (Proc. 7th Intern. Conf.), p. 189. Dunod, Paris and Academic Press, New York, 1965.

1 1. PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

477

and the existence of pinching. The high velocity and density resulting from these assumptions raise the frequency given by the drifted helicon theory from <0.1 Gc to -2 Gc. No justification for the assumption of pinching is given, however, nor do the conductivity curves, Fig. 59, exhibit the usual enhanced resistivity associated with pinching; see Figs. 31 and 32. As these figures show, the pinch onset is postponed to much higher currents in the large longitudinal magnetic fields required in both the microwave emission experiments and this explanation of the effect. It is well known that a transverse magnetic field also inhibits pinching and causes loss of plasma, as the data in Fig. 59 verify.2,8 2 , 84 The application of transverse magnetic fields has produced a number of other oscillatory effects. Toda,I4’ using the microwave probe technique described in Section 12, found plasma density waves in n-type InSb. With the sample a t 210°K he observed damped oscillations in both the voltage and the density, and found their frequencies essentially proportional to the magnetic field intensity and independent of current. His spatial measurements showed that a running wave moved in the direction of hole flow with the wave front at approximately 45” to the current flow. In larger B,, such that w,z z 1, the oscillations are undamped and become random at still higher fields. With the sample at 77°K he found evidence of standing waves. Nakashima et ~ 1 . reported l ~ ~ oscillations in n-type Ge at 77°K in the 2-30 Mc range in B , from 400 to 1600 G ; their cause is unknown. Oscillations in p-type InSb at 77”K, at electric field strengths greater than the impact ionization magnitude, and in n-type InSb before breakdown (no nonequilibrium plasma present) have been observed by Ikoma et al. 14’ Avalanche breakdown quenches these oscillations in n-type InSb. Obviously quite different mechanisms are required to explain these effects in InSb subjected to large transverse magnetic fields. Komatsubara et al. 146 have done experiments similar to Koenig and Brown’32 (Section 16), using very highly compensated p-type Ge and adding a transverse magnetic field. Their oscillations are difficult to understand because several possible sources are involved simultaneously, including relaxation and recombination. Extensive studies by Glicksman and Hi~inbothem’~’ on velocity saturation in n-type InSb in the presence of transverse magnetic fields may help 144

145

14‘ 14’

S. Nakashima and Y . Noguchi, Japan. 1. Appl. Phys. 2, 307 (1963); S. Nakashima and Y. Miyai, J . Phys. Soc. Japan 18, 12 (1963). H. Ikoma. I. Kuro, and K. Hataya, J . Phys. SOC.Japan 19, 238 (1964); H. Ikoma, J . Phys. SOC.Japan 19, 419 (1964). K. Komatsubara, N. Takasugi, and H. Kurono, Japan. J . Appl. Phys. 3, 72 (1964). M. Glicksman and W. A. Hicinbothem, Jr., in “Plasma EKects in Solids” (Proc. 7th Intern. Conf.), p. 137. Dunod, Paris and Academic Press, New York, 1965.

478

BETSY ANCKER-JOHNSON

clarify some of these effects. The Gurevi~h-Ioffe’~~ theoretical work discussed in Section 14 may be directly applicable to these experiments in B,, since their assumed gradient in the density is produced directly by a B,. 18. TWO-STREAM INSTABILITIES

Two-stream instabilities are high frequency instabilities associated with coherent excitation of plasma oscillations (Section 4). The acoustic mode (see Section 4) of collective motion in a current-carrying plasma becomes unstable when the relative drift velocity of the two kinds of carriers exceeds the phase velocity of the plasma oscillations. Pines and S ~ h r i e f f e r ’have ~~ used the Boltzmann equation for single-particle distribution functions, with interactions among plasma constituents taken into account by means of a self-consistent field. They assume the plasma response is characterized by two distinct time scales, so that the drift velocities of charge carriers change adiabatically with respect to the time characteristics of collective plasma effects (note the similarities in assumptions made in one of the pinch theoriess8). Pines and S ~ h r i e f f e r ’show ~ ~ that a dense plasma possessing ~ 0 97 1,~ where w is the frequency of the instability and T + are the hole and electron scattering lifetimes, will produce a two-stream instability in the long wavelength limit when the relative drift velocity uD is approximately equal to the electron thermal velocity given by

More specifically, if the electron and hole temperatures are equal, the instability threshold is achieved when OD =

0.93~-[1+ ( m - / ~ + ) ” ~ ] .

-

(83)

For InSb this threshold is 1 . 2 ~ _If. the ratio of the temperatures T-/T+ is large, the critical velocity decreases toward (rn-/rn+)1’2v- or, for InSb, toward - 0 . 2 7 ~ . Pines and Schrieffer estimate that growth of the instability in InSb is possible if wp+z+3 15. Glicksman and H i ~ i n b o t h e mhave ~ ~ measured Hall drift velocities as high as 1 x 1OScm/sec in n-type InSb. They estimate that the electron thermal velocity is 8 x lo7 cm/sec, hence even for equal electron and hole temperatures a two-stream instability could be produced in this plasma. They also estimate that in their experiment the wp+z+product is within a factor of two of the growth condition. Tentatively, they attribute an observed sharp decrease in drift velocity with increasing electric field strength

-

D. Pines and J. R. Schrieffer, Phys. Rev. 124, 1387 (1961).

1 1 . PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

479

to the onset of a two-stream instability. Later work14’ discusses other possible explanations of the experimental observations. McWhorter and May’49 are seeking evidence for the propagation of the acoustic plasma wave in bismuth. Harrison’ 5 0 has developed a theory of the two-stream instability for degenerate plasmas at zero temperature. He also uses a self-consistent field technique, with neglect of scattering, to observe that an acoustic mode of oscillations results if the Fermi velocity of one kind of carrier exceeds a given magnitude and is substantially larger than the velocity of the other carrier type. The frequencies of plasma excitation result from the modification of the plasma frequency of the low drift velocity holes by the dielectric properties of the large drift velocity electrons. Thus the electrons form a polarizable medium which mediates the hole interaction. Instability growth occurs when the average electron drift velocity is greater than the phase velocity of an incipient wave. The electrons arrive at the potential minimum of the wave only by transitions to states of lower velocity, thus transferring energy into the collective motion. In contrast to the two-stream instability theories already described, Harrison’ 5 0 investigated a collision-dominated plasma by employing the moment equations in which average velocities rather than a velocity distribution are assumed. Using this two-fluid model he considers the phase difference between electron and hole velocities in the long wavelength limit for the acoustic mode. In this limit of short scattering times, and for an intrinsic semiconductor, the velocities of the density perturbations are nearly in or out of phase with the density perturbations induced by the applied E. This phase difference produces a mechanism for growth which operates if diffusion and electron-hole recombination do not dominate. The requirements for growth are a short dielectric relaxation time, which implies a high initial density of carriers, a short transit time, and long carrier lifetimes. Tosima and Hirota”’ have also described a two-stream instability which is assisted by collisions. The same two-fluid model was used to examine a low frequency mode of plasma oscillation. They found that with sufficiently short collision times the carriers’ inertia causes phase differences between electron and hole perturbation velocities and the density perturbation. These phase differences can instigate growth of the perturbation if damping caused by diffusion and recombination are overcome. Temporal growth (instability) and spatial growth (amplification) of transverse waves, in contrast to the longitudinal waves just described, have been studied by Bok and Nozieres66 and by Misawa.” A plasma with 14’

150

A. L. McWhorter and W. G . May, IBM J . Rex Develop. 8, 285 (1964). M. J. Harrison, J . Phys. Chem. Solids 23, 1079 (1962). S. Tosima and R. Hirota, J . Appl. Phys. 34, 2993 (1963).

480

BETSY ANCKER-JOHNSON

n = p is assumed to be drifting parallel to a magnetic field. These are just the conditions for Alfve'n wave propagation in Doppler-shifted fields, as discussed in Section 11. The appropriate dispersion relation is solved for Im(o) = 0 to find the threshold electric and magnetic field strengths. The threshold electric field decreases with increasing electron and hole mobilities, and has a minimum for an applied magnetic field inversely proportional to the product of the electron and hole mobilities. Amplification is predicted by Bok and Nozieres66 when the electron drift velocity exceeds the phase velocity of the wave. The threshold electric field varies in the limit of small magnetic fields as B-'/', and B3/' in the high field limit. Bok and Nozieres66 predict amplification of frequencies up to 600 Mc in impact ionization plasmas of InSb at 77°K with B = 1500G and E = 200V/cm. Misawa70 claims amplification is not possible because this spatially growing wave is evanescent.

