Semiconductors and Semimetals: Transport Phenomena v. 10

SEMICONDUCTORS AND SEMIMETALS Edited by R. K . WILLARDSON COMMCO AMERICAN INCORPORATED ELECTRONIC MATERIALS DIVISION SPOKANE, WASHINGTON

ALBERT C. BEER BATTELLE MEMORIAL INSTITUTE COLUMBUS LABORATORIES COLUMBUS. OHIO

VOLUME 10 Transport Phenomena

ACADEMIC PRESS New York San Francisco London A Subsidiary of Harcourt Brace Jovanovich, Publishers

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Library of Congress Cataloging in Publication Data Willardscin, Robert K ed. Semiconductors and semimetals. Bibliographical footnotes. CONTENTS:-V. 1-2,4. Physics of 111-V compounds.v. 3. Optical properties of 111-V compounds.v. 5. Infrared detectors. [etc.] 1. Semiconductors. 2. Semimetals. I. Albert C., joint ed. 11. Title. QC611.W59 537.6’22 65-26048 ISBN 0-12-752110-0 (v. 10)

PRINTED M THE UNITED STATES OF AMERICA

Beer,

List of Contributors Numbers in parentheses indicate the page on which the authors’ contributions begin.

ROBERTL. PETERSON, * Quantum Electronics Division, National Bureau of Standards, Boulder, Colorado (221) D. L. RODE,Bell Telephone Laboratories, Inc., Murray Hill, New Jersey (1) G. E. STILLMAN, Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts ( 1 7 5 ) J. D. WILEY,Bell Telephone Laboratories, Znc., Murray Hill, New Jersey (91) C . M . WOLFE,~ Lincoln Laboratory, Massachusetts Znstitute of Technology, Lexington, Massachusetts (175)

Present address: Cryogenics Division, National Bureau of Standards, Boulder, Colorado.

t Present address: Department of Electrical Engineeringand Laboratory for Applied Electronic Sciences, Washington University, St. Louis, Missouri.

vii

Preface

Since the inception of this treatise some nine years ago, the sophistication of measurement of most properties of solids and the interpretation of resulting data in terms of fundamental parameters characterizing the solid have advanced greatly. Corresponding progress has also taken place in the theoretical treatment of the basic properties of semiconductors and semimetals. The present volume presents clear evidence of these achievements. The first article shows that electron mobility in many common direct-gap semiconductors can now be calculated with surprising success from basic material parameters, taking into account the established band structure and realistic scattering mechanisms. Favorable comparisons with measured results are seen to occur. The second article deals in a rather similar fashion with p-type materials, concentrating on the 111-V compounds. In this case, the complexities of the valence band structure prevent so satisfying a theoretical treatment as is possible for the n-type conduction. Nevertheless, it is seen that simplifying assumptions made to render the calculations tractable do yield quite satisfactory results for many materials. Furthermore, it is possible to specify in what respects the theory might be improved. Certain hazards, often af€ecting the measurement and subsequent interpretation of data taken on specimens containing inhomogeneities, are discussed in the third article. It is shown both theoretically and experimentally that certain kinds of material inhomogeneitiescan produce effective mobility values that are actually higher than those representative of the bulk material. This behavior is in contrast to the more commonly reported examples where a decreased effective mobility occurs. The final article in the volume is concerned with an increasingly interesting type of phenomenon where interactions in the solid produce oscillatory behavior. The example treated in detail, namely the magnetophonon effect, is found to be a useful tool for studying various properties of semiconductors and semimetals. It provides information on effective masses, including anisotropies, band structure characteristics, phonon energies, and chargecarrier scattering processes. Effects of stress and charge-carrier heating are considerations of special interest.

ix

X

PREFACE

The editors are indebted to the many contributors and their employers who make this series possible. They wish to express their appreciation to Cominco American Incorporated and Battelle Memorial Institute for providing the facilities and the environment necessary for such an endeavor. Special thanks are also due to the editors’ wives for their patience and understanding.

R. K. WILLARDSON ALBERT C. BEER

CHAPTER 1

Low-Field Electron Transport D . L . Rode I . INTRODUCTION . . . . . . . . I1 . FORMAL TRANSPORT THEORY. . . . 1 . Bandstructure . . . . . . . 2. Boltzmann Equation . . . . . 3. Drgt Mobility . . . . . . . 4 . Thermoelectric Power . . . . . 5 . Time-Dependent Effects . . . . I11. ELECTRON SCATTERING. . . . . 6. Ionized Impurities and Heavy Holes . 1. Piezoelectric Acoustic Modes . . . 8 . Deformation-Potential Acoustic Modes 9 . Polar Optical Modes . . . . . 10. Intervalley Modes . . . . . . IV. RESULTS . . . . . . . . . . 11 . II-VI Crystals . . . . . . . 12. III-VCrystals . . . . . . . 13. Group IV crystah . . . . . . V . SUMMARY. . . . . . . . .

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7 1

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24 26 . . . . . . . . . 21 . . . . . . . . . 34 . . . . . . . . . 31 . . . . . . . . . 38 . . . . . . . . . 41

. . . . .

46

47 63 19 86

“I call them Crystals. tho’ Opaque. because angular and of one constant Figure.” Lister‘

.

I Introduction

This chapter is devoted to electron transport in crystals subjected to small driving forces. that is. low-field transport . We will study cases in which the driving forces are steady or time-dependent electric fields. or temperature gradients. In the case of steady forces. the corresponding transport coefficients fundamentally related to the crystal are mobility and thermoelectric power . These quantities can be measured with relative ease. and they have been widely studied in Si and Ge for over two decades. More recently. extensive data have become available on many of the 111-V and * M . Lister. “A Journey to Paris in the Year 1698.” Univ . Illinois Press. Urbana. Illinois. 1967.

2

D. L. RODE

11-VI compounds. Since low-field transport coefficients are characteristic of microscopic properties of the crystals exclusive of the particular field strengths involved, detailed comparisons between theoretical and experimental results permit the construction of a generally satisfactory picture of the electron physics of these crystals. In particular, more recent improvements in transport calculations allow quuntitative comparisons with experiments which are helpful in elucidating electron scattering mechanisms and in refining our knowledge of basic material constants. Naturally, these more accurate findings have dictated the elaboration of transport theory in a fashion consistent with band theory. Some of the earlier, and less accurate, transport calculations allowed only qualitative identifications of electron scattering mechanisms. Now, however, one can achieve analyses for isotropic direct semiconductors which are as accurate as many experiments. Having laid out our problem and its significance, we will find it useful to briefly retrace the evolution of transport theory for semiconductors. The newer work and its significancecan thus be retrieved from the haze of algebra which necessarily follows. Several reviews of transport work have appeared over the years.’-’ There is no need for us to discuss in detail the earlier work, but we may note that much of the previous work (excepting that of Devlin’) relies heavily on the relaxation-time concept. The resulting simplified equations for transport coefficients are quantitatively useful in restricted cases, and they are indispensable to a general understanding of the functional dependences of mobility, say, on effective mass, temperature, etc. It is, however, well known6 that a relaxation time cannot be defined for nonrandomizing inelastic electron scattering. Hence, the large class of polar crystals exhibiting significant amounts of scattering by polar optical modes’ is only poorly described by the relaxation-time method. Variational calculations have been applied to overcome this obstacle. When the Boltzmann equation is properly manipulated, accounting for inelasticity, one derives a linear finite difference equation’ describing the electron distribution function. This finite difference equation can be solved by various approximation

’ W. Shockley, “Electrons and Holes in Semiconductors.” Van Nostrand-Reinhold, Princeton, New Jersey, 1950. ’ H. Brooks. Adt3an. Electron. Electron Phys. 7 , 85 (1955).

A. C. Beer. “Galvanomagnetic Effects in Semiconductors.” Academic Press, New York, 1963.

’J .

’

M. Ziman, “Electrons and Phonons.” Oxford Univ. Press (Clarendon), London and New York, 1960. S. S. Devlin, in “Physicsand Chemistry of 11-VI Compounds”(M. Aven pnd J. S. Prener, eds.), Chapter 1 I . Wiley, New York, 1967. C. Herring and E. Vogt, Phys. Reu. 101, 944 (1956). H. Ehrenreich, J . Appl. Phys. Suppl. 32,2155 (1961). D. J. Howarth and E. H. Sondheimer. Proc. Roy. SOC.London A219, 53 (1953).

1.

LOW-FIELD ELECTRON TRANSPORT

3

schemes to some desirable order of accuracy. Kohler' proposed a variational technique which was later used by Howarth and Sondheimer,* Ehrenreich," and D e ~ l i n In . ~ principle, any degree of accuracy in the transport coefficients can be achieved by the variational method. In practise, however, improved accuracy demands a high price in mathematical tedium. This situation arises because the algebraic form of the solution consists of products of matrices with rank equal to the order of the approximation. The elements of the matrices themselves are fairly elaborate expressions. Nevertheless, accurate numerical solutions have been obtained this way by Devlin5 up to the twelfth order of approximation. We are willing to admit numerical analyses in the interest of accuracy. In this way, we will find that direct iterative solutions of the finite difference equation are superior in several First of all, this method (described in Part 11) permits straightforward physical interpretation of the exact transport equations. Second, the relative simplicity of the formalism discussed in Part I1 permits considerable generalization to includeI4 (a) Fermi statistics, (b) energy band nonparabolicity, (c) s-type and p-type electron wave function admixture, (d) arbitrary time dep,endence, and (e) combination of various scattering mechanisms at the differential probability (matrix element) level. Important advantages accrue also from the fact that these iterative solutions to the Boltzmann equation with Fermi statistics are cast in the form of contraction mapping^.'^ The contraction mapping principle guarantees existence and uniqueness of the solution and shows that the iterative sequence converges exponentially. Hence, numerical convergence and stability properties of this technique are very good. The iterative approach represents a substantial improvement over variational techniques which. require an algebraic formalism proportionately extensive with the described accuracy of the calculated results. In addition, galvanomagnetic and thermomagnetic effects can be easily calculated iteratively,' although this chapter is concerned only with galvanic and thermoelectric effects. The great advantage of choosing an accurate model of electron transport for which exact results can be obtained is that detailed comparisans with experiment now lead to refinements in the physical theory rather than to lo I'

l3

l4

Is

M . Kohler, Z . Phys. 125,679 (1949). H. Ehrenreich, J . Phys. Chem. Solids 2, 131 (1957); 9, 129 (1959). N . N. Grigorev, I . M. Dykrnan, and P. M. Tomchuk, Fiz. Tverd. Tela 10,1058 (1968) [English Transl.: SOP.Phys.-Solid State 10, 837 (1968)l. A. Fortini, D. Diquet, and J. Lugand. J. Appl. Phys. 41. 3121 (1970). D. L. Rode, Phys. Reo. E 2, 1012 (1970). D. L. Rode, Phys. Rer. 5 3 , 3 2 8 7 (1971). D. L. Rode. Phys. Status Solidi 55. 687 (1973).

4

D. L. RODE

misgivings about the numerical calculation. For this reason, the iterative method presented in Part 11 is used exclusively for later comparisons with experiments in Part IV. The usual approximate f o r m ~ l a s ~ ,for ~ , drift '~ mobility, etc., follow from the equations of Section 3 as successive iterative approximations, e.g., the relaxation approximation4 corresponds to the first iteration ;succeeding iterations provide results similar to those derived by the variational method." An obvious disadvantage to the iterative method is the lack of an analytical solution, but the agreement evident in Part IV in detailed comparisons with many experimental results offers sufficient consolation. The theory of Parts I1 and I11 is applicable to a wide range of transport phenomena in various crystals. However, matters of expedience and the availability of accurate data on electrons in crystals, experimental as well as theoretical, limit the present chapter to only a few of the tetrahedrally coordinated crystals. In particular, we discuss the elemental group IV crystals Si, Ge, and a-Sn with the (cubic) diamond structure; the binary 111-V semiconductors GaN, Gap, GaAs, GaSb, InP, InAs, and InSb; and the binary 11-VI crystals ZnO, ZnS, ZnSe, ZnTe, CdS, CdSe, CdTe, HgSe, and HgTe. Crystals of GaN, ZnO, CdS, and CdSe generally (but not always) assume the (hexagonal) wurtzite structure.' All these crystals are semiconductors, except for the zero-gap semimetals a-Sn, HgSe, and HgTe. 17,18 The pertinent elements lie near the metal-nonmetal boundary of the periodic chart as shown in Fig. 1. The aforementioned materials are bounded in Fig. 1 by a few crystals we now mention but which we shall not discuss. For the 11-VI variety, we will discuss the mercuric compounds that are semi metal^'^ (HgSe and HgTe'*) but not HgS, which tends to a crystalline structure not encompassed by the band theory2' of Section 1. The lighter-mass compound CdO and the various compounds of Mg tend to the rocksalt structure' 6 9 2 2(octahedrally coordinated) and are not included. Beryllium compounds are tetrahedrally coordinated but are relatively u n e ~ p l o r e d .Bounding ~~ the 111-V semiconductors at heavy masses are semimetals of TI and Bi, which are not discussed here. InSb has the smallest band gap to be studied. GaSb is only briefly considered because multivalley J . A. Van Vechten, Phys. Rev. 187, 1007 (1969). S. Bloom and T. K. Bergstresser, Phys. Srarus Solid 42, 191 (1970). I s These materials possess the inverted band structure r7-r6-rS and can be accommodated, to a limited extent, by the present theory with proper attention paid to the electron wave functions. J. G . Broerman, Phys. Reo. B 1,4568 (1970). R. Zallen, G. Lucovsky, W. Taylor, A. Pinczuk, and E. Burstein, Phys. Rec. B 1,4058 (1970). zo R. Zallen and M . Slade, Solid Stare Commun. 8, 1291 (1970). zL E. 0. Kane, J . Phys. Chem. Solids 1,249 (1957). 2 2 J . C. Phillips, Phys. Toduy 23, 23 (1970). " D. J. Stukel, Phys. Rev. B 2 , 1852 (1970). Ib

l7

1. LOW-FIELD ELECTRON TRANSPORT IIB

IEA

YEA

PA

5

=A

FIG. I . A portion of the periodic chart showing the elements from which the groups IV, Ill-V, and 11-VI tetrahedrally coordinated crystals are composed.

transport in nonequivalent valleys dominates at room temperat~re.’~ This phenomenon is not discussed in Part I1 (see, however, Basinski et Insufficient experimental data are currently available on pure compounds of A1 which, in any case, are h y g r o s ~ o p i c .The ~ ~ latter property relegates these crystals to a position of lesser practical importance than GaN, which is interesting for optical applications because of its relatively wide energy gap corresponding to the near ultravoilet. At the heavy-mass end of column IV in Fig. 1 lies the semimetal a-Sn, which can be approximately described by the work of Parts I1 and 111.” Diamond closes the boundary of excluded materials around our list of 19 semiconductors and semimetals to be studied in Part IV. It must be emphasized that the band theoretical model2’ and subsequent transport theory of Sections 1 and 2 apply accurately to only isotropic direct 24

M. Averous, G. Bougnot, J. Calas, and J. Chewier, Phys. Status Solidi 37, 807 (1970).

24*J. Basinski, S. D. Rosenbaum, S.L. Basinski, and J. C. Woolley, J . Phys. C Solid State Phys. ”

6,422 ( I 973). Some A1 compounds may be more stable than originally expected. M. Ettenberg, A. G . Sigai, A.Dreeben, and S. L. Gilbert, J . Electrochem. Soc. 118, 1355 (1971).

6

D. L. RODE

semiconductors typified generically by the sphalerite structure. However, they remain reasonably precise for the uniaxial wurtzite structure because anisotropy is, in fact, not too severe for these materials.26The remainder of this chapter is arranged as follows. A band theoretical picture specialized from the more general theory by Kane” is presented in Section 1. This model is used for all calculations discussed subsequently, except where explicitly noted for indirect semiconductors, and is specificallyappropriate for isotropic direct semiconductors and semimetals. These energy bands are maintained without approximation throughout the reduction of the Boltzmann equation for classical transport with Fermi-Dirac statistics.l 4 Formal equations are derived for the electron distribution function and the transport coefficients, drift mobility, and thermoelectric power. Finally, the transport equations are generalized to include arbitrarily time-dependent electric fields in Section 5 . In Part 111, the five more common electron scattering mechanisms are discussed : (1) ionized flaws (i.e., charged impurities and heavy holes), (2) piezoelectric modes, (3) deformation-potential acoustic modes, (4) polar optical modes, and (5) intervalley scattering. This collection of scattering mechanisms is sufficient for descriptions of most crystals over wide ranges of temperature. For impure crystals, we limit ourselves to cases well described by the ionized-impurity scattering theories of Dingle” and Brooks and Herring,28which apply to dilute concentrations of ionized centers. Extensive comparisons between experiment and results calculated by the methods of previous sections are presented in Part IV. The microscopic theory of Part I1 evidently is accurate for most cases concerning direct semiconductors. Of course, at sufficiently high temperatures where conduction occurs in nonequivalent valleys, the theory based on one set of equivalent valleys also fails. Regarding indirect semiconductors, good agreement obtains for lattice scattering only, even under the assumption of spherical constant-energy surfaces for the ellipsoidalvalleys. This approximation does not fare well, however, when impurity scattering dominate^.'^ When the theory is applied to pure direct crystals, for which it was originally designed,’ one finds excellent agreement from the lowest temperatures exhibiting lattice-limited conduction up to near the melting point, for example, in InSb. 26

M. V. Kurik, Phys. Len. 24, A742 (1967).

’’ R. 8. Dingle, Phil. Mug. 46, 831 (1955). 28

29

H. Brooks, Advan. Electron. Electron fhys. 7 , 85 (1955) and C. Herring, unpublished. The actual derivation and requisite assumptions of the Brooks-Herring theory remained unpublished for several years, but they can be found in the following reference: L. M. Falicov and M. Cuevas, fhys. Rev. 164, 1025 (1967). D. L. Rode, Phys. Status Solidi 53.245 (1972).

1. LOW-FIELD ELECTRON TRANSPORT

7

11. Formal Transport Theory

Our object here is to choose a realistic picture for electron transport which, at the same time, can be reduced without further approximation to quantities directly comparable to experiment. The relatively large class of isotropic direct semiconductors is especially useful in this sense since all necessary material parameters can be measured by independent experiments. Thus, we are left with no adjustable parameters in the calculated transport coefficients in this case. For indirect semiconductors, the theory has not yet caught up with experiment, and we must be content with a somewhat empirical approach. 1. BANDSTRUCTURE

The band structure of diamondlike and sphaleritelike crystals near the center of the first Brillouin zone has been discussed in detail by Kane.30(For other points of high symmetry, see C a r d ~ n a . ~ ’The ) results of Kane’s analysis by the k * p method describe, among other bands, the conduction band with far greater precision than is necessary for the present transport for application to calculations. Hence, we will specialize these results” isotropic direct semiconductors. In this way, we obtain a formulation suitable for direct crystals exhibitingany one of the tetrahedrally coordinated forms: diamond, sphalerite, or wurtzite. The energy level scheme at F, the center of the Brillouin zone, is shown in Fig. 2 along with the notation used in the following discussion for the sphalerite structure. The “small-gap’’ approximation3’ includes spin-orbit splitting and accounts for interactions between TlCconduction bands and rl5 v valence k p interaction leaves all these bands bands. Inclusion of only the rlC-rl5” isotropic. The conduction band rlc,in this approximation, is doubly degenerate due to spin, i.e., rlcconsists of a Kramers doublet. The valence band consists of three Kramers doublets-the heavy-hole band, the lighthole band, and the split-off band. The k * p interactions with more remote bands, such as rl5 c , are restored by first-order perturbation theory3’ (the previous interactions having been treated exactly). These higher bands split the Kramers doublets off principal axes of symmetry and warp the surfaces of constant energy away from spherical. This latter effect gives rise to a small amount of anisotropy in transport phenomena which, however, y 3 0

-

’’ E. 0. Kane, in “Semiconductors and Semimetals” (R. K . Willardson and A. C. Beer, eds.), 31

Vol. I , Chapter 3. Academic Press, New York, 1966. M. Cardona, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 3, Chapter 5. Academic Press, New York, 1967.

8

D. L. RODE

FIG. 2. Labeling convention at the center of the first Brillouin zone (r)for various energy levels. The corresponding irreducible representations are given without parentheses. Double group notations are enclosed by parentheses. Superscripts in the figure are replaced by subscripts in the text.

can be observed only by the most sensitive techniques in InSb,32for example. We will neglect such small effects after establishing quantitatively the plausibility of doing so. 32

Anisotropy of the Shubnikov-deHaas frequency has been applied for this purpose by D. G. Seiler, Phys. Left. 31, A309 (1970).

1.

9

LOW-FIELD ELECTRON TRANSPORT

Fawcett and R u c have ~ ~ considered ~ the band structure of 111-V semiconductors in the light of Kane’s the01-y.~’They have compared the “smallgap” approximation for finite spin-orbit coupling A with (a) the approximation involving only Tlc-T1s v interactions without spin-orbit effects (A = 0), and (b) the “small-gap” approximation including interactions with T l U ,and Tll (sphalerite notation). The authors33 higher bands such as conclude that splitting of the Kramers doublet for Tlc is truly negligible regarding electron transport phenomena. As a worst case example, they find the energy split to be33 2 meV for electrons 500 meV above the conduction band edge in GaAs. The comparison between approximations (a) and (b) and the “small-gap’’ approximation shows that the inclusion of spin-rbit coupling on the one hand and higher lying bands on the other hand leads to corrections which nearly cancel one another. The results33for GaAs are reproduced in Fig. 3. Happily, then, we find that the neglect of spin-orbit coupling as well as k p interactions with higher lying bands represents a

-*-

,

EXACT TWO-BAND INCLUDING HIGHER BANDS EXACT TWO-BAND APPROX. TWO-BAND PARABOLIC BAND

t

(3

a W

z W

0

2

4

6

8

10

WAVE VECTOR (106CM-’)

FIG. 3. Conduction band of GaAs near I-,. The solid curves include spin-orbit splitting and rlsc band interactions which make the conduction band slightly anisotropic. The dashed curve includes finite spin-orbit effects [Eq. (I)]. Note that the dotted line, neglecting spin-orbit splitting [Eq. (411, is closer to the solid curves than the dashed curve. The parabolic band is drawn for the same effective electron mass at r,.. (Courtesy of W. Fawcett and J. G. R u c ~ . ~ ~ ) 33

W. Fawcett and J. G. Ruch, unpublished (1969).

10

D. L. RODE

very good approximation to the TlCconduction band.34 This conclusion should hold for low-field transport phenomena at temperatures up to the melting points of the present crystals.35 For the remainder of this chapter, we will assume the Kane" description with only rlc-rlSv interactions for A = 0. There are several calculational advantages to this assumption : (a) The conduction band is isotropic and spin degenerate, (b) the neglect of spin-orbit coupling extends the sphalerite formalism directly to the wurtzite structure, (c) the energy-momentum dispersion relation is quadratic and can be inverted exactly, and (d) the electron wave function consists of a simple admixture of T l C(s-type) and rlsV (p-type) basis functions. The discussion of this section has thus far been concerned with only the energy-momentum dispersion relation, whereas, we see now from (d) of the previous paragraph that admixed wave functions are also involved in the band picture. Effects of wave function admixture become manifest in electron scattering rates through an overlap integral between initial and No comparisons have appeared in the literature on electron final states. scattering with and without higher lying band interactions. However, it seems likely that their inclusion would yield negligible corrections, in view of the minimal influence that higher lying bands have on the dispersion relation itself. In this connection, it is worthwhile noting that one cannot consistently assume nonparabolic bands and pure s-type electron wave functions. If the conduction band is nonparabolic, then the wave functions follow directly 30 as admixtures of s-type and p-type functions. This fact has been occasionally disregarded in the literature,' *37 whereas Fawcett and Ruchj3 show that wave function admixture has an effect on electron scattering which is comparable in magnitude to nonparabolicity. The corr e c t i o n ~due ~ ~ to the various approximations are illustrated in Fig. 4 for electron scattering by polar optical modes in InSb. Having chosen the foregoing band structure, we now present details appropriate to the TlC conduction band minimum of tetrahedrally coordinated crystals. For k - p interactions between rlcand r15" bands, Kane 30 derives the following relation between electron energy d (measured 34

Although we will accordingly assume A = 0, it turns out that choosing A =a: yields an identical band structure. These two limits do nor lead to identical overlap integrals for scattering rates (see Part 111). However, the difference is of second order in the wave function admixture, and we may expect accurate results for materials with large spin-orbit splitting (e.g.. InSb) as well as for small spin-rbit splitting (e.g.. ZnO). 35 R. R. Senechal and J. C. Woolley, Phys. Status Solids 19, 251 (1973),have discussed this situation3' for InAs. 36 E. M. Conwell and M. 0.Vassell, Phys. Rev. 166, 797 (1968). 3 7 P.A. Kazlauskas, Fiz. Tekh. Poluprov. 3,1224 (1969)[English Transl.: Soo. Phys. Semicond. 3, 1025 (197O)J.

1.

11

LOW-FIELD ELECTRON TRANSPORT

SCATTERING RATE IN

- -----

-----

insb

A

-- ---

PARABOLIC BAND NON-PARABOLIC BAND WITHOUT p-FUNCTION ADMIXTURE NON-PARABOLIC BAND WITH p-FUNCTION ADMIXTURE

I

0

1

0.I

0.2

I

0.3

ENERGY (ev) FIG. 4. Electron scattering rate by polar modes in room-temperature InSb. Parabolic band theory (dashdot curve) underestimates the true scattering rate shown as a solid curve (A = 0.89 eV). The highest curve includes nonparabolicity but neglects wave function overlap. and clearly is inaccurate above 0.1 eV electron energy. We will use the A = 0 theory as in Fig. 3. (Courtesy of W. Fawcett and J. G. R ~ c h . ~ ~ )

upward from rlC)and crystal momentum hk = (hk113:

(8- BOHb - 8,

+ 8,Hb - bo + 8,+ A)

- B,b,(B - go + 8,+ $A)

=0,

(1)

where I = 0 at k = 0. Here B, is the effective-mass energy gap," to be discussed, and A is the spin-orbit splitting at rl5 v . The energy 8, = 2mp2/h2

(2)

is proportional to the k * p interaction matrix element3' called P , which is calculated empirically from experimental values of the electron effective mass at k = 0 [see Eq. (S)] and m is the.electron mass in vacuum. Crystal momentum hk enters Eq. (1) through the energy go,where

(3) Equation (1) is cubic in electron energy and reduces to a quadratic form when A = 0 or when A =a.In either case, the b ( k ) relation is identical since &, in the former case is adjusted empirically to 28J3 in the latter case. This result demonstrates why the A = 0 approximation is quite good not only for small-A crystals but also for large-A crystals. In practice, the A = 0 approximation is preferable since it leads to simpler wave functions.1°.13.14 When A = 0,Eq. (1) can be reduced explicitly to either &(k) or k(b), both 8 , = h2k2/2m.

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D. L. RODE

of which forms are required in transport calculations :

In the limit k = 0, we fix €, in terms of the effective mass m* at the band edge since 6 ( k ) becomes parabolic in k here. From Eq. (4),this procedure gives

m 2m & _ - -limT m*

h2 k=Ok

=1

+ &9 , 8,

(5)

from which 8, is determined by the experimental quantities m* and €,. Note that m* throughout this chapter is regarded as the effective mass at k = 0. The effective mass may itself depend upon temperature, however" [see Eq. (22)]. Since &(k)is not parabolic, it follows that the group velocity of the electron wave function is not linearly proportional to crystal momentum. The group velocity v is needed in Section 3 for mobility calculations : v = (l/h) V k € .

(6)

Combination of Eqs. (4)and (6) leads to

m

(7)

It is evident that k and v are parallel no matter how complicated the &(k) relation may be with respect to nonparabolicity, provided only that d be an isotropic function of k. Equation (7) shows that group velocity for a given momentum hk is always less than hk/m*, which is the group velocity for an isotropic and parabolic3* conduction band. Besides Eq. (4)and (7) for band structure and group velocity, we will need an expression for the electron wave function in terms of s-type and p-type basis functions. When A = 0, the otherwise split-off valence band becomes interactions degenerate with the heavy-hole band. Inclusion of rlC-rl5" couples the conduction band with the light-hole band so that the electron wave function results as a linear combination of s-type and p-type basis respectively. For the conduction band, the wave functions [is] and function is'3,30 ( P k p = 4S11' + C[Z11', (8)

[a,

38

This state of affairs, isotropic and parabolic, has come to be known in the literature by the misshapen term "spherical and parabolic" band.

1.

LOW -FIELD ELECTRON TRANSPORT

13

where p is a spin index denoting spin orientation or 7 . Primes indicate coordinate and spin transformations from principal crystallographic axes to the direction of k.The real, positive coefficientsa and c provide normalization and wave function admixture consistent with nonparabolicity, a‘

+ c2 = 1,

(9)

For two electron states characterized by momenta hk and hk’, the overlap integral is”

G(k, k’) =

4b’B 1

ll

q&&)qkS(r) dr

ji

,

(11)

where the asterisk denotes complex conjugation and the integral extends over a unit cell of the crystal. qkSis normalized to a unit cell volume. The necessary coordinate and spin transformations are given by Kane.” Taking k parallel to a principal symmetry axis, straightforward substitution of the transformed wave functions into Eq. (1 1) yields

G(k, k) = (aa’ + cc’x)’,

(12)

where a = a(k), a‘ = a(k‘), etc., and x is the cosine of the angle between k and k’,the wave vectors before and after the scattering event. In the present band model, Eq. (12) applies to any direction for k since anisotropy does interactions. not appear for only rlc-r15v Equations (4), (7), and (10) are rewritten here in terms of the more familiar parameters m* and 8,.The matrix element term 6, is eliminated through use of Eq. ( 5 ) :

R’k’ 8 ( k ) = __ 2m

+ &,(a 2- 1)

9

m l+a a2(k)= -, 2a

where a 2 1. To make the connection with a parabolic band, we let k

14

D. L. RODE

z),

approach zero (a parabolic band also obtains if we let 8,approach infinity) : a -, 1

-(

+ h2k2 1 m*6,

&+-

(

I')*:

h 2 k 2 [1 - - - h2k2 1 - 2m* 2m*6,

.

Hence, a and a [of Eq. (8)] approach unity while c [ of Eq. (8)] approaches zero to yield a wave function of type s only. Equation (18) shows that the group velocity for a given crystal momentum, in general, is less than its counterpart hk/m* in a parabolic band. Energy 6 is a subparabolic function of momentum, in general, as shown previously in Fig. 3, so that the density of states function increases more rapidly with energy in a nonparabolic band than in a parabolic band. We denote this relative enhancement by the quantity d, where l/d = (m/h2k)ablak, l/d = 1 + [(m/m* - l ) / a ] .

(20) (21) In the limit k = 0, d = m*/m, the relative effective mass; d appears as a Jacobian in Part 111 for converting integrals over energy space to integrals over momentum space. Equations (12)-(16) and (21) complete the required band structure and wave function formalism. The particular material to be described enters through the two parameters m*, the effective mass at the conduction band edge, and 8,, the effective-mass energy gap. Of course, one cannot measure 8,directly, but Ehrenreich" has suggested how 8,can be related to measured optical and thermal energy gaps, which we discuss now. Kane's theory,30 strictly speaking, applies only at zero temperature. To find the effective-mass energy gap versus temperature, Ehrenreich' argued that &(, 7')could be related to that at zero temperature gg(0)by accounting for lattice dilatation alone. It is known that the optical energy gap, determined from optical absorption experiments at the fundamental edge,39 decreases with increasing temperature because of lattice dilatation and electronphonon c ~ u p l i n g . ~Dilatation ~-~' alone can explain only about one-third of the temperature-dependent shift of the gap. The remainder is due to electron-phonon interactions. If the electron-phonon coupling resembles 39 40

4'

M. B. Panish and H. C. Casey, Jr., J . Appl. Phys. 40,163 (1969). H. D. Vasileff, Phys. Rev. 105,441 (1957). E. N. Adarns, Phys. Rev. 107.671 (1957).

1.

LOW-FIELD ELECTRON TRANSPORT

15

that of Frohlich's weak polaron theory,42 then we expect the predominant effect of finite temperature to be a downward shift of the conduction band edge, the electron-phonon interaction itself yielding only a much smaller influence on band The effective mass will nevertheless decrease because of increased k p coupling between TlCand TISvbands. In this case, Ehrenreich's'' formula should be C,(T) = gK(0) - 31T(dbK/6'P),/K,

(22)

where 1 is the linear coefficient of thermal expansion, (8fK/6'P),is the pressure and K is the compressibility-all assigned average, rate coefficient of gK, temperature-independent values for the present work. The Tlc-T15 v matrix element 8, of E q . (5) is also assumed to be temperature independent since it is known to be relatively insensitive to lattice dilatation, and m* is allowed to vary consistently with temperature [see E q . (5)]. The band structure model chosen in this section will be used for the remainder of the chapter to analyze electron transport phenomena. The model applies very well to direct semiconductors insofar as the transport coefficients can be calculated with no adjustable parameters. Furthermore, all the necessary material parameters can be determined from independent (of transport phenomena) measurements. There is only one further approximation to be employed between the choice of a band structure and the final results-the Born appr~ximation.~.~' The Born approximation allows calculation of electron scattering rates in Part I11 with considerable ease and also is quite valid for a wide range of material conditions. Its shortcomings have been reviewed by Moore and Ehrenrei~h.~'

2.

BOLTZMANN

EQUATION

For the present class of problems involving galvanic and thermoelectric effects in isotropic energy bands, the electron probability distribution function maintains axial symmetry. In general, the distribution function can be expanded in surface zonal harmonics, even in the case of high-field H. Frohlich, Advan. Phys. 3,324 (1954). Weak polaron theory4' predicts a downward shift of TICat zero temperature by just the electron-phonon self-energy. There is relatively little distortion of band curvature, however, in first order. 44 One of the more careful experimental tests of Eq. (22)can be found in the following reference, which, although it does not confirm the dilatation model rigorously, indicates only small departures for our present purposes. L. Eaves, R. A. Stradling, S. Askenazy, J. Leotin, J. C. Portal, and J. P. Ulmet, J. Phys. C Solid Srare Phys. 4,L42 (1971).See also Part V. 44aInthe spirit of the previous ~ o r k , see " ~ also E. S. Koteles and W. R. Datars, Phys. Rev. B 9, 568 ( 1974). " E. J. Moore and H. Ehrenreich, Solid Srure Commun. 4,407(1966).See also Section 6. 42

43

16

D. L. RODE

t r a n ~ p 0 r t . For l ~ low-field transport, only the first two terms of the series need be retained. This form of the solution is exact,13 although it occasionally is termed the “diffusion approximation” in the literature. Accordingly, the total distribution function is (23) f,(k) = f + xi?, where x is the cosine of the angle between k and the vanishingly small driving force (e.g., electric field or temperature gradient), f = f ( k ) is the equilibrium part of the distribution, and g = g (k)is the perturbation part of the distribution, which is of first order in the driving force. The Boltzmann equation for electrons, with Fermi-Dirac statistics, describes classical transport phenomena with which we are presently concerned exclusively. (Quantum effects have been discussed earlier in this series.46)Allowing for an electric field F and a spatial gradient parallel to F, we have e (24) * ‘fT + -h ’ ‘,fT = /[s’f<(l - fT) - Sf,(l - .fill dk’,

”

where f.,-’= f,(k’), s = s(k, k’), and s’ = s(k’, k) is the differential scattering rate for an electron in the state characterized by k’ to make a transition into the state characterized by k. The details of the differential scattering rates form the subject matter of Part 111, but certain general properties of s are used here to simplify the Boltzmann equation. Substitution of Eq. (23) into Eq. (24) and integration over x of the resulting equation multiplied respectively by unity (the zeroth-order Legendre function) and x (the first-order Legendre function) leads to two equations, “sy’(l - f) - sf(1 - f’)]dk

and

aj +eF af -=

u-

a2

ii ak

-g

s

=0

[s(l - f’)+ s‘f’] d k

where x’ is the cosine of the angle between k and F, and F lies along the z axis. The explicit angular coordinate x can now be removed from the last term of Eq. (26). Since the conduction band is isotropic, s depends upon only k, k’, and X , the angle between k and k , i.e., the crystal has no preferred 46

L. M. Roth and P. N. Argyres, in “Semiconductors and Semimetals”(R. K. Witlardson and A. C. Beer, eds.), Vol. 1, Chapter 6, Academic Press, New York, 1966.

17

1. LOW-FIELD ELECTRON TRANSPORT

direction. The following general formula then becomes appli~able’~.~’ : IxLZt(X)dk‘ = x

I

Xd(X)dk’,

(27)

where d(X)resembles s(k’, k) and is an arbitrary function of X but not a function of x or x’ alone. Application of this result to Eq. (26) yields eF u-+--=

aZ

n

ak

sXg’[s’(1 - f)

+ sf] dk’ - g

s

+

[s(l - f’) s’f’]

a‘.(28)

Equations (25) and (28) satisfy the Boltzmann equation through first order in the driving forces and are the exact solutions for low-field transport phenomena. Once the integration over k is performed, Eq. (28) does not contain any angular coordinates and is a function of k alone, as is required if Eq. (23) is to be an acceptable solution, i.e., g is isotropic. Equation (28) is of first order in the driving forces; Eq. (25) is of zero order in the driving forces and involves only the equilibrium portion f of the total distribution function. The differential scattering rates lead us directly to the Fermi-Dirac distribution; assume the states k and k correspond to energies d and d’, so that4* s‘ exp( -~ ’ / K T = )s exp( - &+/K T ).

(29)

Then Eq. (25) is satisfied identically (as it must be) by [f’/(l- f’)lexP(E/KT) = [f/(l - f)IexP(&/KT)

(30)

for all k’and k. Equation (30) must be true for any temperature T and therefore both sides must equal some constant y( T ) independent of the electron states:

or The Fermi energy 6, is defined such that f (8,)= i, and we obtain the Fermi-Dirac distribution : 1 (33) f = ,(&-&F)/KT + 1 . 47 48

E. Jahnke and F. Emde, “Tables of Functions,” p. 115. Dover, New York, 1945. The appearance of the Boltzmann factors here precludes nonequilibrium phonon effects (e.g., phonon drag) by this theory.

18

D. L. RODE

Note that Eq. (33) follows if and only if b' # I ,that is, there must be at least one inelastic scattering mechanism contained in s which is in thermal equilibrium so that Eq. (29) is valid. Otherwise, Eq. (25) vanishes identically (since 8' = I ,s' = s, and f ' = f ) for any function f. This result demonstrates that f is unaffected by elastic scattering. It also provides the essential difference between degenerate and nondegenerate transport phenomena, which disappears if we simply assume elastic ~cattering.~ The remaining equation [Eq. (28)] for the perturbation g contains all information regarding transport effects. The important point for the present theory is that all integrals appearing in Eq. (28) can be performed analytically without approximation and including the Kane bands3' of Section 1. The arguments of the integrals depend explicitly upon the scattering mechanisms considered in Part 111, but we can manipulate Eq. (28) formaffywith a few definitions which take advantage of general properties common to all differential scattering rates. The differentialscattering rates comprises inelastic scattering mechanisms, and elastic scattering mechanisms, denoted by seI: denoted by sinel, s(k, k') = Sine#, k')

+ sel(k,k'),

(34) where we now must regard the various s as linear operators since obviously k' # k for an inelastic process whereas k' = k for an elastic process. In addition, sineland selmay be sums of several different partial scattering rates associated with various scattering mechanisms, e.g., the elastic processes of acoustic modes, piezoelectric modes, and ionized impurities. Combination of Eqs. (28) and (34) leads to (recall that s:, = seI,etc.)

The total influence of elastic scattering is collected under the second integral on the left-hand side, which we recognize as precisely the relaxation-time formula? In fact, when there is no inelastic scattering, Eq. (35) reduces to the standard relaxation approximation (without spatial gradients)49 g =

eFT af - ----,

where the relaxation time T is given by VeI 49

=

l/r

=

I

D. L. Rode, J . Appl. Phys. 40,4123 (1969).

h ak

(1 - X)se,dk'

(37)

1.

19

LOW -FIELD ELECTRON TRANSPORT

and v,, is called the elastic scattering rate. Approximate relaxation schemes in the presence of inelastic scattering can be defined at low temperature, as Frohlich has shown,42by neglecting the integral on the right-hand side of Eq. ( 3 9 , which yields scattering into the differential volume element dk by inelastic processes. At high temperatures, the corresponding approximation has been discussed by Ehrenreich,” who sets g’ = g so that g’ can be removed from the integral operation. In general, the relaxation scheme is useful only for qualitative descriptions, and we will solve Eq. (35) without approximation after further analytical reduction. The leftmost and rightmost integrals in Eq. (35) physically correspond to scattering O U ~of and scattering into the differential volume element dk in momentum space. For elastic processes, both these terms involve only the electron energy b ( k )and have already been combined into a relaxation rate, Eq. (37). On the other hand, these terms cannot be combined for inelastic scattering, which involves different energies b ( k ) and 6(k’). Hence, we define’ the scattering out rate by inelastic processes So:

(38) The scattering in rate13 Si must be regarded as a linear operator since the process connects g(k) with g(k’):

Finally, Eqs. (359, (38), and (39) yield14

which is simply a statement of the detailed balance between influent and effluent probability fluxes at the differential element d k . Note that the equilibrium distribution f appears in So and Si,in contrast to nondegenerate situation^,'^ where f = 0. Solutions of Eq. (40)will allow us to calculate transport coefficients as will be shown. Given the various scattering rates evaluated in Part 111, we solve Eq. (40) numerically by an iterative procedure. The (i + 1) iteration g i + is taken to satisfy gi+l =

S,(g,’) - v(af/az) - (eF/hMaf/ak) S o + Vel

9

(41)

m

D. L. RODE

where we arbitrarily choose (it really does not matter, since limi=mg i turns out to be unique; see Section 5) go = 0.

(42)

When all nonrandomizing inelastic scattering vanishes, S&') = 0 and g, is the exact solution. In general, g, is a relaxation approximation [see Eq. (36)]. The convergence properties of equations having the general form of Eq. (40) are well known to numerical analysts-Eq. (40) is simply a contraction mapping of g ; more abstruse treatments have appeared for The recognition that Eq. (40) is a contraction high-field mapping allows a firm mathematical basis for the theory. Let it suffice that Eq. (41) is easily solved, and we will see why this must be so in Section 5. We will use several iterations (usually less than five) to solve Eq. (41) with whatever precision is desirable for comparisons with experiment in Part IV. In Sections 3 and 4, g is related to the low-field transport coefficients, drift mobility, and thermoelectric power. In Section 5, Eq. (40) for the perturbation distribution is generalized to include arbitrarily time-dependent driving forces. 3. DRIFT MOBILITY

The group velocity v of an electron with energy &(k) is parallel to the crystal momentum hk, as mentioned in Section 1, and is given by Eq. (14). Drift mobility P is defined as the average drift velocity per unit electric field in the limit of zero electric field.3 Furthermore, there are no additional forces (such as temperature gradient) present, and the crystal is assumed to be uniform and isotropic. The spatial gradient of f in Eq.(40) therefore vanishes and

For high-field transport phenomena, we could extend the expansion [Eq. (23)] to include more terms," or we might find it much easier to do a Monte Carlo simulation of the oneelectron problem. On the other hand, it is clear from the contraction mapping principle discussed in Section 5 that the Boltzmann equation (24) can be rewritten as (neglecting degeneracy and spatial gradients) fT

5' 52 53

=

s'fT'dk' - (eF/h) * VJ,

so+ *el

and solved similarly to Eq. (41), which is Rees's m e t h ~ d . ~ ' H. D. Rees, J. Phys. Chem. Solids 30,643 (1969). M. 0. Vassell, J . Math, Phys. 11,408 (1970). P. C. Kwok and T. D. Schultz, Phys. Rec. B 3, 1180. 1189 (1971).

1.

21

LOW-FIELD ELECTRON TRANSPORT

Now the drift mobility, allowing for arbitrary electron degeneracy, becomes

p = f svk’(g/F)

dk ,

(45)

which is independent of field strength since g is proportional to F from Eq. (43), i.e., Ohm’s law is valid. From Eqs. (14) and (21),

Once g has been determined by successive iteration of Eq. (43), the drift mobility follows from numerical integration of Eq. (46). In the exceedingly simple, but unrealistic, case of a constant relaxation rate v = l/z in a parabolic band, Eq. (46) reduces to the commonly known3 formula p = er/m*, which is helpful in indicating the general trend of mobility with effective mass and scattering rate. In a very approximate fashion, t can be regarded as the mean time between scattering events or as the time constant with which the electron population responds to time-dependent disturbances (see Section 5). In fact, t is so small (10-’2-10-’3 sec near room temperature in GaAsS4)as to be beyond easy direct measurement. Hence, we will simply refer to mobility as the more meaningful physical quantity which is calculated without approximation from Eqs. (43) and (46). POWER 4. THERMOELECTRIC For the measurement of thermoelectric power Q, one determines the ratio of electric field F to temperature gradient VT across an open-circuited c r y ~ t a li.e., , ~ the electron current density J is set equal to zero. Theoretically, one cannot calculate Q in precisely the same fashion since “electric field” is not a transport quantity. We could calculate Q from Kelvin’s relations” (more generally, Onsager’s reciprocal relations), but this approach is less transparent than the following. What we shall do instead is calculate the short-circuit current density from the Boltzmann equation with the help of Poisson’s equation. Then the thermoelectric power follows from the generalized current density formula for arbitrarily degenerate electrons. The current density in the presence of electric field F and temperature

’

54

’’

E. M. Conwell, “High Field Transport in Semiconductors.” Academic Press, New York, 1967. S. M. Puri and T. H. Geballe, in “Semiconductorsand Semimetals” (R.K. Willardson and A. C. Beer, eds.), Vol. 1, Chapter 7. Academic F’ress, New York, 1966.

22

D. L. RODE

gradient V Tin an isotropic crystal iss6 J = o[F - (V&F/e)- Q VT],

(47)

where o is the conductivity and &, is the Fermi energy. Equation (47) is valid for the small driving forces considered here, for which o and Q are independent of the field strengths. When J = 0, as in the open-circuit measurement of Q, the crystal maintains equilibrium so that V&F = 0 and

Q = Fi(aT/az),

(48) which is the defining equation for Q. The temperature gradient is taken parallel to the z axis consistent with the formulation of Section 2 [see Eq. (4011. Since all driving forces are small, the transport coefficients o and Q are constant and Eq. (47)yields Q also in the short-circuit case when F = 0:

Hence, Q is determined theoretically from Eq. (49) and experimentally from Eq. (48). The spatial gradient a&,/& in Eq. (49) must be related to dTlaz through Poisson’s equation, V F = 0, which requires the absence of space charge. The free-electron concentration n is therefore spatially uniform,14

-

an/&

=

I

(a/az) (k/.)zf dk

= 0,

(50)

which allows for conduction band nonparabolicity. Combining Eqs. (33), (49) and (50), we obtain

56

F. J . Blatt, “Physics of Electronic Conduction in Solids.” McGraw-Hill, New York, 1968.

1.

LOW-FIELD ELECTRON TRANSPORT

23

The quantity in square brackets is specified fully from the given free-electron concentration n, which determines 8,. The current density J in Eq. (52) is the short-circuit current to be calculated from the Boltzmann equation, or rather, from Eq. (40)with F = 0, so that

The gradient terw aflaz can be expressed in terms of aT/az,which is now the only driving force. From Eqs. (33) and (51), we have

d

aZ

J k 2 f ( l - f)(&//KT)dk jk2f(l - f ) d k

(54)

Equation (54) shows that g and, thus, J are directly proportional to 8T/az, so that Eq. (52) yields Q as being independent of temperature gradient, as we require for low-field transport. Equations (52)-(54) provide the thermoelectric power, although the appearance of cr = enp in Eq. (52) shows that the drift mobility must be known before one can calculate thermoelectric power. Since the overall dependence of Q upon various material parameters may not be evident from the precise expressions just given, it is worthwhile to record the solution for the simple case of a nondegenerate parabolic band with all scattering described by a relaxation rate. The relaxation rate varies 8'. We have with energy as v

-

af

v g = ---,

az

(55)

aZ

Equation (57) is the well-known formulas6 for thermoelectric power, which shows that (a) Q has the algebraic sign of the mobile charges (gF< 0 dominates in the nondegenerate regime), (b) Q depends more sensitively on the functional behavior of v than on the absolute value of v (as opposed to mobility), (c) Q increases in magnitude as the free-electron concentration decreases. The first property is experimentally useful in determining the conductivity type (n or p) of semiconductors by the hot-probe method. Since the relaxation approximation is not generally valid, Eqs. (52)-(54) have been used to calculate results for InP, InAs, and InSb in the literature. l4

24

D. L. RODE

5. TIME-DEPENDENT EFFECTS

Equation (40) for the perturbation distribution g can easily be adapted to include arbitrarily timedependent driving forces, e.g., microwave and far-infrared electric fields, and pulsed electric fields. Such problems are of interest regarding free-carrier absorption5’ and plasma reflection at high frequency, and also in connection with the frequency response of microwave diodes5’ and transistors. The following method was originally n ~ t i c e d , ’ ~ in connection with low-field transport, from the resemblance of the convergence of Eq. (41) to a time-dependent response of electrons to a stepped-on electric field. An essentially identical (in principle) method for high-field The important simplification is transport was developed earlier by Rees.’ that successive iterations can be made equivalent to steps in time so that the time coordinate need not appear explicitly in the Boltzmann equation. This is also a naturally convenient place to show how the contraction mapping principles9 guarantees a unique solution for the perturbation distribution. We delete spatial gradients from the development since these are not easily manipulated experimentally at high frequencies. To include time dependence, we first note that the addition of a constant R to the scattering-our rate and a term Rg to the numerator of Eq. (43) has no effect on the solution g, provided g exists and is unique. That is, if there exists a unique solution g, of

’

v5’

then g, is independent of R and, furthermore, equal to g,(R = 0), which is the solution of Eq. (43). This conclusion is obvious by inspection of Eq. (58) since terms involving R can be removed if gi+ approaches gi which approaches g,, and Eq. (58) then becomes identical to Eq. (43) for g,. Hence, we need only prove that g exists and is unique. The operator (Si + R)/(S, + ve, + 0) is continuous (recall that So v,, + R is a scalar), although its first derivative in general is not (see Part 111). In this case, the contraction mapping prin~iple’~.~’ states that a unique solution g exists for Eq. (58) if, for 0 < 8 < 1,

+

llqg1’) - Sikl’) + n(g, - gJll G Wl(S0 + ”

’’

’’ 6o

V,I

+Rk,

- gz)ll9

(59)

H. Y. Fan, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 3, Chapter 9, Academic Press, New York, 1967. H. D. Rees, IBM J . Res. Develop. 5, 537 (1969). L. B. Rall, “Computational Solution of Nonlinear Operator Equations.” Wiley, New York, 1969.

Similar results have been obtained for the coupled transport equations which describe galvanomagnetic and thermomagnetic phenomena.

’

1.

25

LOW-FIELD ELECTRON TRANSPORT

and then Eq. (58) is called a contraction mapping of the closed ball U(go,r) which contains g , and g2. When Si arises from polar mode scattering (the only nonrandomizing inelastic process considered in Part 111), it has been shown a p p r ~ x i m a t e l y 'that ~ ~ ~Eq. ~ (58) is a contraction mapping so that g exists uniquely for any R 2 0. (A more rigorous proof than that given by RodeI5 would be desirable.) If R is negative, Eq. (58) does not, in general, converge (although it may do so under certain condition^^^). Therefore, the constant R >, 0 does not affect the solution g. Rees5*has called R the "selfscattering rate." As an added bonus, the contraction mapping principle tells us that the iteration sequence {g,} converges e ~ p o n e n t i a l l y .Hence, ~~ the procedure outlined in Section 2 requires very few iterations for any reasonable accuracy. The condition R > 0 lowers the convergence rate of the sequence { g , } sinceg,, approachesg, as Q approaches infinity. From Eq. (58), we see that ki+l

- gilR>o <

lgi.1 - gilR=o'

(60)

lim(gi+l - g,) = 0.

(61)

R= m

R(gi+ I - g , ) to ag/dt as follows. From

However, we can relate Eq. (W,

,

where the last equation follows from the fact that g i + is indistinguishable from g ifor large R. All the terms of zeroth order which were eliminated from the Boltzmann equation may now be restored since f is still the equilibrium Fermi-Dirac distribution, which is independent of time. Recalling the definitions Si, So,and vet of Eqs. (37)-(39), we have the Boltzmann equation lim R(xgi+

R=m

- Xg,) =

s

[Slf;i(l

- fTJ

-

e

- f;Jl dk' - F * VkfTi, (64)

where f T j is the total distribution function f + xgi. The left-hand side of Eq. (64)is simply identified with af,/at at the time i/R, whereby we see that l/R is the time increment corresponding to an i t e r a t i ~ n . ~Therefore, ' the sequence {gi}yields f, versus time provided R is large compared to So + v e t .

26

D. L. RODE

The foregoing situation with F independent of the iteration index i describes stepped-on driving forces, and the initial condition is contained entirely within go. For example, go = 0 indicates the absence of any driving force for t d 0. Equations (62)-(64) may also be adapted, with slight modification, to include arbitrarily time-dependent electric field F . In this instance, F becomes a function of time through the iteration index i so that gi+l

=

q g , 3 - (eFi/W!f/w

so+ Y e , + R

+ Qgi

9

(65)

where again l/Q is the time increment between successive iterations. Equation (65), for example, is periodically (with respect to i) convergent if F is periodic. Harmonically varying fields offer an exceptionally simple problem since af,/at is also harmonic and Eq. (65) separates into two quadrature component equations. Some results for stepped-on fields in GaAs are presented in Fig. 25 in Section 12. (See p. 73.) It follows from Eq. (58) that the electron distribution relaxes in a steppedon field disturbance exponentially in time only if a relaxation approximation obtains, i.e., S,(g') = 0. The time constant So v,, is energy dependent, so that different portions of fT respond at different rates. The scattering rate So + v,, plays a similar role to a viscosity in real space.49 In general, the scattering-in term Si(g') does not vanish, and, contrary to the popular impression, the electron population does not relax exponentially with time even for the smallest disturbances!

+

111. Electron Scattering

In order to adapt the work of Part I1 to specific crystals, we must identify the essential electron scattering mechanisms. These appear in the Boltzmann equation through the differential scattering rates. Hence, it remains only to connect the scattering mechanisms to the differential scattering rates by some means. The Born approximation allows us to accomplish this feat in a simple manner, i.e., the differential scattering rate is proportional to the absolute square of the matrix element of the scattering p ~ t e n t i a l By . ~ and large, we will take the results from existing literature for the matrix elements, but for scattering by Coulomb centers the calculation is presented in detail in Section 6 with the appropriate modifications for Kane3' bands. This approach seems appropriate since the theory of ionized-impurity scattering is the weakest part of the present-day theory of electron transport due to failure of the Born approximation6' and the occurrence of multiple scattering,62 as will be discussed. b'

bz

E. J . Moore, Phys. Rev. 160, 607,618 (1967). D. L. Rode and S. Knight, Phys. Rer. B 3, 2534 (1971).

1 LOW-FIELD ELECTRON TRANSPORT a

27

6. IONIZED IMPURITIES AND HEAVY HOLES The theory of electron scattering by ionized centers has been developed by several workers, with emphasis on various topics. included space charge screening of the centers in a way suitable for impure metals. This formulation obtains only for degenerate electron systems (i.e., ThomasFermi model) and is not useful for the wider range of semiconductor problems. With semiconductors in mind, Conwell and WeisskopF4 introduced a cutoff procedure for the Rutherford cross section which describes Coulomb scattering. Small-angle scattering, which leads to divergence of the total cross section, was cut off from the integral of the differential cross section to yield a finite scattering rate. However, space charge screening (known to be important) was neglected. Furthermore, the small-angle cutoff is related64 to the interimpurity distance, and the simultaneous inclusion of space charge screening leads to an excessive amount of ~creening.~’ We conclude that one cannot assign a meaningful geometric size to individual charged

center^.^^.^'

The formulation of Brooks and Herring2* is most widely quoted in the literature and is quite useful for nondegenerate transport problems. For the Brooks-Herring theory, one must know not only the free-electron and ionized-impurity concentrations but also the neutral donor concentration (in the case of n-type material). The latter quantity is often unknown experimentally, and we will therefore use as our starting point the simpler theory of Dingle,27includingarbitrary electron degeneracy.The distinction between the Brooks-Herring theory and Dingle’s theory is occasionally overlooked, and in fact the differences are usually of theoretical interest only, i.e., Brooks and Herring account for the screening of impurities by other, nearby impurities, whereas Dingle does not. Hence, the concentrations of neutral as well as ionized centers appear in the screening formula for the former theory. Dingle assumes that all impurities are ionized as well as dilute and form a uniform background of immobile charge. Only the concentrations of mobile charges (electrons and holes) provide screening. It turns out that b3 64 .55

66

N . F. Mott. Proc. Combridge Phil. Soc. 32,281 (1936). E. M. Conwell and V. F. Weisskopf, Phys. Reu. 77. 388 (19SO).

D. L. Rode, unpublished. We might have anticipated this result from the form of the Rutherford cross section. The geometric area of the scattering center should be proportional to the area over which the potential energy exceeds the kinetic energy E vz of the incoming electron. The area u for a Coulomb center with cp l/r scales as (for cp = E ) l / v 4 so that the scattering rate Y uv I/o3 I/T3I2.The ionized-impurity mobility thus is wry roughly proportional T3”, as is often stated in the literature. When the Born approximation fails, to [see Eq. (W)] the electron can tunnel through the potential barrier and the mobility is relatively higher (see, for example, the phase-shift calculations by Blatt67).The lesson is that geometric size of an impurity is a poorly defined quantity. F. 1. Blatt, Solid State Phys. 4, 199 (1957).

-

67

,.,

-

-

-

28

D. L. RODE

the Born approximation is valid only for dilute impurity Concentrations anyway6'S6' and, in this case, the two theories give practically identical numerical results. (The exception occurs for electron freezeout. See Section 12c.) In addition, we always assume the Born approximation since the theory of electron scattering then becomes exceedingly straightforward. We present here a variation on Dingle's theory2' which is appropriate to n-type material, strictly speaking, but which also describes intrinsic material with fair accuracy. This formulation is used for the calculated results of Part IV. All ionized impurities and holes are assumed to form a spatially uniform background of charge. The holes are, in the main, heavy and sufficiently immobile during electron-hole scattering events to be treated as fixed." Consistent with these assumptions, we take the spatial hole distribution to be chaotic so that holes do not contribute to screening of other charged centem6' In this manner, only electrons perform the screening function through localized space charge. In the case of the Brooks-Herring formulation, the impurity potentials overlap one another and screening is enhanced correspondingly. Hence, in the region of electron freezeout at low temperatures where the charged-impurity concentrations greatly exceed the electron concentration, one must use the Brooks-Herring theory. Poisson's equation for the potential cp surrounding the impurity at location r = 0 with charge q is now

where 6n represents the departure from uniformity of the free-electron concentration, E~ is the low-frequency dielectric permittivity, and 6(r) is the Dirac function. In general,

For dilute impurities, ecp/KT is small in an average sense and Eq. (67) can be expanded to first order in the potential2' (when this assumption fails, one must consider multiple scattering6' *62): (68) 68

Dingle" allows for the finite mass of holes and their screening ability. However, the added complications of the formulation are rarely useful (except for minority carrier transport) since validity of the Born approximation guarantees insensitivity of the results to the screening length.

1.

LOW-FIELD ELECTRON TRANSPORT

29

where f is the Fermi-Dirac distribution, Eq. (33). From Eq. (66).27 (69)

cp = (q/4mor)e-fir,

where the inverse screening length B is given by

8’

= (e2/EoicT)J(k/n)’f( 1 - f )dk

The distribution f is related to the free-electron concentration through the Fermi level, ”

n = J (k/n)’f dk. Equations (69) and (70) are valid for dilute impurity concentrations in the sense that the deBroglie wavelength be much smaller than the screening length. More accurately, we require [see Eq. (9O)J that 4k2/p2 1. For nondegenerate GaAs at room temperature, this condition is easily met since 4k2/j3’ x 3.3 x 1O’’ln [ ~ r n - ~ ] . The Born approximation allows the differential scattering rate s(k, k) of Eq. (24) to be related, through the golden rule, to the matrix element of cp between states k and k in the following manner. First, we note that the potential, measured at r, of an ionized center located at R is

w(r

- R) = (q/4ncOlr- RI) exp( -Plr

- RI).

(72)

The total potential measured at r due to a randomly distributed assembly of identical centers is

u(r) = C w(r - R). R

(73)

The scattering process can be regarded as elastic since very little energy exchange occurs between the scattered electron and the heavy ionized center during a single scattering event. Accordingly, the Born approximation yields s(k, k) = (271/h)((kp( - eu(kp)12p,a(& - &’),

(74)

where pk is the density of states function in momentum space and /3 is a spin index for the basis state (kp) given by Eq. (8). The matrix element (k’p I - eul kp) connects the states characterized by admixed wave functions of the Bloch type normalized to the crystal volume V, lkS> = ( l / f l ) c p & ) eMik r),

(75)

30

D. L. RODE

where qkpis given by Eq. (8). Hence, summing over spin directions, we have

x

-

[exp i(k - k’) r]

(76)

lr - Rl

The ( p k are cell-periodic and lend themselves nicely to Fourier expansion in reciprocal lattice vectors k,, where m = (mx,my,m,) represents a triplet of quantum numbers; k, = 0 and k, = mn/a, where 2a is the lattice constant. We have q$(pk =

m

-

BPk exp( - ik, r),

(77)

Substitution in Eq. (76) gives

(79) where the integral can be evaluated in spherical coordinates by defining Km = k - k‘ - k m a n d r ’ = r - R :

= 2n[exp(iK,

-

*

R)] Jomr’dr‘

f, exp(iK,r’x dx

- pr’)

(81)

-

47r exp(iK, R).

+ p2

Km2

The r‘ limit in Eq. (81) has been extended to infinity since the crystal volume is assumed large and the wave functions have been normalized. In addition, k and k’ must be much smaller than a reciprocal lattice vector in order for a normal type of scattering event to be valid. This condition is easily satisfied in practice, and therefore terms beyond the first [with m = (0, 0 , O)]

31

1. LOW-FIELD ELECTRON TRANSPORT

in the sum over m may be neglected. Dropping the superscript on BfGk, we have 4 "" exp[i(k - k') R] . (k'JvJk) = __ E ~ JVk- k(' + f12

-

Multiplication of the matrix element of Eq. (83) by its complex conjugate converts the summand to unity at the randomly located impurity sites. The remainder of the sum vanishes and the sum becomes equal to the total number of scattering centers N,V in the crystal with charge q. The concentration N , of these centers is constant for the uniform distribution,

The density of states function in momentum space is V/8n3 (since a sum over spin has already been included) so that from Eq. (74), s(k', k) =

e2q2N , IBk,kl2 4 7 ~ ~ ~ , ~ h( IkI2 k'

+

/?2)2

a(& - E ) .

Note that s(k', k) = s(k, k ) , in agreement with detailed balance [Eq. (29)J. The coefficient IBkrk12 is the overlap integral evaluated in Eq. (12). We see also that the differential scattering rate is independent of the sign of the charge q, so that all singly ionized impurities with charge & e can be collected into N,: N, = N = N +

+N- +p,

(86)

where N is defined as the total concentration of electron scattering centers, and N + , N - , and p are the respective concentrations of ionized donors, acceptors, and holes. Finally, the differential scattering rate for ionized centers is

The overlap integral G ( k , k) is unity for parabolic bands and less than unity for Kane bands. This fact suggests diminished ionized-impurityscattering in narrow-gap crystals. Ehrenreich" has pointed out that G tends to decrease the relative amount of large-angle scattering [see Eq. (12)]. Elastic scattering processes enter the transport equations only through the relaxation rate y e , of Eq. (37). The reduction of the differential scattering rate from an integral over k' proceeds without approximation. Let vii equal the portion of v,, due only to ionized centers. Expressing k' in spherical

32

D. L. RODE

coordinates (k’, X, cp), we obtain ‘I

Jo

+’

e4N 2nsO2hf_,

v.. = ___

(1 - X ) G ( k , k ) 6(1 - &“)(k‘)’dk‘ d X , (Ik’ - kI2 p’)’

+

(88)

where X is the cosine of the angle between k’ and k, and the integral over rp has been performed. Since the argument of the Dirac function requires k‘ = k, conversion from energy coordinates to momentum coordinates by the Jacobian d of Eq. (20) yields the following integral of standard form:

The relaxation rate for ionized centers is therefore62 vii

where

D

=1

=

(e4Nmd/8nsO2h3k3)[D In(1

+ 4k2/p2)- B ] ,

+ (2/3’c2/k2)+ (3fl4c4/4k4),

4k2fp’ B = 1 + 4k 2/ p2

/I2 + 8

+ 2k2

p2 + 4k2 c2 +

3p4

+ 6b2k2 - 8k4 c4.

(p2 + 4k2)k2

(90) (91) (92)

Equation (90) is to be substituted directly into Eq. (40) for the perturbation distribution g. The quantities D and B contain corrections for Kane bands. When the energy gap is large, the conduction band becomes parabolic and c = 0, so that D = 1 and B = 4k’/(B2 + 4 k 2 ) , which is simply the BrooksHerring,” Dingle,” or Con~ell-Weisskopf6~ result, with suitable choices for the screening length. In general, c is slightly greater than zero, but less than unity, and the Born approximation criterion (4k2/p2$. 1) shows that D x 1 while B z 1 + 4c2, which does decrease ionized-impurity scattering slightly in nonparabolic bands. The relaxation rate vii is very approximately proportional to l / k 3 and leads to a T312mobility dependence, aside from screening factors. Screening effects appear only insensitively through the logarithmic term and through the B term of Eq. (90). When the Born approximation, 4k2/b2 $ 1, is valid, noticeable errors in p usually introduce small errors in v i i . This fact allows rather than that of Brooks and us to use Dingle’sz7simpler expression for /I Herring,28 which requires knowledge of neutral impurity concentrations. For the Brooks-Herring theory, one simply adds to p2 of Eq. (70) the term (e2/.zoKT)(ND- N - - n)(n + N - ) / N , , where N- is the ionized acceptor concentration and N D is the total donor concentration. Besides the Born approximation, the other major assumption of the foregoing description regards binary scattering,62 i.e., the electron interacts with only one ionized center at a time. This assumption is implicit to Eqs.

1. LOW-FIELD ELECTRON TRANSPORT

33

(73) and (74). The Ilk3 dependence of v i i ensures that multiple scattering corrections will eventually dominate at low temperature, for the following reasons. For all practical cases, even n-type crystals contain a few compensating acceptors. At low temperatures, n decreases due to carrier freezeout while p i i falls approximately as T3I2.Hence, the electron deflection time in the impurity potential (see below) becomes greater than the mean time elapsed between scattering events, i.e., the electron senses several scattering centers simultaneously. Now that we have an explicit formula for v i i , the validity of the binary scattering assumption can be specified more accurately. The time required for the deflection of the electron by the ionized center must be smaller than the mean time between collisions,

<

(93) The deflection time T~ is approximately that necessary for the electron to travel J2 x (screening length). For a nondegenerate semiconductor with a parabolic band, the screening length is the Debye length (i.e., 1/B) and thus 7D

1/Vii.

Eom*/ne2, (94) which shows that zD = l/cop, the inverse of the plasma frequency. Evidently, Eq. (93) requires the free-electron plasma to be lightly damped : 7D2 =

cop > V i i .

(95) A more convenient criterion results from the partial mobility due to ionizedcenter scattering alone, pii = e/m*vii, so that Eq. (93) yield^^^,^' p i > Eofm*n.

(96)

For example, Eq. (96) reduces to (piiin MKS units) pi

> 2 x 1oi5/n [cmP3]

-

l/m*, SO that Eq. (96) is for GaAs. Furthermore, Eq. (90) for vij gives p i implicitly almost independent of effective mass. Generally, the Born approximation (4k2/B2 9 1 is also known as the incoherent Born approximation6’) fails at high electron concentrations, whereas multiple scattering occurs at low temperatures, in modification of earlier conclusions6’ in connection with phase-shift calculations. When the We could have deduced a related criterion from the required smallness of ecp/tiT after Eq. (67), but this approach is not extendable to compensating impurities. For example, N - = 0 for an uncompensated n-type crystal and Eq. (96) requires T3/nto be large in the nondegenerate limit. Similarly, requiring ecp/KT to be small at a Debye length distance from the impurity leads to large T’/n; requiring ecp/KT to be small at a mean interimpurity distance leads to large T3/n.Interestingly, requiring many electrons in a Debye sphere also yields the same limit.

34

D. L. RODE

present theory fails, one has recourse to the work of Moore6' on multiple scattering, electron-impurity dressing, and higher Born approximations, although our present criterion, Eq. (96), is considerably less restrictive than that indicated by Moore. Considering the good agreement with experiment, however (see Section 12), Eq. (96) may not be sufficient to predict failure of the Brooks-Herring theory. 7. PIEZOELECTRIC ACOUSTIC MODES In cubic crystals lacking inversion symmetry (e.g., sphalerite), the piezoelectric stress tensor is nonvanishing. Arlt and Quadflieg7' have measured piezoelectric constants for a variety of materials and have proposed the microscopic origins of piezoelectricity as being due to (a) ionic polarization, (b) strain-dependent ionicity, and (c) electronic polarization. One of the surprises of recent years (in view of the old Born theory of piezoelectricity) is that the piezoelectric constants in some 111-V crystals are comparable in size to many of the more ionic 11-VI crystals. Phillips and Van Vechten71 were able to explain theoretically not only the change in algebraic sign of the piezoelectric constants between 111-V and 11-VI crystals, but also the approximate magnitudes of these quantities. Their results suggest that effect (b), which accounts for the strain-induced flow of covalent charge between sublattices, may be the dominant source of piezoelectricity in 111-V compounds. These results lead to the importance of this interaction in transport theory. At low temperatures ( 250"K),lattice scattering of electrons is which causes only elastic dominated by the piezoelectric interaction,' scattering due to the relatively low energy of acoustic phonons. Because of impurity scattering, however, piezoelectric scattering is usually not of major practical importance, except in high-purity crystals. Piezoelectric coupling occurs with acoustic modes of long ~ a v e l e n g t h . ~ ~ , ~ ~ Both transverse and longitudinal vibrations contribute, but only the longitudinal electric fields of these vibrations are significant.7 5 This interaction can be considered separately from the deformation-potential interaction of Section 8 since the perturbing potential is proportional to strain in the latter case. The piezoelectric interaction potential is proportional to the strain gradient. The resulting phase quadrature of these potentials allows them to be considered separately in the Born approximation. 3p72

G. Arlt and P. Quadflieg, Phys. Status Solidi 25, 323 (1968). J . C. Phillips and J . A . Van Vechten, Phys, Rev. Lett. 23, 1 1 15 (1969). l 2 D. L. Rode, Phys. Reo. B 2,4036 (1970). '3 H. J. G. Meijer and D. Polder, Physica 19,255 (1953). 74 W. A. Harrison, Phys. Rea. 101,903 (1956). 7 5 A. R. Hutson, J . Appl. Phys. Suppl. 32, 2287 (1961).

'O

"

1. LOW-FIELD ELECTRON TRANSPORT

35

The potential matrix element for piezoelectric scattering has been derived for parabolic band^,^^.^^ and Hutson and White76 have given the piezoelectric coupling coefficientappropriate to sphalerite and wurtzite structures. Allowing for anisotropy in the effective mass, permittivity, and piezoelectric ~ provided expressions suitable to parabolic bands. interaction, 2 0 0 k ~has These expressions involve integrals which cannot be reduced analytically if the effective mass is anisotropic. But we recall that mass anisotropy is ordinarily negligible for direct gap semiconductors (see Section 1). Hence, we generalize Zook’s treatment, in the case of isotropic effectivemass and permittivity, to the Kane” bands described in Section 1 by inclusion of the overlap integral G(k, k), Eq. (12). Electron degeneracy is automatically accounted for since Fermi statistics do not affect elastic scattering rates [see Eq. (35)l. In this case, the differential scattering rate for piezoelectric scattering is

where P is a dimensionless piezoelectric coefficient whose form depends upon crystal structure. P resembles the spherically averaged electromechanical coupling coefficient discussed by Hutson7’ and Z 0 0 k . ~P~ is isotropic for the sphalerite structure with one independent element in the piezoelectric stress tensor hI4, P 2 = ht4&,,[(12/c,)

+ (16/c,)]/35

(sphalerite),

(98)

where the spherically averaged elastic constants for longitudinal and transverse modes are respectively c1and c,. These constants are given by Z 0 0 k ~ ~ in terms of the three independent elastic constants cll,c 1 2 ,and c44,7 8

+

cL = (3c1, + 2c1, 4c4,)/5 c, = (el1 - c12 + 3c44)/5

(sphalerite), (sphalerite).

(99) (100)

The quantities h,, and cij have been measured and are assumed, to good approximation, to be temperature independent. Equations (99) and (100) show that scattering by transverse modes is much more effective than scattering by longitudinal modes. The ratio of the respective scattering 7h 77

A . R. Hutson and D. L. White, J . Appl. Phys. 33,40 (1962). J . D. Zook, Phys. Rev. 136. A86931964).

’’ The plausibility of Eqs. (99) and (100) is fully evident when we note that nearly the same results obtain when we choose an average longitudinal sound speed u , weighted for six ( 100). twelve ( I lo), and eight (1 1 I ) directions, i.e., c, t pu,* = ( 2 2 c , , + 17c,, + 3Jc4,)/39, and similarly for transverse modes, c, = P U , ~= (17c,, - 17c,, + 44c4,)/78. Theagreement is e.xacI for central-force atomic interactions.

36

D. L. RODE

rates for isotropic crystals whose atoms interact only by central forces (giving7’ c l Z= c44 = c11/3)is (16/c1)/(12/cJ = 4, which is approximately true for sphalerite crystals discussed in Part IV. We assign an average isotropic value to the low-frequency permittivity E, for the uniaxial wurtzite structure. Anisotropy in the mobility has occasionally been observed, however, and we retain this feature in our description in the following way. 7 7 Anisotropy in the piezoelectric interaction accounts for most of the observed anisotropy in transport phen0mena.7~Thus, we use for drift mobility measured with electric piezoelectric coefficientsPl and P,, field perpendicular or parallel to the unique c axis of the crystal. For the wurtzite structure, there are three independent elements of the piezoelectric stress tensor, h15, h31, and l ~ , , , ~ ’

+ 6h1,hx + h,’)/105c1 + ~,(2lh:, - 24h,,hx + 8hx2)/105c, PI,’ = b0(2lh:, + 18h,,hx + 5hx2)/105c, + ~,(63h$,- 36h,,hx + 8hX’)/105c,

PL’ = 4&,(21h:,

hx

= h33

- ‘31

-

2h15

(wurtzite),

(101)

(wurtzite),

(102)

(wurtzite).

(103)

The spherically averaged elastic constants are related to only four of the six independent elastic constants since not all acoustic modes are piezoelectrically active,77

+ 4c1, + 3c,, + 8c4,)/15 c, = (2c1, - 4c1, + 2c,, + 7c4,)/15

cI = (8c1,

(wurtzite),

(1 9

(wurtzite).

(105)

The piezoelectric constants h i jand elastic constants c.. are assigned tempera‘’. ture-independent values in Part IV. Direct experimental measurement shows this to be a good as~umption.’~~’’ The piezoelectric interaction is elastic, and its total influence on transport effects enters through the relaxation rate of Eq. (37). This portion of v,, will be called vpe :

sf, lo

cc (1

: &vpe : = ; ___ ‘

- X)G(k‘,k) 6(6 - S’)(k‘)’dk‘ dX , [k’- k(’

(106)

where Xis the cosine of the angle between k’ and k. From Eqs. (9), (12), and

’’ F.Seitz. “Modern Theory of Solids.” McGraw-Hill, New York, 1940.

1.

LOW -FIELD ELECTRON TRANSPORT

37

(201, we obtain Vpe

eZKTP2md - 4nh3s0k

-

(aZ

+ c2X)' dX

- eZKTPZmd(3

- 6c2 + 4 ~ ~ ) . 6nh3&,k For parabolic bands, we have c = 0, but in general c is slightly greater than zero and Eq. (108) shows that nonparabolicity decreases the scattering rate vpe as was also the case for ionized-impurityscattering. Piezoelectric scattering is proportional to T/k, so that the corresponding partial mobility is proportional to T-'". 8. DEFORMATION-POTENTIAL ACOUSTIC MODES Electron scattering in semiconductors by the deformation-potential interaction with long-wavelength acoustic vibrations was proposed by Bardeen and Shockley.'.*' The theory has been discussed by many aut h o r ~ , ~ , ~and , ' ' we will simply adapt the appropriate matrix element to our present model. The interaction is elastic for temperatures above a few degrees Kelvin, o and the law of equipartition is valid in this case, i.e., there are ~ T l h phonons per mode, where h a is the phonon energy.54We may safely assume that movements of the conduction band cause no noticeable change in effective mass, despite Eq. (5) relating effective mass to energy gap, provided the electron energy is much smaller than the energy gap." If this condition is not well satisfied (for example, in InSb at high temperatures), the material is highly intrinsic (i.e., large free-electron concentration) so that electronhole scattering dominates the transport behavior anyway. Therefore, the matrix element given in the literature' should be valid for all cases studied in Part IV. Including nonparabolic bands, the differential scattering rate for deformation-potential acoustic interaction is s(k, k) = (e2rcTE,2/4nZh~,)G(k', k) 6(1 - 8),

(lo91

where c1 is the spherically averaged elastic constant for longitudinal modes, Eq. (99) or (104). The deformation potential E , (units of eV per strain or simply eV) is equal to the distance the conduction band edge shifts (in eV) per unit strain due to the acoustic vibration. Note that s(k, k) is independent of the angle between k' and k for parabolic bands where G(k', k) = 1. This is the property of randomizing scattering discussed by Herring and Vogt.6 J . Bardeen and W. Shockley, Phys. Rev. 80, 72 (1950). This criterion is slightly less restrictive than that stated earlier.'4 See C0nwe11,~~ p. 108.

