Upon substituting this relation into the collision integral ofeq. (1.31) ..... 1.2, that the symmetric part of the phonon distribution function for q > 2j~has the form ofeq.
PHYSICS REPORTS (Review Section of Physics Letters) 181, No. 6 (1989) 327—394. North-Holland, Amsterdam
THE ELECTRON-PHONON DRAG AND TRANSPORT PHENOMENA IN SEMICONDUCTORS Yu.G. GUREVICH Institute of Radiophysics and Electronics, Ukr. SSR Academy of Sciences, Kharkov, USSR
and O.L. MASHKEVICH AM. Gorki University of Kharkov, Kharkov, USSR Received March 1989
Contents: 0. Introduction 1. Kinetics of nonequilibrium electrons and phonons in semiconductors 1.1. Electron and phonon kinetic equations taking account of heating and mutual drag 1.2. Strong phonon—phonon interaction 1.3. Strong phonon—electron interaction 1.4. Nonequilibrium optical and acoustic phonons 1.5. Electron—phonon drag and the Onsager relations 2. Heat conduction and thermomagnetic effects: the influence of drag 2.1. Kinetic coefficients and drag criteria
329 330 330 342 347 357 363 364 364
2.2. Drag-dependent classification of charge carriers according to energy 2.3. Anomalous temperature fields and thermoelectric phenomena in finite-size semiconductors 2.4. Thermomagnetic effects 3. Current transport in finite-size semiconductors 3.1. Nonlinear current—voltage characteristics of semiconductor specimens 3.2. The Hall field and current in the presence of electron— phonon drag 4. Conclusion References
367 368 372 379 379 387 392 392
Abstract: This paper presents a review of the present-day knowledge of the electron—phonon drag effects and their role in transport phenomena. It describes approaches, based on the Boltzmann kinetic equation, to the study ofinteracting subsystems of electrons and acoustic and optical phonons in semiconductors characterized by an arbitrary degree of degeneracyof the charge-carrier gas and non-equilibrium of electrons and phonons in the energy—momentum space. Novel drag mechanisms associated with nonuniform quasiparticle heating (thermal drag) and intense interaction between acoustic and optical phonons (two-stage acoustooptical drag) are discussed. A fundamentally new phenomenon of spatial classification of carriers according to their energies, owing to the electron—phonon drag is considered. The approach employed allows the study of strong electron—phonon drag and some entirely new effects in drag-related transport phenomena. These include primarily an essential effect of drag on current transport, involving in particular sign reversal in the magnetoresistance of isotropic semiconductors. Under certain conditions, drag effects may result in anomalous temperature fields of charge carriers. Validity of the Onsager relations is analyzed for the case of electron—phonon drag.
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THE ELECTRONPHONON DRAG AND TRANSPORT PHENOMENA IN SEMICONDUCTORS
Yu.G. GUREVICH Institute of Radiophysics and Electronics, Ukr. SSR Academy of Sciences, Kharkov, USSR and
O.L. MASHKEVICH A.M. Gorki University of Kharkov, Kharkov, USSR
I
NORTH-HOLLAND
-
AMSTERDAM
Yu.G. Gurevich and 0 .L. Mashkevich, The electron—phonon drag and transport phenomena in semiconductors
329
0. Introduction In a system of two or more subsystems of interacting quasiparticles*), a directed flow appearing in
either of these brings forth a partial transfer of quasimomentum to the other. As a result, a directed flow of quasiparticles appears in the other subsystem too. Obviously, the collision frequency of the quasiparticles belonging to the two subsystems decreases, thus changing essentially the character of transport processes in the system. Clearly, the process described, known as electron—phonon drag, is suppressed in case one of the subsystems transfers its momentum preferentially to a third object, capable of rapidly discharging it to the enclosing thermostat. This is the case, e.g., when the principal relaxation mechanism of the electron and/or phonon momentum is due to collisions with impurities. The electron—phonon drag effect was first predicted theoretically by L.E. Gurevich in 1946 [1,2]. As
early as in 1953 it was revealed experimentally by Frederikse [3](see also refs. [4,5]). Since that time, various manifestations of the effect have been studied both theoretically and experimentally by many authors (see, e.g. refs. [6—38]).As a result, by the early 1980’s, it became a common belief that the electron—phonon drag, while playing an important part in heat transfer, is always negligible in charge transfer processes. This conclusion followed from the three principal approximate approaches to the appropriate kinetic equations that were employed in all the theoretical papers mentioned. In the case of degenerate semiconductors or semimetals, the solution was restricted to the zeroth order with respect to degeneracy [14—17,19—21, 26—281, which is insufficient, e.g., for the analysis of the effects associated with the energy dispersion of the relaxation time (such as magnetoresistance or the Nernst—Ettingshausen effect [39])* *)~In the case of nondegenerate semiconductors, it was usually assumed that the drag of electrons by a flow of electron-dragged phonons (mutual drag) is a “small effect of second order” [39].Accordingly, the term representing the mutual drag was either neglected [3, 11, 22—24, 30, 32, 39], or used as an iteration parameter [9, 10, 12, 13, 20, 21]. In other words, the consideration was restricted to the case ofweak drag effects. Meanwhile, it seems obvious that with a developed drag the term may not be small at all. Finally, the authors quoted succeeded in solving the kinetic equations for the case of classically strong magnetic fields, where drag effects become effectively weak because ofthe smallness of the initiating electron flows (magnetized electrons) [21].Thus, previous approaches were only adequate for treating effectively weak drag phenomena. The new approach developed over the last decade is suitable for treating electron—phonon drag effects of any degree. On the one hand, it has stimulated revision of many established concepts (including those mentioned above), while on the other it has led to the prediction ofa substantial range of new drag mechanisms and a class of new physical effects. To this point, we have concentrated on the effects associated with quasiparticle distributions that were nonequilibrium with respect to momentum. However, it is well known that different external agents can readily produce in a semiconductor an essential increase in the average energy of charge carriers and phonons (energy nonequilibrium) [42, 43], which can be most conveniently described in terms of heating of the charge carrier and phonon gas (i.e. increase of their respective temperatures Te and Tn)”. Since heating is known to exert a pronounced influence on transport phenomena in semiconductors, being largely dependent on the character of transport itself [43],it is quite understandable that the heating and mutual drag of electrons and phonons should be essentially interdependent. s) Most
frequently, the subsystems in question are electrons and phonons, hence the term “electron—phonon drag”.
An attempt to take into account first-order effects with respect to degeneracy, made in refs. [40, 41J, led to erroneous results (for more details see subsection 2.3). t) Note that the possibility of phonon deviation from equilibrium was first considered by R. Peierls [44]. **)
330
Yu. G. Gurevich and 0. L. Mashkevich, The electron—phonon drag and transport phenomena in semiconductors
The present review is aimed at systematizing modern concepts of transport phenomena in finite-size semiconductors (both degenerate and nondegenerate), taking into account arbitrary heating and mutual drag of charge carriers and phonons. 1. Kinetics of nonequlfibrium electrons and phonons in semiconductors 1.1.