VII. Conclusions Over one-half of the papers published to date on plasma effects in semiconductors and semimetals have appeared during 1963 and about the first two-thirds of 1964. The growth is greater than exponential. The interest in plasmas in solids is primarily motivated by two desires. The first (chronologically) is to learn more about the structure of solids. The work in Part 111 was undertaken on equilibrium plasmas in quest of energy band parameters, and was only incidentally associated with plasma effects as it was being done. Since their propagations have been elegantly demonstrated, interest in helicon and AlfvCn waves has been shifted to the interactions of these waves with other excitations in solids (Section IV). The other primary motivation for research on plasmas in semiconductors and semimetals is the wish to learn more about the plasma state of matter. Since nonequilibrium plasmas are more independent of the properties of the solid in which they are produced than equilibrium plasmas, they are the natural choice for investigating plasma effects. Agreement between experiment and theory on equilibrium plasmas is, for the most part, very satisfying. In contrast many nonequilibrium plasma experiments are difficult to interpret, and theory, for the most part, lags far behind. Exceptions are the theories on the pinch effect and on temporal and spatial growth of helical density waves (Sections 12 and 14). Spatial growth of helical density waves also occurs in equilibrium plasmas, but nonequilibrium plasmas have the virtue of possessing much more promise of practical application. Quantum effects in all three types of plasmas in solids are sure to generate more interest in the near future.

1 1. PLASMAS IN SEMICONDUCTORS AND SEMIMETALS

481

NOTATION a A b0

B B Z

Be C

d D

e

E

f h H I I, {PI

J k

k meff "0

n no Po

R RH S

m*m,

Radius of a plasma Absorption coefficient Impact parameter Applied magnetic field in Gauss Axial magnetic field Azimuthal magnetic field Velocity of light Path length Diffusion constant Electronic charge Electric field strength Frequency Planck's constant Internal magnetic field in oersteds Current Critical current for pinching Plasma current Current density Boltzmann's constant Extinction coefficient Effective mass Mass of free electron Density of plasma Equilibrium density of electrons Equilibrium density of holes Reflection coefficient Hall coefficient Sound velocity

Temperature Average electron velocity Thermal velocity Alfvbn velocity Helicon velocity Angle between magnetic field and direction of propagation Mode index Phase shift Energy Dielectric constant of medium in absence of plasma Index of refraction Angle of rotation of plane of polarization Wave vector Debye length Electron mobility Hole mobility Collision frequency Plasma frequency Ellipticity Mass density Conductivity Time between collisions Nonequilibrium plasma lifetime Plasmon lifetime Radian cyclotron frequency Radian plasma frequency

Original manuscript submitted October 23,1964; and in final form, January 15, 1965.

This Page Intentionally Left Blank

Author Index Numbers in parentheses are footnote numbers and are inserted t o enable the reader to locate those cross references where the author’s name does not appear a t the point of reference in the text.

A Abe, Y., 473 Abeles, B., 265 Abraham, M. S., 150, 151 Adams, E. N., 160, 161, 175, 176, 181, 183, 184, 185, 190, 192, 193, 194, 195, 196, 198, 205, 218, 219(5a), 222(23), 223(5a), 224, 241(25), 246, 253(5a), 289, 290(1), 297, 362 Adler, S. L., 390 Aigrain, P., 128,416 Ainslie, N., 152 Akramov, G., 431 Alekseev, A. I., 113, 122(31) Alexander, F. B., 150 Aleksandrova, M. V., 261, 263(56) Alfvtn, H., 417, 419 Allis, W. P., 417 Allred, W. P., 271, 327, 340(11) Arnirkhanov, Kh. I., 198, 359, 360, 364, 365, 366, 367, 368 Ancker-Johnson, B., 382, 383(2), 384, 385, 386, 435(2), 436, 437(2), 440(2), 441, 442, 444(2), 445, 446(86), 447, 449, 450, 451, 452, 453(94), 460, 461, 463(3), 464, 465, 466,467,469(5), 474,475,477(2) Ando, K., 413,474 Andreev, V. V., 176 Anselrn, A. I., 205, 207(5e) AntonEik, E., 58 Appel, J., 176, 348 Arai, T., 391, 392 Arakawa, E. T., 394 Argyres, P. N., 175, 176, 177(53, 55, 58, 61), 180(53), 183(55), 184(55),205,218,222(23), 223(5f), 253(5f), 297 Askerov, B. M.,205, 207(5e) Astrom, E., 419 Attard, A. E., 12 Aukerman, L. W., 283

Austin, B. J., 58, 59(36) Azaroff, L. V., 12

B Babiskin, J., 160 Bagguley, D. M. S., 10, 76, 89, 93, 95, 145 Bardeen, J., 103, 241 Barraud, A,, 472 Barrie, R., 175, 283, 286(52), 341, 342, 344, 347 Bartelink, D. J., 419, 429, 430 Bashirov, R. I., 198, 359, 360(47), 364(47), 365(60), 366, 367(60), 368(60) Bassani, F., 41, 46, 52, 53(ll), 57, 58, 59(11), 66, 67(24, 34), 68, 69, 72(23), 73, 74, 190 Bate, R. T., l97,198,206,224(6b),227,268,270, 271(18), 322, 326, 327, 337, 340(11), 357, 361 Beattie, A. R., 386 Becker, W. M., 5, 185, 187, 189, 190(69), 191, 192, 272, 278, 279, 280, 281, 282, 371, 372 Beer,A. C.,67, 160,197, 198(81), 199,204,206, 224(6b), 227, 267, 268, 270, 271(18), 326, 337, 351, 357, 361(43), 364, 372 Bell, D. J., 32 Bell, J. C., 270, 271(18), 326 Bellova, N. A,, 142 Bemski, G., 243, 412,413 Bennett, W. H., 433, 442 Berg, M. F., 385, 386(4), 469(5) Berman, R., 249 Bers, A,, 417 Bevacqua, S. F., 152,469,470 Bielan, C. V., 152 Birkhoff, R. D, 394 Black, J. F., 152 Blatt, F., 214 Blount, E. I., 76, 99, 165, I71

484

AUTHOR INDEX

Bogoliubov, N. N., 105, 126 Bohm, D., 387, 388(8), 389, 394 Bok, J., 429, 430, 431, 455, 476, 479, 480 Bolger, D. E., 18 Bonch-Bruevich, V. L., 105, 107, 108(16), 109, 110, 113(11), 118, 119, 120, 121(29), 122(11, 29), 129, 136(67), 140(11), 141 (66, 67), 142(43),388,472 Borofia, H., 13 Bouckaert, L. P., 24, 29, 77 Bowers, R., 77, 100(9), 173,419 Boyle, W. S., 160, 408, 409 Brailsford, A. D., 408, 409(40), 417, 422(49), 424(49) Braunstein, R., 81, 150, 151(17), 156, 267 Brickell, W. S., 17 Brinkman, W., 73 Broom, R. F., 10, 270, 283,284, 286, 358, 364 Brown, M. A. C. S., 311, 313 Brown, R. D., 472,477 Brueckner, K. A,, 122 Brust, D., 68, 73 Buchsbaum, S. J., 414, 415(44), 416, 417, 418(51), 422(48),431, 432, 475 Burdiyan, I. I., 154, 155 Burgess, R. E., 452 Burstein, E., 214 Busch, G. A., 190,450 Button, K. J., 83, 214 Bychkov, Yu. A,, 169, 182

C Callaway, J., 46, 67 Callen, H. B., 209, 210(12) Canel, E., 394 Cardona, M., 4, 6, 14, 67, 77, 96(13), 164, 165, 472 Carlson, R. O., 153 Carruthers, P., 239 Casella, R. C., 97 Casimir, H. B. G., 249 Celli, V., 52, 57, 58, 59, 67(34) Cetlin, B. B., 417,422(49), 424(49) Chambers, R. G., 160, 345,419 Champness, C. H., 274, 276,277, 342, 351,365 366 Chasmar, R. P., 92,277,365,366 Chen, C. Y., 287