38

D. L. RODE

Their theory of multivalley conduction shows the importance of transverse modes for deformation-potential scattering. For a single isotropic T l C valley, only longitudinal modes contribute to Eq. (109). In general, s(k’, k) is not completely randomizing, because of the overlap integral G(k’, k), which emphasizes forward scattering. The relaxation rate follows immediately from Eq. (37) and (109), ‘ac

=

e 2 K T E 1 2J:, 2nhc,

1:

(1 - X)G(k,k) 6(6 - 6’)(k’)’ dk‘ dX

(1 10)

- e 2IC TE mdk (3 - 8c2 + 6 ~ ~ ) . 3nh3c,

The scattering rate vac is added to ve, of Eq. (40). Acoustic scattering, from Eq. (1 12), is proportional to k and leads to a nearly constant mean free path. The corresponding partial mobility is proportional to T - 3’2. All the parameters appearing in va,, characteristic of a given material, are known from independent experiments. The deformation potential can be related to the pressure rate coefficient of the energy gap,’”’ provided the corresponding valence-band shift is negligible. Ehrenreich’s arguments suggest this may be so, and calculations in Part IV for GaAs (sphalerite structure) tend to support this conclusion. 9. POLAR OPTICAL MOD=

In the heteropolar 11-VI and 111-V compounds, longitudinal optical modes have an associated electric polarization wave. This electric field accompanying the polar mode provides the dominant electron scattering mechanism near room temperature in direct semiconductors.’4*72Furthermore, the energy of optical phonons is comparable to K T at room temperature, and the inelastic nature of this process must be included in any quantitative theory. According to the discussion of Section 2, polar mode scattering cannot be assigned a relaxation rate. Instead, we calculate the inelastic scattering operators Si and So of Eq. (40) directly. It is due to polar mode scattering that the equation for the perturbation distribution g becomes a finite difference equation since the electron distribution at energy 8 now becomes related to the distribution at energies 8 polar phonon energy. The matrix element of the perturbing potential appears in the literat ~ r e , ~ and ’ , ~ Ehrenreich” ~ has shown that the generalization to nonparabolic bands simply multiplies the matrix element by the overlap integral

1.

LOW-FIELD ELECTRON TRANSPORT

39

G ( k , k) [see Eqs. (85) and (87)]. The differential scattering rate for polar modes in the Born approximation is

+

+

6(& - 8‘ ~ w , ) ( N ~ ~l), 6(& - &’ - hOpo)Npo,

emission, absorption,

(113)

where E , and E~ are the respective high-frequency and low-frequency lattice permittivities, and hapois the bngitudinal optical phonon energy, which we assume constant (since only small k vectors relative to a reciprocal lattice vector are of concern”) and equal to the value at the center of the first Brillouin zone. The Npoterm corresponds to scattering by phonon absorption, and the Npo+ 1 term corresponds to scattering by phonon emission; N , is the phonon occupation number (i.e., the average number of phonons contained by a vibration mode) and is assumed to have the equilibrium Bose-Einstein distribution : 1 Npo

= exp(hWpJKT) - 1

The use of Eq. (1 14) excludes phonon-drag and acoustoelectric effect^.^.^ The relative factor Npo/(N, + 1) in Eq. (1 13) for absorption/emission rates precisely reflects the Boltzmann factor appearing in Eq. (29), which allows us to prove that the isotropic distribution f is the equilibrium Fermi-Dirac distribution since N,/(N, + 1) = exp( - ho,/KT). For arbitrary electron degeneracy,the value off affects the rate of inelastic scattering through the Pauli exclusion principle. This fact is evident from Eqs. (38) and (39) for the scattering-out and scattering-in rates So and Si containing the function$ From Eq. (1 13), it is clear that s(k’, k) # s(k, k‘) since reversal of the order in time of the occurrence of states k and k’ changes an emission event [proportional to ( N , + l)] into an absorption event (proportional to Npo)and vice versa. Hence, we prefer to write14 So =

I+

+ 1 - f - ) A o - + (Npo+ f + ) A o +

[s(k, k ) (1 - f’) $k’, k)f‘] dk’ ,

so= (Npo

(115)

(1 16) where superscripts plus and minus, corresponding to scattering-out by absorption and emission, indicate that the function is to be evaluated at the energy b + hw,. In this manner, we can work with the energy-dependent scattering rates lLo+ and lbo-,which do not contain phonon occupation

40

D. L. RODE

numbers. We have from Eqs. (1 13) and (1 16), -

’’

e * opomd = 47th’k

+

A + = aa+

($

-

1

A+cc+ - aa+cc+ ,

i),

(117)

+ k2 cc+, + (k+)’ 2k+k

and similarly for L,-(k). When the electron energy is less than ho,, phonon emission is not possible, and the Lo- term of Eq. (1 16) is understood to vanish. The additional terms appearing due to nonparabolicity obscure the functional behavior of S o , but for parabolic bands where c = c- = c + = 0 etc., the scattering-out rate is simply proportional to

Lo+

-

&-I!’

+ + &/how)’/2],

ln[(&/hop0)*~’ (1

( 120)

which tends to a constant nonzero value as & approaches zero. This is the same constant value that appears in previous approximate expressions for polar m~bility.~’ In general, we use the full expression, Eq. (1 17). The corresponding scattering-in operator, from Eq. (39), is Si(g‘)= fXg‘[$k’, k)(1 S,(g’) = (Npo+ f1Ji-g-

f) + s(k, k ) f l dk’,

+ Wpo+ 1 - m i + g + ,

(121) (122)

where, as before, L i - and J i + are energy-dependent scattering rates aside from phonon occupation numbers. The roles of the superscripts plus and minus are interchanged with respect to Eq. (116) and now correspond to scattering-in by emission (+) and absorption (-). We have

and similarly for I,,-(/?). Equation (123) is interesting in connection with the relaxation approximation at low temperatures T % h w p o / ~For . example, in a parabolic band at low energy, Eq. (123) shows that

The right-hand side vanishes proportionally as & ‘ I 2 as energy approaches zero. This is a very important result for low-temperature mobility calculations since the vanishing of scattering-in (independently of phonon occupa-

1.

LOW-FIELD ELECTRON TRANSPORT

41

tion number) guarantees the success of the relaxation approximation. The physical cause for this behavior is evident from Eq. (1 13), which shows that s(k, k) becomes fully randomizing when k becomes small; for a fully randomizing process, scattering-in always vanishes,6 whether or not the process is inelastic, as is epitomized by intervalley scattering discussed in the following section. Unless explicitly noted, we use only the general formula [Eq. (122)] in Part IV so the description should be accurate for high temperatures as well as low temperatures.

MODES 10. INTERVALLEY The development of the transport problem thus far in this chapter has been tailored to apply to direct semiconductors with an isotropic band. There is also considerable interest in the indirect crystals (e.g., Ge, Si, and Gap), and in fact the following simplified model allows a reasonable, albeit semiempirical, description of indirect materials, provided we include inelastic intervalley ~cattering.~’ The only additional assumption needed for this purpose is that each of the varioqs equivalent, indirect minima are isotropic and parabolic. Actually, the minima are ellipsoidal,4 but the dominant lattice-scattering mechanisms are randomizing6 so that the “spherical band” approximation allows a very good empirical fit between experiment and theory on pure crystals. This approach fails for scattering by ionized centers.29 For example, Si has six A l Cminima, at the conduction band edge, located at82 0.83KIo0, where KIOOis the zone-edge lattice vector extending along (100) as shown in Fig. 5. These minima are surrounded by surfaces of constant energy in momentum space which are ellipsoids of revolution with their major axes (also the axes of revolution) lying along the six respective ( 100) axes. Besides the various intravalley scattering processes discussed earlier, there will be some additional scattering between valleys which is called equivalent intervalley scattering (the valleys being equivalent to one another in this case). Since acoustic intravalley scattering (Section 8) and intervalley scattering are randomizing processes, as evidenced by their constant matrix elements,83the assumption of isotropy is a reasonable approximation insofar as the functional dependence of scattering is concerned. We cannot expect to independentlyderive absolute values of transport quantities from the model. Hence, we have the “semiempirical” designation mentioned earlier for the present description of indirect crystals. The remaining scattering mechanisms emphasize forward scattering (see Sections 6-9) and are only poorly described by the isotropic model. 82

83

A. K . Hochberg and C. R. Westgate. J . Phys. Chem. Solids 31,2317 (1970). Nonpolar optical scattering is also randomizings4 and occurs in Ge, for example.

42

D. L. RODE

FIG. 5. First Brillouin zone for electrons in Si. The six equivalent A l e minima possess ellipsoidal constant-energy surfaces and are located along
However, piezoelectric and polar scattering (possible only in heteropolar substances) in Gap, for example, are not dominantz9above 20"K, and some error in this respect seems tolerable. For ionized impurities, the isotropic model furnishes only qualitative results as discussed by Rode.29 Hence, pure indirect crystals are fairly amenable to the assumption of isotropy. To include intervalley scattering, we must first identify the momentumconserving phonons involved. Group theoretical analyses84 are usually used in this respect to identify not only the phonon symmetry but also the phonon polarization, namely transverse or longitudinal, optical or acoustic. On the other hand, a more lucid discussion can be constructed by the graphical approach, depicted in Fig. 6, which identifies the phonon symmetry and energy. This method also demonstrates very simply which of the intervalley transitions are of the umklapp type. For the phonon polarizations, we refer 84

C. Herring, J . Franklin Insr. 233,525 (1942).

Id)

lb)

FIG. 6. (a) Labeling convention for the first Brillouin zone of the sphalerite structure. Greek and English letters at high-symmetry points respectively indicate points interior to the zone and on the zone boundary. The characters themselves correspond to the irreducible representations of the orbital states at the given locations. Noting that several such zones may be close-packed, it becomes obvious that the points K and (Iare equivalent, as are also the lines Z and S, which map onto one another. (b) Intervalley scattering among the three equivalent X minima (X,X', and XI) occurs via K,,, phonons as shown for an electron at X, for example. This is an umklapp process which occurs in GaP and in the XI,satellite valleys of GaAs. (c) Gelike minima at L points also require intervalley phonons of momentum K,,, as in (b). Unlike the case shown in (b) however, the phonon shown here is still K,,, (a reciprocal lattice vector) even considering transitions to second Brillouin zones. Hence, this process is just on the boundary between being an umklapp and a normal process. (d) Scattering between Si-like minima (see Fig. 5 ) consists of two distinct types, l a n d g, as illustrated here. From valley I , transitions to valley 2 @-scattering) require an X-directed phonon participating in an umklapp process. For transitions to the remaining valleys 3-6 (f-scattering), the phonon labeled f in the figure extends beyond the first Brillouin zone. Hence, reduction of this same phonon to the first Brillouin zone yields the phonon labeled S. The latter phonon lies along Sin (a). Thecorrespondingfand g phonon energies are shown in Fig. 7.

43

44

D. L. RODE

to the results of group We consider only crystals with the cubic sphalerite structure. The labeling convention for the first Brillouin zone is shown in Fig. 6a, where points interior to the zone are named by Greek letters and points at the zone edge are named by English letters. Figure6b applies to three ellipsoidal minima occurring at X points. In order for the electron to scatter to either of the remaining two valleys, a phonon with momentum hKlo0 is required. The corresponding phonon energy h a is given by phonon dispersion curves such as those shown ins8 Fig. 7.

,

' 0

0 2 0 4 0.6 0 8 101.0

--ccoo]

06

0.6

0.4

Porcl-

02

0' 0 01 0.2 0.3 0.4 0 5

REDUCED WAVE VECTOR COORDINATE

-[!. 5 !.I

FIG. 7. Phonon dispersion curves for Si at 296°K as determined by inelastic scattering of neutrons. Longitudinal (L) and transverse (T) polarizations are shown out to the edge of the first Brillouin zone along [[OO] or Xdirections, [([O] or Z directions, and [[[[I or L directions. Note that the points U or K in Fig. 6a lie at the dashed line [0, 0.75, 0.751. The extension of [Or[] to [OI 13 is just the line labeled S i n Fig. 6a.Intervalley phonons for Figs. 6b and 6c lie at [IOO]. Intervalieygphonons lie at [0.34,0,0] while f phonons lie at [0.83,0.83,0] corresponding to Fig. 6d for Si. Solid points denote undetermined polarization. (After Dolling.")

The arrangement shown in Fig. 6c corresponds to Ge with four equivalent minima at L points. Scattering from a given valley to any of the remaining three valleys again requires a phonon very near X. For Si, with minima at six equivalent A points, scattering across the Brillouin zone between valleys 1 and 2 in Fig. 6d differs from the diagonal case connecting valley I with valleys 3-6. The first process is called g-scatteringS9; the latter process is calledf-scattering. If the wave number k, at the minimum J. L. Birman, Phys. Rev. 127, 1093 (1962). J. L. Birman, M.Lax, and R. Loudon, Phys. Rev. 145,620 (1966). H. W. Streitwolf, Phys. Srarus Solidi37, K47 (1970). G. Dolling, Proc. Symp. Inelastic Scattering Neutrons in Solids,Liquids, Chalk River, 1962, Vol. 11, p. 37. IAEA, Vienna, 1963. 8 9 F. J. Morin, T. H. Geballe, and C. Herring, Phys. Reu. 105, 525 (1957).

86

''

I.

LOW-FIELD ELECTRON TRANSPORT

45

is less than K10,/2, then g-scattering requires a phonon wave number Q < K,,,andtheprocessisnotoftheumklapptype.Otherwise,k, > Kloo/2 and g-scattering from valley 1 to valley 2 involves a transition between distinct Brillouin zones, i.e., umklapp scattering. Similarly,f-scattering from valley 1 to valleys 3-6 requires a 2 phonon no matter how large k, may be, butf-scattering is not of the umklapp type if k, < 3K,,,/4. In this case, the electron remains in the same Brillouin zone after the transition. When k, = Kloo, we have the case shown in Fig. 6b. In fact,82k, = 0.83K1,, in Si, and the phonon wave vector extends outside the first phonon Brillouin zone (umklapp process). The phonon is, however, parallel to Z and subtraction of a reciprocal lattice vector shows that this phonon lies at a point S separated from the X point by 0.68 of the distance between X and K on a square face of the phonon Brillouin zone. The phonon energy and polarization are discussed in Part IV for particular materials. Electron effective masses for indirect crystals are sufficiently large that the Fermi energy rarely rises noticeably far above the band edge. For this situation, the various indirect valleys are very nearly parabolic, and the overlap integral does not appear explicitly in the scattering matrix element (being equal to unity). The matrix element given in the literatures4 leads, through the Born approximation, to the following differential scattering rate for equivalent intervalley scattering2' :

where hoe is the phonon energy, p is the mass density of the crystal, 2 is the number of equivalent valleys, and D, is the intervalley deformation potential (units of electron volts per meter). The lattice is in thermal equilibrium for low-field transport and N,, the phonon occupation number, assumes the Bose-Einstein distribution :

N, =

1 exp(hwe/lcT)- 1 *

The phonon equivalent temperature h w , / ~is usually comparable to room temperature. Thus, the electron energy changes considerably during an intervalley transition. This process does not, however, lead to a scattering-in operator Si like that appearing in Eq. (39). Because of the remoteness of the valleys, the differential scattering rate [Eq. (125)] is independent of the angle between k' and k so that Si = 0, i.e., intervalley scattering is fully randomizing in the sense of Herring and Vogt6 (see Section 2). Therefore, the scattering-out rate can be combined with the elastic scattering rate veI of Eq. (40). Denoting this portion of veI by ve and including arbitrary degeneracy in the

46

D. L. RODE

parabolic valley, we have

+ s(k', k)f'] dk', ve = ( N e + 1 - f - ) I e - + ( N e + f + ) I e + , v, = J[s(k, k')(l - f ' )

(127)

(128) where superscripts plus and minus correspond to phonon absorption and phonon emission, respectively. Superscripted functions are evaluated at energies 8' = 8' k h o e , as in Section 9. The energy-dependent scattering rates I.,+ and A,- are independent of phonon occupation numbers :

I,'

- l)m*k+/2aphZo,,

= e2De2(Z

(129) and similarly for I,,-, which is understood to vanish for 8 < h o e when phonon emission by the electron is not possible. The intervalley scattering rate is proportional to k' and resembles intravalley acoustic scattering [proportional to k,see Eq. (1 12)]. At low temperatures, compared to h o , / ~ , intervalley scattering is dominated by the exponential term of N e ,and the mobility rises rapidly with decreasing temperature. At high temperatures, intervalley scattering mimics acoustic scattering and leads to a TA3/'mobility dependence. This property generally explains the T - " mobility dependence of Si and Ge (see Section 13) where xis slightly greater than $ at room temperature. The results of this section complete the theoretical picture. Formulas derived here for electron scattering are combined with Eq. (40) of Section 2 to yield the electron distribution function. The various modifications discussed in Sections 3-5 then yield the calculated results presented in Part IV along with experimental results. IV. Results

In the following sections, the electron transport properties of particular crystals are reviewed. Each presentation is accompanied by a brief discussion, but a general discussion based on overall trends of the comparisons is deferred to Part V. The experimental results are interpreted theoretically within the framework of Parts I1 and 111. Observations on these large groups of crystals suggest where theoretical improvements might usefully occur and which additional experiments are needed. To characterize the crystal, we need several parameters, all of which can be obtained from experiments independent of transport properties in the case of direct semiconductors. Thus, there are no adjustable parameters in comparisons between theory and experiment for direct crystals. Parameters that yield the band structure are (1) the effective-massenergy gap and (2) the effective mass or polaron mass. If these quantities are rather small (e.g., in

1. LOW-FIELD ELECTRON TRANSPORT

47

InSb), the temperature dependence of energy gap should be included through Eq. (22), which requires (3) the pressure rate coefficient of the energy gap, (4) the compressibility, i.e., elastic constants, and ( 5 ) the thermal expansion coefficient. The remaining material parameters pertain to scattering mechanisms; (3) and (4) appear in acoustic and piezoelectric scattering rates. In addition, we need (6) the piezoelectric stress tensor, (7) the high-frequency dielectric constant, (8) the low-frequency dielectric constant, and (9) the LO phonon frequency at r. Values used for carrier concentration are measured intrinsic values for lattice-limited transport or are given for specific cases of impurity doping. For indirect crystals, the deformation potentials [Eqs. (112) and (129)] are derived empirically. The good agreement with experiment obtained with only two adjustable parameters suggests that the correct scattering mechanisms have been identified.

1 1.11-VI CRYSTALS The 11-VI crystals with I1 = (Zn, Cd, Hg) and VI = (0,S , Se, Te) comprise a large and varied group of properties. Some of the low-atomicmass materials, such as ZnO, CdS, and CdSe, usually possess the hexagonal wurtzite structure. The anistropy of transport in these crystals is only slight and the direct gap is amenable to description by the Kane band structure discussed in Section 1. Crystals of intermediate mass (ZnSe, etc.) possess the zinc-blende structure and have been widely studied for their electrical and optical properties. The heavy-mass mercury chalcogenides(HgSe and HgTe) are zero-gap semimetals. But even here, nature has been kind enough to allow electron transport in a l-8 symmetry band not unfamiliar to us. Indeed, only minor modifications to the work of Parts I1 and I11 are necessary for a rather good theoretical description of these zinc-blende semimetals, as Broerman has shown.I8 a. ZnO

ZnO is a direct wurtzite crystal with a small amount of uniaxial electron mass ani~otropy.’~ Relatively speaking, ZnO was one of the earliest crystals (after Ge, Si, and InSb) to be prepared in rather pure form. Unfortunately for electroluminescence applications, only n-type conduction appears possible. During the 196O’s, detailed measurements on good quality crystals were made in connection with piezoelectric and acoustoelectric device applications. At low temperatures, the direct gap lies in the ultraviolet at”,’’ 3.435 eV, and we can take this value as equal to the effective-mass gap since it is so 91

D. G. Thomas, J . Phys. Chem. Sohds 15, 86 (1960). Y . S. Park, C. W. Litton, T. C. Collins, and D. C. Reynolds, Phys. Rev. 143,512 (1966).

48

D. L. RODE

large. The high-frequency effective mass was determined by Baer9’ from room-temperature Faraday rotation to be m*/m = 0.24. His results assume unity for the Hall scattering factor, whereas our own calculations” for pure material suggest rH z 1.21 (see subsequent discussion and the results given in Fig. 9). Correcting for the latter value, we find m*/m z 0.264. The acoustic agrees to deformation potential E, = 3.83 eV found e~perimentally~~ 8% with the theory94 of the pressure rate coefficient of the direct gap. We derive piezoelectric coefficients from the data of Crisler et uI.,~’which differ considerably from earlier value^.^'*'^ In order to reduce their data9’ for piezoelectric strain constants dij to piezoelectric stress constants h y , we distinguish between dielectric constants measured at constant strain ~ ~ )those measured at (e.g., probably those of Heltemes and S ~ i n n e y and constant stress (e.g., probably those of Crisler et ~ 1 . ~The ~ ) results . for P [see Eqs. (101) and (102)97] are shown in Table I along with other material parameters necessary for electron transport calculations. The dielectric constants for constant strain are derived from the Lyddane-Sachs-Teller relation,96 and the polaron mass mp follows from Frohlich’s f o r m ~ l a . ~ ~ . ~ ~ A comparison between theory (from Parts I1 and 111) and experiment98is shown in Fig. 8, using the polaron mass. Mobility is measured with electric field perpendicular (pl) or parallel (p,,)to the c axis of the crystal. The agreement is satisfactory above 200°K considering that there are no adjustable parameters. Actually, the calculated curves are drift mobilities p, while the experimental data98 are Hall mobilities ,uH;in Fig. 9, we plot the calculated Hall scattering factor rH = pH!p at low magnetic field, whereby the agreement in Fig. 8 can be somewhat improved.98a Below 100”K, piezoelectric scattering dominates and the calculated mobility anisotropy agrees only approximately with experiment. H ~ t s 0 n . s ~ ~ piezoelectric constants yield agreement with anisotropy but give ,u too low by a factor of -2. Below 60”K, the experimental data are affected by impurity scattering. Of course, one could get excellent agreement at all temperatures for lattice scattering if the piezoelectric coefficients were empirically adjusted. However, it seems more instructive to maintain a rigorous calculation so that the weaknesses of the theory become apparent. The roomtemperature mobility is dominated by polar optical modes. W. S. Baer, Phys. Rev. 154,785 (1967). R. L. Knell and D. W. Langer, Phys. Lert. 21, 370 (1966). 94 F. Cerdeira, J . S. DeWitt, U . Rossler. and M. Cardona, P h p . Srorus Solidi41, 735 (1970). 9 5 D. F. Crisler, J . J. Cupal. and A. R. Moore, Proc. lEEE56,225 (1968). 9 h E. C. Heltemes and H . L. Swinney, J . Appl. Phys. 38,2387 (1967). 9 7 Note that the definition of piezoelectric coefficients P , and P,, includes an average dielectric constant E~ which cancels with c0 in the differential scattering rate [Eq. (9711. This procedure ensures the correct allowance of piezoelectric anisotropy according to Zook’stheory.” 98 A. R. Hutson, J . Phys. Chem. Solids 8,467 (1959); personal communication. 98aP. Wagner and R. Helbig, J . Phys. Chem. Solids 35, 327 (1974), also give ZnO mobility data.

92

93

1.

10'

20

40

49

LOW-FIELD ELECTRON TRANSPORT

60

100 200 TEMPERATURE, T

400 600

1000

(OK)

FIG.8. Theoretical electron drift mobility (solid curves) of pure ZnO compared to experimental Hall mobility (0,pL and 0 , pa) determined by Hutson.'* Since ZnO is uniaxial, mobility is measured with electric field perpendicular (flJ or parallel (pl,) to the crystalline c axis.

b. ZnS This is an allotropic crystal capable of assuming several stru~tures,9~ much to its disadvantage and ours, but the cubic sphalerite form is not uncommon and is stable at room temperature. We limit our discussion to the sphalerite form of ZnS, which also exhibits only n-type conduction. The direct band gap in cubic ZnS isloo 3.799 eV for low temperatures and can be taken as equal to the effective-mass gap. There is a small amount of effective-mass anisotropy in hexagonal ZnS."' From magnetooptical absorption, Miklosz and Wheeler"' find m* = 0.28m in hexagonal ZnS, in good agreement with Lawaetz's value' O2 of 0.267, calculated, however, 99

la'

Io2

W. L. Roth, see Devlin,' Chapter 3. B. Segall and D. T. F. Marple, see D e ~ l i n Chapter ,~ 7. J. C. Miklosz and R. G . Wheeler, Phys. Rev. 153,913 (1967). P. Lawaetz, Phys. Rev. B 4, 3460 (1971).

50

D. L. RODE

I

I

I

'

I

I

I

1

'

I

z no

I.o

20

I

40

I 60

I

I 100

I 200

TEMPERATURE, T

I I I I 400 600 1000 (OK)

FIG. 9. Theoretical Hall factor rH = pH/p of pure ZnO. Low-temperature values of rH near 1 . 1 result from piezoelectric scattering. Polar mode scattering becomes dominant above 200°K.

for cubic ZnS from a five-level k * p analysis. The difference in m* between these structures must be negligible since the spin-orbit and crystal-field parameters are small in either case while the direct gap is almost unchanged (within 3%100*101).For the elastic and piezoelectric constants in Table I, we have used the values of Berlincourt et uZ.'03 measured at 77°K since electron scattering by acoustic modes is significant only at low temperature^.^' In Fig. 10, we compare the calculated electron drift mobility for pure cubic ZnS to some experimental data104-106on hexagonal crystals which are somewhat impure. The comparison below 200°K is hampered by ionizedimpurity ~cattering."~Near room temperature, the calculated mobility of D. Berlincourt, H. Jaffe, and L. R. Shiozawa, Phys. Rer. 129, 1009 (1963). F. A . Kroger, Physiru 22,637 (1956). I o 5 M. Aven and C. A. Mead, Appl. Phys. L e l t . I, 8 (1965). '06 F. Matossi, K. Leutwein, and G . S. Schmid, 2.Nuturforsch. 21, A461 (1966).

Io3

'04

51

-

AA

10'

I

I

I

I

I

I

I

D

FIG.10. Theoretical electron drift mobility (solid curve) of pure ZnS compared to experimental Hall mobility: & I o 4 O,lo5 x

219 cm2/V-sec (Hall mobility is 269 cm2/V-sec) lies above the various values of from 140 to 193 reported in the l i t e r a t ~ r e . ' ~ ~Since ' ~ ' data on highquality crystals are not available, it seems useless to pursue the discrepancies in Fig. 10 meticulously. It does appear interesting that the experimental data'04 above 500°K take a marked downtrend which is reminiscent of multivalley conduction, although band structure calculations do not suggest nearby minima in cubic'o8 or hexag~nal'~'ZnS (for which the dataIo4 were taken). However, the conventional muffin-tin approximation to lattice potential used in the band structure calculation'o8 is subject to noticeable errors in energy levels.' l o lo'

lo*

log 'lo

C. S. Kang, P. B. P. Phipps, and R. H. Bube, Phys. Rev. 156,998 (1967). P. Eckelt, Phys. Starus Solidi 23,307 (1967). U. Rossler, P h p . Rev. 184, 733 (1969). E. 0. Kane, Phys. Rer. B4, 1910, 1917 (1971).

52

D. L. RODE

c. ZnSe

At one time there was considerable interest in using ZnSe for a blueemitting source (the direct gap is 2.78 eV). However, it has never been shown that this cubic crystal can be heavily doped p-type, due to native defect compensation. Therefore, efficient electroluminescence from pn junctions in ZnSe does not appear feasiblealthough optical or electron-beam pumping may be used. The effective mass and high-frequency permittivity were determined by We derive a low-frequency dielectric constant e0 of 9.2 from the Lyddane-Sachs-Teller relationship. and the phonon measurements of Riccius."3 Berlincourt et d l o 3 find c0 = 9.12, in reasonable agreement with our result. We assume that eO/e, is temperature independent, as has been shown for other crystals previ~usly,'~although Aven allows. e0 to vary while E ; is assumed to be constant with temperature. '' Other material parameters are listed in Table 1. The calculated electron drift mobility, shown as a solid curve in Fig. 11, agrees with measured Hall mobility' ' 14*' from 150 to 700°K. At lower temperatures, the experimental results' '' are affected by impurity scattering, which decreases p with decreasing temperature [see Eq. (90)]. The Hall factor at room temperature is' 1. I3 for pure ZnSe and thus direct comparisons between Hall mobility and drift mobility are reasonable. At high temperature (T > 800"K), the data of Smith'l4 fall noticeably below the calculated curve. One can obtain agreement even at these elevated temperatures by assuming either a temperature-dependent e O / e , ratio or a large acoustic deformation potential El , neither of which appears justifiable. At this time, the discrepancy remains unresolved although multivalley conduction may occur (cf. the similar situation in CdSe and CdTe, as seen in Figs. 14 and 15). Measurements on high-purity ZnSe at low temperatures would be useful for estimating the accuracy of the piezoelectric scattering theory discussed in Section 7.

' ''

'

'*' '

d. ZnTe Unlike ZnO, ZnS, and ZnSe, only p-type conduction is easily attainable in ZnTe. Low-conductivity n-type material has been achieved, however, by flourine implantation.' Hence, we have very little data on electron transport in ZnTe.

''

'"

M. Aven, J . Appl. Phys. 42, 1204 (1971). D. T. F. Marple, J . Appl. Phys. 35, 1879 (1964). 'I3 H. D. Riccius, J . Appl. Phys. 39,4381 (1968). l 4 F. T. J. Smith, SolidState Commun. 7, 1757 (1969). ' I 5 Y . Fukuda and M. Fukai. J . Phys. SOC. Japan 23,902 (1967). 1 1 6 S. L. Hou, K . Beck, and J. A . Marley, Jr., Appl. Phys. Lerr. 14, 151 (1969), and private communication. lZ

1.

LOW-FIELD ELECTRON TRANSPORT

53

t *\0ooO

lo2

20

\x

40

60

100 TEMPERATURE, T

(OK)

FIG.11. Theoretical electron drift mobility (solid curve) of pure ZnSe compared to experimental Hall mobility: 0,"' A l4 x . I Comparing the results to Fig. 8 at high temperatures, it may be. that multivalley conduction is occurring above 800°C in ZnSe rather than that the theory is failing.

,'

'

The direct gap of cubic ZnTe lies in the visible portion of the spectrum at

2.34 eV (corrected approximately to room temperature) which we take as the

effective-mass gap.72 The electron effective mass has not been determined experimentally. The value in Table I has been estimated from Kane's t h e ~ r y . The ~ ~ dielectric .~~ constants"' shown in Table I agree with the Lyddane-Sachs-Teller relationship. Elastic constants were measured by Berlincourt et ~ 1 . " ~The calculated electron drift mobility is shown in Fig. 12. At room temperature, the drift mobility and Hall mobility are respectively 797 cm2/V-sec and 858 cm2/V-sec. Fischer et ~ 1 . ' 'measured ~ Hall mobility 'I7

'"

M. Balkanski, in "11-Vl Semiconducting Compounds" ( D . G. Thomas, ed.), p. 1007. Benjamin, New York, 1967. A. G. Fischer, J. N. Carides, and J. Dresner, Solid Stare Commun. 2, 157 (1964).

54

D. L. RODE

X

1021

40

I

60

I

1

100

I

I

I

I

200

400

600

TEMPERATURE, T

1

(OK)

FIG. 12. Theoretical electron drift mobility (solid curve) of pure ZnTe compared to experimental Hall mobility: x,'16 0.'" Ordinarily, only p-type conductivity is obtained in ZnTe. Hence, there are very few data available on electron transport.

up to 340 cm2/V-secon Al-doped ZnTe. Hou et ~ 1 . " reported ~ values from 180 to 540 cm2/V-sec on F-implanted layers, although later work gave a value as high as 1000 cm2/V-sec.116These points are shown in Fig. 12. The overall purity of the experimental material is unknown at this time, and a more complete discussion seems unwarranted without measurements at various temperatures.

1.

55

LOW-FIELD ELECTRON TRANSPORT

e. CdS A great deal of work has been done on the optical and electronic properties of CdS. This hexagonal crystal has been available in fairly pure form for many years, and it is also the only crystal which has shown what appears to be dominant piezoelectric scattering at low temperatures.' l9 The direct effective-mass energy gap at room temperature is approximat el^^^ 2.52 eV. Several measurements of electron effective mass m* and/or polaron mass rn, have given values of m* or rn, f r ~ m ' ~ ~ 0- .'165 * ~m to 0.208m. Unfortunately, it is unclear whether a given experiment yields m* or m,. Baer and Dexter have shown that the piezoelectric polaron effect could be used to correct their cyclotron resonance mass value of 0.165m to m* = 0.2m, although a precise theory of piezoelectric polarons is not available.' 2 2 The measurements of Vella-Coleiro' 2o by Landau level splitting and those of Hopfield and Thornasl2' by the exciton Zeeman effect agree (respectively 0.20m and 0.205m). On the other hand, Henry and Nassau'23 find m,/m = 0.190 0.002 and m,,/m= 0.180 +_ 0.01 by the donor Zeeman effect. In none of these cases has the polaron effect been accurately calculable, although the correction is appreciable for our purposes (mJm* x 1.1). The Faraday effect,12' when the Hall measurements are corrected for the Hall factor,' also yields mpin excess of 0.2m.Therefore, we neglect the small amount of mass anisotropy and take'20 mp = 0.208m. Note that constant-energy surfaces are oblate, as opposed to those of indirect semiconductors. The dielectric constants listed in Table I have been derived from measured cO1 and E ~ I I and the Lyddane-Sachs-Teller relationship.124,125 The calculated mobility in Fig. 13 agrees with the experiments of Podor er al.' 26 and Fujita et al. ' above 100°K. The Hall factor' at room temperature is rH = 1.15. Mobility is measured with electric field perpendicular (pl)or parallel (I(,,) to the c axis. At low temperatures, the data of Fujita et exhibit piezoelectric scattering72with p T - ' / 2 .The calculated mobility, using the piezoelectric coefficients of Berlincourt et al.,' O3 lies considerably above the data at low temperature. The reason for this discrepancy is unclear at present. The calculated anisotropy ratio p1/pl1agrees better with experiment if mass anisotropy is not ignored.77 Note that the

'

'

-

' l9

I*'

H. Fujita, K. Kobayashi, and T. Kawai, J . Phys. SOC.Japan 20, 109 (1965). G . P. Vella-Coleiro, Phys. Rec. Len. 23, 697 (1969). J . J. Hopfield and D. G . Thomas, Phys. Rec. 122, 35 (1961). W. S. Baer and R. N . Dexter. Phys. Reo. 135, A1388 (1964). C. H. Henry and K . Nassau, Phys. Rec. B 2,997 (1970). A. S. Barker, Jr. and C. J. Summers, J . Appl. Phys. 41,3552 (1970). C. A. Arguello, D. L. Rousseau, and S. P. S. Porto, Phys. Rev. 181, 1351 (1969). B. Piidor, J. Balaza, and M. Harsy, Phys. Status Solidi Wa), 613 (1971).

56

D. L. RODE

\

1

I

I l l

CdS

I

I

I

,

I

100

10

TEMPERATURE, T

(OK)

FIG. 13. Theoretical electron drift mobility (solid curves) of pure CdS compared to experimental Hall mobility: 0 ,p i , ’ l 9 0 ,pll , 1 1 9 x . 1 2 6 These data”’show the only case available in the present crystals of what appears to be intrinsic piezoelectric scattering with p T-’”.

-

acoustic deformation potential El = 3.3 eV derived from the pressure rate coefficient of the energy gapiz7would allow moderately good agreement with experiment near 100°K if the piezoelectric coefficients were readjusted upward. The use of m, = 0.208m has been helpful in this respect and partly explains the paradoxical requirement of El in excess of 10 eV for agreement with mobility c a l ~ u l a t i o n s ~whereas ~ * ’ ~ ~ the “normal” El (from the pressure coefficient) suffices in the case of ZnO.

f. CdSe CdSe ordinarily assumes the hexagonal wurtzite structure, although the cubic form has been achieved.99 We consider the former structure. The room-temperature effective-mass energy gap is direct at7’ 1.77 eV. Measure-

”’ D. L. Camphausen, G. A. N. Connell, and W. Paul, Phys. Rev. Lett. 26, 184 (1971). lzS

M. Saitoh, J. Phys. SOC.Japan 21,2540 (1966).

1. LOW-FIELD ELECTRON TRANSPORT

57

ments of effective mass"' yield values from 0.12m to 0.15m. Eaves et a1.''' have considered polaron corrections and band nonparabolicity in deriving the effective mass from the magnetophonon effect (similar corrections also apply to the measurements by Vella-Coleiro'20 on CdS). Such corrections are difficult to make precisely and, indeed, it is probable that different values of mass may be associated with different scattering mechanisms. Therefore, we take the optical polaron mass equal to the average of five values listed by ~ ~ =~ 0.130~1,which lies within 5% of the particular values Eaves et U I . ,mp ' remaining material parameters are listed measured by Eaves et ~ 1 . ' ~The in Table I (see p. 84). I

1

I

'

+

1

cdse

-

m

P

"tb,

lo3-

hn,

-

I o2

20

-

%\s : 00

I

40

I

60

I

I

100

I

200

I

400

I

600

I

I lo00

58

D. L. RODE

In Fig. 14, we compare the calculated drift mobility with experi130-1 32 Agreement is satisfactory from 120 to 800"K, considering that the Hall factor at room temperat~re,'~ for example, is 1.07. At the highest temperatures (8W1200"K), the data of Smith'30 fall progressively farther below the calculated curve, in resemblance to ZnSe shown in Fig. 1 1. It is not known whether nearby satellite minima come into play at these temperatures, but the results in Fig. 14 strongly suggest such a possibility. Sufficiently pure CdSe to exhibit dominant piezoelectric scattering below 40°K is not available. The data131in Fig. I4 below 100°K are influenced by impurity scattering. merit. 5

I

g. CdTe

CdTe ordinarily takes the zincblende form99 and has attracted a great deal of attention during the past decade because both n- and p-type crystals are available,133 as well as fairly pure crystals suitable for nuclear detectors. 34 The microwave Gunn effect also has been observed in CdTe. ' 35 Kanazawa and Brown136have determined the polaron mass mp = 0.096m at liquid helium temperature. CdTe has considerably smaller correction factors due to the piezoelectric polaron effect compared to CdS, and this value is probably accurate to 6%. MarpleI3' has measured E,/E = 7.21 at room temperature. The Raman data by Mooradian and Wright138 lead to E ~ / E= 10.76 through the Lyddane-Sachs-Teller relationship. Other material parameters are given in Table 1. Figure 15 compares the calculated electron drift mobility to experimerit. 133,139,140 At room temperature, the Hall factor' is 1.04and excellent agreement obtains between measured Hall and calculated drift mobility. The agreement is good from 80 to 400°K. The purest sample shown139 (open circles) is affected by impurity scattering below 80°K. Possibly the most interesting portion of Fig. 15 lies above 650"K, where the high-temperature data of Smith140seems to show multivalley conduction in L , , minima.72The onset of multivalley conduction should become apparent at temperatures as low as 500400°K if further experiments are done here. I3O 13' 13'

134

135

'36 13'

F. T. 3. Smith, Solid Srate Commun. 8, 263 (1970). R. A. Burmeister, Jr., and D. A. Stevenson, Phys. Srarus Solid; 24,683 (1967). P. Hoschl and S . Kubalkova, Czech. J . Phys. B 18, 897 (1968). B. Segall, M. R. Lorenz, and R. E. Halsted, Phys. Rev. 129,2471 (1963). W. L. Brown, Proc. Inr. Symp. CdTe, Centre Rech. Nucl., Strasbourg. June 29-30. 1971. G. S . Picus, D. F. DuBois, and L. B. Van Attd, Appl. Phys. Lerr. 12,81 (1968). K. K. Kanazawa and F. C. Brown, Phys. RPI'.135, A1757 (1964). D. T. F. Marple, J . Appl. Phys, 35, 539 (1964). A. Mooradian and G. B. Wright, Proc. Inr. Conf. Phys. Semicond., Moscow, 1968, p . 1020. R. Triboulet. loc. cit. Brown.134 F. T. J. Smith, Me/. Trans. 1, 617 (1970).

1. LOW-FIELD ELECTRON TRANSPORT

59

CdTe

TEMPERATURE, T

(90

FIG.IS. Theoretical electron drift mobility (solid curve) of pure CdTe compared to experiThere is little doubt that the downward trend in the 0,’j9 mental Hall mobility: x data by Smith’40 indicates multivalley conduction above 650°K.

This onset occurs at temperatures as low as that in GaAs (see Section 12) and suggests the rlcto L l cseparation may be much less than the 0.5-1.5 eV proposed in the literature.141 h. HgSe

Under normal conditions of temperature and pressure, both HgSe and MgTe are semimetals with the zinc-blende Because of symmetry, the energy gap is zero and the band structure is “inverted,’”’ as discussed for 14’

J. G. Ruch, Appl. Phys. Leu. 20, 253 (1972), and personal communication. Note the insensitivity of high-field electron transport to the assumed value of the rlcto L , , separation.

60

D. L. RODE

a-Sn by Groves and HgS is also a zero-energy-gap zinc-blende semimetal at temperatures above 6 W K , but ordinarily HgS possesses a trigonal structure which we shall not discuss.20Electron conduction in the zinc-blende semimetals is approximately described in terms of the formalism of Parts I1 and I11 if we consider the Tssymmetry of the lowest conduction The electron band, i.e., the Groves-Paul inverted band structure. wave functions are predominantly of p symmetry, while the energy dispersion relation is still given by Kane's formalism, Eq. (13). Hence, we need only modify the overlap integral [Eq. (12)] by including spin-orbit ~ p 1 i t t i n g . l ~ ~ This description is approximate insofar as RPA (random-phase approximation) corrections to the dielectric constants and quantum corrections to the electron scattering operators are not included.' 89144 Nevertheless, RPA and quantum corrections are frequently of opposing algebraic sign, so that the classical formulation is more accurate than one might a priori suppose. '44 Material parameters for HgSe are given in Table I. The experimental rn~bility'~~ at-4.2 ' ~ ~and 77°K is given in Fig. 16 and compared to theory (solid curves). RPA and quantum corrections are discussed by Rode and Wiley. 144 The agreement between theory and experiment is rather poor (factor of two) for some of the data'47 but rather good for other^.'^^,'^^ Note that mobility is quite insensitive to temperature from 4.2 to 77°K for highly doped material because of the extreme degeneracy of the electrons. cm-3 in Fig. 16b falls substantially below the The datum'48 at n % calculated curve. By itself, one might disregard this discrepancy, but equally interesting is the o b s e r ~ a t i o n ' ~5 0~ that " mobility in HgSe falls rapidly with increasing temperature above 200°K-so rapidly, in fact, that we could not describe, even approximately, the mobility temperature dependence near room temperature by electron-hole and lattice scattering (see Fig. 17).'44 Since we are able to describe mobility in HgTe near room temperature (see Fig. 18) and since a rapid decrease in mobility with temperature may signal multivalley conduction (see Fig. 15), we suggest there are X , , minima lying approximately 0.25 eV above T8 at low temperature in HgSe.'44 1443145

S. H. Groves and W. Paul, Phys. Reu. Lett. 11,194 (1963). H. Overhof, Phys. Status So/idi43,221 (1971). 144 D. L. Rode and J. D. Wiley, Phys. Srarus Solidi 56,699 (1973). 14' J. G. Broerman, Phys. Rev. B 2, 1818 (1970). 1 4 6 C. R. Whitsett, Phys. Reo. 138, A829 (1965). '41 D. G. Seiler, R. R. Galazka, and W. M. Becker, Phys. Rev. B 3,4274 (1971). 14' T. C. Harman, loc. cit. Devlin,' Chapter 15. 149 R. F. Brebrick and A. J . Straws, loc. cit. Balkanski,"' and private communication. l S o H. Gobrecht, U. Gerhardt, B. Peinemann, and A. Tausend, J . Appl. Phys. Suppl. 32,2246 ( 1961). 142

143

1.

LOW-FIELD ELECTRON TRANSPORT

61

FREE ELECTRON CONCENTRATI0N.n (cm-31

FIG. 16. Theoretical electron drift mobility (solid curves) of doped HgSe (zinc-blende semiO14'; the two points metal) at (a) 4.2"K and (b) 77°K.Experimental mobility: (a) 0,146 were suggested by W h i t ~ e t tto ' ~be ~ anomalous; (b) 0 , 1 4 *

Measurements on purer HgSe than is presently available may allow elucidation of the details of the nearby band structure and of the reasons for the discrepancies in Fig. 16.150a IS0'S. L. Lehoczky, J. G. Broerman, D. A. Nelson, and C.R. Whitsett, Phys. Rev. B 9, 1598 (1974). These workers report electron mobility in HgSe in agreement with Gobrecht et ~ 1 . ' ~ ~ The discrepancy between theory and experiment near room temperature is removed by considerations of nonpolar optical scattering. However, unlike previous results,150 these results are not extended above room temperature where satisfactory agreement is more difficult to achieve.

62

D. L. RODE

IX105 HgSe

*

L

t

0

I40

0

0

I

60

I

I

I

I

80 100

200 TEMPERATURE, T

400

600

(OK)

FIG. 17. Experimental Hall mobility of intrinsic HgSe: O,I4’ O.’” The rapid decrease of mobility with increasing temperature above 300°K may be due to multivalley conduction (compare Fig. 18) in nearby XI, minima.

2x1041 10’

I

I

I

I

.,”* 1

I02

1

I

I

I

I03

TEMPERATURE ,T (OK)

FIG. 18. Theoretical electron drift mobility (solid curve) of pure HgTe (zinc-blende semimetal) compared to experimental Hall mobility: ~ . l ’ z a

1. LOW-FIELD ELECTRON TRANSPORT

63

i. HgTe

As discussed in the previous section, HgTe is a zinc-blende semimeta1.’43”5’ Material parameters are given in Table I. The r6-r8energy gap and Tseffective mass vary substantially with temperature, and this fact is taken into account in the following results. 144 The temperature dependence of electron mobility in pure HgTe is given in Fig. 18. The experimental measurements’ 5 2 ~ 1 5 2 a agree well with theory above 40°K. where RPA and quantum corrections are small. Below 40”K, large, positive RPA corrections dominate smaller, negative quantum corrections and both corrections together explain the discrepancy between theory and experiment at low temperatures in Fig. 18.’44 For pure HgTe, intrinsic electron-hole scattering is dominant up to room temperature, where polar mode scattering becomes slightly more frequent. Agreement between theory and experiment for thermoelectricpower on doped n-HgTe at room t e m p e r a t ~ r esuggests ’~~ the absence of nearby XI,or L 1c minima to within 0.3 eV. 12. 111-V CRYSTALS

The 111-V crystals with I11 = (Ga, In) and V = (N, P, As, Sb) are, for the most part, covalently bonded 153 and posses the zinc-blende structure, with the exception of GaN’54 and (not discussed here) InN, which are wurtzite structures. GaP is indirect and GaSb is barely direct with a Tlc-L I c separation of -0.07 eV.lS5 Otherwise, the 111-V compounds are decidedly direct and therefore are well suited to description by the methods discussed in Parts I1 and 111. Aside from these general properties of 111-V semiconductors, there are specific optical and electronic properties which make these crystals tremendously important as an economically viable technology base. For example, the availability of p- and n-type material allows fabrication of pn junctions for light-emitting diodes covering wavelengths from medium infrared through the green visible portion of the spectrum-in some cases by the use of alloyed mixtures of the binary compounds. 5 6 Junction lasers employing GaAs and Al,Ga, -,As are being widely studied at present.lS7

’

A. Saleh and H. Y. Fan, Phys. Status Solidi 53, 163 (1972). V. 1. Ivanov-Omskii, B. T. Kolomiets, V. K. Ogorodnikov, and K. P. Smekalova, Fir. Tekh. Poluproo. 4,264 (1970) [English Transl.: Sou. Phys. Semicond. 4, 214 (197011. l S z a R .A. Stradling and G. A. Antcliffe, J . Phys. SOC.Japan Suppl. 21,374 (1966). A. E. Attard, J . Solid State Chem. 5, 360 (1972). Is4 H. P. Maruska and J. J. Tietjen, Appi. Phys. Lett. 15,327 (1969). A. Ya Vul’, L V. Golubev, T. A. Polyanskaya, and Yu. V. Shmartsev, Fiz. Tekh. Poluprov. 3, 301, 786 (1%9) [English nansl.: Sou. Phys. Semicond. 3,256,671 (1%9)]. H. C. Casey, Jr. and F. A. Trumbore, Mom. Sci. Eng. 6.69 (1970). M. B. Panish, in “Progress in Solid State Chemistry” (H. Reiss and J. 0. McCaldin, eds.), Vol. 7, Chapter 2. Pergamon, Oxford, 1972.

lS2

64

D. L. RODE

Equally important is the application of 111-V semiconductors (espially GaAs and InP) to microwave generation and detection by means of Impatts, Gunn, and LSA diodes, field-effecttransistors, and Schottky barrier mixers. This field has been reviewed by Copeland and Knight in this series.I5* a. GaN

GaN has the wurtzite structure and a direct energy gap of 3.39 eV at room Since the energy gap lies in the ultraviolet, one might anticipate making a blue-emitting diode from GaN.ls4 However, this has not been possible by means of p n junctions since p-GaN does not seem achievable. There are rather few electron transport measurements available on GaN, with fewer than approximately 10" ionized impurities per cubic centimeter. In addition, several material parameters are also undetermined as yet. Therefore, we have estimated some of the parameters, as indicated, in Table1 by comparisons to other nearby materials such as ZnO, InP, etc. 160,160a The calculated electron mobility versus temperature for pure GaN is shown in Fig. 19.l6' Electron scattering at room temperature is predominantly by polar optical modes. The experimental data by Ilegems and Montgomery16' in Fig. 19 are Hall mobilities measured with electric field transverse to the c axis of the crystal. The experimental data'61 lie below the calculated curve by about a factor of four at temperatures above 200°K. Obviously, impurity scattering decreases the experimental mobility at lower temperatures. It is not known at present whether impurities are affecting the experiments above 200°K or whether the chosen material parameters in Table I are grossly inaccurate. Clearly, the shapes of the theoretical and experimental curves in Fig. 19 are so similar that one could force agreement by (unjustifiably at present) varying the effective mass, coupling constants, etc. This is not our object, however, and only further measurements can resolve the discrepancy. (The comparison in Fig. 19 is not unlike that for impure samples of Gap, as shown by the solid triangles in Fig. 20.) Note that room-temperature mobility in GaN is predicted to be considerably higher than that in ZnO, i.e., by a factor of approximately six, due mainly to the lower effective mass and polar mode coupling strength in GaN. J. A. Copeland and S. Knight, in "Semiconductors and Semimetals" (R. K. Willardson and A. C. Beer, eds.), Vol. 7, Part A, Chapter 1 . Academic Press, New York, 1971. S. Bloom, J . Phys. Chem. Solids 32,2027 (1971). I6O D. L. Rode, unpublished. 160"Sincethe original writing, A. S. Barker, Jr. and M. Ilegems, Phys. Rev. B 7, 743 (1973) have reported several parameters for GaN in agreement with the estimates16' in Table I. For example, the effective mass and polaron mass are found to be 0.2 m and 0.216 m, whereas the polaron mass estimated in Table I is 0.218 m. M. llegems and H . C. Montgomery, J . Ph.vs. Chem. Solids 3 4 , 8 8 5 (1973). ISs

65

1. LOW-FIELD ELECTRON TRANSPORT

-

0

0

-

0

10'

0 0 0.

I

I

I

I

I

I

I

I

I

30

FIG.19. Theoretical electron drift mobility (solid curves) of pure GaN compared to experimental Hall mobility (0,pl)determined by Ilegems and Montgomery.'" Hall measurements16' suggest the donor concentration is approximately 1.6 x 1018cm-3.

6 . GaP At the present time, GaP is commercially one of the most interesting 111-V semiconductors because of its application (along with GaAs,P, -,) to electroluminescence.' GaP possesses the zinc-blende structure and an indirect rly-XIcenergy gap.zs.'62 The various material parameters needed for calculations of mobility are given in Table I. The acoustic and intervalley deformation potentials, as indicated, are derived empirically from mobility data. Hence, the calculations for indirect crystals are not quite as satisfying 16'

Although some workers suggest the lowest conduction band minima lie at A,, (based on analogies to Si), this writer feels the direct and indirect evidencez9 for X,,minima is sufficiently strong that only direct evidence to the contrary is worthy of consideration.

66

D. L. RODE

TI--Go P

x x b

A

I

I

1

I

I

I

I

1

TEMPERATURE, T (OK)

FIG.20. Theoretical electron drift mobility (solid curve) of pure GaP derived empirically by comparison to Hall mobility: 0,163 A,166x,I6’ A.16aDeformation potentials are derived by comparison between the data of Casey er a/.166and the calculated drift mobility (dashed line) including measured166electron and acceptor concentrations. Hall measurements on the purest sample (Ol6’) indicate 8.2 x 1015donors/cm3and 1.3 x lot4 acceptors/cm3.

as those for direct crystals, in which case there are no adjustable parameters. Nevertheless, the good agreement obtained with only two adjustable parameters does suggest we have identified the physical mechanisms most responsible for electron scattering.

1. LOW-FIELD ELECTRON TRANSPORT

67

In Fig. 20, we show data on mobility versus temperature as determined by various experiment^.'^^-'^* Since there is considerable scatter in the data, we have used the following approxim&tedevice to retrieve the lattice contribution to electron scattering from these data. The data (open triangles) of Casey et were fitted, including impurity scattering in the Dingle27 formulation, by adjustment of the acoustic and intervalley deformation potentials, for electron and acceptor concentrations determined by Hall measurements,166as shown by the dashed line in Fig. 20. The fit is to be where electron freezeout is relatively regarded as valid only above 100“K, negligible. Then the calculation is repeated without impurity scattering to find the fundamental lattice mobility shown by the solid curve in Fig. 20. The resulting good agreement between this latter curve and the highest’ ~ ~ the purity sample (open circles) measured by Craford el ~ 1 . indicates validity of this procedure. The deformation potentials El = 13 eV and D, = 1.2 x lo9 eV/cm derived in this fashion seem reasonable, although we hasten to point out that a choice of El = 25 eV and D, = 8 x lo8 eV/cm gives deceptively good agreement with the average of the various data in Fig. 20. With the former values for E , and D,,we find that intervalley scattering, acoustic mode scattering, and polar mode scattering are respectively the dominant, next most dominant, etc., scattering mechanisms in GaP at room t e m ~ e r a t u r e . ~ ~ c. GaAs

Regarding device applications, GaAs is currently one of the most versatile semiconductors in use. Applications range from microwave sources (Gunn diodes, Impatts, FET’s) and detectors (Schottky mixers) to infrared sources (LED’S,lasers) and integrated optical c ~ m p o n e n t s . ’ ~The ~ ~ high ’ ~ ~level ~’~~ of interest in this material is perhaps evidenced nowhere so well as by the title of a biennial conference.’” Physically, GaAs is especially interesting as a prototypal direct semiconductor. The cubic zincblende structure and

’’‘

M. G . Craford, W. 0.Groves, A. H. Herzog, and D . E. Hill, J . Appl. Phys. 42,2751 (1971). R. C. Taylor, J . F. Woods, and M. R. Lorenz, J. Appl. Phys. 39, 5404(1968). 16’ A. S. Epstein, J . Phys. Chem. Solids 27, 1611 (1966). 1 6 6 H. C. Casey, Jr., F. Ermanis, L. C. Luther, L. R. Dawson, and H. W. Verleur, J . Appl. Phys. 42,2130 (1971). 16’ R. Nicklin, A. W. Russell, and P. C. Newman, Electron. Lett. 3, 363 (1967). M. Toyarna, M .Naito, and A. Kasami, Jap. J. Appl. Phys. 8 , 3 5 8 (1969). l b 9 S. M. Sze, “Physics of Semiconductor Devices.” Wiley, New York, 1969. I7O C. H. Gooch, ed., “Gallium Arsenide Lasers.” Wiley, New York, 1969. 17’ Inr. Symp. GaAs and Related Compounds, Ist-4th. Inst. Phys., London, 1966, 1968, 1970, 1972. 17’ D. Jones and A. H. Lettington, Solid Stare Commun. 7, 1319 (1969). 163

Ifi4

68

D. L. RODE

moderately large direct energy gap (- 1.43 eV optical gap)173with no nearby satellite minima”4 to within 0.38 eV allow precise description of electron transport by the methods of Parts I1 and 111. Extensive measurements of material parameters and transport quantities (such as mobility and thermoelectric power) on rather pure GaAs have been made. The material parameters shown in Table I have been sifted from a large body of literature with emphasis given to overall consistency in a choice of a particular reported value. For example, by the following argument, I think we can specify the low-frequency lattice dielectric constant to even though some reported values range several percent from the within i”/o ’ that the ratio of longivalue shown in Table I. First, Chang et ~ 1 . ’ ~showed tudinal to transversephonon frequenciesw w / o ,at r is essentially independent of temperature. Lu et found that E@ has a temperature coefficient of 10-4/”K. The mean of four values discussed by Seraphin and Bennett’77 gives the high-frequency dielectric constant E,/E = 10.91 at room temperature. Therefore, the Raman data by Hass178combined with the LyddaneSachs-Teller relationship give E ~ / E= 12.87 _+ 0.06 at room temperature, ’ ~ ~ or E ~ / E= 12.50 & 0.06 at low temperature from the data by Lu et ~ 7 1 . on temperature dependence. This last value (12.50 k 0.06) compares favorably ’ ~ ~ we with E ~ / E= 12.56 k 0.04 measured by Stillman et ~ 1 . Therefore, suggest E ~ / E= 12.53 0.05 at low temperature, and E ~ / E= 12.91 & 0.05 at room temperature. Although such detailed considerations may at first appear superfluous, the reader should note from Eq. (1 18) that a 1 % error in yields a 7% error in the polar scatteiing rate which is the dominant room-temperature lattice scattering mechanism in GaAs.62 of GaAs is shown in Fig. 21. In this figure, we have The Hall chosen to plot Hall mobility (at 5 kG, in agreement with the experiments shown) rather than drift mobility since experimental measurements sufficiently accurate to show effects due to the scattering factor rB = pB/p are available. The subscript B denotes quantities at a specified, finite magnetic field. The agreement between theory (solid curve, calculated for pure, M. B. Panish and H. C. Casey, Jr., J . Appl. Phys. 40,163 (1969). G. D. Pitt and J . Lees, Phys. Rev. B 2,4144 (1970). 1 7 5 R. K. Chang, J . M. Ralston, and D. E. Keating, in Proc. Int. Conf. Light Scattering Spectra Solids. Springer, New York, 1969. 17’ T. Lu, G. H . Glover, and K. S. Champlin, Appl. Phys. Lett. 13,404 (1968). 1 7 7 B. 0.Seraphin and H. E. Bennett, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 3, Chapter 12. Academic Press, New York, 1967. 1 7 8 M. Hass, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 3, Chapter 1 . Academic Press, New York, 1967. 17’ G. E. Stillman, D. M. Larsen, C. M. Wolfe, and R. C. Brandt, Solid State Commun. 9, 2245 (1971).

173

I 74

69

1. LOW-FIELD ELECTRON TRANSPORT

2

4

10

20

40

100

200 400

1000

TEMPERATURE, T ( O K )

FIG.21. Theoretical electron Hall mobility (solid curve) of pure GaAs compared to experiTo illustrate the effect of ionized impurities, mental Hall mobility: 0,180 A ,''' we also plot theoretical Hall mobility including measured'" impurity concentrations; the dashed curve utilizes the Dingle theoryz7and the dash-dot curve utilizes the Brooks-Herring theory.28 Electron freezeout occurs below18o 10°K.where the two theories lose agreement. Above 6WK, multivalley conduction in X,,minima takes place. Hall measurements (0180) indicate 5.2 x 1013donors/cm3 and 2.2 x 10" acceptors/cm'.

intrinsic GaAs) and e ~ p e r i r n e n t ' * ~ -is' ~fairly ~ good for the temperature interval 60400°K. The theoretical curve lies about 10% above some of the data. This unexplained discrepancy may be due to inaccuracies in the chosen

''I

C. M. Wolfeand G. E. Stillman, loc. cit. 3rdInt. Symp.,171 1970; and G. E. Stillman, private communication. H. G. B. Hicks and D. F. Manley, SolidSrate Commun. 7, 1463 (1963); and D. F. Manley, private communication. D. M.Chang, private communication from J. Barrera. P.Blood, Phys. Rev. B 6,2257 (1972).

70

D. L. RODE

material parameters or to inadequacies of the theory on this fine scale.183aAt higher temperatures, electrons are thermally stimulated into higher lying indirect minima at XI, (0.38 eV above rlc).' 74,1*3*1s3bThe high-purity samples studied by Hicks and Manley"' (solid dots) and by Wolfe and Stillman180 (open dots) show evidence of ionized-impurity scattering below 60°K. To illustrate this effect quantitatively, we have plotted the dashed curve (Dingle theory") and dash-dot curve (Brooks-Herring theory") in Fig. 21, which were calculated by including the donor and acceptor concentrations measured by the Hall effecton the sample of Stillman's' (open dots). The agreement is within the estimated"' 20 % accuracy of the acceptor concentration measurements for the Brooks-Herring theory. At lower temperature, the Dingle theory is inadequate due to neglect of impurity screening by ionized acceptors and donors. Even though Eq. (96) is violated at temperatures below 10"K, it is interesting that the Brooks-Herring theory yields fairly good agreement with experiment. Since the donor ionization energy (- 5 meV) in GaAs is rather small, one must make Hall measurements well below 15°K to accurately determine donor and acceptor concentrations from freezeout statistics. This is usually a laborious procedure requiring considerable instrumentation. On the other hand, it is quite easy to make Hall measurements at room temperature and at liquid nitrogen temperature. Therefore, we have plotted mobility at 300 and 77°K in Figs. 22 and 23 for large ranges ofelectron concentration." From N-)/n, these curves, one can determine the compensation ratio (N' and hence, acceptor concentration, from measured values of n and p. These curves have been used by DiLoren~o"~and PanishIs5 to discuss impurity incorporation mechanisms in GaAs grown respectively by vaporphase and liquid-phase epitaxy. Miki and Otsubo' have measured 77°K mobilities up to 244,000 cm2/V-sec. Both drift mobility and Hall mobility are shown in Figs. 22 and 23. The ratio rH = p H / p ,called the zero magnetic field (subscript H) Hall scattering factor, is given for pure GaAs in Fig. 24 as a function of temperature.15,'83a

+

''

183aC. M. Wolfe, G. E. Stillman, D. L. Spears, D. E. Hill, and F. V. Williams, J . Appl. Phys 44,

732 (1973), and C. M. Wolfe and G. E. Stillman (see Chapter 3) show the effects of dopant nonuniformity on experimental Hall mobility. Comparison of Fig, 4 of the former work with Fig. 6 of Rode'' indicates agreement between experimental and theoretical Hall factor to within 1 % when nonuniformities are minimized. '83bJ. W. Orton, Brit. J . Appl. Phys. 6, 851 (1973). J. V. DiLorenzo, J . Crysral Growrh 17, 189 (1972). M. B. Panish, J . Appl. Phys.44,2659 (1973). H. Miki and M. Otsubo, Jup. J . Appl. Phys. 10,509 (1971). In a private communication to this writer, these authors have shown ~ ~ ( 7 7 ° K = )262,000 (422,000)cmz/V-sec for magnetic fields from 0.1 to 7.0 k G on their highest purity sample. These results are consistent with the assumption of uniform doping.

1. LOW-FIELD ELECTRON TRANSPORT

71

DRIFT MOBILITY HALL MOBILITY

FREE-ELECTRON CONCENTRATION,n ( ~ r n - ~ )

Fic. 22. Theoretical electron drift (solid cunes) and Hall (dashed curves) mobility of doped GaAs at 300°K.

-

105

u u) W

J m

8

I o3

FIG.23. Theoretical electron drift (solid curves) and Hall (dashed curves) mobility of doped GaAs at 77°K.

In Section 5, we presented the solution to the Boltzmann equation for timedependent driving forces. We give here some results for electrons in GaAs doped with 3 x l O I 5 ~ r n -donors ~ and 1 x 10'' acceptors, i.e., n = 2 x lo'' ~ m - Prior ~ . to application of a stepped-on, steady electric

72

D. L. RODE

I

: 1.20

Go As

I

L

d 0 t

V

2 -J

-I U

I

1.10

I .oo

20

40

60

100

200

TEMPERATURE, T

400

600

(OK)

1000

FIG. 24. Theoretical Hall factor rH = pH/p of pure GaAs. Below 5"K, rH tends toward 1.1 characteristic of piezoelectric scattering. Above IOO"K, polar mode scattering is dominant, and large values of rHreflect the rapidly varying momentum dependence of this scattering mechanism up to temperatures above the polar phonon temperature (419"K), where the momentum dependence is relatively slight. A similar effect occurs from 20 to 80"K, where piezoelectric, acoustic, and polar scattering combine to yield a scattering rate weakly dependent on momentum and, consequently, smaller values of Hall f a c t ~ r . ' ~

field F a t time r = 0, the electron distribution is in thermal equilibrium and the perturbation distribution g = 0 [see Eq. (58)].At times t > 0, the steady electric field causes the perturbation distribution g to grow as shown in Fig. 25. In the lower part of the figure, the curve labeled So vel is the polar mode scattering-out rate plus elastic scattering rate appearing in Eq. (58). The rapidly decreasing component of So v,, at small momenta is due to piezoelectric scattering [Eq. (108)].The nearly constant scattering rate at small to intermediate momenta corresponds to scattering by polar phonon absorption. The rapid increase in So + ye, at intermediate momenta coincides with the characteristic polar phonon energy hwpo 419°K. For large momenta, scattering occurs primarily by polar phonon emission. Of course, this discussion applies only to scattering-out terms and not to scattering-in

+

+

-

73

1. LOW-FIELD ELECTRON TRANSPORT DISTANCE, SIF ( t o - ~ O c m 2 / v ) 5 10 I

-~

0.1

0.2

15 I

I

0.3

TIME, t (psec)

0.4psec

GaAs, 300°K

S o + ve/

c------

MOMENTUM, h k ( L INEAR SCALE) FIG. 25. Time-dependent response of the electron distribution to a stepped-on electric field F. The scattering-out rate and perturbation distribution are given in the lower part of the figure. The average electron drift-velocity u, normalized to F,is given as a function of time and normalized distance in the upper part of the figure.

terms.13 The perturbation distribution g is shown in Fig. 25 for times 0.08, 0.2, and 0.4 psec after application of the stepped-on field. The velocity versus time and velocity versus distance for an average electron are shown in the upper parts of the figure. The velocity v/Fapproaches the low-field drift mobility (7600 cm2/v-sec) as t approaches infinity. The velocity versus distance curve is interesting insofar as it allows one to determine the distance required for an average electron to reach a given velocity. For example, an average electron in a field of lo3 V/cm reaches a velocity of 5 x lo6 cm/sec in a distance of cm, which is essentially instantaneous considering the size of the electron wave packet.

74

D. L. RODE

d. GaSb GaSb possesses the cubic zincblende structure and a direct energy gap.' However, the four equivalent L Ic minima lie energetically very near the lowest conduction band minimum (-0.07 eV above rlc),155 so that most ~" of this electrons populate L valleys at room t e m p e r a t ~ r e . ~Because complicated transport situation, and because of the relatively small energy gap (0.8 eV) at r, GaSb has not been widely utilized in technological applications. Nevertheless, the band structure of GaSb does allow various interesting physical experiments to be performed on intervalley transfer mechanisms and the nature of L and X,,minima in the Ga series of 111-V crystals. Some of the material parameters applicable to electron transport in TlCare given in Table I. l 3 Further parameters concerning L minima are discussed by Heinrich and Jantsch,"' who made high-field measurements on GaSb. Near room temperature, the Hall mobility of n-type GaSb lies between 10oO and 7700 cm2/V-sec, depending upon the level of purity.24a9' 90a Of course, one can determine drift mobilities applicable separately to the two sets of conduction band minima rlcand Llc,24aor, indeed, to the X , , minima. 9'

,,

,,

"-'

e. I n P InP is a direct-gap semiconductor possessing the zincblende structure. and Because of moderately nearby L,, minima (0.4 eV above r1c)192 X,, minima (0.7eV above rlc),192 this crystal exhibits the Gunn effect useful in microwave devices.' s8,193 Consequently, the quality of InP crystals has been greatly improved through intensive studies in connection with device development during the past few years. Many of the material parameters of InP given in Table I have been carefully redetermined since the mid-1960's following initial studies in Germany and the United States during the 1950's. The direct measurements of effective IBi

H. Heinrich and W. Jantsch, Phys. Starus Solidi 38, 225 (1970). A. 1. Blum, Sou. Phys. Solid State 1, 6% (1959).

H. J . McSkimin, A . Jayaraman, P. Andreatch, Jr., and T. B. Bateman, J . Appl. Phys. 39, 4127 (1968). I9O M. Averous, G. Bougnot, J. Calas, and J . Chewier. Phys. Sratus Solid. 37, 807 (1970). 190aH.Miki, K. Segawa, and K. Fujibayashi, Jap. J . Appl. Phys. 13, 203 (1974). B. B. Kosicki, A. Jayaraman, and W. Paul, Phys. Rev. 172,764 (1968). '91 G. D. Pitt, Solid State Commun.8, 1119 (1970), and J . Phys. C (Solid State Phys.) 6, 1586 (1973). Values quoted in the former work are questioned but seemingly unresolved in the latter. 19' C. Htlsum and H. D. Rees, Proc. X Inr. Con$ Phys. Semicond., Cambridge, Massachusetts, August 1970.

1.

LOW-FIELD ELECTRON TRANSPORT

75

mass by cyclotron resonance performed by Chamberlain et al. 94 illustrate an important point about the k p matrix element P appearing in Eq. (5). Namely, P is substantially larger for GaAs than for InP. Consequently, even though GaAs has a larger energy gap than has InP, the effective mass m* is larger in InP (0.082m)than that in GaAs (0.066m). 194 The calculated drift mobility of pure InP is plotted as a solid curve in Fig. 26.14 Experimental at temperatures from 150 to 600°K generally lie 15% below the calculated curve. The reasons for this discrepancy are not known in detail but may be related to impurity scattering, which is below 100°K.Above 800”K,the data19s=198 clearly evident in the data195*196 fall significantly below the calculated curve. This behavior can be ascribed to electron conduction in L , , minima which become populated at elevated temperatures. l4 Measured Hall mobility at room temperature typically falls from 4200 to 5400 cm2/Vsec195-198for pure InP, while the calculated Hall and drift mobilities are 6370 and 5150 cm2/V-sec.At room temperature, the dominant scattering mechanism occurs by polar modes.’98a

J ZnAs The relatively small direct energy gap (0.46 eV at low temperature’999200 1 and resulting high mobility of cubic (zinc blende) InAs have made this crystal interesting for use in Hall detectors and little else in the way of practical applications, save as a base for alloy systems such as Ga,In, -,As. However, InAs is interesting from a theoretical point of view insofar as its small energy gap allows a fairly sensitive test of transport theory including Kane’s nonparabolic bands.30 Furthermore, the lowest-lying subsidiary L , , minima are sufficiently remote from rlcwith respect to energy (0.84 eV)’O1 that single-valley transport theory should be adequate up to the melting point of InAs at 1215°K. Material parameters of InAs are listed in Table 1. Because of the substantial temperature dependence of the small energy gap, we include the J. M . Chamberlain, P. E. Simmonds, R. A. Stradling, and C. C. Bradley, J . f h y s . C Solid State fhys. 4, L38 (1971). 1 9 5 V. V. Galavanov and N . V. Siukaev, Phys. Status Solidi 3 8 , 5 2 3 (1970). 196 M. C. Hales, J. R. Knight, and C. W. Wilkins, Int. Symp. GaAs. Inst. Phys., London, 1970. 19’ 0. G . Folberth and H. Weiss, 2. Naturforsch. lOa, 615 (1955). ‘ 9 8 H. Wagini, Z . Naturforsch. Zla, 1244 (1966). 1988P. Blood and J. W. Orton, J. Phys. C (Solid State Phys.) 7,893 (1974). The 700°K mobility in this work agrees with previous ~ o r k ’ ~ ’ . ’and ~ * lies below the theory in Fig. 26. 1 9 9 0. Madelung, “Physics of III-V Compounds.” Wiley, New York, 1964. C. Hilsum and A. C. Rose-Innes, “Semiconducting III-V Compounds.” Pergamon, Oxford, 1961. J. E. Smith, Jr., and D. L. Camphausen, J . Appl. Phys. 42, 2064(1971). 19‘

76

D. L. RODE

FIG. 26. Theoretical electron drift mobility (solid curve) of pure InP compared to experimental Hall mobility: .,Ig5 A,196 O,I9' A."* Above 800"K, multivalley conduction in L , , minima occurs. Hall measurements (A196) give the electron concentration as 1.7 x IOl5 at room temperature.

temperature variation of effectivemass and energy gap measured by Stradling and Wood.'O' The solid curve in Fig. 27 shows the calculated drift mobility of pure InAs. Above 150°K,the experimental data203-205on Hall mobility agree quite well with the calculations, considering the Hall factor, which is 1.11 at room temperature, for example. Below 150"K,the data 2 0 3 * 2 0 5 '02

'03 '04 '05

R. A. Stradling and R. A. Wood, J. Phys. C Solid Srace Phys. 3, L94 (1970). T. C. Harman, H. L. Goering, and A. C. Beer, Phys. Rev. 104, 1562 (1956). 0. G. Folberth, 0.Madelung, and H. Weiss, Z. Naturforsch. 9a, 954 (1954). G. R. Cronin and S. R. Borrello, J. Electrochem SOC. 114, 1078 (1967).

1. LOW -FIELD ELECTRON TRANSPORT

77

FIG.27. Theoretical electron drift mobility (solid curve) of pure InAs compared to experiA.205Agreement is good up to 934"K,where KTis several mental Hall mobility: A,203.,'04 times hmp0.Comparisonszo5between calculated and measured mobility (Azo5) give 6.5 x lo1' donors/cm3 and 2.5 x lOI5 acceptors/cm3.

are affected by impurity scattering. Note the good agreement with the data (solid dots) of Folberth et ul. '04 at 934"K, where KTis 2.77 times the polar phonon energy. Thus, it appears that no signzjicunr "multiphonon" scattering occurs in InAs, despite occasional veiled references to the contrary in the literature concerning other compound semiconductors. g. ZnSb

For physicists interested in electronic properties of crystals, InSb has served as the historical dragon since the pioneering work of Kane2'S3O on band structure and of Ehrenreich" on electron transport. InSb possesses the zincblende structure and a rather small direct gap so that the effects of

78

D. L. RODE

band nonparabolicity and admixed wave functions are quite evident in transport proper tie^.'^ The relatively large variations with temperature of energy gap (0.258 - 0.00029TeV)205aand effective mass allow one of the most sensitive tests of whether the effective-mass energy gap appearing in Eq. (22) is indeed the dilatational gap, as we have written it, or the optical gap. Stradling and Woodzo2have shown the dilatational gap is the more accurate choice for InSb, although the choice is less clear for other

material^.^^.^^^

Fairly pure InSb has been available for nearly two decades and, consequently,many of the material parameters, as listed in Table I, are accurately known. We include temperature variations of effective mass202and energy gap in the calculated drift mobility (solid curve) shown in Fig. 28. The agree-

\

104

I

20

I

40

InSb

I

60

I

I

100

I

1

200

400

b

600

loo0

TEMPERATURE, T(OK)

FIG.28. Theoretical electron drift mobility (solid curve) of pure InSb compared to experimental Hall mobility: O,Z"bA , '' A.zo8Below 300"K, polar mode scattering is dominant. Above 4WK, intrinsic electron-hole scattering is dominant. Comparisonszo6between calculated and measured (azo6) mobility suggest -9 x I O l 4 donors/cm3and relatively few acceptors.

205'R. W. Cunningham and J . B. Gruber, J . Appl. Phys. 41,

1804 (1970).

1.

LOW-FIELD ELECTRON TRANSPORT

79

ment between theory and experiment206-208is satisfactory from 200 to 600°K. Clearly, ionized-impurity scattering affects the data of Hrostowski et aL206below 100°K.Just below room temperature, polar mode scattering is dominant. Above 400"K,intrinsic electron-hole scattering is dominant. The rapid downward trend in the data206*207 above 600°K is probably due to electron transfer into L,, minima located209 0.40-0.55 eV above rlc. This phenomenon is discussed in more detail in the l i t e r a t ~ r ein ' ~connection with thermoelectric power measurements by Blum and Ryabtsova'lO through the melting point of InSb at 780°K. 13. GROUPIV CRYSTALS

-

The elemental group IV crystals Si, Ge, and a-Sn are nonpolar and cubic. Silicon and Ge are extremely important semiconductors technologically, in device applications to transistors, memories, microwave generators, optical detectors, Hall detectors, etc.' 69 Silicon has enjoyed the longest, most intense study of semiconductor physics (since the 1930's) compared to other crystals we have discussed.211The cubic form of Sn, called a-Sn, was the first crystal shown to possess the inverted band structure with a symmetry-induced zero energy gap.'42 Since Si and Ge are indirect, our theoretical description of electron transport in these materials is empirical insofar as the intravalley deformation potential El and intervalley deformation potential D,are derived by comparisons between measured and calculated m~bility.~'Nevertheless, the good agreement achieved with only two adjustable parameters does show that the physical mechanisms of electron scattering have been identified. Germanium is available in extremely pure form, and this crystal displays very clearly the effect of acoustic modes on electron transport. a.