Electron and phonon kinetic equations taking account of heating and mutual drag
We shall consider, for simplicity, a cubic monopolar (electronic, for definiteness) semiconductor with a quadratic and isotropic energy—momentum relation e=p2/2m,
(1.1)
where e, p and m are, respectively, the electron energy, quasimomentum and effective mass. Note that non-parabolicity could only change the numerical coefficients or sometimes give rise to new effects. The dispersion laws of acoustic and optical phonons can be written as w=sq/h,
(1.2)
12=12
2q2/112). (1.3) 0(1—aa Here cv and fi are, respectively, the acoustic and the optical phonon frequencies, a the lattice parameter, the dispersion constant a 41, s is the sound velocity, and q the phonon quasimomentum. The kinetics ofelectrons and acoustic and optical phonons in a heating electric field E, magneticfield H and a temperature field is described by the set of coupled Boltzmann equations [39,45, 1, 2, 14,46] (1.4)
~+v~+e(E+!vxH)~=Sep+Sed+See+Seo, + Vq•VN= Spe+ S~~+ SPd+ S~ 0,
(1.5)
~fl\f + v°~VN°= Soe+ 500+ Sod+Sop•
(1.6)
Here f, N and N°are the distribution functions of electrons, acoustic and optical phonons, respectively; V, V~and V°q are the respective velocities; e is the electron charge, c the velocity of light, and S~kare collision integrals, with the first subscript relating to the scattered object and the second to the scatterer; the subscripts e, p, o and d denote electrons, acoustic and optical phonons, and structural defects and boundaries, respectively. Effects like recombination, impact ionization, etc. have been neglected. Naturally, the inequalities ensuring applicability of the kinetic equation method have been assumed satisfied (see, e.g. refs. [43,471). The magnetic field is non-quantizing, i.e. hWH4Te~
(1.7)
Yu. G. Gurevich and 0.L. Mashkevich, The eleco’on—phonon drag and transport phenomena
in semiconductors
331
where WH = IeIH/mc is the electron gyrofrequency. The temperature is expressed in terms of energy units throughout the paper. With the exception of subsections 1.4 and 1.5, a nonpolar semiconductor is considered throughout, in which the excitation of optical phonons may be neglected, and hence eq. (1.6) is not included in the analysis. As was shown in ref. [48](see also ref. [42]),the scattering of electrons by acoustic phonons at phonon temperatures above 1 K is a quasielastic process (hw 4 i). Electron scattering by static defects is also elastic [42,48]. Under certain conditions which we assume to be satisfied, the electron scattering by optical phonons is also quasielastic [49][fordetails, see the discussion after eq. (1.29)]. Then, in view of the assumed quasielasticity of scattering, the electron distribution function may be sought in the diffusion approximation [42,43] f=f0—Vpdf0/~9e,
(1.8)
where f0 is the isotropic part of the distribution function and V(e) pdf0/ôe is the small anisotropic part of the distribution function. Using the same approximation, the phonon distribution functions can be written as N= N0 N°=
u~qôN0/c9(flw),
(1.9)
N0° u°~ q9N~°Id(tia),
(1.10)
—
—
where N0 and N0° are the isotropic parts of the acoustic and optical phonon distribution functions. As is well known, the form of the electron distribution function depends essentially on the relative magnitudes of the frequencies of electron—electron collisions, Vee~momentum relaxation, v, and the electron energy relaxation, i~ (see, e.g. refs. [50, 42, 511). In what follows, we shall assume that the frequencies obey the following inequality (partial or energy control [52]): v(s)~~ Vee(8)~’ i’8(e).
(1.11)
In this case the charge carrier system may be considered in the temperature approximation, i.e. taking for f0(e, r, t) the Fermi (Maxwell) function [521*)
f0(s, r, t) = {1 exp[e —
—
M(r, t)/Te(r, t)]}’
,
(1.12)
where /L is the chemical potential. Indeed, with the condition (1.11) satisfied, the largest term in eq. (1.4) is the electron—electron collision integral whose order of magnitude is v~f0[47].Therefore, the solution for!0 may be sought as *) Indeed, the condition p~~‘ s~is sufficient for the establishment of a temperature in the electron subsystem (S~is the principal collision operator) only if the phonons are in equilibrium. Otherwise, additional criteria like s’~Iavia,Iare needed, owing to the fact that the kinetic equation (1.4) involves a term with a spatial derivative. The corresponding nonuniform phonon heating can lead (because of the electron—phonon interaction in energy space) to nonuniform heating of electrons over characteristic lengths shorter than the pure electronic. With v~,~ (aVIar~,
clearly i’ ~ IaV/a~1,which inequality ensures that the anisotropy off remains small even with nonequilibrium phonons.
332
Yu. G. Gurevich and 0. L. Mashkevich, The electron—phonon drag and transport phenomena in semiconductors
a series in powers of i.~ ‘~ee• Then, in the zeroth approximation with respect to this parameter, we have See ( f0) = 0. The solution of this equation is the function (1.12). The inequalities (1.11) relate to the case when electron—electron collisions affect energy transfer from electrons to scattering centres, while having no influence on the transfer of electron momentum. This is the case of highest flexibility in describing the behaviour of semiconductors. Indeed, a substantial class of results are independent of the amount of control degree, whereas the mathematical formalism of this approximation is very convenient (see refs. [42, 53, 54])*). We will further assume that in the case of a nondegenerate electron gas all the characteristic lengths in the analysis (like lengths of temperature variation of charge carriers and phonons, 2n)112,where n is thespecimen electron dimensions, etc.) are much larger than the Debye radius d = (~oTeI4ire number density and ~ is the dielectric constant. With this condition satisfied, the plasma may be regarded as quasineutral. In the degenerate case d should be the Thomas—Fermi screening length. For a semiconductor with a single type of charge carriers, this means that the electron charge density is equal to its equilibrium value at any point (provided that processes like impact ionization, change of the recombination rate in a field, etc. are neglected). Hence, the electron number density n in a homogeneous impurity semiconductor is position-independent. To solve eqs. (1.4)—( 1.6), we have to know explicitly the collision integrals involved. The collisions of electrons with acoustic phonons are described by the operator [49] Sep= ~
Jd3qw(q)
x {[f(p+ q)(1—f(p))(N(q) + 1)—f(p)(1 —f(p+q))N(q)] ô(i~(p+q)— e(p)—hco(q)) + [f(p
—
q)(1 —f(p))N(q) —f(p)(1 —f(p
—
q))(N(q) + 1)] ~~(p
—
q)
—
~(p) + hw(q))}. (1.13)
Here w( q) is determined by the matrix element of the electron transition from the state with the quasimomentum p to the state with the quasimomentum p ± q through interaction with a phonon with the quasimomentum q; Dirac’s delta functions in the integrand ensure the fulfilment of the energy conservation law, i.e. r(p±q)=e(p)±hw(q).
(1.14)
The collision probability depends on the quasimomentum of the interacting phonon as [42] w(q)=wo(sqIT)2K,
(1.15)
with K = ~ in the case of electron scattering by the acoustic phonon deformation potential (DA scattering), and K = ~in the case of scattering by a polarized potential (PA scattering). For interaction with optical phonons, K =0 in the case of scattering by the deformation potential and K = 1 in the case of scattering by the piezopotential. The values of w 0 are listed in table I of ref. [42]. To transform eqs. (1.13) and (1.14) to the case of optical phonons, N should be replaeed by N°and w(q) by (1(q). Since the electron—defect collisions are quasielastic, the integral Sed may be written in the T —
~ The effect of electron—electron collisions on momentum redistribution has been thoroughly studied in ref. [41].