Chih-Ch’ao, Lien, 353 Chynoweth, A. G., 142, 435, 436, 440(78), 475 Clark, M. Jr., 433 Cohen, M. H., 57, 160, 171 Cohen, R. W., 382,383(2), 384(2), 385(2),435(2), 436(2), 440(2), 444(2), 477(2) Conklin, J. B., 81, 100(16) Conwell, E. M., 120 Coulson, C. A,, 52, 54(29), 55, 63(29) Courant, R., 164 Creuzburg, M., 394 Cuevas, M., 103, 142 Cuff, K. F., 14, 74, 81, lao(l5) Cunnell, F. A,, 10, 283 Cunningham, R. W., 386 Cusano, D. A., 153

D Danilychev, V. A., 310, 311 Davydov, B., 176, 181 de Haas, W. J., 159 Deigen, M. F., 110 Dickey, D. H., 406, 407(37) Diesel, T. J., 224 Dimigen, H., 392(21), 393, 394(21) Dingle, R. B., 168 Dobrego, V. P., 469 Doring, W., 29 Drabble, J. R., 270, 319 Dresden, M., 210 Dresselhaus, G., 24, 25, 70, 75, 77, 82, 88, 89, 90,93,94,96,267,410,411,412

Dresselhaus, M. S., 160, 406, 407137) Drickamer, H. G., 283 Drummond, J. E., 388, 391(10), 437, 441, 444(81), 445, 446, 447, 449, 450, 451, 452, 453(94) Duga, J. J., 268, 373 Dunlap, W. C., 10 Durkan, J., 297 Dyson, F. J., 102

E Ecker, G., 446 Edelman, V. S., 418, 419(53), 426(53) Edmond, J. T., 10, 283 Edwards, A. L., 283

AUTHOR INDEX

485

Edwards, S. F., 124 Galt, J. K., 408,409(40),416,417,422(48, 49), Effer, D, 17,18 424 Bfros, A. L., 200,364,368 Gatos, H. C., 271,327,34q12) Ehrenreich, H., 5,9,10(9),14(20), 15,77,96(13), Gauthe, B., 393 126,127,144,145(3), 148(3),150,234,237, Geballe, T. H., 199,201,205,206,207,209(8),

249,266,276(7),279(7), 283,373,390,392, 393 Elliot, R. J., 29 Ellis, B., 9,145 Emel’yanenko, 0. V, 286,373,374(78) Evans, J. A., 150, 151(18), 153 Evans, W., 17(46),18 Ewald, A. W., 186,188,191

212,214(4), 224(8), 230,232(10),236,238 (17, 33), 245(10), 254(17, 33), 258(4c), 259(4c), 261(4c), 271,276 Gebbie, H., 214 Geiger, J. 393 Cell-Mann, M., 122 Gershenzon, M., 5 Gillett, C. M., 150,151(18),153, 154(28),156 Ginster, J., 351 Glasko, V. B., 107,108 F Glauberman, A. E., 109 Glicksman, M., 144,270, 278, 280(17), 283, Fakidov, I. G, 224 323,370,3’12,382,383(2), 384(2),385(2), Falicov, L. M, 160 435(2), 436(2), 437(2),440(2,79),441(79), Falkovskii, L. A., 418,419(53),426(53) 442,444(2), 79,88), 447,456,459,462, Fan, H. Y.,5, 10, 89,185, 187,189,190(69), 466(104),474,477(2),478(88), 479(179) 191,237,278,279,280,281,282,283,371, Gobeli, G. W., 89 372(67), 373,403,404 Goering, H.L., 154,155(30), 364 Feher, G., 83 Gombas, P.,58 Feldman, W. L., 142,273,275(35),281(35) Goodman, B., 74 Fenner, G. E.,6,152,153 Goodwin, D. W., 307 Ferrell, R.A., 390,394 Gordeev, G. V.,460 Firsov, Yu.A,, 161,182,183,199,200,201,205, Greenaway, D. L., 6,67 214(4), 255,258(4a), 261,263,360 Greene, R.F., 270 Fitchen, D. B., 152 Gremillet, J., 420 Flood, W. F., 92 Grimes, C. C., 431,432 Folberth, 0. G., 149,150, 151 Groves, S., 190 Foner, S., 92 Gubankov, V.N.,313 Foster, E. L.,249 Gubanov, A. I., 52,54(29),55(29),63(29) Fray, S. J., 437 Gurevich, L. I?., 176, 183,239,241(41), 252, Fredkin, D. R.,165 364,457,458(109a-b-c-d),459,478 Frederikse, H. P.R., 160, 185,186,198,216, 228. 243(22), 257(22),274,275,276,277, Gurevich, V. L., 161,182, 199,200,201,205, 214(4), 255,258(4a), 261(4d),263(4d), 360 284,292,294,3$2,357,359,364,367,368 Guseva, G. I., 361 Friedel, J., 63 Frisch, H. L., 128,206,224(6c) Fritsch, D., 373,374 H Fritzsche, H., 103,142, 284,285,352,353 Frohlich, H., 390 Haga, E., 237,249 Fulton, T. A., 152 Hall, G. G., 54,63 Furdyna, J. K., 401,420,421,422 Hall, R.N.,127 Ham, F. S., 46 Hambleton, K., 13 G Harman, T. C., 271,327,340(12), 351,364,406, Galavanov, V. V., 8, 292 407(37) Galitskii, V.M., 118 Harrison, M. J., 479

486

AUTHOR INDEX

Harrison, V. A,, 58, 59(36) Hartl, W., 394 Hasegawa, H., 176, 180(51),246, 253(52), 312 Hashitsume, N., 169, 176, 180(51), 181(32), 182(32), 183(32), 185(32), 194(32), 195(32), 196(32) Haslett, J. C., 197, 198, 224, 303 Hass, M., 13 Hataya, K., 477 Hatoyama, G . M., 284 Hattori, T., 440, 453, 454(98) Hebborn, J. E., 297 Hebell, L. C., 397, 398, 399, 400, 414, 415(44) Heine, V., 57,58, 59(36), 70 Hellmann, H., 58 Hensel, J. C., 83, 243 Henvis, B. W., 13, 408, 409(39), 416(39) Herickhoff, R. J., 394 Herman, F., 14, 46, 67, 68, 73, 74, 81, 100(15), 144 Herring, C., 28, 29, 46, 55, 199, 206, 210, 211(15), 212, 213, 215, 220, 224(6a), 225(6a), 230(15), 232(33), 236, 237, 238(15, 17, 21, 33), 239, 242, 254, 265, 271, 330 Hicinbothem, W. A,, 447, 475, 476, 477, 478, 479( 147) Hieronymus, H., 327, 328, 330(10), 343 Hilbert, D., 164 Hilsum, C., 13, 15, 144, 145(5), 147(5), 149(5), 341, 342,344,347,450 Himes, R. C., 154, 155(30) Hinkley, E. D., 186, 188, 191 Hirota, R., 430, 453, 454(99), 479 Hodby, J. W., 145, 156 Hoh, F. C., 451, 462 Holonyak, N. Jr., 152, 469, 470 Holstein, T. D., 176, 181, 183, 184(52), 185, 190(52), 192(52), 193, 194, 195, 196, 205, 219(5a), 223(5a), 224, 246,253(5a) 456, 459, 466(106) Holter, 0., Hopfield, J. J., 94 Horing, N. J., 176 Hosler, W. R., 185, 186, 198, 274, 275, 276(38), 277, 284(38), 292, 352, 357, 359, 364, 367, 368 Howard, P. E., 312 Howard, W., 16, 17 Howland, L. P., 53 Hrostowski, H. J., 92, 276 Hsu, F. S. L., 160

Hull, G . W., 236 Hulme, K. F., 271 Hurwitz, C. E., 461, 462, 463, 468, 469

I Ichikawa, Y. H., 457 Ikoma, H., 477 Ioffe, I. V., 457,458,459,478 Isenberg, I., 270 Ivanov, I:L., 455, 463(101)

J Jan, J. P., 208 Jerde, D. A,, 443 Johnson, F. A,, 437 Johnson, L. E., 81, 100(16) Johnson, L. F., 5 Jones, B. K., 419 Jones, R. H., 307,437