Si

At this time, Si is by far the most commonly used semiconductor which we shall discuss. This crystal is the basis of nearly all of today's integrated circuit technology, part of which awakened me this morning, transported H. J. Hrostowski, F. J . Morin, T. H. Geballe, and G . H. Wheatley, Phys. Reu. 100, 1672 (1955). ' 0 7 G . B u s h and E. Steigmeier, Helc. Phys. Acta 34, I (1961). N.1. Volokobinskaya, V. V. Galavanov, and D. N. Nasledov, Fiz. Tuerd. Tela 1,756(1959) [Engtish Transt.; Sou. Phys. Solid State 1,687(1959)l. '09 J . C. McGroddy, M. R. Lorenz, and T. S. Plaskett, Solid State Commun. 7 , 901 (1969); and J. E.Smith, Jr., private communication. ' l o A. I. Blum and G . P. Ryabtsova, Fiz. Tuerd. Tela 1,761 (1959)[English Transl.: Sou. Phys. Solid State 1,692(1959)l. '"C.Weiner, IEEE Spectrum 10,24(1973). '06

80

D. L. RODE

me from my home, and analyzed the data to be discussed here. Nevertheless, following nearly 35 years of study,211 there does remain an interesting controversy of detail concerning electron scattering mechanisms among the six Alc minima in Si.29I will not discuss this situation extensively here, but briefly, the question is whether g-scattering depicted in Fig. 6d can occur via low-energy LA phonons or via high-energy LO phonons alone as predicted by selection r ~ l e ~ .In ~accordance ~ , ~ with ~ ~selection - ~ ~ ~ we assume the latter in the following discussion. The material parameters are listed in Table I. Intravalley scattering occurs through deformation-potential coupling to acoustic modes.29 Intervalley scattering shown in Fig. 6d occurs through mixed LA TO phonons ($scattering) and through LO phonons (g-~cattering).~~ The energies of these phonons are similar to one another (see Fig. 7) and we do not expect to see large, separate effects due to these two different mechanisms. Consequently, we presume all intervally scattering takes place by a single phonon-in particular, the 47.4-meV phonon observed by O n t ~ n , ~which " seems to be the LA phonon near X.'lSa At high temperatures, intrinsic electron-hole scattering occurs. The theoretical assumes isotropic conduction minima, which is a good assumption for lattice scattering (randomizing) but not for electron-hole scattering (nonrandomizing). Hence, the theory cannot be applied at high temperatures. The deformation potentials El and D, are derived empirically by fitting calculated and measured mobilities. The calculated drift mobility (solid curve) and m e a s ~ r e d ~ ' ~Hall -~ mobility of intrinsic Si is shown in Fig. 29. Below 80"K,mobility is proportional to T-3'2 due to acoustic mode scattering. At higher temperatures, intervalley scattering sets in and the mobility decreases somewhat more rapidly with temperature. Near room temperature, intervalley and intravalley electron scattering are about equally freq~ent.~'

+

'"M. Lax and J. L. Birman, Phys. Status Sotidi49, K153 (1972).

H. W. Streitwolf. Phys. Status So/idi 37, K47 (1970). M.Costato and L. Reggiani, Phys. Status Solid; 38, 665 (1970). 'I5 A. Onton, Phys. Rec. Lett. 22,288 (1969). '"'P. Norton, Phys. Reo. B 8,5632 (1973) has neatly interpreted the data obtained by Onton215 in terms of intervalley scattering by emission of a 59-meV TO phonon near X rather than a 47.4-meV phonon. Use of the TO phonon in the present work would shift the knee of the curve in Fig. 29 from near 130 to near 160°K.High-purity mobility given here agrees within a few percent with Fig. 29. 'I6 F. J . Morin and J. P. Maita, Phys. Rev. 96, 28 (1954). G. W. Ludwig and R. L. Watters, Phys. Rev. 101, 1699 (1956). 'I8 D. Long and J. Meyers, Phys. Rec. 115, 1107 (1959); 120, 39 (1960). 'I9 R. A. Logan and A. J. Peters, J . Appt. Phys. 31, 122 (1960). 'I3

'I4

'"

81

I. LOW-FIELD ELECTRON TRANSPORT

I

I

20

40

I

60

I

I

1

100

400

200

TEMPERATURE,f

I

600

I

1000

(OK)

FIG. 29. Theoretical electron drift mobility (solid curve) of pure Si derived empirically bycomparison toexperimental Hallmobility: A,”’ A,21’ x , 218 0.z19 Acousticmodescattering is dominant below 80°K. Intervalley scattering is comparable to acoustic scattering at 300°K.Hall measurements ( O Z i 9indicate ) 1 . 1 x lOI4 donors/cm3 and 4.0 x 10l2 acceptors/ cm3.

6. Gq

Germanium is undoubtedly the purest crystal available of those we are considering, because of intense development for nuclear detectors.’” The four L , , minima comprise the conduction band edge and are well characterized (see Table I). Intervalley scattering between L,, minima occurs by longitudinal phonons at X as shown in Fig. 6c.’13 Intravalley scattering occurs by deformation-potential coupling to acoustic modes and by nonpolar L. H. deLaet, W. K. Schoenmaekers, H. J . Guislain, and M. Meeus, Symp. Semicond. Derect. Nucl. Radial., 2n4 Munich, September 1971.

82

D. L. RODE

optical mode scattering.221Both nonpolar and intervalley scattering involve phonons of similar energies, and we combine the description of these phonons into a single, average phonon of Debye temperature 382°K as shown in Table I.29 The deformation potentials El and D, are derived empirically by fitting theoretical results to experiment. The theoretical model assumes isotropic minima.29 Hence, while lattice scattering in pure Ge is adequately described by this assumption, electron-hole scattering is not and the calculations cannot be extended to high temperatures. The calculated electron drift mobility (solid curve) is compared to experimental Hall mobility220*222-224a in Fig. 30. The remarkable consistency of the experimental data (gathered in the two decades since 1953) attests to the quality of the crystals and the measurements. The agreement with the calculated mobility is excellent over nearly three decades in mobility. Clearly, the T - 3 / 2mobility dependence below 60°K is indicative of acoustic mode scattering [see Eq. (1 12)]. Above 100”K, intervalley scattering and nonpolar optical scattering cause a slightly more rapid decrease in mobility with temperature. At room temperature, acoustic mode scattering comprises about three-fourths of the total scattering rate. Due to the similarity of the phonon energies involved, it has not yet been possible to determine the relative strength of nonpolar optical scattering compared to intervalley scattering. c. a-Sn

Although Sn is stable as tetragonal, white Sn (or fl-Sn) above 286°K the cubic (diamond structure) form called a-Sn or gray Sn has been prepared by solution growth and by annealing at lower temperature^.^^^.^^^ a-Tin is a zero-gap (at I?,) semimetal with the inverted Groves-Paul band structure similar to that in HgSe and HgTe.226Broerman has shown the effect of this band structure on electron wave functions and, consequently, electron transport properties in a-Sr~.’~’He also has given RPA corrections to the dielectric constant. In addition, there are comparable quantum corrections to the Born approximation for electron scattering matrix elements.144 W. A. Harrison, Pbys. Rev. 104, 1281 (1956).

’*’J . L. Blankenship, Pbys. Rev. B 7 , 3725 (1973); and private communication.

M. B. Prinqe, Phys. Rev. 92,681 (1953). P. P.‘Debye and E. M. Conwell, Phys. Rev. 93,693 (1954). ZZ4*P. Norton and H. Levinstein, Phys. Rev. B 6, 470 (1972) also report mobility in high323 1 2 ‘

225

226

227

purity Ge. A. W. Ewald and E. E. Kohnke, Pbys. Rev. 97,607 (1955). S. H. Groves, C. R. Pidgeon, A. W. Ewald, and R. J. Wagner, Proc. Int. ConJ Pbys Semicond., 9th. Leningrad, 1968. Nauka, Leningrad. J . G. Broerman, J. Pbys. Chem. Soiids 32, 1263 (1971).

1.

83

LOW-FIELD ELECTRON TRANSPORT

I06

0 0) v)

4

‘ N

E

105

0

a

*-

c_

1 m

0

r

104

103

2

4

10

20

40

TEMPERATURE, T

100

200

400

lo00

(OK)

FIG. 30. Theoretical electron drift mobility (solid curve) of pure Ge derived empirically A,2230.224 All the experimental by comparison to experimental Hall mobility: x ,220 0,222 mobilities are dominated by intrinsic acoustic mode scattering. A small amount of intervalley scattering takes place above 100°K.Hall measurements ( O Z z 2 indicate ) 7.8 x 10l2donors/cm3 and 4.9 x 10” acceptors/cm3.

Since the T8-L,,energy separation is only -0.09 eV, multivalley condtlction - ~lower temsets in above 80°K in pure a-Sn or for n > 6 x 1017~ r n at peratures.”” Lavine and Ewald2’* have discussed transport properties in this latter regime. The mobility of electrons in a-Sn at 4.2”K is shown in Fig. 31. Material parameters are given in Table I. For electron concentrations below 2 x 10l6 cm - 3, the mobility calculated from classical theory (solid line) without 228

C. F. Lavine and A. W. Ewald, J . Phys. Chem. Solids 32, 1121 (1971).

84

D. L. RODE TABLE I

Crystal

ZnOb

Low-frequency dielectric constant,

8.12

3.72

ZnSe'

ZnTe'

CdS'

CdSeb

CdTe'

8.32'

9.20

9.67

8.58

9.40

10.76

25.6'

5.13'

6.20

7.28

5.26

6.10

7.21

I2 (Y

ZnS'

Hgse"

Cole

High-frequency dielectric constant, E,/f

Polar-phonon Debye temperature, T, ( W Longitudinal elastic constant, C , ( I O ~N/m2)" ~ lntravalley deformation potential, El ( e W Piezoelectric coefficient P or P , / P , ,

837

506

360

428

303

246

268'

20.47

12.82

10.34

8.40

8.47

7.37

6.98

8a.J

3.8

4.9

4.2

3.5

3.3

3.7

4.0

43.'

0.076

0.024

0.0146

0.022

0.022','

I

I

I

0.2I /0.36

Number of equivalent valleys, Z

I

I

I

Lattice mass density

-

-

-

Intervalley-phonon Debye temperature,

-

-

Intervalley deformation potential, D, (eV/cm)

-

Effective-mass energy gap 4 (ev) Polaron or density of states effective mass,

0.143/0.192' 0.104/0.148 I

I

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

3.43

3.80

2.78

2.34

2.52

I .77

I .54

0.233'

0.318'

0.312'

0.183'

0.208'

0.13W

0.096'

0.0265'

P (g/cm3)

T,W)

297

-0.159'

m*/m

' Several quantities listed in the table are significantly temperature dependent, although there are circumstances in which the temperature dependence yields fortuitously little influence on transport quantities. For example, the dielectric constantsofGaAsvary by 3%from0to30O0K, while thetranspon-relevant ratio&,/&, remainssensiblyconstant.Usually, we have given 300°K values of the dielectricconstants and neglected their temperature variations in the calculations of the preceding sections. The effective-mass energy gaps listed in the table correspond either to low temperatures (4.2K. 77°K. etc.) or to dilatation gaps at room temperature when the diRerence causes a significant etTect (2%) on transport quantities. The temperature dependence of theenergy gap through lattice dilatation alone is included in the calculations of the precedingsections when thisdependenceis significant, i.e., for small-gap materials such as InAs, InSb, HgSe, and HgTe, but not for ZnO, CdS, etc. Temperature dependence for the etTective mass is correspondingly included. [See Eqs. ( I ) and ( 5 ) modified for finite spin-orbit splitting.] The effective mass given in the table usually corresponds to low temperatures. Piezoelectric and elastic tensors and remaining quantities are assumed constant for calculational purposes. The reader i s referred to the text and original references for details on temperature dependences. * Direct energy gap. wurtzite structure. ' Direct energy gap, zincblende structure. Zero energy gap, zincblende structure with spin-orbit splitting of 0.45 eV (HgSe) and I eV (HgTe). Indirect energy gap, zincblende structure. Indirect energy gap, diamond structure. Zero energy gap, diamond structure with spin-orbit splitting of 0.7 eV. Rode72:Berlincourt el a/.'0'

-

I(

85

1, LOW-FIELD ELECTRON TRANSPORT MATERIAL PARAMET& HgTe'

GdNb

Gap

GaAs'

20.0-

9.87.

ll.lW

12.91

14.W

5.80'

9.11'

10.91

199"

6.12"

1 W

26.53'

4'

8.4

580'

419

GaSV

InP

InAs'

InSb'

15.W

12.38

14.54

17.64

11.70

15.98

14.44'

9.55

11.74

15.75

-

-

-

-

-

-

346

497

337

274

Gel

24'

13.97

10.38

12.10

9.98

7.89

19.02

15.03

8.44

13.0'

8.6

8.3

6.8

5.8

7.2

6.5'

9.V

3.2'

0.0131

0.0168

0.027

-

-

I

I

6

4

2.33

5.32

0.036'

0.052

-

I

3

I

I

4.13

-

-

-

-

-

I

-

I

-

359

-

-

-

-

-

542

-

1.2 x 109'

-

-

-

-

-

3 x 108' 3 x

0.290'

3.39

-

1.54

0.80

1.42

0.46

0.232

-

0.0244'

0.218'.'

0.365'

0.066'

0.042

0.082

0.025

0.0125

0.3V

-

a-Sn'

16.61.

0.022' 0.118/0.152'

-

Si'

I

-

382

108'

-

-

0.413""

0.2Zv

0.0236'

' Estimated.

'Rode and Wiley."'

' rh-r8separation, temperature dependent. ' Whitsett,"'

temperature dependent. Harman."n Rode.'"' ' D. D.Manchon. Jr., A. S. Barker, Jr, P. J. D u n , and R. B. Zcttentrom, Solid Stare Conntwn. 8, 1227 (1970)give E ~ I=E5.8.SeealsoE. Ejder. Phys. StutusSolidi6.445(1971). whogiwsE,/~ = 5.24. R.Dingle (unpublished)obtained V J =~ 726 cm- and w, = 556 c n - I .

'

' Empirically derivedzg using density of states mass m4 per valley. Use of conductiviry mass m, implies deformation potentials larger by factor (mr/D1,)'". 'A. Onton and R. C. Taylor, Phys. Rec. B 1,2587(1970).

' Stradling and Wood.zoz " Hass.1'8

' 6. J. Roman, unpublished. "Groves et a/.zz" "Camphausen er al."' Polaron mass.

86

D. L. RODE

FREE ELECTRON CONCENTRATION,nicm-9

FIG.3 1. Theoretical electron drift mobility (solid curve) of doped a-Sn (semimetal) at 4.2”K compared to experimental mobility: 0,228 m,2290,230 A.231

RPAZ2’ and quantum c o r r e c t i o n ~ agrees ’ ~ ~ fairly well with the experimental data.228-231In the region of n = 2 x 10l6cm-j, RPA and quantum corrections are about equal and of opposite algebraic sign, so that the For smaller n, RPA corrections dominate overall correction is and the curve in Fig. 31 should be shifted upward. At larger n, quantum corrections dominate and the curve should be shifted downward. These trends are in agreement with the experiments shown in Fig. 31. Multivalley conduction occurs for n > 6 x l O ’ ’ ~ m - ~and mobility is significantly enhanced by increased screening due to electrons in the heavier mass L minima.228 V. Summary

Seventeen years ago E. 0. KaneZ1wrote, “There is now a considerable amount of experimental and theoretical information available concerning the properties of indium antimonide . . . ,” and while this is still true, the same statement applies to many more crystals today. Hence, I intend to summarize here some of the more significant qualitative advances made in 229

230

*”

S. H. Groves and W. Paul, in “Physics of Semiconductors” (Proc. 72h Ini. Conf.). Dunod, Paris and Academic Press, New York, 1964. E. D. Hinkley and A. W. Ewald, Phys. Rev. 134, A1261 (1964). 0. N. Tufte and A. W. Ewald, Phys. Rea. 122, 1431 (1961).

1. LOW-FIELD ELECTRON TRANSPORT

87

the last few years rather than discuss the refinement and extension of our knowledge of specific material parameters. Since Ehrenreich’s discussion” of the effective-mass energy gap Q, [Eq. (5)], there have been several attempts made to determine if 8, is the dilatation gap [Eq.(22)] or the optical gap. While not answering the key question, experiments such as those by Akselrod et al.232do show that the effective mass scales with energy gap changes induced by pressure. Probably the closest approach to resolving the issue is that reported by Stradling and Wood,202who have deduced m* from magnetophonon magnetoresistance measurements on InSb, InAs, and GaAs from 40 to 290°K. InSb, with the smallest energy gap, allows the most sensitive test. The most remarkable result reported by Stradling and Wood is that m* increases as temperature increases from 40 to -60°K in agreement with the dilatation (Recall that the InSb lattice contracts with increasingtemperatures below 57.5”K.233) Moreover, the observed change in m* (9%) for temperatures frbm 40 to 260°K agrees fairly well with that induced by the dilatation gap (7+%) and not with that induced by the optical gap (22%).202The results2” for InAs and GaAs are much less clear, however, and further measurements will be required to determine more than qualitative agreement. Furthermore, these analyses2” presume temperature independence for the spin-orbit splitting and matrix element P appearing in Eq. (1) and (2). There is still an occasional tendency in the literature to explain mobility results on one particular sample by assigning questionable values to material parameters. Judging from the variability of some of the experimental results in Part IV, one can expect absurd results by this procedure for insensitive parameters-to wit, the acoustic deformation potential El. Throughout Part IV, we have employed values of El determined independently from elastic tensors and pressure rate coefficients of the band edge of direct crystals, assuming the valence band is nearly stationary with strain.l o Generally, El lies between 3 and 9 eV and the agreement apparent in comparisons between theory and experiment in Part IV supports these values. Values of E, between 20 and 30 eV occasionally appear in the literature and these values give scattering rates (proportional to El 2, one to two orders of magnitude larger than those reported here. In view of the results in Part IV, it seems doubtful that such large El values are correct. Regarding the accuracy of the comparisons in Part IV, one may wonder why we generally plot theoretical drift mobility for comparison to experimental Hall mobility. First, Hall factor corrections” are usually less than 232

13’

M. M. Akselrod, K . M. Demchuk, I. M. Tsidilkovski, E. L. Broyda, and K. P. Rodionov, Phys. Status Solidi 27, 249 (1 968). S. 1 . Novikova, Fiz. Tverd. Tela 2, 2341 (1960) [English Transl.: Sor. Phys. Solid State 2, 2087 (1960)l.

88

D.

L. RODE

20% except for temperatures near h o J 2 ~and the theory is probably no more accurate than 20% in most cases because of material parameters, etc. Second, the experimental data are taken for many different values of magnetic field and when the product pl? is larger than unity, the Hall factor rB is very nearly unity. Thus, while the experimental results may approximate pHat higher temperatures (lower mobility), they may lie close to p at lower temperatures (high mobility). Other notable features evident from Part IV are that (a) multiphonon processes do not appear to be significant for polar mode scattering, in agreement with selection rules for the harmonic lattice, (b) the finding of Chang et ~ 1 . ' ' ~that E&, is nearly temperature independent indicates the validity of using the Lyddane-Sachs-Teller relationship in calculating polar mode coupling strengths which are regarded as temperature independent, (c) the equipartition assumption for acoustic modes in Ge is accurate to temperatures as low as IWK, and (d) the Brooks-Herring theory" of ionized-impurity scattering is superior to the Dingle theory" when the concentration of neutral donors is comparable to, or larger than, the free-electron concentration. In closing, I would like to point out some trends and potential directions in current research on transport properties of semiconductors. Overall, 111-V materials have permitted the growth and application of useful new technologies involving light emitters, microwave generators and detectors, junction lasers, etc. On the other hand, 11-VI materials have found disappointingly little application, particularly because of the limited levels and types of electrical doping which are available. Consequently, the quest for greater variety in optical and electronic properties has led in recent years to the emergence of 111-V ternary and quaternary alloys such as GaAs, -,P,, In,Ga,-,As, and AI,Ga,-,As,-~,,. Work on these materials can be expected to continue at an accelerated pace as further applications are developed.

In regard to research on fundamental transport properties, I find two recent findings to be particularly exciting. Dingle et U I . ~ ~have " succeeded in measuring the onedimensional quantum states of electrons confined to very thin layers (65-500A) of GaAs sandwiched between layers of AI,Ga,-xAs. The technique may prove helpful in future work on band structure measurements. Concerning electron-phonon scattering, there has been little hope in the past of determining the strength of a particular mechanism in the presence of other equally influential mechanisms. However, T s ~ has i made ~ ~ the ~ first direct measurement of the LO phonon emission 234

R. Dingle, W.Wiegmann, and C. H.Henry, Phys. Reu. Lett 33,827 (1974).

"' D.C. Tsui, Phys Rev. B Id, 5088 (1974).

1. LOW-FIELD ELECTRON TRANSPORT

a9

time for electrons in InAs by analyzing tunneling currents through InAsoxide-Pb junctions. The experimentally determined emission time is 5.1( f0.3) x sec whereas theory predicts 5.3 x sec, in excellent agreement. The emission time is related to the polar scattering rate given by Eq. (116). Extensions of these techniques to other materials, such as ternary and quaternary alloys, could greatly improve our quantitative understanding of the electron-phonon interaction. ACKNOWLEDGMENTS I am very grateful to Mrs. N.J. Firestone and Miss S. Miller for their pleasant assistance and typing of the manuscript, and to Drs. F. H. Blecher, J. A. Copeland. B. C. De Loach, J. E. Kunzler, and D. G. Thomas for their support and encouragement. It is a privilege to acknowledge enlightening discussions with and assistance from Drs. J. L. Blankenship, J. V. DiLorenzo. W. Fawcett, S. Knight, A. R. Hutson, D.F. Manley, J. G. Ruch. G. E. Stillman, R. A. Stradling, J. D.Wiley, and C. M. Wolfe.

CHAPTER 2

Mobility of Holes in III-V Compounds J . D. Wiley 1. INTRODUCTION . . . . 11. VALENCEBANDSTRUCTURE.

. . . . . . . . . . . . . . . . . . . . . . . . 111. SCATTERING MECHANISMAND MODELS . . . . . . . . . 1. General Feaiures of Hole Transport 2. Intrinsic Scarrering Mechanisms . 3. lonized-lmpuriiy Scattering. . .

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

126 139

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

141 141 144 149 155

1V. EXPERIMENTAL HOLEMOBlLtTIEs . . . . . . . . . . . 4. AIP, AIAs, and AlSb . . . . . . . . . . . . . 5. 6. 7. 8. 9. 10.

GaP GaAs GaSb InP InAs InSb

91 95 110 110

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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v. SUMMARY . . . . . . . . . . APPENDIX . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

160 163 165 170 171

I. Introduction Since the early 1950’s, when interest in elemental and compound semiconductors first began its rapid expansion, carrier mobilities have been among the most important indices used in studying, characterizing, and assessing the quality of semiconducting crystals. This is so because the mobility’ is, in principle, quite easy to measure and it can, either by itself or in conjunction with other measured parameters, yield valuable information about the impurity content, degree of compensation, carrier scattering mechanisms, doping homogeneity, and other factors of experimental and theoretical importance. The literature of electron and hole mobilities is now so large that even a relatively restrictive topic such as the present “Mobility of Holes in III-V Compounds” is represented by several hundred papers. Any review of such a large literature must necessarily be limited to papers of recent vintage and, with few exceptions, to work that has received

’ When not otherwise stated, “mobility” in this chapter will be taken to indicate Hull mobility

since this is the quantity most frequently measured and quoted. The relationship between calculated drift mobilities and experimental Hall mobilities will be discussed in Part 111.

91

92

J . D. WILEY

the greatest amount of recognition and discussion. Willardson and Goering2 have given an exhaustive bibliography covering all aspects of 111-V compounds up to and including the year 1960. The books by Hilsum and RoseI n n e ~M , ~a d e l ~ n gand , ~ Putley’ also contain reviews of the early literature and extensive bibliographies. A more recent, but highly selective, bibliography is contained in the handbook compiled by Neuberger.6 After two decades of experimental and theoretical work, the present status of our understanding of carrier mobilities in elemental and 111-V compound semiconductors can be summarized as follows. For n-type direct-gap materials (InSb, InAs, InP, GaSb, and GaAs) there is excellent agreement between theory and e~periment.’~’In these materials, the carriers have predominantly s-like wave functions and spherically symmetric dispersion relations. Theoretical models for scattering by ionized impurities and acoustic phonons give simple analytic expressions for the mobilities. Because of the spherical symmetry of the electron wave functions, optical phonons affect the mobility only through lattice polarization [i.e., there is no nonpolar optical (NPO) mode scattering]. Ehrenreichg has treated polar mode scattering and given a closed-form expression for the polar mobility. It is possible to obtain approximate agreement between theory and experiment by using these analytic expressions and combining mobilities according to Matthiessen’s In order to obtain the best agreement between theory and experiment, however, it is necessary to include several scattering mechanisms simultaneously and to treat the wave functions and dispersion relations exactly. When this is done (usually through numerical solution of the Boltzmann equation’), one obtains excellent agreement between theory and experiment. The situation of n-type indirect-gap materials (Ge, Si, Gap, AlSb, AlAs, and Alp) is complicated by the many-valleyed nature of their lowest conduction band This allows intervalley scattering by acoustic and optical phonons in addition to intravalley scattering. In this case it is no longer possible to write closed-form expressions of plausible accuracy,

* R. K. Willardson and H. L. Goering (eds.),

’

‘Compound Semiconductors,” Vol. 1, Preparation of 111-V Compounds. Van Nostrand-Reinhold, Princeton, New Jersey, 1962. C. Hilsum and A. C. Rose-Innes, “Semiconducting111-V Compounds.” Pergamon, Oxford, 1961. 0. Madelung, “Physics of 111-V Compounds.” Wiley, New York, 1971. E. H. Putley, “The Hall Effect and Related Phenomena.” Butterworth, London, 1960. (Republished as “The Hall Effect and SemiconductorPhysics.” Dover, New York, 1968.) M. Neuberger, “111-V Semiconducting Compounds, Handbook of Electronic Materials,” Vol. 2. Plenum Press, New York, 1971. D. L. Rode, Phys. Rev. B 2, 1012 (1970). D. L. Rode, Low Field Electron Transport, Chapter 1, this volume. H. Ehrenreich, J. Phys. Chem. Solids 2, 131 (1957); 8, 130 (1959); 9, 129 (1959).

2. MOBILITY

OF HOLES IN 111-V COMPOUNDS

93

and one must resort to computer calculation of the Aside from some uncertainty in the deformation potentials for intervalley scattering, the agreement between theory and experiment is satisfactory for the latticelimited mobility. The proper treatment of ionized-impurity scattering remains a problem. l o In attempting to analyze the results of early experiments on p-Ge and Si, it was quickly realized that a proper theoretical treatment of electrical transport in the valence bands would be quite formidable. Cyclotron resonance experiments’ and band structure calculations’2 ~ 1 3had shown that the thermally occupied region of the uppermost valence band was of r8+ symmetry (atomic plike character, fourfold degenerate including spin) and consisted of two sheets or energy surfaces, customarily referred to as the light- and heavy-hole bands. These two bands were shown to be degenerate at k = 0 and to have energy contours which resemble warped spheres12 for finite values of k. Thus, in a proper calculation of the hole mobility, one would have to include (1) the use of Ts+(p-like) wave functions to calculate scattering matrix elements, (2) anisotropic dispersion relations for the two sets of carriers, and (3) the simultaneous presence of two interacting bands of carriers. (Later work14 showed that it is also necessary to include the nonparabolicity of the bands and the energy dependence of their warping.) Note that the presence of two types of carriers means that the experimentally accessible quantity, the measured mobility, is actually an “effective” mobility containing contributions from each set of carriers. Because of these complications, it is quite hopeless to expect to find simple analytic expressions giving the effective hole mobility as a function of material parameters and experimental conditions. Even for the simplest realistic models of transport in the valence bands, one must resort to numerical (computer) calculations. A fully realistic calculation incorporating all important scattering mechanisms and accurate models for both bands would be extremely difficult even in its formulation. Lax and MavroidesI5 were the first to incorporate the warped energy surfaces into a calculation of low-field transport coefficients inp-Ge, but theirs was essentiallya classical calculation (see Part 111 for a discussion of the assumptions implicit in various models of hole transport). The full complexity of transport in the valence bands of Ge and Si was first discussed by HarrisonI6 and by lo

I1

l3 l4

l6

D. L. Rode, Phys. Status Solidi 53,245 (1972). For a complete review of the early work on cyclotron resonance in Ge and Si, see B. Lax and J. G . Mavroides, Cyclotron Resonance, Solid Stare Phys. 11 (1960). G. Dresselhaus, A. F. Kip, and C. Kittel, Phys. Rev. 98, 368 (1955). F. Herman, Physicu 20,801 (1954). W. Bernard, H. Roth, and W. D. Straub, Phys. Rev. 132,33 (1963). B. Lax and J. G. Mavroides, Phys. Rev. 100, 1650 (1955). W. A. Harrison, Phys. Rev. 104, 1281 (1956).

94

J . D. WILEY

Ehrenreich and Overhauser’ in 1956. Other authors’8-” have also treated this problem in elaborate detail. Excellent reviews of the situation as of 1963-1964 have been given by Beerz3and Paige.24The most recent theoret(the best understood and ical examination of ho!e mobility in p-Ge2Z*Z5 most thoroughly studied material) concludes that, even in this case, nontrivial theoretical problems remain to be solved. In view of the great similarities between the valence bands of group IV and 111-V semiconductors, one should expect hole transport to be nearly identical in these two groups of materials, with the 111-V compounds having the added complications of polar mode and piezoelectric scattering. It is somewhat surprising, therefore, to find that, until quite recently, most workers used only the simple analytic formulas (derived for nondegenerate s-like bands) in fitting Hall mobility data inp-type 111-V compounds, without even mentioning the complications which had already been found in p-Ge and Si. One can only speculate as to how this situation developed, but it seems plausible that the reason lies in the following historical accident. Early interest in 111-V compounds centered largely on the n-type materials with small, direct gaps (primarily on InSb, InAs, and GaAs), because of their extremely high mobilities. Ehrenreich showedg that, in these materials, polar modes provide the dominant lattice scattering .mechanism, and he derived a formula for polar mobility which was highly succe~sful~-~ in fitting the observed mobilities. Largely because of this success, and despite warnings by Ehrenrei~h~.’~ the notion became widespread that polar mode scattering is the dominant lattice scattering mechanism for all carriers in 111-V compounds. This notion was given greater currency by the fact that it is (perhaps unfortunately) quite easy to obtain qualitative agreement between calculated and experimental hole mobilities using the simple but demonstrably inappropriate theoretical expressions. Even quantitative agreement can often be obtained if, as is usually the case, one or more paramH. Ehrenreich and A. W. Overhauser, Phys. Reo. 104, 331, 649 (1956).

G.E.Pikus and G. L. Bir, Fiz. Tverd. Tela 1, 1642,1828 (1959)[English Transl.: Sou. Phys.-

Solid State 1, 1502,1675 (1959)l. G. L. Bir and G. E. Pikus, Fiz. Tuerd. Tela 2, 2287 (1960) [English Transl.: Sou. Phys.Solid Stare 2,2039 (196O)l. 2o G. L. Bir, E. Normantas, and G. E. Pikus, Fir. Tverd. Telu 4, 1180 (1962).[English Transl.: Sou. Phys.-Solid State 4,867 (1962)l. M.Tiersten, IBM J . Res. Develop. 5, 122 (1961); J . Phys. Chem. Solids 25, I151 (1964). 2 2 P.Lawaetz, Phys. Rev. 166,763 (1968); 174,867(1968); 183,730(1969). * 3 A. C. Beer, “Galvanomagnetic Effects in Semiconductors.” Academic Press, New York, 1963. 24 E. G. S. Paige, “The Electrical Conductivity of Germanium.” Wiley, New York, 1964. ” P. Lawaetz, private communication. 26 H. Ehrenreich, Phys. Rev. 120, 1951 (1960). l9

2.

MOBILITY OF HOLES IN 111-V COMPOUNDS

95

eters are sufficiently uncertain to allow reasonable adjustment of their Indeed, if the simple formula for polar mobility alone is applied uncritically to the valence bands, using an average effective mass (approximately equal to the heavy-hole mass), one finds reasonably good agreement3.27between calculated and experimental mobilities at 300°K without any adjustment of parameters! It is now known that this agreement is purely coincidental,28but it is only in the most recent that some of the complexities mentioned earlier for p-Ge and Si have been introduced into models of transport in p-type 111-V compounds. Thus, with the theoretical situation in a rapid state of flux, the emphasis in this chapter will rest heavily on summarizing experimental work. Recent theoretical results will be included, but it is too early for their comprehensive review or for extensive reinterpretation of data. Part I1 contains a brief discussion of those features of the valence band structure that are relevant to hole mobility, Although little formal use will be made of the material in Part 11, it is felt that an appreciation of the nature of the valence bands is vital for even a qualitative understanding of hole transport. In Part I11 the most important intrinsic and extrinsic scattering mechanisms and the general features and results of recent models for hole transport are discussed. Part IV contains summaries of hole mobility data for the most important 111-V compounds. A few brief summarizing remarks and observations are contained in Part V. 11. Valewe Band Structure

All of the 111-V compounds with which we shall be concerned in this chapter crystallize in the zinc-blende lattice structure. The zinc-blende lattice for a compound AB consists of two interpenetrating FCC sublattices (each sublattice containing only A or B atoms) displaced along the (1 1 1> direction by a distance of one A-B bond length. If all atoms on the B sublattice were replaced by A atoms, one would obtain a lattice of slightly higher symmetry: the diamond structure of the group IV semiconductors. C. Hilsum, Some key features of 111-V compounds, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 1. Academic Press, New York, 1966. J. D. Wiley and M. DiDomenico, Jr., Phys. Rec. B 2,427 (1970). 29 J. D. Wiley, Phys. Rev. B4,2485 (1971). ’O M. Costato, G. Gagliani, and L. Reggiani, Lett. Nuouo Cimento 4, 171 (1972). 31 D. Kranzer, Phys. Srarus Solidi 50, K109 (1972). 32 M. Costato, C. Jacoboni, and L. Reggiani, Phys. Status Midi 52, 461 (1972). 33 D. Kranzer, J . Phys. C: Solid State Phys. 6,2967: 2977 (1973). 34 M. Costato and L. Reggiani Phys. Status Solidi 58,471 (1973). 35 M. Costato and L. Reggiani, Phys. Status Solidi 59,47 (1973). 27

96

J . D. WILEY

The additional symmetry gained in going from zinc-blende to diamond is known as inversion symmetry (the symmetry operation in which r is replaced by -r). The first Brillouin zone for diamond and zinc-blende lattices is shown in Fig. 1 with the high-symmetry points and lines labeled according

FIG.1 . The first Brillouin zones for diamond and zinc-blende lattices, showing the principal symmetry points and lines labeled in the conventional notation.

to standard n0tati0n.j~The center of the Brillouin zone r is the point of highest symmetry, possessing octahedral (0,)and tetrahedral (T,)symmetry for diamond and zinc blende, respectively. The spacing and symmetries of the energy levels at r are illustrated in Fig. 2, using the specific case of GaAs as an example. For convenience in comparing the notations used by various authors, Fig. 2 includes both the single-and double-group notations. A full discussion of the distinction between single and double groups can be found e1~ewhere.j~ Briefly stated, the double group is used when spin-orbit effects are included in the energy band calculations. 36

D. Long, “Energy Bands in Semiconductors.”Wiley (Interscience), New York, 1968.

2. MOBILITY Et

OF HOLES IN 111-V COMPOUNDS GROUP 'ION (ZB) rl2

10

DOUBL GROUP NOT 'ION ( 0 ) (ZB)

re- re r6

r7

r.2'

0

97

t

CONDUCTION BANDS

r25

VALENCE BANDS

r6

FIG.2. Energy level structure of r showing the single-group(spin-orbit splittingignored) and double-group (spin-orbit splitting included) notations for diamond (D) and zinc-blende (ZB) lattices. The level spacings, ordering, and energy scale were chosen to correspond to GaAs.

Figure 3 shows a portion of the GaAs band structure (from r to X and L along the lines A and A, respectively)as calculated by Pollak et aL3' using a full-zone k p approach and including spin-orbit splittings. This figure serves to illustrate the major features of the band structures of all the materials considered in this chapter. The band structures of other group IV and 111-V semiconductors differ from GaAs only in quantitative detail, and can 37

F. H. Pollak, C. W. Higginbotham, and M. Cardona, Proc. Int. Conf. Phys. Semicond., Kyoro, 1%6. J . Phys. SOC.Japan Suppl. 21 (1966).

98

J. D. WILEY

> -4

I

Ls

At?

-6

/

FIG.3. A portion of the energy band structure of GaAs. (After Pollak et 01.”)

~ - ~i ~ n c l ~ d i n g ~ ’and - ~ ~e x ~ l u d i n g ~ ~ - ~ ~ be found in the l i t e r a t ~ r e ~both the effects of spin. The various techniques used in calculating the energy bands in the references just cited have been reviewed e l s e ~ h e r e . ~ ~ . ~ ’ . ~

’

M. Cardona and F. H. Pollak, Phys. Ret.. 142, 530 (1966). M. Cardona, F. H. Pollak, and J. G. Broerman, Phys. Leu. 19,276 (1965). 40 F. H. Pollak and M. Cardona, J. Phys. Chem. Solids 27,423 (1966). 4 1 E. 0. Kane, J . Phys. Chem. Solids 1,82 (1956). 4 2 E. 0. Kane, J . Phys. Chem. Solids 1,249 (1957). 4 3 M. Cardona, J . Phys. Chem. Solids 24, 1543 (1963); 26, 1351 (1965). 44 M. L. Cohen and T. K. Bergstresser, Phys. Reo. 141,789 (1966). ” J. P. Walter and M.L. Cohen, Phys. Rev. 183, 763 (1969). 46 T. C. Collins, D. I. Stukel, and R. N. Euwema, Phys. Reo. B 1,724 (1970). 4 7 D. J. Stukel and R. N. Euwema, Phys. Rev. 186,754 (1969); 188, 1193 (1969). D. J. Stukel and R.N. Euwema, Phys. Rev. B 1, 1635 (1970). 49 D. J. Stukel Phys. Rev. B 1, 3458,4791 (1970). 5 0 F. Bassani, in “Semiconductors and Semimetals” (R.K . Willardson and A. C. Beer, eds.), Vol. 1. Academic Press, New York, 1966. 5 1 E. 0. Kane, in “Semiconductors and Semimetals” (R.K. Willardson and A. C. Beer, eds.), Vol. 1. Academic Press, New York, 1966. 39

2.

99

MOBILITY OF HOLES IN 111-V COMPOUNDS

By comparing Figs. 2 and 3, the following picture emerges for the top of the valence band. In the absence of spin-orbit splitting, the highest point of the uppermost valence band is at r and has rI5symmetry (atomic p-like; sixfold degenerate including spin degeneracy). When spin-orbit effects are included, this band split into a fourfold degenerate T8 band and a doubly degenerate r, band which is separated from Tsby an energy A,, (the zonecenter spin-orbit splitting). For points away from k = 0, the symmetry is lower and the T8 band is further splits2 into two doubly degenerate bands called the heavy-hole (or u l ) and light-hole (or u 2 ) bands. The band that has r, symmetry at the zone center is known as the split-off (or u3) band. This notation will be used throughout this chapter and is summarized in Fig. 4. 4E

f

re

k

FIG.4. Features of the energy band structure relevant to hole transport.

Although numerous techniques are available for calculating the gross features of the energy bands of s e m i ~ o n d u c t o r s ,the. ~ ~technique ~ ~ ~ ~ ~ most suitable for investigating detailed features near band edges is the so-called k p technique. It will be assumed in what follows that the reader is already familiar with the basic ideas involved in the k p method, and only the briefest summary will be given. The papers" *42 and review article' by Kane contain the details omitted here, along with extremely lucid explanations of the overall technique. Briefly, one divides all electron states at k = 0 into two classes such that class A contains the states of primary interest, as well

-

'*

Strictly speaking, the lack of inversion symmetry in zinc-blende crystals causes even more splitting of the energy bands than described here or shown in the figure^.^^^^^^^' The energy bands near k = 0 contain terms which are linear in k and which further split the Tsband and move the valence band maxima slightly away from k = 0. These effects are so small, however, that they have never been shown to have any effect on hole transport, and will be ignored throughout this chapter.

100

J. D. WILEY

as any other states which interact strongly with them. The remaining states are in class B and, by selection, interact only weakly with states in A. The A-B interactions are first removed (or partially removed) using perturbation theory. This results in a renormalized Hamiltonian in which A-A interactions have been modified by the A-B perturbations. Finally, the A-A interactions are handled by exact diagonalization of the renormalized A-A Hamiltonian. The method of including spin-orbit perturbations and the choice of which states to include in class A depend on the relative sizes of the energy gaps and spin-orbit splittings at r. As more states are added to class A, the accuracy of the final results increases, but the difficulty of diagonalizing the renormalized A-A Hamiltonian increases as well. Kane4' has calculated the valence bands of Ge and Si by starting with only the six for zinc-blende) states in class A. This results in the degenerate r2, (r15 following secular equation for the energies of the three valence bands of Fig. 4: H l I H 2 2 H 3 3+ 2 H I 2 H 2 , H l 3 - H 1 1 H 223 - H'2 2 H 213

- ~ A o ( M I 1 H 2+, H I 1 H i 3+ H;2H;3

where

- w33H:2 - H:2 - H:3 - H:3)

Hii = Hii + (A2/2m,)k2 - E , .

(2)

The H,j are elements of the matrix H,., =

[

Lk,'

+ M(k; + kZ2) NkXkk, NkXkZ

NkZk,

Lk,Z

+ M(k: + kz2) Nkykz

(1)

= 0,

NkA

1.

Nk,k* (3) LkZ2 M(kX2 kY2)

+

+

The quantities N,L,and M contain all interactions between the valence bands and other states at r (i.e., all A-B interactions). Expressions for N, L,and M in terms of interband matrix elements can be found in the and in appendix to this chapter. Since the rl conduction band T I cis the nearest state which was included in the class B, the ratio [E(Tlc) - E ( q 5 ) J / A o is an index of the expected reliability of Eqs. (1)-(3). = E(rlC) - E(T;,), and Table 1 lists Ao(TSy - r,"), E,,(rlc E,'(Tf, - Ti,) = E(Tf,) - E(c5) for Si, Ge, and the 111-V comIt "can pound~.~~,~ ~ ' be seen from Table I that only InSb, InAs, and GaSb "

P. Lawaetz, Phys. Reo. 84,3460 (1971).

" D. E. Aspnes and A. A. Studna, Solid State Commun. 11, " D. E. Aspnes, Phys. Reu. Lett. 31,230 (1973). 'I "

1375 (1972).

D. D. Selland P. Lawaetz, Phys. Rev. Lett. 26, 311 (1971). D. E. Aspnes and A. A. Studna, Phys. Rev. B 7,4605 (1973).

2.

MOBILITY OF' HOLES IN 111-V COMPOUNDS

101

TABLE I VALUB FOR THE ZONE-CENTER SPIN-ORBIT SPLITTlNG OF 'THE VALENCE

BAND,AND THE TWO PRINCIPAL DIRECT GAPSEo

Si Ge AIP AlAs AlSb GaP GaAs GaSb InP InAs InSb

0.04 0.297' 0.05

0.28 0.75 0.078d O.34le 0.77 0.13 0.38 0.81

4. 185b 0.887c 5.12 3.06 2.30 2.884* 1.518' 0.81 1.42 0.42 0.237

AND

Eo"

3.37w 3.006' 5.18 4.66 4.73 5.33 4.8 1 3.69 5.10 4.40 3.49

Unless otherwise noted, all values are taken from a recent tabulation by Lawaetz. Aspnes and S t ~ d n a . ' ~ A~pnes.'~ *Sell and L a ~ a e t z . ' ~ Luttinger."

''

have E,,/A,, ratios smaller than that of Ge. Based on estimates by Kane4' and later work by F a ~ c e t t , ~Eqs. * (1)-(3) are known to provide excellent approximations for the u1 and u2 band of Ge,and a reasonably good approximation for the u3 band. Therefore, it is to be expected that they will provide equally good or better results for all the materials in Table I except InSb, InAs, and GaSb. In order to illustrate the valence band structure of typical 111-V compounds, Eqs. (1)-(3) have been used to calculate energy contours for the valence bands of GaAs and Gap, and the results are shown in Figs. 5-8. The values used for N,L,and M were calculated from tables of the Luttinger valence band parameters" recently reported by LawaetzS3 Equations relating the commonly used valence band parameters are given in the appendix, along with a table of values. Similar calculations for Si and Ge60 reveal that the valence bands of these materials are quite similar to those of GaP and GaAs, respectively. This is also confirmed by comparisons with

'*

W. Fawcett, Proc. Phys. SOC.London 85,931 (1%5). J. M. Luttinger, Phys. Rev. 102, 1030 (1956). 6o J . D. Wiley, unpublished work. 5g

P

W

W

tx

I

50 mev

(b)

FIG.5. Energy contoursfor the heavy-hole( u , ) band in GaAs: (a) for a (100) plane and (b) for a ( I 10) plane. The scale for k values is marked in units of 0.01~~; I , where a. is the Bohr radius (0.529 A).

102

FIG.6. Energy contours for the heavy-hole ( 0 , ) band in Gap: (a) for a (100) plane and (b) for a (1 10) plane. The k scale is in the same units as in Fig. 5.

103

104

J . D. WILEY

FIG. 7. Energy contours for the light-hole ( u 2 ) band in GaP for a (1 10) plane. The k scale is in the same units as in Fig. 5.

published contours for Gel‘ and Si.61*62 Spicef3has shown the heavy-hole (ul) contours for GaAs in a (100) plane over the entire Brillouin zone, Figure 8 shows all three valence bands in GaP for two principal directions, and averaged over all direction^.^^ Along the (100) and (1 1 l), directions the o1 and o2 bands are parallel over most of the Brillouin zone. From Fig. 8b it is clear, however, that this situation does not prevail for general directions. In view of the extreme anisotropy of the u1 band in sphericaliy M.Asche and J. von Boneszkowski, Phys. Status Solidi 37,433 (1970). M.Costato and L. Reggiani, Lett. Nuovo Cimento 3, 239 (1970). 63 W. E. Spicer, in “Optical Properties of Solids” (F. Abeles, ed.). North-Holland Publ., 61

Amsterdam, 1972. J. D. Wiley and M.Di Domenico, Jr., Phys. Rev. B 3 , 375 (1971). ” The severe warping of the u1 band in GaP is largely a result of a strong renprmalized inter, action between u , and uj. If the valence bands of GaP are recalculated using a fictitious spin-orbit splitting of 300 meV, the uI contours are much less warped and resemble those of GaAs. 64

2.

105

MOBILITY OF HOLES IN 111-V COMPOUNDS

k<'l.l>

k<,,>

(a<''

(02)

0.10

0.05

0

0.05

0.10

0.10

0.05

k (a,') 0

0.05

0.10

FIG.8. Valance bands of Gap: (a) as calculated for the (100) and ( 1 11) directions and (b) averaged over all directions. (After Wiley and Di Domenic0.6~)

106

J . D. WILEY

averaged bands (and effective masses) are of questionable utility and are shown in Fig. 8b only for purposes of comparison with Fig. 8a. The valence bands of the narrow-gap semiconductors InSb, InAs, and GaSb cannot be handled by the foregoing technique. For these materials, the interaction between the valence bands and the lowest r6conduction band is too strong to be treated as a perturbation, and must be treated exactly. Kane” has given a general prescription for this case which leads to an 8 x 8 secular determinant. This could, of course, be solved numerically but the following approximate approach4’ is usually taken instead. One first ignores all k. p interactions except those among the r7’,Tsv light hole, heavy-hole band does not interact and r6‘ states. To first order, the rsv with the other three bands, so it is omitted. This leads to the following interaction Hamiltonian and secular equation :

and

+

E(E‘ - E,) (E‘ A,) - k2P2(E

+ 2A0/3),

(5)

where

-

The k p interaction between conduction and valence bands is contained in P [which is often reported as an equivalent energy E, = (2rno/h2)P2]. In this approximation, the heavy-hole band is left parabolic, with an effective mass of unity. Corrections from higher bands must be included in order to obtain a realistic approximation for this band. For purposes of this chapter, the most interesting feature of the narrowgap model is that it is the simplest model which provides expressions for the valence band wave functions, including the admixture of s-like symmetry from the conduction band. According to Kane,42 the wave functions are

where subscript i refers to the conduction, v 2 , and v3 bands. Note that the heavy-hole band retains its pure p-like symmetry in this model, since it

2.

107

MOBILITY OF HOLES IN 111-V’COMPOUNDS

does not interact with the conduction band. The coefficients a i ,bi;and ciare given by ai = kP(Ei’ + 2 A 0 / 3 ) / N ,

(JzA$)

(E; - EJ/N ci = (E; - E,) (E; + 2 A 0 / 3 ) / N ,

bi

=

(8)

9

where E,’ is the appropriate root of Eq. (5), and N is a normalizing factor equal to the square root of the sum of the squares of the numerators. At 0.8

0.6

a

0.4

0.2

0

0

2

8

6

4

k2 (

10

12

14

aG2 1

FIG.9. The wave function coefficient (I for the light-hole band as a function of wave vector squared (k is in units of reciprocal Bohr radii). The three materials shown (1, InSb; 2, GaAs; 3, Gap) are taken to be representative of the narrow-, intermediate-, and wide-gap 111-V compounds, respectively.

108

J. D. WlLM I .o

0.9

0.8

-b

0.7

0.6

0 .s

0

2

4

6

6

10

12

14

k2 ( a b1 FIG.10. The wave function coefficient b for the light-hole band. Note that b < 0 for this band. The three materials shown are: I, Gap; 2, GaAs; 3, InSb.

k = 0 the coefficients become6’’ U,

= 1,

b, = C, = 0,

= 0,

bu2= - (1/3)”’, bu3= (2/3)“’,

uU3= 0,

cV2= -(2/3)”’,

(9)

cU3= -(1/3)”’.

6saNotethat Eq. (16) of Kane4* contains a typographical error. The coefficients b,, and should be negative.

c”,

2.

109

MOBILITY OF HOLES IN 111-V COMPOUNDS

Although the narrow-gap model is expected to provide rather poor approximations for the energy bands in intermediate- and wide-gap materials, it should provide a fair measure of the admixture of conduction and valence band wme fictions. Therefore Eqs.(4)-(8) have been used to calculate u, b, and c for the light-hole bands of InSb, GaAs, and Gap. The results are shown in Figs. 9-1 1. From Eqs. (7) and Fig. 9, it is seen that the valence bands of the intermediate- and wide-gap material remain almost totally plike, even for relatively large k. This will be seen to result in considerable simplifications in Part 111.

I.o

0.8

-C

0.6

0.4

0.2

0

2

8

6

4 k2 (

10

12

14

a# 1

FIG.I I. The wave functioncoefficient c for the light-hole band. Note that e < 0 for this band.

The numbered curves are for same materials as in Fig. 10.

110

1. D. WILEY

111. Scattering Mechanisms and Models

As discussed briefly in Part I, the transport properties of the valence bands of 111-V compounds are considerably more complex than those of the conduction bands. It is the purpose of this part to discuss these complexities in more detail, and to review the approximations and models which have been used by various authors in calculations of hole mobilities. It will be found that the models fall into two broad categories :(1) detailed calculations in which the mobility is obtained by numerical solution of a pair of coupled Boltzmann equations, and (2) phenomenological models which, through numerous approximations of doubtful validity, yield fairly simple, accurate expressions for the effective hole mobility. Although the phenomenological models do little to advance our understanding of hole transport, they are quite useful in fitting and systematizinghole mobility data, and extrapolating from existing data into unmeasured experimental regimes. Both types of theoretical approaches will be discussed in the present section, but an attempt will be made to maintain a clear distinction between conclusions based on exact calculations66 and conclusions based on simple phenomenological models. FEATURES OF HOLETRANSPORT 1. GENERAL a. Hole Wave Functions

When an electron or hole is scattered from state k in band i to state k’ in band j , the scattering matrix element contains a multiplicative factor given by9,29,67,68

where the sum is over spin states and the functions u(r) are the cell-periodic parts of the Bloch functions. The function G is known as the “overlap function” since it represents the overlap between initial- and final-state wave function^.^^ For spherically symmetric wave functions, the overlap is unity for all k and k’. For anisotropic wave functions, however, the overlap is always < 1. It is easy to see qualitatively that this reduced overlap between There are, of course, no “exact” calculations of hole mobility since even the most elaborate models involve numerous approximations. The term “exact” is used here and in later sections to denote calculations that involve numerical solutions of the coupled Boltzmann equations as opposed to those that simply attempt to “patch up” the simple expressions for mobility which were derived for nondegenerate s-like bands. D. Matz, J . Phys. G e m . Solids 28,373 (1967); Phys. Rev. 168,843 (1968). 6 8 M. 0.Vassell, A. K. Ganguly, and E. M. Conwell, Phys. Rev. B 2,948 (1970). 6 9 The overlap function is denoted variously as G(k, k), G(y), %(/?A g(k, k , y), g, and G.

b6

’’

2.

MOBILITY OF HOLES IN 111-V COMPOUNDS

111

initial- and final-state wave functions will reduce the scattering rates and increase the mobilities of carriers with anisotropic wave functions. Using the wave functions given by Eqs. ( 7 ) and (8), Wiley2’ has shown that the overlap functions for intraband and interband scattering of holes are given by

G, ,(k, k’) = 31 + 3 C O S ~y),

+

G2,(k, k‘) = a2(k)a2(k’) 2a(k)a(k’)[b(k)b(k‘)

+ c(k)c(k‘)]cos y + [b(k)b(k’)+ c(k)c(k’)]cos2 y + (ib(k)b(k’)- [b(k)c(k‘)+ b ( k ’ ) ~ ( k ) ] / fsin2 i ) ~y , G I ,(k, k ) = &k’) + f l c(k’)I2sin2y , G,,(k, k ) = :[6(k) + 8c(k)I2sin2 y ,

(1 1)

where G,(k, k’) involves transitionsfrom state k in band i to state k’ in band j , and y is the angle between k and k’. The wave function coefficients a, 6, and

c are given by Eqs. (8) and are shown in Figs. 9-1 1. The simplicity of the expression for G,,(y) is a consequence of the fact that the wave functions used were obtained from a simplified model in which the heavy-hole band was not coupled to any other bands, and remained purely p-like [Eqs. (7)]. In the limit k, k‘ + 0 the admixture of s and p symmetries vanishes (Figs. 9-1 1) and the expressions for the overlap functions reduce to

G,,(y) = G22W = $1

+ 3 cos2Y),

G,,(y) = G21(y) = ;sin2 y .

(12)

The extent to which Eqs. (12) are useful approximations, even for fairly large k , is shown in Figs. 12-15, where G2,(k, k ) and G,,(k, k ) have been plotted for InSb, GaAs, and Gap, using k values typical of those encountered in scattering processes at 300°K. By careful comparison of Figs. 12-15 with Eqs. (ll), it is possible to identify the sources of various features in the angular dependences of G2,(k, k’) and G2,(k, k’). Thus, for example, the reduction of G,, for y > 742 is particularly pronounced for InSb and increases with increasing k. This is due to the cos y term in Eqs. (1 l), which contains a multiplicative factor a(k)a(k’).From Eqs. (7), it is seen that the coefficient a is a measure of the admixture of s-like symmetry. For the intermediate and wide-gap materials, a is quite small and there is very little reduction in the large-angle scattering. Similarly, the reduction of G,, at all angles can be attributed to the admixture of s-like symmetry. In this case the reason is less obvious since a does not appear explicitly in the expression for G,,, but manifests itself through the normalization parameter N in Eqs. (8). As a increases, b and c are decreased and the magnitude of G,, is reduced.

112

J. D. W1L.EY

1.4

1.2

-

-

1. k2

a 0

2. k

= I x lo-' ao-2

3. k2 = 2 x I O - ' O ~ * 4. k 2 = I O - ~ O ~ ~

1.2

.-

-

I. k 2 = O

2. k2 = 2 x

10-4a<2

s.

J

N

N

0

0.6

0.4

0.2

--

01 0

I

a/4

I

a/2 SCATTERING ANGLE,

I 3 lr/4

y

FIG. 13. Similar to Fig. 12. The curves shown here are for GaAs, which is representative of the intermediate-gapmaterials GaAs, Ge, AISb, and InP.

2.

113

MOBILITY OF HOLES IN 111-V COMPOUNDS

1.4

1.2

1.0 c

0.8

J M

0.6

W 0.4

0.2 0

0

T/4

3m4

7r/ 2

SCATTERING ANGLE,

y

B

FIG. 14. Similar to Fig. 12. The curves shown here are for Gap, which is representative of the wide-gap materials Gap, Si, AIP, and AIAs. k2 = O

1.

2. GOAS, k 2 * 14 X to-'ai2 3. Gap,

k 2 x 14 x lo-4~,'2

4. insb, k 2 = 1 4 X I O - ~a0-*

0

7T /4

.IT/ 2

SCATTERING

3 K/4 ANGLE,

lr

y

FIG.15. The overlap function G,,(k, k') as a function of scattering angle y.

114

J . D. WILEY

From the foregoing discussion and Figs. 12-15, it is seen that Eqs. (12) should be quite acceptable as approximations for the intermediate- and wide-gap materials. If Eqs. (12) were used for the narrow-gap materials, the primary effect would be to overestimate the scattering rate for light holes, and thus to underestimate the light-hole mobility. As will be seen later, the light holes make a relatively small contribution to the overall effective mobility, so that it is probably safe to use Eqs. (12) even for the narrowgap materials. The manner in which the overlap functions enter into and affect calculations of the mobility will be discussed in a later section since the precise effect is different for different scattering mechanisms. b. Two-Band Transport

The presence of two interacting bands of carriers introduces considerable complications into any calculation of hole m~bility.~’ The correct way of handling this problem is by solving a set of coupled Boltzmann equations. 14.17-23.30-33 0ne thereby obtains the perturbed carrier distribution functions from which the currents and transport coefficients can be cal~ u l a t e d The . ~ ~ results and conclusions of calculations of this sort will be discussed later in connection with specific scattering mechanisms. For the present, the consequences of two-band transport will be discussed in terms of simple models which have been found to be useful (and even accurate) in special cases. The simplest model results from assuming that u1 and u2 are decoupled (i.e., there is no interband scattering). The conductivities of the two bands are then simply additive and it is easy to show that

where perf is the effective (or measured) mobility,” and the total hole concentration isp = p1 + p 2 . If the bands are assumed to be approximately spherical, thenp,/p, = (m,/rn,)3’2 and Eq. (3) can be written as

There are, in the literature, numerous calculations of the transport coefficients for cases of two or three noninteracting (decoupled) bands. These models are not generally applicable to the valence bands (although some of them have been so applied) and will not be discussed here. For discussions of these models see Hilsum and R~se-Innes,~ Madelung: Putley,’ and Beer.23 ” Here we are tacitly assuming the Hall and drift mobilities to be related by p,,/pd = 1. The relation between p M and pd will be discussed more fully in a later section. ‘O

2. MOBILITY OF HOLES

IN 111-V COMPOUNDS

115

where

When p2 and pl are calculated using the standard expressions for the mobilities, the ratio p2/p1 reduces to a simple power of r, depending on the mass dependence of the scattering mechanism under consideration. Thus, for ionized-impurity, polar phonon, and nonpolar phonon scattering, the The decoupled band approximaratio is r1I2, r3I2,and r5I2, tion is known to be extremely poor in the cases of phonon scattering,17-22.30-33 but may have some validity for ionized-impurity scattering.34*72 For phonon scattering, the assumption of decoupled bands leads to an overestimation of the mobility of the light holes, and it is necessary to use an approximation which acknowledges the presence of interband scattering. In this approximation, one writes"' -

1

= - 1+ -

1

71

711

712

and

_1 -- -1

1 + - 9

722

=2

721

where rijis a relaxation time associated with scattering from band i to bandj. If the 7ij differ primarily in the density of final states, then transitions with final states in the heavy-hole band will be dominant and we can write

In this approximation, we have TI

(19)

z T2.

Using the approximation of equal relaxation times, pl = ez/m, , p2 = er/m, ,and p2/pl = r. Thus Eq. (14) becomes Perf =

r

+ r3I2

p1,

71

= TZI

(20)

l 2 H . Brooks, Advan. Electron. Electron Phys. 7,85 (1955). '*'A much more sophisticated discussion of interband scattering can be found in Appendix B of Bernard ef a/.'*

116

J. D. WlLEY

Note that in the T: = z2 approximation, the bands are coupled only to the extent that coupling is implied by the conditions in Eq. (18). No explicit account has been taken of interband scattering. Indeed, if the bands are truly coupled, then Eq. (13) is invalid since it implies a simple linearly additive relation between the conductivities, t7 = c1 + 0 2 , which is only valid if the bands do nor interact. The approximate nature of Eq. (20) can also be seen by considering the limiting case r -, 1. In this limit, one obtains, in effect, a single, fourfold degenerate band. The mobility in such a band should be p1/2due to the doubling of the density of final states. Thus Eq. (20) has the wrong limit as r + 1. Nevertheless, Eq.(20) is surprisingly accurate for the case of acoustic and nonpolar optical (NPO) scattering. Figure 16 shows the error incurred by using the = 7 2 and decoupled band approximations for calculating the effective mobility due to acoustic and NPO scattering. The exact results were obtained by Costato et aL3' and will be discussed in more detail later. It is seen from Fig. 16 that Eq. (20) is in error by about 20% for AlP and Si, and by less than 10% for all other materials. The decoupled band approximation is high by a factor of three in the best case, and is clearly not a satisfactory model for phonon scattering. The m,/m, values used in Fig. 16 are given in Table 11, and are taken from a

TABLE I1 CALCULATED VALUES FOR THE AVERAGE BAND-EDGE E m c n v E MASSES OF THE 0 , . U 2 , AND U 3 BANDS' Material Si Ge AIP AIAs AlSb GaP GaAs GaSb InP InAs InSb

r = m,/ml

m1

m2

m3

0.53 0.35 0.63 0.76 0.94 0.79 0.62 0.49 0.85 0.60 0.47

0.16 0.043 0.20 0.15 0.14 0.14 0.074

0.24 0.092 0.29 0.24 0.29 0.24 0.15

3.3 8.2 3.0 5.0 6.7 5.6 8.4

0.046 0.089

0.14

10.7 10.5 22 31

0.027 0.015

0.17 0.089 0.107

See also Fig. 4. These masses are taken from a tabulation by J ~ w a e t z 'and ~ are the result of a fivelevel k p calculation.

-

2.

MOBILITY OF HOLES IN 111-V COMPOUNDS

117

I

FIG.16. The error incurred in using the (1) decoupled band and (2) T , = r 2 approximations for estimating the effective mobility in the presence of acoustic and nonpolar optical scattering.

table of calculated valence band parameters given by Lawaet~.”*~’ When energy bands are as warped and nonparabolic as the valence bands, the concept of a simple scalar effective mass loses s i g n i f i ~ a n c e , 4 * ” and *~~~~ one must define different masses for different effects. For the sake of simplicity and consistency, the masses in Table I1 will be used throughout this chapter and no distinction will be drawn among the various types of masses. In Part IV, it will be shown that Table I1 is in reasonable agreement with various measured masses.

c. Warping of the Energy Surfaces In Part 11, it was shown that the valence bands of the group IV and 111-V semiconductorsexhibit considerable warping and that the warping becomes more severe at high energies. Warping of the energy surfaces has numerous interesting consequences for hole transport, but for present purposes, the most important effect is its consequence on the ratio of Hall to drift mobility. This is important because most calculations yield a drift mobility, whereas the experimentally measured quantity is usually the Hall mobility. The two mobilities are related by a quantity known as the Hall coefficient See also the appendix to this chapter. 2.Koped, Acfa Pbys. Pol. 19,295 (1960). 7 5 J. Kolodziejczak and R. Kowalczyk, Acra Phys. Pol. 21, 389 (1962). 76 J. Kolodziejczak and S. Zukotynski, Pbys. Sfatus Solidi 5, 145 (1964). 73

74

118

J . D. WlLEY

For nondegenerate, spherically symmetric bands, it can be shown thatZ3 rH > 1 and that (22) where T is the scattering time and ( ) indicates a thermal average over the Even when a scattering time does not distribution of carrier energie~.’~ exist (as, for example, in the case of polar mode scattering), the Hall coefficient factor is still greater than ~ n i t y for ~ . isolated, ~ spherical bands. When the bands are warped, however, the Hall coefficient factor depends on the degree of warping as well as the scattering mechanism, and can be written as rH

= /(T>’,

(23) where rs, the scattering factor, is given by Eq. (22) and r A is the anistropy factor. Shockley,” Allgaier,81-83 and others23,84,85have given detailed physical explanations of the origin of r A. Briefly, the anisotropy factor depends on both scattering anisotropy and anisotropy of the energy surfaces. Allgaier summarizes the general behavior of rA (for all models investigated to date) as follows83: (a) If T alone is anisotropic, r A 2 1; (b) if the energy surface alone is anistropic, rA < 1 for most cases; (c) if both types of anisorH = rsrA,

’’ Where there is chance of confusion, the following notation will be used for mobilities in the

remainder of this chapter: For single-band mobilities calculated using s-like wave functions we write p:,sc where i = 1,2 for v I or u2, and sc = AC, NPO, PO, I1 denotes the specific scattering mechanism involved. For eflective mobilities we write p z , where sc has the same meaning as before, and eff implies that account has been taken of both bands as well as the correct wave functions. For effective Hall mobilities, we write pi:‘:,. When there is no chance of confusion, or when nonspecific mobilities are intended [as in Eq. (21)], the cumbersome subscripts and superscripts will be dropped in favor of a simpler, self-explanatory notation. ” Here, in the spirit of the other approximations used in this section, the distinction between the Hall coefficient factor rH= neR and the Hall-to-drift mobility ratio r = p Jpd is intentionally ignored in the interest of simplicity. For a discussion, see Beer.” It will be seen later that there is no adequate theory for rH or r inp-type material, so it is pointless to burden the present discussion with unnecessary detail. l 9 Price79ahas recently given a more general formula in terms of a vector mean free path. 79sP. J. Price, Phys. Rev. B 6, 4882 (1972). See also P. J. Price, IBM, J . Res. Develop. 1, 239 (1957); 2, 200 (1958).

W. Shockley, “Electrons and Holes in Semiconductors,” Sect. 12.9, pp. 336-341. Van Nostrand-Reinhold, Princeton, New Jersey, 1950. R. S. Allgaier, Phys. Rev. 158,699 (1967); 165, 775 (1968). *’ R. S. AIlgaier and R. Perl, Phys. Rev. B 2, 877 (1970). 83 R. S. Allgaier, Phys. Rev. B 2,3869 (1970). 84 C. Goldberg, E. Adams, and R. Davis, Phys. Rev. 105, 865 (1957). ” H. Miyazawa, Proc. Int. Con$ Semicond. Phys., Exeter, p. 636, Inst. Phys. Phys. Soc.,London, 1962.

2.

MOBILITY OF HOLES IN 111-V COMPOUNDS

119

TABLE I11 THEHALL COEFFICIENT ANISOTROPY FACTOR FOR VARIOUS SHAPES OF ENERGY SURFACES“ Shape of energy surface

rA

1

Sphere Regular octahedron

2 3 I -

cube

Cube with rounded edges ~

a

2

$R

~~

See Allgaier,” Allgaier and Perl,s2

and Goldberg et aLE4

tropy are present, r A can be > 1 or < 1, but when the scattering anisotropy becomes sufficiently extreme, it always dominates the shape anisotropy and causes rA > 1. Table 111gives values of rA calculated by Allgaier81-83for various shapes, assuming isotropic scattering. We now investigate the extent to which r A differs from unity for the valence bands discussed in Part 11. Dresselhaus er al.” have shown that the energy bands which result from diagonalizing Eq. (3) are given by

E(k)= -(h2/2m0){Ak2 & [BZk4+ C2(kX2k; + ky2kz2 + kzZkxZ)]1~2), (24)

where A , B, and C are valence band parameters which are discussed and tabulated in the appendix. These bands (the upper sign corresponds to u2 and the lower sign to u l ) are approximations to the more accurate bands obtained from Eq. (l), and are only valid near k = 0.86 Lax and Mavroides,15 Beer,23 and Stirn” have shown that Eq. (24) can be rewritten in the following approximate form:

E(k) x -(h2k2/2m0) ( A & B’) { 1 - r[(kx2ky2+ k;kz2 where and

86

+ kzzkx2)]/k4]+ . - .) , (25)

B’ = (B2+ C2/6)’/’

r = T c ~ / [ ~ Bf ’~’11. (A

(26) (27)

Actually, theform of Eq. (24) is valid for larger values of k. It is the method used for calculating A, B, and C which restricts Eq.(24) to small values of k. This is intimately related to the “variable warping” of the bands. See Bernard er a/.14 and Kane4’ for further discussion. R.J. Stirn, in “Semiconductors and Semimetals,” (R. K. Willardson and A. C. Beer, eds.), Vol. 8. Academic Press, New York. 1972.

120

J.

D. WILEY

Again, the upper signs refer to the u2 band and the lower signs to u l . Using Eqs.(25)-(27). L a x and Mavroides were able to show thats8

By comparing Eqs. (28) and (23), it is seen that ’A = ada 121’;

1

Note that the a coefficients are different for u1 and v2 so that each band has its own rA. Equations (26), (27), (29), and (30) have been used, together with valence band parameters given in the appendix, to construct Table IV.