Yu.G. Gurevich and 0.L. Mashkevkh, The electron—phonon drag and transport phenomena in semiconductors
333
approximation [42],i.e.
—f(p)J
~ed = Ved(E)[fO(S) Ved(S) =
(1.16)
,
Ve°d(8IT)~
(1.17)
.
Here ~ is a constant value depending on the specific form of the scattering potential; the values of i are presented in table II of ref. [42].The integral Spe is [42] sp~=fd3pw(q){f(p+q)[1-f(p)][N(q)+ 1]-f(p)[1-~f(p+q)}N(q)} (1.18) The transition to the case of optical phonons can be made as in eq. (1.13). Since the process of phonon scattering by defects is elastic, the collision integral SPd generally is taken in the r approximation [55,56, 14], i.e. SPd
N0]. (1.19) 1~,dis the phonon—defect coffision frequency. Here The general form of the phonon—phonon collision integral is rather complex [46],therefore, we are going to consider it below for various limiting cases only. Before starting to solve the kinetic equations, let us transform the coffision integrals Sep~Seo, Spe and Soe. Assuming the anisotropic parts of all the distribution functions to be much smaller than the isotropic (the necessary criteria for the phonons will be presented below), substitute eqs. (1.8)—(1.10) into the respective collision integrals and expand these in the small anisotropic corrections. Then, using quite standard methods (see, e.g. ref. [42]),we can find for Sep Sep =
—
v~d[N(q)
~
—
[
~ + ~:
Trg(s)(v(~)
_(vepv+~j~fdqq3w(q)u
~
(1.20)
where g(s) = 4V~~m312~112 is the density of states, T the thermostat temperature introduced for convenience,
J
p 6
=
2dq,
qw(q)(N0 + fl(~w)
v: =
~
f
qw(q)hcv dq,
(1.21)
are frequencies, in particular the energy frequencies of electron relaxation owing to phonons. Finally
~ep =
(2m)”2s312
f
q3w(q)(N
0+
~)dq
is the frequency of the electron momentum relaxation owing to phonons.
(1.22)
334
Yu.G. Gurevich and 0.L. Mashkevich, The electron-phonon drag and transport phenomena in semiconductors
To transform Spe (see ref. [41]),we shall use the identity valid for the Fermi distribution function
f0(e), viz. f0(E
+
hw)[1
—
f0(s)] = NTe[fO(S) f0(s —11w)], —
where NT is the Planck distribution function with temperature NTe =
[exp(hwIT6)
(1.23)
1]1.
—
Te
Representing eq. (1.23) as 2w(q)
Spe = (NTe
+
—
N) 2~m
J de [f~(e) f0(s + 11w)] —
6min
J(
(N
0+ 1)2trmw(q) q~6min
~i~)V(r)dr
(1.24)
with Emin =
(1I2m)(~q mIlwIq)2 —
we can ascribe a very clear physical meaning to the first term of (1.24), namely that electrons can interact with (and transfer energy to) only those phonons whose temperature is different from T~. Accordingly, the factor multiplying NT N in (1.24) has the meaning of the phonon relaxation frequency owing to interaction with electrons, which we shall denote by ~ q). The integration yields —
vp~(q)=
[2irm2w(q)Iq]T~mEl +f
0(~q)IN~].
For a nondegenerate electron gas (f0 41) and with allowance for the quasielasticity of the coffisions (ff—. Te hw) we have N~ Tel 11cv ~> 1, and hence the logarithm may be expanded in powers of ~>
f0(~q)IN~,
ln[1
+
f0(~q)lN~]
(11WITe)f0(~q).
(1.25)
If the electron gas is degenerate, then f0(~q)=1, q2p~,
where p~is the Fermian quasimomentum. Then ln[1+fØ(~q)/N~]=(hw/T6)f0(~q).
(1.26)
Yu.G. Gurevich and O.L. Mashkevich, The eleciron—phonon drag and transport phenomena in semiconductors
335
The electron temperature Te may be either higher or lower than the phonon energy 11w (the average electron energy e ~t however is always much larger than 11w). Taking account of eqs. (1.25) and (1.26), Vpe becomes v~~(q)=2irm2w(q)(11w/q)f0(~q),
(1.27)
and the phonon—electron collision integral takes the form
~ J(
2w(q)(NO+ L
Spe=
1/pe(NT6_N)+2~JTm
~)V(6)dE.
(1.28)
e(q/2)
In eq. (1.24) we have introduced the value SmIfl specifying the lowest energy for the electron to be able to interact with a phonon of quasimomentum q. The corresponding lowest electron momentum is Pmjn =
Ilq
—
mhwIqf.
In all of the above equations, the second term of this momentum has been neglected, i.e. we assumed q218m and Pmin ~q. As follows from the energy and momentum conservation laws, this corresponds to the quasielastic scattering (ms2l Te 41). This also implies that electrons are scattered by phonons having momentum values of the order of the electron momentum. As can be seen from eq. (1.27), electrons interact preferably with those phonons which obey the condition q 2
-~
1i
As is known for polar semiconductors, quite often their electrons are scattered mainly by optical phonons [57].If their characteristic energy is 1111 4 T~,then the electron scattering by optical phonons remains quasielastic as before. In this case, expressions for S~0and S06 are readily obtained from (1.20) and (1.28) [takingaccount of eqs. (l.21)—(l.23) and (1.27)] by substituting N°for N and (1~for w(q). If 11(1~T~,
(1.29)
as is the case at and below the liquid nitrogen temperature, e.g., in Ge (11(1~= 430 K), InSb (11f2~= 284 K), GaAs (hfl~= 421 K) and Si (11fl~= 735 K) [58],then, following ref. [49],the electron— optical phonon collisions may also be regarded as quasielastic. This approach is based on the following simple physical reasons. Assuming the optical phonons to be equilibrium and Planck-distributed, we can obtain, owing to the condition (1.29), N°= exp(—11110/T)4 1. Since the time of induced phonon absorption is r where r0 is the time of spontaneous phonon emission, and the time of induced emission is r~ + 1) 1 ~0, then r ~ r÷. Thus an electron is much more likely to emit a phonon than to absorb. However, in the “passive” region of the electron momentum space, where 0< ~ 11Q~,lasting for the time r~,the electron cannot change its momentum either through collisions or under the action of an external field. Since the dispersion of optical phonons is weak, the electron upon emission of a phonon returns practically to the same constant-energy surface in the passive region, where it was before it left for the active one. Treating the absorption and emission of an optical phonon as a single collision event, we have to assume it to be quasielastic inasmuch as the optical phonon dispersion is weak. An event like this may be regarded as a four-particle interaction in which an electron of momentum p and a phonon of momentum q change their momenta to p’ and q’, respectively (see fig. 1), with (p p’)Ip 4 1. If optical phonons are in nonequilibrium at Te 411(1, then we recall that the principal relaxation mechanism of optical phonons (both in the energy and momentum space) is their decay into acoustic phonons [59]. The lifetime of a nonequilibrium optical phonon before decay into acoustic phonons varies from 1 to lOps for various materials [60]. Therefore, the leading term in eq. (1.6) is the optical—acoustic phonon collision integral, S0~,.As follows from the energy conservation law and related considerations [46],among the three-particle processes only the decay of an optical phonon into two acoustic phonons and the reverse fusion can contribute to S0~,.Thus, the integral S0~,can be written as —
S~=
~3J
3q’ 1M 2{N(q q’)N(q’)[N°(q) + 1] d 55 x ~(1(q) w(q’) w(q q’)), —
—
—
—
N°(q)[N(q’) + l][N(q
—
q’) + 1]) (1.30)
—
where M
55~is the matrix element of the interaction considered, and the delta function is responsible for the energy conservation. Assume that the principal interaction for acoustic phonons consists of phonon—phonon collisions
~li i
U
—i-—
i1 1 1
Fig. 1. Reduction of two inelastic three-particle processes of electron—optical phonon interaction to a quasi-elastic four-particle process ~
is the “virtual” electron quasimomentum).
