Table Of ContentProblems of Linear Electron
(Polaron) Transport
Theory in Semiconductors
BY
M. I. KLINGER
TRANSLATED BY
J. B. SYKES
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First edition 1979
British Library Cataloguing in Publication Data
Klinger, M I
Problems of linear electron (polaron) transport
theory in semiconductors.
- (International series in natural philosophy: vol. 87)
1. Electron transport 2. Semiconductors
1. Title 11. Series
537.622 QC176.8.E4 78-40821
1 SBN 0-08-018224 0
In order w make this volume available as economically
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Preface
The usual theory of electron transport (that is, transport due to elec-
trons) in a crystal is actually based on three fundamental concepts: (1) that
the current-carriers are quasi-particles with a definite quasi-momentum and
dispersion relation; (2) that the non-zero electrical resistance is due to the
scattering of carriers by dynamic and static defects in the ideal structure of
the crystal, and depends on the mean free path between two scattering events;
(3) that the carrier mean free path is large compared with the de Broglie wave-
length (in this sense there is 'occasional' scattering of the carriers).
Consequently, the behaviour of the carriers is described by a probabilistic
quasi-momentum distribution function, which is determined as the solution of
the familiar Boltzmann-Bloch transport equation. This may be regarded as a
kind of equation of continuity that takes account of the change in the carrier
distribution function in the crystal both because of the applied external
forces and because of the redistribution of quasi-momenta in scattering. In
this sense (and neglecting the quantum nature of the dispersion relation and
the scattering probabilities for carriers), the theory of such transport
effects, frequently referred to nowadays as Boltzmann transport, is essentially
classical and involves no specifically quantum features. The principles and
numerous applications of this theory, which has a very wide range of applic-
ability in crystals, have been explained and discussed in several well-known
and by now classic monographs.
During the last fifteen years, the general ideas of transport theory have been
quite considerably developed by means of studies of the fundamental properties
of dissipative systems in which irreversible transport occurs (Bogolyubov,
Van Hove, Prigogine) and by means of a general formulation of the theory of
linear transport phenomena in a dissipative system subject to the action of
weak external interactions (Kubo, Feynman, Peierls, Greenwood, Lax), and also
by means of the study of new carrier types in solids and carrier dynamics in
unusual extremal conditions. This development has already had a considerable
effect on the theory of transport in crystalline semiconductors, giving rise
to new ideas and areas: polaron transport, low-mobility transport in a
xi
X11 PREFACE
semiconductor, electron transport in a strong (quantizing) magnetic field, and
electron transport in magnetic crystals and in disordered systems (disordered
solid solutions and glasses).
In this book an attempt is made to summarize and critically discuss the develop•
rent of new areas in electron transport theory in semiconductors mainly up to
the beginning of 1973, having regard principally to transport phenomena in
fairly weak perturbations, i.e. linear transport, except in sections 11.4 and
1111.5. The main attention is given to the consideration of fundamental
aspects of the theory and the essential physical nature of the transport pro-
cesses. On the other hand, a very detailed account is given only for certain
selected problems of the theory, whose evolution is still far from complete;
the critical discussion of these in the available reviews relates only to
certain aspects. The author has taken as such problems in linear electron
transport theory in semiconductors those of small-polaron (low-mobility) con-
duction in crystals and polaron (electron) conduction in disordered structures.
Some differences in the treatment arise from the fact that the development of
these areas of the theory, especially the second one, is still by no means
complete. Problems of linear transport theory for large polarons in crystals
and of polaron and electron conduction in strong quantizing magnetic fields in
their most fundamental aspects have been considered in various review articles
and monographs, which are cited at the appropriate places in the book. It has
therefore appeared possible to give only a fairly brief treatment of these
problems.
The book deals mainly with electron (polaron) transport theory in ordinary
three-dimensional solids, in the 'single-particle' picture. Such fundamentally
many-electron correlation effects as the Hubbard model, transport phenomena
near the Mott dielectric-metal phase transitions, and the very interesting
specific properties of one-dimensional or quasi-one-dimensional structures
with Peierls instability are not specially discussed here, except for some
aspects of the Hubbard model and of many-electron systems with a low-
temperature charge-ordered phase, in sections 11.3 and 1111.7; see, e.g., the
reviews by Adler (1968), Doniach (1969), Mott and Zinamon (1970), and (for
quasi-one-dimensional systems) Fröhlich (1954b), Bardeen (1973), Shchegolev
(1972) and Zeller (1973).
In this book there is no discussion of physical phenomena in semiconductors
in electromagnetic fields having frequencies close to or exceeding the thres-
hold value for intrinsic electron absorption, since they are optical rather
xiii
PREFACE
than transport phenomena. Moreover, it is traditional to regard the study of
the behaviour of ferromagnetic and antiferromagnetic crystals, where collec-
tive effects play a significant role, as a separate branch of solid-state
theory (see, e.g., Vonsovskii 1971); electron transport in magnetic crystals
is therefore not specially considered here, apart from some comments in sec-
tions 11.5 and 1111.5. Moreover, the important specific properties of electron
transport phenomena in external electric fields and sound waves (acousto-
electric effect, sound generation and amplification, non-linear effects, etc.)
are also not specially discussed; see, e.g., the review by Einspruch (1965).