’

TABLE IV

VALUES OF THE LAXAND MAVROIDES’ ANISOTROPY PARAMETERS [Eqs. (26)-(3 I)] CALCULATED

USINGVALENCE BANDPARAMETERS GIVEN IN THE APPENDIX u2 Band

u1 Band

Material

r

a,

~

Si Ge AIP AlAs

AlSb

GaP GaAs GaSb InP InAs lnSb

2.571 2.482 2.623 2.790 3.064 2.648 2.621 2.754 2.867 2.679 2.763

1.251 1.238 1.259 1.286 1.332 1.264 1.260 1.280 1.300 1.269 1.282

r,

a,, 1.374 1.346 1.390 1.448 1.551 1.400 1.390 1.433 1.477 1.410 1.437

‘d

‘11

‘12

‘A

~~~~~

1.012 1.010 1.013 1.018 1.028 1.014 1.013 1.017 1.020 1.015 1.017

0.673 0.687 0.659 0.626 0.570 0.652 0.659 0.638 0.605 0.645 0.630

-0.858 -0.349 -0.950 -0.668 -0.562 -0.563 -0.368 -0.311 -0.357 -0.141 -0.0994

0.969 0.985 0.966 0.974 0.977 0.977 0.984 0.986 0.984 0.993 0.995

1.016 0.999 1.021 1.007 1.004 1.004 0.999 0.999 0.999 0.998 0.999

1.033 1.008 1.039 1.021 1.016 1.016 1.009 1.007 1.009 1.003 1.002

0.997 0.959 0.966 0.974 0.997 0.996 0.996 0.998 0.997 0.997 0.997

p21p1 0.149 0.042 0.167 0.089 0.058

0.076 0.041 0.029 0.033 0.009 0.005

The notation used here is that given by Beer.’’ See also Stirn” for a very concise, lucid explanation of the method of Lax and Mavroides.ls

2.

MOBILITY OF HOLES IN 111-V COMPOUNDS

121

It is seen from Table IV that the anisotropy of u1 results in a considerable reduction in rA for this band but that rA x 1 for u 2 . To the extent that the transport properties of the valence band are dominated by u l , we can therefore expect an effective r A somewhat less than unity. Also listed in Table IV is the ratio of light to heavy holes p 2 / p 1as calculated using yet another expression of Lax and Mavroides' 5 , 2 3 * 8 7

where ad,is ad for band i. By comparing p2/pl from Table IV with values of p 2 / p 1 given for simple spherical bands p 2 / p 1= (m2/ml)3i2= r - 3 i 2 (see Table 11), it is found that the errors incurred by using the latter expression are too small to be of any consequence, and this approximation is well justified. The entire discussion of warping given in this section serves only to provide a qualitative insight into the complications which arise in the presence of scattering and energy band anisotropies. There has been, to date, no calculation of the hole mobility in any 111-V compound that includes simultaneously all relevant scattering mechanisms and anisotropic bands. Until such a calculation is performed, there is little justification for assuming that rHdiffers from unity. When comparing theory and experiment, it must be borne in mind that this uncertainty hangs over whatever agreement is obtained. d. Allowed Scattering Mechanisms

One of the principal reasons for the great interest in carrier mobilities is that a careful study of the temperature dependence of the mobility can, under favorable conditions, provide valuable information about the relative importance of the various allowed scattering mechanisms. It should be emphasized that the temperature dependence of the mobility alone is a very poor and potentially misleading indicator on which to base conclusions about carrier scattering. This is because there are often several competing scattering mechanisms which vie for importance in different temperature ranges, as well as band structure effects which alter the temperature dependence predicted by simple models. Needless to say, both of these difficulties are present in the case of the p-type 111-V compounds. Since there is no a priori reason for ruling out any known scattering mechanism, the greatest progress is likely to be made by insisting that similar materials be viewed, not as isolated special cases, but as members of a class within which all variations are slight and systematic. Thus, before investigating the details of scattering inp-type 111-V compounds, it is appropriate to see what can be learned from experience already gained in the much more thoroughly

122

J . D. WlLEY

studied group IV materials. In view of the great similarity between these two groups of materials, it is reasonable to expect that any scattering mechanisms which have been found to be important in Ge and Si will be important in the 111-V compounds as well. Pursuing this line of reasoning, Wiley and Di Domenico” found that the lattice mobilities of holes in Ge, Si, Gap, GaAs, InP, and AlSb are, indeed, strikingly similar both as to magnitude and temperature dependence. A11 of these materials have hole mobilities which can be approximated by p a T - 8 with fl x 2.2-2.4 in the 100-400”K temperature range. Figure 17 shows the temperature dependence of the hole

GaP

Klllo

’

0

I

I

I00

1 1 1 ’ 1 1 ‘

I

1 I I I

1000

TEMPERATURE ( O K )

FIG. 17. A comparison between experimental (open circles) and calculated (solid lines) mobilities assuming only acoustic and nonpolar optical mode scattering,and adjustingonly the acoustic deformation potential. The dashed lines have a slope of T-”’ and correspond to pure acoustic mode scattering. (After Wiley and Di Domenico.’*)

2.

MOBILITY OF HOLES IN 111-V COMPOUNDS

123

mobilities in Ge,89 GaAs,” and All three sets of data have been fit using the same theoretical (a simplified model involving acoustic and NPO scattering), and adjusting only the acoustic mode deformation potential. On the basis of this work, Wiley and Di Domenico concluded that acoustic and NPO scattering are quite important and probably dominant in the p-type 111-V corn pound^.^^ Figure 18 shows a comparison of the temperature dependence of polar mode scattering with that of combined acoustic and NPO scattering.2eIt is

0.I

1.0 T/B

10

FIG. 18. A comparison between the temperature dependence of the polar mobility, and the mobility obtained by combining acoustic and NPO scattering. (After Wiley and Di Domenico.”) D. M.Brown and R. Bray, Phys. Rev. 127, 1593 (1962). D. E. Hill, J. Appl. Phys. 41, 1815 (1970). 9 1 H. C. Casey, Jr., F. Ermanis, and K. B. Wolfstirn, J. Appl. Phys. 40,2945 (1969). 9 2 As an interesting aside, Tsui has seen extremely direct evidence of hole-TO phonon coupling in tunneling measurementsonp-GaAs. [See D. C. T s ~ i . ~Coupling ~’] between holes and TO phonons can only occur via the (NPO) deformation potential interaction. The size of the effects which are observed suggests that the TO and LO phonons are of roughly comparable importance. In agreement with the selection rules for NPO scattering (Harrison),’6 the TO structure is seen only in p-GaAs, while the LO structure is seen both in n-GaAs and p-GaAs. 92aD.C. Tsui, Phys. Reu. Leu.21,994 (1968). 90

124

J . D. WILEY

seen that the temperature dependenceg3of the polar mobility weakens considerably above T = 0/2 and is in considerable disagreement with the experimental data in Fig. 17. Polar mode scattering is, of course, not ruled out as an important hole scattering mechanism, but on the basis of temperature dependence alone, one must conclude that it is less important than acoustic and NPO scattering. The final scattering mechanism which will be considered in this chapter is ionized-impurity scattering.34 ,72*94 This mechanism is of great practical importance since the mobility at a fixed temperature (usually 77 or 300°K) is often used as a measure of the purity of a crystal. So far, there have been only two model^^^.^^ of ionized-impurity scattering which are applicable to the valence bands. The model of Brooks72 essentially follows the decoupled ~i~~ band approximation outlined in Section 1b. Costato and R e g g i a ~have solved the Boltzmann equations and given a detailed account of the interband and intraband scattering. Neither of these theories is applicable at high impurity densities, due to the importance of multiple scattering and quantum effects not included in the simple semiclassical Numerous other scattering mechanisms have been treated in the literature and are important in special cases, but will not be discussed in this chapter. Among these are piezoelectric,97-’00dipole,’” carrier-carrier,’” inhomogeneity,lo3 and ~pace-charge”~ scattering as well as any scattering due to crystalline defects such as dislocations or grain boundaries. O 5

’

e. Summary

of “Exact” Models

Beforeproceeding to detailed discussions of specificscattering mechanisms, it seems appropriate to summarize the implicit and explicit assumptions contained in the major “exact” models of hole mobility, since the results of The temperature dependence of pm in Fig. 18 was obtained using Ehrenreich’s theory (Ehrenreich’). Exact calculations for p-type 111-V compounds (Kranze?’ and Costato cf ~ 1 . ~exhibit ’) the same temperature dependence. This point will be discussed further in a later section. 94 P. P. Debye and E. M. Conwell, Phys. Rev. 93,693 (1954). 9 5 E. J. Moore and H. Ehrenreich, Solid Stare Commun. 4,407 (1966). 96 E. J. Moore, Phys. Rev. 160,607,618 (1967). 9 7 W. A . Harrison, Phys. Rev. 101,903 (1956). 9 8 A. R. Hutson, J . Appl. Phys. Suppl. 32,2287 (1961). 99 A. R. Hutson and D. L. White, J. Appl. Phys. 33,40 (1962). l o o J. D. Zook, Phys. Rev. 136, A869 (1964). l o ’ J. Appel and W. B. Teutsch, J . Phys. Chem. Solids 23, 1521 (1962). l o * J. Appel, Phys. Rev. 125, 1815 (1962). C. Herring, J. Appl. Phys. 31, 1939 (1960). L. R. Weisberg, J. Appl. Phys:33, 1817 (1962). H. Matare, “Defect Electronics in Semiconductors.” Wiley (Interscience), New York, 1971.

93

h,

TABLE V A SUMMARY OF THE ASSUMPTIONS USED IN THE PRINCIPAL “EXACT”MODELSFOR HOLETRANSPORT Ehrenreich and Overhauser” Type of scattering: I1 AC N PO PO Warping: Spherical Constant Variable E(k): Parabolic Nonparabolic

J J

Pikus Tiersten2’ et u / . ’ ~ - ~ O

J J J J

Lawaetz22

J J

Kran~er~’.’~

el

Costato a[,3 0 . 3 2 , 3 4 , 3 5

J J

J

J

J J

J

126

J . D. WILEY

these models will be used to provide guidance and justification for various approximations used in later sections. Table V contains such a summary. All of the models in Table V utilize the Boltzmann equation approach, and take correct account of the hole wave functions in calculating scattering probabilities.‘ 0 6 A few of the more interesting phenomenological models are also worth summarizing: Lax and Mavr~ides’~ consider I1 and AC scattering using the 71 = 72 approximation and parabolic bands with constant warping. Bernard et ~ 1 . consider ’ ~ I1 scattering in the 7 1 = f2 approximatiton for nonparabolic bands with variable warping. This is apparently the only model to date ’ ~ include, in an appenwhich includes variable warping. (Bernard et ~ 1 . also dix, an elegant discussion of interband scattering.) Brooks7’ has discussed I1 scattering in the decoupled band approximation for spherical, parabolic bands, as well as AC and NPO scattering with interband transitions.

2. INTRINSICSCATTERING MECHANISMS In the present section, attention will be focused on the scattering of holes by phonons (often called intrinsic or lattice scattering). In nonpolar materials, the electron-phonon interaction has been discussed in terms of the d e f o r m a b l e - i ~ n , ’ ~ ~rigid-ion’Og *’~~ and deformation-potential’ l o models. Of the three models, the last has been shown to be the most realistic”’ and is nearly always used in modern calculations of nonpolar electron-phonon scattering in semiconductors. The original deformationpotential theory of Bardeen and Shockley’ l o considered only the interaction between electrons in spherically symmetric, nondegenerate bands and longitudinal phonons. The generalizations needed for the treatment of degenerate bands and transverse phonons have been discussed by several aUthors.18-20,72.1 12 Lawaetz22 has recently reviewed the deformationpotential approach and discussed the limits of its validity. In compound semiconductors,the LO phonons cause an electric polarization which leads to additional scattering of the charge carriers. This scattering mechanism, known as polar mode scattering, was first discussed by Io6

There are numerous differences among these models which are not indicated in Table V.

Thus, for example, Ehrenreich and Overhauser” use the “deformable-ion” and “rigid-ion”

models rather than a more appropriate deformation-potential approach. There are also differencesin the treatment of scattering anisotropies, the inclusion of transverse phonons, and other details. lo’ F. Bloch, Z . fhys. 52, 555 (1928). F. Seitz, Phys. Rev. 73, 549 (1948). ‘09 L. Nordheim, Ann. Phys. 9,607 (1931). l L o J. Bardeen and W. Shockley, fhys. Rev. 80,72 (1950). J. M. Ziman, “Electrons and Phonons.”Oxford Univ. Press, London and New York, 1960. G . D. Whitfield, fhys. Rer. Leu. 2 , 2 0 4 (1959); fhys. Rev. 121,720 (1961).

’’’

2. MOBILITY

OF HOLES IN 111-V COMPOUNDS

127

Frohlich113 and Callen’14 in the context of electric breakdown of ionic crystals. The theory of polar mode scattering in semiconductors was first given by Howarth and Sondheimer115 for electrons in simple parabolic bands, and later generalized by Ehrenreich’ to include nonparabolicity, screening, and the effects of mixed s and p wave functions. More recently, several authors have discussed the further modifications necessary for In the next two subtreating the polar mode scattering of holes.29*3’-33,35 sections, these nonpolar and polar intrinsic scattering mechanisms will be discussed more fully. a. Acoustic and Nonpolar Optical Phonon Scattering For electrons in nondegenerate, parabolic, spherically symmetric bands, the acoustic mobility is given by3-5

where p is the material density, uI is the velocity of longitudinal sound waves, m* is the effective mass of the charge carriers, and E , is the acoustic deformation potential. This can be conveniently rewritten as PU,Z __ 1 p i c = 3.1727 x lo-’ (33) (m*/mo)5/2E12T3” ’ where p i c is in cm2/V-sec,El is in eV, p is in g/cm3, u1 is in cm/sec, and T is in OK.The dependence of pic on effective mass, deformation potential, and temperature should be noted since it will later be seen that these dependences are preserved even in cases involving more complex band structures. Harrison16 has shown that, to first order, electrons with spherically symmetric wave functions have no deformation-potential interaction with optical phonons. The simplest case of electron-NPO phonon interaction involves electrons in nondegenerate ellipsoidal bands such as are found in n-Ge and n-Si. For this case, C ~ n w e l l ” ~ has ~ ” ~shown that the NPO mobility may be written in the form PNPO

=

x l3

I4 I Is

8 ~ c ’ / ~ e h ~ pu12(ee’T- 1) (m*/mo)5/2E~,9T’/’

3fik;12,,,;/2

JOm + (1

xe-x dx B/xT)li2 e8/T(1-

+

e/XT)1/2’

H. Frohlich, Proc. Roy. SOC.A160.230 (1937). H. B. Callen, Phys. Rev. 76, 1394 (1949). D. J. Howarth and E. H. Sondheimer, Proc. Roy SOC.A219, 53 (1953). E. M. Conwell, Sylvania Techno/. 12,30(1959).

(34)

E. M.Conwell, “High Field Transport in Semiconductors.” Academic Press, New York, 1967.

128

J. D. WILEY

where 0 is the characteristic temperature of the optical phonons (k,0 = ho), EN,, is a suitably defined optical phonon deformation potential,”’ and all other quantities have the same meaning as in Eq. (32). Comparing Eqs. (32)-(34), it can be seen that the numerical prefactor involving fundamental physical constants in Eq. (34) is 6.345 x lo-’ if pNpois expressed in cm2/Vsec and EN,, is in eV. Note that the temperature dependence of pNpois more complicated than that of pAcat low temperatures (0/T 5 1) but approaches T - 3 ’ 2at higher temperatures. The contrast between this behavior and the high-temperature behavior of the polar mobility has already been shown in Fig. 18. Turning now to hole mobilities, it can be seen from Table V that there are several models which include the AC and NPO scattering of holes. While it is not feasible to review the details of these calculations, there are several major conclusions to be drawn from them, which have proved useful in formulating simpler phenomenological models. The exact models are in agreement on the following points: (1) Both transverse and longipdinal phonons participate in the scattering, the longitudinal phonons being somewhat more important. (See also footnote 92 for independent confirmation of hole-TO phonon coupling.) (2) The angular dependence of the scattering matrix elements and the degree of interband scattering differfor longitudinal and transverse phonons and for light and heavy holes. In general, light holes tend to be scattered preferentially in the forward directions, and heavy holes in the backward directions. (3) Despite the anisotropy in the individual scattering matrix elements, the final calculated relaxation times are nearly isotropic, and one is justified in assuming z to be a function only of energy. (4) The assumption of spherical energy bands is a poor one for pure acoustic mode scattering in Ge,21 giving rise to a 20% overestimate in the calculated effective mobility. Since many of the 111-V compounds have valence bands which are more severely warped than those of Ge, this is an important conclusion to keep in mind. ( 5 ) The ratio of light-to-heavy hole mobilities is approximately p2/p1 w m , / m 2 , lending substantial support to the z, = z2 approximation. Tiersten2’ states that f l = z2 is not a good approximation, but he bases this on a 30%differencebetween z l and r,-clearly a small difference compared to what would be calculated for decoupled bands. (6) The temperature dependences of the mobilities calculated in the exact models are the same as those given in Eqs. (32)-(34). From a practical or computational point of view, the last conclusion is the most important. It says that if one could define appropriate “effective”

2.

MOBILITY OF HOLES IN 111-V COMPOUNDS

129

deformation potentials, all of the complications of the degenerate valence bands could be absorbed into these parameters, and the simple expressions given in Eqs. (32)-(34) could be used.118 LawaetzZ2has shown that this is indeed possible. He defines an effective acoustic mode deformation potential given by22*"9 (35) qrr = a2 (C,/C,)(b2 H2),

+

+

where a, b, and d are the fundamental valence band deformation potentials introduced by Pikus et u1.18-20 and C, and C, are spherically averaged elastic coefficients given by

c, = i(3C11 -k 2c12+ 4c4,),

c, = +(ell - c,, -t 3c4,).

(36)

The parameter Serfcan be related to the phenomenological acoustic deformation potential E A C used by Wiley and Di Domenico2*by the following expression' 1 9 ~ 1 2 0:

where fl = C,/C,. Similarly, the phenomenological optical deformation potential EN, of Eq. (34) can be related to the optical deformation potential doof Pikus ef u1.'8-20 In this case one

where M , and M, are the masses of the atoms in the unit cell, o,is the angular frequency of zone-center LO phonons, and a, is the lattice constant. The deformation potentials appearing in Eqs. (35)-(38) all have dimensions of energy. Detailed discussions of the effects of stress on the degenerate valence ~ ~ will - " ~not ~ 'be~ repeated ~ bands are available in the l i t e r a t ~ r " ~ - ~ ~ *and here. Conwell' " and Costato and Reggiani'" have also given useful discussions of the relationships among various deformation potentials which appear in the literature. Severe nonparabolicity of the heavy-hole band would tend to steepen the temperature dependence of the effective mobility and invalidate conclusion (6). This effect has been discussed by Asche and von Borzeszkowski,61 who introduced a temperature-dependent effective mass as an approximate correction factor in psi. The only materials for which severe nonparabolicity is likely to present a problem are those with extremely small spinorbit splittings (Si, AIP, and Gap). In these materials, there is a strong energy-dependent warping and nonparabolicity which causes the average effective mass to increase sharply for a particular range of carrier energies (see also footnote 65). J. D. Wiley, Solid Stare Commun. 8, 1865 (1970). P. Lawaetz, private communications. M. Costato and L. Reggiani, Lett. Nuoco Cimenro 4, 848 (1970).

130

J. D. WILEY

Using the phenomenological deformation potentials, EAc and ENpo,it is possible to show that the combined AC and NPO mobility is given by

(39)

where p is in cm2/V-sec,E,, is in eV, p is in g/cm3, and ii, an average sound velocity defined in Eq. (43, is in cm/sec. The function S is given byz8

where C = (e/T)q/2(ee/’- 1)

(41)

and (42) The expression given in Eq. (39) differs from one given earlier by Wiley and Di Domenico2’ by a factor (1 + r-3/’)-1 which was shown by Costato et u1.30*122 to take explicit account of interband scattering. The r factor in 1=

’

r=ml/m2

FIG.19. The r factor which occurs in Eq. (39) (solid curve) and a similar factor given earlierz8 for a model which ignored interband scattering (dashed curve). See also Fig. 16 and related discussion.

122

M. Costato, G. Gagliani, C. Jcoboni, and L. Reggiani, J . Phys. Chem. Solids 35, 1605 (1974).

2.

131

MOBILITY OF HOLES IN III-V COMPOUNDS

Eq. (39)and the earlier expression of Wiley and Di Domenico2*are shown in Fig. 19. (See also Fig. 16 for the error involved in neglecting interband transitions in the T~ = T~ approximation.) Figure 20 shows the temperature

r

0

02

0.4

06

0.8

10

1.2

1.4

1

1.6

T/B

FIG.20. Temperature dependence of the function S(0, q , T)given by Eq. (40).

dependence of the function S. For purposes of hand calculation, S can be conveniently approximated (to approximately 1 % accuracy for T/8 5 1.5) by s x (1 A q ) - l , (43) where

+

A = 1.34(8/T)/(~?/~ - 0.914).

(44) Further discussion of the approximation of S can be found in the appendix of Wiley and Di Domenico." The overall temperature dependence of Eq. (39) is shown in Fig. 21 for various values of q . At low (T/8 < 0.1) and high (T/B > 1) temperatures,

132

J. D. WILEY

TI8 FIG.21. Temperature dependence of the combined AC and NPO mobility for various values of q . PAC.XPO approaches a T -jIz temperature dependence. In the transition region, the temperature dependence can be approximated by T d 8 with = 1.8, 2.0,2.2,2.3, and 2.4 for = I , 2, 3,4, and 5, respectively. Tables VI and VII give values for the deformation potentials and other physical parameters needed in Eq. (39). In Table VI, the parameters b and d were obtained from the literature, while a and do were calculated using a method first suggested by Lawaetz.'20*'23All of the quantities in Table VII are self-explanatory with the exception of u, which is an averaged sound velocity given by

P. Lawaetz, unpublished work.

2.

133

MOBILITY OF HOLES IN 111-V COMPOUNDS

TABLE VI POTENTIALSFOR Ge, si, VALUES FOR THE VALENCE BANDDEFORMATION AND THE 111-V COMPOUNDS~ Material Si Ge AIP AlAs AlSb GaP GaAs GaSb InP InAs InSb

a 2.1 2.0 2.9 2.6 2.7 3.0 2.7 2.2 2.9 2.5 2.0

-b

-d

2.2' 2.2' 1.6' 1.6' 1.35d

5.3' 4.5'

4.4'

4.4' 4.3d 1.4' 4.4' 1.7/ 4.4/ 2.OP 4.6' 1.55h 4.4' 1.8' 3.6' 2.0' 4.9'

7.5 6.5 6.7 6.6 6.4 6.6 6.6 6.9 7.0 6.1 7.5

E,,

do

4.0 3.5 3.5 3.5 3.4 3.5 3.5 3.6 3.6 3.2 3.9

40 40 43 42 37

44

41 39 42 42 39

ENpo

6.4 6.2 5.8 6.3 5.6 6.7 6.5 5.9 6.3 5.7 5.6

q

2.6 3.2 2.6 3.2 2.9 3.6 3.6 2.7 2.9 3.2 2.0

The values given for b and dare experimental. All other values were calculated as described in the text. * Costato and Reggiani!21 Assumed similar to GaAs and Gap. L. D. Laude, M. Cardona, and F. H. Pollak, Phys. Rev. E 1, 1436 (1 970). I. Balslev, J. Phys. SOC.Japan Suppl. 21, 101 (1966). Corrected for errors in elastic constants by P. Lawaetz, private communication. J I. Balslev, Solid Stare Commun. 5, 315 (1967). * C. Benoit a la Guillaume and P. Lavallard, J. Phys. Chem. Solids 31, 411 (1970). *A. Gavini and M. Cardona, Phys. Rev. B 1,672 (1970). ' P. Y. Yu,M. Cardona,and F. H. Pollak, Phys. Rev. 83,340(1971).

where ul and u, are the velocities of longitudinal and transverse (shear) sound waves, respectively. From Table VI it is seen that the deformation potentials for the group IV and 111-V compounds are quite similar, confirming the intuitive expectation that the nonpolar scattering mechanisms should be comparable in these materials. Two points should be emphasized with regard to the deformation potentials in Table VI : (1) They are merely estimures, and did not result from fitting experimentaldata. (2) The quantities =AC, EAC,ENm, and q are phenomenological parameters which allow us to ignore the full complexity of these scattering mechanisms and use simpler, otherwise inapplicable expressions for the hole mobility. In this regard, it is worthwhile to point out specifically that the phenomenological deformation potentials already contain the effects of the overlap function G(k,k). Thus it is unnecessary to treat these effects separately as has been done in some recent work.'22 [Treating G(k, k') separately does not lead to any "error"

-

134

I , D. WILEY

TABLE VII PARAMETERS NEEDED IN EQS. (35)-(44)” NUMERICAL VALUESFOR PHYSICAL a0

Material Si Ge AIP AlAs AlSb GaP GaAs GaSb InP lnAs InSb

(A) 5.43 5.66 5.463 5.661 6.136 5.450 5.642 6.094 5.869 6.058 6.479

P O ii (g/cm3) (“K) (lo5cmlsec)

2.33 5.32 2.40 3.598 4.26 4.130 5.307 5.614 4.787 5.667 5.775

730 430 725 550 493 582 42 1 347 498 350 284

6.82 3.63 (5.95) (4.55) 3.72 4.76 3.90 3.22 3.81 3.09 2.83

C,,

C,,

C,,

C,

C,

18.85 6.804 7.95 6.39 16.56 15.03 5.636 4.83 12.88 6.71 (13.2) (6.30) (6.15) (15.36) (5.07) (12.5) (5.34) (5.42) ( I 3.40) (4.49) 3.392 8.939 4.425 4.155 10.46 6.253 7.047 16.61 14.12 5.804 14.03 4.864 11.88 5.94 5.38 3.554 8.839 4.033 4.316 10.38 4.60 12.12 3.652 5.76 10.22 9.975 3.136 8.329 4.526 3.959 7.875 3.021 6.669 3.645 3.020

Unless otherwise noted, all values for the 111-V compounds were taken directly or inferred from other parameters given in the tabulation by Neuberger.6 Values for Si and Ge were similarly obtained from P. Aigrain and M. Balkanaski, “Selected Constants Relative to Semiconductors.” Pergamon, Oxford, 1961. Numbers in parentheses were estimated using empirical relations given in R. W. Keyes, J . Appl. Phys. 33, 3371 (1962). All C values are x 10’’ dyn/cmz.

in the final results; it is simply redundant and will ultimately lead to a somewhat larger fitted value for the phenomenological deformation potentials.]

b. Polar Optical Phonon Scattering The expression normally used for the polar mobility of electrons in nondegenerate, s-like bands is given by’

pio

= 0.199(~/3~)”Z(e/e,*)z(mo/m*)3~2

x (1OZ2M)(lOZ3u,)(10-’30,) (el - 1)G(z)

(46)

where p is in cm2/V-sec,e,* is the Callen effective charge, M is the reduced mass of the unit cell in g, u, is the volume of the unit cell in cm3 (u, = aO3/4), o,,is the angular frequency of zone-center LO phonons [a,, = 1.309 x 10”8(“K)]. z = ho,,/k,T, and the function G(z) is given graphically by Ehrenreich, with and without screening effects.’ Hammar and Magnushave recently recalculated the unscreened G(z) using an iterative solution of the Boltzmann equation rather than the less accurate variational calculation of Ehrenreich’ and others.”’ The results are given in Table VIII. C. Hammar and B. Magnusson, Phys. Scripfa 6,206 (1972).

2. MOBILITY OF HOLES IN 111-V COMPOUNDS

135

TABLE VllI THE FUNCTION G(z) WHICH APPEARS IN EQ.(46)”

I

G(z)

z

G(z)

0.0 0.2 0.4 0.6 0.7 0.8 1.o 1.2 1.4 I .6 1.8 2.0 2.2

1 .o 0.8957 0.8102 0.7524 0.7340 0.7219 0.7146 0.7263 0.7528 0.7909 0.8378 0.8911 0.9490

2.5 3 .O 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 11.0

1.041 1.194 1.353 1.495 1.621 1.733 1.919 2.065 2.188 2.296 2.394 2.487

“ A s calculated by Hammar and Magnusson.’24

For cases where screening of the electron-phonon interaction is important, one must still use the results of Ehrenreich.’ From Eq. (46) and Table VIII, it can be seen that the temperature dependence of & is not simple except at high temperatures, where it approaches T - ’ / 2(see also Fig. 18). As was the case for AC and NPO scattering, the temperature dependence of pio will be shown to be approximately preserved in the degenerate p-like valence bands. When &, is written in the form given by Eq. (46), the electron-phonon coupling parameter is contained in the effectivecharge, which is given by

(e,*)2= (M~02~J47M1/&,)- (1/&0)1, (47) where E~ and E, are the low-frequency and high-frequency relative dielectric constants, respectively, and all other quantities have the same meaning as in Eq. (46).Table IX gives values for c0, E ~ and , e,* for the 111-V compounds. The error bars quoted for e,* are based on an assumed 2% uncertainty in E~ and cC4and lie in the range 10-25%. Thus it is seen that small errors in the dielectric constants can lead to very substantial errors in the calculated mobility. In the derivation of Eq. (46), Ehrenreich’ discussed the effects of the overlap function on the final mobility,and showed that the mobility is a minimum for carriers with pure s-like wave functions. The first quantitative discussion of overlap effectson the polar mobility of holes was given by wile^,^^ who

136

J. D. WILEY

TABLE IX VALUES FOR THE DIELECTRIC CONSTANTS AND CALLENEFFECTIVECHARGESOF THE 111-V

COMPOUNDS' Material AIP AlAs AlSb GaP GaAs GaSb InP lnAs lnSb

e0

E l

9.9' 10.9 12.04 11.1 12.9 15.69 12.35 14.55 17.88

7.6' 8.5 10.24 9.09 10.9 14.44 9.52 11.8 15.68

e, *

0.302 f 0.036 0.278 f 0.028 0.212 f 0.021 0.241 .f0.034 0.201 k 0.028 0.128 f 0.033 0.270 k 0.027 0.221 k 0.027 0.159 k 0.027

"The error bars quoted for e,* are based on an assumed 2% uncertainty in the dielectric constants. Unless otherwise noted, all dielectric constants were obtained from the compilation given by Neuberger.6 Estimated using the LST relation together with E , . ' B. Monemar, Solid State Commun. 8, 1295 (1970).

'

showed that, for carriers with p-like wave functions, the polar mobility is approximately twice that given by Eq. (46). In reaching this conclusion, use was made of the relaxation-time approximation, which is invalid except at high t e m p e r a t ~ r e . ~Nevertheless, -~ by direct numerical solution. of the Boltzmann equation, Kranzer3' has shown that 0.500 < &/&, < 0.525, where the upper limit is achieved for z = 6. Thus, the conclusion based on the relaxation-time a p p r o ~ i m a t i o nis~ ~(somewhat surprisingly) accurate to within 5 %. Although it was not specifically stated in their paper, Rode and Wiley reached a similar conclusion in their study of electron transport in zinc-blende semi metal^,'^^ where the electrons move in nondegenerate, p-like conduction bands. From Table V it can be seen that there have been two calculations of the polar mobility of holes in 111-V compounds. K r a n ~ e r ~has l . ~used ~ an iterative technique, and Costato et a1.32,35 have used a Monte Carlo technique for solving the coupled Boltzmann equations under the assumptions given in Table V. The major conclusions of these calculations are as follows: lz5

D. L. Rode and J. D. Wiley, Phys. Status Solidi 56,699 (1973).

2.

137

MOBILITY OF HOLES IN 111-V COMPOUNDS

(1) As was the case in nonpolar phonon scattering, the most important scattering rates are intraband scattering of heavy holes and interband scattering of light holes. Costato et ~ 1 . have ~ ~ given ~ ~ particularly ’ interesting and detailed discussions of the behavior of the various scattering probabilities. (2) Due to strong interband scattering, the light-hole mobility is dramatically reduced from the value which would be calculated if the bands were decoupled. Because of this, the contribution of the light holes to the overall effective mobility is approximately 10-30% depending on the temperature and material. (3) The temperature dependence of & is approximately (though not exactly) the same as that given by Eq. (46). As a result of conclusions 2 and 3, it is possible to write an approximate expression for the effective polar mobility of holes. Using the notation introduced earlier,” the effective mobility is given approximately by P;: = ~KP:,,, (48) where the factor of 2 arises from the p-like symmetry of the wave functions, K is a correction factor of order unity which accounts for the contribution from the light holes, and pi is calculated from Eq. (46) using the heavyhole effective mass. The correction factor K has been given graphically by K r a n ~ e and r ~ ~is shown in Fig. 22 as a function of r for various values of z 1.3

: 1

1

I

I

1

I

1

1 1 ,

I

I

[

I

I

I

I l l

2.5

3

1.2

-

s ; r

Y

1.1

I .o

-

-

I

I

I

I

I

I

,

I

,

(

I

I

I

I

I

1

1

1

1

138

J . D. WILEY

(where, as before, r = m1/m2and z = hw,/k,T = O/T).From Fig. 22 it is clear that the degree of interband coupling is temperature dependent. Using values of r and 8 from Tables I1 and VII, the information presented in Fig. 22 has been replotted in Fig. 23 as a function of temperature for 1.3

-

I

AQP

I

1

I

I

I

I

I

I

1

1

I

I

I

1

APSb

GaSb

-

1.1

I .o

0

I

I I00

1

200

300

400

500

T (OK)

FIG. 23. The factor K used in Eq. (48) shown as a function of temperature for specific materials.

specific materials. By using Eq. (48) together with Eq. (46), Table VIII, and Fig. 23, one can calculate reasonably accurate values for the polar mobility of holes in 111-V compounds. The error incurred by using this procedure is almost certainly less than that caused by uncertanties in the numerical values of rn, and e*. Thus, for example, if one calculates the effective mobility in p-GaAs at 300°K using realistic error bars for m, and e* (m,= 0.5 & 0.1 and e* = 0.2 0.03), one obtains pg(300) = 950 & 350 cm2/V-sec. In order to facilitate similar estimates for other materials, Table X has been constructed using Eq.(48), but normalizing out the effects of m, and e*. Finally, it must be emphasized again that there has been, to date, no calculation or estimate of the effects or warping and nonparabolicity on the polar mobility of holes. These effects could alter the magnitude and temperature dependence of the mobility, and have already been shown to have a strong effect on the relation between Hall and drift mobilities. In this connection, it is worth pointing out that Kranzer3j has calculated the Hall scattering factor rHfor polar mode scattering of holes, including the effects of

2. MOBILITY

139

OF HOLES IN III-V COMPOUNDS

TABLE X VALUES FOR pg AT 300°K CALCULATED USING (48), BUT WITH THE EFFECrs OF m, A N D e* NORMALIZED AS SHOWN

m.

p 3 r n ,/m0)3‘2

AIP AlAs AlSb GaP

34.65 17.25 17.04 20.06 13.48 12.38 16.83 12.43 11.70

GaAs

GaSb InP InAs InSb

601.6 223.2 379.1 345.4 333.7 755.6 230.9 254.5 462.8

interband scattering. In agreement with the earlier discussion of the general behavior of rH (Part 111, Section Ic), Kranzer finds that rx > 1 in all cases, and that it depends strongly on m&m2 and on temperature. While these results are interesting and important, they are nevertheless incomplete without information concerning the compensating effect of the anisotropy factor.

3. IONIZED-IMPURITY SCATTERING When electrons in nondegenerate s-like bands are scattered by ionized impurities, the mobility is given approximately by the so-called BrooksHerring formula’

’

Pi, =

1 2 8 ( 2 1 t ) ” ~ ~ , ~ k ~ /ksZT3/’ ’ e3m;I2 Ni(m*/m0)1’2

+ b) - l + b 1

‘

3

(49)

where b = 24mO k R ‘ E 0- m*T2k, e2h2 m,n’ ’

n‘ = n

+ (N,

Ni = n

+ 2N

and

- N, - n) (n + N,)/N,,

(51)

(52) The static dielectric constant has been denoted k, to avoid confusion with the free-spacepermittivity e0 . If the numerical prefactors in Eqs. (49) and (50) A’

140

J. D. WILEY

are evaluated, one obtains piI = 3.284 x 1015

ksZT3’2

N i(m */m,)I/’

[In(l

+ b) - -]-I l+b

(53)

and m*T’kb b = 1.294 x 1014-, mOn’

(54)

where p is in cm’/V-sec, k, is dimensionless, T is in OK,and N,and n‘ are in ~ m - ~ . In considering the effect of ionized-impurity scattering on hole mobility, Brooks7’ argued as follows. Ionized-impurity scattering is expected to be rather ineffective in causing interband transitions since most scattering events are elastic and involve only small scattering angles. Thus, to a first approximation, the light- and heavy-hole bands can be considered independently (the decoupled band approximation). Furthermore, it can be seen from Eq. (49)that plIhas a very weak dependence on m*,so that the mobility of the light holes is only slightly greater than that of the heavy holes (a factor of 2-5; not nearly enough to compensate for the much smaller concentrations of light holes). The consequence of this is that the light holes make only a small contribution to the effective mobility. Exact69calculations of the interband and intraband scattering rates have shown that, except at very low temperatures or high impurity concentrations, the two bands are indeed d e c ~ u p l e d . Therefore, ~ ~ . ~ ~ the effective hole mobility is given approximately by

where W is a factor of order unity (1 < W Q 2) which accounts for the p-like symmetry of the hole wave functions, and the r factor accounts for the presence of light holes.lZ6 Equation ( 5 5 ) should be regarded as a heuristic expression which is useful for estimating the magnitude of the hole mobility, but which cannot possibly give quantitative agreement with experimental data. This is because the magnitude of Wand the degree of coupling of the bands depend on the temperature and doping c o n ~ e n t r a t i o n . Never~~*~~ theless, the errors incurred in using Eq: (55) with W % 1.5 are not significantly worse than those incurred by neglecting hole-hole scattering”’ or multiple scattering and quantum correction^.^^.^^ To date, there has been no critical evaluation of ionized-impurity scattering in p-type materials. The r factor in Eq. (55) approaches unity as r -* 1 and r ‘ 0 0 , and has a peak value of 1 . 1 184 at r % 2.82. Thus the light holes can be expected to increase the effective mobility by only about 10% or less.

2. MOBILITY

OF HOLES IN 111-V COMPOUNDS

141

IV. Experimental Hole Mobilities

In this part, the best available hole mobility data will be reviewed for the 111-V compounds. As has been mentioned at several points in this chapter, most of the mobility data have been deduced from Hall effect measurements, and there is presently no adequate theoretical treatment of the relationship between Hall and drift mobilities inp-type 111-V compounds. The theoretical analyses which accompanied these data in the original literature are thus necessarily based on inadequate models, and any reinterpretation must be left as a topic for future research. Nevertheless, it is tempting to compare the observed Hall mobilities with calculated (or estimated) drift mobilities. When such comparisons are made, it must be borne in mind that we are z 1. implicitly assuming, without theoretical justification, that CleHf7~;~~ 4. AIP, AlAs, AND AlSb

There have been no reported measurements of hole mobilities in AlP or AIAs. Both of these materials are difficult to prepare in large single-crystal form,’ and decompose upon exposure to room air.127*1’8In the absence of any experimental data, the expressions and tabulated parameters of Part I11 can be used to obtain rough estimates of the effective hole mobilities in AlP and AlAs. The results are as follows. For AlP at 300”K, pg r 1200 cm2/Ve 720 cm2/V-sec. Using Matthiessen’s rule gives an sec. and pLeAf,NpO approximate lattice mobility of pi::, w 450 cm2/V-sec. The corresponding results for AlAs are &A x 500 cm2/V-sec, p:i,NpO = 350 cm2/V-sec, and p ; z t r 200 cm2/V-sec. In view of the fact that the effective masses and deformation potential used in these estimates are purely theoretical, the estimated mobilities have an uncertainty of at least f50%. AlSb is also rather difficult to prepare in high-purity, single-crystal form, because it tends to react with most crucible materials, producing rather heavily doped (normally p-type) material.2”29.’30When exposed to room air, AlSb tarnishes or slowly decomposes depending on the purity of the crystal, its surface preparation, and the water vapor content of the air.’30 Despite these difficulties in preparation and handling, there have been several reported measurements of hole mobilities in A1Sb.129-’33Figure 24 lZ8

13’

13’

H. G. Grimmeiss, W. Kischio, and A. Rabenau, J . Phys. Chem. Solids 16,302 (1960). M. Ettenberg, A. G . Sigai, A. Dreeben, and S. L. Gilbert, J . Electrochem. SOC.118, 1355 (1971). F.J. Reid and R. K. Willardson, J . Electron. Contr. 5, 54 (1958). W. P.Allred, B. Paris, and M. Censer, J . Electrochem. SOC.105,93(1958);107,117 (1960). A . Herczog, R. R. Haberecht, and A . E. Middleton, J. Electrochem. SOC.105. 533 (1958). D. N. Nasledov and S. V. Slobodchikov, Zh. Tekh. Fiz. 28, 715 (1958)[English Tronsl.: Sou. Phys.- Tech. Phys. 3,669 (1 958)l. R. J . Stirn and W. M. Becker, Phys. Rec. 148,907 (1966).

142

J. D. WILEY

A

p ( 3 0 0 ° K ) = l o f 6 cm-3

0 p(300'K) =

I

I

I

I

lo1? cm-3

1

TEMPERATURE (OK)

FIG.24. Temperature dependence of the Hall mobility in p-AISb. The straight-line portion of the upper curve has a slope of -2.2 and passes through p = 450 cmz/V-sec at 300°K. (Data of Reid and Willardson.129)

2. MOBILITY OF HOLES IN

143

111-V COMPOUNDS

shows the temperature dependence of the Hall mobility as obtained by Reid and W i l l a r d ~ o n . 'From ~ ~ the upper curve in Fig. 24, one obtains a latticelimited mobility of -450 cm2/v-sec at 300°K and a temperature dependence of approximately T - 2 . 2These . numbers are in excellent agreement with the measurements of Allred et who point out specifically that the magnitude and temperature dependences of the lattice mobility in p-A1Sb are quite similar to those in p s i . This point has also been emphasized by Wiley and Di Domenico?* and discussed in Part 111 of this chapter. Stirn and B e ~ k e rhave ' ~ ~obtained similar results for a somewhat less pure sample (N, - ND x 3 x 1OI6 ~ m - ~(300°K) ~ ; = 330 cm2/V-sec; p oc T - ' . 9 5 ) and have analyzed their result using the model of Lax and Mavroides." In analyzing their results, Stirn and B e ~ k e r 'use ~ ~ the valence band parameters of Cardona4' (A = 5.96, B = 3.36, and C2 = 23.2, all in units of h2/2mo) and find them to be in good agreement with experiment. These parameters imply a heavy-hole effective mass of ml*/mo x 0.5. If this value is used to estimate the polar mobility at 300"K, one obtains pg % 1100 cm2/V-sec. If the remainder of the scattering is attributed to acoustic and nonpolar optical modes one obtains (using Matthiessen's rule) p2&pg x 750 cm2/V-sec. This, in turn, implies an acoustic mode deformation potential of EAc x 3 eV, in good agreement with the estimate given in Table VI.

G

t

600

>

\

-E

N

0

400

> k

2

m 0

200

W J

0 I

0 1015

1016

10"

1018

HOLE CONCENTRATION, p (cm-3)

FIG. 25. Concentration dependence of the Hall mobility in p-A1Sb at 300°K. Data: 0 , Allred et ~ l . ' ~ ' 0, ; Allred et o I . ' ~ ' ;A,Reid and Willardson'29;A, Stirn and Becker.'33

144

I . D. WILEY

Thus one obtains a very reasonable and self-consistent picture if the value ml/m, = 0.5 is used.134 The dependence of mobility on hole concentration is shown in Fig. 25. The solid line was calculated using the Brooks-Herring formula together with a lattice mobility of 450 cm2/V-sec.As explained in Part 111, however, such a fit has no real theoretical significance and must be regarded as an empirical fit to the data. 5. GaP

A number of investigators have reported measurements of hole mobilities in GaP.91,135-144a One of the earliest reports was that of Alfrey and Wiggins, who reported p = 66 cm2/V-secfor material withp x 10l8 cm-3 at room temperature. The temperature dependence of the mobility was found to be roughly T - 3 / 2from 100 to 400°K.Cherry and Allen136obtained a rather similar temperature dependence above 200°K but reported mobility values approximately twice as high as those of Alfrey and Wiggins @ x 4 x 10” cm-3 and p x 150 cm2/V-secat 300°K). A more extensive series of measurements was reported by Cohen and Bedard,I3’ who studied GaP single crystals which had been grown epitaxially on GaAs substrates. The room-temperature hole concentration and mobilities were in the ranges 7 x 1015-7 x 1Ol8 ~ r n and - ~ 10040 cm2/V-sec,respectively. All of their crystals were found to be highly compensated, possibly due to contamination from the GaAs substrate. Taylor et a[.’38 found that GaP samples which had been vapor-deposited on GaP substrates were more uniform and showed higher mobilities than otherwise similar samples which were deposited on GaAs substrates. The most extensive series of measurements reported to date is that of ’ obtained Hall and resistivity data for Zn-doped GaP over Casey et u I . , ~who In view of this agreement, one must certainly question the value given for m,/rno for AlSb in Table 11. The masses given in Table I1 were obtained from a k . p calculation53 using the latest and best values for the energy gaps. As will be seen throughout this section, most of the masses in Table I1 are in good agreement with experiment. 135 G. F. Alfrey and C. S. Wiggins, Z. Naturforsch. 15a, 267 (1960). 136 R. J. Cherry and J. W. Allen; J. Phys. Chem. Solids 23, 163 (1962). 1 3 ’ M. M. Cohen and F. D. Bedard, J. Ap@ Phys. 39,75 (1968). L 3 8 R. C. Taylor, J. F. Woods, and M. R. Lorenz, J. Appl. Phys. 39, 5404 (1968). 13’ L. M. Foster, J. F. Woods, and J. E. Lewis, Appl. Phys. Lett. 14, 25 (1969). 140 R. C. Taylor, J. Electrochem. SOC.116,383 (1969). 1 4 1 V. V. Ostroborodova and P. Dias, Fiz. Tekh. Poluprou. 3, 1573 (1969) [English Transl.: Sou. Phys.-Semicond. 3, 1319 (1970)l. 142 Yu. L. Win, V. S. Sorokin, and D. A. Yas’kov, Neorg. Mater. 6, 1232 (1970). 143 D. Diguet, Solid Stale Eleczron. 13, 37 (1970). 144 S. F. Nygren, C. M. Ringel, and H. W. Verleur, J. Electrochem. Soc. 118,306 (1971). 144aD. P. Bortfeld, B. J. Curtis, and H. Meier, J. Appl. Phys. 43, 1293 (1971). 134

2.

145

MOBILITY OF HOLES IN 111-V COMPOUNDS

the temperature range 4.2-775°K. The Zn concentrations (as determined by neutron activation analysis) ranged from 6.7 x loi6 to 2.1 x 1019cm-3. The temperature dependence of the mobility is shown in Fig. 26. The mobility of the purest crystal is quite accurately described by p = 135[300/T(OK)]2.2 for temperatures above 150°K. As the temperature is lo4

I

I

I

I

I

I l l

I

1

I

I

I

I l l

6

8

8

-

6-

0

. 0

lo3 8 -

-

c

v

6 -

Q

In

N =,,

1.9 x l o t 7

N =,,

6.7 x 10"

NZ,=2.9

>

-

\

N

E

I

I

I I I E J

X

10"

NZn= 3.8 x 10"

0

N =,,

1.2 x 1019

A

N =,,,

2.1 x 1019

4 -

I

I

NZn=6.7x10i6

4-

2-

I

2-

0

v

:

I02

i

t

8 --

='

6-

0

4 -

m 2

W -I

0

I

-

2-

10

-

0 -

c-

*

4 -

-

2 -

1.0

I .o

2

4

6

8

2

4

10 Io2 TEMPERATURE, T ( O K )

2

4

6

8

lo3

FIG.26. The temperature dependence of the Hall mobility in Zndojxd Gap. (After Casey er a1.97

146

J. D. WILEY

lowered, most of the samples exhibit a maximum mobility followed by a sharp drop in the neighborhood of 20-40°K. This is characteristic of impurity c o n d u ~ t i o n ~ ' in * ' ~which ~ conduction takes place by the phononassisted hopping of holes from neutral to ionized acceptors. As the Zn concentration is raised above 1.2 x 10'' cm-j, one encounters a sharp transition from thermally activated conduction to metallic impurity conduction, in which the band of Zn impurity levels has become so broadened by mutual interaction that it has merged with the valence band. This effect is most strikingly observed in the resistivity data, shown in Fig. 27. From Fig. 27, it is seen that a change in Zn concentration from 1.2 x 1019 to 2.1 x 1019 cm-j causes the resistivity at 4.2"K to fall by nearly eight orders of magnitude. This effect has been discussed in detail by Mott and Two~e.'~' One of the most interesting features of the data presented by Casey et d9' is the following. By careful comparison of the calculated hole concentrations (calculated using p = r/eR,) and the measured Zn concentrations, Casey et aLgl found that no single value of r was consistent with their data over the entire range of temperature and doping. Thus, for low Zn concentrations, they obtained r x 1, whereas for 5 x lo" 5 Zn 5 5 x lo'* cm-j, they obtained r = 0.6. At extremely high Zn concentrations (21019cm-j), they again obtained r x 1. A similar effect has also been reported by Foster et al.,'j9 who found that a value of r < 0.7 was necessary to explain their data over a considerable range of Zn concentrations. These data remain unexplained, although both authorsg1 j 9 mention valence band anisotropy as a possible explanation for r 1. (See also the discussion of r in Part I11 of this chapter.) One further anomaly in the mobility of holes in Zn-doped GaP has been noted by Nygren et al. 144 They find that Czochralski-grown ingots simultaneously doped with Zn and Ga,O, show higher mobilities than ingots doped with Zn alone (as much as 25% higher for hole concentrations in the 1017-1018cm-j range). This effect may be due to ion pairing of the Zn and 0 impurities. 146:!47 Nearest-neighbor donor-acceptor pairs give rise to dipole scattering1Q'which is considerably weaker than ionizedimpurity scattering. Similar effects have been observed in Ge which was simultaneously doped with Li and Ga.146 The dependence of hole mobility on hole concentration is shown in Fig.. 28. There are very few data available for hole concentrations below 1016-m73.Taylor et have reported a few samples with hole concentrations ip,tlp3!lfJ'2~1013 cm-3 range and mobilities ranging from 65 to 140 cm2/V-secat 300°K. In view of the extremely low hole concentrations,

-=

145 146

14'

v1

N.F.Mott and W. D.Twose, Advan. Phys. 10, 107 (1961). H. Reiss, C. S. Fuller, and F. J. Morin, Bell Sysr. Tech. J. 35, 535 (1956).

J . D. Wiley, J . Phys. Chem. Solids 32,2053 (1971).

2. MOBILITY OF

HOLES IN 111-V COMPOUNDS

NZn = 6 . 7 x

loi6

X

Nzn'1.9

IOl7

o

NZn -2.9 x l0le

o

N ~ , =1.2 x loi9

A

Nz"'2.I

X

x

147

lo"

FIG.27. The temperature dependence of the resistivity in Zn-doped Gap. Note, particularly, the dramatic reduction in p at the onset of metallic impurity conduction in the neighborhoodof 1-2 x l O I 9 (After Casey et ~ 1 . ~ ' )

one would assume that these samples were quite heavily compensated, although the value of 140 cm2/V-sec appears to be approximately the latticelimited mobility at 300°K.

148

J. D. WILEY I

I

I

0

l

l

I I I I I

)

I

I

l

l

1

I

I'II

0

HOLE CONCENTRATION,

P (cm-3)

FIG.28. The dependence of hole mobility on hole concentration for GaP at 3000°K (.,91 0,137 ~

~

1 04, 1 4 2 4 0140).

In attempting to compare the observed lattice-limited mobility with theoretical estimates, one is faced with the difficulty of specifying an appropriate value for the heavy-hole effective mass. This is particularly difficult in GaP since, if the k p calculations are to be believed, the u1 band is severely warped and nonparabolic (see Part 11, Figs. 6 and 8). Using the pseudopotentials of Cohen and Berg~tresser,~~ Fau1kne1-l~~ has obtained valence band effectivemasses ofm,/m, = 0.88, m,/m, = 0.16, and mJm, = 0.26 (averaged over all directions in k space). Lawaetzs3 has obtained rather similar values (0.79, 0.14, and 0.24, respectively) from a k * p calculation. Both of these sets of numbers are in excellent agreement with the experimental results of L ~ r i m o r , ' ~who ' obtained m' = (m1m2)1/2= (0.35 & 0.02)m0from analysis of the infrared reflectivity of heavily doped (>3 x 10" ~ m - p-Gap. ~ ) Unfortunately, the infrared reflectivity is rather insensitive to m* at lower doping levels. Recent cyclotron resonance measurements by S~hwerdtfeger"~yielded ml/m, = 0.67 f 0.04 and m2/mo = 0.17 _+ 0.01. Similar measurements by Bradley et u L ' ~ ' gave ml/mo =

-

149

R. A. Faulkner, private communication. (See also Wiley and Di D ~ m e n i c o . ~ ~ ) 0. G. Lorimor, J . Appl. Phys. 41,5035 (1970). C. F. Schwerdtfeger, Solid State Commun. 11, 779 (1972). C.C. Bradley, P. E. Simmonds, J. R. Stockton. and R. A. Stradling, Solid Srafe Commun. 12,413 (1973).

2. MOBILITY

OF HOLES IN 111-V COMPOUNDS

149

0.54 & 0.05 and mJm0 = 0.16 +_ 0.02. The cyclotron resonance experiments were both performed at low temperatures (1.6 and 77"K, respectively) with B along a (111)direction. Thus, the reported masses represent averages over a cyclotron orbit in the plane perpendicular to (1 1l), rather than averages over all directions. Nevertheless, it is difficult to reconcile these rather low values for m, with the values obtained by Faulkner and Lawaetz. Pollak et aL3' have given a set of valence band parameters which yield better agreement with the cyclotron resonance results.' 51*152 Assuming the effective lattice mobility of holes at 300°K is p x 135-140 cm2/V-sec,and using rn, as an adjustable parameter, one obtains the following results [using the fitting procedure described in the section on AlSb and reporting results in the order (m,/rno,EAc,,:p p2&p0)]: (0.6,7.8,750,165), (0.7, 6.3, 595, 175), (0.8, 5.2, 485, 185), (0.9, 4.3,410,200), (1.0,3.6, 350,220). Three points should be made concerning these results: (1) All of these sets of values give good fits to the magnitude and temperature dependence of the lattice mobility for the purest sample of Casey et aL9' This emphasizes the danger in relying on such fits as the sole basis for drawing conclusions about the relative importance of scattering mechanisms, or for obtaining numerical values for poorly known parameters. (2) Keeping conclusion (1) in mind, it is noted that for mJmo = 0.8 f.0.2, the polar mobility is considerably larger (less important) than the combined acoustic and NPO mobility. (3) It is only the larger values of m , which allow the value of E , to be as low as the theoretical values given in Table VI. 6. GaAs As was found to be the case for Gap, there have been numerous reports of hole mobility measurements in GaAs. 153-168 Rosi et al. 5 3 have reviewed The valence band parameters obtained by Fa~lkner,"'~L a w a e t ~and , ~ ~Pollak et are ( L = -8.2, M = -2.9, N = -9.0), ( L = -9.14, M = -3.23, N = -9.93), and ( L = - 10.76, M = -3.20, N = -9.25), respectively, all in units of h2/2m,. Although they do not appear to be significantly different, the set ofvalues, given by Pollak et al. leads to lower effective masses and less valence band anisotropy. (See also the appendix.) F. D. Rosi, D. Meyerhofer, and R. V. Jensen, J . Appl. Phys. 31, 1105 (1960). 154 0. V. Emel'yanenko, T. S. Lagunova, and D. N . Nasledov, Fiz. Tuerd. Tela 2, 192 (1960) [English Transl.: Sou. Phys.-Solid State 2, 176 (1960)l. Is' 0. V. Emel'yanenko, T. S. Lagunova, and D. N. Nasledov, Fiz. Tverd. Tela 3, 198 (1961) [English Transl.: Sou. Phys.-Solid Slate 3, 144 (1961)l. D. E. Hill, Phys. Rev. 133, A866 (1964). 15' F. Ermanis and K. Wolfstirn, J. Appl. Phys. 37, 1963 (1966). "* H. Ikoma, J. Phys. Soc. Japan 25, 1069 (1968). F. E. Rosztoczy, F . Ermanis, I. Hayashi, and B. Schwartz, J . Appl. Phys. 41,264 (1970). 160 D. Diguet, Solid Slate Electron. 13, 37 (1970). 16' D. E. Hill, J. Appl. Phys. 41, 1815 (1970). F. E. Rosztoczy and K. B. Wolfstirn, J. Appl. Phys. 42,426 (1971).

150

J . D. WILEY

the experimental situation through the end of 1959, and have reported the results of measurements on p-GaAs prepared from high-resistivity n-GaAs by Cu diffusion. Because of the method of preparation, these samples were necessarily compensated. Nevertheless, the best samples showed mobilities as high as 350-370 cm2/V-sec. This is reasonably close to the lattice limit Emel'yanenko estimated by R o s i e ~ a f .tobep ' ~ ~ = 418(300/T)2.3cm2/V-sec. et al.' 5 4 ~ 1 5 5 have reported measurements on GaAs samples which were heavily doped with Zn and Cd. Their results show that for impurity con~ , impurity conduction is centrations greater than -5 x lo'* ~ m - metallic observed. Ermanis and Wolfstirn' 5 7 performed Hall and resistivity measurements on p-GaAs samples obtained by a variety of growth techniques (Bridgman, Czochrabki, and float-zoned) and doped with Zn in concentrations ranging from l O I 7 to 10'9cm-3. Several of their samples showed anomalies in the p/Zn ratios similar to those already discussed for Gap. Anomalies of this nature have also been reported for Zn-doped GaAs by Ruehrwein and Epstein.'68 Rosztoczy er af. 5 9 and Rosztoczy and Wolfstirn'62 investigated the behavior of Ge in solution-grown GaAs and found that, although it is amphoteric, Ge enters the GaAs lattice preferentially as an acceptor. 69 More recently, Vilms and Garret'66 have reported measurements on Sn- and Ge-doped GaAs samples grown by liquid-phase epitaxy. Their samples showed room-temperature hole concentrations of 10' cm- and mobilities which are among the highest ever reported for p-GaAs [as high as 442 cm2/V-secfor one sample withp(300"K) = 1.68 x 1015].It is interesting to note that, in contrast to Ge, Sn enters the lattice preferentially as a d ~ n o r . ' ~ ~ , ' ~ ~ In recent years, considerable effort has been expended toward the production of extremely high-purity GaAs by epitaxial growth techniques. As a result of this work, there have been several reports of samples which appear to have hole mobilities very near to the lattice-limited mobility for temperatures above 77°K.'61~'63~'64~'66~167 Figure 29 shows the temperature dependence of the mobility for three high-purity samples grown by vaporphase epitaxy.'61*'649'67Based on the behavior of his purest sample

'

V. L. Dalal, A. B. Dreeben, and A. Trians, J. Appl. Phys. 42, 2864 (1971). A. L, Mears and R. A. Stradling, J . Phys. C4, L22 (1971). 165 Sh. M. Gasanli, 0 . - V . Emel'yanenko, V. K. Ergakov, F. P. Kesamanly, T. S. Lagunova, and D. N. Nasledov, Fiz. Tekh. Poluprou. 5, 1888 (1971) [English Trans/.: Sou. P h p Semicond. 5, 1641 (1972)l. 166 J. Vilms and J. P. Garrett, Solid Stare Elecrron. 15,443 (1972). 16' K. H. Zschauer, in Proc. 4th Inr. Symp. GaAs and Relared Compounds, Boulder, 1972. Inst. Phys., London, 1973. 1 6 8 R. A. Ruehrwein and A. S. Epstein, J. Electrochem. SOC.109,98C (1962). 169 Further references to work on the behavior of amphoteric dopants (Si, Ge, Sn) in GaAs are cited by Rosztoczy and Wolfstirn,'62 Vilms and Garrett,'66 and Zschauer.I6' 163

2.

10

MOBILITY OF HOLES IN 111-V COMPOUNDS

151

I00 TEMPERATURE

(OK)

FIG.29. The temperature dependence of the Hall mobility for high-purity p-GaAs '6I,)( 0,164 0 ' 6 ' ) .

@ = 3.32 x 1014 cm-3 at 300°K) above 100"K, deduces a latticelimited mobility of /A x 400(300/T)2.41.Comparison of this formula with other data153*163*164,167 suggests that the prefactor may be somewhat low, and the temperature exponent somewhat high. A temperature exponent of 2.3 gives excellent agreement with experimental data (including those of Hill' 61), and is more in line with the observed behavior of other 111-V compounds.28 Although their sample appears to be of very high quality, the high-temperature data of Mears and Stradling164do not follow the usual T-fl temperature dependence. This causes their mobility values between 100 and 250°K

152

J . D. WILEY

to lie somewhat above the "lattice limit" as deduced from data on other samples. The results shown in Fig. 29 represent Hall mobilities obtained in the lowmagnetic-field limit. All three a ~ t h o r s ' ~ ' ~ reported ' ~ ~ ~a' reduction ~~ in the Hall coefficient with increasing magnetic field, indicating the absence of inhomogeneities of the type reported by Wolfe et al.'70-171a The magnetic field dependence of the Hall coefficient as obtained Mears and Stradling is shown in Fig. 30. The solid lines are based on a theoretical model for conduction in two independent bands, using the relaxation times as fitting parameters. The best fit to the data was obtained with ( T , ) / ( T ~ ) = 1.5. It should be reemphasized, however, that this fit must be regarded as empirical since no one has yet given a proper theoretical treatment of the magnetic field dependence of the Hall coefficient in p-type 111-V compounds (see also Part I11 of this chapter). If the bands are truly decoupled, then, for temperatures at which ionized-impurity scattering is dominant, one would expect 1 .o

0.9

-

0.8

-

0.7

-

0.6

-

0.5

-

0.4

I

c

0

v

I

a

2

I

I

a

I 0.125

I

2.5

I

I

I

10 20 MAGNETIC FIELD, B ( k G ) 5

I

40

J

80

FIG.30. The magnetic field dependence of the Hall coefficient R,(B), normalized to the value at B = 0. The dashed lines simply connect data points taken at the same temperature. The solid lines are based on a phenomenological model discussed in the text. (After Mears and Stradling.'64) C. M. Wolfe and G. E. Stillman, Appl. Phys. Lerr. 18, 205 (1971). C. M. Wolfe, G. E. Stillman, and J. A. Rossi, J. Elecrrochern. SOC.119, 250 (1972). '7'aC.M. Wolfe and G. E. Stillman, Chapter 3, this volume. I7O

17'

2. MOBILITY

OF HOLES IN Ill-V COMPOUNDS

153

( zl)/( T ~ )!z (ml/m2)''2 z 2.5 f 0.3. This is to be compared with a ratio of (tl)/(z2) !z 2.0 obtained from cyclotron resonance linewidths at 50°K. There have been several e ~ p e r i m e n t a l ' ~ ~ ~and ~ ~t ~h -e' o' ~r e t i ~ a l ~ ~ ' ~ ~ , ~ ~ determinations of the effective masses of holes in GaAs. The most accurate determination is that of Mears and Stradling,'64 who obtained m,/m, = 0.475 & 0.015 and mJmo = 0.087 0.005 by cyclotron resonance at 50°K with Bll( 100). Since the light-hole band is fairly isotropic (see Part II), different experiments give similar values for mz/mo (m2/mo= 0.068 & 0.015'72 and 0.082 _+ 0.006173). If one uses valence band parameters . ~ ~ , ~ with ~ a formula given by calculated by the k . p m e t h ~ d , ~ ' together Dresselhaus et al." for the cyclotron mass with B )I(loo), one obtains m2/mo = 0.077,370.084,43and 0.077,53in good agreement with the cyclotron resonance164and magnetooptical' 73 results. The heavy-hole band is more anisotropic, and the results for this case are more widely scattered. In addition to the cyclotron resonance value already quoted, the following values have been reported for m,/m, . From low-temperature magneto~ ' ~ ~ mJm, = optical experiments with B )I(100) or (1 1l), V r e h e ~ obtains 0.45 & 0.05. Walton and M i ~ h r a report ' ~ ~ a value of mJm, = 0.50 2 0.02 based on Faraday rotation at 300°K with Bll(111). By fitting the infrared absorption spectrum of p-GaAs, Balslev has deduced valence band parameters of A = -6.77, B = -4.55, and C2 = 37.45 A2/2m,. These parameters imply ml/mo = 0.45 for kII (100) and 0.99 for k(l(ll1). Again using the formula of Dresselhaus et d.'' together with Balslev's valence band parameters gives ml/mo = 0.52 for BII (100) and 0.77 for B )I(1 11). Similarly, the valence band parameters of Pollak el al.37 yield (0.45, 0.54) and those of L a ~ a e t yield z ~ ~(0.42, 0.67), respectively. Lawaetz gives a value for ml , averaged over all directions, of mllmo = 0.62. Choosing average values for the effective masses of ml/rno = 0.50 f 0.05 and mz/mo = 0.08 & 0.01, the polar mobility at 300qK is estimated to be 950 f 150, where the error bars reflect only uncertainty in ml/m,. GaAs is the only material for which we have exact calculations of the polar mobility of hole^,^^,^^ and thus affords an excellent opportunity to check the reliability of Eq. (48) for estimating polar mobilities. Figure 5 of Costato et al.32appears to give a polar mobility of about 1100-1200 cm2/V-secat 300"K, indicating that our estimate may be on the low side. Closer examination shows that the difference lies entirely in the choice of parameters. Costato et a1.32 have used mJmo = 0.5, m2/mo = 0.068, E, = 11.5, E , = 10.2, and t3 = 408"K, whereas our estimate is based on ml/mo = 0.5, A. K. Walton and U. K. Mishra, J . Phys. C 1, 533 (1968). Q. H. F. Vrehen, J. Phys. Chem. Solids 29, 129 (1968). 1. Balslev, Phys. Rev. 177, 1173 (1969).

154

J. D. WILEY

m2/m, = 0.08, E~ = 12.9, E , = 10.9, and 8 = 420"K.'75 When the para: = 1110 meters of Costato et al.32 are used in Eq. (48), one obtains & cm2/V-sec. Thus, Eq. (48) appears to give excellent agreement with the results based on numerical solution of the Boltzmann equations. Assuming the lattice-limited mobility at 300°K to be -400 cm2/V-sec, and using our estimate of ,u$A z 950 k 150 cm2/V-sec, the combined AC and NPO mobility is p'&Nw z 690 T 70 cm2/V-sec. This implies an acoustic deformation potential of EAcx 4 f 1 eV, in good agreement with the theoretical estimate in Table VI. Figure 31 shows the dependence of mobility on hole concentration at 300°K. The solid line was calculated using the Brooks-Herring formula together with an assumed lattice-limited mobility of 400 cm2/V-sec. As has been emphasized before, this procedure has no real theoretical significance, and simply provides a curve which follows the general trend of the data. It should be noted that the data in Fig. 31 span six orders of magnitude in hole concentration. In this respect, GaAs is practically unique among the

400

-

0

c

0

$ 300 I

-

, i ->5 2 0 0 -c-1 m I

0

-I -I

100

-

6

I

I

I

I

.%I

lot5

I

,

,.I

I

loi7

1

,*I

I

10'~

* # , I

*

. *la 1O2O

HOLE CONCENTRATION (c rn-3~

FIG.31. The concentration dependence of the Hall mobility in p-GaAs at 300°K (0,154 ~ 7 , 1 6 50 , 1 5 6 .,I61

0 , 1 5 3 +,I59

166).

The most serious difference in parameters is in the dielectric constants. Their values for E,, and E , imply an effective charge of e*/e = 0.178 compared to our value of 0.201. This difference alone causes a 25 % difference in the calculated mobilities.

2.

MOBILITY OF HOLES IN Ill-V COMPOUNDS

155

p-type 111-V compounds since most of these materials run into problems of residual impurities or intrinsic conduction below 10l6 cm-3. Thus, p-GaAs should provide an excellent opportunity for comparing theory with experiment when an adequate theory is developed for &.

7. GaSb Gallium antimonide exhibits two difficultieswhich were not present in any of the materials discussed so far: (1) Because of its small band gap (0.7 eV at 300”K), GaSb begins to show complications due to intrinsic conduction at relatively low temperatures, and (2) as-grown crystals contain extremely large densities of native acceptor defects, making it difficult to obtain samples with room-temperature hole concentrations less than -2 x lo” ~ m - ~ . Thus, despite a considerable number of papers reporting electrical measurements in p-GaSb,176-189we still know very little about the transport properties of holes in “perfect” GaSb. The native defect responsible for the high level of background acceptors has been identified as an antistructure defect consisting of a combination of Ga vacancies and Ga atoms substituted on Sb lattice sites’**[conventionallydenoted by (V,,, Ga,)]. Although no attempt will be made to give a chronological review of the work which has led to this identification, a few of the more important bits of evidence bear directly on the topic of this chapter and will therefore be mentioned. By the year 1963, there was fairly convincingevidence that the troublesome background acceptors were related to native defects rather than chemical impurities.I8’ Experiments involving ion pairing between Li and the unknown acceptor showed the acceptor to be doubly ionized, further suggesting a native defect since it was felt that no doubly ionizable chemical defect could have escaped detection by mass-spectrographic analy~is.’~’ This led Reid et ~ 1 . to ’ ~perform ~ a series of experiments in which GaSb crystals were H. N. Leifer and W. C. Dunlap, Jr., Phys. Rev. 95, 51 (1954). D. P. Detwiler, Phys. Rev. 97, 1575 (1955). 178 D. F. Edwards and G. S. Hayne, J. Opt. Soc. Amer. 49,414 (1959). R. N. Hall and J. H. Racette, J. Appl. Phys. 32, 856 (1961). W. M. Becker, A. K. Ramadas, and H. Y. Fan, J . Appl. Phys. Suppl. 32,2094 (1961). V. 1. Ivanov-Omskii, B. T. Kolomiets, and Chou-huang, Fiz. Tuerd. Telu. 4, 383 (1962) [English Transl.: Sou. Phys.-Solid State 4, 276 (1962)l. D. Effer and P. J. Etter, J. Phys. Chem. Solids 25, 451 (1964). R. D. Baxter, R. T. Bate, and F. J. Reid, J . Phys. Chem. Solids 26, 41 (1965). 184 M. H. van Maaren, J. Phys. Chem. Solids 27,472 (1965). F. J. Reid, R. D. Baxter, and S. E. Miller, J . Electrochem. Soc. 113, 713 (1966). Y. J. van der Meulen, J . Phys. Chem. Solids 28,25 (1967). M. S. Mirgalovskaya, G. V. Kukuladze, and V. A. Kokoshkin, Neorg. Muter. 4,694 (1968). J. Allegre, M. Averous, and G. Bougnot, Cryst. Lurr. Defecrs 1 , 343 (1970). A. Ya. Vul’, L. V. Golubev, and Yu. V. Shmartsev, Fiz. Tekh. Poluprov. 5, 1208 (1971) [English Transl.: Sou. Phys.-Semicond. 5, 1059 (1971)l. I”

”’

156

J . D. WILEY

grown from nonstoichiometric melts. Some of the results are shown in Fig. 32, where the 78 and 300°K hole concentrations are plotted as functions of the stoichiometry of the melts from which the samples were grown. As the Sb concentration in the melt is increased above 60% or so, one obtains significant reductions in the level of background acceptors, indicating that the acceptor defect involves either an excess of Ga or a deficiency of Sb in the crystal. Reid et a/.' 8 s suggested that the most likely candidate was G a on an Sb site. Growth from nonstoichiometric melts can be achieved (with increasing difficulty)up to Sb concentrations of about 87 %, at which point one reaches a Ga-Sb eutectic. By growing slightly on the Ga side of the eutectic, Reid et aI.lSs were able to produce GaSb crystals with p x 2-3 x 10l6

50

60

70

ATOMIC PERCENT A N T I M O N Y I N GaSb MELT

FIG.32. The hole concentration in GaSb as a function of the stoichiometry of the melt. Growing off stoichiometry in the Sb direction reduces the concentration of native-defect acceptors. (After Reid et

2.

157

MOBILITY OF HOLES IN 111-V COMPOUNDS

-

cm-3 at 300°K which had Hall mobilities of 800 cm2/V-sec at 300°K and up to 6000 cm2/V-sec at 78°K. Several workers have reported larger values for hole mobilities at 300"K, and a representative number of these results are shown in Fig. 33. The highest value reported for ~(300°K)is that of Edwards and Hayne,17' who find p = 1420cm2/v-sec for a sample with p = 7.5 x 10l6 ~ m - Such ~ . a high value must certainly be regarded cautiously. Mirgalovskaya et al?' have shown that anomalously high mobilities in GaSb can sometimes be correlated with sample inhomogeneities (see also Wolfe and Stillman' 7 0 and Wolfe et al.' 71), and both these workers'" and van der Meulen 86 have found that as-grown GaSb is often extremely inhomogeneous.

'

1200

I000

0

3

800

5 \

0)

-5 t

600

I!

m

0

z -I

r

400

2 00

0 1018

I0'9

I020

HOLE CONCENTRATtON (CffT3)

FIG.33. The dependence of Hall mobility on hole concentration in p-GaSb at 300°K. The .,182 0,l8 A,187 ' .IE6). solid line has no theoretical significance

158

J . D . WILEY

The temperature dependence of the mobility is shown in Fig. 34 for a typical sample'76 with N A x 10l7 cm-j. (This figure is a composite of Figs. 2 and 3 of Leifer and D ~ n l a p ' ~The ~ . ) high density of background acceptors prevents us from observing lattice-limited mobility below 2003 W K , and the onset of intrinsic conduction causes significant deviations in the 300400°K range. At a temperature of approximately 630"K, the sample converts from p-type to n-type due to the rapidly increasing intrinsic carrier concentration and the larger mobility of the electrons. The exact temperature at which this conversion takes place depends, among other things, on the acceptor concentration and is lower for lower densities of acceptors. Figure 35 shows the intrinsic carrier concentration as a function of temperature for the narrow-gap 111-V compound^.'^^ It is seen that GaSb achieves an intrinsic carrier concentration of ni z 10' cm- in the neighborhood of 600°K. 104,

I

I

I

1

TEMPERATURE

I

I

I

1

(OK)

FIG.34. The temperature dependence of the Hall mobility for GaSb with N, zz 10'' (A composite o f Figs. 2 and 3 o f Leifer and Dunlap.176) H . C. Casey, Diffusion in the Ill-V compound semiconductors, in "Atomic Diffusion in Semiconductors" (D. Shaw, ed.), p. 426. Plenum, New York, 1973.

IYo

2.

159

MOBILITY OF HOLES IN 111-V COMPOUNDS

0

500

1000

TEMPERATURE ("C)

FIG.35. The intrinsic carrier concentration as a function of temperature for the narrow-gap

111-V compounds. (After Casey.'")

In the absence of any significant information on the lattice-limited mobility in p-GaSb, it is particularly interesting to attempt a theoretical estimate. There have been several measurements of hole effective masses in l ~ given ~ an excellent review of the GaSb.172*'91-193Reine et ~ 1 . have present status of our knowledge of the valence band of GaSb, including an extensive tabulation of effective mass values. For present purposes, it R. A. Stradling, Phys. Lerr. 20,217 (1966). M. Reine, R. L. Aggarwal, and B. Lax, Sorid State Cornrnun.8, 35 (1970) 193 M. Reine, R. L.Aggarwal, and 8. Lax, Phys. Rez.. B 5 , 3033 (1972).

19'

19*

160

J . D. WILEY

suffices to say that there is reasonablygood agreement on the valuesrn,/m, = 0.35 & 0.05 and mz/rno = 0.045 & 0.005. Using these values, the polar : z 3800 cm2/V-sec. Assuming an acoustic mobility is estimated to be & deformation potential of EAc z 3.5 eV gives z 1500 cmz/V-sec. Combining these mobilities according to Matthiessen’s rule gives p;i:t NN 1100 cm2/V-sec, in excellent agreement with the value of 1200 cm’/V-sec suggested by Fig. 33. Before leaving GaSb, attention should be drawn to some very recent work by Metzler and B e ~ k e ron ’ ~stress-induced ~ decoupling of the valence bands of GaSb. Upon application of uniaxial stress, the degeneracy of the Tsvalence band is lifted, causing the light- and heavy-hole bands to separate in energy. Under compressive stress, the light-hole band moves upward (toward lower hole energies) and the heavy-hole band becomes progressively depopulated. Metzler and B e ~ k e r have ’ ~ ~ measured the hole mobility and the magnetic field dependence of the Hall coefficient(as a measure of the importance of two-band conduction) as functions of stress. The dramatic effects which they observe mark this as a very powerful technique for studying the valence bands of III-V compounds. 8. InP

-

Crystals of InP grown from starting materials of the highest available purity are invariably n-type,’ with residual donor concentrations of 1OI6 cm-3. InP also has a strong tendency toward the formation of growth twins so that ingots are frequently found to be heavily twinned or even polycrystalline.’ Thus, although p-type InP can be obtained by doping with Zn or Cd, the crystals are necessarily compensated and often of rather poor quality. Figure 36 shows the temperature dependence of the mobility for the ]. purest samples reported to date [p(300”K) cz 2-3 x 1OI6 ~ m - ~ Above 2 W K , thesedataare fit quite well by the empirical formulap = 150(300/T)2.2 cmz/V-sec. This represents a lower limit for the lattice-limited mobility of holes in InP. After attempting to correct for ionized-impurity scattering, Glicksman and Weiser19’ obtained a lattice-limited mobility of p = 148(300/T)’.4.The samples shown in Fig. 36 reached mobilities of 1200 cm2/V-sec at 77°K (the lowest temperatures at which data were taken) and had not yet reached their low-temperature maxima. Galavanov et have given the temperature dependence of the mobility in several heavily doped samples [p(3WK) z 1018-1019crn-j]. These samples showed maximum mobilities of 30-60 cm2/V-secin the 200-300°K range, followed by

-

194

19’ 19’

R. A. Metzler and W. M . Becker, Solid Srate Commun. 12, 1209 (1973). M. Glicksrnan and I(.Weiser, J . Phys. Chem. Solids 10,337 (1959). V. V. Galavanov, S. G. Metreveli, N. V. Siukaev, and S. P. Starosel’tseva, Fiz. Tekh. Poluproc. 3, 120 (1969) [English Transl.: Sou. Phys.-Semicond. 3,94 (1969)l.

zooor--2.

1000

0

Om

-

0.

0

-

400

0

0. 0.

-

om 0.

a In l

8

?

-

161

MOBILITY OF HOLES IN 111-V COMPOUNDS

N

E

d

0

-

200

-

e

9

=!

m

0

0

I -I J

p

100

40

1

0

-

20 I

40

0

InP

0 0

0 0

I

I

1

I 400

100

I

000

TEMPERATURE (OK)

FIG.36. The temperature dependence of the Hall mobility in InP for two samples with p(300'K) = 2-3 x 1016~ r n ( -0 , ~1 9 ' 0196 ).

a drop in mobility at lower temperatures due to the onset of impurity conduction. The dependence of the mobility at 300°K on hole concentration19 s - 1 9 7 is . shown in Fig. 37. Once again, the solid line has no theoretical significance, but shows the general trend of the data. 19'

D. N. Nasledov, Yu. G. Popov, N. V. Siukaev, and S. P. Starosel'tseva,Fiz. Tekh. Poluproc. 3,454 (1969) [English Transl.: SOL'.Phys.-Semicond. 3,387 (1969)l.

162

J. D. WILEY

>

J

J

a

=

o 10"

1018

H O L E C O N C E N T R A T I O N (cm-3)

FIG.37. The dependence of Hall mobility on hole concentration for InP at 300°K (0,196

0,195 0 1 9 7

1.

There have been no reliable measurements of the effective masses of holes in InP. Values reported in the literature for the average effective mass of holes range from 0.2 to 1.0 (0.2-0.8,1950.8,'96 l.0,198and l.0'99). Nasledov et al.'97 introduce a temperature-dependent effective mass (m*= 0.4 at low temperatures, 0.8 at 300"K, and some unspecified higher values above 300"K), in order to force agreement between experimental mobility data and Ehrenreich's formula' for polar mobility. Theoretical masses obtained from k p calculation^^^-^^ are in excellent agreement on a value for the light-hole mass (m,/m, = 0.08643and 0.08953)but are in disagreement on the heavyhole mass (ml/mo = 0.543 and 0.8553).Rejecting the extremes, a value of m l / m o = 0.65 5 0.15 seems reasonable. Using this number, the polar mobility at 300°K is estimated to be p g : x 440 cm2/V-sec.Combining this with a lattice mobility of 160 cm2/V-sec gives p$Npo z 250 cm2/V-sec which, in turn, implies EAc z 4.5 eV. While these are all reasonable numbers, it must be emphasized that the error bars are large, due primarily to the large uncertainty in M , . In this regard, it is perhaps worthwhile to point out that the best samples of p-InP have low-temperature mobilities which are sufficiently high to allow cyclotron resonance to be observed. (In fact, InP presents somewhat more favorable conditions for the observation of cyclotron resonance than Gap, which has already been studied by this technique.) 198 199

0. G. Folberth and H. Weiss, Z . Nnturforsch. 10A,615 (1955). V. V. Galavanov, S. G . Metreveli, and S. P. Starosel'tseva, Fiz. Tekh. Poluproo. 3, 1391 (1969) [English Transl.: Soa. Phys.-Semicond. 3, 1159 (1970)l.

2.

163

MOBILITY OF HOLES IN 111-V COMPOUNDS