K Kadomtsev, B. B., 456, 466(105) Kahn, A. H., 160, 182, 206, 216, 243(22), 257 (22), 294 Kalashnikov, S. G., 470, 471, 472, 474 Kampschulte, G., 18 Kanai, Y., 224, 420, 421, 422(59) Kane, E. O., 70, 76, 81, 86, 99(6), 129, 138(71), 142(70), 243, 244, 267, 273(10), 348, 350, 438 Kaner, E. A., 418, 419, 431(54) Karasik, V. R., 224 Keldysh, L. V., 126, 129, 136, 459 Kern, R., 190 Kesamanly, F. P., 8 Kessler, F. R., 327, 340(13) Kessler, J . O., 466 Keyes, R. J., 92 Keyes, R. W., 160, 161, 198, 223, 224(25), 237, 241(25), 289, 290(1), 292, 297, 362 Khaikin, M. S., 418, 419(53), 426 Khvoshchev, A. N., 437, 438, 440, 447(82), 451, 477(82) Kikuchi, M., 473 Kimmitt, M. F., 311, 313 Kimura, H., 237, 249 Kinch, M. A,, 307 Kip, A . F., 75, 90(3), 410, 411(41), 412(41)

487

AUTHOR INDEX

Kirsch, J., 427, 428 Kirzhnitz, D. A., 113, 122(32), 130(32) Kittel, C., 75, 90(3), 110, 410, 411(41), 412(41) Kjeldaas, T., 172, 173 Klauder, J. R., 120 Kleiner, W. H., 10 Kleinman, L., 52, 57(28), 58(28), 67, 68, 73(45), 74 Klemperer, O., 391 Klinger, M. I., 176, 195, 205, 214(4), 255 Klotyn’sh, E. E., 8 Knight, J. R., 17(46), 18 Knox, R. S., 41, 53(11), 59(11), 74 Kobayashi, A,, 416 Koenig, S. H., 472,477 Kogan, Sh. M., 307, 308, 313 Kohn, W., 46, 76, 107, 109, 110, 120, 172, 173, 243, 244 Koike, T., 460 Kolm, H. H., 92 Kolodziejczak, J., 361 Kolomiets, B. T., 155 Komatsubara, K., 477 Konovalenko, B. M., 469 Konstantinov, 0. V., 416 Kopaev, Yu. V., 126 Korenblit, L. L., 139 Kortum, R. L., 14, 74, 81, lOO(15) Kosevich, A. M., 171, 176, 177(60) Kosman, M. S., 473 Koster, G. F., 24, 31(4), 39, 52(10), 64 Kovalev, A. N., 142 Kravchenko, A. F., 5,283, 373 Ku, S.-M., 152 Kubo,R., 118,169, 176, 180, 181, 182, 183, 185, 194, 195, 196,246,253(52) Kuglin, C. D., 14,74, 81, lOO(15) Kuhrt, F., 320 Kunz, C., 393 Kunzler, J. E., 160, 212, 230, 238(17, 33), 254(17, 33), 271 Kuro, I., 477 Kurono, H., 477 Kurova, 1. A,, 470,471,474 L Lagunova, T. S., 285, 286,366(62), 367,368(62) Lampert, M. A,, 473 Landau, L. D., 107, 159, 162,354

Langenberg, D. N., 431 Langer, J. S., 125, 141(55) Langmuir, I., 386 Larach, S., 8 Lark-Horovitz, K., 284, 285, 352, 353 Larrabee, R. D., 455,475,476 Lavine, M. C., 271, 327, 340(12) Lax, B., 76, 77, 83, 85(7),92, lOO(7, 9), 164, 203, 214, 244, 282, 395, 396, 402,403, 405, 406, 407(33, 36), 408(36),410, 411 Lax, M., 118, 128 Legendy, C., 419 Lehnert, B., 457,462 Leifer, H. N., 10 Leman, G., 63 Levinger, B. W., 120 Libchaber, A,, 305,420,427 Liebman, W. K., 18 Lifshits, T. M., 307, 313 Lifshitz, E. M., 107 Lifshitz, 1. M., 102, 160, 171, 172, 176, 177(59, 60) Lin, L., 190 Lippmann, H . J., 320 Listvin, V. N., 313 Liu, L., 52, 72 Lloyd, S. P., 128 Logan, R. A., 142 Long, D., 144, 145(4), 146(4), 148(4) Love, W. F., 197, 198,224, 303 Lowdin, P., 61, 77, 79 Lubowski, S. J., 152 Luttinger, J. M., 76, 110, 115, 120, 138(35),169, 243, 244

M Mackintosh, I. M., 353 Madelung, O., 144, 145(5a)

Maeda, H., 83 Mahan, G. D., 94 Mal’tsev, Yu. V., 8 Mamaev, S., 8 Maradudin, A. A,, 102 March, N. H., 297 Marechal, Y., 439, 443 Mashovets, D. V., 261, 263 Matossi, F., 10 Mattheiss, L. F., 46 MatyaS, M., 282, 372

488

AUTHOR INDEX

Maurer, J., 427 Mavroides, J. G., 160, 203, 282, 395, 396,403, 406,407(37), 410,41 l(27) May, W. G., 479 Mazur, P., 102 Mead, C. A,, 5,6, 152 Meiboom, S., 265 Meijer. H. J. G., 192, 242 Mel’nik, P. G., 307, 313 Mermin, N. D., 394 Merritt, F. R., 417, 422(49), 424(49) Messiah, A,, 74 Metzger, H. J., 327, 340(13) Michel, R. E., 412, 413, 414(43) Migdal, A. B., 115, 118, 140 Miller, J. F., 154, 155 Miller, P. B., 427, 428 Miller, R. C., 5 Miller, S. C., 298 Mina, R. T., 418, 419(53), 426(53) Mironov, A. G., 107, 120 Misawa, T., 430, 457, 460, 466, 469(121), 479, 480 Mitchell, M., 110 Miyai, Y., 477 Miyake, S. J., 169, 176, 181(32), 182(32), 183 (32), 185(32), 190(54a), 194(32), 195(32), 196(32) Miyazawa, H., 83 Montroll, E. W., 102 Moore, A. R.,466 Mooser, E., 63 Morin, F. J., 276 Morita, A., 63 Morrison, J. A., 206, 224(6c) Moss, T. S., 9, 77, 100(9), 145 Mott, N. F., 103 Miiller, A,, 327, 330, 338, 340(14), 341(14), 357 Muller, E. K., 143, 147(1) Mullin, J. B., 17, 271 Muravskii, B. S., 473 Murray, A. A., 435,436,440(78) Muzhdaba, V. M., 258 MacDonald, D. K. C., 214 McWhorter, A. L., 461,462,463,468,469,479

N

Nad, F. J., 469 Nachtrieb, N. H., 442

Nakashima, S., 477 Nanney, C., 430 Nasledov, D. N., 8, 285, 286, 292, 353, 366(62), 367, 368(62), 373, 374 Nedlin, G . M., 176, 239, 241(41), 252 Nedospasov, A. V., 456,466(105) Nelson, R. F., 5 Nii, R., 416 Noguchi, Y., 477 Nozieres, P., 113, 122(33), 387, 388(8), 394(8), 429,430,476,479,480 Nran’yan, A. A,, 52, 54(29), 55(29), 63(29) Numata, H., 416

0 Obraztsov, Yu. N., 205,207,213,232(5~) Okamoto, F., 460 Olekhnovich, N. M., 12 Onsager, L., 166 Osipov, B. D., 310, 311, 437, 438, 440, 447(82), 451, 477(82) Oswald, F., 149, 155(15)

P Palik, E. D., 76, 77, 89(8), 93(8), 95(8), 100(9), 401, 402, 408, 409, 416(39) Parfen’ev, R. V., 200, 201(89), 205, 214(4), 258. 261(4d), 263{4d, 56), 360 Parrnenter, R. H., 24,25,77,82, 88(12),89(12), 93(12), 94(12), 96(12), 120, 144, 267 Patton, V. A,, 145 Paul, B., 340

Paul, W., 5, 16, 17, 190, 283 Pauling, L., 13 Pearson, G . L., 103, 265, 269(1), 273, 275(35), 281(35), 371 Pearson, W. B., 63 Peierls, R. E., 161, 170, 210 Pekar, S. I., 109, 110, 126 Pelzer, H., 390 Perel’, V. I., 416 Peschanskii, V. G., 160 Peskett, G. D., 237 Phelan, R. J., 303 Philipp, H. R., 4, 390, 392, 393 Philips, J. C., 14, 52, 57(27, 28), 58, 67, 68, 73, 74,76, 128