= p+ q
Yu.G. Gurevich and 0. L. Mashkevich, The electron—phonon drag and transport phenomena in semiconductors
337
capable of quickly mixing the energy and momentum within the subsystem. Therefore (see subsection 1.2), the isotropic part of the acoustic phonon distribution may be assumed to be a Planck function with temperature T~and the phonon drift velocity u independent of the quasimomentum q. Proceeding from this assumption and using eqs. (1.9) and (1.10), we find from the condition that the collision integral (1.30) should vanish, that the isotropic part of the optical phonon distribution is a Planck function with the same temperature T~(optical and acoustic phonons “merged” in energy), and that the functions u° and u characterizing the anisotropic parts of the optical and acoustic phonon distribution functions respectively, are equal (phonon “merging” in momentum). The interaction of optical phonons with electrons can be responsible only for small corrections to the phonon temperature and the magnitude of These are the facts to be remembered in the analysis of “electron—optical phonon” and “optical phonon—electron” collision integrals. Following the procedure developed in ref. [49],we shall neglect in the coffision integral the incoming terms with phonon absorption and the outgoing terms with emission for electrons with energies in the interval [0,11IIOJ, while for electrons whose energies are within [11(1~, 211(1~},we shall retain in the kinetic equation only the term responsible for the exchange with the former energy range, i.e. v + e(E + v X H) ~?L c
f
~ d3q w(q){f(p
=
+
q)[1 —f(p)][N°(q) +
1] —f(p)[1
x6[s(p+q)_E(p)_hfl(q)]+See+S~~+Sep,
f
3q w(q){f(p q)[1 f(p)]N°(q) f(p)[1 f(p 0= d x ö[g(p q) ~(p) +h(l(q)], hD~< r > r(eav)*:~whose drift velocity in *)
Here ~ denotes the energy of the electrons for which the Lorentz force is balanced by the Hall force, so that they move along a rectilinear
trajectory.
382
Yu.G. Gurevich and O.L. Mashkevich, The electron—phonon drag and transport phenomena in semiconductors Table 3 Current—voltage coefficients of a nondegenerate semiconductor specimen
q
State of the Lw
phononsubsystem
0,jf
~
Tnv,j3° K=l
i=—~ i=~
K=?i
bnv,’y° K——i
equilibrium LW phonons (ref. [98]) 0.5 heated LW phonons, (ref. [40]) —0.5 heated LW phonons —0.042 heated and dragged LW phonons 0.364
—0.5
—0.5u(1 — 7~)Ie.1,2
0.5~(1— Y~)Ik=1,2
1.5uQy~I,j,2
—1.5 —0.5o(1 — v~)Ik—1,2 —1.5o(1 — y~)I~..112 0.5oQy~I,.,,2 —1.093 —0.042u(1 — y~)Ie~i,, —1.093o(1 — v~)Lt.._i,2 0.958uQy~I,.112 —0.868
1.387o(1 —
0.5o-Qy~I,_112 —0.5oQ’y°~j~.112 —0.093rQy~j~._112
—2.368o(1 — )‘~d)Ik~_1/2(12.944/F)uQy°~dI,I/2 (2.127/F)uQ7~d~k.._l,2
~‘~d)lR~1,2
an electric field is higher than the average, are displaced by the Lorentz force on one side of the specimen, while those with T< electrons is greater, they produce, under the action of the Lorentz force, a larger flow along the electric field than the T< electrons against the field. Accordingly, in the case ofequilibrium phonons, the size-dependent magnetoresistance is negative [98]*)~
a) ~ Fig. 6. Interpretation of the size.dependent magnetoresistance.
e~
~
~(&aw)
Fig. 7. Interpretation of the volume magnetoresistance.
*) The mechanism responsible for the conventional magnetoresistance is shown in fig. 7, representing electron trajectories for r> and r > T(Cav) are turned in a magnetic field to the direction of the Lorentz force and those with T< . However, the latter are characterized by a lower frequency of collisions with phonons, and therefore the flow of dragged phonons proves greater in the Z-axis direction (i.e. edt) than in the opposite direction (~d1). In their turn, the dragged phonons involve more electrons in the motion along the positive Z axis than in the opposite direction. The magnetic field rotates these flows in the directions shown in the figure. For these reasons, the electron flow in the direction of the external electric force eE proves greater than in the opposite direction (negative magnetoresistance). The mechanism described is involved in a competition with the normal mechanism of a positive magnetoresistance and does not necessarily exceed that, hence the net magnetoresistance can remain
I
®1T
-
~(i~)
P~>
Fig. 8. Interpretation of the negative volume magnetoresistance associated with mutual electron—phonon drag.
Yu.G. Gurevich and O.L. Mashkevich, The electron—phonon drag and transport phenomena in semiconductors
384
positive (though lower than in the absence of drag effects). Yet it proves more efficient for the mechanism of momentum relaxation promoting the drag (via the “tail” of the electron distribution). Note one more feature of the magnetoresistance reflected in (3.19) and (3.20). As is easy to see, with K = ~(the electron free path time grows with energy) the magnetoresistance varies twice as much as with K = ~ (the electron free path time decreases with growing energy). The reason is that energetic electrons spend a part of their momentum “inefficiently”, viz, they transfer it to SW phonons not involved in drag motion. As a result, for K = ~(see fig. 8) the flow of i’> electrons becomes weaker and the above described magnetoresistance-reducing mechanism is more efficient; with K = ~, it is the T< electron flow that is reduced, and accordingly the mechanism is less efficient. Turning to the discussion of the size-dependent magnetoresistance, we note first of all that the main contribution to the variation of the magnetoresistance due to inclusion of drag is given by the coefficient Q of the transverse Nernst—Ettingshausen effect*) which was discussed in detail in the preceding section. The relationship between the Nernst—Ettingshausen effect and the size-dependent magnetoresistance is as follows: the latter is associated with the electron flow in the direction of the X axis produced by the gradient V~Te owing to the energy sorting of charge carriers. In fact the appearance of such an electron flow is the essence of the Nernst—Ettingshausen effect for the case of closed contacts. Making use of the numerical values presented in table 2, we can obtain for the size-dependent magnetoresistance —
—
=
=
(17.12IF2)(6°’y?dIy~)Ik=l/2 for PA phonons, 2)(6°’y?d/y?)Ik=_l/ (54.271F 2, for DA phonons.