Thus, there is reason to expect that, with the recent increase in interest in InP for device applications, we will soon have more reliable information on the valence band structure and transport properties of InP. 9. InAs

High purity, as-grown InAs is always found to be n-type with a concentration of 1-2 x loi6 cme3 residual donors.z00-z04One must therefore resort to rather heavy doping with Zn or Cd in order to obtain p-type samples.z05-z09 Most work on the electrical characterization of p-InAsz05-z07~z10-z15 has been concerned with various anomalies which have been summarized by DixonZo7as follows. For heavily doped material k(300"K) 2 2 x l O I 7 ~ m - the ~ ]behavior of the Hall coefficient is that of a "normal" extrinsic semiconductor. For lower doping levels, however, the behavior is anomalous, showing double reversals in the sign of R , as a function of t e m p e r a t ~ r e , ~and ~ ~peculiar - ~ ~ ~ annealing e f f e ~ t s . ~Zl'* ~ ~ ~ ~ ' DixonZo7attributes this anomalous behavior to the presence of microscopic inhomogeneities-possibly small n-type regions associated with dislocations. RupprechtZ'O has shown that the anomalous behavior of p-InAs can be eliminated by etching the samples in nitric acid and can be reintroduced by grinding or polishing the surface. On the basis of these results (which, apparently, are not always reproduciblezo7), Dixon suggests the existence of an n-type skin on the surfaces of otherwise p-type samples. Further evidence for n-type surface layers has been given by other w ~ r k e r s . ~ ' ~ * ~ '

'

Loo T. C. Harman, H.

L.Goering, and A. C. Beer, Phys. Rev. 104, 1562 (1956). R. H. Harada and A. J. Straws, J. Appl. Phys. 30, 121 (1959). D . Effer. J. Electrochem. Soc. 108, 357 (1961). '03 G . R. Cronin, R. W. Conrad, and S. R. Borrello, J. Electrochem. Soc. 113, 1337 (1966). '04 G . R. Cronin and S. R. Borrello, J. Electrochem. SOC.114, 1078 (1967). '05 0. G. Folberth, 0. Madelung, and H. Weiss, Z . Naturforsch. 9A, 954 (1954). 0. G . Folberth and H. Weiss. Z . Naturforsch. l l A , 510 (1956). 'O' J. R. Dixon, J. Appl. Phys. 30, 1412 (1969). * 0 8 N . V. Zotova and D. N. Nasledov, Fiz. Tverd. Tela. 4, 681 (1972) [English Transl.: Sou. Phys.-Solid State 4,496 (1962)l. '09 M. P. Mikhailova, D. N. Nasledov, and S. V. Slobodchikov, Fiz. Tverd. Tela 5, (1964) [English Transl.: Sou. Phys.-Solid Stare 5, 1685 (1964)l. 'lo H. Rupprecht, Z . Naturforsch. 13A, 1094 (1958). 'I1 J. R. Dixon and D. P. Enright, J. Appl. Phys. 30, 753, 1462 (1959). "'J. T. Edmond and C. Hilsum, J. Appl. Phys. 31, 1300 (1960). ' I 3 S. Kawaji and Y. Kawaguchi, Proc. Int. Conf. Phys. Semicond., Kyoto, 1966, in J. Phys. SOC.Japan Suppl. 21,336 (1966). 'I4 C. S. Fuller and K. B. Wolfstirn, J. Electrochem. SOC.114,856 (1967). 'I5 V. V. Voronkov, E. V. Solov'eva, M. I. Iglitsyn, and M. N. Pivovarov, Fiz. Tekh. Poluprov. 2, 1800 (1968) [English Transl.: Sou. Phys.-Semicond. 2, 1499 (1969)l. '01

164

J . D. WILEY

The net result of this is that residual donors and as-yet-unexplained anomalous behavior prevent the study of lightly doped p-InAs. Work is therefore limited to heavily doped, compensated material. In addition the intrinsic carrier concentration reaches 10'' cmW3at a temperature of only 280°C (see Fig. 3 9 , by which time the high-mobility electrons totally dominate the electrical transport processes. This explains the paucity of data on hole mobility in InAs. Figure 38 shows the temperature dependence of the mobility for two samples of Zn-doped I ~ A S . ' ~One ' sample had N A x 2 x 10'' cmV3,a mobility of 150 cm2/V-secat 300"K, and a conversion fromp-type to n-type at 350°K. By extrapolating their data to higher temperatures (the dashed line in Fig. 38), they obtain a temperature dependence of T - 2 . 3 ,although, considering the impurity concentration in this sample, it is difficult to believe that one is truly observing lattice-limited mobility. The other sample shown , of 100 cm2/V-secat 300"K, in Fig. 38 had N A x 7 x lo'* ~ m - a~mobility and a conversion from p-type to n-type at 475°K.A few results on more heavily doped samples have been reported by Zotova and Nasledov208and Mikhaiiova et aL209 400

c

0

>

200

\

-

w

E

0

>

c 2

m

g

100

-I

-I

a

I

40 100

200

1000

4 00

TEMPERATURE

(OK

)

FIG.38. The temperature dependence of the Hall mobility in p-InAs. (After Folberth er .1.*05)

2. MOBILITY

OF HOLES IN 111-V COMPOUNDS

165

As was the case for InP, there have been no direct measurements of the valence band effectivemasses in InAs. Assuming the values given by Lawaetz5 (m,/m, = 0.6 and m2/m, = 0.027), one obtains p g x 500 cm2/v-sec at 300°K.Assuming the same masses, together with EAc = 3.2 eV (from Table VI) gives p2&Np0x 400 cm2/V-sec, for a combined lattice-limited mobility of 220 cm2/V-sec at 300°K (subject, of course, to the large uncertainties in mJmo and ,FA,-).

10. InSb InSb is probably the most thoroughly studied member of the 111-V semiconductors. Willardson and Goering’ list over 460 references to work on InSb prior to 1961. Additional references can be found in the review articles by Moss216 and Hulme and Mullin217 (the latter being devoted primarily to methods of preparation and device applications of InSb). A recently published compendium of the physical properties of 111-V compounds6 lists over 120 references for InSb, with the emphasis placed on the best and most recent work. No attempt will be made in the present section to review or comment upon all, or even a significant fraction, of the papers which have dealt with the mobility of holes in InSb. Rather, experimental results will be presented from a few representative papers, together with a brief discussion of the most recent theoretical work. In addition to considerations of time and space, there are several reasons for this approach. InSb has the smallest band gap of any of the 111-V semiconductors (0.18 eV at 300”K), causing the intrinsic carrier concentration to be quite large (ni 1017, and 10l8cm-3 at temperatures of 0, 120, and 400°C, respecti~ely’~~). This, combined with the extraordinarily high electron mobility (p, z 8 x lo4 cm’/v-sec at room temperature and > 6 x lo5cm2/ means that one is dealing with a three-carrier system in V-sec at 77”K3-’), nearly all experimental regimes. A considerable amount of effort has gone into the development of theoretical models for two-carrier (electron-hole or light hole-heavy hole) and three-carrier (electron-light hole-heavy hole) transport. These models have been extensively reviewed e l s e ~ h e r e ~ - ~ ~ ~ ~ * ’ and will not be discussed here except to say that they all assume simple (spherically symmetric, parabolic, s-like), noninteructing bands.’ Although they give qualitatively correct predictions concerning some of the more striking features of multiband transport (for example, the magnetic field dependence of the Hall coefficient), it is felt that the model assumptions are very poorly satisfied for p-InSb, and that conclusion’s based on fitting these ’I6

T. S. Moss, Progr. Semicond. 5, 189 (1960).

’”K. F. Hulme and J. B. Mullin, Solid Sfate Elecrron. 5, 21 1 (1962). ’’* An interesting exception is the model developed by Appel’O’ to account for electron-

electron and electron-hole scattering. Appel includes interband transitions by solving a set of three coupled Boltzmann equations.

166

J. D. WILEY

models to experimental data are potentially very misleading. Thus, for example, Fischer’ l 9 has given very extensive and high-quality Hall data for a sample of p-InSb, and has shown that the experimental points can be fit almost perfectly by adjustment of the parameters in a three-band model.220 The parameters obtained from this fit, however, imply a light-hole mobility which increases with temperature and a negative concentration of light holes above 150°K. Fischer rather implausibly attributes this to the neglect of polar mode scattering. Galavanov2” has recently given an excellent review of the experimental data on hole mobilities in inSb, and Figs. 3 9 4 1 show some of the results which he has presented. Figure 39 shows the dependence of Hall mobility at 77°K on hole The shaded region above 10l8 c ~ n - ~ contains a high density of experimental points, the references for which are given by Galavanov.”’ Similar results are shown in Fig. 40 for data taken at

I

o

1021

I

lo1* toi3

1

toi4

5

1

toi5

I

1

10’~

10”

l

I

d8

I

loi9

1

lo2o

HOLE CONCENTRATION ( ~ r n - ~ )

FIG.39.The dependence of mobility on hole concentration for InSb at 77°K. The shaded region contains a large number of experimental points ( 0 , 2 2 AZz2 ’ ). G. Fischer, Helu. Phys. Acta 33,463(1960). R. G. Chambers, Proc. Phys. SOC.(London) A65,903 (1952). V. V . Galavanov, Fiz. Tekh. Polupror. 4,853 (1970)[English Transl.: Sor. Phys.-Semicond. 4,723 ( I 970)]. 2 2 2 A. J. Strauss. J . Appl. Phys. 30, 559 (1959).

219

220

2.

lo2

10'~'

MOBILITY OF HOLES IN Ill-V COMPOUNDS

I 0l5 HOLE

167

lo1' CONCENTRATION,

p (crn-j)

FIG.40. The dependence of mobility on hole concentration for lnSb at 290°K (After GalaoZz5). vanov,221+ , 2 2 3 0,224

room The solid lines in Figs. 39 and 40 were calculated by Galavanov" l using a combination of ionized-impurity, acoustic phonon, and polar mode scattering, together with mJm, = 0.4, c0 = 17, e*/e = 0.13, and E,, = 4 eV. Aside from small corrections due to the presence of light holes, the principal errors in this procedure arise from using an inappropriate expression for ppo (see Part 111, Section 2b) and neglecting nonpolar optical mode scattering. The effects of these two errors are in opposite directions, and it is their near cancellation which gives the agreement between theory and experiment shown in Figs. 39 and 40. From the previous discussion, it should be apparent that it is difficult to obtain hole mobility data for InSb over a wide range of temperatures. Above 200"K, one must make large corrections for the intrinsic electrons, and the resulting hole mobilities are not very reliable. Galavanov has collected results on the temperature dependence of the hole mobility from a number 223 224 225

K. I. Vinogradova, V. V. Galavanov, and D. N . Nasledov, Fiz. Tuerd. Te/u 4, 1673 (1962) [English Transl.: Sou. Phys.-Solid Siaie 4, 1230 (1962)l. H. Schonwald, Z . Nururforsrll. 19A. 1276 (1964). H. J. Hrostowski, F. J. Morin, T. H. Geballe, and G. H. Wheatley, Phys. Rev. 100, 1672 (1955).

168

J . D. WILEY

FIG.41. The temperature dependence of the hole mobility in InSb. The vertical bars indicate the ranges observed for a large number of experimental points. The solid line has a slope of -1.8.

of sources, and the general trend of these data can be seen from Fig. 41. The vertical bars indicate the spread of experimental points and the solid line shows the average temperature dependence (approximately T - I.*). Individual authors have also obtained temperature dependences in this general range, or somewhat steeper ( T - 8 with fl = Z.l,2252.1,226 2.0,227

’’’ G . Busch and E. Steigmeier, Helv. Phys. Acfa 34, 1 (1960). 227

Ya. Agaev, 0. Mosanov, and 0. Ismailov, Fiz. Tekh. Poluproo. 1, 855 (1967) [English Transl.: Sou. Phys.-Semicond. 1, 71 1 (1967)l.

2. MOBILITY OF HOLES IN

Ill-V COMPOUNDS

169

and 1.81228*229). These results seem to be in reasonable agreement with the general behavior of the other p-type 111-V compounds, and indicate a lattice mobility of approximately 850(300/T)1.8cm2/V-sec. The effective masses of holes in InSb have been measured by several workers and are found to be as follows. For the light-hole mass, there is nearly universal agreement (m2/mo= 0.016,2300.021 f 0.005,2310.016,232 0.016,43and 0.01653).Even for the heavy-hole mass, there is unusually good agreement, although different experiments yield slightly different averages. Pidgeon and report mJmo = 0.44, 0.42, and 0.32 for k 1) (1 1l), (1 lo), and (loo), respectively. Bagguley et ~ 2 1 . ’ find ~ ~ cyclotron effective massesofmJm, = 0.45 f 0.03,0.42 f 0.03,and0.34 f 0.03forBII ( l l l ) , (1 lo), and (loo), respectively. C a r d ~ n and a ~ ~LawaetzS3calculate average effective masses of ml/m0 = 0.39 and 0.47, respectively. Thus, the effective masses seem to be fairly well established as mJm, = 0.40 f 0.05, and m2/mo = 0.016 f 0.002. Using these values, the polar mobility at 300°K is estimated to be g . 5 ~1800 f 400 cm2/V-sec, where the error bars are estimated solely on the basis of uncertainty in ml/mo. Once again, this result can be used together with the observed lattice mobility to estimate E,,. The results are p$,Npo x 1600 f 300 cm2/V-sec and E,, x 2.5 f 0.5 eV. Several papers have appeared in the Russian applying top-InSb. Since this is the first application of such the theory of Bir et a1.18-20 a detailed theoretical model to the 111-V compounds, it is of interest to summarize the findings here. E r m o l o v i ~ hconsiders ~ ~ ~ , ~only ~ ~ acoustic and ionized-impurity scattering at low temperatures and obtains the following results. For acoustic mode scattering, interband transitions are quite effective in reducing the mobility of the light holes so that their overall contributions to the effective mobility is only about 12%. The contributions of I

R. W. Cunningham, E. E. Harp, and M. Bullis, Proc. Inr. Conf. Phys. Semicond., Exeter, p. 732. Inst. Phys. Phys. SOC.,London, 1962. 2 2 9 R. W. Cunningham and J. B. Gruber, J . Phys. Chem. Solids 31,2017 (1970). 2 3 0 E. D. Palik, S. Teitler, and R. F. Wallis, J. Appl. Phys. Suppl. 32, 2132 (1961). 2 3 1 D. M. S. Bagguley, M. L. A. Robinson, and R. A. Stradling, Phys. Lett. 6, 143 (1963). 2 3 2 C. R. Pidgeon and R. N. Brown, Phys. Rev. 146, 575 (1966). 233 Yu. B. Ermolovich, Fiz. Tverd. Tela 11, 533 (1969) [English Transl.: Sou. Phys.-Solid State 11,429 (1969).]. 23 4 Yu. B. Ermolovich. Izv. Akad. Nauk SSSR Neorg. Muter. 7 , 697 (1971) [English Transl.: Inorg. Marer. 7, 604 (1971)l. 2 3 5 V . V. Galavanov and F. M. Gashimzade, Fiz. Tekh. Poluproo. 5, 2316 (1971) [English Transl. :Sou. Phys.-Semicond. 5, 2024 (1972)l. 236 References to additional articles, some of which appear only in the untranslated Russian literature, are cited by E r m o t ~ v i c h . ~ ~ ~ 228

170

1. D. WILEY

longitudinal and transverse phonons are found to be 81% and 19%, respectively, for the heavy holes and 67% and 33 %, respectively, for the light holes. For scattering by ionized impurities, E r m o l ~ v i c hfinds ~ ~ ~that interband transitions are of negligible importance and that the light holes raise the effective mobility by only about 7 %. These results lend quantitative to the arguments which were used in Part I11 to justify the approximate expressions which have been used to estimate mobilities throughout this chapter. Galavanov and G a ~ h i m z a d emodified ~ ~ ~ the theory of Bir et af.18-20to include mixed scattering by ionized impurities, acoustic phonons, and polar optical phonons, as well as valence band anisotropy. They then used this theory to calculate the magnetic field dependence of the Hall coefficient for p-InSb at 77°K. They obtained good agreement with experimental results for a relaxation time ratio of z1/z2 = 1. From the information presented,235 it is not possible to asses the relative importance of the various model assumptions, but this is clearly the type of calculation which is needed for just such assessments. V. Summary

Although we now have a reasonably clear qualitative or semiquantitative understanding of the factors which determine hole mobilities, there are numerous questions which can only be answered by extending the work summarized in Table V. A few of the more important unresolved problems which require numerical solution of the Boltzmann equations for their resolution are : (1) The importance of anisotropies in the energy surfaces and in the scattering probabilities. (2) The effects of nonparabolicity (particularly the effect of nonparabolicity on the temperature dependence of the mobility). (3) The overall effective mobility under realistic conditions of mixed scattering. (4) The magnetic field dependence of the Hall coefficient and Hall mobility. (5) The relationship between Hall and drift mobilities, taking account of all relevant scattering mechanisms, valence band warping, and two-band conduction. The results of these detailed numerical calculations will undoubtedly also suggest improvements in the phenomenological expressions presented in this chapter, and allow an assessment of the use of Matthiessen’s rule in combining hole mobilities. 237

If Eqs. (39) and (55) are used to estimate the contribution of light holes to the acoustic and ionized-impurity mobilities, one obtains 18% and 7%, respectively. These values are in excellent agreement with the values (12%and 7%) obtained by E r r n o l ~ v i c h . ~ ~ ~

2.

MOBILITY OF HOLES IN 111-V COMPOUNDS

171

Even the most elaborate theoretical models cannot be fairly tested until we have more reliable values for some of the material parameters. The most outstanding examples of poorly known parameters are the valence band parameters (see the appendix for further discussion), the heavy-hole effective masses, and the deformation potentials. Having said this, however, it should be emphasized that the general ranges of the material parameters are fairly well defined (for example, 0.4 < m,/m, < 1.0 and 2 5 E,,, 5 5 eV for all the 111-V compounds reviewed here), and attempts to force agreement between theory and experiment by using extreme values for these parameters are certainly unjustified. Turning to the experimental results, there is virtually 90 information on hole mobilities in p-type A1P or AlAs. Any work on these materials would thus represent a valuable contribution. Present crystal growth technology (particularly vapor-phase epitaxy) should allow the growth of high-quality films, although sample preparation and handling are made difficult by the high reactivity of these materials. Based on data for the remaining 111-V compounds, it is clear that the lattice mobility of holes is limited by a combination of acoustic, nonpolar optical, and polar optical phonon scattering, and that these scattering mechanisms are of roughly equal importance at room temperature. The temperature dependence of the lattice mobility , fl values fairly well established near and above room temperature is T P 8with for AlSb (2.25), GaP (2.17), GaAs (2.28), InP (2.3), and InAs (2.3). The value of p for InSb is less well known but appears to be between 1.8 and 2.1. With the exception of GaAs and InSb, the electrical properties of the p-type 111-V compounds have been studied only over a narrow range of ) , there has impurity concentrations (approximately 10'6-1019~ m - ~ and been no critical test of the theory of ionized-impurity scattering. This is an area in which there is room for further work in materials preparation as well as experimental and theoretical evaluation.

Appendix

The valence band parameters L, M , and N appearing in Eq. (3) are defined in terms of interband matrix elements as follows'2*41:

+ 2G, H , + H,,

L =F

M=

N=F-G+H,

-H2,

('43)

172

J . D. WILEY

where

In Eqs. (A4)-(A7), the sums are over all states of the indicated symmetries, E, is the energy of the TZ5. valence band edge, mo is the free-electron mass, and F, G, H , , and H2 are in units of h2/2m0.Dresselhaus et al.l 2 have given an approximate expression for the light- and heavy-hole energy surfaces in terms of these valence band parameters:

E(k) = Ak2 & [B2k4 + C2(kX2ky2 + yY2kz2+ kz2kx2)]1'2,

(A8)

where A

= ;(L

+ 2M) + 1 ,

B = +(L- M ) ,

c2= +"2

- ( L- M)2].

('49) (A101 (All)

Although Eq. (AS) is quite general?' Eqs. (A9HA11) are only valid very near k = 0. For larger values of k, the coefficients A, B, and C become energy dependent (markedly so in materials with small spin-orbit splittings) and energy surfaces exhibit a variable (energy dependent) ~ a r p i n g . ' ~ , ~ ' In his extensive tabulation of the valence band parameters of cubic semiconductors, Lawaetzs3 has used parameters which are similar to those first introduced by L ~ t t i n g e r , 'but ~ which include correction terms arising from the spin-orbit splitting of the rl conduction band59-238,239 : 7 5 8 .

y1 = -$(F =

-k(F

73 =

-aF

7,

K =

+ 2G + 2 H , + 2 H 2 ) - 1 + $ q , + 2G - HI - H , ) - &, - G + H , - H2)+ 44,

-A(F-G-H,

+ H )2 - ' 1 3- - 9 4q3

(A12) (A13) ('414) (A15 )

238

S. H. Groves, C. R. Pidgeon, A. W. Ewald, and R. J. Wagner, J . Phys. Chem. Solids 31,

239

2031 (1970). J. C. Hensel and K. Suzuki, Phys. Rec. Leu. 22, 838 (1969).

2. MOBILITY

173

OF HOLES IN 111-V COMPOUNDS

where q is the correction term, given by

TABLE XI VALENCE

Material

Si Ge

AIP

AlAs AlSb GaP GaAs GaSb InP lnAs InSb

BANDPARAMETERS

FOR

si, Ge, AND THE 1II-V

y,

y2

ys

q

-L

4.22 13.35 3.47 4.04 4.15 4.20 7.65 11.80 6.28 19.67 35.08

0.39 4.25 0.06 0.78 1.01 0.98 2.41 4.03 2.08 8.37 15.64

1.44 5.69 1.15 1.57 1.75 1.66 3.28 5.26 2.76 9.29 16.91

0.01 0.07 0.01 0.03 0.07 0.01 0.04 0.13 0.01 0.04 0.15

6.80 31.5 4.73 8.21 9.30 9.14 18.4 29.1 15.6 54.2 98.9

-M 4.43 5.75 4.34 3.44 3.03 3.23 3.77 4.55 3.11 3.87 4.58

COMPOUNDS'

-N

-A

-B

Cz

8.61 33.9 6.87 9.33 10.3 9.93 19.6 31.2 16.5 55.6 101.0

4.22 13.3 3.47 4.03 4.12 4.20 7.63 11.7 6.28 19.7 35.0

0.790 8.57 0.130 1.59 2.09 4.97 4.86 8.19 4.17 16.8 31.4

27.84 163.4 15.7 21.4 22.2 21.2 56.7 122.6 38.9 186.5 437.5

,

" Values for y ,yz, y3, and q are from a tabulation given by L a ~ a e t zThe . ~ ~remaining values were calculated using Eqs. (Al)-(A3), (A9)-(AI I), and (A12)-(A14).

Table XI contains values for yl, y2, y 3 , and gas given by L a ~ a e t ztogether ,~~ with the equivalent values of L, M,N , A, B, and C2.The numbers given in using the Table XI are based on a semiempirical five-level k p cal~ulation'~ latest values for the energy gaps, but no claim is made that they are in anyway definitive. Section IV contains numerous references to other sets of valence band parameters. Equations (Al)-(A16) are appropriate when the Tzs, valence band states are the only states in class A (see Part 11) and all interactions with higher lying states are treated as perturbations. If the lowest r2,conduction band is also included in class A, the definition of F must be changed by omitting this Tz,state from the sum. The resulting quantity is denoted F' and is given by

-

F' = F - [EJ(E, - E J ] , where E, is the energy of the lowest r2,conduction band and E, = (2/m0)1 (x Ipx 1 r2,)1'. The consequences of this change have been discussed thoroughly by Kane.'

174

J. D. WILEY

ACKNOWLEDGMENTS It is a pleasure to acknowledge numerous helpful discussions with H. C. Casey, M. Di

Domenico, Jr.. P. Lawaetz, D. L. Rode, D. C. Tsui, and S. H. Wemple during the course of this work. I should also like to thank M. Costato, D. E. Hill, R. Kaplan, D. Kranzer, K. L. Ngai, S. F. Nygren, R. A. Stradling, and K. H. Zschauer for communication of experimental and theoretical results, in some cases prior to publication.

CHAPTER 3

Apparent Mobility Enhancement in Inhomogeneous Crystals* C.M. Wove and G.E. Stillman I. INTRODUCTION . . . . . . . . 1. Low- and Average-Mobility Models . 2. High-Mobility Observations . . . 11. MAGNETOCONDUCTIVITY THEORY . . . 3. Bolrzmann's Equarion . . . . . 4. Porenrial Equation . . . . . . 111. SINGLE CONDUCTING INHOMOGENEITY . . 5. BasicModel . . . . . . . . 6. Conductivity Discontinuity . . . . 7. Conducticity Gradient . . . . . 8. Experimental Verpcation . . . . IV. MULTIPLE CONDUCTING INHOMOGENEITIES. 9. Isolated Cylindrical lnclusions . . . 10. Isolated Spherical Inclusions . . . 11, Qualitative Experimental Verijication . V. CONCLUSIONS.. . . . . . . . I 2. Applicability . . . . . . . . 13. Characteristic Features . . . . . VI. SUMMARY.. . . . . . . . .

. . . . . . . .

. . . . .

I15 116 182 185 I85 188

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

190 190 I 92 199 201 202 203 210 21 1 215 2 15 216 219

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . . . . . . . .

I. Introduction Several of the chapters in this volume are devoted to the study of transport phenomena in homogeneous crystals, taking into account the relevant lattice and impurity scattering mechanisms. These efforts are useful, not only for the light they shed on the basic properties of materials, but also as a guide for the selection of suitable materials for various applications. When such studies are used in conjunction with transport measurements (which are commonly used to evaluate materials), the intrinsic properties of a material can be established. Unfortunately, all crystals deviate to a greater or lesser extent from homogenenity in some significant manner, and it has been the crystal-growers' hope to achieve a sufficiently close approximation to the ideal crystal to satisfy solid-state physicists and device engineers.

* This work was sponsored by the Department of the Air Force 175

176

C. M. WOLFE AND G . E. STILLMAN

Probably the most commonly used transport measurements for characterizing the quality of a material are resistivity and Hall coefficient measurements, since it is well known that two samples of the same material with similar impurity content can have significantly different values of mobility. In the past it has been customary to assume that the sample with the higher mobility is of the higher quality, since departures from ideality would be expected to scatter charge carriers and thus lower mobility. Although this is generally true for homogeneous crystals, in inhomogeneous crystals the mobility value determined from resistivity and Hall measurements is not necessarily a good approximation to the real mobility of the charge carriers. That is, depending upon the type and relative extent of the inhomogeneity, the measured mobility value can be lower than the carrier mobility, some average value, or, as we have shown,' even higher than the carrier mobility. It is the purpose of this chapter to discuss the effects which result in an apparently high measured mobility. For this purpose we will first review the inhomogeneity models which result in a low or average value for mobility due to carrier scattering or the averaging inherent to resistivity and Hall measurements. We will then discuss some observations of measured mobilities which are anomalously high and thus cannot be explained by these models. This should serve as a prelude to the remainder of the chapter, which is devoted to a discussion of inhomogeneity models which result in a high apparent mobility. 1. Low- AND AVERAGE-MOBILITY MODELS

For this discussion it is convenient to divide inhomogeneities into three categories2 which are based on the relative size of the inhomogeneities: microscopic, intermediate, and macroscopic. Microscopic inhomogeneities are those of a size comparable to the carrier mean free path or the Debye length. Intermediate-sized inhomogeneities are large compared to the carrier mean free path but small compared to the size of the sample. Macroscopic inhomogeneities are of sufficient size that the sample geometry must be taken into consideration. a. Microscopic Inhomogeneities

The effects of microscopic inhomogeneities on the transport of charge carriers are very difficult to analyze theoretically since this requires the solution of an atomistic transport problem retaining both momentum and position variations. Thus, little has been done in this area. Frisch and Morrison3 have used the classical Boltzmann equation for a microscopically

' C. M. Wolfe and G . E. Stillman, Appl. Phys. Letr. 18, 205 (197.1). C . Herring, J. Appl. Phys. 31, 1939 (1960). H. L. Frisch and J. A. Morrison, Ann. Phys. 26, 181 (1964).

3.

APPARENT MOBILITY ENHANCEMENT

177

stratified medium to study the nonsaturation of the magnetoresistance at high magnetic fields. Also, McKenna and Frisch4 have examined a quantum mechanical formulation of the same problem. Although resistivity and Hall constant have not been examined in sufficient detail to result in quantitative prediction, qualitatively, inhomogeneities of this type are expected to produce carrier scattering and lower mobility.

b. Intermediate Inhomogeneities The effects of inhomogeneities of intermediate size have been examined theoretically in more detail. These problems are more tractable since they can be handled analytically by classical macroscopic techniques in a general sense without the necessity of taking into account specific sample geometry. Juretschke et al.’ have examined the effects of noninteracting, nonconducting cavities in a conducting material on the resistivity and low magnetic field Hall constant, taking into account the charge induced at the cavity surfaces. Their results for nonconducting cylinders parallel to the magnetic field and perpendicular to the applied electric field are as follows. The apparent Hall constant RApp(corresponding to that which would be measured) is equal to the Hall constant of the conducting material; that is,

dal

(1)

where ,u and (i are the true mobility and conductivity of the conducting material, respectively. The apparent conductivity is OAPP

= a(1

- f)/(l+ f),

(2)

where f is the volume fraction of the sample occupied by the nonconducting regions (which need not be circular in cross section). The apparent mobility for this case is then ~ A p p= RAppaApp =

A1 - f)M1

+ f).

(3)

For cylinders perpendicular to the magnetic field and parallel to the applied electric field, they find RAPP

= P/dl -

f),

J. McKenna and H. L. Frisch, Ann. Phys. 33, 156 (1965). H. J. Juretschke, R. Landauer, and J. A. Swanson, J . Appl. Phys. 27,838 (1956).

(4)

178

C. M. WOLFE AND C. E. STILLMAN

For spherical cavities they find

or The effects of inclusions with finite conductivity in a conducting sample have also been examined. In a manner similar to that used by Juretschke et al.,' Herring' calculated the low magnetic field Hall constants for a medium of conductivity a and mobility p which contains cylindrical inclusions of conductivity go and the same mobility p as the medium. His results for cylindrical inclusions parallel to the magnetic field are RAP, = p(1 + 4f/9)/0 RApp= p(1 - 8f/9)/. RApp = PU(l - 4f)/o

for a0/0= 1/2, for o0/a = 2 , for ao/a = 0 0 .

(10) (1 1) (12)

For cylindrical inclusions perpendicular to the magnetic field, Herring obtains (13) RAPP = (do){' + f[' for any ao/a. The problem of conducting spherical inclusions has been considered by Voronkov et aL6 Their result for the apparent conductivity is 3(a0 a.

+ 20

1

(14)

The low magnetic field Hall constant is determined from Eq. (14) and

Equations (14) and (15) result in PAPP = d1+ 3 f )

for a,/@

= 00,

(16)

and PApp

= p(i

- sfpy(i - 3 j - p )

ror ao/o = 0 .

(17)

This latter result agrees only qualitatively with the nonconducting spherical cavity result of Eq. (9). In many instances inhomogeneities such as those discussed are surrounded by space charge regions which can enhance the effects of the inhomogeneities

' V. V. Voronkov, G. 1. Voronkova, and M . I . Iglitsyn, Fi:. [English Transl.: Sor. Phys.-Semicond. 4, 1949 (1971)l.

Tekh. Poluproc. 4, 2263 (1970)

3. APPARENT MOBILITY ENHANCEMENT

179

by effectively increasing their relative size and thus the value of 5 These effects have been discussed for small disordered regions produced by neutron bombardment by Crawford and Cleland’ and Gossick.* Following their analysis, the effective volume fraction of a sample with spherical inclusions of vdlume fraction f is approximately

*

where is the potential difference between the center of the inclusion and the surrounding medium, E is the dielectric constant, e is the electronic charge, n is the concentration of carriers in the surrounding medium, and A is the cross-sectional area of the inclusion. Carrier scattering from disordered regions has also been investigated by Weisberg,’ Flanagan,” and Dzhandieri and Tsertsvadze.’ For the scattering of electrons in n-type material with N disordered regions which are either p-type or intrinsic, the latter authors obtain

where vo is the volume of a disordered region, v is the volume of the sample (Nvo/v = f ) , N A is the concentration of acceptors in a disordered region, and N D is the concentration of donors in the surrounding medium. All of the models discussed so far have been concerned with the electrical properties of materials in which isolated regions are separated from the surrounding medium by distinct boundaries and in which the inhomogeneities and the medium can have sizeable differences in properties. Small, continuously variable fluctuations in electrical properties have been examined by Brown,” N e d ~ l u h a , ’and ~ Herring.’ For a medium in which the local conductivity, though spatially varying, is isotropic at each point, the effective conductivity is where u is the local conductivity and ((u - (u))’) is the mean square deviation of the local conductivity from the average conductivity ( 0 ) . If

’ J. H. Crawford, Jr. and J . W. Cleland, J . Appl. Phys. 30, 1204 (1959). ’ L. R. Weisberg, J . Appl. Phys. 33, 1817 (1962). * B. R. Gossick. J. Appl. Phys. 30,1214 (1959).

lo

T. M. Flanagan, IEEE Trans. Nucl. Sci. NS15 (6), 42 (1968). M. Sh. Dzhandieri and A. A. Tsertsvadze, Fiz. Tekh. Poluproil. 5,1445 (1971) [English Trans/.: SOC.Phys.-Semicond. 5, 1264 (1972)l. W. F. Brown, Jr., J . Chem. Phys. 23, 1514 (1955). A. Nedoluha, Z . Phys. 148,248 (1957).

*’ l3

180

C. M . WOLFE AND G . E. STILLMAN

the fluctuations are of carrier concentration n only, which is a good approximation for semiconductors, Herring' obtains for the low magnetic field Hall constant RAPP

= (p,/Cc)/e(n) 9

(21)

where pH and p are the Hall and drift mobilities, respectively. Since the apparent mobility is just the apparent Hall constant times the apparent conductivity, small fluctuations in n reduce the effective mobility by the same factor as the effective conductivity. The carrier mobility in materials with one-dimensional periodic distribuet a1.,16 tions of impurities has been analyzed by K ~ r n y u s h i n , ' ~ Vinetskii .'~ and Vinetskii and Kukhtarev." Although we will not reproduce their results, the effect is to reduce the mobility along the direction of the inhomogeneity. c. Macroscopic Inhomogeneities

Materials with macroscopic inhomogeneities cannot be analyzed in any general sense since it is necessary to take into account the boundary conditions at the outer surfaces of the specific measurement sample, including the contacts, as well as the boundary conditions at the inhomogeneitymedium interfaces. For this reason, only relatively simple configurations can be treated analytically without great difficulty. Bate and Beer' * have analyzed a standard Hall measurement sample with a gradient in carrier concentration along the direction of the current flow. To obtain a separable potential equation [see Eq. (56)] for this problem, it is necessary to assume an exponential variation of carrier concentration with distance. In this case the measured Hall constant is just the Hall constant corresponding to the carrier concentration at the Hall voltage contact. The measured resistivity depends on the position of the resistivity voltage contacts in a similar manner. A Hall sample with a planar discontinuity in conductivity and mobility in the direction of current flow has been examined by Bate et a/." They Yu. V. Kornyushin, Fiz. Tekh. Poluproc. 1, 1121 (1967) [English Transl.. SOP.Phys.-Semicond. I , 939 ( 1968)l. l 5 Yu V. Kornyushin, Fiz. Tekh. Polupror. 1, 1214 (1967) [English Transl.: Sot.. Phys.-Semicond. 1, 1066 (1968)l. l 6 V. L. Vinetskii, N. V. Kukhtarev, and A. K. Semenyuk, Fiz. Tekh. Poluprou. 6, 1007 (1972) [English Transl.:SOC.Phys.-Semicond. 6 , 879 (1972)l. " V. L. Vinetskii and N. V. Kukhtarev, Fiz. Tekh. Poluproc. 6, 1029 (1972) [English Transl. : Sor Phys.-Semicond. 6, 896 (1972)l. R. T. Bate and A. C. Beer, J . Appl. Phys. 32, 800 (1961). l 9 R . T. Bate, J. C. Bell, and A. C. Beer, J . Appl. Phys. 32, 806 (1961). l4

3 . APPARENT MOBILITY ENHANCEMENT

181

consider the case where the Hall voltage contacts are at the discontinuity between a region with conductivity and mobility a and p and a region with values a. and p,,. One resistivity contact is in each homogeneous region at equal distances from the boundary. If the discontinuity is one of carrier concentration only, so that the mobilities are equal, Beer2’ has shown that the apparent mobility is

Variations in carrier concentration n and mobility p in the direction of the magnetic field have received a good deal of attention, since this is the configuration most suitable for analyzing samples with significant surface regions, diffused layers, or epitaxial layer variations. Various aspects of the problem for the standard Hall configuration have been considered by Petritz,21Subashiev and Poltinnikov,22T ~ f t eH, l~a ~~ n i kand , ~ ~Kravchenko et af.25and for the van der Pauw configuration by Pavlov.26For arbitrary magnetic field B in the z direction and arbitrary variations of a and p in the z direction, the apparent Hall constant in the standard Hall sample is24

The resistivity in a magnetic field is

where

A. C. Beer, “Galvanomagnetic Effects in Semiconductors,” p. 320. Academic Press, New York, 1963. 2 1 R. L. Petritz, Phys. Rev. 110, 1254 (1958). V. K. Subashiev and A. S. Poltinnikov, Fiz. Tverd. Telu 2, 1169 (1960) [English Trans/.: Sou. Phys.-Solid State 2, 1059 (1960)l. 2 3 0. N. Tufte, J . Eleccrochern. SOC.109,235 (1962). 24 1. Hlasnik, SolidSrare Electron. 8, 461 (1965). A. F. Kravchenko, B. V. Morozov, and E. M. Skok, Fiz. Tekh. Poluprov. 6, 300 (1972) [English Transl. : SOD.Phys.-Semieond. 6,257 (1972)l. 2 b N. I. Pavlov, Fiz. Tekh. Poiuprov. 4, 1918 (1970) [English Trunsl.: Sou. Phys.-Semicond. 4, 1644 (1971)l. 2o

*’

’’

182

C. M. WOLFE AND G. E. STILLMAN

and r is the thickness of the sample. The apparent mobility is then

which is in effect a weighted average for the sample.

2. HIGH-MOBILITY OBSERVATIONS All the inhomogeneity models discussed in the previous section lead to an apparent mobility which is either low or an average for the sample. Although low mobilities are commonly observed experimentally and have been discussed rather extensivelyin the literature, mobility values which appear to be anomalously high have also been reported. In retrospect, it is interesting to find numerous reports of high mobility values in the literature being used to support contentions of higher quality material, when many of the high mobility values appear to be anomalously high. a. Surface Accumulation Layer

To our knowledge the first report of a high measured mobility which was clearly recognized as being anomalous was that by Colman and KendaLz7 The object of their work was to determine the effects of surface preparation on resistivity and Hall measurements on silicon. For this purpose they examined both n- and p-type high-resistivity samples. Their results for the apparent resistivity and the apparent mobility of the n-type silicon sample are shown in Figs. 1 and 2, respectively. The initial measurements were made with an “as-received’’ polished surface. Then, with the contacts intact the sample was given an HF rinse, measured, and successively rinsed in boiling, deionized water for the times shown in the figures, and measured after each rinse. This process was then repeated on the sample after the surfaces were sandblasted. The change in the measured resistivity and mobility produced by the HF rinse is quite striking. The apparent resistivity is decreased by almost two orders of magnitude, while the apparent mobility is increased by a factor of about seven to a value of 10,000 cm’/V-sec (the apparent Hall constant is decreased by slightly over an order of magnitude). Since a room-temperature lattice-limited mobility for silicon of about 1500cm2/V-sec is well established,’* this value is clearly anomalous. (Similar anomalies were observed for the p-type sample with polished surfaces.) In contrast, this behavior was not observed for the same sample after the surfaces were sandblasted. Thus, the anomalously high mobility was attributed to the accumulation

’’ D. Colman and D. L. Kendall, J . Appl. Phys. 40,4662 (1969). M. B. Prince, Phys. Reu. 93, 1204 (1954).

3. I

I

tnlt

183

APPARENT MOBILITY ENHANCEMENT I

I

HF

5r

I

I

I

I

I

I

I

I

I

l

l

I

I

I

I

I

I

+5s

+5r +5r +5r +1m +3Om PROCESSNO STEPS

+58

+58

'20

FIG.I . Effect of surface treatment on the apparent resistivity of n-type silicon (0,polished surface; A, sandblasted surface). (After Colman and Kenda11.27)

layer which is induced on n-type silicon by an HF rinse.29 However, it can be shown from the macroscopic inhomogeneity model appropriate for this configuration [Eqs. (25) and (26)] that a simple continuous accumulation layer extending to the contacts cannot account for these results. b. Metallic Inclusions or Precipitates

High measured mobilities which are clearly anomalous have also been observed in several other materials, including GaAs, PbSnTe, and polycrystalline silicon. A common feature of the compound semiconductors for which high mobilities have been measured appears to be growth under metal-rich conditions or the presence of inclusions or precipitates. For example, mobility values from 9000 to 25,000 cm2/V-sechave been obtained from resistivity and Hall measurements at room temperature on GaAs 29

T. M. Buck and F. S. McKim, J . Electrochem. Soc. 105, 709 (1958).

184

C. M. WOLFE AND G. E. STILLMAN

I

I

lnlt

I

I

HF

l

I

55

l

I

I

I

I

I

I

I

I

I

I

+5r +5r +5s +5r +5a PROCESSING STEPS

I +Sr

I

1

I

I

+lm +30m t

FIG.2. Effect of surface treatment on the apparent mobility of n-type silicon (0, polished surface; A, sandblasted surface). (After Colman and Kenda11.27)

samples grown under Ga-rich conditions. Since a room-temperature, latticelimited mobility for GaAs of about 8000 cm2/V-sec is reasonably well e ~ t a b l i s h e d , ~ ~these . ~ ' values are clearly anomalous. However, the inhomogeneity models appropriate for inclusions or precipitates in a sample as represented by Eqs. (9) and (16)-(18) do not account for these results. To explain these anomalously high mobility observations, a simple macroscopic model''32 which leads to apparent mobility enhancement in inhomogeneous crystals was developed by the authors. The purpose of this chapter is to discuss the inhomogeneity effects which can produce a high apparent mobility value. To achieve this end, the theoretical foundation for analyzing an inhomogeneous material is presented in Part 11; the single H. Ehrenreich, J . Phys. Chem. Solids 8, 130 (1959). D. L. Rode, Phys. Rev. 8 2 , 1012 (1970). 32 C. M. Wolfe. G. E. Stillman, and J. A. R o s i , J. Electrochem. Sor. 119, 250 (1972). 30 3'

3.

APPARENT MOBILITY ENHANCEMENT

185

macroscopic inhomogeneity model is reviewed in Part I11 ; multiple, intermediate-sized inclusion models which produce apparent mobility enhancement are developed in Part IV; and, finally, the applicability and salient features of the models are discussed in Part V. 11. Magnetoconductivity Theory

Before analyzing specific inhomogeneity problems, we first derive the basic transport equations which are appropriate for inhomogeneous isotropic isothermal materials with simple energy band structure. To avoid unnecessary complications in the analyses of specific problems, a number of simplifying assumptions regarding the basic properties of the material are made. Thus, the resulting equations are not applicable to the detailed behavior of a variety of materials in any general sense. The reader is referred to the work of Beer33 for a more general treatment, including the case of anisotropic solids with thermal gradients. Although the appropriate transport equations can also be derived from the equations of motion for the charge carriers in a solid using a method similar to that used by we will use Boltzmann’s equation, since this method tends to emphasize the basic assumptions used in the analysis.

3. BOLTZMANN’S EQUATION a. Basic Considerations

In thermal equilibrium the distribution function for electrons in a crystal is given by the conduction band density of states and the Fermi function for the occupancy of these states. No transport of charge occurs in thermal equilibrium since the probability that an energy state with wave vector k is occupied is the same as that of an energy state with wave vector - k. Under the influence of applied external forces F, the distribution function can be shifted in momentum and position space, and transport of charge can occur. Let f(k, r, t ) be the probability that a state with wave vector k at position vector r is occupied at time t . (Variations off with rare over distances of the order of the lattice spacing.) Under forces F the wave vector k will change at a rate (d/dt)k = h - F , (27) and the position vector r will change at a rate (d/dt)r = Y. 33 34

(28)

A. C. Beer, “Galvanomagnetic Effects in Semiconductors,” p. 18. Academic Press, New York, 1963. H. Brooks, Advan. Electron. Elecrron Phys. 7, 85 (1955).

186

C . M . WOLFE AND G . E. STILLMAN

Thus, an electron which had wave vector k and position vector r at time t will have wave vector k A - ’ F dt and position vector r + v dt at time t dt. The function f is then given by

+

+

k dt, r

+ dtd r d t , t + dt -

and the total rate of change o f f is

d 4 af d - = - + - k grad, f + - r - grad, f dt at dt dt af = - + h - ‘F - grad, f + v * grad, f, at where grad,f denotes the gradient off with respect to k and grad,f denotes the gradient o f f with respect to r.

b. Relaxation Time Approximation Under steady state conditions

aflat

=o

(31)

and the total rate of change in the electron distribution produced by F, which is dfldt, is equal and opposite to the rate of change produced by the relevant electron scattering mechanisms in the crystal, which is dfldtl,, . If the change in distribution produced by the relevant scattering mechanisms is not large compared to its initial value, it is reasonable to set the total derivative equal to the incremental change in f, which is fo - f (since scattering tends to restore f to its equilibrium value f,) divided by the incremental change in time, which is t

- to = 7 .

(32)

Thus,

where 7 is the relaxation time of the electrons to their equilibrium distribution. This process depends on the relevant scattering mechanisms. Although it is well known that this “relaxation time” analysis is not a good approximation for many scattering m e c h a n i ~ r n sit, ~introduces ~ considerable simplification in the distribution function and should be adequate for our purposes. 35

C. Herring, Bell Sys. T d . J . 34,237 (1955).

187

3. APPARENT MOBILITY ENHANCEMENT

Under this assumption, Boltzmann's equation is

f c.

= fo

- zh-'F-grad, f - rv-grad, f .

(34)

Approximate Solution

If we assume the crystal to be homogeneous over distances of the order of a lattice spacing (not microscopically inhomogeneous), grad, f = 0. (35) Then, since the force on an electron in applied electric and magnetic fields is

F = -e(E

+v x

B),

(36)

where e is the magnitude of the electronic charge,

f - fo

= eTh-'[E*grad,f

+ (v X

B)*grad,f].

(37)

To solve Eq. (37) for f, it is necessary to make certain approximations. Since we are primarily concerned with the magnetic field dependence of the transport properties, we will limit the solution to first-order terms in electric field E; that is, we will not be concerned with non-Ohmic effects. This enables us to obtain a more general solution in higher-order terms of B. Under the earlier assumption that the difference between f and fo is not large, we set f = fo in the electric field term of Eq. (37). Then, since

h-' grad, fo = h-'(grad,b) afo/db,

(38)

and

v = h-' grad, 8, where d is the electron energy,

f - fo

= ez(E

- v)(dfo/dd)

+ erh-'(v

(39)

-

x B) grad,

f.

(40)

Assuming a scalar electron effective mass m* and restricting ourselves to a magnetic field parallel to the z direction in a rectangular coordinate system, we obtain the solution

a f o + ero,E, + 1 e2rZB/m* + (erB/m*)2(uyE, - u E ) a6 x

y

afo -.

a8

(41)

Using this expression for f, we can obtain a relation between the current density J and the electric field E.

188

C. M. WOLFE AND G. E. STILLMAN

4. POTENTIAL EQUATION

a. Conductivity Tensor The current density due to electrons in the conduction band is given by J = (-e/4n3)

I

fvdk,

(42)

where the integral is taken over the first Brillouin zone. Substituting f from Eq. (41) into Eq. (42), we find

+

--I

e3B 4n3m*

e2 J r(u,E, vyEy)8fo -V dk 472 I + (ezB/m*)2 d 8 r2(uyEx- v,Ey) d j o v dk 1 + (ezB/m*)2 db

(43)

~

-e2 / r v , E , afo v dk. ~

4n3

a8

The first integral in Eq. (43) vanishes because fo is an even function of k while v is an odd function of k. Then, since the integrals involving the products uxuy, vyu,, and u,v, do not contribute to conduction, we obtain e3BEy

4n3m*

I

1

T2VX2

+ (ezB/m*)2d b 1

+ (ezB/m*)’ af, d b dk]y ?2VY2

where x, y, and z are the unit vectors in the rectangular coordinate system. Since for a conduction band with spherical constant-energy surfaces

s

u2h(S)dk = (2/3m*) bh(b)dk,

(45)

in cases where the relaxation time does not vary with energy, Eq. (44)can be put in the form J = QE, (46) where a is a magnetoconductivity tensor with component matrix given by

0

(47)

3.

APPARENT MOBILITY ENHANCEMENT

189

where (48)

= (ez/rn*)B = p B ,

o = ( - e2r/6n3rn*)f q d f o / a & ' )dk .

(49)

For a general inhomogeneous crystal c and /3 depend on position in the crystal. b. Field Relations

Using the relationship between the current density and the electric field given by Eqs. (46)-(49), a general equation can be obtained for the potential Vat any point in the crystal, in the following manner. Since

E = -grad

(50)

then

J

=

-agradV

Then, in the steady state, to obtain continuity of current, div J

=

(52)

0,

and the differential equation which determines the transport of charge is div(a grad V ) = 0 .

153)

c. Curvilinear Coordinates

To determine the transport properties of samples with macroscopic inhomogeneities, it is necessary to obtain an expression for the potential at all points in the sample. The problem is uniquely determined by a solution of Eq. (53) which satisfies all the bpundary conditions. Since there is usually a coordinate system in which the boundary conditions can be most simply expressed, it is desirable to formulate and solve Eq. (53) in this coordinate system. For this purpose we will represent Eq. (53) in orthogonal curvilinear coordinates, from which the equation in the coordinate systems with which we are concerned can be obtained. In orthogonal curvilinear coordinates 1 av A av I av grad V = u -- + u - _ _ + u - _ _ ' h , du, h, au, h, au, '

'

(54)

where ui are unit vectors, u iare the coordinates, and h iare the metric factors. For rectangular coordinates u1 = x, u2 = y, u3 = z, and h , = h, = h, = 1.

190

C. M. WOLFE A N D G. E. STILLMAN

For cylindrical coordinates u , = r, u2 = q, u, = z, h , = 1, h, = r, and h, = 1. With Eq. (54) and

(56) Since the magnetoconductivity tensor was derived for B in the z direction in a rectangular coordinate system, Eq. (56) is valid only for orthogonal coordinate systems where the magnetic field can be set in the direction of u3. We can now use Eq. (56) to analyze specific inhomogeneousmeasurement configurations. 111. Single Conducting Inhomogeneity

The simplest model which leads to apparent mobility enhancement in inhomogeneous crystals was developed'*32 for a macroscopic conducting region in a measurement sample. For this problem it is necessary to take into account the specific sample geometry and the boundary conditions at the sample perimeter. Thus, to obtain tractable quantitative results, it is necessary to consider the simplest possible geometry.

5. BASICMODEL As shown in Fig. 3, this simple resistivity and Hall configuration consists of a cylindrically symmetric van der P a ~ w measurement ,~ sample of radius a, conductivity a, and mobility p, which has a conducting inhomogeneity in the center of radius b, conductivity g o , and mobility po that extends

throughout the sample thickness t. In general, 6,p, oo, and po are position dependent. The apparent resistivity is obtained by passing a current Il2 through contacts 1 and 2 and determining the voltage V,, induced between contacts 3 and 4 in zero magnetic field. The apparent resistivity is then36

36

L. J. van der Pauw, Philips Res. Repr 13, I (1958).

3. APPARENT

191

MOBILITY ENHANCEMENT

2

3 FIG.3. Cylindrically symmetric van der Pauw measurement configuration with a macroscopic conducting mhomogeneity. The contacts are indicated by 1 4 . (After Wolfe and Stillman.')

4

The apparent Hall constant is obtained by passing a current Z24 through contacts 2 and 4 and determining the voltage V,, induced between contacts 3 and 1 by a magnetic field B. The apparent Hall constant is then

R,,, =

v,,t/l,,B.

(58)

Since it is necessary to simplify the problem as much as is realistically possible to obtain tractable quantitative results, we assume that the contacts have only one dimension in the direction of the sample thickness r. Consider a current 1 through any contact along the perimeter r = a. The current density at the contact is J,(a, cp)

=

l/@aA d ,

(59)

where a Acp is the width of the contact along the perimeter. A one-dimensional contact is obtained by letting A(p approach zero : lim JJa, rp) = lim (I/ta Acp) = (l/ta) d(cp - p0), A(p- 0

(60)

A(p-0

where 6((p - 9,)is the Dirac delta function and ( p o is the angular position of the contact. Thus, the boundary conditions at the perimeter of the sample for the apparent resistivity calculation are

and for the apparent Hall constant calculation,

With this simplification it is possible to obtain an exact solution of Eq. (56) for this simple model.

192

C. M. WOLFE AND G. E. STILLMAN

Obviously, this problem can be most easily analyzed in a cylindrical coordinate system. With these coordinates Eq. (56) for the potential becomes

6. CONDUCTIVITY DISCONTINUITY a. Exact Analysis

If the conductivity and the mobility are assumed to be discontinuous at r = b, having values uo and p, for 0 < r d b and values o and p for b < r < a, then Eq. (63) reduces to 1 azv r2 av2

iav r ar

-+ - - + - -+ ( 1 a2v

ar2

a2v + p 2 ) __ = 0, az2

which is Laplace's equation in cylindrical coordinates for

avlaZ= const.

(65)

Equation (64) is thus separable with solutions of the form t o ( r ,cp) = A,'

for 0


+ r"(Anocos n q + B,'

sin ncp)

(66)

6, and

V,(r,p)= A ,

+ r"(A, cos n p + B, sin np) + r-"(C, cos nv + D, sin n v )

(67)

for b < r < a. At the boundary (b, cp) between the two regions we require that the tangential components of the field be equal,

E,(b, cp) = E,O(b, CD)

9

(68)

and that the radial components of the current density be equal,

J,(hcp)

= Jr0(b,cp).

Substituting Eqs. (47) and (54) into Eq. (51), we obtain

(69)

3.

APPARENT MOBILITY ENHANCEMENT

This gives us for the potential at any point (r, cp) in the outer region b ,< r

193


where

U=-

Is

1

+

(74)

p2’

and

It can be shown that Eq. (71) and the equivalent equation for the potential of the inner region satisfy Eqs. (64) and (65) and boundary Eqs. (68) and (69). Then A,, and B, are determined from the boundary conditions at the outer perimeter of the sample (a, cp). Mathematically, this is classified as an interior Neumann boundary value problem so that V(r, cp) can be uniquely determined to within the constant A , .37 To determine the apparent resistivity in zero magnetic field (P = 0), the terms A, and B, in Eq. (71) are evaluated from the boundary conditions at (a, cp) given by Eq. (61). This results in a potential given by

+ sin

2 sin ncp]

i],

where u = b/a

(77)

1. N. Sneddon, “Elements of Partial Differential Equations,” p. 151. McGraw-Hill, New York, 1957. 37

194

C. M. WOLFE AND G. E. STILLMAN

The potential difference V34is then given by V(a,a) -

v

( 2) a,-

=-

[i - a"y

1 1 + a2"y

act 21 ,=

(

'1.

nx cos - - cos na) 2 n

(78)

Simplifying Eq. (78) and substituting into Eq. (57), the apparent resistivity in zero magnetic field is

We note in passing that W

1 [( - I)"+'/n]

n=

1

= In 2,

so that when there is no inhomogeneity (a = 0 or y = 0),

For the apparent Hall constant determination, the terms A, and B, in Eq. (71) are obtained from the boundary conditions at (a, rp) given by Eq. (62). This results in the potential

+ a'"[) - a2"/?2E(1- a2"q)]sin nrp + [(I - a2"y)(l + bZ(I - a'" 49'1 [(I + where

r =y +

sin(nn/2)

p2&

and ? = y - E .

As before, y and

is then

w, a) - b, 0)

E

(84)

are given by Eqs. (72)-(75). The potential difference V , ,

3.

APPARENT MOBILITY ENHANCEMENT

195

Simplifying Eq. (85) and substituting into Eq. (58), the apparent Hall constant h

Note that

c 2n-

71

(-1Y+1

,,=I

-=-

1

Thus, with no inhomogeneity (a = 0 or y,

4' E,

C, and q = 0)

(The /3 argument is maintained for notational consistency.) The exact expression for the apparent mobility is PAPP('9

8,

= RAPP(a9

p, oO/')/pApp(a~

ao/o),

(89)

where RApp(a,P, o,,/o) and pApp(a,oo/o) are given by Eqs. (86) and (79), respectively. As we will show, this results in

6. detallic Approximation Unfortunately, the exact expressions for the apparent resistivity and Hall constant, Eqs. (79) and (86), are rather cumbersome for purposes of discussion. To gain some physical insight into the problem, these equations can be simplified by assuming that the conductivity of the inhomogeneity oo is much greater than the conductivity of the surrounding medium a. From Eqs. (72), (741, ( 7 9 , and (79) we find that in the limit

196

C. M. WOLFE AND G. E. STILLMAN

From Eqs. (72)-(75), (83), (84), and (86), in the limit,

+

+

(1 p2)[(1 - C14n-2)/(i u4”-’ 1 + B”(1 - u4n-2)/(1 a 4 n - 2)] 2n - 1

+

Using Eqs. (92) and (94), we can obtain a qualitative understanding of the effects of the inhomogeneity on the resistivity and Hall measurements. When there is no inhomogeneity (a = 0) the series in Eq. (92) reduces to In 2, and the apparent resistivity is l/a. The series in Eq. (94) reduces to n/4,and the apparent Hall constant is do. The calculated mobility is then equal to p, the real mobility of the homogeneous sample. In the low magnetic field limit (b’ G 1) both the apparent resistivity and Hall constant decrease as the relative size of the inhomogeneity a increases. However, the resistivity decreases faster than the Hall constant, giving a higher apparent mobility. Physically, this is due to the geometry of the sample, where the current in the Hall configuration tends to spread around the inhornogeneity more than the current in the resistivity configuration. Thus, the resistivity voltage is decreased more than the Hall voltage by the metallic inhomogeneity. For an inhomogeneity of constant size, as the magnetic field is increased, the apparent Hall constant increases. Since the apparent resistivity is measured with no magnetic field, the apparent mobility increases. The apparent Hall constant continues to increase with field until the high-field limit (p’ % 1) is attained. In this regime the apparent Hall constant is equal to p / o , which is just the Hall constant of the medium surrounding the inhomogeneity. Physically, this is a result of the current being distorted out of the inhomogeneity, until in the high-field limit there is no current flow into the inhomogeneity. This is a direct consequence of the interdependence among Eqs. (68)-(70) which was first pointed out by Herring.’ c. Quantitative Results

Quantitative results for the dependence of the apparent resistivity, the apparent Hall constant, and the apparent mobility on the relative extent of the inhomogeneity u, the normalized magnetic field p, and the ratio of the conductivity of the inhomogeneity to the conductivity of the surrounding medium ao/o have been calculated from Eqs. (79), (86), and (89), respectively. Numerical results for the apparent resistivity and Hall constant were obtained by treating the series in Eqs. (79) and (86) as N

m

1 f(4A “,/a, M

n= 1

n )=

1 fb,A

n=l

“Ob,

n)g(n) +

1 s(4 - c s(4 ”= N

W

1

n=l

(95)

3.

APPARENT MOBILITY ENHANCEMENT

197

where N was chosen large enough to ensure that

f(& 8, ao/a, N ) = 1 to the number of significant figures used in the calculation. Numerically, it was determined that an N of 40 easily satisfied Eq. (96) to six significant figures for all practical values of u, p, and a,/o. To this extent these series were evaluated exactly. Since the calculations indicated that the results are dominated by the conductivity discontinuity a,/o rather than the mobility discontinuity po/p, in the results to follow it was assumed that Po = 8 to avoid unnecessary complications. The dependence of the normalized apparent resistivity pAPp(a,a,/o)a on u and a,/a is shown in Fig. 4.In addition to the decrease with increasing u previously discussed, the major decrease for small values of u occurs for values of ao/a between one and ten. Thus, relatively small values of a,/o can significantly affect the resistivity measurement.

FIG.4. Theoretical dependence of the normalized apparent resistivity pAPp(a,a,/a)a on the relative extent of the inhomogeneity a = 6/a and the ratio of the conductivity of the inhomogeneity to the conductivity of the surrounding medium a,/o. (After Wolfe er a/.")

198

C. M. WOLFE AND G . E. STILLMAN

Figure 5 shows the dependence of the normalized apparent Hall constant, RAP,(@,fi, a,/o)a/p on a, fi, and ao/a. Here we see the increase with fl previously discussed. However, these quantitative results show that the apparent Hall constant approximates rather closely the Hall constant of the medium surrounding the inhomogeneity for values of /? as low as about ten for all values of a. Also, for values of a greater than about 0.2 (which corresponds to 0.04 of the sample volume) relatively small values of a,/a have a significant effect on the Hall constant measurement as well as on the resistivity measurement.

0

,B,

P

B

FIG.5. Theoretical dependence of the normalized apparent Hall constant &.(a, 8, u,/u)u/p on the normalized magnetic field 8 = p B and a for several values of uo/u.(After Wolfe e?~ 1 . ~ ' )

The dependence of the normalized apparent mobility pApp(tl, /?,ao/a)/pon fi, and o,/o is shown in Fig. 6. For small values of c1 and fl the apparent mobility increases with increasing a as previously discussed. However, for rather large values of a the apparent mobility begins to decrease. This is because at low magnetic fields the relative shunting effect of the inhomogeneity on the Hall voltage compared to the resistivity voltage begins to decrease. At high magnetic fields, however, the apparent mobility continues to increase with increasing tl since almost all the current is in the medium surrounding the inhomogeneity regardless of the size of the inhomogeneity. M,

3.

199

APPARENT MOBILITY ENHANCEMENT

0

12

3)

PAPP(".B*

r

lo'

loo

162

loo

1 6

B

10'

FIG.6. Theoretical dependence of the normalized apparent mobility pmp(a, b, a,/a)/p on a and /3 for several values.of ao/a.(After Wolfe er ~ 1 . ~ ' )

Also, for constant a rather large increases in apparent mobility are observed for increasing B and o,/a. Thus, depending on the various parameters, Fig. 6 shows that the apparent mobility can be greater than the real mobility by as much as several orders of magnitude. 7. CONDUCTIVITY GRADIENT If the conductivity has a radial dependence only and the mobility is considered uniform, then, using Eq. (as),Eq. (63) reduces to -a2v + - - +i-a-v+ - - 1

ar2

r ar

a2v

r2 d'p2

i d o av o dr ( a r

gav = o r 8'p)

'

(97)

For a radial dependence of the form o = oo(b/r)2'

(98)

so that (110) da/dr

= - 2l/r,

(99)

200

C. M. WOLFE AND G. E. STILLMAN

Equation (97) is separable. The potential at any point (r, cp) in the outer region is V(r, cp) = A , + ,.te-t@v

f

r-~(4n2+

n= 1

x (A,, cos[((4n2 - p’)’/’(p]

1)1/*

+ B,, sin[<(4n2 - p’)”’(p])

(LOO)

for 2n > p and g positive. (An expression for arbitrary 13 would also have terms involving exp[ - (pep f ((p2 - 4n’)’ ”(p] for 2n < #I and exp( - rpcp), cp exp( - (pep) for 2n = 8.)To obtain A,, and B,, from the boundary conditions at (a, cp), it is necessary to find an orthogonal set of solutions to Eq. (97). Such a set constructed from the terms in Eq. (100) becomes very cumbersome after the first few terms, so we make the simplifying assumption that -g 1. Now the terms in Eq. (100) are orthogonal on the interval 0 to 271 with respect to the weighting function exp(e&). Then, to approximate the radial conductivity dependence observed experimentally for the “facet effect” in Czochralski-grown ~ r y s t a l s ,we ~ ~take , ~ ~5 = $ so that

o2

co(b/r). (101) For the apparent resistivity determination, Eq. (100) and the boundary conditions of Eq. (61) give for the potential difference V34 (T

=

(102) Simplifying Eq. (102) and substituting into Eq. (57), the apparent resistivity is

1 a =

(6)

4

+

+

[(16n2 1)”’ - (4n2 1)”’l + l)’/’ - 1][(4n2 + 1)’” - 11

[(16n2

(103) For the apparent Hall constant determination, Eq. (100)and the boundary conditions of Eq. (62) give for the potential difference V , ,

38

39

J. B. Mullin and K. F. Hulme, J . Phys. Chem. Solids 17, 1 (1960). M. D. Banus and H. C. Gatos, J . Electrochem. Soc. 109, 829 (1962)

3.

APPARENT MOBILITY ENHANCEMENT

201

Using the approximations cosh2(fiz/4) x 1 and sinh2(pn/4) x 0 for fi2 -4 1 and simplifying, we substitute Eq. (104) into Eq. (58) to obtain an apparent Hall constant

For the apparent mobility Eqs. (103) and (105) indicate that

’

PAPP/P (106) Thus, it is not necessary to have a discontinuous change in conductivity to obtain apparent mobility enhancement in inhomogeneous crystals. 8. EXPERIMENTAL VERIFICATION

Since the cylindrically symmetric configuration shown in Fig. 3, on which both of the foregoing analyses were based, represents an idealized inhomogeneous measurement sample, several commonly used measurement configurations were investigated experimentally to test the generality of the model. Inhomogeneities were intentionally introduced into thin epitaxial layers of GaAs which were grown on high-resistance substrates. Conducting regions were obtained by alloying tin, which forms a regrown n+ region, into the surface. Metallic inclusions, which could be obtained in growth under Ga-rich conditions, were simulated by alloying gallium into the surface. This was expected to be a reasonable approximation of the theoretical model, since the layers were thin compared to the size of the alloyed inhomogeneities. Also, as is discussed in Section 9b, the current flow becomes more two dimensional with increasing magnetic field. In this manner several van der Pauw measurement configurations, including cloverleaves, crosses, squares, and circles, as well as a rectangular Hall configuration were investigated. The real mobility was determined from measurements before the inhomogeneities were introduced, while the apparent mobility was determined from resistivity and Hall measurements after the inhomogeneities were alloyed into the surface. In all configurations the apparent mobility, after introducing the inhomogeneities, was greater than the real mobility. Some of these results are plotted in Fig. 7, where the ratio of the apparent mobility to the real mobility is given as a function of the relative extent of the inhomogeneity. The parameter a for the experimental samples was estimated from the alloyed area and the measured area. The various symbols in the figure refer to the several different sample configurations. Since the experimental values of fl were between 0.5 and 1.0, the apparent mobility curves calculated for these values from the metallic approximation [Eqs. (92)

2Q2

C.

M. WOLFE AND G . E. STILLMAN

I

0

0.1

I

0.2

I

0.3

I

0.4

I

0.5

I

0.6

I

0.7

I

0.8

I 3

a FIG. 7. Experimental dependence of pApp(a,B)/p on a. Theoretical curves for the metallic approximation bracket the values of /Iapplicable to the experimental data. Different symbols indicate different measurement configurations.(After Wolfe et

and (94)] are also shown. It can be seen that in all configurations apparent mobility enhancement was obtained by introducing the inhomogeneities. Although there is some scatter in the experimental data, most of it is in reasonable agreement with the theoretical model. In Fig. 8 the dependence of the apparent mobility on the normalized magnetic field p is shown for the same measurement configurations as in Fig. 7. Values of p were varied either by changing the magnetic field or by changing the real mobility of the sample (measurements taken at 300 or 77°K). Since the estimated values of c1 for the samples were between 0.4 and 0.5, calculated curves for these values from Eqs. (92) and (94) are also shown. As can be seen from the figure, the apparent mobility for these samples increases with /Iin , reasonable agreement with the model. IV. Multiple Conducting Inhomogeneities

The general features of the macroscopic inhomogeneity model are also expected to be applicable to smaller inhomogeneities. We now consider the effects of many smaller conducting regions on resistivity and Hall measurements in inhomogeneous crystals and discuss models developed4' 40

C. M. Wolfe and G . E. Stillman (unpublished).

3.

APPARENT MOBILITY ENHANCEMENT

a = 0.5

7 01 10-2

I

I

I

lo-’

loo

10’

lo2

I3 FIG.8. Experimental dependence of pApp(a,P)/p on B. Theoretical curves for the metallic approximation bracket the values of a applicable to the experimental data. Different symbols have the same meaning as in Fig. 7. (After Wolfe er ~ 1 . ” )

for isolated inclusionsof intermediate size which have a uniform conductivity different from the conductivity of the surrounding medium. In these analyses we can use the macroscopic concepts developed in Part I1 without having to consider the specific sample geometry. Relatively simple results can be obtained if interactions among inhomogeneities are ignored. Thus, the results discussed are applicable only for inclusions which are sufficiently small and sufficiently dilute to not interact with each other and to not affect the boundary conditions at the outer perimeter of the sample. 9. ISOLATEDCYLINDRICAL INCLUSIONS

a. Two-Dimensional Current Flow

First consider an isolated cylindrical inclusion of radius a placed in a uniform electric field Eo in the x direction with the magnetic field B in the z direction as shown in Fig. 9. It is assumed that the conductivity uo and the mobility po of the inclusion are uniform and that there is an abrupt change from the conductivity u and the mobility p of the surrounding medium at the perimeter of the inclusion. The height of the cylinder is assumed to extend throughout the sample, so that the current flow is two dimensional. At distances far from the inclusion a Hall field BE, is induced

204

C . M. WOLFE AND G . E. STILLMAN

FIG.9. Isolated intermediate-sized cylindrical conducting inhomogeneity in a sample with uniform electric field Eo and magnetic field B at distances removed from the inhomogeneity. (After Wolfe and Stillman.40)

in the material by the applied fields E, and B. Thus, Laplace's equation [Eqs. (64) and (65)] is applicable to this problem and the potential inside the inclusion, V o ,has the form given by Eq. (66),while the potential outside, I/, has the form given by Eq. (67). At the boundary (a, cp) between the two regions the tangential components of the electric field are equal and the normal components of the current density are equal. Thus, the potential in the outer region is given by Eqs. (71)-(75) except that b in Eq. (71) is replaced by a. The potential inside is m

,@'/I

+ C I"{ [ A , - ( ~ , y- B,@)] cos ncp + [B, - (B,y + A,Bc)I sin ncp),

cp) = A ,

I=

1

(107)

where y and E are given by Eqs. (72)-(75). For distances sufficiently removed from the inclusion ( r large), we require that the potential in the outer region be given by V(r,rp)

=

V, - Eor cos cp

+ /3E,r sin cp.

(108)

This ensures that the current density in the y direction is zero and that the current density in the x direction is aE, at distances removed from the

3.

APPARENT MOBILITY ENHANCEMENT

205

inclusion. From this boundary condition we obtain A,=

V,,

A, = - E

B, =BE,,

0’

A, = B, = 0

(109)

for n 2 2. Thus, the potential outside the inclusion is

and the potential inside the inclusion is VO(r, 44 =

v, - E,r(cos

CPNl

- 0+ PE,r(sin

4 w - q),

(111)

where and 9 are given by Eqs. (83) and (84). Equation (1 11) shows that the x component of the field inside the inclusion is uniform and given by

E,O = E,(l - 0, (112) while they component of the field inside the inclusion is also uniform and of magnitude E,O = - BEo( 1 - q) . (113) We now proceed to average over the entire sample using a method suggested by Landau and Lif~hitz.~’ With no magnetic field the relationship between the electric field and the current density is E=pJ, (114) where p is a suitably defined resistivity tensor for the inhomogeneous sample. We then write Eq. (1 14) in the form

where, as before, a is the scalar conductivity of the region outside of the inclusions, which is independent of position. Averaging over the entire volume of the sample v, we obtain

By definition Eq. (1 16) becomes a 41

V

L. D. Landau and E. M. Lifshitz, “Electrodynamicsof Continuous Media,” p. 45. Pergamon, Oxford, 1960.

206

C . M. WOLFE AND G . E. STILLMAN

where (J) and (E) denote average values of J and E over the sample volume. Then, since (p - c-') vanishes everywhere except within the inclusions, where the conductivity is a. , we obtain

where N is the total number of inclusions of volume vo in the sample volume v. (All inclusions are assumed to have the same volume.) From Eq. (1 12) the field (and therefore the current) within the inclusions is uniform and Eq. (1 18) becomes

where the fraction of the sample volume occupied by inclusions is

f

=

( 120)

N(v,/v).

If we assume that the current density in the sample sufficiently far away from an inclusion aEox is equal to the average current density in the sample (J) (this is a zeroth-order approximation, setting (J) = aEox + . . .), then the apparent resistivity (E)/(J) is

In a similar manner, with an applied magnetic field, we have E = pJ + R(J x B)

(123) (124)

a

where R is the Hall tensor for the inhomogeneous sample. Averaging over the sample volume, we find

(E)

=

-51(J)

+ 5-(J P

x B)

+ AS, ( R - ;)(J V

+V

(p - ;)Jdv

x B)(iIl,

where (J x B) is the average value of J x B for the sample. Since (p - a- ') and (R - pa-') vanish everywhere except within the inclusions, where, from Eqs. (1 12) and (1 13), the fields (and therefore the currents) are uniform,

3.

APPARENT MOBILITY ENHANCEMENT

207

and if the discontinuity is one of conductivity only, we obtain

If we assume as before that aE,x is approximately equal to (J) and that

- oE0& is approximately equal to (J x B) ,then

(E) = (J)-

Q

1-2

00

- a)2 - /32(ao - a)2 + 4 2 + p ( o 0 - a)2

If I

a

The apparent Hall constant is thus

In the low magnetic field limit Eq. (128) gives results identical to those of Juretschke et al.’ [Eq. (l)] and Herring’ [Eqs. (10)-(12)]. Defining the limit as o,/a approaches infinity of Eqs. (122) and (128) as pAPP(f)and RApp(f,fl), respectively, we find that 1 PAPP(f) = - 2f) ( 129)

a

9

Qualitatively, it can be seen that Eqs. (129) and (130) exhibit the same behavior as Eqs. (92) and (94), respectively, which were obtained for the single macroscopic conducting inhomogeneity. That is, from Eq. (1 29) the apparent resistivity decreases as the fraction of the sample volume occupied by inclusions f increases. In low magnetic fields, the apparent Hall constant also decreases with increasing f. For a given value o f f , the apparent Hall constant increases with /I. In the high-field limit the apparent Hall constant is equal to p/a, which is the Hall constant of the material surrounding the inclusions. Since the apparent mobility is given by pAPP(f 9

8) = R A P P ( f ,

pAPP(f 9

8) = Pu[(’ + B’) - 4fi/(i+ B2)(l - 2f),

8)/pAPp(f),

(131) (132)

C. M . WOLFE AND G. E. STILLMAN

208

this model for multiple cylindrical inclusions results in apparent mobility enhancement for the inhomogeneous sample for sufficiently large magnetic fields. The apparent mobility, determined from Eq. (132), is plotted as a function of the volume fraction of the sample occupied by cylindrical inclusions for two values of fi in Fig. 10 and as a function of for two values off in Fig. 11 40

3.0

/’

I / 4

%

/

c

/

2.c

/

/

/’

Q

3

/HcO

1c

--. “,p.1

C

I

I

I

I

f

FIG. 10. Theoretical dependence of the normalized apparent mobility pAPp(f.B)/p in the metallic approximation on the volume fraction of the sample occupied by intermediate-sized cylindrical inclusions; inhomogeneitiesffor two values of the normalized magneticfield (--, - _ -, spherical inclusions). (After Wolfe and S t i l l ~ n a n . ~ ~ )

(the case of the spherical inclusions is treated in the next section). In contrast to the single macroscopic inhomogeneity model, this intermediate model predicts that the apparent mobility will be less than the real mobility for values of jl less than one, equal to the real mobility for fi equal to one, and greater ‘than the real mobility only for jl greater than one. However, the quantitative value of this model should be considered questionable because of the assumptions ((J) = cEox, etc.) used in the derivation and because it is not known at what value off the model breaks down. Thus, this analysis is useful only to the extent that it demonstrates the possibility of apparent mobility enhancement in inhomogeneous samples with multiple cylindrical inclusions. b. Three-Dimensional Current Flow

When the height of the cylinders does not extend throughout the sample the current flow is three dimensional and an analysis similar to that given

3. APPARENT

MOBILITY ENHANCEMENT

209