489

AUTHOR INDEX Picus, G, 214 Pidgeon, C. R., 401 Pilkuhn, M., 152 Piller, H., 145 Pines, D., 109,387,388(8), 389,394,478 Pogorelov, A, 172 Polder, D., 192, 242 Pomeranchuk, I., 176, 181 Powell, C. J., 392(20), 393 Powlus, R. A., 437,440(79), 441(79), 444(79) Pratt, G. W. Jr., 81, lOO(16) Priestley, M. G., 160 Putley, E. H., 290, 292, 293, 295,298, 299, 303, 305(5), 311, 313, 353, 362 Puri, S. M., 199, 201, 205, 206, 207, 209(8), 212(10), 214(4), 224(8), 225(8b), 228(8b), 232(10), 235(42), 239, 240(42), 241(42), 244(42), 245(10,42), 246(42), 253(42), 258 (k), 259(4c), 261(4c)

Q Quarrington, J., 128 Quelle, F., 67, 68(43), 70 Quinn, J. J., 431

R Racette, 1. H., 127 Raether, H., 394 Ramdas, A. K., 5, 10, 191, 278, 279, 280, 281, 282, 371, 372(67) Rashba, E. I., 97 Rtdei, L. B., 52, 54(29), 55(29), 63(29) Reid, F. J, 10 Reitz, J, 46 Richards, J. L., 143, 147(1) Ridley, B. K., 472 Ritchie, R. H., 390 Roberts, V., 128 Robins, J. L., 393 Robinson, J. E., 74, 190, 191, 281 Robinson, M.L. A., 10, 76, 89(8), 93(8), 95(8), 145 Rodrigueq S., 190, 191,281,431 Rollin, B. V., 237, 307 Romanov, Yu. A., 390 Rose, A,, 473 Rose, D. J., 433 Rose, F., 419

Rose-Innes, A. C., 144, 145(5), 147(5), 149(5), 450 Rosenblum, B., 412, 413,414(43) Rosi, F. D., 150, 151(17) Ross, I. M., 283, 286(52) Rostoker, N., 46 Roth, L. M., 76, 83, 85, 100, 164, 165, 166, 173(29), 176, 177(53), 180(53), 205, 214, 223(5f), 244,253(5f), 400 Rozman, R., 119, 142(43) Ruppel, W., 472 Rupprecht, H., 152, 271, 277, 321, 328(4), 329, 330(4), 341, 342, 359, 360(4) Russell, B. R., 270 Ryvkin, S. M., 455, 463(101), 469

S Sagar, A., 5, 371 Sakamoto, J., 54 Samoilovich, A. G., 139 Sasaki, W., 284, 363,364 Sasakura, Y., 457 Sayer, M., 474 Schiff, L. I., 86, 99(19) Schillinger, W., 453, 454, 455 Schmidt, H., 120 Schmidt-Tiedemann, K. J., 403,404(35), 405 Schonwald, H., 345, 346, 347, 348, 349, 350, 351,356, 357, 375 Schrieffer, J. R., 74, 387, 388(8), 394(8), 478 Schrieffer, R. S., 74 Seitz, F., 46, 75, 269 Shalyf S. S., 200, 201(89), 205, 214(4), 258, 261(4d), 263(56,4d),285,360,364,367, 368 Sham, L. J., 58, 59(36) Shibuya, M., 265, 372 Sheka, V. I., 97 Shepherd, J. P. G., 391 Shockley, W., 75, 80, 204, 241 Shoenberg, D., 160, 170 Shrader, R. E., 8 Shtivel’man, K. Ya., 91 Shubnikov, L., 159 Silin, V. P., 205, 207(5d), 232(5d) Simon, F. E., 249 Sirota, N. N., 12 SkBcha, J., 282, 372 Skillman, S., 67, 68, 73 Skobov, V. G., 182,418,419,431(54)

490

AUTHOR INDEX

Sladek, R. J., 16, 160, 161(17), 185, 187, 189, I90(67), 191, 197, 198, 224, 237, 290, 292, 293, 295(3), 296, 298(3), 299(3), 357, 358, 362, 366(65), 368, 369 Slater, J. C., 39, 46, 52(10), 64, 68, 73, 110 Slykhouse, T. E., 283 Smith, G. E., 414, 415, 424, 425, 426(61) Smith, R. W., 470 Smith, S. D., 77, 100(9), 401 Smoluchowski, S., 24, 29, 77 Sondaevskii, V. P., 470 Sosnowski, L., 361 Spitzer, L., Jr. 433 Spitzer, W. G., 5,6, 144, 152,237,283,403,404 Stern, E. A., 390, 429 Stern, F., 10 Stafeev, V. I., 470 Stark, R. W., 160 Stasyuk, J. V., 109 Steele, M. C., 160, 382, 383,435,440, 453, 454, 455,473,476 Steigmeier, E., 450 Stelzer, E. L., 283 Stepanova, G. I., 102 Stephen, M. J., 395 Stevens, K. W. H., 353 Stirn, R. J., 272 Stocker, C. T., 8 Stocker, D., 52, 54(29), 55(29), 63(29) Stradling, R. A,, 10, 76, 89(8), 93(8), 95(8), 145 Stratton, R., 92 Straughan, B. W., 17 Strauss, A. J., 271, 327, 340(12), 371, 406, 407 (37) Sturge, M. D., 5, 145

Suhl, H., 265, 269(1), 371 Suris, R. A,, 124 Suzuki, K., 83 Swanson, J. A., 21 1 Szymanska, W., 351 T

Taft, E. A., 4 Takahashi, V., 54 Taksugi, N., 477 Takeno, S., 102 Tanenbaum, M., 273,275(35), 281 Tauc, J., 14 Taylor, K. W., 77, lOO(9)

Teitler, S., 76, 77, 89(8), 93(8), 95(8), 100(9), 401,402, 408,409(39), 416(39) Theriault, J. P., 10 Tinkham, M., 70 Titeica, S., 175 Toda, M., 430, 438, 440(84), 476, 477(84) Tomchuk, P. M., 389 Tonks, L., 386 Tosirna, S., 453, 454(99), 460, 479 Toyozawa, Y., 266, 284,285(6) 354 Tsidil’kovskii, I . M., 361 Tufte, 0. N., 283 Twose, W. D., 103 Tyablikov, S. V., 105, 109(11), 110, 113(11), 118, 121(29), 122(11,29), 126, 140(11),388

U Uemura, Y., 416 Ukhanov, Yu. I., 8 Upadhyaya, U. N., 173 V van Alphen, P. M., 159 Veilex, R., 305, 420, 455 Vinogradova, K. I., 292 Vladimirov, V. V., 109 Vogt, E., 242 Vonsovski, S. V., 105, 109 Vosko, S. H., 125, 141(55) Vystavkin, A. N., 307, 3 13 W Wagini, H., 350, 361 Wallis, R. F., 76, 77, 89(8), 93(8), 95(8), 100(9), 401, 402, 408, 409(39), 416(39) Walton, A. K., 9, 145 Wannier, G. H., 165, 173 Ward, J. C., 115, 138(35) Warner, J., 8, 9 Watkins, T. B., 472 Weber, R., 271, 321, 328(4), 329(4), 330(4), 341(4), 342(4), 359(4), 360(4), 366, 367, 368 Wei, W. F., 224 Weisz, G.. 160 Weiss, G. H., 102 Weiss, H., 149, 150, 151, 271, 272, 275, 277, 319, 321, 322, 324, 325(5), 326, 327, 328(4), 329(4), 330(4, 10). 331(5), 332, 335, 336 339, 340, 341(4), 342(4), 343, 351, 359(4), 360(4), 365, 366, 367, 368, 373, 374

491

AUTHOR INDEX

Welker, H., 149, 150, 151, 319, 323, 324 Wheatley, G. H., 92, 276 Whelan, J. M., 283 Whitaker, J., 18 Wick, R. F., 270 Wigner, E., 24, 28, 29, 46, 77 Wilhelm, M., 327, 330, 338, 339, 340(14), 341 (14), 357 Willardson,R. K., 10,67,197,198(81),224,268, 283, 351, 357, 361(43), 373 Williams, G. A., 424, 425, 426(61) Wilson, A. H., 170, 195, 267 Wolfe, R., 270, 319 Wolff, P. A., 120, 397, 398, 399 Woodall, J. M., 18 Woodruff, T. O., 46, 55(17) Woods, J. F., 18,281 Woolley, J. C., 8, 9, 143, 147(1), 150, 151, 153, 154(28), 156 Wright, G. B., 77, 100(9),402,405,406,407(33, 37)