(3.21)
As follows from comparison of eqs. (3.19), (3.20) and (3.21), the size-dependent magnetoresistance 2). If 8 ~ with allowance drag, for is much higher than theowing volume (roughlyrelaxation by a factorchannel, of 1/F the net should even be for positive the bulk specimen, to resistance some momentum magnetoresistance of a thin plate will become negative. The extra factor 1 / F2 in the size-dependent magnetoresistance is associated with the transverse temperature gradient V~Te, i.e. with the “thermal drag”, which generally is more important in nondegenerate semiconductors than the “true” drag
governing the magnetoresistance of bulk specimens. We recall that in the absence of drag 1601 and 6011 are of the same order of magnitude (with b —*0, the two values are equal [101]).As we see, drag effects in thin specimens result in a kind of “overcompensated” rather than compensated magnetoresistance. Numerically calculated unreciprocal coefficients and those of non-ohmic behaviour obtained from eqs. (3.14) and (3.15) with the help of tables 1 and 2 are presented in table 3, where 0’ is the electric conductivity in the absence of heating and drag and with H 0 [39].Comparison of these values with the results for equilibrium phonons [98]and for heated phonons, as calculated according to ref. [40], shows that allowance for the “tail” of the electron—phonon interaction in the formula for v~markedly affects the magnitudes of the coefficients in the case of heated phonons not involved in drag. For example, in the case of PA phonons, taking account of the tail reduces the non-ohmic factor almost by a factor of 12. Drag effects are even more important. The unreciprocal coefficient increases in magnitude by approximately a factor of exp( I ITe)’ and besides changes its sign for scattering by DA phonons. The non-ohmic factor, while retaining the same order of magnitude with respect to the parameter *)
As follows from our analysis, even with v~,independent of the energy e(K = 0), the Nernst—Ettingshausen coefficient (as well as the volume
magnetoresistance) is nonzero, in contrast to the case of no drag [681.
Yu.G. Gurevich and O.L. Mashkevich, The electron—phonon drag and transport phenomena in semiconductors
385
exp( j.~I Te) as in the absence of drag, still increases by roughly 2.5 times in the case of DA scattering. For scattering by PA phonons, it changes its sign and rises in absolute value by a factor of over 30, compared to the case of heated phonons not involved in drag motion. As we see, drag effects can alter essentially the current—voltage characteristic of a semiconductor specimen, the thermal and the mutual drag being equally important. In nondegenerate semiconductors, thermal drag is more important as capable of changing the kinetic coefficients (the size-dependent magnetoresistance and the unreciprocal coefficient) by orders of magnitude; meanwhile, mutual drag also yields correction factors of the order of unity, which are small as compared to the contribution of thermal drag. In case the physical effect under discussion is not affected by temperature gradients, and hence thermal drag is not essential (e.g., çlectric conductivity, volume magnetoresistance, non-ohmic behaviour), then the mutual drag itself can substantially alter the result, both quantitatively, and even qualitatively (changing the sign). Therefore, allowance for mutual drag is unquestionably necessary in both situations. While analyzing current—voltage characteristics of semiconductor specimens in high magnetic fields, we shall restrict ourselves, for simplicity, to the case of non-heating electric fields, i.e. fields of such intensity that the right-hand side of eq. (3.4) may be neglected*). Then, the boundary conditions (3.5) imply T’(z) -~E with arbitrary magnetic fields, therefore by linearizing eqs. (3.4) and (3.5) in E, we can find that the energy balance equation reduces to eq. (2.32) and the corresponding boundary conditions to eq. (3.7), where the superscript 0 in the electric conductivity, thermal conductivity and the Nernst—Ettingshausen coefficient should be omitted. Upon solving this equation, we shall arrive, as above, at an expression for the averaged conductivity in an arbitrary magnetic field, i.e. &
=
~~[1
6d
—
+
(1/~r2)6~],
(3.22)
with 0
= (0’d
=
0
o’d)/o’d,
—
(3.23)
0’d~dTH2~2tanh kdb Y1d~
(3.24)
kdb For asymptotically high magnetic fields acting upon nondegenerate semiconductors with total electron—phonon drag, one can obtain, using the results of ref. [71]:
ÔIH_ d
—
1—6~’ ~1d k+1/2~ F(~—K,x 0)~ xex I xo
x
*)
325
8(3—2K)F(~ K,xo)B?o ~
d
(J
2[F(1 K, x
tanhk~’b H k~’b Yld
—
dx exx~
0)
—
x~2~F(_ ~, x0)]
—
4(3—2K) F( ~ k, x0)). —
The results can be easily extended to the case of heating fields of moderate intensity by applying iterations in E [102].
(3.26)
386
Yu.G. Gurevich and O.L. Mashkevich, The electron-phonon drag and transport phenomena in semiconductors
As follows from the results obtained, the size-dependent magnetoresistance grows rapidly upon the onset of drag (approximately, by a factor F _2) and besides changes its sign in case the electrons are scattered by the deformation potential. However, the high phonon thermal conductivity “cuts down” the size-dependent parts ofthe effects, despite the fact that the cooling length k~is increased compared with the case of non-heated phonons. As regards the volume magnetoresistance, which is contributed to by “true” drag alone, its magnitude in the absence of drag is 6H = 0.115 for K = ±~. It is increased by drag effects to 0.160 for K = ~and to 0.491 for K = ~. In other words, in the case of scattering by the deformation potential, the magnitude is increased by a factor higher than 4. This is likely to be observable experimentally. With the above results in mind, it can be stated that in the range of intermediate fields, ~° — 1, total electron—phonon drag should result in at least one extremum (maximum) on the curve of the bulk sample conductivity versus the magnetic field. Figure 9 represents corresponding dependences for the PA and DA scattering, as well as a curve for the case of no drag [39].Note that in the latter case the K = ±~curves practically coincide on the scale chosen. An important point of the discussion in the present subsection is that drift velocities of electrons and LW phonons do not tend to infinity even in the case of total mutual drag. This result follows from the assumption that SW phonons are in equilibrium and comprise a spatial energy and momentum thermostat. Such a thermostat is absent for strong phonon—phonon interaction. Accordingly, the effects due to electron—phonon drag should be better pronounced. However, it should be remembered that the absence of a spatial energy thermostat results in high nonuniform heating of both electrons and —
6’o
I’
1/
‘5
I~J\i_ ~i~’~_TJiii
\
\
\ 2
Fig. 9. Dependence of the volume magnetoresistance (o, — r°)Io’°on the magnetic field (curve 1 is for PA scattering, complete drag; curve 2 for DA scattering, complete drag; curve 3 for DA and PA scattering without drag).