FIG.1 1. Theoretical dependence of pApp(f,/?)/pon /? for two values off. Note that for cylindrical inclusions (solid curves), pApp(f,1) = p and that for spherical inclusions (dashed curves), p A p p ( J $2) = p for any$ (After Wolfe and Stillman.40)

for two-dimensional flow becomes much more complex. The problem of an isolated cylindrical inclusion surrounded on all sides by a material with different electrical properties has been examined by Herring.’ For this analysis Eq. (63) can be used to determine the potential in the sample. Taking the limit of Eq. (63) as j? approaches infinity, we find that (i?/az)(J,) = 0 .

(133)

From Eqs. (94) and (130) we know that in the high-field limit, current does not flow through the surface of the cylindrical inclusion that is along the magnetic field or z direction. Then, from Eq. (133) the resulting distortion of the current must continue for some distance above and below the circular ends of the inclusion in the z direction. As j? approaches infinity, this current distortion should extend throughout the sample. Herring presents a more detailed analysis’ which shows that a strong current distortion exists above and below the ends of the cylindrical inclusion for a distance in the z direction which is of the order of

Az

-

afl/ln /?.

(134)

This shows that the current flow around an isolated cylindrical inclusion becomes more and more two dimensional as fl is increased. Thus, for large

210

C . M. WOLFE A N D G. E. STKLMAN

values of fl the analysis presented for cylindrical inclusions that extend throughout the sample should be valid for cylindrical inclusions that are surrounded by the sample medium. 10. ISOLATED SPHERICAL INCLUSIONS We next examine the problem of isolated spherical inclusions of radius a, conductivity uo ,and mobility po surrounded by a medium of conductivity u and mobility p. When this spherical inclusion is placed in a uniform electric field which is orthogonal to an arbitrarily large magnetic field the analysis is much more complicated than for the equivalent cylindrical inclusion. Equation (47) for the magnetoconductivity tensor and Eq. (56) for the potential are no longer applicable, since the magnetic field cannot be defined along only one direction in a spherical coordinate system. It is therefore necessary to proceed from Eq. (40) and determine a magnetoconductivity tensor for spherical coordinates, where none of the tensor components is zero. Although the resulting equations are rather complex, the analysis proceeds in roughly the same manner as for the isolated cylindrical inclusion discussed earlier. Under the same assumptions used to obtain Eq. (122), we find that for an inhomogeneous sample with a volume fraction of spherical inclusions f the apparent resistivity is given by

If the discontinuity is of conductivity only, the apparent Hall constant in an arbitrarily large magnetic field is

Defining the limit, as ao/a approaches infinity, of Eqs. (135) and (136) to be p A p P Oand RAp,Cf,B), respectively, we find

These equations are entirely analogous to Eqs. (129)and (130)for cylindrical inclusions, differing only in the weight placed on the volume fraction f . The apparent mobility for spherical inclusions, P ~ ~ ~= ( PW ~ J +) b2) - 9

m

+ p2)(1 - 3.0,

( 139)

3.

APPARENT MOBILITY ENHANCEMENT

211

is shown in Figs. 10 and 11, where it can be compared to the values for cylindrical inclusions under the same conditions. As can be seen from these figures, spherical inclusions have a more pronounced effect on the apparent mobility than cylindrical inclusions. That is, for small B the apparent mobility is decreased more and for large fi the apparent mobility is enhanced more by spherical inclusions. This is because the current distortion around spherical inclusions is larger than that around cylindrical inclusions. Again, the quantitative results of this model are questionable, the analysis being useful to the extent that apparent mobility enhancement is demonstrated. 11. QUALITATIVE EXPERIMENTAL VERIFICATION

A sample which exhibits apparent mobility enhancement presumably because of metallic inclusions has been discussed by Wolfe et aL4’ This sample was a high-purity, 20-pm-thick GaAs epitaxial layer which was grown on a (100)-oriented.semiinsulatingGaAs substrate and ultrasonically cut into a cross-shaped van der Pauw measurement configuration. This sample exhibited a measured room-temperature mobility of 15,200 cm’/V-sec, which exceeds recent theoretical estimates of the lattice-limited mobility at this temperature3’ ,43 by a factor of about two. It should be emphasized that this was an accurate experimental value for the mobility and there was no possibility for experimental error of this magnitude. The parts of the epitaxial layer remaining after the cross-shaped sample was cut were also measured. However, they exhibited normal mobility values. a. Inhomogeneity Determination

The sample was first investigated for macroscopic variations in conductivity by observing the photoconductivity as the sample was scanned44 with a 2-mil optical beam. To monitor the entire thickness of the epitaxial layer, the photon energy of the beam was selected to give 20% transmission through the sample.This measurement indicated that, to within the resolution of the beam, the lateral conductivity did not vary by more than 10%. Thus, the apparent mobility enhancement was not caused by a macroscopic variation in conductivity similar to that discussed in Part 111. To examine the sample for inhomogeneities smaller than the 2-mil resolution of the optical probe, infrared microscopy with transmitted light was used. This technique revealed a rather high density of opaque spots which appeared to be precipitates randomly dispersed throughout the epitaxial layer. Although the precipitates were too small to be identified by 42

43 44

C. M. Wolfe, G. E. Stillman, D. L. Spears, D. E. Hill, and F. V. Williams, J . Appl. Phys. 44, 732 (1973). D. L. Rode and S. Knight, Phys. Rev. B 3,2534 (1971). D. L. Spears and R. Bray, J. Appl. Phys. 39, 5093 (1968).

212

C. M. WOLFE AND G. E. STILLMAN

analytical techniques, they were of sufficient size and density to produce a large strain in the sample. The presence of strain was evident in far-infrared photoconductivity measurementsof the shallow donor levels at low temperat ~ r e The . ~ observed ~ spectra clearly exhibited evidence of an anisotropic compressive strain in the epitaxial layer. The nature of the strain was consistent with the presence of gallium inclusions, since gallium would expand upon freezing, leaving the sample in compression. Thus, considering the spherical inclusion model discussed earlier, it seems likely that the. apparent mobility enhancement observed for this sample was caused by gallium precipitates.

b. Electrical Measurements To examine the salient electrical characteristics of this sample, detailed resistivity and Hall data were obtained. The temperature dependence of the Hall constant RH,measured at 5 kG, is shown in Fig. 12. The experimental data are represented by solid circles, and an analysis of the data

4 0 20

10

8

6

4

d4

r

a lo"

Id0 to9 1000/T

(OK)

FIG. 12. Temperature dependence of the Hall constant for the anomalous GaAs sample (solid circles)and a normal GaAs sample (open circles) selected to have similar values for N, and N, from analyses of the data (curves). (After Wolfe et d4') 45

G . E. Stillman, C. M . Wolfe, and J . 0.Dimmock, SolidSrare Commun. 7, 921 (1969).

3.

APPARENT MOBILITY ENHANCEMENT

213

with single donor statistics is given by the corresponding curve. Values for the donor concentration ND, the compensating acceptor concentration N A ,and the thermal activation energy of the donors EDas determined from this analysis are also shown. The open circles and curve marked “normal” are for an epitaxial GaAs layer with normal mobility, which was selected to have values of NDand NAsimilar to those determined for the “anomalous” sample. Although values of N D for the two samples are virtually the same, it can be seen that the values of ED are quite different. The donor concentration dependence of ED is well known for high-purity G ~ A sThis . ~ depend~ ence shows that NDand ED for the normal sample are in good agreement, whereas ND and ED for the anomalous sample do not agree. Apparently, the analysis of the Hall constant for the anomalous sample does not give appropriate values for N,, ED,and, thus, N,. Figure 13 shows the temperature dependence of the resistivity p for these two samples. As shown, p for the anomalous sample is approximately a factor T 40 20

10-2

0

I 50

10

8

I 100

(OK)

6

I 150

4 I

I

200

I 250

30

i000/T (OK)

FIG.13. Temperature dependence of the resistiviiy for the anomalous and normal GaAs samples. (After Wolfe er ~ 1 . ~ ’ ) 4b

C. M. Wolfe and G. E. Stillman, Proc. 3rdInt. Symp. GuAs, Aachen, 1970, p. 3. Inst. Physics, London, 1971.

214

C. M. WOLFE AND G . E. STILLMAN

of two lower than p for the normal sample over the temperature range from 300 to about 20"K, even though the values of RH from Fig. 12 are reasonably close for the two samples over the same temperature range. However, if RH for the anomalous sample is approximately equal to the Hall constant of the homogeneous part of the sample (as we will show later), then this difference in p between the two samples is in qualitative agreement with the high predictions of the inclusion models discussed in Sections 9 and 10. The temperature dependence of the mobility for the two samples as determined from R,/p is shown in Fig. 14. As can be seen, the anomalous sample has a higher apparent mobility than the normal sample over the entire temperature range. Also, the normal sample has a change in the temperature dependence of the mobility in the range from about 20 to 10"K, whereas the anomalous sample does not. This change in temperature dependence, which is determined by the ratio of ionized acceptors N A to ionized donors ND in the sample, arises because ionized-impurity scattering is dominant in GaAs at these temperatures. That is, if N A is substantially less than N , , there will be a significant change in the number.of ionized impurities, and thus in the mobility, as the ionized donors are neutralized by carrier freezeout. Since the anomalous sample shows no such change in

t

loo

I

I

I I l l l l l

I

I

I I I Ill1

lo'

102

I

I

I I l l l j

ic

T ("K)

FIG. 14.Temperaturedependence of the mobility as determined from R,/p for the anomalous and normal GaAs samples. (After Wolfe er a / . 4 2 )

3.

APPARENT MOBILITY ENHANCEMENT

215

temperature dependence, the number of ionized acceptors and donors must be approximately the same in this sample, even though the analysis of the Hall data indicates values similar to the normal sample. Thus, the multiple inclusion models do not predict all the detailed features of this anomalous sample of GaAs. However, the apparent mobility enhancement resulting from a lower apparent resistivity is in qualitative agreement with the analyses for multiple conducting inhomogeneities. V. Conclusions 12. APPLICABILITY The inhomogeneity models developed and discussed in this chapter should be applicable, at least qualitatively, to a number of commonly encountered inhomogeneities in a variety of materials. Although the single macroscopic conducting inhomogeneity model discussed in Part I11 was derived for samples with circular symmetry, qualitatively similar results should be directly applicable to arbitrarily shaped samples with extended doping variations. Such variations can easily lead to measurementsof high apparent mobility. One obvious and commonly observed impurity variation to which this model directly applies is the so-called “facet effect”38939 which is observed in Czochralski-grown crystals. For cylindrical measurement samples cut normal to the growth direction quantitative comparisons with the model can be made. The results of this model may also be valid for polycrystalline samples with large grains, since crystallites of different orientation can incorporate impurities at different rates during the growth process.47The apparent mobility enhancement obtained from this macroscopic model may also explain the accumulation layer results of Colman and KendallZ7on silicon, which were discussed in Section 2a. The results of Herring,’ which were outlined in Section 9b, would enhance the current distortion due to the accumulation layer. However, to obtain apparent mobility enhancement for this sample, it is necessary that the Hall and current probes not be in direct contact with the accumulation layer, otherwise the layer model discussed in Section l c would apply. From the experimental procedure used, it is possible that the accumulation layer did not extend to the contacts. The multiple intermediate-size conducting inhomogeneity models developed in Part IV are independent of sample geometry and should be applicable to arbitrarily shaped samples with random doping fluctuations and to arbitrarily shaped samples with metallic inclusions or precipitates. 47

R. N. Hall, J. Phys. Chem. 57, 836 (1953).

216

C. M. WOLFE AND G. E. STTLLMAN

Fine-grained polycrystalline samples could exhibit apparent mobility enhancement for the same reason indicated for large-grained samples or due to preferential accumulation of dopant at the grain boundaries. High apparent mobility may be most commonly observed in compound semiconductors which are grown under metal-rich conditions resulting in inclusion or precipitate formation. Results for one such sample were presented in Section 11. The cylindrical model should be directly applicable to samples that have precipitated accumulations of dopant or excess metal atoms around dislocations, while the spherical model should apply to samples with randomly dispersed precipitates or inclusions. Apparent mobility enhancement can occur in samples with any of these inhomogeneity types.

13. CHARACTERISTIC FEATURES Since it has been established that high apparent mobility measurements can be obtained as a result of both macroscopic and intermediate-sized conducting inhomogeneitiesin a crystal, it is apparent that the homogeneity of the material must be positively determined before the mobility can be used as a figure of merit for the material. Although there are a number of techniques for detecting inhomogeneities,4* they may not yield positive results for a particular inhomogeneity. Materials with conducting inhomogeneities, for which the models discussed in this chapter apply, are expected to have several characteristic features. First, the observation of obvious apparent mobility enhancement may not be very reproducible. That is, only an occasional sample may exhibit enhancement to such a dramatic extent that it becomes obvious. Other samples may appear to have somewhat better than average mobility values. Resistivity and Hall measurements for samples with conducting inhomogeneities may indicate less compensation than is actually present. For measurements taken at low magnetic fields, this can arise because the apparent carrier concentration deduced from the Hall constant is higher than the real carrier concentration. However, an apparently low compensation is also indicated for measurements in high magnetic fields (even though the apparent carrier concentration approaches the real concentration) because in this case the apparent mobility is higher. Mobility enhancement in pure samples with conducting inhomogeneities may result in apparent mobility values which exceed well-established theoretical lattice scattering limited values. This was observed in the results of Colman and Kendall” for silicon, where a room-temperature mobility of 10,OOO cm2/V-sec was measured, and in the results of Wolfe et ul.32,42for 48

R. T. Bate, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 4, p. 459. Academic Press, New York, 1968.

3. APPARENT

MOBILITY ENHANCEMENT

217

GaAs, where a room-temperature mobility of 24,000cm2/V-secwas measured for a sample with intentionally introduced inhomogeneities and 15,200 cm2/V-secwas measured for a naturally occurring inhomogeneous sample. One method for determining the presence or absence of conducting inhomogeneities is to examine the magnetic field dependence of the Hall constant. If the sample is homogeneous, the Hall constant should vary in magnetic field with the expected dependence of the Hall coefficient or scattering factor. This dependence can be estimated or calculated from the Hall coefficientfactors for the appropriate scattering mechanism^.^^'^^ Conducting inhomogeneities in a sample are expected to cause the Hall constant to increase with magnetic field which may reduce or eliminate the expected variation due to the Hall coefficient factor. Thus, if the magnetic field dependence of the Hall constant is anomalous, the electrical properties of the sample are probably dominated by inhomogeneities. This effect is quite pronounced for the intentionally introduced conducting inhomogeneities discussed in Section 8. The magnetic field dependence of the normalized Hall constant is shown in Fig. 15 for a GaAs sample before and after an inhomogeneity was introduced. The upper curve shows the variation expected due to the magnetic field dependence of the Hall coefficient factor4’ and the open circles are the experimental data points for the homogeneous sample. The lower solid circles show the experimental dependence obtained after an inhomogeneity was introduced. The lower curves indicate the dependence [Eq. (94)] expected from the macroscopic conducting inhomogeneity model developed in Part 111. This behavior is not quite as pronounced for samples with intermediatesized conducting regions as for macroscopic conducting regions. Figure 16 shows the magnetic field dependence of the Hall factor neR, for the normal and anomalous samples discussed in Section 11. The carrier concentration n for both samples was obtained from the highest magnetic field measurement of R,. The dependence for the normal sample is in agreement with the expected variation of the Hall coefficient factor. The lower curve shows the qualitative dependence expected from a conducting inhomogeneity model. Although neR, for the anomalous sample does not increase with magnetic field as expected from the model, it does demonstrate unusual behavior. This behavior apparently reflects a combination of Hall factor and inhomogeneity effects, and is probably typical of the type of behavior to be expected for samples with inclusions or precipitates. We can also note that the carrier concentration for the anomalous sample as determined from l/eR, at low magnetic field is fairly accurate. 49

S. S. Devlin, in “Physics and Chemistry of 11-VI Compounds” (M. Aven and J. S. Prener, eds.), p. 558. North-Holland Publ., Amsterdam, 1967.

218

C. M. WOLFE AND G. E. STILLMAN

0.2 10-2

100

10-1

101

10

B

FIG.15. Theoretical and experimental dependence of R,,(a, b)c/fi on b for a homogeneous sample (upper curve and open circles) and a sample with a macroscopic intentionally introduced inhomogeneity (lower curves and solid circles) for a between 0.4 and 0.5. (After Wolfe et d3’) 1.20 1.i51.10-

r

[L, c

1.05-

1.00---

0.950.90

/

Model I I I I I III

loo

I

I I IIlllI 10’

I

I1111l11

io2

I

I

I I1

B (kG) FIG.16. Experimental dependence of the Hall factor determined from neR, on magnetic field B for the anomalous and normal GaAs samples at 295°K.The lower curve is the approximate dependence expected from a conducting inhomogeneity model. (After Wolfe et

3.

APPARENT MOBILITY ENHANCEMENT

219

VI. Summary Previous analyses of the effects of various types of inhomogeneity have shown that the mobility of inhomogeneous materials should be either low or some average value due to either carrier scattering or the averaging inherent in resistivity and Hall measurements. Thus, observations of anomalously high measured mobilities are not explained by these treatments. To explain these high values, we have developed several models' ,32*40 for inhomogeneous measurement samples which lead to a high apparent mobility. These models are discussed in this chapter and the appropriate equations for the apparent Hall constant and resistivity in the metallic approximation are summarized in Table I. The analysis of a single, macroTABLE. I

SUMMARY OF INHOMOGENEITY MODELS

(o,la

-+

co) ~

Model

Equation

~~

Eo. No.

(exact)

Multiple intermediate cylindrical inclusions (approximate)

Multiple intermediate spherical inclusions (approximate)

h p p ( f >

(L - $)

8) =

8) =

R ~ p p ( f 3

PAPP(f)

=

1

;(I

- 3f)

scopic conducting inhomogeneity yields quantitative results for apparent mobility enhancement in a crystal. These results are directly applicable to samples with extended doping variations such as those observed for the "facet effect" in Czochralski-grown crystals and those obtained in largegrained polycrystals.The analyses of multiple, intermediate-sized conducting inhomogeneities give similar mobility enhancement results. These results for cylindrical and spherical inhomogeneities are directly applicable to crystals with dopant or excess metal inclusions or precipitates and smallgrained polycrystals.

220

C. M. WOLFE AND G . E. STILLMAN

Aside from the strictly geometric effect discussed in Section 6b, the features of these models and of real inhomogeneous materials which are necessary for the observation of apparent mobility enhancement are as follows: (1) The inhomogeneities must have a higher conductivity than the surrounding medium. (2) A single macroscopic inhomogeneity must not extend to a current or Hall probe. That is, the inhomogeneity must not be directly contacted. (3) Multiple macroscopic or intermediate-sized inhomogeneities may extend to a current or Hall probe, provided a sufficient number are surrounded by a medium of lower conductivity. Under these conditions, apparent mobility enhancement can be explained qualitatively in the following manner: (1) In the low magnetic field limit the Hall constant measurement gives an average value for the inhomogeneities and the surrounding medium. Since the conductivity of the inhomogeneities is higher than the conductivity of the surrounding medium (assuming the conductivity difference is larger than the mobility difference), the measured Hall constant will be lower than the Hall constant of the surrounding medium. (2) As the magnetic field is increased (to some value commonly used for Hall measurements), the current flowing into the inhomogeneities decreases because of the magnetic-field-induced interactions between the boundary conditions [Eqs. (68) and (69)] at the inhomogeneitymedium interface. Thus, the Hall constant measurement tends to preferentially measure the Hall constant of the medium surrounding the inhomogeneities, and the measured Hall constant is larger than the Hall constant measured in the low magnetic field limit. (3) In the high magnetic field limit there is no current flow into the inhomogeneities, and the measured Hall constant is that of the medium surrounding the inhomogeneities. (4) The resistivity is measured with no applied magnetic field, so the resistivity measurement gives an average value for the inhomogeneities and the surrounding medium. Since the conductivity of the inhomogeneities is higher than the conductivity of the surrounding medium, the measured resistivity will be lower than the resistivity of the surrounding medium. (5)The mobility as determined from this measured resistivity and the Hall constant measured in (2) or (3) will be higher than the mobility of the medium surrounding the inhomogeneities. It is in this manner that the inhomogeneity models indicate that a high apparent mobility can be obtained and the essential features of apparent mobility enhancement predicted by these models have been verified experimentally. Thus, a high mobility cannot be used as an indication of the quality of a crystal without an unambiguous determination of homogeneity. ACKNOWLEDGMENT We would like to express our deep appreciation to Margaret R. Southard for an energetic and meticulous preparation of the manuscript.

CHAPTER 4

The Magnetophonon Effect Robert L. Peterson I. INTRODUCTION

. . . . . . . .

.

. . . . . . . 22 I

. . 11. HISTORICAL OVERVIEW A N D PHYSICAL DISCUSSION 111. OHMIC REGIME . . . . . . . . . . . . 1. General Theoretical Considerations . . . . . . 2. Early Theoretical Results. . . . . . . 3. Theory of Transverse Magnetoresistance . . 4. Theory of Longitudinal Magnetoresistance . . 5. Theory of Magneto, Hall, and Seebeck Eflects . 6. Experimental Results . . . . . . . . IV. HOT-ELECTRON REGIME. . . . . . . . . 7. Theory . . . . . . . . . . . . 8. Heating by Electric Fields . . . . . . 9. Heating by Irradiation . . . . . . . v. EFFECTSOF STRESS . . . . . . . . . VI. FINALREMARKS. . . . . . . . . . APPENDIX . . . . . . . . . . . .

. . . . . . . .

. . . . . . . . . .

. . . .

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

.

.

.

. . . .

. _ . .

. . _ . . . . . . . . .

224 231 . . 23 1 . . 235 . . 237 . . 242 . . 247 . . 249 . . 273 . . 213 . . 278 . . 284 . . 285 . . 287 . . 288

I. Introduction As with most scientific phenomena, the magnetophonon effect could have been predicted or observed many years before its actual discovery. Experiments showing the characteristic oscillatory structure of the magnetoresistance of certain semiconductors can be made simple enough that some have been performed in undergraduate laboratories.' Transport theory general enough to include quantum mechanical effects in crossed electric and magnetic fields had been worked out in the 195O's, particularly in the work of Kubo and co-workers,24 Adams and H o l ~ t e i nand , ~ Argyres and Roth.6 And of course the Boltzmann equation, which can be used for

' R. A. Stradling, J . Phys. E. Sci. Instrum. 5, 736 (1972).

R. Kubo, J. Phys. SOC.Japan 12, 570 (1957). R. Kubo, H. Hasegawa, and N. Hashitsume, Phys. Rev. Lett. I, 279 (1958); J . Phys. SOC. Japan 14, 56 (1959). R. Kubo, S. J. Miyake, and N. Hashitsume, Solid State Phys. 17,269 (1965). E . N. Adams and T. D. Holstein, J. P h p . Chon. Solids 10, 254 (1959). P. N. Argyres and L. M. Roth, J. Phys. Chem. Solids 12.89 (1959).

221

222

ROBERT L. PETERSON

parallel fields, has been around for a very long time. But not until the 1961 paper of Gurevich and Firsov’ was it pointed out that when the electron scattering is inelastic, as for example on optical phonons, will there be resonant interactions with the quantized orbital energy of the electrons in a magnetic field B. These resonanceswould be expected to give rise to an oscillatory structure of transport properties, nearly periodic in l / B . Figure 1 shows an example of such oscillations in the transverse magnetoresistance of n-InSb.

I

I

I

I

I

I

1

0 I / B(i‘)

FIG. 1. Transverse magnetoresistance of n-InSb at 77°K ( n = 8.9 x 10’’ showing the periodic structure of magnetophonon oscillations. The vertical scale is proportional to the second derivative of the resistance. (After S e i l e ~ ~ ’ )

The quantization of the electron orbital motion into Landau levels produces a large density-of-states region near the bottom of each level, so that there is a high probability per unit time for scattering of a carrier into these regions. The magnetophonon resonances predicted by Gurevich and Firsov’ occur when the optic phonon energy matches the separation between any two Landau levels of the electrons (or holes). For parabolic bands, this resonance condition is N w ~ = w ~ ,N = l , 2 ,...,

(1)

where the cyclotron frequency o,is given by o,= eB/m*,

’ V. L. Gurevich and Yu. A. Firsov, Zh. Eksp. Teor. Fiz. 40, 198 (1961) Phys.-JETP 13, 137 (1961)l.

(2) [English Transl.: Sou.

4.

THE MAGNETOPHONON EFFECT

223

m* is the carrier effective mass, and o, is an optical phonon frequency. Spin splitting of the Landau levels will lead to a doublet structure when nonparabolicity of the bands is important, and has been resolved in some materials. Optical-phonon-induced transitions between spin-up and spindown states have not yet been shown definitely to play a role in the magnetophonon effect, although some observed structure is possibly caused by such transitions. The magnetophonon oscillations are to be distinguished from ShubnikovdeHaas oscillations (see, e.g., the review articles of Roth and Argyres,* Kahn and Frederikse,’ and Landwehr”), which are also nearly periodic in 1/B. The latter are characteristic of the degenerate regime, and occur whenever the bottom edge of a Landau level passes through the Fermi level. The two phenomena are readily distinguishable experimentally because, in addition to the fact that they occur mostly at different temperatures and carrier concentrations, the ShubnikovAeHaas period depends upon carrier concentration, whereas the magnetophonon oscillations do not (except indirectly; e.g., the effective mass depends to a small degree on carrier concentration’ ’). Transverse and longitudinal magnetoresistance are by far the two most investigated properties in which the magnetophonon effect is manifested. Studies of the field positions of the extrema in the transverse configuration have allowed an accurate determination of the band-edge effective mass and its dependence on temperature and pressure. The pressure studies have also permitted determination of various deformation potentials associated with the conduction and valence bands, and have provided confirmation of theories of the band structure. The hot-electron regime is rich in structure; some is due to the usual transitions between Landau levels; other structure has been correlated with transitions to impurity levels and with transitions involving zone-edge phonons ; still other structure has yet to be identified. Thus, the magnetophonon effect has proved to be a useful and exciting tool for studying the properties of semiconductors. In Part 11, a very brief historical overview of the subject, together with some physical discussion, is presented. Part I11 reviews the theoretical work and experimental observations in the Ohmic regime. The experimental work is grouped into subsections by material. The hot-electron regime is L. M . Roth and P. N. Argyres, in ‘Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol,. 1, p. 159. Academic Press. New York, 1966. A. H. Kahn and H. P. R. Frederikse, Solid SIale Pkys. 9,257 (1959). l o G . Landwehr, in “Physics of Solids in Intense Magnetic Fields” (E. D. Haidemenakis. ed.), p. 415. Plenum, New York, 1969. ” E. D. Palik and G . B. Wright, in “Semiconductors and Semimetals” (R..K. Willardson and A. C. Beer, eds.), Vol. 3, p. 421. Academic Press, New York, 1967.

224

ROBERT L. PETERSON

covered in Part IV, stress work in Part V, and concluding remarks are given in Part VI. The experimental studies discussed in Parts IV and V are also referenced under the materials headings of Part 111, so that all experimental work on a given material can be found under these headings. All papers published through 1973 on the magnetophonon effect (any omission is unintentional) are referenced and discussed. Other reviews on various aspects of the magnetophonon effect can be found in the work of Stradling12*13and Harper et 11. Historical Overview and Physical Discussion

Shortly after the first theoretical papers of Gurevich and Firsov (GF),’,15 predicting magnetophonon oscillations in the Ohmic transverse magnetoresistance, two groups independently reported seeing such oscillations in n-InSb, in the longitudinal as well as the transverse magnetoresistance, and also in the longitudinal magnetothermal emf (magneto-Seebeck effect). 6-22 Theoretical analyses were then published for the longitudinal proper tie^.^^.^^ The few experimental results to date on the longitudinal magnetothermal emf are not in good agreement with the early theory.24

‘’ R. A. Stradling, Proc. Summer School Semicond., McGill Unic. Wolters-Noordhoff-Holland, 1972.

l3

R. A. Stradling, Proc. X I Inr. Conf. Phys. Semicond., Warsaw, p. 261. Polish Scj. Publ.,

Warsaw, 1972. P. G. Harper, J. W. Hodby, and R. A. Stradling, Rep. Progr. Phys. 36, l(1973). I s Yu. A. Firsov and V. L. Gurevich, Zh. Eksp. Teor. Fiz. 41, 512 (1961) [English Transl.:Soo. Phys.-JETP 14, 367 (1962)l. l 6 S . M. Puri and T. H. Geballe, Bull. Amer. Phys. SOC.8, 309 (1963). S. S. Shalyt, R. V. Parfen’ev, and V. M. Muzhdaba, Fiz. Tuerd. Tela 6, 647 (1964) [English Transl. :Soo. Phys.-Solid Srate 6, 508 (1964)l. V. M. Muzhdaba, R. V. Parfen’ev, and S. S. Shalyt, Fiz. Tcerd. Tela 6 , 3194 (1964) [English Transl. :Sou.Phys.-Solid State 6 , 2554 (1965)l. l 9 R. V. Parfen’ev, S . S. Shalyt, and V. M. Muzhdaba, Zh. Eksp. Teor. Fiz. 47, 444 (1964) [English Transl.:Sov.Phys.-JETP20, 1131 (1965)l. S. S. Shalyt, R. V. Parfen’ev, and M. V. Aleksandrova, Zh, Eksp. Teor. Fiz. 47, 1683 (1964) [English Transl.:Sou. Phys.-JETP 20, 1131 (1965)l. Yu. A. Firsov, V. L. Gurevich, R. V. Parfen’ev, and S. S. Shalyt, Phys. Reo. Lett. 12, 660 (1964). 2 2 V. L. Gurevich, Yu. A. Firsov, R. V. Parfen’ev, and S. S. Shalyt, Proc. 7th Int. Conf. Phys. Semicond., Paris, p. 653. Academic Press, New York, 1964. 2 3 V. L. Gurevich and Yu. A. Firsov, Zh. Eksp. Teor. Fiz. 47, 134 (1964) [English Transl. : SOC. Phys.-JETP 20, 489 (1965)l. 24 S. T. Pavlov and Yu. A. Firsov, Fiz. Tverd. Tela 6, 3608 (1964) [English Trans. :So?. Phys.Solid Srate 6, 2887 (196511. ” I. M. Tsidil’kovskii and M. M. Aksel’rod, Proc. 81h Int. Con$ Phys. Semicond., Kyoto, J . Phys. Soe. Japan Suppl. 21, 362 (1966). 2 6 N. T. Sherwood and W. M. Becker, Phys. Lett. 27A, 161 (1968). I4

4.

THE MAGNETOPHONON EFFECT

225

Additional experimental r e s ~ l t s on ~ ~ longitudinal ,~~ magnetoresistance, consistently showing minima near the GF resonance fields instead of the maxima23 which were predicted for most cases, made clear the need for further theoretical work there also. Subsequent observations at high temp e r a t u r e ~ , ~ ~which - ~ ’ showed the development of additional minima at a, = 2~11,and between the minima at the GF fields, further demonstrated this. Theories incorporating transitions between spin states with emission or absorption of an optical phonon were d e ~ e l o p e d ~to~ explain * ~ ~ - the ~~ new longitudinal minima, but were generally unsuccessful in that the implied g-factors were not very close to values obtained by other methods. Later, it was noticed that the extra minima together.with the minima near the GF fields lay approximately at fields given by No, = 2a0,34 and this k d to the notion that two optical phonons were i n ~ o l v e d , ~ through ~ . ~ ’ some form of second-order process. It was later shown36 that the extra structure is a natural consequence of first-order optical phonon scattering. The theoretical analysis also led to a clearer understanding of the structure at - ~ ’ the discussion at the GF fields in the longitudinal c ~ n f i g u r a t i o n ~ ~(see end of this section). These results were achieved by investigating nonpolar optical phonon scattering, which is considerably simpler than the polar case considered by the early workers. Even though the polar case has not been worked through in the same mathematical detail, it is clear physically

’’ I. M. Tsidil’kovskii,

M. M. Aksel’rod and V. I. Sokolov, Fiz. Tverd. Tela 7, 316 (1965) [English Transl. :Sou. Phys.-Solid State 7 , 253 (19631. 2 8 D. V. Mashovets, R:V. Parfen’ev, and S. S. Shalyt, Zh. Eksp. Teor. Fiz. 47, 2007 (1964) [English Transl.:Sou. Phys.-JETP 20, 1348 (1965)l. z 9 M. M. Aksel’rod, V. J. Sokolov, and 1. M. Tsidil’kovskii, Phys. Status Solidi8, K15 (1965); 9, K91 (1965). 30 D. V. Mashovets, R. V. Parfen’ev, and S. S. Shalyt, Zh. Eksp. Teor. Fiz. Pis’m Red. 1 (No. 3), 2 (1965) [English Transl.:Sou. Phys.-JETP Lett. 1, 77 (1965)l. 3 1 I. M. Tsidil’kovskii, M. M. Aksel’rod, and S. 1. Uritsky, Phys. Status Solidi 12, 667 (1965). 32 S. T . Pavlov and Yu. A. Firsov, Fiz. Tuerd. Tela 7,2634 (1965) [English Transl.:SOU.Phys‘.Solid State 7,2131 (1966)l; Zh. Eksp. Teor. Fir. 49, 1664 (1965) [English Transl.:Sac. Phys.JETP22, 1137 (1966)l. 3 3 S. T. Pavlov and Yu. A. Firsov, Fiz. Tuerd. Tela 9, 1780 (1967) [English Transl.:Sou. Phys.Solid Stare 9, 1394 ( 1967)l. 34 R. A. Stradling and R. A. Wood, J . Phys. C . Solid Stare Phys. 1, 171 1 (1968). 35 M. M. Aksel’rod and I. M. Tsidil’kovskii, Zh. Eksp. Teor. Fiz. Pis’rna Red. 9, 622 (1969) [English Transl.:Sou. Phys.-JETP Lett. 9, 381 (1969)]. 36 R. L. Peterson, Phys. Rev. Lett. 28,431 (1972); Bull. Amer. Phys. Sor. 17,281 (1972). 37 R. L. Peterson, Phys. Rev. B 6 , 3756 (1972). 38 R. L. Peterson, Phys. Rev. B 7 , 5405 (1973). 39 G . I. Kharus and I. M. Tsidil’kovskii, Fir. Tekh. Poluprov. 5,603 (1971) [English Transl. :Sou. Phys.-Semicond. 5, 534 (1971)l.

226

ROBERT L. PETERSON

that the longitudinal structure exists for the same reason as in the nonpolar materials. The early experimental work used “dc” techniques; that is, the quasistatic voltage drop across the sample was measured in a quasi-static magnetic field, produced either by electromagnets or by pulsed systems. Typically, the results would be in the form of small oscillations superposed on a rapidly increasing monotonic background, which made accurate measurement of extremal positions rather uncertain. Accuracy was later improved by bucking out most of the monotonic background with use of a Hall probe. More recently, derivative techniques have been incorporated which have greatly improved the sensitivity. Two methods are in use. In one, the second derivative of the resistance with respect to time is measured with the use of two RC circuits, while the magnetic field is swept linearly with time’’40; the measurement is thus that of the second derivativewith respect to magnetic field. In the second method, the magnetic field or the current through the sample is modulated, and phase-sensitive detection is used. First, second, and third derivative measurements have been reported. Magnetophonon oscillations in the non-Ohmic regime were reported as early as 1966,41and this area has now been investigated aimost as extensively as the Ohmic regime. At very low temperatures, where no oscillations can be seen in the Ohmic regime because of the paucity of thermal optical phonons, oscillations can be brought out by application of a larger electric field (a good survey of the British work is given by Stradling et d.14*42), or by irradiation with energetic photons (e.g., Parfen’ev et either of which makes some of the carriers energetic enough to emit optical phonons. Typically, the extrema do not match those of the Ohmic region. Some extremal series can be correlated with transitions to an impurity level rather than to the lowest Landau Other series have been identified with the simultaneous emission of two zone-boundary transverse acoustic p h o n o n ~ . ~ ~ Still other structure has yet to be identified. The hot-electron region thus can provide much information, and has only begun to be exploited. Another area of importance is stress work, in which the variation of magnetophonon peak positions, and hence effective mass, with stress is 40 41

R. A. Stradling and R. A. Wood, J . Phys. C Solid State Phys. 3, 2425 (1970). N. Kotera, K. F. Komatsubara, and E. Yamada, Proc. V l l l In[. Conf: Phys. Semicond.,

Kyoto, J. Phys. SOC.Japan Suppl. 21,411 (1966). R. A. Stradling, L. Eaves, R. A. Hoult, A. L. Mears, and R. A. Wood, Proc. X Int. Con/ Phys. Semicond., Cambridge 1970, p. 816. U.S. Atomic Energy Commission, Oak Ridge, Tennessee. 4 3 R. V. Parfen’ev, 1 . I. Farbshtein, and S. S. Shalyt, Zh. Eksp. Teor. Fir. Pis’ma Red. 5, 253 (1967) [English Transl.: Sou. Phys.-JETP Lett. 5, 203 (1967)l; Zh. Eksp. Teor. Fiz. 53, 1571 (1967) [English Transl.: Sou. Phys.-JETP 26,906 (1968)l.

42

4. THE

MAG~~ETOPHONON EFFECT

227

Besides confirming aspects of the KaneSo and Bir and Pikus' theories regarding band structure, this technique has shown itself capable of yielding values of several deformation-potential parameter~.~~.~~ To date, the magnetophonon effect has been seen in some dozen materials, and the list is growing steadily. Table I lists these materials and the properties in which the magnetophonon effect has been seen. We turn now to a qualitative, physical discussion of the reason for magnetophonon structure in the Ohmic magnetoresistance. The explanation is quite different for the transverse and longitudinal cases. In the transverse configuration, the conductivity is essentially a k, integral over transition probabilities (Part 111, Section 3), where hk, is the carrier quasimomentum in the direction of B. Since the transition probabilities are usually additive, the transverse conductivity is obtained by summing the contribution to the conductivity from each scattering process separately. (When level broadening is important, this is not true.) Figure 2 shows three Landau level diagrams for magnetic fields near the w, = w, resonance. The diagonal and vertical arrows represent optical phonon emission and absorption processes which end at k, = 0, the infinite density-of-states region when Landau level broadening is not taken into account. At resonance (Fig. 2b), the initial as well as the final state is at k, = 0, producing a strong singularity in the scattering probability, which results in a conductivity maximum. In the absence of Landau level broadening, the conductivity is logarithmically divergent at the resonance points, in the Born approximation. The principal effects of level broadening can be deduced on physical arguments : At the lower fields, the broadening will be a greater fraction of the spacing between Landau levels; thus, not only will broadening make the transverse conductivity maxima finite, but the oscillatory character will tend to become more sinusoidal with lower field. This is seen in Fig. 1, although the phasesensitive detection technique used there modulates the damping somewhat. v4'

D. G. Seiler, D. L. Alsup, and R. Muthukrishnan, Solid State Commun. 10, 865 (1972); R. Muthukrishnan and D. G. Seiler, Phys. Sraius Solid; 54, K83 (1972). " D. G. Seiler, Proc. Int. ConJ Appl. High Magn. Fieids Semicond. Phys., Univ. of Wiirzberg, Germany (1974). D. G. Seiler and F. Addington, Rev. Sci. Instrum. 43, 749 (1972); Bull. Amer. Phys. Soc. 17, 281 (1972). 4 7 D. G. Seiler, T. J. Joseph, and R. D. Bright, Phys. Rev. B9, 716 (1974). '* E. S. Itskevich, V. M. Muzhdaba, V. A. Sukhoparov, and S. S. Shalyt, Zh. Eksp. Teor. Fiz. Pis'ma Red. 2, 514 (1965) [Engiish Transl.: Sou. Phys.-JETPLert. 2, 321 (1965)l. 49 M. M. Aksel'rod, K. M. Demchuk, I. M. Tsidil'kovskii, E. L. Broyda, and K . P. Rodionov, Phys. Status Solidi 27, 249 (1968). E. 0. Kane, J. Phys. Chem. Solids 1, 249 (1 957). '' G . L. Bir and G . E. Pikus, Fiz. Tverd. Tela 3, 3050 (1961) [English Trans/.;Sou. Phys.-Solid State 3, 222 1 (1 962)]. 44

''

''

228

ROBERT L. PETERSON TABLE I TRANSPORT PROPERTIE s I N WHICHTHE MACNETOPHONON EFFECT BY MATERIAL HASBEENOBSERVED.

Material

Properties" ~~~

n-lnSb p-InSb n-InAs n-Ge p-Ge n-GaAs p-GaAs p-Te n-InP n-CdTe n-CdSe n-CdS n-PbTe n-HgCdTe

~

OTMR, OLMR, HETMR, HELMR, LMTEMF, TNE, PC, PME, Hall, AG, ME OTMR, OLMR OTMR, OLMR, LMTEMF, HELMR, Hall OTMR, OLMR, LMTEMF OTMR OTMR, OLMR, HETMR, HELMR, Hall OTMR OTMR, OLMR OTMR, OLMR OTMR, OLMR, HELMR, Hall OTMR, OLMR Raman, PC OTMR OLMR

' Code: OTMR (OLMR)

-Ohmic transverse (longitudinal) magnetoresistance. HETMR (HELMR)-hot-electron transverse (longitudinal) magnetoresistance. LMTEMF -longitudinal magnetothermal emf (magneto-Seebeck). TNE -transverse Nernst effect. PC -photoconductivity. -photomagnetic effect. PME -Hall effect. Hall -acous toelectric gain . AG -microwave emission. ME -Raman scattering. Raman

t

FIG. 2. Landau level diagrams for w, near w,, showing the important transitions ending at k , = 0. The origin is placed at the bottom of the N = 0 level for simplicity. (a) w, < 0,; (b) w, = w.; (c) 0,> coo.

4. THE

MAGNETOPHONON EFFECT

229

Structure occurs in the longitudinal case for reasons. quite different from the transverse case. Here, since the electric field and consequently the current are in the z direction, the “vertical” transitions of Fig. 2b do not carry any weight since there is no momentum transfer in such collisions. At first, it may seem that there wouId be no resonance extrema at all in first order of scattering, since the singularity involved when only the final state lies at k, = 0 is removable, being an inverse square root, and further, since optical phonon transitions ending at k, = 0 can occur for all magnetic fields (see Fig. 2). However, in the longitudinal case the different scattering processes “interfere” with each other, in the sense that the longitudinal conductivity is a k, integral over the inverse transition probabilities. In the language of relaxation times, the conductivity is found by integrating over some effective relaxation time z(k,) in each Landau level, where l/z(k,) is found (approximately) by adding the inverse times for each scattering ) discontinuously go to zero (when level broadening process. Thus, ~ ( k ,will is ignored) for each kZ for which a transition (whether inelastic or elastic) ending at the infinite density-of-states region is possible. Consider Fig. 2a, for which the field is just below the w, = w, field. The carriers that contribute dominantly to the mobility are those of energy less than that of the carrier involved in the absorption a ending at k, = 0. Carriers of energy just higher than this one are not as important since they are able to make transitions to near k, = 0, thus keeping z(k,) small. Carriers of yet higher energy are discriminated against by the Boltzmann factor. Now as B increases, the mobility tends to increase: more carriers can contribute because the absorption a does not occur until higher energies are reached. However, as o, increases through w o , absorption a is no longer important because emission a‘ now “limits” the mobility (Fig. 2c). With further increase of B, the mobility tends to remain constant since the carrier involved in emission a’ remains at the energy h a , . The reader can readily show by drawing diagrams similar to Fig. 2 that there is a change in the “mobility limiting” mechanism from optical phonon absorption to emission at all fields given by

Lo, = 200, L = 1,2,. . . , (3) for parabolic bands. Notice that the GF resonance fields coincide with even values of L. So far, the elastic transitions such as c in Fig. 2 have not been mentioned. They are important at the GF fields [although not for the fields corresponding to odd values of L in Eq. (3)]. This may be seen from Fig. 3, showing the important transitions for fields near o,= w0/2. Here the interchange of absorption a with elastic transition c is the major factor affecting the mobility. The same is true for all GF fields with N 2 2 in Eq. (1) ;N = 1 is an exception.

230

ROBERT L. PETERSON 3

( 0 )

lb)

FIG.3. Landau level diagrams for wc near wJ2, with the origin placed at the bottom of the N = 0 level. (a) w, < 0 , / 2 ; (b) w, > w,/2.

Thus, longitudinal structure at the GF fields is caused principally by the replacement of optical phonon absorption by elastic transitions as mobility limiters, as B increases through these fields. The exchange of optical phonon absorption with optical phonon emission is not important at the GF fields, except for N = 1,since the carriers involved lie at considerably higher energy. However, structure at the fields given by odd values of L in Eq. (3) is caused by this exchange. These fields have been called “pseudoresonance fields” in order to distinguish them from the GF field^.^^-^* The pseudoresonances appear only at the higher temperatures since the relevant carriers lie at energy hwo, whereas the carriers involved at the GF fields lie closer to the bottom of the band, with the exception of N = 1. The pseudoresonances are also washed out more readily by increasing elastic scattering than are the GF resonances, since the structure at the latter fields is created in large part by the elastic scattering. These features are observed.34 The longitudinal structure thus is not characterized by divergences, as in the transverse case in the absence of level broadening, but by slope discontinuities, since the various limiting mechanisms depend upon B in different ways. The discontinuities are always in the direction of mobility maxima (resistance minima), which are asymmetric. This structure is superposed on a background which itself is oscillatory but off-resonant. The off-resonance oscillations are the result of the onset of LO absorptions of type b in Figs. 2c and 3b, which occur when the magnetic field is on the highfield side ofa GF field. Carriers on the left side of b tend to increase the mobility, eventually causing the mobility to reverse direction. At low temperatures,

4. THE MAGNETOPHONON EFFECT

231

most carriers are close to the band edge, and so the mobility minima occur close to the true GF fields. These two types of structures tend to merge because of level broadening and nonparabolicity of the bands, and it is clear why the magnetophonon structure in the longitudinal case is experimentally often less regular than in the transverse case. 111. Ohmic Regime

1. GENERAL THEORETICAL CONSIDERATIONS Most theoretical work to date on the magnetophonon effect has invoked the simplifying assumptions of sphericity of the carrier effective mass, and plane wave carrier states as modified by Landau quantization. These approximations permitted the mathematics to proceed to the point where the magnetophonon effect was predicted and a reasonably clear physical picture emerged. When it was recognized that resonance structure should be seen in transport properties whenever the optical phonon energy matched the separation between any two Landau levels (the other resonance conditions are discussed later), the tensor character of the effective mass and the nonparabolic character of the energy bands were readily inserted into the resonance conditions to provide a more accurate description of the extremal positions. In this section, some of the basic theory and formulas are given. A more detailed description of Landau quantization is given by Kubo et a[.: Roth and Argyres,* and Kahn and Frederikse.’ The Hamiltonian describing a “free” carrier of momentum p, spin s, spherical mass m*,charge - e with e > 0, and interacting with the lattice is

H

= H,

+ H, + e E - r + j

Vj,

(4)

where r is the carrier position, E is an electricfield, HLis the lattice Hamiltonian, and V j is the carrier scattering term of typej. With the magnetic field B in the z direction, the carrier Hamiltonian H , in the Landau gauge is

where ocis the cyclotron frequency defined in Eq. (2), p Bis the Bohr magneton, and g is the effective g-factor. The energy eigenvalues of Ho are

+

E,,Jk,) = (h2kZ2/2m*) hw,(N

+ i)+ sgpBB,

(6)

232

ROBERT L. PETERSON

corresponding to eigenfunctions

$ d k y , k,; r) = (P& - X)(exp ik,y)(exp ik,z)lL,L,,

(7) and spin states 1s = +$). The q N are harmonic oscillator wave functions centred at X = -l2k, = -hk,/eB. (8) The length 1 defined by Eq. (8) is the classical cyclotron radius, and L, and L, are sample dimensions. The scattering terms V j may be written

V, =

4

Cj(q)b,exp iq. r

+ Hermitian conjugate

(9)

for the carrier-phonon interaction, in which the summation is over phonon wave vectors q in as many phonon branches and polarization states as relevant for the type of interaction. Cj(q) is the coupling coefficient, and b, is a phonon destruction operator. Deformation-potential scattering on the optical phonons occurs in any material in which optical branches exist. In nonpolar materials, this is the only mechanism, and for the long-wavelength phonons, only the longitudinally polarized branch is involved, characteristic of the deformation potential. For this case,52

I C(q)J2= ~:,,ho,,,/2Qpu,~

(nonpolar, optical), (10) where Elopis a deformation potential energy, Q is the volume of the sample, and p is its mass density. Customarily, a directionally averaged longitudinal sound velocity u, is used, and the optical phonon frequency oois taken as dispersionless. This approximation, together with the fact that q does not appear elsewhere in the coupling coefficient, makes the nonpolar optical phonon scattering quite tractable, and for this reason, the most detailed analyses have been carried out with this In polar materials (most of the materials in which the magnetophonon effect has been observed to date are in this category), the polarization of the crystal due to the vibrations of the longitudinal optical branch produces another scattering mechanism for the caniers. This polar optical scattering is usually assumed to be more important than the deformation-potential optical scattering in polar materials, although this is not definitely established in most of the materials. The coupling coefficient may be written52 where

52

~c(q)(' = 4xah(Aw0)3'2q-2(2m*)-i'2n- (polar, optical),

E. M. Conwell, Solid Stare Phys. Suppl. 9 (1967).

(11)

4.

233

THE MAGNETOPHONON EFFECT

is the Frohlich parameter, and K , and K~ are the high-frequency and static dielectric constants, respectively. Thus, the longitudinal optical (LO) phonons are involved in both the polar and nonpolar scattering. Interactions with the acoustic phonons must also be taken into account. The deformation potential produces scattering with the longitudinal modes, with coupling ~ o e f f i c i e n t ~ ~

1 C(q)12= E , 2h2q2/2pRw4

(deformation potential, acoustic), (13)

where El is a deformation potential energy, and oqis customarily taken to be equal to qu,. In polar materials, there is also a piezoelectric interaction, which produces scattering on both transverse and longitudinal acoustic modes. The coupling coefficient may be writtens3vs4 (piezoelectric, acoustic),

(14)

where K is the dimensionless electromechanical coupling, and rs is the screening length given by rs = KcokT/e2nc, (15) where n, is the carrier concentration. In the hot-electron regime, the lattice temperature T would be replaced by the carrier temperature, provided the latter is reasonably well defined. Averaging over directions, again, is practically a necessity in making the subsequent calculations tractable. 54,5 Impurity scattering is usually very important in the magnetophonon effect, particularly at the lower temperatures where thermal excitation of phonons is small. The scattering interaction between a carrier and N , randomly distributed, screened ionized impurities at positions R may be written56 e2

ymp= f l Z K E O ~

c exp[iq +(rrS-’R)] *

q,R

q2

-

‘

By analogy with the phonon scattering terms, Eq. (9), one can define a coupling coefficient (ionized impurity),

(17)

obtained by averaging the squared matrix elements of Eq. (16) over impurity positions. 53 54 55

56

A. R. Hutson, J . Appl. Pbys. Suppl. 32,2287 (1961). H. J. G. Meijer and D. Polder, Physica 19, 255 (1966). R. L. Peterson, Phys. Rev. B5,3994 (1972); Nat. Bur. Std. Tech Note No. 614. U.S. Govt. Printing Office, Washington, D.C. 1972. P. N. Argyres and E. N. Adams, Phys. Reu. 104,900 (1956).

234

ROBERT L. PETERSON

The GF resonance condition (1) involving scattering between Landau levels at k, = 0 follows immediately from Eq. (6). If the spin state changes as well, as could happen if there is sufficient spin-orbit coupling, one has the socalled spin-magnetophonon resonance condition NW

= W,

& gpBB.

(18)

However, as mentioned earlier, there is as yet no definite indication of spin transitions in the magnetophonon effect. Particularly with the narrow-band-gap semiconductors, such as InSb, nonparabolicity of the bands is important." In these cases, the carrier energy in a magnetic field is not given by Eq. (6), but by a more complicated expression. No one expression enjoys universal acceptance. For InSb, the conduction band energy is given with reasonable accuracy by34,57

where E, is the energy gap, and mo*and go are the band-edge effective mass and effective g-factor, respectively. Theoretical expressions for mo*and go in terms of their free-electron values and the band parameters are available.J'~57The generalized GF resonance condition, including the possibility of spin-flip transistions, is

EN.s(0)- EL,J0) = R w ~ , N - L = 1,2,. . . . (20) There are a number of consequences of nonparabolicity. One is that the spacings between adjacent Landau levels of the same spin s are unequal. Another is that for a given pair of Landau levels, the distance between the s = $ levels is not equal to that between the s = levels. Thus, a given magnetophonon extremum consists of a number of contributions, generally unresolved. Still another consequence of nonparabolicity is that the carrier effective mass has a small dependence upon carrier concentration, for the concentrations typically used. A careful calculation of mo* from data must take these contributions into account, together with the proper thermal weighting from the Boltzmann f a ~ t o r . ~ ~ , ~ ' Further, in the polar materials, account should be taken of the polaron effect,59 i.e., of the fact that near resonance, the electron-optical phonon coupling becomes very strong. Each Landau level is, in effect, smeared into a continuum of states, with the result that the magnetophonon extrema are shifted a small amount to higher magnetic fields, a relative increase of

-:

ST

58

59

E. D. Palik. G. S . Picus, S. Teitler. and R. F. Wallis, Phys. Rev. 122, 475 (1961). R. A. Stradling and R. A. Wood, J . Phys. C Solid Stare Phys. 3, L94 (1970). D.M.Larsen, Phys. Rev. 135, A419 (1964).

4. THE MAGNETOPHONON EFFECT

235

r o ~ g h l y0.3cr, ’ ~ where a is the Frohlich coupling constant given in Eq. (12)- No detailed theoretical account of the polaron influence on the magnetophonon effect has yet appeared in the literature, although a dissertation by Palmer6’ is r e p ~ r t e dto ’ ~have considered it. A volume by Emin6’ in the current series will be devoted to the polaron effect. Finally, although mobilities or conductivities are the quantities usually computed, the measured quantities are usually voltage changes, which are proportional to the tensor resistivities in the Ohmic regime. The relations connecting the Ohmic resistivities and conductivities of interest here are

In the lowest order of scattering, it is well known that oyx= ne/B, where n is the carrier concentration. When two-carrier conduction is not important, it is then generally true that oyx9 ox, in the region of quantizing magnetic fields. Thus, the resistivity and conductivity magnetophonon oscillations are reciprocally related in the longitudinal configuration, but nearly linearly related in the transverse case. Magnetophonon oscillations in the Hall component pxy may occur either through ox, or through higher order contributions to oyx(Section 5). 2. EARLYTHEORETICAL RE~ULTS In their first papers, Gurevich and Firsov7*15studied the Ohmic transverse magnetoresistance(OTMR), using the Kubo formali~m.~ They treated scattering on LO phonons in polar materials, using the isotropic, parabolic band model. It was known by then that in the absence of level broadening, the Born approximation gives divergent results in the OTMR for elastic scattering of carriers, but finite results for inelastic scattering. Gurevich and Firsov showed that at the resonance condition (l), the inelastic LO phonon scattering would produce a logarithmic divergence, indicating OTMR maxima, at the resonance fields. At almost the same time, Klinger62 made similar, although less detailed, theoretical statements about the possibility of observingmagnetophonon oscillations.Gurevich’andFirsov also studied’ the effect of the Coulomb interaction between the carriers, and the effect of the dispersion of the LO phonons, in limiting the amplitude of the resonance 6o 61

R. J . Palmer, D. Phil. Thesis, Oxford Univ., 1970. D. Emin, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.). Academic Press. New York (to be published). M. I. Klinger, Fiz. Tuerd. Telu 3, 1342 (1961); 3, 1354 (1961) [Enghh Trunsl.: SOC.Phys.Solid Srure 3, 974, 983 (1961)J

236

ROBERT L . PETERSON

peaks. Neither of these is now considered to be the dominant limiting mecha n i ~ mbut , ~rather ~ the broadening of the Landau levels due to collision^.^^-^^ Efros6’ extended the GF analysis to the case of degenerate statistics, showing that the OTMR magnetophonon maxima still exist, together with the further discussed the Shubnikov4eHaas oscillations. Gurevich et differences in the two types of oscillations, and the experimental conditions under which the magnetophonon oscillations should be observable. Essentially, the latter requires that the temperature be high enough and the impurity concentration low enough that scattering on the LO phonons is a significant fraction of the total scattering. Further, the Landau levels must be reasonably well defined, meaning that O,T > 1, where T is some effective relaxation time. This relation can also be expressed in terms of the mobility ,u as p B > 1 , by using Eq. (2) and the well-known relation p = er/rn*. In 1964, Gurevich and F i r ~ o developed v~~ a theory for Ohmic longitudinal magnetoresistance (OLMR), which showed magnetophonon extrema. In this case, the Boltzmann equation can be used to determine the carrier distribution function for each Landau level. As earlier, GF considered LO phonon scattering in polar materials. This scattering mechanism is difficult to work with analytically because of the wave vector dependence of the scattering matrix element. As a consequence, their analysis had to incorporate several approximations, which did not allow a clear physical picture to emerge. Gurevich and Firsov concluded that resistance maxima would occur at resonance for pure LO phonon scattering, but that with increased competition from elastic scattering, minima would exist at the resonance points. It is now known that this conclusion is only partially (see Part I1 and Section 4). B r y ~ k i nsubsequently ~~ extended the Gurevich-Firsov analy~is,’~ using their model but considering only LO phonon scattering. He pointed out that the OLMR maxima actually lie to the high field-sides of the resonance points, and that the displacements increase with temperature. He also called attention to the existence of a broad OLMR minimum at fields near o,= 1.70,. This minimum is not a pseudoresonance. L.Dworin, Phys. Rev. 140, A1689 (1965).

M. Nakayama, J . Phys. SOC. Japan 21,636 (1969). J. R. Barker, Phys. Lett. 33A, 516 (1970). b6 J . R. Barker, J . Phys. C Solid State Phys. 5 , 1657 (1972). 67 A. L. Efros, Fiz. Tverd. Tela 3, 2848 (1961) [English Transl.: Sou. Phys.-Solid State 3, 2079 ( 1962)l. 6 8 V. L. Gurevich, Yu. A. Firsov, and A. L. Efros, Fiz. Tverd. Tela 4, 1813 (1962) [English Transl.: Sou. Phys.-Solid Stare 4, 1331 (1963)l. b9 V. V. Bryksin, Fiz. Tverd. Tela 9, 232 (1967) [English Transl.: Sor. Phys.-Solid State 9, 171 ( I 967)]. 64

65

4. THE MAGNETOPHONON EFFECT

237

Gantsevich and Gurevich7’ evaluated the probabilities for scattering of carriers between nonequivalent valleys ; they found that these scattering probabilities would have maxima at the resonance condition (l), in which o,, is replaced by the frequency of the appropriate “intervalley phonon.” Pavlov and fir so^^^ analyzed the magnetophonon oscillations in the longitudinal magnetothermal emf (LMTEMF), and other thermomagnetic kinetic coefficients. This paper is similar in analysis to that of Gurevich and fir so^,'^ and contains additional physical discussion; the interested reader may find it helpful to read these two papers together. Briefly, the thermal emf or Seebeck tensor7’Q is defined as the coefficient of the temperature gradient, giving rise to an electric field when an electric current is prevented from flowing: E

=

(24)

Q(B)VT.

Since in general the current J can be expressed as72 (25)

J = u(B)E - P(B)VT,

the Seebeck coefficient Q depends both upon the conductivity t s and the diffusion tensor p. Pavlov and FirsovZ4predicted the appearance of LMTEMF maxima at the GF fields.

3. THEORY OF TRANSVERSE MAGNETORESISTANCE The crossed-field configuration necessitates a quantum-theoretic transport approach for the magnetophonon effect.’Themost direct, and exact, starting ~.~ other approaches are also possible, point is the Kubo f ~ r r n a l i s m ,although as discussed by Roth and Argyres.8 The electric conductivity tensor is expressed generally as a,,(o)

= R-

’

dt e-’”’ Jop d l (J,( - iM)J,(t)),

p, v

=

x, y , z ,

(26)

for an electric field oscillating at frequency o.The J, are current operators, and

S. V. Gantsevich and V. L. Gurevich, Fiz. Tuerd. Tela 6, 2871 (1964) [English Transl.: SOL-.Phys.-Solid State 6, 2286 (1965)l. ” S. M. Puri and T. H. Geballe, in “Semiconductors and Semimetals” (R. K . Willardson and A. C. Beer, eds.), Vol. 1, p. 203. Academic Press, New York, 1966. 72 G. I. Guseva and P. S. Zyryanov, Phys. Status Solidi 25, 775 (1968). ’O

238

ROBERT L. PETERSON

where H is the Hamiltonian for the system. The angular brackets represent an average over an ensemble at a temperature indicated by B = l/kT. Ordinarily, the proper reduction of the correlation function in Eq. (26) is the most difficult part of the theoretical problem, because the Hamiltonian in the time-development operators of Eq. (27) contains scattering terms. However, in the transverse case, the scattering terms may be omitted in the time-development operators in the lowest order of scattering, for the following reason. As Kubo et point out, the current operator J,, which appears in the expression for the static transverse conductivity B,,(O), can be written for electrons as

-4ri

+ X,,),

(28) where 5 is the relative coordinate of the cyclotron motion, and Xopis the center coordinate. For the simple band model, the eigenvalue of ,Yo, is given by Eq. (8), and in general = x - X,,. Kubo et u I . ~show for the static case and a closed Fermi surface that the 5 terms make no contribution in Eq. (26). Further, from the Heisenberg equation of motion, J, =

r

fix,, = i [ H , x,,]= i[Y X,,] = (eB)-' aV/dy,

(29) where V is the sum of the scattering interactions. Thus the correlation function of ax, begins at second order in the scattering, and to this order, the scattering terms may be dropped in the time-development operators, permitting a great simplification. Kubo et aL4 show that for inelastic phonon scattering (including elastic scattering as a special case), the transverse conductivity expression then reduces to m

a,,(O)

= (e2B/2Q) x

x

J-m

dE

1 [f(E - ho,) - f(E)1(r2qJZ(2./h) q

l W l 2 q ~ +, 1) tr{G(E - H,)(exp iq * r) G(E - ho, - H,)(exp

- iq

- r} ,

(30)

wheref(E) is the Fermi function

f ( E ) =(eptE-C) + I)-',

and i is the Fermi energy. The trace operation in Eq. (30) is over electron states, the lattice states already having been summed over, yielding the Planck factors 2, for the number of thermal phonons: 6, = [exp(Bho,) - I]-'

.

(32)

Further reduction shows that Eq. (30) is identical to the expression obtained

4.

239

THE MAGNETOPHONON EFFECT

by Argyres and Roth6 and Roth and Argyres’ :

c f(E,)[1

ox, = (e28/2Q)

.,a‘

-

f(EaJlw,,.(x,- XaJ2

9

(33)

where the set of one-electron states is designated by a, and

1 IC(q)1’(riq+ 1)I (a1 exp iq - rla’) 1’

Wua,= (2n/h)

S(E. - E., - hw,) (34)

cl

is the transition probability per unit time between states a and a’. In the simple band model, the matrix elements of exp iq r are conventionally written

-

where

Spin indices are here suppressed since the scattering mechanisms indicated by Eqs. (9) and (16) contain no provision for spin transitions. The quantity ~J,,,,,~ is a function of the single variable 1 = 1’qL2/2,and possesses simple integraP and recurrence73 properties useful for computations. In particular, IJ,,12 and

=

n! n’!An-’“e-’

[C0

s! (n! - s)! (n - n’ + s)!

y,

n 2 n’,

(37)

Thus, in the Born approximation, oxxgoes as the transition probabilities, and if more than one type of scattering is occurring, ox, is found by simply adding the separate contributions. Unfortunately, for elastic processes, the right side of Eq. (30) or (33) diverges, and even for inelastic processes, there are divergences at the GF resonance fields. This is because the initial and final density-of-states factors can become identical for these situations, resulting in a singularity giving a logarithmic divergence. This is seen by a 73

E. Yamada and T. Kurosawa, J . Phys. SOC.Japan 34,603 (1973).

240

ROBERT L. PETERSON

further reduction of Eq. (30) to the form

c,.. =

1

nn’s

Jb”

d1 11Jnn,(A)12 S$E JdE’

where

+ 1)

G(1, k, - kz‘, E, E‘) a 1 C(q)12nq(riq

x [f(E - h o q ) -

f(E)]6(E - E

- ho,).

(40)

Here, G is assumed to depend on q only through its magnitude ( q L 2 + qz2)’/2= [(2L/Z2) (k,- ~k,’)~]’/~, in which k, and kZ‘ are understood to be replaced by E and E‘, respectively, through Eq. (6). Thus, for elastic collisions (oq+ 0),the singularity occurs in each term for which n’ = n, for all fields B. When wq is a finite constant coo, as is generally assumed when considering scattering on optical phonons, the conductivity expression is finite except at fields for which nhw, = ho, n’ho,, which is just the GF magnetophonon resonance condition (1); the singularity occurs at k, = k,‘ = 0. For nonparabolic bands, the resonance condition is generalized to Eq. (20). There are no “pseudoresonances” (see Part I1 and Part 111, Section 4) in the transverse configuration. The necessity of level broadening considerations in the OTMR was recognized in the first papers on the magnetophonon e f f e ~ t . ~Kubo . ’ ~ et aL4 and Roth and Argyres* discuss in some detail both formal and practical means for improving upon the Born approximation, through collisional broadening techniques. However, they did not apply these techniques to cases in which optical phonon scattering is significant. D ~ o r i nwas ~ ~the first to do this, considering the effect of electron-LO phonon scattering alone in broadening the levels in polar materials. He developed an expression for the finite transverse conductivity at the o, = o,resonance, which, of course, does not correlate with the experimental fact that the oscillation amplitude depends upon impurity content. N a k a ~ a m ageneralized ~~ Dworin’s work to include fields in the neighborhood of the N = 1 GF field. A numerical calculation was performed for the oscillatory part of the OTMR, which showed a temperature-insensitive shift in the peak position to higher field. The temperature dependence of the amplitude showed qualitative agreement with data of Parfen’ev et al. l 9 on n-InSb, although the magnitude of the calculated amplitude was too small by a factor of about 1.5. N a k a ~ a r n a ~ ~ also suggested that the o b ~ e r v a t i o nin~ n-CdTe ~ of a shift from the peak

+

+

74

A. L. Mears, R. A. Stradling, and E. K. Inall, J . Phys. C Solid Srate Phys. 1, 821 (1968).

4.

THE MAGNETOPHONON EFFECT

241

position as deduced by using the low-frequency cyclotron resonance effective-mass value in Eq. (1) may be due to collision broadening as well as to the polaron e f f e ~ t . ' ~ *However, ~ ' * ~ ~ in this material as well as in r ~ - C d S ethe ,~~ polaron correction to the effective mass as deduced by Palmer6' appears to account for the shift adequately. Barker65.66carried the collisional broadening analysis of the OTMR further, by applying the type of analysis used so successfully with the ShubnikovdeHaas and deHaas-van Alphen e f f e ~ t s . ~ ,This * , ~ entails use of the Poisson summation formula m

C

n=O

f(n

+ +) =

JOm

dz f(z)

+2

m

( - 1)' r= 1

JOm

dz f(z) cos(2arz)

(41)

to replace a summation over Landau levels. In the absence of level broadening, there is little or no advantage in using the Poisson formula, since the summation over r usually converges slowly, if at all. However, with broadening, a damping factor appears, so that only one or two terms need be kept. Dingle76 was apparently the first to use such a technique in magnetics problems. He assumed that the influence of collisions would be to broaden and shift the infinitely sharp energy levels by constant amounts r and A, respectively, into a Lorentzian shape. The damping factor so obtained is exp( - 2nrl-/hw,). Barker66 considered several scattering processes separately, in some detail, in determining the effects of level broadening on the magnetophonon oscillations in the transverse magnetoresistance. Each scattering mechanism treated lead to a conductivity expression similar to that obtained under the assumption of constant level widths r and shifts A. The oscillatingpart of the transverse conductivity was found to be of the form

-C a0

oo,,

r= 1

r - '[cos(2lrroJoc)]e-

Znrq,

(42)

where q is a damping factor whose detailed form depends upon the damping process considered, and is difficult to evaluate. However, in the constantwidth approximation, q takes the Dingle form Tlho,. For impurity broadening, Barker finds that r is proportional to oo.One sees from Eq. (42) that if 27rq $= 1, only the r = 1 term is important (the strong damping limit), yielding an expression equivalent to that deduced empirically by Stradling and Wood34 for the OTMR of GaAs. Observations on many materials since that time have shown that a damped sinusoidal shape is a common result as the magnetic field becomes progressively smaller; deviations are L. Eaves, R. A. Stradling, S. Askenazy, G. Carrere, J. Leotin, J. C. Portal, and J.-P. Ulmet, J. Phys. C Solid Stare Phys. 5, L19 (1972). 76 R. B. Dingle, Proc. Roy. SOC.(London) A211, 517 (1952).

75

242

ROBERT L. PETERSON

greatest at the highest fields (see, e.g., Fig. 1). The expression for the strong damping limit is accurate66 if (1) r 9 hw,/2a; ( 2 ) r < 2kT; and ( 3 ) hw, 2kT. Physically, these conditions mean that level broadening must be comparable to the spacing between Landau levels, and that the electrons must be distributed over many Landau levels. The strong damping expression does not hold in the quantum limit.

+

4. THEORY OF LONGITUDINAL MAGNETORESISTANCE

The magnetophonon theory for OLMR was first considered by Gurevich and fir so^,'^ who considered the effect in polar materials. They used the Boltzmann equation in the isotropic, parabolic band model, without level broadening. As mentioned earlier, the wave vector dependence of the polar coupling constant creates some computational disadvantages relative to the nonpolar case. It prevents the existence of a simple relaxation time, and prevents the use of the integral reduction formula (38) which greatly simplifies the final mobility expression. As a consequence, the true character of the longitudinal magnetophonon structure was not recognized until considerably later, when Kharus and Tsidil’k~vskii~~ and P e t e r ~ o n ~ ~ - ~ * worked out the details for nonpolar materials. The latter papers were, in fact, anticipated by an interesting, but littlereferenced paper of Pomortsev et al.77 which treated pure LO phonon scattering in degenerate nonpolar materials. Solving the Boltzmann equation in the longitudinal configuration for arbitrary degeneracy, they concluded that in the degenerate limit, there should be OLMR maxima at magnetic fields satisfying

where [ is the Fermi energy. The condition (43) corresponds to transitions between the Fermi level and the Landau levels, with LO phonon absorption and emission, and is thus a combination GF-ShubnikovdeHaas resonance condition. These extrema will ordinarily be competing with the usual ShubnikovAeHaas oscillations, and can be expected to be difficult of observation. Nevertheless, Pomortsev et al.’ claim to have seen such maxima in n-HgTe and n-GaSb (Section 6 ) .The partially degenerate regime deserves additional study, both experimentally and theoretically. Although to date most magnetophonon experiments have been made on polar materials, the theory for the magnetophonon oscillations in the OLMR will be presented briefly here for nonpolar materials. Even with combined acoustic and optical phonon scattering, no approximations need be made in ”

R. V. Pomortsev, A. 1. Ponomarev, G. I. Kharus, and I. M . Tsidil’kovskii, Proc. I X Int. Con$ Phys. Semicond., Moscow, p. 720. Nauka, Moscow, 1969.

4.

243

THE MAGNETOPHONON EFFECT

this case within the parabolic model, and the expressions are simple enough to permit an easy physical understanding. The results for polar materials should be the same except for some difference of detail because of the fact that the polar LO scattering favors small momentum transfers during collisions [cf. Eqs. (10) and (1 l)]. Similarly, the acoustic scattering should be representative of elastic scattering generally. The steady-state Boltzrnann equation for a spatially homogeneous material may be written

where CI = (n,k,, kz). The Lorentz term (v x B) VJ, is absent because the only direction for free translational motion is parallel to B. Equation (44) is to be solved for@,). The magnetic field enters the solution both through the equilibrium value for f,(k,) and through the transition probability w,,.. If more than one type of scattering is being considered, the wQQ,are summed over the different transition probabilities. For scattering on phonons, w,.. = (2rr/h)

c I C ( d l Z I < exp ~l 9

i q - r l a >IZ

+ h ~ , +) (A, + 1) d(E, - E,,

x [R, 6(EQ- E,,

- ha,)].

(45)

The reverse transition probability w,,, is obtained from waQ,by interchanging ff, and B, 1. For both acoustic and optical phonon scattering of carriers via the deformation potential, Eq. (38) may be used to simplify wQa,and Linearization of Eq. (44)in E then allows immediate solution for the linear correction to the equilibrium distribution function, since the first term on the right in Eq. (44)vanishes for both of these mechanisms. The longitudinal mobility is then

+

P = ( - e/m*)

where

I/&)

=

1 n

J-a m

dk, k z ~ n ( ~ z ) ( a / ~ k , ) f n o ( k z ) Y

1 [w,.,(acoustic) + w,.,(optical)].

(46)

(47)

Q‘

For nondegenerate statistics, the equilibrium distribution function is f,O(kZ)= (/3hZ/27crn*)”*(1 - exp[ - y,]) exp( - ny, - BhzkzZ/2rn*), (48)