Y Yafet, Y., 77, 100(9), 161, 173, 244, 289, 290(1), 362

Yager, W. A,, 417, 422(49), 424(49) Y amada, T., 460 Yamanouchi, C., 284, 363, 364 Yaroshetskii, 1. D., 469 Yep, T. O., 185, 190, 192 Yoshimine, M., 52, 66, 67(24), 68, 69

2

Zakiev, Yu. E., 198, 359, 360(47), 364(47), 365(60), 366, 367(60), 368(60) Zallen, R., 5 Zavadskii, E. A,, 224 Zehler, V., 29 Zerbst, M., 13 Zil’berman, G. E., 176 Ziman, J. M., 179,249 Zoeller, O., 446 Zotova, N. V., 285, 286, 366, 367, 368 Zvyagin, I. P., 120, 139(51) Zubarev, D. N., 113, 118 Zwerdling, S., 10, 76, 83, 85(7), 92, 100(7), 164, 214,244,402 Zyryanov, P. S., 205, 207(5d), 232(5d)

Subject Index A Absorption, see Free carrier absorption, Magnetoabsorption Activation energy, see Ionization energy Alfvkn waves, 416-433 dispersion relation, 417-420 extraordinary, 418,427 ordinary, 418,427 phase velocity, 418 Alloys, see Mixed crystals Aluminum antimonide band structure, 6, 7, 96, 98, 146-148, 273 effective charge, 13, 16 effective masses, 9, 10 energy gap, 5-7, 148 mobility of carriers, 15, 16 spin-orbit splitting, 14 Aluminum arsenide band structure, 6, 7 energy gap, 5-7 Aluminum phosphide band structure, 7, 69 energy gap, 5-7 Augmented plane wave method, 46 B Band conduction, 103, 120 impurity, 103, 108, 120, see also Impurity levels conduction, 266,284-287 tailing, 102, 133-138 Band approximation, 22ff. Band calculations, 21-74 Band structure, 4-8, 143-156, see also specific listings of compounds, Energy spectrum calculations, 21-74 effect of alloying, 68, 143-1 56 of pressure, 68 on plasmas, 381 from Faraday rotation, 401 from magnetoplasma effects, 406-416 from magnetoresistance, 265-287 warped surfaces, 267,282, 283

Birefringence, 403405 stress, 404,405 deformation potential constant, 404 Bismuth Alfvh waves, 419,422427 Doppler-shifted resonance, 428 hybrid resonances, 414-416 pinch effect, 453455 reflection coefficient, 409 reflectivity, 399,400 Te-doped, helicon waves, 423,424 Block functions, 31ff., 76 symmetrized combinations, 31ff. Bohr-Sommerfeld quantization condition, 166 Bond energy, 55 Born-Oppenheimer approximation, 22ff. Brooks-Herring theory, 297, see also Scattering, ionized impurity C

Charge carrier, see also Crystal momentum, Density of states concept, 108-116 degeneracy, 107, 129, 139 density fluctuations, see Inhomogeneities energy spectrum, 109 freeze-out, 161,289ff. screening 106-108, 112-115, 126, 141, 297, 298 Charge carrier absorption, see Free carrier absorption Cohesive energy, 63 Collision broadening, see Level broadening Collisional damping AlfvCn waves, 419,420 helicon waves, 419, 420 plasma, 389, 398 Collisions, see also Scattering randomizing, plasma, 381 Compensated plasma, 381 Complex conductivity, 118, 119 Conduction states, 53, 73, 74 impurity band, 266, 362, 363 negative magnetoresistance, 284-287, 353, 354

492

493

SUBJECT INDEX

Conductivity, 175 enhancement, plasma, 382-386 gradient, 271, 324340 magnetoconductivity, 180, 181 divergence, 181,257 mixed, see Two-band conduction Corbino disk, 319-323 Core states, 53, 60, 61, 68, 7&74 Correlation effects, 22, 74 screening, 74 Crystal Hamiltonian, 39ff. potential, 39ff., 68, 73, 74 Crystal momentum, 104-106, see also Charge carrier Current lines, magnetic field, 318, 323, 324 Cyclotron frequency, 162-164, 355 complex bands, 165-167, 172 Cyclotron resonance, 397 tilted orbit. 416

D Debye length, 380, 383 Debye potential, 131 Debye shielding distance, see Debye length Degeneracies bands, 80, 86 Kramers’ doublet, 86, 88 Kramers’ type, 86 charge carrier, 139, see also Charge carrier time reversal, 28, 29, 77 de Haas-van Alphen effect, 169-172, see also Magnetic susceptibility Density matrix, 110, 111, 176-181 Density of states, 108, 116-118, 121, 127, 128-142, 162-164,255-257,294,295 electron-electron interaction, 115, 119-123 electron-impurity interaction, 123-126, 128-131 electron-phonon interaction, 126128 magnetic field, 162-164, 255-258, 294, 295 collision broadening, 181, 182, 185, 253, 254, 258 complex bands, 165-167,258 divergence, 163,164, 255-258 polaron effects, 126, 127 tail, 102, 133-138 Dielectric constant, 107, 290, 291, 307, 380, 390,417419

Dingle temperature, 171, 188, 191 Direct transition, 4-8, 148 Divergence density of states, 163, 164,255-258 magnetoconductivity, 181, 257 Doppler-shifted phenomena, 427430 Alfvtn waves, 429 cyclotron resonance, 427 helicon waves, 429 polarization rotation, 429, 430 Double refraction, see Birefringence

E Effective charge, 10-15 Callen, 16 Szigeti, 13 Effective mass approximation, 110, 121 Effective masses, 8-10, 82, 85, 89-93, 95, 114, 164, 165 density of states, 126 from Faraday rotation, 401 from magnetoplasma effects, 406416 from Voigt effect, 402 optical, 126 Electric breakdown, see Impact ionization, Injection Electron-electron interactions, see also Density of states, Scattering Gurevich-Firsov oscillations, 200, 205, 214, 255-264 Ellipticity, 408, 409 Energy gap, see also Band structure, specific listing of compounds relation to cohesive energy, 63 to ionicity, 63 Energy spectrum, 108-115, 126, see also Band structure dispersion, 106, 113-115, 117, 118, 122-127 Energy surfaces, see also Band structure extrema, from magnetoresistance, 265-287 extremal loops, 97 nonparabolic bands, 87-95 symmetry relationships, 270 toroidal, 97 warped, 267,213, 282,283 Equipotential lines, magnetic field, 318 Extinction coefficient, 391

494

SUBJECT INDEX

F Fabry-Parot dimensional resonance, helicon waves, 420427 Faraday rotation, 400-402,407 Fermi energy, see Fermi surface Fermi level, see Fermi surface Fermisurface, 114,115,117,120,127,138-140, 294, 295 cross-sectional area, 172 magnetic field, 171 Field plate, 323, 324 Free carrier absorption, 307-31 1

G g-Factor, 162-1 65,294, 364 Gallium antimonide band structure, 5-7, 83, 96, 146-148, 272, 273, 370-372 density of states mass, 279 effective charge, 13, 16 effective masses, 8-10, 188, 190, 191, 281 heavy hole, 282 light hole, 282 energy gap, 5-7, 148 magnetoresistance, 278-283, 371,372 mobility of carriers, IS, 16 oscillatory magnetoresistance, 185-1 90 piezoresistance, 283, 371 Seitz coefficients, 280, 282, 371, 372 spin-orbit splitting, 14 subsidiary band conduction, 191, 272, 279281, 370-372 Gallium arsenide band structure, 4-7, 62, 83, 96, 146-148, 272 effective charge, 13, 16 effective masses, 8-10 energy gap, 5-7, 148 magnetoresistance, 283, 372-375 mobility of carriers, 16, 17 negative magnetoresistance, 286, 287, 375 special growth techniques, 17, 18 spin-orbit splitting, 14 subsidiary bands, 283 Gallium phosphide band structure, 5-7,96, 146-148,273 effective charge, 13, 16 effective masses, 9 energy gap, 5-7, 148 mobility of carriers, 15, 16

spin-orbit splitting, 14 Galvanomagnetic effects, see Hall effect, Magnetoresistance, Seitz coefficients Germanium band structure, 6 plasma effects helical instabilities, 455, 456 oscillations, recombination, 469-472 relaxation, 472474 quantum region magnetoresistance, 223-232 thermoelectric effects, 223-232 Green’s function, 111-115, 124, 126, 128, 130, 131, 140 Grey tin band structure, 6, 190, 191 effective mass, 188 oscillatory magnetoresistance, 186-188, 190 Groups double group, 25-28 irreducible representation, 24ff. point group &, 23ff. simple group, 24-28 Gurevich-Firsov oscillations, 199-202, 205, 214. 255-264