Yu.G. Gurevich and O.L. Mashkevich, The electron-phonon drag and transport phenomena in semiconductors
387
phonons (which becomes higher as the sample becomes thicker), depending essentially on the electron—phonon drag. This dual manifestation of drag effects markedly complicates the analysis of current—voltage characteristics, while introducing nothing new to the physics involved. Some aspects of this problem have been considered in refs. [86,110]. 3.2. The Hall field and current in the presence of electron—phonon drag We now turn to the study of transverse galvanomagnetic effects, particularly their dependence on the specimen size, using the methods developed in refs. [96]and [103].We shall assume, as before, that ~‘pe~> i’rn, and retain the geometry of the preceding subsection. By multiplying eq. (3.1) linearized in T’(z), by dz and integrating it between —b and b, we shall find, with the use of eqs. (3.8) and (3.10), the dimensionless transverse emf e~= V5/2bE for a weak magnetic field in the form [69] 0
/ ~
0’
tanh kd°b ~ \
+ Rd
=
0
kdb
lid) +
aTdtanh k°db 0 Pfllcdb
(3.27)
where 2H, Rd°’ = aTdPd, R~= I20IeI~°~ are the volume and the size-dependent Hall factors, respectively, and
(3.28)
(3.29)
P°d=TQ°d/K°d
is the Ettingshausen coefficient. To derive eq. (3.27), we have excluded the external electric field by means of the linearized eq. (3.3) for the longitudinal current. The resulting term proportional to the electric current represents the transverse size-dependent thermo-emf due to the asymmetry in the cooling of electrons and LW phonons at the Hall boundaries of the specimen [the last term in eq. (3.27)]. Note that the size-dependent factors corresponding to the Hall effect and the transverse thenno-emf are the same as the respective factors in the size-dependent magnetoresistance (3.13) and the unreciprocal coefficient (3.14). In order to estimate the contribution of drag to the effects under study, let us calculate the kinetic coefficients found for the case of nondegenerate semiconductors with complete drag (a —*0) using the data of table 1. The results are summarized in table 4, along with the coefficients calculated forthe case when phonons are heated but not involved in drag motion. Table 4 Galvanomagnetic kinetic coefficients in low magnetic fields Scattering mechanism Coefficient
R°
without drag with drag
1.104/nec
piezoacoustic oscillations (PA,K*~) R°’ ea~+ ~u/T~R° 2aa..
1.332/nec
(0.341/nc)F 0.701/nec
acoustic oscillations (DA,K-’—j) R°’ eaT + 1.~/T~
3
1.178/nec
(0.231/nc)F2aa~
2
1.632/F
2.204/nec
0.842/nec
1.876/F
Yu.G. Gurevich and O.L. Mashkevich, The electron—phonon drag and transport
388
phenomena
in semiconductors
All that was said in subsection 2.4 on the relation between the volume and the size-dependent part of the transverse Nernst—Ettingshausen coefficient fully applies to the volume R°and the size-dependent R°’part of the Hall coefficient. Note that the size-dependent transverse thermo-emf does not vanish even in bulk specimens. Since it increases sharply, by a factor of exp(—~/T)upon the onset of drag, it is likely to be observable experimentally even in large specimens of sufficient purity. It should be emphasized that the increase of the volume Hall coefficient (almost two-fold in the case of scattering by the deformation potential) is entirely due to the “true” drag, really an important factor to be allowed for in solving kinetic equations. As in the analysis of the current—voltage characteristics, we shall consider implications of high magnetic fields, for the sake of simplicity, ~olelyfor moderate, non-heating electric fields. Then, 2kdb]lid) cr~JH{R~ + R~[tanh(kdb)/~’C
=
(3.30)
,
where Rd = I
20Ie(I~0+ I~0)H,
(3.31)
and R~=
aTdPd~’.
(3.32)
For asymptotically strong magnetic fields applied to nondegenerate semiconductors with complete drag, we can obtain, with the use of I~of ref. [71] (3.33)
R~’=1Inec, R~H=
R~(3
x
~
—
2K)4x~2(9~JdtettV2~)
(3
—
2K)F( ~ K, x0) —
—
J
tt ~2 [F(1 K, x —
dt e
0)
—
$12_kf(_
~,
x0)]).
(3.34)
It can be seen from these expressions that in high magnetic fields the behaviour of the size-dependent Hall emf with the onset of drag is similar to that of the size-dependent magnetoresistance, while the volume Hall coefficient does not change. Note that the total thermal conductivity of electrons and LW phonons without drag amounts to that of the LW phonons alone. Therefore, in the absence of drag, R’ 4 R. The thermal drag essentially enhances the electron thermal conductivity, which approaches, in order of magnitude, the level of the phonon thermal conductivity (the latter is increased in the presence of drag by several times, as a result of the “true” drag* ). Thus, Rd and R~become values of the same order of magnitude. *) Note that the mutual drag which until recently was often neglected in nondegenerate semiconductors in fact can change by several times not only K but also 8’ (which also reverses sign) and o-~.In the case of total mutual drag the latter value grows by a factor of 2.7 with DA scattering or almost 3.8 with PA scattering [68].
Yu.G. Gurevich and O.L. Mashkevich, The electron—phonon drag and transport phenomena in semiconductors
389
It is quite clear now that the size-dependent part of the Hall emf may be much greater than the volume part, provided that ~ tpe~ Indeed, it is the case when the temperature Te characteristic of the electron subsystem alone (in which the quasiparticles are classified in the magnetic field according to their energies), and hence K L should be replaced in the Ettingshausen coefficient for the absence of drag by Ke~ which is Ke 4 K ~ [36~.The development of drag, while changing Ke only slightly, results in an essential increase in Q, so that R’ can greatly exceed R, R’ ~ R. We have thus far considered the case of open Hall contacts. Meanwhile, the experiments aimed at studying the transverse galvano- and thermomagnetic effects most often involve voltage measurements with probes (see, e.g. refs. [104]and [83]) allowing a current in the Hall direction. Consider this situation for ~pe ~ ~‘pp [62]. The appropriate set of equations consists of the Maxwell equations (2.17), the energy balance equation for energetically unified electrons and LW phonons, eq. (2.20), explicit expressions for the electron currentj, eq. (1.104), the total heat flux Q of electrons and LW phonons, eq. (1.105), and the relation expressing the second Kirchhoff law, i.e. °(T) =
+
V~p~a.~ —
ciTe +
R~dHE)dl,
(3.35)
where R~d= I
20II~°~H is the Hall current factor. The integration in eq. (3.35) is along the closed electric circuit contour (the Z axis), including the semiconductor specimen and a 21-long ballast metal conductor of conductivity ~ The boundary conditions for the current and the heat flux are [105] ±(Q2~
=
[±A±J~±~
+ ~ext)1z+ ~±(Te
—
T)]1Z±b
(3.36)
and ~JzIz=±b
=
9~(~ — ~ext+ P±(Te—
T))1Z±b.
(3.37)
Here ~ and ~ext are, respectively, the electrochemical potentials of the specimen and of the ambient medium; ç the electric potential of the specimen, A the surface Peltier factor, 0 the surface conductivity, and p the surface thermo-emi obeying the Kelvin relation (3.38)
P=AITe.
The surface characteristics have been defined as in ref. [43]. It follows from the Maxwell equations (2.17) and the one-dimensionality of the problem that the Hall current )~ =j0 =const.