where

Y,

= /3koc.

(49)

244

ROBERT L. PETERSON

Equation (46) reduces to3’

where B0 is the thermal number of optical phonons, and y = Bhmo, D$(z’, 7,) =

c n‘

[z2

(51)

- yc(n’ - 41- l”,

All quantities x-lI2 are defined as zero for x < 0. The D r and Dzp are proportional to the respective transition probabilities, and z is a normalized form of J k , J . Equations (50)-(53) describe longitudinal structure at fields given by Eq. (3). The physical picture, given briefly in Part 11, is easily constructed from inspection of these equations. The radicands are equal to the final state z” through energy conservation. The term containing e7 in D;, corresponds to optical phonon emission, while the other term is absorption; the single term in describes both absorptive and emissive processes, because equipartition for the acoustic phonons has been used. The integrand of Eq. (50) goes discontinuously to zero at each value of z for which the finalstate momentum z’ vanishes. The positions of some of these zeros vary with magnetic field. Thus the magnetic field dependence of the mobility is controlled principally by whichever transition happens to be the dominant “cutoff” mechanism for the mobility. For example, as yc passes through the pseudoresonance point 2y, there is a change from the (n = 0, n’ = 1) LO absorption to the (n = n’ = 0) LO emission in controlling the magnetic field behavior of p, while the acoustic term plays no critical role. The result is a slope d i s c o n t i n ~ i t y ,which ~ ~ ? ~is~ always in the direction of a mobility maximum, although it may not be a true local maximum. When nonparabolicity of the bands is important, Eqs. (1) and (3) must be generalized. The longitudinal structure determined by the change from LO absorption to elastic transitions in controlling the mobility occurs at fields given by Eq. (20), whereas structure due to the change from LO absorption to LO emission is given by

EN,JO)- E , , . ( O ) = 2)iO0,

N - L = 1,2,.

..

(54)

245

4. THE MAGNETOPHONON EFFECT

(including the possibility of spin transitions; see below). Notice that fields corresponding to even values of N - L in Eq. (54) do not generally coincide with the fields given by Eq. (20). Magnetoresistance curves have been ~ a l c u l a t e d ~for ~ . ~several ’ temperatures and ratios of acoustic to optical phonon scattering, showing the development of the longitudinal structure. The pseudoresonances are found to develop only at the higher temperatures, and are more easily erased by increasing elastic scattering than are the features at the G F fields. These effects are observed e~perimentally.’~.~~ Figure 4 shows the calculated structure at several temperatures, for a given ratio of acoustic to optical phonon scattering. 0.6

0.4 -

0.2

0

-0.2

0.2

1 .o

0.5

1 .5

%/WO

FIG.4. Ohmic longitudinal magnetoresistance calculated from the Boltzmann equation

for combined optical and acoustic phonon scattering in nonpolar materials, at (El,p/E,)2 = 6. Numbers on curves are values of y = hw,/kT. Vertical arrows show the pseudoresonance positions; the remaining kinks lie at the GF resonance positions. Several scales are used to separate the curves. (After Peter~on.~’)

Recently, Magnusson’* has solved the Boltanann equation numerically

for combined LO and acoustic phonon scattering in polar materials. He

calculated the Ohmic longitudinal magnetoresistanceat several temperatures

and ratios of acoustic to LO phonon scattering, and found the same general

features as described earlier for the nonpolar case, such as the development of pseudoresonance minima with increasing temperature. B. Magnusson, Phys. Status Solidi 56, 269 ( I 973).

246

ROBERT L. PETERSON

Thus, it has been established theoretically from the Boltzmann equation, in the absence of Landau level broadening, that (1) OLMR slope discontinuities, in the direction of minima, exist at the GF and pseudoresonance fields, with the pseudoresonances being more sensitive to temperature and elastic scattering; and (2) the amount of elastic scattering, although affecting the magnitudes of the discontinuities, does not change their signs or positions. Finally, this structure is superposed on an oscillatory background which is not quite periodic in 1/Band whose maxima lie on the high-field sides of the fields defined by Eq. (1). Barker78ahas recently published an analysis of the effect of Landau level collision broadening on the OLMR magnetophonon structure. He first uses an elegant resolvent analysis of the Kubo expression, Eq. (26), for the conductivity, showing that in the longitudinal case, the conductivity expression following from use of the Boltzmann equation is reproduced. After analyzing the no-broadening limit, confirming the results of Kharus and Tsidil’k~vskii~~ and Peter~on~’.~’ at the GF fields, he considers the strong damping limit (see the discussion in Section 3). His detailed analysis uses the approximation of constant level widths r and shifts A in a Lorentzian form for the levels. The result in this limit is pzz

=

xo +

(ne22Jm*)Xt COS[(2.rrOJW,) - (87c/2)]’

(544

where z, , xo, and xz are slowly varying functions of the magnetic field, and 6 varies slowly from unity at high temperatures ( k h w , / k )to at very low temperatures (recall, however, that the strong damping limit is not valid at low temperatures; see Section 3). The pseudoresonance structure does not appear in Eq. (54a) because one of the approximations used by Barker consists of the neglect of LO phonon emission processes, without which the pseudoresonancescannot be described, as discussed earlier in this part and in Part 11. See also Peter~on.~’ Experimental OLMR results on very polar materials show about a 4 2 shift, in agreement with Eq. (54a), although the less polar materials typically show a shift ofjust a few percent (see Section 6). This is an area where further theoretical study, including numerical work, will be of great value. Other properties affecting the structure are (1) nonparabolicity, which will give rise to “fine structure” within each major feature; (2) dispersion of the optical phonons, because finite-momentum phonons are involved in the longitudinal case; and (3) the polaron effect. In principle, transitions between levels of opposite spin can also affect the structure. This was the initial explanation for the pseudoresonance structure. However, since no structure

5

’“J. R. Barker, J. Phys. C Solid Srure Phys. 6, 880 (1973).

4.

THE MAGNETOPHONON EFFECT

247

attributable to such transitions has yet been seen unambiguously, and the maximum cross sections for such processes have been shown in InSb to be relatively very the t h e o r i e ~ ~ ' established -~~ for these spinmagnetophonon transitions will not be developed here. Mak~vskii'~ has recently pointed out in a short note that magnetophonon oscillations should be observable in the current produced by momentum transfer from incident photons to the carriers. Assuming that the carriers involved in the drag current are those removed from shallow donor impurities by the photons, Makovskii treats the longitudinal configuration, and establishes the resonance condition

ho - Ei f h o , = ( N

+ $ho,

(55)

where w is the photon frequency and Eiis the donor ionization energy. The signs and other characteristics of the extrema were not discussed. No experimental data are yet available to test Eq. (55). 5 . THEORY OF MAGNETO, HALL,AND SEEBECK EFFECTS

Magnetophonon oscillations have been seen in the Hall voltage of n-InSb, n-InAs, and n-GaAs (Section 6). The theory for the appearance of magnetophonon oscillations in the Hall effect is not well developed, however. Only a phenomenological approach has been used to date,80 which has several restrictions. The relevant magnetoresistance tensor component is given by Eq. (25). Phenomenologically,the conductivities may be expressed as8'

where the angular brackets indicate a suitable average in energy space. The magnetophonon oscillations in this formulation appear through the relaxation time r. The condition for the observability of magnetophonon oscillations, o,t %. 1, thus implies a, %- ox,. In lowest order of scattering, a, = ne/B, independent of the scattering. Thus, by Eq. (23), oscillations in p,, are due to ox, directly, or to higher order terms in oyx. When two-carrier conduction is important, the iowest order contribution to oyxis (n - p)e/B, 79

L. L. Makovskii, Fir. Tekh. Poluprou. 6, 985 (1972) [English Transl.: Sou. Phys.-Semibond. 6, 860 ( 1 972)]. M. M. Aksel'rod, V. A. Vilisov, A. P. Podkolzin, and I. M. Tsidil'kovskii, Fiz. Tekh. Poluprou. 4, 2217 (1970) [English Transl.: Sou. Phys.-Semicond. 4, 1910 (1971)l. 1. G. Lang, S . T. Pavlov, and P. V. Tamarin, Fiz. Tuerd. Tela 13,3654 (1971) [English Transl.: Sou. Phys.-Solid Stare 13,3083 (1972)l.

248

ROBERT L. PETERSON

where p is the hole c o n c e n t r a t i ~ n .It~ ~can then happen that Joyxl< ox.. For E IB, an electron-hole pair gradient develops in the direction of the Hall field, producing diffusion currents which can also give rise to magnetophonon oscillations. Aksel’rod et a1.” developed a phenomenological theory for this case, assuming n x p . They concluded that for InSb with 477°K) x 1014 cm-3 and T > 130”K, the Hall field E, should vary essentially as the hole contribution to ox., and that the influence of scattering on uyxwould not be important. The observed minima” in the Hall voltage of InSb (Section 6) at the G F resonances were argued to be in qualitative agreement with their theoretical expressions. However, agreement was not reached for the relative sizes of the voltage oscillations in the x and y directions. Further, Wood et al.” observed Hall voltage maxima in GaAs (Section 6). A more detailed theoretical understanding of the magnetophonon oscillations in the Hall effect will probably require a first-principles calculation, perhaps along the lines developed by Guseva and Zyryan~v.~’ Two recent theoretical paperssza*8zb have treated the magnetophonon aspects of the longitudinal magnetothermal emf. The quantity of interest, the longitudinal magneto-Seebeck coefficient Q,, , is equal to /3zz/o,, by Eqs. (24) and (25). In those cases where a relaxation time .r,(k,) exists, further reductionszbbrings it to the form

where 6 is the Fermi energy, and En and Loare given by Eqs. (6) and (48), respectively, in the isotropic, parabolic model. Arora and Petersonazbhave shown in the no-broadening limit that the overall oscillating structure is very similar to that observed, especially by Puri and Geballe.” They findEzbthat the maxima discussed by Pavlov and Firsovz4 are shifted a few percent to the high-field sides of the G F resonance fields, the shift being greater as either temperature or amount of elastic scattering increase. The shift is not as great as observed, however. The analysis also reveals kinks at the G F and pseudoresonance fields. Barker’ 2a considered the opposite limit, in which level broadening is comparable to the spacing between Landau levels. In this limit, the oscillations of Q,, are quite sinusoidal, the maxima being shifted by as much as 7c/2 higher than the GF fields, an amount greater than observed. Since the two limiting theories seem to bracket the experimental results, the magnetothermal emf oscillations appear to be reasonably well understood. R.A. Wood, R.A . Stradling, and 1. P. Molodyan, J . Phys. CSolidStare Phys. 3, L154(1970). J. R. Barker, J . Phys. C Solid State Phys. 6, L52 (1973). 82b V. K. Arora and R. L. Peterson, Phys. Rev. B 9,4323 (1974).

4. THE MAGNETOPHONON EFFECT

249

6. EXPERIMENTAL RESULTS Many characteristics of the magnetophonon effect are common to all materials, and some of these will be mentioned here. There is always an optimum temperature at which the oscillations can be seen, which of course varies with the material. That is, at very low temperatures, the optical phonons are not generated thermally in sufficient numbers to allow optical phonon absorption to be a significant fraction of the total scattering. On the other hand, Landau level broadening increases with increasing temperature, obscuring the oscillations. Another feature in common for all materials is that the oscillations are most readily seen in high-mobility samples, the reason being, of course, that a large amount of elastic scattering obscures the optical phonon effects. Also, no dependence of the oscillation period upon carrier concentration has yet been reported. Indeed, this independence is one of the means for confirming that the oscillations are of magnetophonon rather than ShubnikovdeHaas type. In principle, the periods should depend to a small extent on carrier concentration (through the effective mass”), and more so as the degenerate region is approached. The positions of the OTMR maxima are generally observed to shift slightly to lower field with increasing temperature (roughly, 5 % over the temperature range in which the magnetophonon oscillations can be seen). In part, this can be attributed to the temperature dependence of the energy gap, and hence effective The behavior of the OLMR peak positions with temperature is considerably more irregular.34 A good portion of this can probably be attributed to distortion of the curves caused by the onset of the pseudoresonances with increasing temperature. Level broadening will certainly also cause small temperature-dependent shifts in both configurations. In the following, the experimental work is grouped by material. An attempt is made to put each report in perspective, and to comment on conclusions which were later changed. As a rule, the later papers contain the most accurate results, and are accorded a fuller treatment. For quick reference, Table I1 is provided, giving the fundamental fields (defined here as the magnetic field corresponding to the N = 1 GF resonance in the OTMR), the magnetophonon masses [defined by combining Eqs. (1) and (2) even when nonparabolicity is important], the LO phonon frequencies, and the optimum temperatures for observing the oscillations. The entries in the table are not meant to be “best values,” but rather are intended to give the reader a quick idea of the magnitudes of fields, masses, frequencies, and optimum temperatures involved. In some cases, the fundamental field is higher than the magnetic fields available to the investigators, and an estimate is made for the table. Division of the fundamental field by the integer N gives the approximate value of the Nth GF peak position. Similarly, in some instances, results are reported at only one or two temperatures, and the table entry is then simply an indication of where the oscillations have been observed.

250

ROBERT L. PETERSON

c,,

TABLE 11

“OPTIMUM” TEMPERATURE , FOR OBSERVING OHMIC MAGNETOPHONON OSCILLATIONS, OPTICALPHONON FREQUENCY o,,FUNDAMENTAL FIELDB,, MAGNETOPHONON MASS m*, AND BAND-EDGE MASSm,* Material

T,,, (“K)

w,(lO” sec-’)

B,(T)

m*/m,

mo*/me

n-InSb p-InSb n-lnAs n-GaAs p-GaAs

104 90 290 140 120

3.7 3.7 4.5 5.5 5.5

0.0134

n-Ge

120

5.7

0.016 0.33 0.027 0.071 0.64 0.11 0.139 0.105 0.362 0.0862 0.221 0.0494 0.319 0.375 0.0484 0.402 0.26 0.26 0.12 0.086 0.11 0.140 0.146 0.084

n-lnP n-CdTe n-CdSe

130 60 80

2.05 2.20 2.20 6.5 3.2 4.0

n-PbTe

77

2.15

3.4 70 6.9 22 200 34 45 34 117 28 72 16 103 122 16 130 30 32 15 32 20 32 33 10

n-Hg,-,Cd,Te 75

0.41

1.5

p-Ge

p-Te

5.7

85

46

0.06

Remarks

-

0.0223 0.0655 -

-

-

-

0.081 0.096 0.115 0.120 0.005

a. n-Type Indium Antimonide

The first successful experiments showing structure attributable to the magnetophonon effect were made with n-InSb, doubtless because of the ready availabilityof samples of high mobility and low electron concentration, e.g., ~ ( 7 7 ° Kx ) 5 x lo5 cm2 V - ’ sec-’ and n x lof4 ~ m - Oscillations ~ . were seen in the longitudinal magnetothermal emf (LMTEMF),’6.’8.22,71*83 the OTMR,l6,’ * 1 9 * 2 1 and the OLMR. 6,19-2 The various measurements were largely in agreement, although some details of interpretation were different. In particular, the dc techniques used (Part 11) not only made

’

83

3”

’v7’

V. M. Muzhdaba, R. V. Parfen’ev, and S. S. Shalyt, Fiz. Tverd. Tela 7,2379 (1965) [English Transl.: Sou. Phys.-Solid State 7, 1922 (196611.

4.

251

THE MAGNETOPHONON EFFECT

measurement of the extremal positions rather inaccurate (small oscillations superposed on a rapidly increasing background), but raised some initial uncertainty as to whether the maxima or the minima were the resonance features. The OTMR indeed showed maxima at about the expected values of magnetic field, based upon prior knowledge of m* and wo, confirming the GF transverse t h e ~ r y The . ~ N = 1 GF maximum lay at about 3.4 T (tesla; see the appendix) and the oscillation amplitudes were found to be largest at a temperature of about 100°K.The OLMR results of the Russian workers showed minima near these positions, and, also in contrast to the transverse case, the positions were apparently temperature dependent, shifting to lower fields as the temperature was raised. Figure 5 is representative of this early work. However, Puri and Geballe71saw no shift with temperature, and also argued that the resonances are marked by maxima rather than minima.

,90 K

,120 K

150 K

200 K

1

2

3

B(T)

FIG.5. Curves of longitudinal magnetoresistance at several temperatures for a sample of n-InSb with n = 4 x I O l 3 cm-3 and p = 4.9 x lo5 cmz V - ’ sec-’ at 90°K. The sloping dashed lines indicate the changing positions of the minima with temperature. The vertical lines labeled with integers locate the positions of maxima which were observed in the transverse magnetoresistance. (After Firsov er a/.*’)

252

ROBERT L. PETERSON

The derivative techniques used more recently (Part 11) have confirmed the general correctness of the early Russian identification, however. This can be seen from Fig. 6a, showing both OLMR and OTMR at 77°K in n-InSb in a recent experiment47 using field modulation, phase-sensitive detection, in which the sample remained fixed and the magnetic field was rotated through 90".This figure also shows well the near coincidence of the OLMR minima with the OTMR maxima in n-InSb at 77°K. I

I

I

I

I

I

I

I

u 5.6 X 10' Pa

1

I

I

1

06

07

08

0.3

I

B (T)

FIG. 6. Reproduction of X-Y recorder traces showing stress shifts in the extrema in the transverse (B Ia // J) and longitudinal (B (1 a (1 J) magnetoresistance at 77°K in n-InSb. The stress a is applied parallel to the [211] direction. (a) a = 0. (b) a = 5.6 x lo* Pa. The detector response is the output of the lock-in amplifier while detecting the second harmonic of the modulation field. The extrema are labeled by the integers N of Eq. (1). (After Seiler er ~ 1 . ~ ' )

The LMTEMF e ~ t r e m a ' * ~ ~ ~were * ' 'found * ~ ~ to be well displaced from the resonance fields as given by Eq. (l),although the period A(l/B) agreed fairly well with that of the magnetoresistance measurements. A later result by

4.

THE MAGNETOPHONON EFFECT

253

Bashirov and Gad~hialiev'~in n-InSb also showed this. Figure 7 shows experimental results on one sample at several temperatures. Muzhdaba et observed a temperature shift of the extrema, while Puri and Geballe" saw none. All LMTEMF results to date have been obtained with dc techniques. Oscillations were also looked for'' but not seen in the transverse configuration, in which the magnetic field is perpendicular to the thermal gradient. The lack of oscillations is in accordance with expectations, since in this case the emf should be independent of carrier ~ c a t t e r i n g . ~ ~ Subsequent studies of the magnetoresistance of n-InSb at higher magnetic field^^^,^',^^ by two groups showed an interesting new result, namely, the existence of an OLMR minimum at about 7.5 T (w, x 2wJ. In fact, this ~

1

.

~

~

9

'

~

B(T)

FIG.7. Curves of longitudinal magnetothermal emf at several temperatures for a sample of n-InSb with n = 3.5 x cm-3 and j~ = 5.6 x lo5 cmz V - ' sec-' at 77°K. The vertical lines labeled by integers denote the GF resonance fields as inferred from transverse magnetoresistance measurements. (After Muzhdaba er a/.'s)

84

R. 1. Bashirov and M. M. Gadzhialiev, Fiz. Tekh. Poluproc. 2, 115 (1968) [English Transl.: SOP.Phys.-Semieond. 2, 96 (1968)l.

254

ROBERT L. PETERSON

minimum is evident in the results of Puri and Geballe,” but it was not recognized as such. Figure 8 shows an example of the growth of this minimum with temperature. Since this feature was not explained in the longitudinal treatment of GF,23 a new explanation was sought. Both groups of workers suggested that the minimum might be due to transitions between the two spin sublevels of the lowest Landau level (0 - + 0 + ). However, when proper account is made for band nonparabolicity, with the use of Eq. (19), the observed position of this high-field minimum is inconsistent with the known value of the g-factor. A subsequent careful search34 at lower fields revealed no structure where spin-flip transitions, with emission or absorption of an

10.5

FIG. 8. Transverse (dashed lines) and longitudinal (solid lines) Ohmic magnetoresistance at several temperatures for a sample ofn-InSb with n = 8.5 x 10l3~ m -showing ~ , the development of the extra minimum at about 7.5 T (w, % 2w0)in the longitudinal configuration. (After Tsidil’kovskii er ~ 7 1 . ~ ’ )

4.

THE MAGNETOPHONON EFFECT

255

LO phonon, could have produced structure. The relative cross section for such spin-flip transitions has been estimated4’ to be so small that detection in magnetophonon experiments is unlikely, at least in n-InSb. Stimulated by the new observations, Pavlov and fir so^,^',^^ and Tsidi1’developed, in somedetail, theories of spin-flip transitions with kovskii et absorption of an optical phonon, based upon the spin-orbit interaction. Both short-range scattering on transverse optical phonons” and longrange scattering on longitudinal optical phonons3’*3 were treated. The laterpapers3’ *33agreed that the OLMRspin-magnetophonon extrema would always be minima. Both groups concluded that the OTMR would show spinmagnetophonon maxima. Tsidil’kovskii et al.’ also presented some new ) measurements on n-InSb, showing a minimum at about 2.4 T (a,z 2 a,,in the OLMR, and a weak maximum at 8.2 T in the OTMR. Both structures were attributed to spin-flip transitions (0- -, 1+, and 0- -,0+, respectively), although the implied g-factor of about - 65 is rather far from the accepted value of about - 50. Baranskii and Gorodnichii” called attention to the fact that different methods of crystal growth can greatly affect results of magnetoresistance measurements in n-InSb. They could detect no magnetophonon oscillations in samples prepared by the zone melting technique, presumably because of inhomogeneities,although electron concentrations were about equal to those in samples obtained by the Czochralski method. Bashirov and GadzhialievE4 very briefly reported magnetophonon oscillations in the transverse Nernst-Ettingshausen effect (the aXyof Pavlov and FirsovZ4)in n-InSb at 77°K. The observations showed two maxima, at 1.7 and 1.0 T, in the emf induced in the y direction, for a temperature gradient in the x direction. The two maxima match fairly well the N = 2 and 3 OTMR maxima in n-InSb, but do not align with the LMTEMF oscillations. Stradling and Wood34 studied both the OTMR and OLMR of n-InSb at several temperatures between 65 and 180°K. Taking into account unresolved structure due to the unequal spin splitting of different Landau levels, they calculated the band-edge effective mass. They laterS8 reinvestigated the OLMR and OTMR using secondderivative techniques. The band-edge effectivemass (see Table 11)and its temperature dependence were recalculated from the transverse results, taking into consideration the several transitions which can contribute to a given resonance peak, and other small corrections. At 180”K, they reported “extra” minima at about 2.3 and 1.4 T in the

’

P. I. Baranskii and 0. P. Gorodnichii, Fiz. Tekh. Poluprov. 2, 854 (1968) [English Transl.: Sou. Phys.-Semicond. 2, 708 ( 1 968)].

256

ROBERT L. PETERSON

OLMR, which they attributed to two-LO processes. An interesting new result was the observation of an increasing asymmetry of the high-field peaks in the OTMR with decreasing temperature. From 135 to 90”K, an additional peak became resolved on the high-field side of the N = 1 GF peak. This doublet was attributed to the unequal spin splitting of the lowest two Landau levels, due to the large nonparabolicity of the conduction band of n-InSb. A calculation of the peak positions associated with the two spinconserving transitions showed excellent agreement with the two peaks at N = 1. Aksel’rod and T s i d i l ’ k ~ v s k ialso i ~ ~ reinvestigated the OLMR and OTMR in n-InSb by the double differentiation technique in magnetic fields to above 10 T, at temperatures ranging from 93 to 144°K. Their earlier results31 were confirmed, including the “extra” OLMR minimum at about 2.5 T and the “extra” OTMR maximum at about 8.2 T. The amplitude of the extra minimum was observed to increase with temperature even when the GF minima began to decrease, as with ~ - G ~ AAksel’rod s . ~ ~ and Tsidil’kovskii attributed the extra structure to two-LO phonon scattering, showing agreement with a nonparabolic calculation based upon Eq. (54).As mentioned in Part I1 and Part 111, Section 4,the “extra” OLMR minima can be understood as due to the replacement of LO absorption with LO emission processes in limiting the mobility as the magnetic field passes through the pseudoresonance values. However, the 8.2-T maximum in the transverse configuration still is explained only in terms of a second-order LO process. Further, in a recent note Aksel’rod and Tsidil’kovskiis5”report a very broad OLMR minimum near 13 T, which they attribute to third-order LO phonon processes. If this minimum is confirmed, and no uther explanation is forthcoming (it cannot be explained as a pseudoresonance) the higher order processes may indeed be occurring. measured the Hall voltage in n-InSb [n(77”K) = Aksel’rod et 1014 ~ r n - ~under ] Ohmic conditions and observed minima at the GF resonance fields for temperatures above 120°K. Wood er d s 2 also observed resonance minima at similar temperatures and weak electric fields (but maxima with n-GaAs). Figure 9 shows some of their results on n-InSb, n-InAs, and n-GaAs. That the observationss2 were indeed of Hall voltage alone was verified by several checks, including the fact that when either the current or the magnetic field was reversed in direction independently, the oscillatory components were unchanged apart from a polarity reversal. Wood et d s 2 also concluded that two-carrier conduction played no role in their results (Section 5). M. M. Aksel’rod and I. M. Tsidil’kovskii, Fiz. Tekh. Poluprov. 7,402 (1973) [English n a n d . : Sou. Piiys.-Semicond. 7,289 (1973)l.

4.

THE MAGNETOPHONON EFFECT

1

0.5

"

n1

0.1

1

1

1

1.5

I

257

J

2.0

FIG. 9. Magnetophonon oscillations in the Hall voltage (with the exception of curves I and J) of three materials. The second derivative of the voltage, which is proportional to the component of the resistivity tensor concerned, is plotted against magnetic field. Each pair of recordings shows the effect of simultaneously reversing J and B. The current is in the same direction for the first recording of each pair, and a maximum in the experimental recordings corresponds to a peak in the Hall resistivity (p,,(. Also included is one set of curves for pxX obtained with GaAs which shows the effect on the transverse magnetoresistance of reversing J and B. Curves A and B: InSb, pxy,77°K; curves C and D: InSb, p x y ,160°K; curves E and F: InAs, p*,, 180°K; curves G and H : GaAs, p x y , 110°K; curves I and J: GaAs, p x s , 110°K. (After Wood et

258

ROBERT L. PETERSON

Other work on n-InSb is discussed in Part IV on hot e l e ~ t r o n s ~ ~ - ~ ~ . ~ ~ and Part V on

6. p-Type Indium Antimonide The first report of magnetophonon oscillations inp-InSb was by Amirkhanov et Using pulsed magnetic fields, they observed OLMR minima at 37 and 22 T at a temperature of 140°K.These were attributed to LO phonon scattering on the heavy holes, corresponding to the N = 2 and 3 GF resonance. Results for the transverse configuration were not reported. Stradling’z*’4 reports a study of the anisotropy of the OTMR peaks in the (110) plane at 90°K. Heavy-hole peaks for N ,< 12, as well as four light-hole peaks were observed. The light-hole series showed an extra peak attributed to the complex Landau level structure near the band edge, arising from the admixture of light- and heavy-hole states. A considerable deviation from periodicity in B - * in the heavy-hole series was also attributed to this effect. An anisotropy of about 20% was reported. Stradling et al.” also reported seeing magnetophonon oscillations in the Hall voltage of p-InSb, due, however, to the intrinsic conduction electrons. The rapid disappearance of the oscillations below 160°K was attributed to freezeout of the intrinsic carriers, and allowed the conclusion that twocarrier conduction was not responsible for the observed Hall voltage oscillations.

C. Hamaguchi, T. Shirakawa, T. Yamashita, and J. Nakai, Phys. Rev. Lett. 28, 1129 (1972); T. Shirakawa, C. Hamaguchi, and J. Nakai, J. Phys. SOC.Jap. 35, 1098 (1973). R. A . Wood and R. A . Stradling, Proc. Inr. Conf. High Magn. Fields Their Appl., Norringham. p. 47. Inst. Phys. Phys. SOC.,London, 1969. 88 R. 1. Lyagushchenko, R. V. Parfen’ev, 1. 1. Farbshtein, S. S. Shalyt, and I. N. Yassievich, Fiz. Tcerd. Tela 10, 2241 (1968) [English Transl.: Sou. Phys.-Solid Stare 10, 1764 (1969)]. 89 M. M. Aksel’rod, V. P. Lugovykh, R. V. Pomortsev, and I. M. Tsidil’kovskii, Fiz. Tverd. Tela 1 1 , 1 13 ( 1969) [English Transl. :Sou. Phys.-Solid Stare 1 1 , 8 1 ( 1969)l. yo V. Dolat and R. Bray, Phys. Ret,. Lett. 24, 262 (1970). 9 ’ N . Kotera, E. Yamada, and K. F. Komatsubara, J . Phys. Chem. SoZids33, 131 1 (1972). S . Morita. S. Takano. and H. Kawamura, Solid Stare Commun. 12, 175 (1973). 93 R. C. Curby and D. K. Ferry, Bull. Amer. Phys. SOC.17, 282 (1972); Proc. XI Inr. Conf. Phys. Semicond., Warsaw, p. 312, Polish Sci. Publ., Warsaw, 1972; Bull. Amer. Phys. SOC. 18, 354 (1973). y3a G . Bekefi, A. Bers, and S. R. J. Brueck, Phys. Rev. Lett. 19, 24 (1967); IEEE Trans. Electron Devices ED-14,593 (1967). 93b H. W. J. M. Niederer and R. G. van Welzenis, Appl. Phys. Leu. 23,41 (1973). 93c W. Racek, G . Bauer, and H. Kahlert,’ Phys. Rev. Lett. 31, 301 (1973). 94 Kh. I. Amirkhanov, R. I. Bashirov, and 2. A . Ismailov, Fiz. Tekh. Polupror. 2, 434 (1968) [English Transl.: SOP.Phys.-Semicond. 2, 356 (1968)l; Proc. I X Inr. Conf. Phys. Semicond., Moscow*,p. 744. Nauka, Moscow, 1969. 86

’*

4.

THE MAGNETOPHONON EFFECT

259

c. n-Type Indium Arsenide

Magnetophonon oscillations in n-InAs were first reported by Parfen'ev et aLi9 in a polycrystalline sample with n(90"K) = 1.25 x 10l6 cm-3 and ~ ( 9 0 ° K= ) 6.4 x lo4 cm2 V - ' sec-'. They observed two weak OLMR minima at 2.2 and 3.3 T at 90"K, corresponding to the N = 3 and 2 G F resonances. A later report3' extended this work to 25 T, uncovering the N = 1 minimum, and another minimum at about wc = 2w, which they attributed to the spin-flip transition O + --t I -, although noting disagreement with a nonparabolic band calculation. They found no OTMR oscillations. At about the same time, Tsidil'kovskii and c o - ~ o r k e r spublished ~~ similar results on single crystals with n(90"K) 2 3 x 10l6 cm-3 and ~ ( 9 0 ° K )d 5.4 x lo4 cm2 V - ' sec-'. Tsidil'kovskii and A k ~ e l ' r o d ~ ~ reported a striking minimum at 300°K in the OLMR of n-InAs, at the very high (pulsed) magnetic field of 48 T, which they attributed to a spin flip within the lowest Landau level; the implied g-factor is acceptable. They reported room-temperature OTMR maxima near 3 and 7 T, corresponding to the N = 2 and 1 G F resonances. The OTMR results were obtained by differentiating the voltage signal with respect to time in pulsed fields. An investigation by Amirkhanov and B a ~ h i r o v , ~who ' used dc techniques at 300°K in polycrystalline samples of quality similar to that of the preceding paragraph, revealed the N = 1 and 2 G F resonances in the OTMR. In another report96 of work at 300"K, they observed OLMR and LMTEMF extrema which approximately coincide, with a minimum in the LMTEMF near the N = I G F resonance. Work on similar material by Waller et ~ 1 . ~ ' showed very weak LMTEMF oscillations, which seemed to be shifted to lower fields than those of Amirkhanov and B a ~ h i r o v . ~ ~ The first relatively accurate OLMR measurements on n-InAs were made by Stradling and Wood34 on an epitaxially grown sample [n(77"K) = 3.9 x I O l 5 c m - j ; ~ ( 7 7 ° K )= 8.1 x lo4 cm2 V - ' sec-'I. The dc technique used showed the N = 1-4 G F minima. The N = 1 position shifted strongly ( 20%) to higher field when the temperature was raised from 77 to 300°K. The N = 2 position was relatively constant, and the N = 3 and 4 positions moved slightly to lower field with increasing temperature. At room temperature, additional minima at about 4.8 T and a little below 3 T were observed. These, and the minimum at w, z 20, observed by the Russian workers, were attributed to two-LO phonon processes rather than to transitions

-

Kh. I . Arnirkhanov and R. I. Bashirov, Fiz. Tekh. Poluproc. 1, 667 (1967) [English Transl.: SOL'.Phys.-Semicond. 1, 558 (1967)l. 9 6 Kh. I . Arnirkhanov and R. I . Bashirov, Fiz. T w r d . Telu 8. 3105 (1966) [English Trunsl.: Sou. Phys.-Solid S f a f e8, 2482 (1967)l. 9 7 W. M. Waller, J. R. Sybert. and H. J. MacKey, Phys. Lerr. 26A,477 (1968). 95

260

ROBERT L . PETERSON

involving spin flips. It now seems likely that at least part of the shifts of the extrema near the G F fields may be due to distortions caused by the developing "extra" minima with increasing temperature. Oscillations in the OTMR were not as well resolved. A later measurement5' using the more accurate second derivative technique permitted the first accurate measurement of the OTMR maxima, from which the band-edge effective mass was calculated (Table 11). Minima in the Hall voltage have been observed at the G F fields in n-InAs by Wood et ~ 1 . 'The ~ reported results were at 180'K on epitaxially grown material of 8.1 x lo4 cmz V - ' sec-' peak mobility and N , - N , = 7 x i015cm-3. Other work on n-InAs is discussed in Part IV on hot electron^.^^.^^ d. n-Type Gallium Arsenide

The first magnetophonon study on n-GaAa was made by Aksel'rod et on bulk samples [n(80"K) = 10'6-10'7 ~ m - ~~( 8; 0 ° K )z 5 x 10' cmz V - ' sec-'I. In the temperature range 200-400°K, they observed no oscillations in the transverse case and a broad oscillation in the OLMR, consisting of a maximum at about 20 T and a minimum at about 22 T. They interpreted this oscillation as showing an N = 1 G F maximum. The field position of this feature increased about 10% with increasing temperature in the range studied. Sherwood and Beckerz6 later reported both OTMR and OLMR oscillations in two samples of high-mobility, epitaxially grown nGaAs [n(77"K) z lOI4 ~ m - ~~( 7;7 ° K )% 5 x lo4 cm2 V - ' sec-'1. Their measurements were made in the temperature range 80-200°K in pulsed fields up to 17 T and revealed OLMR minima approximately coinciding with OTMR maxima at the N = 2, 3, and 4 G F resonances. From their results, shown in Fig. 10, one could expect to find an N = 1 OLMR minimum at ~ ~ about 22 T, showing that the feature observed by Aksel'rod et U I . should have been interpreted as a resonance minimum. The oscillation amplitudes were largest at about 140°K. At about the same time, Stradling and Wood34 published an extensive analysis for three materials, including epitaxial n-GaAs. The temperature and mobility dependences of both the OLMR and OTMR magnetophonon oscillation amplitudes were studied in four samples [ N , - N , = 5 x lOI3 to 4 x 10'' cm-'; ~ ( 7 7 ° K )% (2-10) x lo4 cm' V - ' sec-'1 at fields up to 14 T. Extrema up to N = 11 were seen. The OTMR oscillations at fields for which wc 5 t w oclosely approximate a damped sinusoid, suggesting a large amount of Landau level broadening, and also showing that electron-

'*

M. M . Aksel'rod. V. I. Sokolov. and I . M . Tsidil'kovskii, Phys. SiarusSolidi9. K163 (1965).

4. THE

MAGNETOPHONON EFFECT

261

~

PO

021

-

0 I6 -

0.15

-

0 12

-

009 006 -

0

3

6

9

12

B(T)

FIG. 10. Reproduction of oscilloscope photographs of the longitudinal and transverse cm- and p = 7.8 x lo4 Ohmic magnetoresistance at 134°K ofepitaxial n-GaAs with n x cm2 V-' sec-I at 77°K. (After Sherwood and Becket.26)

electrcn scattering plays no large role in limiting the transverse amplitudes." As with the Sherwood and Becker results,26 the oscillation amplitudes were found to be largest at about 140"K,and generally to increase with increasing mobility, although the latter relation is not a simple one. The N = 1 maximum was observed to shift 2*% to lower fields as the temperature was raised from 65 to 320°K in the transverse configuration; an opposite trend occurred in the longitudinal case, with a greater sample-to-sample variation. The band-edge effective mass at 65 and 300°K was calculated after taking into account corrections due to nonparabolicity and the polaron effect, and the resuIts were shown to agree well with those from other measurement techniques. Extra minima appeared in the OLMR of the highmobility samples at high temperatures, and were attributed to two-LO processes. Stradling and Wood5*later used the second derivative technique, allowing a more accurate determination of the OTMR peak positions. No asymmetry (i.e., no spin splitting) of the high-field peaks was observed, as expected, since the conduction band of GaAs is quite parabolic. These measurements

262

ROBERT L. PETERSON

of the peak positions permitted a more accurate determination of the bandedge effectivemass than earlier (Table II), and possibly constitute the most accurate effective-massmeasurements to date by any technique. Askenazy et aLg9 studied both the OTMR and OLMR of epitaxial n-GaAs, using dc techniques in pulsed fields up to 30 T and temperatures between 63 and 340°K. They observed several extrema, including the N = 1 G F resonance somewhat above 20 T. The OLMR positions increased by more than 10% from 63 to 340"K, while the OTMR maxima were not observed to shift with temperature. These results are in qualitative agreement with those of Stradling and Wood.34 Neifel'd and Tsidil'kovskii' O0 have recently investigated the OTMR of bulk n-GaAs [n(300"K) = 4 x 10'' ~ m - ~~(300°K) ; = 5 x lo3 cm2 V - sec- '3 in pulsed magnetic fields for temperatures from 300 to 400"K, using a contactless microwave method. Double time-differentiation of the signal from the wave reflected from the sample increased the resolution of the magnetophonon oscillations, but introduced an uncertainty in the peak positions estimated at 10%. They observed several "extra" maxima, which they attributed to two- and three-LO phonon scattering. No curves were shown. By inference, the extra structure is observed because of the avoidance of Ohmic contacts. Recall that pseudoresonances (Part I1 and Section 4) are not expected to occur in the transverse configuration. Wood et aLa2saw magnetophonon oscillations in the Hall voltage in two samples of epitaxially grown n-GaAs, of ND - N A FZ 10" and 1014 ~ m - ~ . Contrary to their results in n-InSb and InAs," maxima were observed at the G F fields. Maxima were also seen in the hot-electron regime. This and other hot-electron work is discussed further in Part IV.40942*101 A recent hydrostatic pressure experimentlo2on n-GaAs is discussed in Part V. e. p-Type Gallium Arsenide In his reviews, Stradling'2-'4 has mentioned unpublished magnetophonon studies in which both light- and heavy-hole peaks have been observed in the OTMR of p-GaAs in pulsed fields up to 30T. The light-hole series showed an extra peak, as in p-InSb, attributed to the complex structure of the Landau levels due to mixing of the light- and heavy-hole states. The fundamental fields and magnetophonon masses given in Table I1 are estimated from the curves published. S. Askenazy, J. Leotin, J.-P. Ulmet, A. Laurent, and L. Holan, Phys. Lerr. 29A,9 (1969); C. R. Acud. Sci. Paris 268, 4 12 ( I 969). l o o E. A. Neifel'd and I. M. Tsidil'kovskii,Fiz. Tekh. Poluprov. 5,2239 (1971) [English Trans/.: Soo. Phys.-Semicond. 5, 1957 (1 972)]. l o ' R . A. Stradling and R. A. Wood, Solid Srure Commun. 6, 701 (1968). Io2 G. D. Pitt, J. Lees, R. A. Hoult, and R. A. Stradling, J. Phys. C: Solid StutePhys. 6,3282 (1973). 99

4. THE MAG~TOPHONONEFFECT

263

f. n-Type Germanium The multivalley conduction band structure of a material such as Ge can be expected to produce a complicated magnetophonon behavior. In addition to the scattering within a given valley, scattering between valleys with emission or absorption of a high-energy, high-momentum phonon is possible. The resonance condition for scattering between valleys a and b can be expressed as EN,- EL, = hwq, where wq describes the relevant "intervalley phonon." In the harmonic oscillator approximation, this relation can be expressed asLo3 wq = $ma

- oa)+ Nu, - Lo,.

(57)

The cyclotron frequencies o,and o,are generally unequal because of the anisotropic mass. If the intravalley structure can be disentangled from the intervalley structure, the magnetophonon effect can provide valuable information about the relative scattering probabilities. Lutskii et af.'04 very briefly reported seeing several very large, nonperiodic magnetoresistance oscillations in n-Ge at temperatures near 400"K, apparently in the transverse configuration. The authors made no attempt to analyze their data, but ascribed the oscillations to the magnetophonon effect. Gluzman and Tsidil'k~vskii,'~~ using second-derivative techniques, were not able to find any oscillations in the OTMR of n-Ge above 340"K,but observed several maxima from 6.5 to 15.5 T below that temperature. The magnetic field was oriented parallel to the [loo] and [110] directions, the former being the direction for which the lowest lying valleys are equivalent. However, even in this direction, they saw more than a single series of oscillations. They correlated the maxima with calculated resonance positions due to both intervalley and intravalley scattering. Sokolov and Tsidil'kovskii'06 had earlier studied the OLMR in n-Ge , temperature range samples with n = 2 x lOI4 to about 10'' ~ m - in~ the 20-80°K,and pulsed fields up to 30 T. For the magnetic field direction along the [ 1 101 axis, they saw 13minima between about 4.4and 21.7 T, and ascribed some of these to transitions between equivalent valleys. The remaining structure was not assigned. Gluzman and Tsidil'ko~skii'~~ observed similar oscillations in the LMTEMF in n-Ge, in the temperature range 25-1 20"K, lo3

lo4

lo'

lo6

N. G. Gluzman and I. M. Tsidil'kovskii, Fiz. Tekh. Poluprou. 2, 1039 (1968) [English Transl.:Sov. Phys.-Semicond. 2, 869 (1969)J V. N. Lutskii, A. A. Zhirnov, and M. 1. Elinson, Fiz. Tverd. Tela 7 , 521 (1965) [English Transl.: Sou. Phys.-Solid Stare 7,415 (1965)l. N. G. Gluzman and I. M. Tsidil'kovskii, Fir. Tverd. Tela 10, 3128 (1968) [English Transl.: Sou. Phys.-Solid Stare 10, 2469 (1969)l. V. I. Sokolov and 1. M. Tsidil'kovskii, Fiz. Tekh. Poluprou. I, 835 (1967) [English Trmsl.: Sov. Phys.-Semicond. 1, 695 (1967)l.

264

ROBERT L . PETERSON

in both the BII [loo] and B(I[I 101 orientations. Correlations between the experimental minima and the theoretically possible minima were made. The large density of possible transitions within valleys and between valleys makes positive identification quite uncertain, however. Using second-derivative techniques, Eaves et d."' studied both the OLMR and OTMR in n-Ge samples of exceptionally high purity (N, N A = 2 x 1OI2 Contrary to Sokolov and Tsidil'kovskii,lo6 they found no OLMR oscillations at all. In the transverse configuration, much structure was found at 120°K (Fig. 11). For B // [loo], the conduction valleys are equivalent, and the observed peaks could all be accounted for by a single with an series (contrary to the result of Gluzman and Tsidil'ko~skii'~~) effective mass of 0.139me, where me is the free-electron mass. For Bll[llO], 109

8

7

6

N

% \

a

m

FIG.1 1 . Experimental recordings of -a2R/aB2 against E for n-type germanium at following orientations of E in the transverse (110) plane: (a) [loo], (b) [I001 + 18", (c) [IOO] 26", (d)[lOO] + 30°,(e)[100] + 38",(f)[lll],(g)[lOO] + 80",(h)[lOO].(AfterEaveset~l.~~')

+

lo'

L. Eaves, R. A. Stradling, and R. A. Wood, Proc. X Inr. Conf. Phys. Semicond., Cambridge, p. 816. U S . AEC, Oak Ridge, Tennessee, 1970.

265

4. THE MAGNETOPHONON EFFECT

two separate series could be identified, corresponding to masses of 0.362me and 0.105me. No peak at any orientation was identified with intervalley scattering, whose scattering probability was inferred to be an order of magnitude smaller than that for intravalley scattering. This implies that the less pure samples used by Tsidil'kovskii and cow o r k e r could ~ ~have ~ had ~ intervalley ~ ~ ~ ~ scattering ~ ~ assisted ~ ~ by impurities, or that there could have been small misalignments. g. p-Type Germanium

made the first magnetophonon investigation on p-Ge. Eaves et Working with samples of exceptionally high purity ( N , - N D = 3 x 10l2~ m - ~they ) , saw no oscillations in the OLMR, but a very rich structure in the OTMR (Fig. 12). Peaks due to transitions within the heavy-hole valence band were observed from the highest field of about 9 T down to about 3.5 T, at which field the N = 35 GF resonance occurs. At fields lower than 3.5 T, a weaker series of peaks was observed, attributed to transitions within the light-hole valence band. A substantial anisotropy (20%) of the heavy-hole mass was observed by rotating the magnetic field in the (110) 20

I

1

I

2

I

I

I

4

6

8

L

B(T)

FIG. 12. Experimental recordings of -d2R/dB2 against B for p-type germanium at following orientations of B in the transverse (100) plane: (a) [lOO], (b) [loo] + 15", (c) [I001 + 25", (d) [I001 + 40", (e) [ I l l ] , (f) [IOO] + 75". The downward arrows mark the position of the N = 20 peak for the heavy-hole series. (After Eaves er a/.107)

266

ROBERT L. PETERSON

plane, transverse to the current direction, while the light-hole mass showed only about 2% anisotropy. The heavy-hole peaks for N ,< 20 showed a wellresolved splitting into two components, for which a definite explanation could not be offered. It was suggested that the splitting may be the result of transitions between different values of k, to points on the warped heavy-hole band where the density of states is large. Stradling1'-14 refers to a subsequent study of OTMR oscillations, using pulsed fields up to about 20T. The splitting of the heavy-hole peaks was observed to develop into a yet more complicated structure at fields above 15 T.

h. p-Type Tellurium Because of its crystal structure, tellurium has several aspects not seen in cubic crystals. The constant-energy surfaces of the valence band are now known to be dumbbell shaped (Fig. 13). For a magnetic field applied perperpendicular to the principal (trigonal) axis C , ,the Landau level energies

FIG. 13. Dumbbell shape of the constant-energy surfaces of the tellurium valence band. showing different types of extremal cross sections at various levels of filling. (After Bresler and Mashovets.'12)

4. THE MAGNETOPHONON EFFECT

267

are nonlinear with magnetic field, and split in two above a certain field. For B C,,the Landau level energy as a function of kZ has a double minimum (Fig. 14). The phonon properties are also complex. The three atoms per unit cell create six optical modes. Near the zone-center r point, the A, and El

)I

E (meV1

FIG. 14. Two valence band Landau levels in tellunum at 3 = I5 T, showing the various types of magnetophonon transitions of holes when the magnetic field is parallel to the C , axis. The inversion asymmetry splitting is not included in the figure. (After Miura et n l . l L 3 )

modes show significant electron-phonon coupling.'08*'09 Further, Raman scattering datal10 have shown that there exist A,-E, coupled modes for phonons whose wave vector q is neither parallel nor perpendicular to C , . Thus, one can expect to find a rich and complex magnetophonon structure. Oscillations in the magnetoresistance of p-Te were first reported by Mashovets and Shalyt"' in samples of hole concentration p from 2 x IOl4 O9

'lo

G. Lucovsky, R.C. Keezer, and E. Burstein, Solid Stare Commun. 5.439 (1967). M. Selders, P. Gspan, and P. Grosse, Phys. Status Solidi 47, 5 19 (1 97 I ). W. Richter, J. Phys. Chem. Solids 11,2123 (1972). D. V. Mashovets and S. S. Shalyt, Zh. Ekspe. Teor. Fiz. Pis'mn Red. 4,362 (1966) [English Transl.; Sou. Phys.-JETP Lett. 4, 244 (1966)l; "The Physics of Selenium and Tellurium" (W. C. Cooper, ed.), p. 371. Pergamon, Oxford, 1967.

268

ROBERT L. PETERSON

to 4 x ( 3 x 1 1 At ~ ~ .77°K and with the current parallel to the principal axis C , , they observed maxima in the OTMR having a periodicity of A(l/B) = 0.03 T-', and corresponding to the N = 1-4 GF resonances. Minima in the OLMR were observed at 4.7, 7.0, 14, and 29 T ; the three lowest features give a periodicity of 0.07 T-', and correspond to N = 1-3. The minimum at 29 T was left unexplained, although spin-flip transitions were considered and rejected. The two distinct series were interpreted in terms of hole scattering on two different LO phonon branches. A more detailed investigation on the same samples was made later by Bresler and Mashovets' l 2 after the valence band structure of tellurium had been more fully investigated. In addition to confirming the earlier measurements,' ' they investigated the variation of several peak positions and amplitudes with angle between Band C, .With these data, and using a theoretical analysis of the coupling between the holes and the various optical phonon modes, together with the dumbbell model of the valence band, they concluded that the holes scatter principally from just the El mode. The large OLMR minimum at 28-29 T (w, x 20,) was considered in terms of a twoLO process, but the authors rejected this explanation on the grounds that no similar peak was seen in the OTMR. It seems likely that this is a pseudoresonance minimum ; recall that no pseudoresonance structure is expected in the transverse configuration. A still more detailed study of p-Te has recently been reported by Miura et af.'l 3 on single crystals for which p x l O I 4 cm-, and ~ ( 7 7 ° K x ) 6 x lo3 cm2 V - ' sec- They studied the OTMR and OLMR for the current J perpendicular as well as parallel to the C , axis. The double-differentiation technique used showed many more peaks than had been resolved earlier. For J (1 C, they observed peaks up to N = 10, in good agreement with the results of Mashovets et af.' '' l 2 in both the OTMR and OLMR. For J I C, 1B, the longitudinal resistivity minima appeared at 4.05, 4.60, 5.35, and 6.40 T. These almost coincided with the maxima observed for J I/ C, I B, but were lower in field than for the transverse maxima with J IC , I B. From consideration of the polarization and coupling strengths of the various phonon modes, they argued that the A , LO mode should provide the dominant contribution for J ( 1 C, ,and the E lmode for J IC,. From their results, Miura ef al. calculated the optical phonon frequencies. The discrepancies ( - 10%) from the phonon energies deduced from other experiments were attributed to the polaron effect. Still other structure was ascribed to scattering on finite-energy acoustic phonons. When B I(C , ,the OLMR oscillations were

'.

,'

"* 'I3

M. S. Bresler and D. V. Mashovets, Phys. Starus Solidi 39,421 (1970). N. Miura, R. A. Stradling, S . Askenazy, G . C a d r e , J . Leotin, J. C. Portal, and J.-P. Ulmet, J . Phys. C. SolidState Phys. 5, 3332 (1972).

4. THE MAGNETOPHONON EFFECT

269

about as large as those in the OTMR. This effect was thought due to the double minimum structure of the Landau levels, which allows several types of LO transitions (Fig. 14). i. n-Type Indium Phosphide

Eaves et al. 'I4 observed magnetophonon oscillations in both the QTMR and OLMR of epitaxially grown n-InP. For the best sample, N,, - N A = 2 x l O I 5 cm-3 and p(77"K) = 6 x lo4 cmz V-' sec-'. The oscillations were quite sinusoidal, except for the longitudinal measurement at the highest temperature (270"K), for which structure at the pseudoresonance fields had developed. The transverse peaks shift to lower magnetic field by about 3% between 77 and 300°K. An interesting feature is that the extrema in the longitudinal configuration do not closely align with the transverse extrema, the phase shift being about 90" (see Fig. 15). The reason for this is not definitely known, although level broadening may be responsible (Section 4). Other quite polar materials (CdTe and CdSe) also show this effect.' l4 The transverse peak positions were used to calculate the band-edge effective mass. This confirmed a cyclotron resonance result which was surprisingly

N

m

P cc

'D

I

1

I

4

6

I

8

B(T)

FIG.IS. Experimental recordings of -d2R/aB2 against magnetic field for fields up to 9 T in n-InP. Curves (i), (ii), and (iii) are taken with one sample in the transverse orientation at temperatures of 290,175, and 85"K,respectively. Curves (iv) and (v) are with another sample in the longitudinal orientation at 180 and 270"K, respectively. The last curve shows the development of extra structure with increasing temperature. (After Eaves er at. ' 14) 'I4

L. Eaves, R. A. Stradling, S.Askenazy, J. Leotin, J. C. Portal, and J.-P. Ulmet, J . Phys. C Solid Srute Phys. 4, L42 (1971).

270

ROBERT L. PETERSON

higher (e.g., 0.0817me at 50°K) than a theoretical estimate (0.0545me)based on the k * p method,' l S which had been expected to be good in view of its success with GaAs. At about the same time, Bashirov et al.' l 6 reported OTMR and OLMR oscillations in an n-InP sample of comparable quality to the best sample of Eaves er a1.Il4 The transverse peak positions agree quite well with those of Eaves et al., but the longitudinal peaks, indicated only by a figure, do not agree well. However, there is still a substantial phase shift between the transverse and longitudinal extrema, and pseudoresonance structure appears at the higher temperatures. Pitt et a1.'02 have recently reported the results of applying hydrostatic pressure to n-InP, causing shifts in the magnetophonon peaks. This is discussed in Part V. j . n-Type Cadmium Telluride

Mean et studied the very polar material n-CdTe, observing several oscillations in both the OTMR and OLMR in samples of maximum mobility greater than 2 x lo4 cm2 V-' sec-', and n(77"K) = 10I2to 2 x 1015 cm- '. The peak positions were found to be independent of temperature and mobility. The OLMR minima were displaced to significantly lower fields relative to the OTMR maxima, as with CdSe and InP. Since the LO phonon frequency, cyclotron mass, and degree of nonparabolicity are accurately known for this material, the authors were able to deduce that the OTMR peak positions were shifted from their expected position. They attributed the shift to the polaron i.e., to the fact that a strong electron-phonon coupling near resonance smears each Landau level into a continuum of states. The 13% increase of the magnetophonon mass over the low-frequency cyclotron mass is equivalent to an a/3 polaron contribution, where a is the Frohlich coupling constant given in Eq. (12). A later c ~ r r e c t i o n 'for ~ the nonparabolicity of the conduction band changed the estimated polaron contribution to 0.8(~(/3),which is close to the theoretical correction of 0.73(a/3)obtained by Palmer6' for the N = 1 peak. Wood et a1.82looked for, but did not find, magnetophonon oscillations in the Hall voltage of n-CdTe. k . n-Type Cadmium Setenide

Eaves et al." observed magnetophonon oscillations in both the OTMR and OLMR of n-CdSe, which is noncubic and has the largest Frohlich

'Is '16

F. H. Pollak, C. W. Higginbotham, and M. Cardona, Proc. V f f I I n r .Conf. Phys. Semicond., Kyoto, J. Phys. SOC.Japan Suppl. 21,20 (1966). R. I. Bashirov, A. Yu. Mollaev, and N. V. Siukaev, Fiz. Tuerd. Tela 13, 3097, 3432 (1971) [English Transl.: Sou. Phys.-Solid State 13,2597. 2892 (1972)J.

4. THE MAGNETOPHONON EFFECT

271

constant a of any material studied to date in the magnetophonon effect. The transverse results were used to deduce the band-edge effective masses and their anisotropy, probably the most accurate determination of these masses to date: 0.115m, for B I C,and 0.12Ome for B (1 C.The material was in the form of platelets deposited from the vapor phase, with a peak mobility at 40°K of 1.1 x lo4 cm2 V-' sec-'. The fundamental field is about 32 T. As with InP and CdTe, the longitudinal minima were shifted to significantly lower fields than the transverse maxima.

i. n-Type Cadmium Surfide Vella-Coleiro' investigated Raman scattering and photoconductivity at high magnetic fields (8-27 T) in high-resistivity crystals of CdS at 77°K. Electron-hole pairs were created by two-photon absorption from a pulsed ruby laser. When the GF resonance condition is met, the electrons so created can fall readily into lower Landau levels by LO phonon emission. The surplus of LO phonons can be expected to enhance Raman scattering as well as affect the conductivity. Vella-Coleiro could match many of the oscillations(peaks in the Raman scattering,minima in the photoconductivity) with the GF resonance condition including spin-flip transitions. From this, he deduced the transverse and longitudinal effective masses and g-factors. The values deduced were close to those of Zeeman effect measurements, but the effective masses were higher than those obtained from cyclotron resonance by about 30%. Other oscillations did not fit the mentioned resonance condition. This type of experiment is clearly capable of providing much detailed information, and deserves to be repeated on this and other materials. m. n-Type Mercury Teiluride and Gallium Antimonide

Oscillations in the OLMR of n-HgTe, a semimetal, and n-GaSb at temperatures from 20 to 80°K in samples in which the electron concentrations were in excess of l O I 7 cm-3 have been reported by Tsidil'kovskii and P ~ n o r n a r e v " ~and ~ ~ Pomortsev '~ et In the early papers,' some of the oscillations were interpreted as magnetophonon and spin-magnetophonon oscillations. Pomortsev et al.77developed a theory applicable to the degenerate case (see Section 4) in which there could be oscillations due to transitions of electrons between the Fermi level and the Landau levels, I17

G . P. Vella-Coleiro, Phys. Rev. Lett. 23,697(1969);Proc. 3rd hi.ConJ Photocond., Stanford, 1%9(E. M. Pell, ed.), p. 351. Pergamon, Oxford, 1971. I. M. Tsidil'kovskiiand A. 1. Ponomarev,Int. Conf: I/-VISemicond. Compounds, Providenee (D. G . Thomas, ed.), p. 1103. Benjamin, New York, 1967. A. I. Ponomarev and I. M. Tsidil'kovskii, Fiz. Tekh. Poluprou. 1, 1656 (1967) [English Trunsl.: Sou. Phys.-Semicond. 1, 1375 (1968)l.

272

ROBERT L. PETERSON

with absorption or emission of an LO phonon, occurring at magnetic fields given by Eq. (43). The theory developed7' suggests that OLMR maxima should exist at these positions, and indeed maxima were observed which roughly correlated with some of the positions given by Eq. (43). However, a more systematic study is needed before these maxima can be identified with certainty. n. n- andp-Type Lead Telluride Very recently Tsui et al.' 2 o have observed magnetophonon resonances in the OTMR of degenerate samples of n-PbTe at temperatures from 57 to 77°K. At 77°K and with a sample of n = 5 x 1017 c mP 3 a d ~ ( 7 7 ° K )= 3.5 x lo4 cm2 V - ' sec-', they saw a single resonance series with three oscillatigns, corresponding to N = 2, 3, 4, and an implied fundamental field of about 10 T for B 1) [I 101and J 11 [liO]. The amplitudes were about 1 % of the magnetoresistance. Rotating the magnetic field in the (110) plane, for which the expression (rn, sig - 1 ___ NeB, = w,(mp,)112 1 +

[

)

can be used, they deduce m , = 0.029me and m , = 8.7m,, assuming that the unscreened LO phonons just away from q = 0 are involved."' In Eq. (58), B is the angle between B and the [I101 direction, and m, and m , are the longitudinal and transverse effective masses of the ellipsoidal valleys at the L symmetry points of the Brillouin zone. These values of m,and M , are in good agreement with the low-frequency cyclotron masses obtained at helium temperatures. Tsui et al. also conclude that there is 00 polaron enhancement of the magnetophonon mass, to within experimental accuracies, in contradistinction to results on n-InP, n-CdSe, and n-CdTe. Sreedhar et a1.lZ2had earlier found oscillations in both the OTMR and OLMR of p-PbTe at 77°K in thallium-doped single crystals of p > lo'* cm-3 and ~ ( 7 7 ° Kx) 1.5 x lo4 cm2 V-' sec-'. Four OTMR ascinations of unusually large amplitude ( - 10%) were seen below 1 T for B 1 [OlO] and J 1) [loo], from which a fundamental field of 3.3 T was deduced. The authors ascribed the oscillations to the magnetophonon effect because no dependence upon carrier concentration was noted. The implied transverse hole mass (0.017rne for the unscreened LO phonons near q = 0, or 0.025m,For the phonons at the X points) is considerably below values quoted from other measurements. Additional studies on the p-type material are warranted. D. C. Tsui. B U N . Amer. Phys. SOC.18, 17 (1973); D . C. Tsui, G. Kaminsky, and P. H. Schmidt, Solid State Commun. 12, p. 599 (1973). I Z 1 R. A. Cowley and G. Dolling, Phys. Rec. Lett. 14,549 (1965); W. Cochran, R. A. Cowley, G. Dolling, and M. M. Elcombe. Ptoc. Roy. SOC.(London)A293,433 (1966). "'A. K. Sreedhar, N . Chaudhuri, P. Wakhaloo. and R. S. Wadha, Phys. Lerr. 29A, 398 (1969). IZo

4. 0.

THE MAGNETOPHONON EFFECT

273

n-Type Mercury Cadmium Telluride

Kahlert and BauerlZzahave reported the observation of OLMR oscillations in n-Hg,-.Cd,Te (x = 0.212) in samples of n(4.2"K) = 1.6 x 10'' cm-3 and p(4.2"K) = 8.56 x lo4 cm2 V - ' sec-'. The band edge effective massmo* = 0.005 m,andg-factor go = - 172werededuced from Shubnikovde Haas measurements at 4.2"K. From these results, and using the nonparabolic energy expression (19), they found that the magnetophonon peaks involved only the LO phonons characteristic of HgTe (17.1 meV). Peaks associated with the other known LO phonon branch characteristic of CdTe (19.6 meV) were not seen. Spin splitting of the N = 1 GF peak was well resolved. The peaks were observed to shift to higher fields linearly with temperature from 50 to 130°K. Kahlert and Bauer analyzed this shift in terms of a positive temperature coefficient of the energy gap (dEJdT = 7.6 x 10-4eV/"K) which however is not in good agreement with other measurements. In view of the factors which influence the peak positions in the longitudinal configuration (merging of the theoretical resonance minima and off-resonance maxima as discussed in Part 11 and Section 4, and possible distortion due to the onset of pseudoresonances at higher temperatures), the shift in peak positions in the transverse configuration may be a more reliable measure of dEJdT.

IV. Hot-Electron Regime 7. THEORY The electron-temperature concept is a popular method for discussing and theoretically analyzing experiments in the non-Ohmic regime. It is powerful in that it permits a substantial amount of theoretical analysis in a reasonably straightforward manner. For example, its use is independent of whether or not a relaxation time exists. Its disadvantage is that one seldom knows just how quantitatively accurate one's analysis will be, since it is very difficult to establish with confidence both necessary and sufficient conditions for the validity of an electron temperature. Conwe115* reviews some of the sufficient conditions, which state in effect that the electrons should interact with each other much more frequently than with any other entities, such as phonons or impurities. Pomortsev and K h a r ~ s " ~made a theoretical study of the magnetophonon effect in non-Ohmic electric fields from the electron temperature point of view. They calculated on this basis the rate at which energy would H. Kahlert and G. Bauer, Phys. Rev. Left.30,121 1 (1973). R. V. Pomortsev and G. I. Kharus, Fiz. Twrd. Tela 9, 1473 (1967) [English Transl.: SOC. Phys.-Solid Store 9, 1150 (1967)l.

lZza

lZ3

274

ROBERT L. PETERSON

flow from the electrons into the LO phonon system. They found that in the absence of level broadening, the power transferred diverges at the GF resonances, accompanied by the electron temperature returning to the lattice temperature. Assuming that the hot-electron longitudinal magnetoresistance varies inversely with some power of the electron temperature, because of an assumed dominance of acoustic phonon or impurity scattering (which give respectively, in the quantum limit), they approximately T,- 'lZ and Te-312, deduced that this case should exhibit resonance maxima. In a following paper, Pomortsev and Kharus"' generalized their considerations to include the possibility that the LO phonon temperature might rise above the background lattice temperature, which removes the divergence in the power transferred at resonance. Yamada and K u r o s a ~ a and ~ ~ YamadalZ6 ~ ' ~ ~ also considered the hotelectron problem theoretically, but avoided the electron temperature concept by assuming that the energy exchange of the electrons with the electric field and the phonons can be described by diffusion and continuity equations representing Brownian motion in energy space. How this technique compares in accuracy with one invoking an electron temperature is not yet clear. The diffusion approximation in the face of scattering by optical phonons makes the validity of the theory difficult to assess. Nevertheless, the diffusion approach is capable of yielding the expected kinks in the electron energy distribution function due to optical phonon emission which has not yet been accomplished in the electron temperature approach. The basic equations, formulated first by K u r o s a ~ a , are ' ~ ~the diffusion equation

48) = -)"

(W4 + 4 o v b ) f ( 8 ) ,

(59)

applicable when energy changes are sufficiently small, supplemented by a continuity equation dl/d& =

ym,abs(g

f hwo)N(gk hwo)f(df hao) -

~m,,bs(&)~v(~)f(&).

(60) Here I(&)is the electron particle flow in energy space, N ( E )is the density-ofstates function, D ( I ) is a diffusion coefficient, v(B) is the electron velocity, and f(8)is the distribution function to be determined. The continuity equation describes large energy changes due to optical phonon scattering.

126

12'

R. V. Pomortsev and G. 1. Kharus, Fiz. Tcerd. Tela 9, 2870 (1967) [English Transl.: Soc. Phys.-Solid Siare 9, 2256 (1968)l. E. Yamada andT. Kurosawa, Proc. I X Inr. Con$ Phys. Semicond.. Moscow. p. 805. Nauka, Moscow, 1969. E. Yamada, Proc. X l l n t . ConL Phys. Semicond., Warsaw. p. 274. Polish Sci. Publ., Warsaw. 1972; Solid Stale Commun. 13, 503 (1973). T. Kurosawa, J . Phys. SOC.Japan 20,937 (1965).

4.

THE MAGNETOPHONON EFFECT

275

Yamada and K u r ~ s a w a ~2 5~ .omit ' the optical phonon absorption processes in Eq. (60), and assume that the distribution functions in the different Landau subbands are all equal, having the distribution f ( B ) in common. Inserting approximate expressions for D(b) and u(&), they calculate the energy distribution function for various strengths of crossed electric and magnetic fields. These distribution functions are distinctly nonMaxwellian, and show slope discontinuities at energies equal to multiples of the cyclotron energy. With these distribution functions, and parameters appropriate to n-InSb, Yamada and Kurosawa calculated that at very high electric fields, the hot-electron transverse magnetoresistance would show minima at the G F resonances. As will be discussed further in the following section, the set of hot-electron magnetoresistance oscillations in n-InSb at 20"K, which can be correlated with LO phonon emission, does show minima in both the transverse and longitudinal orientation^.^^*^' However, transverse maxima are seen in n-InSb at 77°K under warm-electron conditions,86 and n-GaAs, n-InP, and n-CdTe all show hot-electron magnetoresistance maximd4 in both transverse and longitudinal orientations. Kotera et a1." extended the diffusion theory to include the longitudinal case. The distribution function and current-voltage characteristics were calculated for n-InSb at several magnetic fields, taking into account scattering on LO and acoustic phonons, and ionized impurities. Although the effect of LO emission was discussed in these calculations, the magnetophonon oscillations were not analyzed. The longitudinal magnetophonon aspects have been treated by Yamada,lZ6 who finds resistance minima as in the transverse case. Ferrylz8 has improved upon the diffusion equation approach of Kurosawa,'" Yamada and and YamadalZ6by removing two approximations used by those workers. He allows for absorption of optical phonons at the low temperatures and for the disturbance of the optical phonon distribution. Both are found to be important in the quantitative aspects of the energy distribution functions. Ferry's analysis treats only the B = 0 case at present, but has important implications for finite B situations. Magnusson'28aavoids use of both the diffusion approach and the electron temperature approach by solving the Boltzmann equation numcrically for parallel electric and magnetic fields of arbitrary magnitude, using a Chambers method generalized by BuddiZsband Rees.128' He considers scattering by optical and acoustic phonons, and ionized impurities, in a polar semiconductor with a single parabolic, isotropic band, and computes the electron D. K . Ferry, Phys. Rev. B 8, 1544 (1973). B. Magnusson, Phys. Status Sofdi52, 361 (1972). H.Budd, Phys. Rev. 158, 798 (1967). '*" H. D. Rees, J. Phys. Chem. Solids 30,643 (1969)

276

ROBERT L. PETERSON

distribution function under various conditions. He finds a considerably more complex structure than found by Yamada.’ 26 However, Magnusson does not use these distribution functions in analyzing the hot electron magnetophonon effect. Further work along these lines would very instructive since this method in principle is exact. Frohlich and Paranjape’” have shown that when an electron temperature T, exists, the electron distribution function for arbitrary electric field in an isotropic material with a parabolic conduction band is Maxwellian with a width measured by T,, and whose mean is displaced by the drift velocity vd: f(v)

- exp[ -m*(

v - ~ , ) ~ / 2 k T. , ]

(61)

In some cases, a nondisplaced Maxwellian at temperature T, is used.52In the presence of a quantizing magnetic field parallel to the electric field, a displaced Maxwellian can be written for each Landau level. In an isotropic material, the distribution function for the nth level takes the form

-

f,(u,) exp[-(nhoJkT,) - m*(u, - uJ2/2kT,]. (62) The displaced Maxwellian has been used for examining hot-electron magnetophonon structure in the longitudinal configuration in both polar and nonpolar material^.^^.'^^,'^' The transverse case cannot be treated in this way since the displaced-Maxwellian distribution function is a “diagonal element” of the density matrix, and “off-diagonal elements” are needed in ) . use the displaced-Maxwellian the transverse case (see, e.g., Kubo et ~ 1 . ~To technique, one writes the steady-state equations for carrier momentum and energy. For E and B in the z direction, these are

where af,(k,)at is the rate of change, due to scattering, of the distribution function for the nth Landau level, given in Born approximation by the right side of Eq. (44). Equations (63) and (64) are just the first and second moments of the Boltzmann equation, and together they determine T, and ud as functions of E, B, and the lattice temperature T. In calculating the longitudinal magnetoresistance for purely LO phonon scattering in nonpolar materials, Peterson’ 30 showed that the resonance 129

I3O

’”

H. Frohlich and B. V. Paranjape, Proc. Phys. SOC.(London) B69,21 (1956). R . L. Peterson, Phys. Rer. B 2 , 4135 (1970). R. L. Peterson, B. Magnusson, and P. Weissglas, Phys. Status M i d i 46,729 (1971).

4.

THE MAGNETOPHONON EFFECT

277

extrema could be maxima or minima, depending on lattice temperature and strength of the electric field. The trend observed was that maxima would occur in the warm-electron regime, which would convert to minima as the electric field increased. The extrema are due primarily to energy relaxation effects,as contrasted to momentum relaxation in the Ohmic case. used the displaced-Maxwellian technique in analyzing Peterson et al. hot-electron phenomena in polar materials, considering the combined effect of scattering on LO phonons and on acoustic phonons via the deformationpotential interaction. A subsequent analysiss5 extended this work by considering scattering on ionized impurities and acoustic phonons via the piezoelectric interaction, in addition to the previous two. Similarly to the nonpolar case, the extrema in the longitudinal magnetoresistance were found to be either maxima or minima, depending upon which scattering mechanism is brought into predominance by means of varying the lattice temperature or electric field. Figure 16 illustrates the behavior at low temperature (20°K for InSb). For the typical amounts of scattering assumed, ionized-impurity scattering dominates at this temperature. The Ohmic curve shows no magnetophonon structure whatever. As the electric field is raised, resonance maxima develop, principally because of the inverse

*c /*o

FIG.16. Ohmic and hot-electron magnetoresistance in a polar material at y = ho,/kT = 14, as calculated from the displaced-Maxwellian distribution function. The hot-electron curves are evaluated for constant-current conditions; the numbers refer to values of the constant ' ~ . carriers are assumed to scatter from optical dimensionless drift velocity ( m * ~ , ~ / 2 h w , ) 'The and acoustic phonons and ionized impurities of concentration 4 x l O I 3 ~ m - The ~ . average value of the magnetic-field-dependent electric field across the sample is about 1 V cm-' for the 0.015 curve, and about lOVcm-' for the 0.21 curve, when InSb parameters are used. (After Peterson.5s)

278

ROBERT L. PETERSON

dependence of ionized-impurity scattering on electron temperature. At yet higher electric fields, the extrema have converted to minima because of the increased importance of LO phonon scattering in determining the electron temperature. The minima become increasingly pronounced with increasing electric field.'31 As noted earlier, this general behavior is seen in n-InSb?0*86*89 How well the experimental and theoretical results correlate has yet to be demonstrated. 8. HEATING BY ELECTRIC FIELDS

The first observation of the magnetophonon effect in the hot-electron regime was by Kotera et af.41who studied both degenerate and nondegenerate samples of n-InSb at 1.5"K at electric fields up to about 1 V cm-'. At these low temperatures, absorption and emission of optical phonons is completely negligible in Ohmic fields; a large electric field can make some of the electrons energetic enough to emit LO phonons, however. In their nondegenerate samples, Kotera et aL4* observed oscillations, having the magnetophonon periodicity, in the "warm-electron'' coefficient fi in the conductivity expression o(E) = a,(l - BE2). A related experiment was ~ on n-InSb, but at 77"K, with later carried out by Hamaguchi et ~ 1 . ' also electric fields of amplitude less than 1 V cm-'. They used a differential technique which measured the third harmonic of a sinusoidal voltage applied to the sample. The third harmonic is proportional to fi, and showed the N = 2-7 resonance minima in the transverse configuration, corresponding to resistance maxima (Fig. 17). The warm-electron peak positions were in general agreement with those deduced from the nonparabolic expression (19), and slightly down-shifted from the OTMR results at 77"K4' Photoconductivity measurements by Kotera and c o - w o r k e r ~ ~have ~*'~~ been made on n-InSb at helium temperatures in which the electrons were heated by a large electric field. The results show a ripple in the photoresponse current at about 1.5-1.7 T. The authors speculate that this may be a manifestation of the magnetophonon effect. Since no other oscillations were seen at different magnetic fields, this remains a conjecture. Stradling and Wood40*87*'0'made the first detailed investigations of magnetoresistance in the hot-electron regime. Their first report"' concerned epitaxial n-GaAs at 20°K in electric fields of 1-10 V cm-' (fields of about 0.05 V cm- ' are typical for Ohmic studies at higher temperatures). Maxima were observed to coincide in the transverse and longitudinal configurations, but occur at about 20% lower field than the high-temperature OTMR peaks. Further, the maxima are not accurately periodic in 1/B.The peak positions 132

N . Kotera, R. Ito, and K. F. Komatsubara, Proc. IX Int. Cot$ Phys. Semicond., Moscow, p. 773. Nauka, Moscow, 1969.

4. THE MAGNETOPHONON EFFECT

279

FIG. 17. Magnetic field dependence of the warm-electron coefficient B in n-InSb at 77"K, with n = 1.8 x 10'4cm-3 and p = 5.8 x 105cm2V-' sec-'. The change A/3 obtained by subtracting the data points from the envelope is shown by the solid curve. (After Hamaguchi er 01.'~)

were identical for the three samples studied [n = 5 x l O I 3 to 2 x 10'' ~ m - ~ ; peak mobilities = (4.0-6.5) x lo4 cm2 V-' sec-'I, but the amplitudes varied. The authors explained their results in terms of LO phonon emission from the Nth Landau level to the ground state of a shallow donor. They postulated that resistance maxima should occur at the resonance condition hw, = N h ,

+E@),

(65)