H Hall angle, 3 15-3 18, 324 Hall effect, 175 plasma region, 382-386 Hartree-Fock equations, 22ff. Hamiltonian, 76, 99, 100 Hartree Hamiltonian, 76, 77, 99, 100 Helical instabilities, plasmas, 440, 455469 oscillistor, 456, 460 Helicon waves, 416-433 dispersion relation, 417420 oscillations, 305 phase velocity, 418 phonon interaction, 431433

Holes heavy (slow), 86-95, 146-148,273,276, 351, 352 density, 350, 375, 376 magnetoplasma resonance, 4 12-414 mobility, InSb, 348, 350 light (fast), 86-95, 146148, 191, 273, 276, 351, 352 density, 350, 375, 376

SUBJECT INDEX

magnetoplasma resonance, 412414 mobility, InSb, 348, 350 pressure effects, light-hole density, 375, 376 Hot carriers InSb, 299-313,435453 plasmas, 381ff., 433ff., 455ff. Hybrid resonances, 414416

I Impact ionization, 301-305 plasmas, 381-386,435440,456 microwave emission, 475 Indium antimonide band structure, 7, 76, 77, 83, 89, 93, 95, 96, 100, 146-148,273 effective charge, 13, 16 effective masses, 8-10, 188, 190 electron scattering, 237, 245-249 ellipticity, 409 energy gap, 5-7, 148 Faraday rotation, 400 freeze-out, magnetic field, 292-297 Gurevich-Firsov oscillations, 201, 202, 205, 214, 255-264 helicon waves, 420 hot carriers, 299-313 impact ionization, 301-305 inhornogeneities, 271, 275,325-340 growth direction, 328,329 growth rate, 328, 341 inclusions, 339,340 magnetoreflectivity, 397 magnetoresistance, 272-276, 319-364 pressure, 375, 376 mobility of carriers, 16 heavy hole, 276,348,350-352 light hole, 276, 348, 35CL352 negative magnetoresistance, 284, 285, 352354 nonparabolic bands, see Nonparabolic bands oscillations, electric, 303-305 plasma, pinch, 440-453 oscillatory magnetoresistance, 185-190, 201,202, 358-364 phonon scattering, 237, 238, 245-255 plasma effects, 383416 edge splitting, 405, 406 helical instabilities, 455467

495

magnetoplasma resonance, 41 1 recombination radiation, 437439 pressure effects, light-hole density, 375, 376 quantum limit magnetoresistance, 196-199 quantum region magnetoresistance, 196199,224, 262-264 spin-orbit splitting, 14 thermoelectric power, 233 magnetic field, 232-238, 259, 260 phonon drag increment, 235, 236, 247253 two-band conduction, 341-349, 362 Voigt shift, 401 Indium arsenide band structure, 7, 83,96, 146-148, 272 effective charge, 13, 16 effective masses, 8-10, 188, 190, 365 energy gap, 5-7, 148 magnetoresistance, 277, 365-370 oscillatory, 185-190, 366, 369, 370 negative, 285 mobility of carriers, 16, 364 special growth techniques, 17 spin-orbit splitting, 14 Indium phosphide band structure, 7, 83,96, 146-148 effective charge, 13, 16 effective masses, 8,9 energy gap, 5-7, 148 magnetoresistance, 278, 370 mobility of carriers, 15, 16 spin-orbit splitting, 14 Impurity band conduction, 362,363 negative magnetoresistance, 266, 284-287, 353,354, 375 Impurity levels, 102-104, 108, see also Band, impurity broadening, 103, 108,292 density of states, 162 ionization energy, 290-299 magnetic breakdown, 160 magnetic field, 160 Indirect transition, 4-8, 148 Infrared sensors, materials, 145 tnhomogeneities, 199, 205, 206, 208, 224-228, 270,271, 324-341 detection, 327-341 InSb crystals, 271,275, 325-340 growth direction, 328, 329

496

SUBJECT INDEX

growth rate, 328, 329 inclusions, 339,340 magnetoresistance, 199, 205, 206, 208, 224228 thermomagnetic effects, 199, 205, 206, 208, 224227 Injection lasers, materials, 145, 150, 152 Injection, 381, 386, 435, 437,456 contacts, plasmas, 381 impact ionization, plasmas, 381-386, 435440 optical, plasma, 381, 386 lonicjty, 63, 64, seealso Effective charge polarity, 55 Ionization energy, magnetic field, 290-299 lrreducible representations, 29ff.

K k . p Method, 70,75-100 k .p representation, 78-83 linear k terms, 93-96 three-band model, 87-91 two-band model, 91-93 with spin-orbit terms at k = 0, 86-96 without spin-orbit terms at k = 0, 83-86 Kelvin relation, 209 Kerr magnetooptic effect, 407409 Kohn-Rostoker variational method, 46 Kubo formula. 180

L Landau damping, plasmas, 389,394 Landau levels, 163, 197, 243, 255-261, 289, 294,295, 354,355,361-364, 397-399 Lattice character diamond, 29 “empty,” 60, 64-66 zinc-blende, 23, 29 Lead telluride helicon waves, 420, 421 magnetoabsorption, 416 Level broadening, 167-169, 191, 200, 253, 254, 258, see also Dingle temperature, Impurity levels collision, 167-169, 191, 253, 254, 258 divergence removal, 253,254 phonon dispersion, 258 nonparabolic band, 258

Linear combination atomic orbitals, 52-54 bond orbitals, 54, 55, 63 Linear k terms, 93-96, 146 Lorentz force, 268, 315 Lowdin perturbation method, 61, 79

M Madelung constant, zinc-blende lattice, 54 Magnetic field, see also Hall effect, Magnetoabsorption, Magnetoreflection, Magnetoresistance, Magnetoresistivity, Thermomagnetic effects carrier freeze-out, 161, 289ff. InSb, 289-299 cyclotron frequency, 162, 172 complex bands, 165-167, 172 density of states, 162-164,255-257,294,295 collision broadening, 181, 182, 185, 253, 254, 258 complex hands, 165-167,258 Fermi energy, I7 I hydrogenic impurity, 289-299 impurity banding, 160 impurity ionization energy, 290-299 Landau levels, 163, 197, 243, 255-261, 289, 294,295, 354,355,361-364, 397-399 level broadening, 167-169, 191, 200 Dingle temperature, 171, 188, 191 orbits, 162-167 plasmons, 394 pinch effect, seepinch effect population changes, 284 quantum effects, 159-202,203-264 quantum limit, 161, 173, 174, 192-199, 230, 294,361-364, 399 relaxation time, 179, 183 spin-ordering, 284, 285 Magnetic susceptibility, 169-174, see also de Haas-van Alphen effect quantum limit, 161, 173, 174 Magnetoabsorption, 395416 quantum effects, 397402 quantum limit, 399 Magnetoconductivity, 180, 181 divergence, 181, 257 Magnetooptic effects, 214, 395416, see also Magnetoabsorption, Magnetoplasma effect, Magnetoreflection oscillations, 214