(3.39)
In the case of weak heating (Te T 4 T), eqs. (2.17), (1.104), (1.105), (3.35)—(3.37) and (3.39) yield, upon solution of the energy balance equation (2.20), 2 R~HE2b o~E 2 0 D 1+ D2, (3.40) xkdKdT X —
390
Yu.G. Gurevich and O.L. Mashkevich, The electron—phonon drag and transport phenomena in semiconductors
where x
= 2(b + leff)Icro =
2M’{(A+
=
1
—
is the total resistance of the Hall circuit, A_) sinh2k~b+ (cosh2k~b 1)[(A~ —
—
4dT)~_
—
(A
—
a~.dT)~+]}, (3.41)
and +
+
leff =
MkdKdRldb
(A_
—
~+ ~ +
(A_
—
[(At
—
a~.dT)(cosh2k~b —1 + ~ sinh2k~b)
aTdT)(cosh2kdb (~-
+
—
1+
~+
~_) + J~0 ~
sinh2k~b)], [(At aTdT) (cosh 2k~b+ ~Lsinh 2k~°b) —
a~.dT)2(cosh2k~b + ~ sinh2k~b)—2(A~ aTdT)(A_ —
—
a~dT)].
(3.42)
As can be seen from eq. (3.42), x is the sum of the resistances of the semiconductor with allowance for heating and LW phonon drag, of the metal, of the probes and of those due to the Pettier effect. The latter contribution goes to zero only if =
aTdT.
(3.43)
Otherwise, this contribution to x is nonzero even in the case of a bulk semiconductor (k°db 1), where ~‘
2b
2!
1
1
x=—+—+~i-+~-, 0o °m u~
(3.44)
the total surface resistance being 1
1
(A—a~.dT)2
345
(.
)
In the case of a thin specimen (k°db4 1) in which 2b
21
(To
0m
1 1 0~ 0_
(A~—A_)2 ~)
(3.46)
kdKd(e+ +
the only condition for the Pettier contribution to be zero is A~= Consider eq. (3.40) representing the Hall current J~.Its dependence on the magnetic field is determined by the term proportional to D 2, which value consists of two terms. The first represents the common Hall current of a bulk specimen [106,39] calculated taking into account phonon heating and drag, while the other is the size-dependent thermoelectric current due to sortjng of electrons in the magnetic field according to their energies. Naturally, with k°db ~, the size-dependent term goes to zero as 1 I k°db.In the case of heated LW phonons, even in very thin samples (kd°b4 1) where —~
00
D2
0
0
1+ K~R~d Qdud (A÷—aTdT)~_+(A..—aTdT)e÷ ~~++~_
.
Yu. G. Gurevich and 0.L. Mashkevich, The electron—phonon drag and transport phenomena in semiconductors
391
the size-dependent term in the absence of drag proves much smaller than the volume term (cf. the relationship between R and R’). Whereas the development of drag brings about a sharp increase in Q° and a ~.(see section 2), thus compensating the increase in thermal conductivity, the latter of the above mentioned terms becomes, as in ref. [106],equal in order of magnitude to the former. However, its sign becomes independent of the mechanism of electron scattering by phonons, as was the case with equilibrium phonons [106].As a result, with total drag and with A.. 4 4dT, becomes D2 = 1.23 for PA 2 corresponds to scattering and D2 heating = 1.38 of forthe DAspecimen scattering. term in Jothermoelectric proportional current. to E Naturally, it is nonuniform Joule and The the associated zero both in bulk specimens showing no transverse temperature gradients because there are no boundaries*), and in thin specimens (k°db4 1), where the temperature gradient is small because of the considerable heat transfer to the boundary. Drag effects give rise to an essential [by a factor of exp(—/LIT)] increase of this term, because of the growth in a~.. Let us derive the current—voltage characteristic of the present type of specimens by integrating eq. (1.104) over the sample cross section and using the above definitions of the kinetic coefficients and expressions for Jo and Te. We have b
)X1J
+
02
~
/ din O~(Te) \(Q~o~ tanh2 k°~b0 R~dDlD 2 EH~T dTe T~=T + ~ k~~K°~ k°~b )‘2d + Xk~K~T d 2 lno-~(T~) ( tanhk~b ~ ________
+ cT~~E 0,02 KdKd
_ip ~51e
Te=T\
,O~ KdO
V3d
L102
0
(3.48)
LDlCd KdIX
We shall analyze the current—voltage characteristic for a nondegenerate semiconductor with complete drag. In the term of eq. (3.48) representing the magnetoresistance, the second member arises from the size-dependent transverse thermal field due to sorting of electrons in the magnetic field according to their energies. Along with the first term, it has been given a detailed analysis in subsection 3.1. The third member is associated with the Hall current. It is always positive and has been shown to grow upon the onset of drag by a factor of 36.4, in the case of PA scattering or, by a factor of 49.8, in the case of DA scattering, thus compensating for the change in sign of the volume magnetoresistance. Thus, closed Hall contacts lead to suppression ofthe negative volume magnetoresistance (see subsection 3.1). Yet, in the case of k°db— 1 and with total drag, the leading role in the magnetoresistance is played, because of the growing Q~,by the term associated with the size-dependent thermal field. It is that term which controls the magnitude and the negative sign of the effect. Note here the essential role of the total resistance y of the Hall circuit, which can even diminish the significance of the Hall current terms, both in the magnetoresistance and in other components of the current—voltage characteristic (the above estimates have been given for 1~ff~b). We have already discussed the essential drag-induced increase in the first member of the term proportional to EH, which is responsible for the unreciprocity with open Hall contacts and arises from energy sorting of carriers in a magnetic field and cooling at the walls. However, if the energy relaxation rates of electrons and LW phonons at the Hall walls are equal (i.e. ~÷ = ~), the term is zero, whatever *)
Boundaries serve as the only source of nonuniformity in a homogeneous specimen.
392
Yu. G. Gurevich and 0.L. Mashkevich, The electron—phonon drag and transport phenomena in semiconductors
the degree of drag. Then the unreciprocity will be associated with the second term proportional to EH which is associated with unequal amounts of the Pettier heat released by the Hall current at the walls z = ±b. Numerical estimates show this term to increase with drag by a factor of 6 or 7. The E2-proportional term of eq. (3.48) is responsible for the non-ohmic specimen performance. The volume non-ohmic factor which is represented by the unity in the parentheses following E2 decreases in this case (which can be interpreted as linearization of the current—voltage characteristic) as a result of both electron and LW phonon cooling at the boundaries (the second term in the parentheses) and the Peltier effect (the third term). In sufficiently thin specimens (k~b4 1), the second term compensates for the unity, while the third one, though small compared with the other two (— k~b),ultimately determines the sign of the effect. This sign can change with A~~ A_, if the rate of energy dissipation at the boundary is not too high. With k°db 1, the second term decreases as 1/ k~band the third one as ~‘