The book accordingly consists of three parts. The first part (Chapter I)
deals with some general topics in the theory of transport phenomena: the
general dynamical theory of linear transport in dissipative systems (Kubo
formulae) and the phenomenological theory. The second part (Chapters 11-11)
is devoted to the theory of polaron transport in a crystalline semiconductor,
including the basic dynamics and scattering properties of polaron carriers
(Chapter III), the transport theory of polarons in a wide-band semiconductor
(Chapter IV), and a fuller discussion of the transport properties of small
polarons governing the conductivity of narrow-band semiconductors with low
mobility (Chapters V and VI). A polaron is a kind of 'bare' quasi-particle
(electron or hole) together with an associated phonon cloud. For this reason
brief information is also given in Chapter II, to the extent that is necessary
for the discussion of polaron transport, concerning the dynamics and scatter-
ing of such 'bare' quasi-particles and phonons in a crystalline semiconductor.
The third part contains a critical account of electron transport in disordered
systems, including amorphous substances, with allowance for polaron effects
(Chapters VII and VIII). Particular attention is given here to the nature of
the carrier states, which is very important in determining the type of trans-
port.
The transport phenomena considered in the second and third parts of the book
have recently been the subject of extensive experimental studies in a great
variety of semiconductors. Here, however, it is mainly the theoretical prob-
lems that are discussed, the appropriate experimental results being mentioned
only in isolated cases by way of illustration.
It should also be noted that the choice of basic problems of the present-day
theory of linear electron transport in semiconductors, as made here, is to a
large extent subjective and determined by the author's personal partialities
xiv PREFACE
and the range of topics he has been concerned with investigating in recent
years. This explains the absence, already mentioned, of any adequate discus-
sion of some equally important problems in linear electron transport theory in
semiconductors. The book is designed for readers familiar with quantum mecha-
nics, statistical physics and the electrodynamics of continuous media, and
also with the standard theory of the solid state as described in various well-
known monographs and reviews cited in the text.
No attempt has been made to compile a detailed bibliography of papers on the
problems discussed; the number of these is large and rapidly increasing.
The references given in the various chapters merely include review articles
(where details of the original papers may be found), together with original
papers that are the subject of direct discussion in the book. For convenience
of exposition, calculations as a rule use the system of units in which the
Planck constant h and the Boltzmann constant k are put equal to unity. It
is quite easy to use dimensional arguments in order to restore these constants
in the final formulae.
The book mainly includes the results of theoretical work on the subjects under
discussion during the 15-20 years up to the beginning of 1973. A brief account
of a small number of more recent results that have become known to the author
was added to the manuscript in 1973.
It is a pleasure to thank many of my colleagues, in particular Professors
N.F. Mott, Yu. Kagan, M. Krivoglaz and M. Deigen for discussions on some of
the problems considered in this book, and Professors N.F. Mitt, R.J. Elliott,
Yu. Kagan, M. Krivoglaz, M.H. Cohen, H. Fritzsche and M. Pollak for sending
preprints of some of their papers mentioned in the book.
Leningrad, 1973 M. Klinger
KEY TO ABBREVIATIONS
BBM basic band model
BCB band of concentration broadening
BFSW band of fluctuation scattered waves
BSSW band of strongly scattered waves
BSW band of scattered waves
BWSW band of weakly scattered waves
CPA coherent-potential approximation
1CP(A) single-centre coherent-potential (approximation)
DBA disordered binary alloy
DDS disordered doped semiconductor
FCA fluctuation cluster approximation
FHIP Feynman, Hellwarth, Iddings and Platzman
GT Gaussian tail
h.c. Hermitian conjugate
HDDS heavily doped disordered semiconductor
HF Bar tree-Fock
lB impurity band
LDDS lightly doped disordered semiconductor
M-CFO Mott-Cohen-Fritzsche-Ovshinsky
'CT non-Gaussian tail
OFA optimal fluctuation approximation
RPM random phase model
SCA single-centre approximation
SDS stationary disordered structures
XV
CHAPTER I
Basic Relations in the Quantum Theory of
Linear Transport Phenomena (Kubo Formulae)
1. General relations in quantum statistical mechanics
1.1 Formulation of the problem. Macroscopic systems
The basic problem in the quantum theory of irreversible transport
phenomena in macroscopic systems departing only slightly from equilibrium
(linear transport theory) consists of the following parts:
(1) To establish linear phenomenological relations between the Fourier
components J(r) (k , w) of the current densities within a macroscopic system
and the forces (perturbations) F(p)(k , w) acting on the system in equilibrium,
and also to establish the range of 'weak' forces in which these relations are
valid.