SUBJECT INDEX

497

InAs-GaAs, 150, 151 Magnetoplasma effect InAs-InP, 149, 150 reflection, 406-410 InSb-GaSb, 153, 154 resonance, 410-416 effective masses shift of cyclotron resonance, 410 InAs-InP, 155 Magnetoreflection, 395416 quantum effects, 397402 InSb-GaSb, 156 spin-orbit splitting quantum limit, 399 Magnetoresistance, 175, 265-287, 315ff., see GaAs-Gap, 156 Ge-Si, 156 also Shubnikov-de Haas effect, specific Mobility, see also specific listings of comlistings of compounds anisotropy pounds polar mobility, 15-17 inhomogeneity, 327,329 longitudinal effect, 333 Gurevich-Firsov oscillations, 199-202, 205, N 214,255-264 inhomogeneity effect, see lnhomogeneities, Nernst effect, 206,207 Surface effects photo, 313 negative, 266, 284287, 326, 321, 352-354, Nonequilibrium plasma, 381-386 357 Nonohmic effects, see Hot carriers, Impact quantum region, 183-192,369,370 ionization, Electric breakdown Ge, 223-232 Nonparabolic bands InSb, 262-264 GaSb, 190 longitudinal, 183-190, 358-364 Hamiltonian, 76 transverse, 183-190,358-364 InSb, 77,92-96, 100,235,276, 361 quantum limit, 192-199 k p calculations, 87-98, 100 sample geometry, see Sample geometry quantum effects, 242-244,397-399 Magnetoresistance coefficients, see Seitz coefficients Magnetoresistivity 0 Gurevich-Firsov oscillations, 199-202, 205, 255-264 Onsager relations, 175, 209, see also Kelvin quantum effects relation longitudinal, 177-179, 184 Optical absorption, 128 transverse, 179-181, 184, 185 Oscillations, see also de Haas-van quantum limit, 192-199 Alphen effect, Shubnikov-de Haas Mass density effect, Plasma oscillations electric field, 303-305 Alfvkn waves, 418 quantum oscillations, 426 magnetic field Mass operator, 112-114, 116, 121-127, 140 density of states, 160, 162-169 Matrix elements, 30ff., 37-52 Gurevich-Firsov effect, 199-202, 205, exchange integrals, 59 214,255-264 overlap integrals, 39ff., 37-52 magnetooptic effects, 214 Mixed conductivity, see Two-band conduction magnetoresistance, 355, 358-364, 369, Mixed crystals 370 band structure p1asma s A1As-GaAs, 7,8 helical instability, 440, 455469 GaAs-Gap, 7, 8, 151-153 recombination, 469472 Gap-InP, 7, 8 relaxation, 472474 GaSb-AISb, 154, 155 Orthogonalized plane wave approach, 22ff. Ge-Si, 144, 145 application, 46ff., 5 5 4 1 , 64-70

.

498

SUBJECT INDEX

P Peltier coefficient, 207, 208 phonon drag, see Thermoelectric power Perturbation approach, 59-61 k .p method, 79, 80 Piezoelectric phenomena, 142 Phase velocity Alfven waves, 41 8 helicon waves, 418 Phonon drag, thermoelectric power, 205-208 boundary effects, 236 extremely high fields, 252 InSb, 234-238,247-253 theory, 238-245 Phonon relaxation time, 215-223 Photoconductivity, submillimeter, 305-3 12 Pinch effect, 382, 383 oscillations, 4 4 0 4 5 3 recombination radiation, 4 3 7 4 3 9 semiconductors, 4 3 3 4 5 3 semimetals, 4 5 3 4 5 5 theories, 441447 thermal, 4 4 7 4 5 3 Plane waves, symmetrized combinations, 32ff. Plasma, 379481, see also Injection, Impact ionization amplification, 462,468 band structure sensitivity, 381 compensated, 381 definitions, 379-386 density gradients, 458 helical instability, 440, 4 5 5 4 6 9 nonequilibrium, 381-386,480 randomizing collisions, 381 oscillations, recombination, 4 6 9 4 7 2 relaxation, 4 7 2 4 7 4 stability, 459 two-stream instabilities, 4 7 8 4 8 0 uncompensated, 381 Plasma cutoff, see Plasma edge Plasma, drifted, see also Two-stream instabilities amplification, 430 Plasma edge, 396, 398,402,403,405-408 splitting, 403, 405408 linear, 407 quadratic, 407,408 Plasma frequency, 380 Plasma oscillations, 386-395 Plasma temperature, 380

Plasmon, 387-395 collective behavior, 388 collisional damping, 389 conduction electron, 386-391 dispersion, 388 energy, 387, 392, 393 frequencies calculated, 392, 393 effective, 393, 394 measured, 392, 393 individual particle behavior, 388 Landau damping, 389,394 lifetime, 391, 394 magnetic field, 394, 395 photon radiation, 394 surface, 390 valence electron, 386-394 Poisson summation, 164 Pressure, magnetoresistance, InSb, 375, 376 Pseudopotential approximation, 52,57-59,76

Q Quantum effects, see also Magnetic field, Magnetic susceptibility, Quantum galvanomagneticeffects, Thermomagnetic effects magnetoabsorption, 397402 magnetoreflection, 397402 quantum limit, 161, 173, 174, 192-199, 213, 230,294, 361-364, 399 Quantum galvanomagnetic effects, 174-202, 230, 255-263, 354-364, 368-370, see also Shubnikov-de Haas effect

R Random phase approximation, 74 Rayleigh interference patterns, helicon waves, 421,422,424 Recombination oscillations, plasma, 469472 Recombination radiation, 142 Reflection coefficient, 395-397 Reflection minimum, 396400,402 Reflectivity, 395400,402-412 determination effective charge, 11-13 Refractive index, 391,395 Relaxation oscillations, plasma, 4 7 2 4 7 4 Relaxation time, seealso Scattering electronic InSb, 363

SUBJECT INDEX magnetic field, 179, 183, 222 quantum limit, 193 from Faraday rotation, 401 from magnetoplasma reflection, 406 phonon, 215-223,236-238,245,250-254 Relativistic effects, 22, 74, 81 Darwin term, 74 mass-velocity correction, 74 spin-orbit coupling, 74 Response time, photoconductive process, 30831 I Responsivity, 306-310

S

Seitz coefficients, 269, 270 GaSb, 280, 282, 371, 372 Shielding, 380 Debye length, 380,383 Shubnikov-de Haas effect, 183-192, see also Oscillations subsidiary minima, 191, 192,280, 281 Silicon band structure, 6, 69 plasmon frequency, 392, 393 Silicon carbide, band structure, 66 Solid state plasmas, 381ff. Spin-orbit interaction, 26-29,70-72,74,81,82 Spin-orbit splitting, 14,70-72, 146-148 mixed crystal, 156 Spin splitting, InSb, 233, 235, 364, 370 Spins, localized, see Scattering Subsidiary band minima GaSb, 191,272,279-281,370-372 Grey tin, 191 Surface effects magnetoresistance, 208, 224, 227, 228, 270, 271,284, 352, 353 thermoelectric power, 208, 224, 227,228 Surface plasmons, 390, 393 Surface polarization, plasmon loss, 390 Symmetries, see also Symmetrized functions diamond and zinc-blende, 83-98 [lo01 and [ 111 J directions, 96-98 r point; k = 0, 83-96 tight-binding wave functions, 84, 85 use of, 82 Symmetrized functions, 30ff. Symmetry relationships, Seitz coefficients, 270

Sample geometry, 270 Hall effect, 270 magnetoresistance, 270, 319-324 Scattering (electron), see also Mobility, Scattering, specific listing of compounds acoustic phonon (deformation potential), 119, 192-197, 222, 234, 237-242, 245-249 Born approximation, 177, 178 electron-electron interactions, GurevichFirsov oscillations, 200, 205, 214, 255264 electron-hole, plasma, 437 inelastic, 182, 183 ionized impurity (elastic), 177,192-197,222, 234,246, 268,297, 300, 373,389 non-Born scattering, 182 optical phonon Gurevich-Firsov oscillations, 199-202, 205,214,255-264 magnetoresistance, 361 phonon, 177 T piezoelectric, 192-196, 237-242, 245-247 plasmon, 389, 390 Thermal pinch effect, 447453 polar mobility, 15-17,237,245-249 Thermoelectric power, 206, 208 spins, localized, negative magnetoelectronic, 207 resistance, 284, 285 high magnetic fields, 210-214 Scattering (phonon), 215-223, 237, 238, 245magnetic field 255 germanium, 223-232 boundary, 221,245,247,248 InSb, 232-238,259,260 impurity, 217-221 phonon drag, 207 phonon-phonon, 220, 221, 238, 245, 250high magnetic fields, 210, 214-223 254 Thermomagnetic effects Screening, see Charge carrier, Shielding quantum region, 160, 199,203-264 Seebeck coefficient, 206,208, see also Thermogermanium, 223-232 electric Dower. Thermomametic effects - ~ ~ - Three-band ~ ~ conduction, 349, 350 v

~~

499

500

SUBJECT INDEX

Tight-binding approach, 22ff. application, 52-55,61-64 Time reversal, 25-29 Herring’s test, 28 Kramers’ theorem, 29 Tunnel diode, excess current, 142 Two-band conduction, 341-349, 362, 371 Two-center integrals, 39ff. Two-stream instabilities, 478480

U Uncompensated plasma, 381

V Valence states, 53, 54, 73, 74 Voigt effect, 401,402

W Wigner-Seitz cellular method, 46 X

X-Ray diffraction, determination effective charge, 12, 13

Z Zinc-blende lattice, see Lattice character

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