where the magnetic-field-dependent activation energy of the donor can be approximated by E,(O) hwJ2, provided the magnetic field is not too large. Since the zero-field activation energy can be calculated or measured, Eq. (65) can be tested. The fit to the observed peak position is very good, not only for n-GaAs,'" but for n-InP and n-CdTe as ell.'^-'^ A later s t ~ d y ~ ' , ~ ~ in n-GaAs showed the appearance of yet another series at lower (but nonOhmic) electric fields. This series was characterized by maxima which were periodic in 1/B, with a fundamental field of 1 1.9 T. They were attributed to transitions between Landau levels involving the emission of two transverse acoustic (TA) phonons from the Brillouin zone boundary (Xpoint). A study on n-CdTe42*87 revealed structure which also could be identified as arising from capture of conduction electrons at the ground state of a shallow donor. In addition, as the electric field was raised while the lattice temperature was maintained below 20"K, many more peaks came into

+

280

ROBERT L. PETERSON

evidence. One series could be identified as the same as that observed in the OTMR at higher temperatures, and another series was tentatively attributed to a process involving emission of two zone-edge phonons. Hot-electron studies on n-InSb [n(77"K) = 2 x I O l 4 ~ m - ~(77°K) ~ ; = 5 x lo5 cm2 V-' sec-'1 by Aksel'rod et ~ 1at temperatures . ~ ~ from 16 to 30°K and electric fields up to 12V cm- ' showed resistance minima in both the transverse and longitudinal configurations, and no significant shift in positions from the Ohmic case, except for the N = 1 GF resonance, which occurred at about 2.9 T in the transverse case, contrasted with their Ohmic result of 3.5 T. Second-derivative techniques were used. Hot-electron measurements on high-purity n-InSb by Stradling and In fact, Wood40,42,87 generally confirmed the results of Aksel'rod et three series were seen, two of which consisted of maxima, and were correlated with emission of two zone-edge phonons from the lowest transverse acoustic (TA) phonon branches. In addition, some isolated minima were observed in the longitudinal configuration, which were not identified. Peterson37 called attention to the fact that this structure lay close to the pseudoresonance positions. Figure 18 shows some of the hot-electron results obtained by Stradling and Wood4' in n-InSb. A complex warm-electron structure, but less well resolved, was also observed in ~ - I ~ A S . Two-TA ~ ~ . ~ ' phonon emission was also identified in the hot-electron spectrum of n-InP. l 4 With each of n-InSb, n-GaAs, and n-InP, the series attributed to twophonon emission were the first identifiable magnetophonon structures to appear as the electric field was increased. The series attributed to LO phonon emission to impurity levels appeared at higher fields. Although the amplitudes of the oscillations varied with electric field, the peak positions did not, and also were found to be independent of lattice temperature and carrier c~ncentration.'~ Dolat and Braygo measured the acoustoelectric gain in n-InSb at 77 and 4.2"K when a magnetic field was applied parallel to a non-Ohmic current in the [110] direction through the sample. This direction is favorable for the acoustoelectric generation of shear-wave phonons, and gain is expected when the electron drift velocity considerably exceeds the acoustic shear-wave velocity. A delayed increase in voltage across the sample under constantcurrent conditions indicated the growth of acoustic flux. The delayed voltage increase was observed to be magnetic-field-dependent and to exhibit maxima near the GF resonance conditions. The results are shown in Fig. 19. Dolat and Bray argued that the acoustic gain should vary inversely as the electron temperature T, , which drops to near the lattice temperature at the resonance fields. Peterson et ~ 1 . provided ' ~ ~ some confirmation of tliese ideas by developing-an expression for the gain in terms of T , , using a Vlasov-like equation. The T, dependence was found to be nearly of the form T,- '.They

4.

0

THE MAGNETOPHONON EFFECT

I

1.0

0.4

281

1.6

B(T)

N

0 \

a ‘p

0

I

FIG.18. Experimental recordings of -dR/dB2 against B in n-InSb at a lattice temperature of 20°K. Th ey gain is different for different recordings. The electric fields down the sample at low magnetic fields are as follows (in mV cm-I): (i) 500 (taken with pulsed electric fields), (ii) 450, (iii) 650 (taken with pulsed electric fields), (iv) 450, (v) 300, (vi) 50, (vii) 20 (with a lattice temperature of 77°K).All recordings with the exception of (vii) were taken in the longitudinal configuration. (After Stradling and Wood.40)

calculated T, numerically with the displaced-Maxwellian approach for combined polar LO and deformation-potential acoustic phonon scattering. The acoustic gain was computed, showing maxima at the GF resonance fields (Fig. 20). Another manifestation of the magnetophonon effect in acoustoelectric amplification is the appearance of Bekefi peaks in the microwave emission from InSb at low temperature^.^^" The explanation for these peaks was given recently by Niederer and van W e I z e n i ~as~a~resonance ~ cooling of

282

ROBERT L. PETERSON

43 2

n

I = n at 77%

FIG. 19. Magnetic-field dependence of the acoustoelectric voltage AV, in n-InSb with x 10'3cm-3andp = 6 x 105cm2V-'sec-'at77"K.Curves(a)and(b)wereobtained

=4

at sampling times sufficiently long after the start of a constantcurrent pulse ( - 5.9 and 6.2 psec) to altow for buildup of acoustic flux. Curve (d) represents the magnetoresistance of the sample measured at a time before the acoustic flux buildup. (After Dolat and Bray.")

the hot electrons. The peak positions were shown to match quite well the resonance condition for capture of electrons at donor sites with emission of an LO phonon. The Hall voltage in GaAs has also been measured under hot-electron conditions*' as well as in the Ohmic regime (see Section 6). Maxima were observed at the magnetic fields at which maxima occured in the longitudinal and transverse hot-electron magnetoresistance, which, as mentioned

4.

0

0

283

THE MAGNETOPHONON EFFECT

0.5

I .o

1.5

2.0

wc/wo

FIG.20. Theoretical acoustoelectric gain as a function of magnetic field for three constant drift velocities, indicated on the curves by values of (rn*o,2/2h~,)1’2. The vertical scale is (hwo/kTJA1’*e-A, where J. = m*cd2/2kT,.The electrons are assumed to scatter on optical and acoustic phonons in polar materials. (After Peterson et al. I 3 l )

above, was 20% down-shifted from the Ohmic transverse results, and interpreted in terms of electron capture at donor sites. Hot-electron Hall voltage oscillations were also sought in InSb, InAs, and CdTe, but were not found.82 Curby and Ferry93have briefly reported results in n-InSb at 77”K, which show that an increasing electricfield causes a significant shift in the rhagnetophonon extremal positions to higher magnetic fields, in both the transverse and longitudinal configurations. The implied increase in the magnetophonon mass, defined through Eqs. (1) and (2), can be explained in a number of ways. The electric field helps to populate the higher Landau levels, which are flatter and thus give rise to a higher average effective mass. Also, the longitudinal case was for B in the [ 1101 direction, and generation of acoustic flux may have affected the results since the electric fields used (up to 170 V cm-’) were of the same order as those of Dolat and Bray,” and sampling times greater than 2 psec were used with some of the data. Still another explanation is that the shift may be due to the natural distortion of the curves as the electric field is increased, as illustrated in Fig. 16 and by

284

ROBERT L. PETERSON

Peterson13' and Peterson et a1.131 No shifts were indicated in the warm. ~ at~77"K, in which the electric electron study of Hamaguchi er ~ 1in n-InSb field amplitude was restricted to less than 1 V cm-'. Recently, Racek et a1.93chave used time-dependent techniques to observe the growth of the hot electron magnetophonon peaks in n-InSb at 11"K, and thus were able to estimate the energy relaxation times associated with emission of two TA phonons and single LO phonons.

9. HEATING BY IRRADIATION ~ ~ and Shubnikov-deHaas oscillations Parfen'ev et ~ 1saw. magnetophonon simultaneously in the photoelectric coefficients of n-InSb at helium temperatures in crystals with n s loi7 cmP3.Several GF minima were observed in the photomagnetic-induced voltage. The amplitudes decreased rapidly with increasing temperature, and oscillations were not observed at 77°K. They also could not be found in samples of n = l O I 4 cm- '. The appearance of the magnetophonon oscillations was explained by Parfen'ev et al.43 as due to heating of the electrons by the incident radiation (of about 0.5-2.8 pm wavelength), so that a significant number of the electrons became energetic enough to emit LO phonons. Lyagushchenko and co-workerss8.' 3 3 developed a theory for explaining the oscillations by assuming the existence of an electron temperature. In this theory, the photodiffusion current depends upon the electron temperature gradient as well as on the electron concentration gradient. The short-circuit current of the "odd" photomagnetic effect (the current, radiation, and magnetic field directions being mutually orthogonal) is then proportional to the Nernst coefficient Q, and is given by

+ c;J/cyyI[T,(O) - TI,

J = -Q M ( c ; y

(66)

where d is the dimension of the sample along B, and T,(O) is the electron temperature on the illuminated surface. The observed GF minima43-88were explained as due to a decrease in the electron temperature, and consequently the photomagnetic current or voltage, at resonance. The photoconductivity of n-InSb at 4.2"K has been measured by Morita et ~ 1 . ~They ' used CO, laser irradiation (10 pm) to heat the electrons, which is efficient for this purpose, since the CO, photon energy (117 meV) is about six times the LO phonon energy in n-InSb (24 meV). Many oscillations were observed, which were attributed to the magnetophonon effect, the explanation being that electrons lifted high into the conduction Landau levels cascade down, emitting LO phonons, when the resonance condition is met. Many of the photoresponse minima correlated well with GF resonances and with 133

R. I. Lyagushchenko and I. N . Yassievich, Zh. Eksp. Teor. Fir. 56, 1432 (1969) [English Transl.: Sou. Phys.-JETP 29, 767 (1969)l.

4.

THE MAGNETOPHONON EFFECT

285

spin-flip transitions with emission of an LO phonon. Two minima were identified as due to two-phonon emission, but in these cases, agreement with the expected positions is not very good. The photoconductivity and Raman scattering experiments of VellaColeiro,’” who created energetic electrons in CdS by radiation from a pulsed ruby laser, have been discussed in Section 6. The irradiation techniques have thus proved capable of bringing out a large amount of magnetophonon structure, in fact more than can be explained at present, and appear to be a promising method of investigation. V. Effects of Stress

The magnetophonon effect has been used for determining the effect of compressive stress on the effectivemass in semiconductors. This is accomplished by observing the change in period of the magnetophonon oscillations, which is e/m*o, in the parabolic band model by Eqs. (1)and (2). When the fractional change of o, with stress is much smaller than that of the energy gap (and hence of m*),44*48 the measurement gives the change of m* with stress rather directly. Figure 6 illustrates the effect of uniaxial stress on the rnagnetophonon oscillations in n-InSb at 77°K. Itskevich et aL4*showed that the resonance peaks in the OTMR of n-InSb shift to higher fields with application of hydrostatic pressure. From the shift of the N = 2 peak, they deduced an increase in the magnetophonon mass from 0.016me at zero pressure to 0.025me at 8 x lo8 Pa (pascal; see the appendix). They also established that the effective mass is directly proportional to the energy gap, confirming Kane’s theory. 5 0 A similar measurement on n-InSb was later made by Aksel’rod et al.,49 who improved the accuracy by studying the N = 1 peak and by using a second derivative technique to emphasizethe oscillations.They also pointed out the superiority of the magnetophonon technique over that of measuring the saturation thermoelectric power for determining the pressure dependence of the effective mass. Seiler et a1.44,45have used uniaxial stress in their magnetophonon study of the band structure of n-InSb at 77°K. Their work used compressive stress up to about 8 x lo8 Pa and incorporated magnetic field modulation, phasesensitive detection techniques. They derive theoretical expressions for the (tensor) effective mass as a function of stress and the various band parameters appropriate to a semiconductor having a diamond or zinc-blende crystal structure. From their data, Seiler et deduce values for the quantities a + C,, b, d, and C, ,where C , is the conduction band deformation potential and Q, b, and d are valence band deformation potentials. The quantity a + C, is the hydrostatic deformation potential of the direct energy

286

ROBERT L. PETERSON

gap, while b and d are related to the splitting of the J = 4 valence band for uniaxial stress in the [loo] and [ I l l ] directions, respectively (Fig. 21). C , takes into account the stress-induced coupling between valence and conduction bands for materials lacking inversion symmetry. The quantities a + C,, 6, and d occur in terms linear in the stress, while C , appears in a quadratic term. The slight nonlinearity of the curves of effective mass versus stress is close to the experimental uncertainty, and so the deduced value of C , (1 eV) is somewhat uncertain. If only linear terms are ~onsidered,~’ the orientation B I J jl G cannot determine a C,, b, and d uniquely. This was later rectified through the development of a new type of stress apparatus,46 enabling B to be applied parallel as well as perpendicular to the stress. The longitudinal stress data allowed a unique determination of a C,, b, and d,47giving values of about -5.5, - 1.4, and -3.2 eV, respectively. These values are in fair agreement with those from other technique^,^^ although a considerable spread in those previous values exists. An independent assessment of the various techniques for measuring the several deformation potentials would seem to be in order. Hopefully, this would also serve to

+

+

Zero Stress

I E9

Compressive Uniaxial Stress

EQ+(o+ c, I E

I

FIG.21. A representationof the stressed and unstressed energy bands in InSb at zero magnetic field. The band separation denoted by X is equal to h for o 11 [lo01 and to I for 0 (1 [Ill]. The vertical scale is electron energy and the horizontal scale is electron momentum in the direction of the applied stress.

4.

THE MAGNETOPHONON EFFECT

287

establish a good value for the deformation potential El [see Eq. (1311, the ' ~ ~ also value for which in InSb is the subject of continuing c o n t r o v e r ~ y (see Petersons5). Pitt et a1.1°2 have recently reported applying hydrostatic pressures up to 15 x 10' Pa to n-GaAs and n-InP, noting shifts in the magnetophonon peaks in both the transverse (pxx)and Hall (p,,) resistivities. After correcting for the change in LO phonon frequency with pressure, they deduce a dm*/dP of 0.70 and 0.62 %, respectively, for n-GaAs and n-InP for each 10' Pa of pressure applied. Agreement with a four-band k p theory was noted.

VI. Final Remarks In the dozen years since its prediction and discovery, the magnetophonon effect has made the transition from a scientific curiosity to a useful tool for studying various properties of semiconductors. With it, carrier effective masses can be estimated probably more simply and quickly than by any other technique. A little more work allows one to deduce the band-edge effective masses with about the same accuracy as by other techniques.The anisotropy of the effective masses is readily measured. More generally, the magnetophonon effect can be used to study the E(k) relations of the conduction and valence bands if the relevant phonon energies are accurately known. Conversely, it has been used to measure the phonon energies in a few cases where the effective masses were already determined. The magnetophonon effect has also proven useful in estimating the magnitudes of the transition probabilitiesfor spin-flip processes with absorption of an LO phonon and similarly for estimating the relative probabilities for intervalley and intravalley scattering. Typically, liquid nitrogen temperatures and higher are suitable for Ohmic studies, and magnetic fields of 1 or 2 T are sufficient, particularly with the narrow-gap materials. The necessary electronic gear is minimal. Hot-electron and stress studies are at present the two areas which appear to offer the most promise for fruitful investigation in the magnetophonon effect. High dc electric fields and laser irradiation have both yielded a detailed magnetophonon structure, only part of which is now understood. The few stress studies made to date have shown the capability of unravelling some of the basic energy band parameters, particularly the deformation potentials. The initial experimentalinaccuraciesin locating the peak positions, and the occasional early ambiguities in deciding whether the maxima or the minima were to be identified with the resonances, have been largely eliminated with the introduction of various modulation and derivative techniques. The least 134

I. U Tsidil'kovskii and K. M. Demchuk, Phys. Status Solidi 44,293 (1971).

288

ROBERT L. PETERSON

accurate determinations at present are those connected with the magnetothermal coefficients.The ac techniques have not yet been applied in that area. There are several areas which need further theoretical analysis. The effects of level broadening, which have been studied principally in the strong damping limit, need to be further analyzed, probably with the aid of a computer, to resolve the important questions concerning the shifts of the extrema, not only in the magnetoresistance but also in the magnetothermal properties. The reason why the strongly polar materials show large shifts in the OLMR oscillations, while the weakly polar materials do not, is not yet understood. The theoretical analysis of the hot-electron regime is still in an unsatisfactory state. Both the electron temperature and diffusion approaches have deficiencies, and might not ever be quantitatively accurate. The electron temperature idea seems to have been carried about as far as it can go in describing the magnetophonon effect. The diffusion approach can be developed further. It would be interesting to see whether it can yield the oftenobserved resistance maxima as well as the already deduced minima. A numerical solution to the Boltzmann equation seems to be the best way of attacking the hot-electron regime in the longitudinal case. The one such study to date'28a offers encouragement that such studies will be fruitful. This method should be capable of shedding light on the important question of whether hot-electron resistance maxima or minima should occur for given values of the electric field and amount of elastic scattering. Finally, no theoretical studies to date have included the possibility of a varying carrier concentration with magnetic field or temperature. Capture of electrons at low temperatures and high magnetic fields is well known, but there is also the possibility that at the various resonance conditions, the electron concentration could decrease because there are fewer energetic electrons for ionizing the donors, or because of the greater recombination rate of electrons with the donors, with emission of an LO phonon. Thus the varying carrier concentration with magnetic field could by itself give rise to magnetophonon oscillations. How important this can be is not yet known.

Appendix The International System of Units'35 is a modification of the MKS systems, and will presumably be in rather general use throughout the world in the not too distant future. The SI unit of magnetic flux density (magnetic induction) is the tesla (T), which is equal in magnitude to lo4 gauss (G). 135

"The International System of Units," translated by C. H. Page and P. Vigoureaux, NBS Spec. Pub. 330 U.S. Govt. Printing Office, Washington, D. C., 1972; NBS Tech. News. Bull., January 1971. U.S. Govt. Printing Office, Washington, D.C., 1971.

4.

THE MAGNETOPHONON EFFECT

289

Workers on the magnetophonon effect have always used kilogauss (kG), which is the field in the sample, or kilooersted (kOe), which is the applied field. In the magnetophonon effect, typical magnetic fields can run into hundreds of kG, so that the tesla is in fact a more logical unit to use. Further, since the tesla has gained acceptance in many disciplines, I have decided to use it in this chapter. This is why the familiar “velocity of light” factors are missing in some expressions, e.g., that for the cyclotron frequency. Mobility seems to be expressed universally in the mixed units cm2 V - ’ sec- , and is so expressed in this chapter. Nevertheless, the SI unit is more convenient; e.g., 20,000 cm2 V - ’ sec-’ = 2 m2 V - ’ sec-’. Further, the condition p B > 1 for well-defined Landau levels is much simplier numerically in SI units. For example, the cgs relation pB/c > 1 is usually written p B > lo8 cm2 Oe V - sec- where the factor on the right has no physical meaning, representing the ratio of the vacuum velocity of light to 300, the conversion factor from statvolts to volts. Further, p and B in such units are often each of the order lo4 numerically in the magnetophonon effect. The advantage of SI usage is obvious. No single unit for stress, or pressure, is in universal acceptance, although dyn cm-2 seems to enjoy the most use at high pressures, followed perhaps by kilobars. The SI unit is the newton per square meter, or pascal (Pa): 1 Pa = 10 dyn cm-2 = lo-’ kbar. Although the pressure magnitudes of interest in this chapter are most conveniently expressed in kilobars, the SI unit is used since no one unit is currently dominant.

’

’

’,

ACKNOWLEDGMENTS I t is a pleasure to thank Dr. D. G. Seiler for many discussions and comments on the manuscript, and several others for useful conversations and correspondence. I also wish to thank Mrs. Sandra McCarthy for her expertness and patience in typing the manuscript.

Author Index Numbers in parentheses are footnote numbers and are inserted to enable the reader to locate those cross references where the author’s name does not appear at the point of reference in the text.

A

Adams, E. N., 14, 118,221,233 Addington, F., 227, 258(46), 286(46) Agaev, Ya., 168 Aggarwal, R. L., 159 Aksel’rod, M.M.,87, 224, 225, 227, 245(35), 247,248,253(25,27), 254(27), 255(31), 256, 258,259,260,275(89), 278(89), 280,285 Aleksandrova, M. V., 224,250(20) Alfrey, G. F.,144 Allegre, J., 155 Allen, J. W.. 144 Allgaier, R. S., 118, 119 Allred, W. P.,141, 143 Alsup, D. L., 227, 258(44), 285(44), 286(44) Amirkhanov, Kh. I., 258,259 Andreatch, P.,Jr., 74 Antcliffe, G. A,, 62(152a), 63 Appel, J., 124, 140(102), 146(101), 165 Arguello, C. A., 55 Argyres, P. N., 16,221,223,231,233,237,239, 240, 241(8) Ark, G., 34 Arora, V. K., 248 A*e, M.,104, 129 Askenazy, S., 15, 56, 7@44), 241, 262, 267( 113). 268, 269, 270(75, 114) Aspnes, D. E., 100, 101 Attard, A. E., 63 Aven, M.,50, 51(105), 52, 53(11I ) Averous, M., 5,74, 155

B Baer, W. S.,48,55 Bagguley, D. M. S., 169 Balhzs, J., 55, 56(126) Balkanski, M.,53,60,61(149), 62(149) Balslev, I., 153

Banus, M. D., 200.215(39) Baranskii, P.I., 255 Bardeen, J., 37, 126 Barker, A. S.,Jr., 55.64 Barker, J. R., 236, 239(66), 241, 242(66), 246, 248 Barrera, J., 69 Bashirov, R. I., 253, 255, 258, 259, 270, 271(116) Basinski, J., 5 , 74(24a) Basinski, S. L., 5, 74(24a) Bassani, F., 98,99(50) Bate, R. T., 155, 180.216 Bateman, T. B., 74 Bauer, G., 258,273,284(93~) Baxter, R.D., 155, 156(185) Beck, K., 52,54(116) Becker, W. M.,60,61(147), 141,143, 155,160, 224,260,261 Bedard, F. D., 144, 148(137) Beer, A. C., 2, 20(3), 21(3), 76, 77(203), 94, 114, 117(23), 118, 119, 120, 121(23), 163, 165(23), 180, 181, 185, 217(33) Bekefi, G., 258,281(93a). Bell, J. C., 180 Bennett, H.E., 68 Bergstresser, T. K., 4, 59(17), 98, 148 Berlincourt, D., 50, 52, 53, 5 5 , 84 Bernard, W., 93, 114(14), 115, 119, 126, 172(14) Bers, A., 258, 281(93a) Bir, G. L., 94, 114(18-20), 115 (18-20). I 25( 18-20), 126( 18-20). 129( 18-20), I 4 0 ( 20). 169, 170,227 Birman, J. L., 44.80 Blankenship, J. L., 82, 83(222) Blatt, F. J., 22, 23(56), 27, 33(67) Bloch, F., 126 Blood, P.,69,70(183), 75 Bloom, S.; 4, 59(17), 64 Blum, A. I., 74, 79

291

292

AUTHOR INDEX

Borello, S. R..76. 77(205). 163 Bortfeld, D. P., 144 Bougnot, G., 5.74, 155 Bradley, C. C., 75, 148, 149(151) Brandt, R.C., 68 Bray, R.,123, 21 1, 258, 280, 282, 283 Brebrick, R.F., 60,61(149), 62(149) Bresler, M. S., 266, 268 Bright. R. D., 227, 252(47), 258(47), 285(47) Broerman, J. G., 4, 5(18), 47, 60, 61, 82, 86(227),98 Brooks, H., 2,6,27,28,32,69,70,88,115,124, 126, 139, 140, 165(72), 185 Brown, D. M., 123 Brown, F. C., 58 Brown, R.N., 169 Brown, W. F., Jr., 179 Brown, W. L., 58, 59(139) Broyda, E. L., 87,227,258(49) Brueck, S. R.J.. 258,281(93a) Bryksin, V. V., 236 Bube, R. H., 51 Buck, T. M., 183 Budd, H., 275 Bullis, M., 169 Burmeister, R.A,, Jr., 57(131),58 Burstein, E., 4, 59(19). 267 Busch, G., 78(207), 79, 168

Chou-huang, 155, 157(181) Cleland, J. W., 179 Cochran, W., 272 Cohen, M. L., 98, 148 Cohen, M. M., 144, 148(137) Collins, T. C., 47, 98 Colman, D., 182, 183, 184,215,216 Connell, G. A. N., 56,85(127) Conrad, R. W.,163 Conwell, E. M., 10, 21, 27, 32, 37, 38(54), 41(54), 45(54), 82, 83(224), f 10, 124, 127, 128(117), 129,232, 233(52), 273,276(52) Copeland, J. A., 64,67(158), 74158) Costato, M., 80, 95, 104, 114(30, 32), I15(30, 32,34), 116, 124, 125, 127(32,35), 129, 130, 133, 136, 137, 140(34), 153, 154 Cowley, R. A., 272 Craford, M. G., 66(163). 67 Crawford, J. H.,Jr., 179 Crisler, D. F., 48 Cronin, G. R..76, 77(205), 163 Cuevas, M., 6, 27(28), 28(28), 32(28), 69(28), 70(28), 88(28) Cunningham, R.W., 78. 169 Cupal, J. J., 48 Curby, R.C., 258,283 Curtis, B. J.. 144

C

D

Calas, J., 5,74 Callen, H. B., 127 Camphausen, D. L., 56, 75,85 Cardona, M., 7, 48, 97, 98, 143. 149(37), 153(37,43), 162(43),169,270,271(115) Carides, J. N., 53,54118) Carrtre, G., 56, 241, 267(113), 268, 270(75) Casey, H. C., Jr., 14, 63, 65(156), 66, 67, 68, 123, 144, 145, 146, 147, 148(91), 149, 158, 159, 165(190) Cerdeira, F., 48 Chamberlain, J. M., 75 Chambers, R. G., 166 Champlin, K. S., 68 Chang, D. M., 69 Chang, R. K., 68,88 Chaudhuri, N., 272 Cherry, R.J., 144 Chewier, J., 5, 74

Dalal, V. L., 149(163). 150, 151(163) Datars, W. R.,15, 78(44a), 87(44a) Davis, R.,118 Dawson, L. R.,66(166), 67 Debye, P. P.,82, 83(224), 124 deLaet, L. H., 81,82(220), 83(220) Demchuk, K. M., 87, 227, 258(49), 285(49), 287 Detwiler, D. P., 155, 157(177) Devlin, S. S., 2, 3, 39(5), 49, 56(99), 57(5), 58(5, 99), 60, 61( 148), 85(148), 217 DeWitt, J. S.,48 Dexter. R.N., 55 Dias, P.,144 DiDomenico, M., Jr., 95, 104, 105, 122. 123. 129, 130, 131, 143, 148. 151(28) Diguet, D., 144, 149 DiLorenzo. J. V.. 70 Dimmock. J. O., 212

293

AUTHOR INDEX

Dingle, R., 88 Dingle, R. B.. 6, 27, 28. 29(27), 32. 67, 69, 70, 88,241 Diquet, D., 3 Dixon, J. R., 163 Dolat, V., 258,280, 282,283 Dolling, G., 44,272 Dreeben,A. B.,5, 141, 149(163), 150, lSl(163) Dresner, J., 53, 54(118) Dresselhaus, G., 93, lOO(12). 104(12). 119, 153. 171(12), 172 DuBois, D. F., 58 Dunlap. W. C., Jr., 155, 158 Dworin, L., 236,240 Dykman, I. M.,3, lO(11) Dzhandieri, M.Sh., 179

E Eaves, L., 15,56,57,78(44), 226,241,258(42), 260(42), 262(42),264,265,269,270,279(42), 280(42) Eckelt, P., 51 Edmond, J. T., 163 Edwards, D. F., 155, 157 Effer, D., 155, 157(182), 163 Efros, A. L., 236 Ehrenreich, H., 2, 3, 4(7, lo), 10(10), 11(10), 12(10), 13(10), 14, 15, 19, 28(10), 31, 37, 38, 39(10), 77, 87, 92, 94, 110(9), 114(17), 115(17), 118(9), 124, 125, 126, 127, 134, 135, 140(9S), 162, 184 Elcombe, M. M.,272 Elinson, M.I., 263 Emde, F., 17 Emel'yanenko, 0. V., 149, 150, 154(154, 165) Emin, D., 235, 241(61), 270(61) Enright, D. P., 163 Epstein, A. S., 67, 149(168), 150 Ergakov, V. K.,149(165), 154(165) Ermanis, F.,66(166),67,123,144(91), 145(91), 146(91), 147(91), 148(91), 149,150. 154(159) Ermolovich, Yu. B., 169, 170 Ettenberg, M.,5, 141 Etter, P. J., 155, 157(182) Euwema, R.N., 98 Ewald, A. W., 82, 83, 85(226), 86, 172

F Falicov, L. M., 6, 27(28), 28(28), 32(28), 69(28), 70(28), 88(28)

Fan, H. Y., 24.63 Farbshtein, 1. I., 226, 258. 284(43, 88) Faulkner, R.A,, 148, 149 Fawcett, W..9, 10, 11, 101. 172 Ferry, D. K., 258,275,283 Firsov, Yu. A., 222, 224, 225, 235, 236, 237, 2 q 7 , 15),242,247(32,33),248,250(21,22), 251, 252(22), 253(24), 254(23). 255, 261(15) Fischer, A. G., 53, 54(118), 166 Fisher, G., 166 Flanagan, T. M., 179 Folberth, 0. G., 75, 76, 77, 162, 163, 164 Fortini, A,, 3 Foster, L. M.,144, 146 Frederikse, H. P. R., 223. 231, 241(9) Frisch, H. L., 176, 177 Frohlich, H., 15, 19, 38(42), 48, 127,276 Fujibayashi, K., 74 Fujita, H., 55, 56(119) Fukai. M., 52, 53(115) Fukuda, Y.,52, 53(115) Fuller, C. S., 146, 163 C

Gadzhialiev, M. M., 253, 255 Gagliani, G., 95, 114(30), 115(30), 116(30), 125(30), 130, 133(122) Galavanov, V. V., 75, 76(195). 78(208), 79, 160, 161(196), 162, 166, 167, 169, 170 Galazka, R. R.,60,61(147) Ganguly, A. K., 110 Gantsevich, S. V., 237 Garrett, J. P., 149(166), 150, 154(160) Gasanli, Sh. M.,149(165), 154(165) Gashimzade, F. M.,169, 170 Gatos, H. C., 200, 215(39) Geballe, T. H., 21, 44, 78(206). 79, 167, 168(225), 224, 237, 248, 250(16. 71). 251, 252(71), 253,254 Censer, M.. 141, 143(130) Gerhardt, V., 60,61(150), 621150) Gilbert, S. L., 5, 141 Glicksman, M.,160, 161(195), 162(195) Glover, G. H.,68 Gluzman, N. G., 263,264,265( 103, 105) Gobrecht, H., 60,61,62(150) Goering, H. L., 76.77(203), 163 Goldberg, C., 118 Golubev, L. V., 63, 74(155). 155

294

AUTHOR INDEX

Gorodnichii, 0. P., 255 Gossick, B. R., 179 Grigorev, N. N., 3, 10(11) Grimmeiss, H. G., 141 Grosse, P.,267 Groves, S. H., 60, 79(142), 82, 85,86, 172 Groves, W.O., 66(163), 67 Gruber, J. B., 78, 169 Gspan, P.,267 Guislain, H. J., 81, 82(220), 83(220) Gurevich, V. L., 222, 224, 225, 235, 236, 237, 240(7, 15), 242, 250(21, 22), 251(7, 21), 252(22), 254(23), 261(15) Guseva, G. I., 237,248

H Haberecht, R. R.,141 Hales, M.C., 75, 76(196) Hall, R.N., 155, 215 Halsted, R. E., 58, 59(133) Hamaguchi, C., 258,275(86), 278,279,284 Hammar, C., 134, 135 Harada, R. H., 163 Harman, T. C., 60, 61(148), 76, 77(203), 85, 163 Harp, E. E., 169 Harper, P. G., 224, 226(14), 23q14). 258(14), 262(14), 266(14), 270(14), 275(14), 279(14), 28q 14) Harrison, W.A,, 34, 35(74), 82, 93, 123, 124, 127 Harsy, M.,55, 56(126) Hasegawa, H., 221 Hashitsume, N., 221, 231(4), 235(4), 237(4), 238(4), 240(4), 241(4), 276(4) Hass, M., 68, 85 Hayashi, I., 149, 154(159) Hayne, G. S.,155, 157 Heinrich, H., 74 Helbig, R.,48 Heltemes, E. C., 48 Henry, C. H., 55,88 Hensel, J. C., 172 Herczog, A., 141 Herman, F., 93 Herring, C., 2, 6, 27, 28, 32, 37, 41(6), 42, 44, 45, 69, 70, 88, 124, 176, 178, 179, 180, 186, 196,209,215 Herzog, A. H., 66(163), 67

Hicks, H. G. B., 69.70 Higginbotham, C. W.,97, 98(37), 149(37), 153(37), 270, 271(115) Hill, D. E., 66(163),67,70, 123, 149, 150(161), 151, 152(161), 154(156, 161), 211, 212(42), 213(42), 214(42), 216(42), 218(42) Hilsum, C., 74, 75, 92, 94(3), 95, 114, 115(3), 118(3), 127(3), 136(3), 163, 165(3) Hinkley, E. D., 86 Hlasnik, I., 181 Hochberg A. K., 41,42(82), 45(82) Hodby, J. W.,224, 226(14), 235(14), 258(14), 262(14), 266(14), 270(14), 275(14), 279(14), 280( 14) Holan, L., 262 Holstein, T. D., 221 Hopfield, J. J., 55 Hoschl, P.,57(132), 58 Hou, S. L., 52, 54 Hoult, R. A., 226, 258(42), 260(42), 262, 270(102), 279(42), 280(42), 287(102) Howarth, D. J., 2, 3, 127, 134(115) Hrostowski, H. J., 78(206), 79, 167, 168(225) Hulme, K. F., 165,200,215(38) Hutson, A. R., 34, 35,47(75), 48.49, 124, 233

I

Iglitsyn, M.I., 163, 178 Ikoma, H., 149 Ilegems, M.,64,65 Il’in, Yu.L., 144, 148(142) Inall, E. K., 240,241(74), 270(74) Ismailov, O., 168 Ismailov, 2.A., 258 Ito, R., 278 Itskevich, E. S.,227,258(48), 285 Ivanov-Omskii, V. I., 62(152), 63, 155, 157(181)

J Jacoboni, C., 95, 114(32), 115(32), 124(32), 125(32), 127(32), 130, 133(122), 136(32), 137(32), 153(32), 154(32) Jafle, H., 50,52(103), 53(103), 55(103), 84(103) Jahnke, E.,17 Jantsch, W.,74

295

AUTHOR INDEX Jayaraman, A., 74 Jensen, R. V., 149, 150(153), 151(153), 154( 153) Jones, D., 67 Joseph, T. J., 227,252(47), 258(47), 286(47) Juretschke, H. J., 177, 178, 207

K Kahlert, H., 258, 273, 284(93c) Kahn, A. H., 223, 231, 241(9) Kaminsky, G., 272 Kanazawa, K. K., 58 Kane, E. O., 4,5(21), 6,7,9, 10,11(30), 12(30), 13, 14, 18, 26, 35, 51, 53, 75(30), 77, 86, 98, 99, 100,101, 106, 108, 119, 171(41), 172(41), 173,227,234,285 Kang, C. S.. 51 Kasami, A., 66(168), 67 Kawaguchi, Y.,163 Kawai, T., 55, 56( 119) Kawaji, S., 163 Kawamura, H., 258,284(92) Kazlauskas, P. A,, 10 Keating, D. E., 68, 88(175) Keezer, R. C., 267 Kendall, D. L., 182, 183, 184, 215,216 Kesamanly, F. P., 149(165), 154(165) Kharus, G. I., 225, 232(39), 236(39), 242, 244(39), 246, 271(77), 272(77), 273, 274 Kip, A. F., 93, 100(12), 104(12), 119(12), 153(12), 171(12), 172(12) Kischio, W.,141 Kittel, C., 93, 100(12), 104(12), 119(12), 153(12), 171(12), 172(12) Klinger, M.I., 235 Knell, R. L.,48 Knight, J. R., 75, 76(196) Knight, S., 26, 28(62), 32(62), 33(62), 64, 67(158), 68(62), 70(62), 74(158), 211 Kobayashi, K., 55,56(119) Kohler, M.,3 Kohnke, E. E., 82 Kokoshkin, V. A., 155, 157(187) Kolodziejczak, J., 117 Kolomiets, B. T., 62(152). 63, 155, 157(181) Komatsubara, K. F.,226,258,275(91), 278 KO@, 2.. 117 Kornyushin, Yu.V., 180

Kosicki, B. B., 74 Koteles, E. S., 15, 78(44a), 87(44a) Kotera, N., 226,258,275,278 Kowalczyk, R., I17 Kranzer, D., 95, 114(31,33), 115(31,33), 124, 125, 127(31, 33), 136, 137, 138, 153(33) Kravchenko, A. F., 181 Kroger, F. A,, 50, 51(104) Kubalkova, S., 57(132), 58 Kubo, R., 221,231,235,237,238,240,241(4), 276 Kukhtarev, N. V., 180 Kukuladze, G. V., 155, 157(187) Kurik, M. V., 6, 53(26) Kurosawa, T., 239,274,275 Kwok, P. C., 20

L Lagunova, T. S.,149, 150, 154(154, 165) Landau, L. D., 205 Landauer, R., 177, 178(5), 207(5) Landwehr, G., 223 Lang, I. G., 247 Langer, D. W., 48 Larsen, D. M., 68,234,241(59), 270(59) Laurent, A,, 262 Lavine, C. F., 83,86(228) Lawaetz, P., 49.94, 100, 101, 114(22), 115(22), 116, 117, 125, 126, 129, 132, 144(53), 148, 149, 153, 162(53), 165, 169, 172, 173 Lax, B., 93, 119, 120, 121, 126, 143, 159 Lax, M., 44,80 Lees, J., 68, 70(174), 262, 270(102), 2871102) Lehoczky, S. L., 61 Leifer, H.N., 155, 158 Leotin, J., 15, 56, 78(44), 241, 262, 267(113), 268,269,270(75, 114) Lettington, A. H., 67 Leutwein, K., 50, 51(106) Levinstein, H., 82 Lewis, J. E., 144, 146(139) Lifshitz, E. M., 205 47 Litton, C. W., Logan, R. A., 80,81(219) Long, D., 80,81(218), 96,98(36), 99(36) Lorenz, M. R., 58, 59(133), 66(164), 67, 79, 144, 146(138) Lorimor, 0.G., 148 Loudon, R., 44

2%

AUTHOR INDEX

Lu, T., 68 Lucovsky, G., 4, 59(19), 267 Ludwig, G. W., 80, 81(217) Lugand, J., 3 Lugovykh, V. P., 258, 275(89), 278(89), 28q89) Luther, L. C., 66(166), 67 Lutskii, V. N., 263 Luttinger, J. M., 101, 172 Lyagushchenko, R.I., 258,284

M McGroddy, J. C., 79 McKenna, J., 177 MacKey, H. J., 259 McKim, F. S.,183 McSkimin, H. J., 74 Madelung, O., 75,76,77(204), 92,94(4), 95(4), 1 14, 1 15(4), 1 17(4), 118(4), 127(4), 136(4), 163, 164(205), f65(4) Magnusson, B., 134, 135, 245, 275, 276, 277( l3l), 278( 13l), 280( l31), 283( 131). 284(131), 288(128a) Maita, J. P., 80, 81(216) Makovskii, L. L., 247 Manley, D. F., 69, 70 Marley, J. A., Jr., 52, 54( 116) Marple, D. T. F., 49, SO(lOO), 52, 58 Maruska, H. P.,63,64(154) Mashovets, D. V.,225, 253(28), 259(30), 266, 267,268 Matark, H., 124 Matossi, F., 50, 51(106) Matz, D., 110 Mavroides, J. G., 93, 119, 120, 121, 126, 143 Mead, C. A., 50, 51(105) Mears,A. L., 149(164), 150,151,152,l53,226, 240,241(74), 258(42), 260(42), 262(42), 270, 279(42), 280(42) Meeus, M., 81,82(220), 83(220) Meier, H., 144 Meijer, H. J. G., 34, 35(73), 233 Metreveli, S. G., 160, 161(196), 162 Metzler, R. A., 160 Meyerhofer, D., 149, 150(153), 151(153), 154(153) Meyers, J., 80,81(218) Middleton, A. E., 141 Mikhailova, M. P., 163, 164

Miki, H., 70,74 Miklosz, J. C., 49, SO(101) Miller, S. E., 155, 156(185) Mirgalovskaya, M. S.,155, 157 Mishra, U. K., 153, 159(172) Miura, N., 267,268 Miyake, S. J., 221, 231(4), 235(4), 237(4), 238(4), 240(4), 241(4), 276(4) Miyazawa, H., 118 Mollaev, A. Yu., 270, 271(116) Molodyan, I. P., 248, 256(82), 257(82), 258 (82). 260(82), 262(82), 270(82), 282(82), 283 (82) Montgomery, H. C., 64,65 Mooradian, A., 58 Moore, A. R.,48 Moore, E. J., 15, 26, 28(61), 33(61), 34, 124, 140(95,96) Morin, F. J., 44, 78(206), 79, 80,81(216), 146, 167, 168(225) Morita, S.,258,284 Morozov, B. V., 181 Morrison, J. A., 176 Mosanov, O., 168 Moss, T. S.,165 Mott, N. F., 27, 146 Mullin, J. B., 165, 200, 215(38) Muthukrishnan, R., 227, 258(44), 285(44), 286(44) Muzhdaba, V. M., 224, 227, 240(19), 250, 252(18, 83), 253, 258(48), 259(19), 285(48)

N Naito, M., 66(168), 67 Nakai, J., 258, 275(86), 278(86), 279(86), 284 (86) Nakayama, M., 236,240 Nasledov, D. N., 78(208), 79, 141, 149, 150, 154(154, I65), 161,162, 163, 164, 167 Nassau, K., 55 Nedoluha, A,, I79 Neifel’d, E. A., 262 Nelson, D. A,, 61 Neuberger, M., 92, 134, 136, 165(6) Newman, P. C., 66(167), 67 Nicklin, R.,66(167), 67 Niederer, H. W. J. M., 258,281 Nordheim, L., 126

297

AUTHOR INDEX

Normantas, E.,94, l14(20), 115(20), 125(20), 126(20), 129(20), 140(20), 169(20), 17q20) Norton, P., 80,82 Novikova, S. I., 87 Nygren, S.F., 144, 146, 148(144) 0

Ogorodnikov, V. K., 62(152), 63 Onton, A., 80 Orton, J. W., 70,75 Ostroborodova, V. V., 144 Otsubo, M., 70 Overhauser, A. W., 94, 114(17), 115(17), 125, 126 Overhof, H.,60,63(143)

P P a Q , E. G. S., 94, 129(24) Palik, E. D., 169, 223, 234, 249(11) Palmer, R. I., 235, 241,270 Panish, M.B., 14,63,68, 70 Paranjape, B. V., 276 Parfen’ev, R. V., 224, 225, 226, 240, 250, 251(21), 252(18, 22, 83), 253(18, 28, 83), 258,259,284 Paris, B., 141, 143(130) Park, Y.S., 47 Paul, W.,56, 60,74,79(142), 85(127), 86 Pavlov, N. I., 181 Pavlov, S.T., 224,225,237,247,248,253(24), 255 Peinemann, B., 60,61(150), 62(150) Perl, R., 118, 119 Peters, A. J., 80, 81(219) Peterson, R.L., 225, 230(36-38), 232(36-38), 233, 236(36-38), 242, 243(37), 244(37, 38), 245, 246, 248,276, 277, 278(131), 280, 283, 284,287 Petritz, R. L., 181 Phillips, J. C., 4, 34 Phipps, P. B. P., 51 Picus, G. S.,58, 234 Pidgeon, C. R., 82, 85(226), 169, 172 Pikus, G. E., 94, 114(18-20), 115(18-20), 125 (18-20), 126(18-20), 129(18-20), 140(20), 169(18-20), 170(18-20), 227 Pinczuk, A., 4, 59(19) Pitt, G. D., 68,70(174), 74, 262,270,287 Pivovarov, M. N., 163

Plaskett, T. S.,79 Podkolzin, A. P., 247,248(80), 256(80) Podor, B., 55, 56(126) Polder, D., 34, 35(73), 233 Pollak, F. H., 97,98, 149, 153, 270, 271(115) Poltinnikov, A. S.,181 Polyanskaya, T. A., 63,74(155) Pomortsev, R. V.,242,258,271,272(77), 273, 274, 275(89), 278(89), 290(89) Ponomarev, A. I., 242,271,272(77) Popov, Yu.G., 161, 162(197) Portal, J. C., 15, 56, 78(44), 241, 267(113), 268,269, 270(75, 114) Porto, s. P. s.,55 Price, P. J., 118 Prince, M. B., 82, 83(223), 182 Puri, S. M.,21,224,237,248,250(16,71), 251, 252(71), 253, 254 Putley, E. H., 92, 94(5), 95(5), 114, 115(5), I18(5), 127(5), 136(5), i65(5)

Q Quadflieg, P., 34

R Rabenau, A., 141 Racek, W., 258,284 Racette, J. H.,155 Rall, L. B., 24.25 Ralston, J. M., 68, 88(175) Ramadas, A. K., 155 Rees, H. D., 20.24, 25,74, 275 Reggiani, L.,SO, 95, 104, 114(30, 32), 115(30, 32, 34), 116(30), 124, 125(30, 32, 34, 35), 127(32, 39, 129, 130, 133, 136(32, 35), 137 (32, 35), 140(34), 153(32), l54(32) Reid, F. J., 141, 142, 143, 155, 156 Reine, M.,159 Reiss, H., 146 Reynolds, D. C., 47 Riccius, H.D., 52 Richter, W., 267 Ringel, C. M., 144, 146(144), 148(144) Robinson, M.L. A,, 169

298

AUTHOR INDEX

Rode, D. L., 3,6,1 1(13,14),12(13), 16(13,14), 17(14), 18, 19(13, 14),20(14),21(14),22(14), 23(14), 24(14, 15). 25,26,27,28(62), 32(62), 33(62), 34, 35(14), 36(14, 72), 38(14, 72). 39(14), 40(49), 41(29), 42,45(29), 48(15,72), 50(72), 51(72), 52(15,72), 53(72), 55(15,72), 56(72), 58(15, 72), 60, 63(144), 64,65(29), 67(29), 68( 15,62), 70,72( 15),73(I3), 74(13), 75(14), 78(14), 79(14, 29), 80(29), 82(29, 144). 84, 85, 86(144), 87(15), 92, 93, 136, 184, 211 Rodionov, K. P., 87,227,258(49) Rossler, U., 48, 51 Rose-Innes, A. C., 75, 92, 94(3), 95(3), 114, 115(3), 118(3), 127(3), 136(3), 165(3) Rosenbaum, S. D., 5,74(24a) Rosi, F. D., 149, 150, 151(153), 154(153) Rossi, J. A,, 152, 157(171), 184, 190(32), 197 (32), 198(32), 199(32), 202(32), 203(32), 216(32), 218(32), 219(32) Rosztoczy. F. E., 149, 150, 154(159) Roth, H., 93, 114(14), 115(14), 119(14), 126 (14), 172(14) Roth, L. M.,16, 221, 223, 231,237,239, 240, 241(8) Roth, W. L., 49, 56(99), 58(99) Rousseau, D. L., 55 Ruch, J. G., 9, 10, 11, 59 Ruehnvein, R. A,, 149(168), I50 Rupprecht, H., 163 Russell, A. W., 66(167), 67 Ryabtsova, G. P..79

S

Saitoh, M.,56 Saleh, A,, 63 Schmid, G. S., SO, 511106) Schmidt, P. H.,272 Schoenmaekers,W. K., 81,82(220), 83(220) Schonwald, H., 167 Schultz, T. D., 20 Schwartz, B., 149, lUy159) Schwerdtfeger, C. F., 148 Segall, B., 49, SO(lO0). 58,59(133) Segawa, K., 74 Seiler, D. G., 8, 60, 61(147), 222, 227, 252, 258(44-47), 285,286(44-47) Seitz, F., 36, 126

Selders, M.,261 Sell, D. D., 100, 101 Semenyuk, A. K., 180 Senechal, R. R., 10 Seraphin, B. 0.. 68 Shalyt, S. S., 224, 225, 226, 227, 240(19), 250, 251(21), 252(18, 22, 83), 253(18, 28, 83). 258,259(19,30), 267,268(11I), 284(43,88), 285(48) Shenvood, N. T., 224,260,26L Shiozawa, L. R., 50,52(103), 53(103), 55(103), 84(103) Shirakawa, T., 258,275(86), 278(86), 279(86), 284(86) Shmartsev, Yu. V., 63,74(155), 155 Shockley, W., 2,37,38(1), 118, 126 Sigai, A. G., 5, 141 Simmonds, P. E., 75, 148, 149(151) Siukaev, N. V., 75,76(195), 160,161, 162(196, 197). 270, 271(116)

Skok,E.M., 181 Slade, M., 4, 60(20)

Slobodchikov,S. V., 141, 163, 164(209) Smekalova, K. P.,62(152), 63 Smith, F. T. J., 52, 53(114), 57, 58, 59(140) Smith, J. E., Jr., 75, 79 Sneddon, I. N., 193 Sokolov, V. I., 225, 253(27), 254(27), 259(29), 260,263, 264,265(106) Solov’eva, E. V., 163 Sondheimer, E. H.,2, 3, 127, 134(115) Sorokin, V. S., 144, 148(142) Spears, D. L., 70, 211, 212(42), 213(42), 214 (42), 216(42), 218(42) Spicer, W. E., 104 Sreedhar, A. K., 272 Starosel’tseva, S. P., 160, 161, 162 Steigmeier, E., 78(207), 79, 168 Stevenson, D. A., 57(131), 58 Stillman, G. E., 68,69,70, 152, 157, 176, 184, 190(1, 32), 191, 197(32), 198(32), 199(32), 202. 203(32), 204, 208, 209, 211, 212, 213, 214(42), 216(32,42). 218132, 42). 219(1, 32, 40) Stirn, R. J., 119, 120, 121(87), 141, 143 Stockton, J. R., 148, 149(151) Stradling, R. A., 15, 62(152a), 63, 75, 76, 78, 85, 87, 148, 149(151, 164), 150, 151, 152, 153, 159, 169,221, 224, 225, 226,230, 234, 235(14), 236(34), 240, 241, 245(34), 247(34,

299

AUTHOR INDEX

Stradling, R. A.-cont. 40), 248, 249(34, 58), 254(34), 255, 256(34, 82), 257(82), 258, 259, 260, 261, 262, 264, 265(107), 266, 267(113), 268, 269, 270( 14, 74. 75, 82, 102, 114), 275(14, 40), 278, 279 (12, 13, 14, 40, 42, 87, IOI), 280, 281, 282 (82), 283(82), 287(102) Straub, W. D., 93, 114(14), 115(14), 119(14), 126(14), 172114) Strauss, A. J., 60,61(149), 62(149), 163, 166 Streitwolf, H.W., 44,80,81(213) Studna, A. A., 100,101 Stukel, D. J., 4, 98 Subashiev, V. K., 181 Sukhoparov, V. A,, 227,258(48), 285(48) Summers, C. J., 55 Suzuki, K., 172 Swanson, J. A., 177, 178(5), 207(5) Swinney, H. L., 48 Sybert, J. R., 259 Sze, S.M., 67, 79(169)

T Takano, S.,258, 284(92) Tamarin, P. V., 247 Tausend, A., 60,61(150), 62(150) Taylor, R. C.,66(164), 67, 144, 146, 148(140) TdyIOr. w., 4, 59(19) Teitler, S., 169, 234 Teutsch, W. B., 124, 146(101) Thomas, D. G., 47,55 Tierstein, N., 94, 1 I4(21), 115(21), 125, 128 Tietjen, J. J., 63,64(154) Tomchuk, P. M., 3, 10( I I ) Toyama, M.,66(168), 67 Trians, A., 149(163), 150, 151(163) Triboulet, R., 58, 59(139) Trumbore, F. A., 63,65(156) Tsertsvadze. A. A,, 179 Tsidil’kovskii, I. M., 87,224,225,227,232(39), 236(39), 242,244(39), 245(35), 246,247,248 253(25, 271, 254, 255, 256, 258, 259, 260, 262, 263, 264, 265, 271, 272(77), 275 (89). 278(89), 280(89), 285(49), 287 Tsui, D. C., 88, 123,272 Tufte, 0. N., 86, 181 Twose, W. D., 146

U Ulmet, J. P., 15,56,78(44), 241,262,267(113), 268,269,270(75, 114) Uritsky, S. I., 225, 247(31), 255(31), 256(31)

V Van Atta, L. B., 58 van der Meulen, Y.J., 155, 157 van der Pauw, L. J., 190 van Maaren, M. H., 155 Van Vechten, J. A., 4,34 van Welzenis, R. G., 258, 281 Vasileff, H.D., 14 10,20, 25(52), 110 Vassell, M. 0.. Vella-Coleiro, G. P., 55,57, 271,285 Verleur, H. W., 66(166), 67, 144, 146(144), 148(144) Vilisov, V. A., 247,248(80), 256(80) Vilms, J., 149(166), 150, 154(166) Vinetskii, V. L.,180 Vinogradova, K. I., 167 Vogt, E.,2, 37,41(6), 45 Volokobinskaya, N. I., 78(208). 79 von Borzeszkowski, J., 104, 129 Voronkov, V. V., 163, 178 Voronkova, G. I., 178 Vrehen, Q. H. F., 153 Vul’, A. Ya., 63, 74(155), 155

W Wadha, R. S.,272 Wagini, H., 75, 76( 198) Wagner, P., 48 Wagner, R. J., 82,85(226), 172 Wakhaloo, P., 272 Waller, W. M.,259 Wallis, R. F., 169, 234 Walter, J. P., 98 Walton, A. K., 153, 159(172) Watters, R. L., 80,81(217) Weiner, C., 79, 80(211) Weisberg, L. R., 124, 179 Weiser, K., 160, 161(195), 162(195) Weiss, H., 75, 76, 77(204), 162, 163, 164(205) Weissglas, P., 276, 277(131), 278(131), 280 (131), 283(131), 284(131)

300

AUTHOR INDEX

Weisskopf, V. F., 27, 32 Westgate, C. R.,41,42(82), 45(82) Wheatley, G. H., 78(206), 79, 167, 168(225) Wheeler, R. G., 49,50(101) White, D. L., 35, 124 Whitfield, G. D., 126, 129(112) Whitsett, C. R.,60, 61,85 Wiegmann, W., 88 Wiggins, C. S., 144 Wiley, J. D., 60,63(144), 82(144), 85,86(144), 95, 101, 104, 105, 110(29), 111, 122, 123, 127(29), 129, 130, 131, 135, 136, 143, 146, 148, 151(28) Wilkins, C. W., 75, 76(196) Willardson, R.K., 141, 142, 143 Williams, F. V., 70, 211, 212(42), 213(42), 214(42), 216(42), 218(42) Wolfe, C. M., 68, 69, 70, 152, 157, 176, 184, 190(1, 32), 191, 197, 198, 199, 202, 203, 204, 208, 209, 211, 212, 213, 214, 216, 218, 219(1, 32,40) Wolfstim, K. B., 123, 144(91), 145(91), 146 (91), 147(91), 148(91), 149, 150, 163 Wood, R.A., 76,78,85,87,225,226,230,234, 236(34), 241, 245(34), 248, 249(34, 58), 254 (34), 255, 256, 257, 258, 259, 260, 261, 262, 264,265(107), 270,275(40), 278,279(40,42, 87, IOI), 280, 281, 282(82), 283(82)

Woods, J. F., 66(164), 67, 144, 146(138, 139) Woolley, J. C., 5, 10, 74(24a) Wright, G. B., 58,223,249(11)

Y Yamada, E., 226, 239, 258, 274, 275, 276, 278(41,91) Yamashita, T., 258,275(86), 278(86), 279(86), 284(86) Yas’kov, D. A,, 144, 148(142) Yassievich, I. N., 258, 284

2

Zallen, R.,4, 59(19), 60(20) Zhirnov, A. A., 263 Ziman, J. M., 2, 4(4), 15(4), 18(4), 26(4), 37(4), 39(4), 41(4), 126 Zook, J. D., 35, 36(77), 48, 55, 124 Zotova, N. V., 163, 164 Zschauer, K. H., 149(167), 150, 151(167), 152 Zukotynski, S., 117 Zyryanov, P. S . . 237, 248

Subject Index polar hole mobility, 138, 139 spin-orbit splitting, 101 subsidiary minima, 101 valence band parameters, 173 anisotropy parameters, 120 effective masses, 116, 143, 144

A a-Sn. see Gray tin Acoustic mode scattering 6, 37, 38, 127, 129-134.233.243-24, seealso Scattering, specific materials coupling coefficient, 233 differential scattering rate, 37 partial mobility, 38, 82, 127, 130, 132 temperature dependence, 38 relaxation rate, 38 Acoustoelectric gain, 280-283 AlAs, 101 deformation potentials, 133 dielectric constants, 136 direct gap, 101 effective charge, 136 hole mobility, 117. 141 material parameters, 134 polar hole mobility, 138, 139 spin-orbit splitting, 101 subsidiary minima, 101 valence band parameters, 173 anisotropy parameters, I20 effective masses, 116 AIP, 101 deformation potentials, 133 dielectric constants, 136 direct gap, 101 effective charge, 136 hole mobility, 117, 141 material parameters, 134 polar hole mobility, 138, 139 spin-orbit splitting, 101 subsidiary minima, 101 valence band parameters, 173 anisotropy parameters, 120 effective masses, 116 AISb, 101 deformation potentials, 133 dielectric constants, 136 direct gap, 101 effective charge, 136 hole mobility, 117, 141-144 intrinsic carrier density, 159 material parameters, 134

B Band gap, see also Energy gap, specific materials direct, 6-1 1 indirect, 6 Band structure, 7-15, 88, 95-110, see also Charge carrier energy, Effective mass, Energy levels, Warped bands, specific materials conduction band, 7-15 coupled bands, 100-1 17, 140, see also Twoband conduction stress effects, 160 energy level notation diamond lattice, 8, 97 zinc-blende lattice, 8.97 energy-momentum relation, 10-15,97-109, 119-121, 234 Wall coefficient anisotropy factor, 118-120 heavy-hole band, 7,99 inverted, 4, 59, 82 light-hole band, 7.99 magnetophonon effect measurements, 227 many valley, 41-46,67, 80-82,92 nonparabolicity, 9-14, 32, 75, 78 one-dimensional quantum states, 88 small gap approximation, 7 split-off band, 99 valence band, 7,95-110 parameters, 116, 120, 171-173 warped bands, 93, 117-121 Boltzmann equation, 2, 15-20, 110, 136-138, 185-187, 221, 243, 245 coupled equations, 110, 136-138 diffusion approximation, 16, 274, 275 distribution function, 16-20, see also Distribution function

301

302

SUBJECT INDEX

hot-electron regime, 275, 276 numerical solution, 275, 276, 288 inhomogeneous specimens, 176, 177 iterative solutions, 2 4 , 134, 275 quantum effects, 221-223, 237-247 relaxation time concept, 2,4, 186 variational calculations, 2 Born approximation, 15, 26, 29, 32-34, 39. 45. 240, 276 conductivity divergences, 227, 235, 239 Bose-Einstein distribution, 39,45 Brillouin zone, 8 , 4 2 4 , 9 6 9 8

C CdS. 4. 55, 56, 271 deformation potential, 56 direct gap, 55 effective mass, 55 electron mobility, 55, 56 Hall factor, 55 magnetophonon effects, 271 photoconductivity, 271 285 Raman scattering, 271, 285 material parameters, 84 CdSe, 4, 56-58.270. 271 direct gap, 56 effective mass correction, 57 electron mobility, 57 Frohlich parameter, 271 Hall factor, 58 magnetophonon mass, 57, 250, 270,271 magnetophonon resonances, 241, 270, 271 material parameters, 84 nonparabolicity correction, 270 polaron mass, 84, 250, 270, 271 anisotropy, 271 CdTe, 4, 58, 59, 271 electron mobility, 59 Frohlich parameter, 270 magnetophonon mass. 250 magnetophonon resonances, 240,270 hot-electron regime, 275, 279, 283 material parameters, 84 nuclear detector application, 58 polaron mass, 58, 84, 250 subsidiary minima, 59 Charge-carrier density, see also Electron concentration, specific materials hole density, warped bands, 120, 121

magnetic field variation, 288 Charge-carrier energy, 11-15, 97-109, 119121, 234, see also Band structure Charge-carrier mobility, see Mobility Charge-carrier temperature, see Electron temperature concept Collision broadening, 236, 240, see also Landau levels Conduction band, see Band structure Conductivity, apparent, 177ff, see also Inhomogeneities cylindrical cavities, 177 cylindrical inclusions, 178 discontinuity, I 9 g l 9 9 current distortion, 196 exact analysis, 192-195 metallic approximation, 195, 196 quantitative results, 197- I99 fluctuations, 179 gradient, 199-201 facet effect, 200 spherical cavities, 178 spherical inclusions, 178 Conductivity tensor, 188, 235, 237-240, 247, see also Magnetoconductivity tensor, Magnetoresistance inversion, 235 warm-electron coefficient, 278 Contraction mapping, 20,2426 Cubic crystal structure, see Diamond structure

D Debye temperature, 84, 85, 128, 134, see also Optical phonon interactions intervalley phonon, 84,85 polar phonon, 84,85 table, 84, 85, 134 Deformation potential, 37, 38, 84-87, 127134,232,233,285-287, see also specific materials acoustic, 127-133, 233 table, 133 “effective,” 128, 129 intervalley, 45, 84, 85 magnetophonon data, 227,285-287 optical, 128-133, 232 table, 133 relation to pressure coefficients, 38,87,285, 286

303

SUBJECT INDEX table, 84, 85, 133 valence band, 127-134 Density of states, 14,222, 239, 274 effective mass table, 84, 85 magnetic field, 222,229,239 Landau levels, 222,223,227-230 nonparabolic band, 14 warped band, 121 Diamond structure, 4,95 Dielectric constant, 39, 55, see also specific materials high frequency, 39, 84, 85, 135, I36 low frequency, 39.84.85, 135. 136 Dipole scattering, 146 Direct band gap, 6-15,41,84-87,92, 101,234, see also Energy gap, specific materials effective mass energy gap, 14.84-87 table, 84,85, 101 Distribution function, 15-19, 185-187, 238, 243, 274, see also Fermi-Dirac distribution diffusion approximation, 16 displaced Maxwellian, 276-278 hot-electron regime, 274-277

E Effective charge, 134-138, see also specific materials table, 136 Effective mass, 12-14, 116, 226, 285, see also Band structure, Magnetophonon masses, specific materials band edge, 12-14, 116,234,250,287 magnetophonon, 222, 226, 227, 249-251, 287 table, 84, 85, 116, 250 valence band, 116, 117 Effective mass energy gap, 11-15. 84-87, 101, 106, 234, see also Direct band gap table, 84, 85, 101 Elastic constants (coefficients), 35, 36, 129, 133, 134 table, 84.85 Electric field relationships, 189-207 curvilinear coordinates, 190 Electron concentration, 22, 23, 29, see also Charge-carrier density Electron energy, see Charge-carrier energy

Electron group velocity, 12-14 Electron temperature concept, 233, 273-277 displaced Maxwellian distribution, 276-278 resonance cooling, 281 Electron transport, see ulso Hall effect, Hole transport, Magnetoresistance, Mobility, Scattering, Thermoelectric power, Thermomagnetic effects, Transport coupled equations, 24, 110 high field, see Hot electron effects inhomogeneous crystals, 175-220 low field, Iff, 16 multicarrier, see Multicarrier conduction Electron wave functions. see also Hole wave functions admixture, 10, 12, 78 group velocity, 12-14 overlap integral, 13, 3 I , 60 p-like, 10, 12 s-like, 10, 12 Energy-gap, 6-11, 14, 84-87, 101, 234, see also specific materials direct, see Direct band gap effective mass, 14,87 electron-phonon coupling effect, 14 indirect, 41-46, 92 lattice dilatation effect, 14, 87 optical, 14, 87 pressure effects, 15, 87, 285-287, see ulso Stress effects table, 84,85, 101 thermal effects, 14, 87 Energy levels, see also Band structure, Chargecarrier energy labeling convention, 8,97

F Facet effect, 200,215, seealso Inhomogeneities Fermi-Dirac distribution 17, 238 Friihlich parameter, 233, 235, 270, 271 G GaAs, 4.67-73, 149-155,260-262 amphoteric dopants, 150 band structure, 9,70,97-I09 deformation potentials, 85, 133, 154 device applications, 67 dielectric constants, 68, 85, 136, 154

304

SUBJECT INDEX

direct gap. 85, 101 effective charge, I36 effective masses, 85, 116, 153, 250,262 magnetophonon, 250,262 electron mobility, 68-71 Hall coefficient factor, 152, 153, 218 inhomogeneous sample, 2 17.21 8 Hall scattering factor, 70, 72 Hall voltage, hot-electron regime, 282, 283 oscillatory, 262, 282 hole mobility, 117, 149-155 impurity conduction, 150 inhomogeneities, 201-203, 217. 218 magnetophonon masses, 250,262 magnetophonon oscillations, 247,248,257, 2W262 Hall voltage, 257, 262, 282 hot-electron regime, 275, 278-280 stress effects, 287 material parameters, 85, I34 nonparabolicity, 261 overlap functions, 1 12. 1 I3 phonon frequencies, 68 characteristic temperature, 85, 134 polar hole mobility, 138, 139 spin-orbit splitting, 101 subsidiary minima, 70, I01 time-dependent fields, 71-73 valence band, 97-109 anisotropy parameters, 120, 153, 173 effective masses, 116, 153, 154, 250 energy contours, 102 warping, 153 wave function coefficients, 107-109 Zn-doping anomaly, 150 Galvanomagnetic effects, 3, 24 GaN, 4.64 direct gap, 64 electron mobility, 65 material parameters, 85 Gap, 4 , 6 5 6 7 deformation potentials, 67, 85, 133 dielectric constants, 85, 136 dipole scattering, 146 direct gap, 101 effective charge, 136 electron mobility, 66 hole mobility, 117, 144-149 material parameters, 85, I34

polar hole mobility, 138, 139 scattering mechanisms, 67 spin-orbit splitting, 101 subsidiary bands, 101 valence band, 97-109, 148, 149 anisotropy parameters, 120. 149, 173 energy contours, 103-105 masses, 116, 148, 149 nonparabolicity, 148 warping, 148 wave function coefficients, 107-109 Zn-doping anomaly, 146 GaSb, 4,74,271 antistructure defects, 155 deformation potentials, 85, 133, 160 dielectric constants, 85, 136 direct gap, 74, 101 effective charge, 136 electron mobility, 74 hole mobility, 117, 157-160 inhomogeneity problems, I57 intrinsic carrier density, 159 magnetophonon oscillations. 242, 27 I material parameters, 85, 134 polar hole mobility, 138, 139 spin-orbit splitting, 101 stress effects, 160 subsidiary minima, 74, 101 valence band, 106 anisotropy parameters, 120, 173 masses, 116, 160 Germanium, 43,8143.263-266 deformation potentials, 82.85. 133 direct gap, 101 effective masses, 85, 116,250, 264 magnetophonon. 250 electron mobility, 82, 83 intervalley scattering, 43, 81, 263-265 magnetophonon effects, 250,263-266 Seebeck coefficient, 263 magnetophonon masses, 250 material parameters, 85, 133, 134 nonpolar optical scattering, 82, 265 spin-orbit splitting, 101 subsidiary minima, 101 valence band, 116, 120,265. 266 anisotropy parameters, 120, 173 masses, 116, 250 G F resonances, see Gurevich-Firsov resonances

SUBJECT INDEX

Gray tin, 4, 5.60.82-86 band structure, 82 dielectric constants, 82, 85, 86 electron mobility, 83,86 material parameters, 85 subsidiary minima, 87 Gurevich-Firsov resonances, 222-225, 229231, 23&231, 240, 242, 245, 246, 248, 251-254, 258, 262, 265, 271, see aiso Magnetophonon effect hot-electron regime, 275, 280, 281, 284 shifts from maxima to minima, 275-277

H Hall coefficient, apparent, see also Inhomogeneities conductivity discontinuity, 181, 192-199, 219 current distortion, 196 exact analysis. 195 metallic approximation, 196, 219 quantitative results, 196-199 conductivity gradient, 180, 199-201 facet effect, 200 cylindrical cavities, 177 cylindrical inclusions, 178 fluctuations, 179, 180 gradients in magnetic field direction, 181 isolated cylindrical inclusions, 207-210,Z I9 three-dimensional current, 209 two-dimensional current, 207, 209 isolated spherical inclusions, 210, 219 experimental verification, GaAs, 212-214 magnetic-field dependence, 217.218 measurement standard, 181 van der Pauw, 181, 191, 201 single conducting inhomogeneity, 191 experimental verification, 20 1-203 spherical cavities, 178 spherical inclusions, 178 surface effects, 182-1 84 Hall coefficient factor, 48, 50, 52-55, 58, 70. 72, 76, 1 18-120. 152. 153, 21 7 anisotropy factor, 118-120 inhomogeneous specimen, 217 scattering factor, 1 I 8 Hall effect, see also Hall coefficient, Hall coefficient factor, Hall tensor

305

magnetophonon oscillations, 228, 235,247, 248,256,257,258,260,262 diffusion currents, 248 hot-electron regime, 282. 283 Hall tensor, 206 Heavy-hole band, see Band structure (HgCd)Te, 273 electron effective mass, 250, 273 energy gap temperature coefficient, 273 magnetophonon mass, 250 magnetophonon oscillations, 250, 273 HgSe. 4. 5 9 4 2 electron mobility, 61, 62 inverted band structure, 59.60 material parameters, 84 semimetal, 59 subsidiary minima. 60 HgTe, 4, 62.63, 271 electron-hole scattering, 63 electron mobility, 62 magnetophonon oscillations, 242. 271 material parameters, 85 subsidiary minima, 63 Hole transport, 110-126. see also Electron transport, Mobility, Transport assumptions. various models, 125 coupled equations, 110 decoupled bands, 114-1 17 "effective" mobility, 93. 114-1 17 Hall coefficient factor, 118-120 hole density, warped bands, 120, 121 interband scattering, 115-1 17, see also I nterband scattering scattering mechanisms. 121-140, see also Scattering two-band conduction, 93. 114-1 17, see also Two-band conduction Hole wave functions, see also Electron wave functions overlap functions, 110-1 14, 135 p-like, 93. 135. 136 s-like, 92, 135 Hot-electron effects, 233,273-288 Bekefi peaks, 281,282 microwave emission, 281 Boltzmann equation numerical solution, 275, 276 diffusion approximation, 274,275, 288 distribution function, 275, 276 displaced Maxwellian, 276-278

306

SUBJECT INDEX

electron-temperature concept, 233, 273276,288 longitudinal magnetoresistance, 274 magnetophonon masses field dependence, 283 magnetophonon phenomena, 223,226,228 shifts from maxima to minima, 275-277 relaxation time, 273 resonance cooling, 281 transverse magnetoresistance, 275 warm-electron coefficient, 278-279 oscillations, 278, 279

I InAs, 4,75-77,259, 260 annealing effects, 163 deformation potentials, 85, 133, 165 dielectric constants, 85, 136 direct gap, 75, 76, 101 effective charge, I36 effective masses, 76, 85, 165,250 electron mobility, 76, 77 Hall factor, 76 hole mobility, 117, 164, 165 intrinsic carrier density, 159 magnetophonon masses, 250 magnetophonon oscillations, 247, 259, 260 Hall voltage, 260, 283 Seebeck coefficient, 259 warm-electron effects, 280, 283 material parameters, 85, 134 nonparabolic bands, 75,259 phonon emission time, measurement, 88,89 polar hole mobility, 138, 139 spin-orbit splitting, 101 subsidiary bands, 101 surface effects, 163, 164 valence band, 106 anisotropy parameters, 120, I73 masses, 116, 165 Indirect band gap, 6,92, see also Energy gap Inhomogeneities, transport effects, 175-220, see also Conductivity, Hall coefficient, Mobility, Resistivity conductivity discontinuity, 181, 192-199 current distortion, 196, 209,220 conductivity gradient, 180, 199-201 facet effect, 200 in magnetic field direction, 181

cylindrical cavities, 177 cylindrical inclusions, 178 disordered regions, I79 facet effect, 200,215 fluctuations, 179, 180 Hall coefficient field dependence, 217, 218 intermediate-size, 176180 isolated cylindrical inclusions, 203.210 current distortion, 209 three-dimensional current, 208-2 LO two-dimensional current, 203-208 isolated spherical inclusions. 210.21 1 current distortion, 21 I experimental verification, GaAs. 21 1-215 mobility enhancement, 21 1 macroscopic, 176, 180-182 microscopic, 176, 177 periodic variations, 180 polycrystalline samples, 216 single conducting inhomogeneity, 190-192 experimental verification, GaAs, 201-203 space charge regions, 178, 179 spherical cavities, 178 spherical inclusions, 178, 179 InP, 4,75-76,269,270 deformation potentials, 85, 133, 162 device applications, 74 dielectric constants, 85, 136 direct gap, 101 effective charge, 136 effective masses, 75,85,250,270 magnetophonon, 250,269,270 electron mobility, 75. 76, 269 hole mobility, 117, 16&162 k p matrix element, 75 magnetophonon mass, 250 magnetophonon oscillations, 269.270 band-edge effective mass. 250, 269 hot-electron regime, 275, 279, 280 stress effects, 287 material parameters, 85, 134 polar hole mobility, 138, 139, 162 pressure effects. 270, 287 scattering, 75 spin-orbit splitting, 101 subsidiary minima, 74.75, 101 valence band anisotropy parameters, 120, 173 masses, 116, 162 InSb, 4. 10, I1,77-79.250-258,277-282

-

307

SUBJECT INDEX

acoustoelectric gain. 28&283 Bekefi peaks, 28 I , 282 hot-electron conditions, 280-283 deformation potentials. 85, 133, 167, 169 dielectric constants, 136, 167 direct gap, 78. 85, 101. 165 effective charge, 136, 167 effective masses, 78, 85. 250, 255 magnetophonon, 250 electron mobility, 78, 79.250 Hall coefficient field dependence, 170, 257. 258 Hall voltage hot-electron regime, 283 magnetophonon oscillations, 228, 250, 256-258.283 hole mobility, 117, 166169 hot-electron effects, 275.277-284 resonance cooling, 281 interband scattering, light holes, 169 intrinsic carrier density, 159, 165 magnetophonon masses, 250,255 magnetophonon oscillations. 222,224, 228, 240,247,248,25&258 Hall voltage, 228, 256258 hot-electron regime, 275, 277-284 Seebeck coefficient, 228, 250-255 stress effects. 285-287 material parameters, 85, 134 microwave emission, 281 J3ekeB peaks, 28 I , 282 nonparabolic bands, 78,256 overlap function, 112. 1 13 photoconductivity hot electron, 284,285 polar hole mobility, 138. 139, 169 scattering processes, 79. 167-170,255,256 spin-orbit splitting. 101 stress effects, 285-287 subsidiary minima. 79. 101 valence band. 106 anisotropy parameters, 120, 173 masses, 116, 169.250 wave function coefficients, 107-109 warm-electron coefficient, 278. 279 lnterband scattering, 114-1 17, 130, 137. 169, 237,265, see also Intervalley scattering, Mobility, Scattering. specific materials relaxation times. 115. 116 Intervalley scattering, 4146, 80-82, 92, 263,

265, 287, see also interband scattering, Mobility, Scattering, specific materials deformation potential, 45 differential scattering rate. 45,46 .$ and g-transitions, 4 3 4 5 partial mobility, 46 temperature dependence, 46 randomizing, 41,45 umklapp process, 4 2 4 5 Ionized impurity scattering, 6, 18-20, 27-34, 93, 139, 140. 233, see also Scattering, specific materials Brooks-Herring, 6.27-32.88, 139, 140 Conwell-Weisskopf, 32 coupling coefficient, 233 differential scattering rate. 29-31 Dingle, 6, 27-32. 88 electron freezeout, 28. 33 electron-hole, 27,62,63, 78-80 hole-hole, 140 inverse screening length, 29 multiple, 26, 28. 32-34 nonparabolic bands, 32 partial mobility. 33 temperature dependence, 33 phase-shift calculations, 27, 33 relaxation rate, 32 screening, 27, 29, 33, 140. 233 two-band effects, 140, see also Two-band conduction effective mobility, 140 Iterative solutions, 3. 19, 134, see also Boltzmann equation

K k - p method, 7-15,97-109 Kane's theory. 6-15, see also k - p method Kelvin relations. 21 Kubo formalism, 237

L Landau levels. 222, 223. 227-230, 231, 234. 24 1,266-268 broadening, 227,236,240.246,249,260,288 nonlinear, 267 split, 256,267 unequal spin splitting, 256 Laplace's equation, 192. 204 Light-hole band, see Band structure

308

SUBJECT INDEX

M Magnetoconductivity tensor, 188, 190, 210, 273-240, 247, see also Conductivity tensor, Magnetoresistance inversion, 235 spherical coordinate system, 210 Magnetophonon effect, 221-289, see also Gurevich-Firsov resonances, Hall effect, Magnetoresistance, Pseudoresonances, specific materials band structure considerations, 227, 23 I, 234 deformation potential parameters. 227, 285-287 effective mass considerations, 226,227,287 GF resonances, see Gurevich-Firsov resonances Hall effect, 228, 235, 247, 248, 256258, 260,262,282,283 hot-electron regime, 282, 283 hot-electron regime, 223, 226, 273-285 magneto-Seebeck oscillations, 224, 237, 248, 252, 253, 259, 263, see also Thermoelectric power masses, 249, 250, see also specific materials electric-field dependence, 283 measurement methods, 226,287,288 derivative techniques, 226, 252 double differentiation, 268 nonparabolic bands, 231,234,240,254.256 Ohmic regime, 226-273 pseudoresonances, see Pseudoresonances stress effects, 226, 227, 285-287 transport properties where observed, 228 Magnetophonon mass, 249, 250, 287, see also Effective mass, specific materials table, 250 Magnetoresistance, see also entries under Magnetoresistance, oscillatory longitudinal theory, 242-247 combination resonances, 242 combined scattering, 242-245 damping, 246 hot electrons, 274-277 level broadening, 246 mobility, 243 transverse, theory, 237-242 damping factor, 241, 242 hot electrons, 275-277 Kubo formalism, 237

level broadening, 240-242 Magnetoresistance, oscillatory, see also Magnetophonon effect, specific materials longitudinal, 229, 236, 242-247. 249-265, 268-273, 288 hot electrons, 274-277 magnetophonon oscillations, 223,228,235247, 250-285 ShubnikovdeHaas oscillations, 223, 236, 241, 242,284 transverse, 229, 235, 237-242. 249-265. 268-270,272,288 hot electrons, 275-277 Material parameters, 46,47, 84, 85. 116 conduction band parameters, 84, 85 tables, 84, 85, 101, 116, 120, 133, 134, 136, 173 valence band parameters, 116, 120, 133, 171-173 Maxwellian distribution hot-electron regime, 276278 Mobility, see also following entries longitudinal magnetic field, 243 magnetophonon phenomena, 230, 231,243 Mobility, apparent, see also Inhomogeneities, Transport conductivity discontinuity, 182, 192-199 exact analysis, 195 metallic approximation, 196 quantitative results, 199 conductivity gradient, 182, 201 enhancement, 182-220 polycrystalline samples, 216 surface accumulation layer, 182-184 isolated cylindrical inclusions, 207-21 0 three-dimensional current, 209. 210 two-dimensional current, 207-209 isolated spherical inclusions, 210 experimental verification, GaAs, 21 1-214 lowering, 176-182 carrier density gradient, 18&182 conductivity discontinuity, 180 cylindrical cavities, 177 cylindrical inclusions, 178 disordered regions, 179 fluctuations, 179, 180 periodic distributions, 180 spherical cavities, 178 spherical inclusions, 178, 179 periodic inhomogeneities, I80

309

SUBJECT INDEX

single conducting inhomogeneity. 190. 191 experimental verification. 201 -203 Mobility. electrons, 20. 87, 92. see ulso Electron transport, Scattering, specific materials, specific scattering mechanisms acoustic scattering, 38, 82 drift, 20. 21. 87 Hall, 87 Hall coefficient factor, 48, 50. 52-55, 58, 70.72, 76, 118 intervalley scattering, 46 ionized impurity scattering, 33 measurement,inhomogeneityenhancement, 182-220 piezoelectric scattering, 37 polar scaitering, 40.94 Mobility, holes, 91-95, 110-140, see also Hole transport, Scattering, specific materials, specific scattering mechanisms acoustic and nonpolar optical scattering. 130, 132 acoustic scattering. 127, 130. 132 assumptions, various models, 125 decoupled bands, 114-1 17, 130, 137, 140 deformation potentials, 127-134 ”effective,” 128, 129 tables, 133, 134 dipole scattering, 146 “effective.” 93. 114-1 17. 140 Hall coefficient factor, 118-120 interband scattering, 130 ionized impurity scattering, 140 nonpolar optical scattering, 127, 130, 132 notation, 118 polar optical scattering, 134-139 two-band conduction, 93,1161 17,130,137 Zn-doping anomaly GaAs. 150 Gap. 146 Mobility, inhomogeneous crystals, 175-219. see also Inhomogeneities, Mobility Monte Carlo simulation. 20, 136 Multicarrier conduction, 51,52.58-62,74-76, 79,86, 165, see also Two-band conduction, specific materials N

Nonparabolic bands, 9-14.32,75,78,93. 117, 148, see also specific materials

magnetophonon phenomena, 231,234,240, 2&246,254.256.26 I, 210 Nonpolar optical mode scattering, 61, 82, 92, 127-134. 232, 233. see also Optical phonon interactions, Scattering, specific materials coupling coefficient. 232 partial mobility, 127, 128

0 Onsager reciprocal relations, 21 Optical phonon interactions, 38-46. 127-134, 232-237, 242. 246. 247, 274-278, see also Magnetophonon effect, Magnetoresistance, Nonpolar optical mode scattering, Polar optical mode scattering, Scattering characteristic temperature, see also Debye temperature tables. 84, 85, 134 hot-electron regime, 274-285 intervalley, 263, see also Intervalley scattering mobility-limiting mechanism, 229, 230 oscllatory effects, see Magnetophonon effects phonon energies (frequencies), 128. 134 spin-state transitions, 225 Optical scattering, see Nonpolar optical mode scattering, Optical phonon interactions, Polar optical mode scattering, Scattering Overlap functions, 110-114, 135, see also Overlap integral Overlap integral, 13, 3 1. 60,see also Overlap functions

P PbTe. 272 magnetophonon masses. 250. 272 magnetophonon oscillations, 250, 272 Phonon emission times direct measurement, 88,89 hot-electron magnetophonon peaks, 284 Phonon energies, 37, 39,44-46, 232, 233, see also Debye temperature, Optical phonon interactions, specific materials characteristic temperature, 84, 85, 134, 250 intervalley, 263, see also Intervalley scattering tables, 84, 85, 134, 250

310

SUBJECT INDEX

Photomagnetic effects magnetophonon phenomena, 228,247,27 1 hot electrons, 278, 284 Piezoelectric constants (coefficients),34-36 tables, 84,85 Piezoelectric scattering, 6, 18, 34-37, 52, 233, see also Scattering, specific materials coupling coefficient, 233 differential scattering rate, 35 interaction potential, 34-36 partial mobility, 37 temperature dependence, 37 relaxation rate, 36, 37 temperature dependence, 37 Poisson summation formula, 241 Poisson's equation, 28 Polar optical mode scattering, 2, 6, 10, 11, 3841, 72,92, 127, 134-139, 232, 233,

Optical phonon interactions, Scattering, specific materials coupling coefficient, 232 differential scattering rate, 39 effective partial mobility, holes, 137-1 39 inelastic scattering operators, 38 multiphonon processes, 88 randomizing case, 41 relaxation approximation 41, 136 scattering-in rate, 40, 41 scattering-out rate, 39,40 Polaron effects, 15, 234, 235, 241. 261, 268, 270, see also specific materials effective mass, 84,85,241 Potential. 188-205, see also Electric field relationships Pseudoresonances, 230, 240, 245, 246, 248, 256, 262. 268-210, 273, see also Magnetophonon effect see ufso

Q Quantum effects, 16, 221-223, 237-246, see also Boltzmann equation electron mobility, semimetals, 60,63, 86 magnetophonon resonances, 221-223 transport theory, 237-248

R Raman scattering magnetophonon effects, 271

Relaxation time, 2, 4, 18, 21, 186, 247, 273, see also Scattering, specific scattering mechanisms acoustic scattering, 38 hot-electron regime, 273 interband scattering, I 15, 116 ionized impurity scattering, 32 low temperature polar scattering, 40.41 magnetophonon effects, 247 piezoelectric scattering, 36, 37 polar scattering (approximation), 41, 136 time-dependent fields, 26 Resistivity, apparent, see also Inhomogeneities conductivity discontinuity, 181, 192-199, 219

exact analysis, 194,219 metallic approximation, 195 quantitative results, 197 conductivity gradient, 181, 199, 200 gradients in magnetic field direction, 181 isolated cylindrical inclusions, 205-210,2 19 threedimensional current, 209 two-dimensional current, 206, 207, 219 isolated spherical inclusions, 210, 219 experimental verification, GaAs, 213 single conductivity inhomogeneity, 190-192 experimental verification, 201. 202 surface effects, 182, 183 van der Pauw measurement, 190 Resistivity tensor, 205-207, 235

S

Scattering, 26-46, 110-140, 146, 232-235, see also following entries, specific materials, specific scattering mechanisms coupling coefficients, 232, 233 Scattering, electron, 2, 6, 2 W , see also Mobility, specific materials, specific scattering mechanisms acoustic mode, 6, 18, 233 deformation potential, 6, 37, 38, 232, 233 elastic, 18-20, 236, 243 electron-hole, 27, 62, 63, 79, 80 inelastic, 18-20, 235 intervalley, 6 , 4 1 4 , 237 ionized impurity, 6, 18-20, 27-34, 233 multiphonon processes, 77, 88, 256, 259, 262,280

multipIe, 26, 28, 32-34

311

SUBJECT INDEX

nonpolar optical, 82, 232, 233 nonrandomizing, 20.25 piezoelectric mode, 6, 18, 34-37, 233 polar optical mode, 2, 6, 10, 11, 25, 38-41, 232 randomizing, 20,41,45 self-scattering rate, 25 Scattering, hole, 121-140, see also Mobility, specific materials, specific scattering mechanisms acoustic, 127, 130 acoustic and nonpolar optica1, 122, 123, 130-133 dipole, 146 interband, 114-1 17, 130, 237 nonpolar materials deformable ion, I26 deformation potential, 126 rigid ion, 126 nonpolar optical mode, 127-1 34 polar optical mode, 123, 126, 127, 134-139 Scattering rate, see also specific scattering mechanisms differential, 16-19, 26, 29-31 in-, 19 Out-, 19 Seebeck coefficient, 21-23, 237, 248, see also Thermoelectric power, specific materials Shubnikov-deHaas oscillations, 223, 236, 241,242,284 Silicon, 4146,7941, 182-184 deformation potentials, 80. 85, 133 direct gap, 101 first Brillouin zone, 42 intervalley scattering, 41-46 material parameters, 85, 133, 134 mobility, 80, 81, 117 anomalously high, 182-184, 216 decoupled band approximation, 117 inhomogeneous specimens, 182484,216 surface effects, 182-184 scattering, 4146,80, 117 spin-orbit splitting, 101 subsidiary minima, 101 valence band, 114-121 anisotropy parameters, 120, 173 masses, 116 Spin-orbit splitting, 7, 9, 99-101, see also specific materials

table, 101 Split-off band, see Band structure Stress effects, 285-287 magnetophonon oscillations, 285 deformation potential information, 285287 shift in mass, 285 Surface effects, see Inhomogeneities

T Tellurium, 266-269 Landau levels, 266-268 nonlinearity, 267 splitting, 267 magnetophonon hole masses, 250,266-269 valence band structure, 266-268 Tetrahedrally coordinated crystals, 4, 5, 7 Thermoelectric power, 1, 21-23, 237, 248, see also Magnetophonon effect, Seebeck coefficient, Thermomagnetic effects, specific materials diffusion tensor, 237 oscillatory, 224, 228, 237, 248, 252, 253, 255,259,263 Thermomagnetic effects, 3, 24, 228, 237, 248, 288, see also Magnetophonon effect, Thermoelectric power, specific materials hot-electron regime, 284 oscillations, 228, 237, 248, 252, 253, 255, 259,263 Time-dependent fields, 24-26 hotclectron magnetophonon peaks, 284 Transport, see also Electron transport, Hole transport, Mobility coupled equations, 24, 110, 117 inhomogeneous crystals, 175-220 multicarrier, see Multicarrier conduction quantum regime, 221-223, 237-248, see also Quantum effects Two-band conduction, 51, 52, 58-62, 74-76, 79,86,93, 114-117, 130, 137, 140,235, 247, see a/so Multicarrier conduction decoupled bands, 114-117, 130, 137, 140 “effective” mobility, 93, 140 stress effects, 160,286 U

Umklapp processes, 4245 Units, 288, 284

312

SUBJECT INDEX

effective mass, 48 electron mobility, 49 Valence band, see Band structure Hall scattering factor, 48 van der Pauw measurement technique, 181, material parameters, 84 191,201,211 Variational calculations, 2, see also Boltzmann ZnS, 4,49-5 1 direct gap, 49 equation effective mass, 49, 50 W electron mobility, 51 Hall factor, 50 Warm-electron coefficient, 278, 279, see also material parameters, 84 Hot-electron effects ZnSe, 4, 52 Warped bands, 93, 117-121, 148, see also dielectric constant, 52 Band structure, specific materials direct gap, 52 density of states, 120, 121 effective mass, 52 Hall anisotropy factor, 120 electron mobility, 53 Hall factor, 52 Z material parameters, 84 Zinc-blende structure, 95-97 ZnTe, 4, 52-54 energy band structure, 97-109 direct gap, 53 ZnO, 4,47 electron mobility, 53, 54 acoustic deformation potential, 48 fluorine implantation, 52 direct gap, 47 material parameters, 84 V

A 5 8 6

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G 1 H 2 1 3

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