For any system which is spatially uniform in equilibrium,t these relations
have the general form
J~r)
(k,w) = L 1," (k,0) F(p')(k,w) , (1.1)
p/
where the superscripts p (=1, 2, ...) and pf label the currents and forces,
which have a particular number of tensor dimensions; the functions Lpp (k,w)
are independent of the F(P and have the same number of tensor dimensions,
and are called transport coefficients.
(2) To establish general formulae for the Lpp (k,w), starting from the
microscopic quantum-mechanical description of the macroscopic system.
(3) To calculate explicitly their values and their dependences on the
temperature R. the chemical potential z of the particles (current-carriers);
the external magnetic field H, and other intensive parameters of the system.
Particular attention will be given to the currents of charge J(1) = Jú1) and
energy J(2) = J(2) and their relation to the force vectors F(p) - F(P), so
The term 'equilibrium' will be used only to denote thermodynamic equilibrium.
These are of course particles whose number is conserved in the processes
under consideration (l1= constant). The analysis will usually be given
in terms of one type of particle only.
1
2 LINEAR ELECTRON TRANSPORT THEORY IN SEMICONDUCTORS
that the transport coefficients
L - Lpp/
UM'
are tensors of rank two, with m, p' =1, 2, 3 labelling the Cartesian coordi-
nate axes. Summation is implied over repeated Cartesian subscripts m and p':
JúP)(k,w) _ LPP (k,w) FúP')(k,w)
P'
FúP')(k,w).
L L LúP,(k,w) (1.2)
r' U'
The quantities Júp)(k,w) will be called simply currents where this can be
done without misunderstanding; the currents corresponding to irreversible
transport of charge (1) and energy (2) are denoted by
(1) J and J(2) - (U) (1.3)
We shall be concerned with macroscopic bodies, i.e. condensed (solid or liquid)
systems containing a large number H of interacting non-relativistic particles
(electrons, ions, etc.) in a large volume V (» a3) with a mean density N and
mean distance between the particles a,
22 3,
N = Yl /V ^ 10 - 1 0 2 3 cm
-g
a= N 1 0-7 - 1 0 cm ,
and such that the ground-state energy E is finite and the typical energy
0
interval d, between adjacent excited levels E (> E ) is so small (as
0
doaR (a3/V)) that the energy spectrum is practically continuous when account
is taken of the broadening y0 of the energy levels of an incompletely iso-
lated system due to its interaction with the environment (y3 >r do) ( Landau and
Lifshitz 1964). It is generally assumed (e.g. by Chester 1963) that an ade-
quate approximation to the behaviour of such a system is given by that of an
asymptotically large system in the limit
V , H , N = = lim = constant ; (1.5)
V-
for brevity, this will be denoted by 'lim ' or 'the V-limit'. According to
V+~
the foregoing description, the energy spectrum of the excited states of the
z
system is continuous in the V-limit (go r 6o « 22/V3 + 0). When the particle
interactions have certain properties (see Chapter IV), such systems are able
to absorb external energy and (under normal experimental conditions) to dissi-
pate it by irreversible transport processes. Such dissipative systems will be
considered here. The current densities J(r), like the (volume) densities A
BASIC RELATIONS IN THE QUANTUM THEORY 3
of the other extensive parameters ~,i of the system, are defined by relations
such as
A= Ci/V = um (1.6)
V *~
so that the V-limit is regarded as being taken in all the resulting formulae
of the theory before any other limits such as T + 0.
1.2 Dynamical and statistical (thermal) forces
A system not in equilibrium, if left to itself in the absence of per-
turbing forces F(p), reaches an appropriate equilibrium state by suitable
irreversible processes in a characteristic relaxation time tR (see Chapter
IV). Experience shows that both the final equilibrium state (denoted by eq)
and the stage of completed relaxation at times t-,t R depend only on certain
averaged microcharacteristics of the system, not en the detailed form of
these characteristics, which is forgotten in a characteristic time tc « tR at
the start of the relaxation; is frequently of the same order as the charac-
cc
teristic times of the characteristic motions within the system. The stage of
completed relaxation is described by the familiar phenomenological relations:
(1) Ohm's lam for the relaxation of the charge density (r,t)
pe t
and the vanishing of the charge current density Ju (r,t):
(
pe(r,t) = ei(r,t) + p eq)(r)
(1 .8)
(e)(
J (r,t) + J ) = 0,
U
where N(r,t) is the number density (concentration) at the
point r in the system, and e is the unit charge (positive
or negative);
(2) Fourier's law of thermal conduction for the relaxation of
the temperature: T(r,t) T(r) ;
(3) the diffusion law for the relaxation of the concentration
N(r , t) or the related chemical potential C(r,t) : N(r ,t) +1(r),
C(r,t)-~r(r).
The parameters which occur in these relations (the electrical conductivity o,
the thermal conductivity A, and the diffusion coefficient D) are directly
The frame of reference used will always be that in which the macroscopic
body is at rest, so that the equilibrium currents are
(p) = (p) (eq) = 0 . (1.7)