1 I(k°db)2.
4. Conclusion The consistent analyses of the electron—phonon drag phenomenon summarized in this review show that, in contrast to the earlier viewpoint, the role played by mutual (true) drag is not less important in kinetic phenomena than that of thermal drag. In the presence of the latter, the additive contribution of the mutual drag kinetic coefficients is negligible, yet the form of the term due to thermal drag depends essentially on the mutual drag. In the absence of thermal drag, the contribution of mutual drag to transport phenomena is essential, sometimes predominant. Therefore, drag effects are important not only for thermomagnetic phenomena, but for current transport as well. Certainly, the phenomena discussed in this review do not exhaust the set of processes sensitive to drag. For example, while studying the propagation of electromagnetic or acoustic waves through solids it may prove necessary to take into account the mutual drag [6, 107]. If, in addition, the amplitudes of such waves are sufficiently high to heat charge carriers, then thermal drag can manifest itself [41,65, 107—109]. A detailed analysis of these effects, however, deserves special discussion. References [1] L.E. Gurevich, Zh. Eksp. Teor. Fiz. 16 (1946) 193. [2] L.E. Gurevich, Zh. Eksp. Teor. Fiz. 16 (1946) 416. [3] H.P.R. Frederikse, Phys. Rev. 92 (1953) 248. [4] T.H. Geballe and G.W. Hull, Phys. Rev. 94 (1954) 1134. [5] T.H. Geballe and G.W. Hull, Phys. Rev. 98 (1955) 940. [6] F.G. Bass, Pis’ma Zh. Eksp. Teor. Fiz. 3 (1966) 357. [7] G.E. Pikus, Zh. Eksp. Teor. Fiz. 21(1951) 852. [8] C. Herring, Phys. Rev. 96 (1954) 1163. [9] J.E. Parrot, Proc. Phys. Soc. B 70 (1957) 590. [10]J.E. Parrot, Proc. Phys. Soc. B 71(1958) 82. [11] V.L. Gurevich and Yu.N. Obraztsov, Zh. Eksp. Teor. Fiz. 32 (1957) 390. [12]J.Z. Appel, Z. Naturforsch. a, 12 (1957) 410. [13]J.Z. Appel, Z. Naturforsch. a, 13 (1958) 386. [14] L.E. Gurevich and I.Ya. Korenblit, Th. Eksp. Teor. Fiz. 44 (1963) 2150. [15] L.E. Gurevich and I.Ya. Korenblit, Fiz. Tverd, Tela 6 (1964) 856. [16] L.E. Gurevich and I.Ya. Korenblit, Fiz. Tverd. Tela 9 (1967) 1195.
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[17]I.Ya. Korenblit, Fiz. Tekh. Poluprovodn. 2 (1968)1425. [18] E. Mooser and S.B. Woods, Phys. Rev. 97 (1955) 1721. [19]V.A. Chuenkov, Fiz. Tverd. Tela 7 (1965) 2467. [20]AG. Sanioylovich and IS. Buda, Fiz. Tekh. Poluprovodn. 9 (1975) 1478. [21] 1G. Lang and S.T. Pavlov, Zh. Eksp. Teor. Fiz. 63 (1972) 1495. [22] N. Perrin and H. Budd, Phys. Rev. B 9 (1974) 3454. [23] V.A. Kozlov and EL. Nagaev, Pis’ma Zh. Eksp. Teor. Fiz. 13 (1971) 639. [24]V.A. Kozlov, N.S. Lidorenko and E.L. Nagaev, Fiz. Tverd. Tela 15 (1973) 1458. [25]~P.J. Baranski, IS. Buda and IV. Dakhovski, Theory of Thermoelectric and Thermomagnetic Phenomena in Anisotropic Semiconductors (Naukova Dumka, Kiev, 1987) (in Russian). [26] V.A. Kozlov and Y.D. Lakhno, Pis’ma Zh. Eksp. Teor. Fiz. 24 (1976) 437. [27] V.A. Kozlov and V.D. Lakhno, Fiz. Tverd. Tela 18 (1976) 1373. [28] V.A. Kozlov and LU. Kreshchinina, Pis’ma Zh. Eksp. Teor. Fiz. 27 (1978) 329. [29] V.N. Galeev, VA. Kozlov, N.Y. Kolomoets, S.Ya. Skipidarov and NA. Tsvetkova, Pis’ma Zh. Eksp. Teor. Fiz. 33 (1981) 112. [30] M.M. Babaev and TM. Gasymov, Fri. Tekh. Poluprovodn. 14 (1980) 1227. [31] B.M. Mkerov, B.I. Kuliev and S.R. Figarova, F1zTverd. Tela 22 (1980) 2426. [32] MM. Babaev and TM. Gasymov, Phys. Status Solidi b 84 (1977) 473. [33]AM. Konin, Litou. Fiz. Sb. 24 (1984) No. 1, 68. [34]T.M. Gasymov and M.Ya. Granovski, Izv. Akad. Nauk. Az. SSR Ser. Fiz.-Tekh. Mat. Nauk. (1976) No. 1, 55 (in Russian). [35]P.1. Baranski, EN. Yidalko and VV Savyak, Fiz. Tekh. Poluprovodn. 18 (1984) 538. [36]AM. Konin, Litov. Fiz. Sb. 19 (1979) 791. [37]A.A. Belchik and VA. Kozlov, Fiz. Tverd. Tela 26 (1984) 1479. [38]A.A. Belchik and VA. Kozlov, Fiz. Tekh. Poluprovodn, 20 (1986) 53. [39] A.I. Anselm, Introduction to the Theory of Semiconductors (Nauka, Moscow, 1978) (in Russian). [40] Yu.G. Gurevich and A.M. Komn, Litov. Fiz. Sb. 15 (1975) 995. [41] M.Ya. Granovski and Yu.G. Gurevich, Zh. Eksp. Teor. Fiz. 68 (1975)127. [42] F.G. Bass and Yu.G. Gurevich, Hot Electrons and High.Amplitude Electromagnetic Waves in Semiconducting and Gas Dischargc Plasmas (Nauka, Moscow, 1975) (in Russian). [43] F.G. Bass, VS. Bochkov and Yu.G. Gurevich, Electrons and Phonons in Bounded Semiconductors (Nauka, Moscow, 1984) (in Russian). [44] R. Peierls, Ann. Phys. (Germany) 4 (1930) 121. [45] EM. Conwell, High Field Transport in Semiconductors (Academic Press, New York, 1967). [46] J.M. Ziman, Electrons and Phonons (Oxford Univ. Press, Oxford, 1960). [47] V.L. Ginzburg, Electromagnetic Wave Propagation in Plasmas (Nauka, Moscow, 1967) (in Russian). [48] B.I. Davydov, Zh. Eksp. Teor. Fiz. 7 (1937) 1069. [49]B.I. Davydov and I.M. Shmushkevich, Zh. Eksp. Teor. Fiz. 10 (1940) 1043. [50]F.G. Bass, Zh. Eksp. Teor. Fiz. 47 (1964) 1322. [51]H. Fröhlich and BY. Paranjape, Proc. Phys. Soc., B 69 (1956) 21. [52]F.G. Bass, Yu.G. Gurevich and MV. Kvimsadze, Zh. Eksp. Teor. Fiz. 60 (1971) 632. [53]l.A. Kazlaushas, LB. Levinson and G.E. Mazuolyte, Litov. Fiz. Sb. 6 (1966) 377. [54] LB. Levinson and G.E. Mazuolyte, Litov. Fri. Sb. 6 (1966) 245. [55]L.E. Gurevich and TM. Gasymov, Fiz. Tverd. Tela 9 (1967) 106. [56]L.E. Gurevich and T.M. Gasymov, Fiz. Tekh. Poluprovodn. 1 (1967) 774. [57]L.M. Dykman and P.M. Tomchuk, Transport Phenomena and Fluctuations in Semiconductors (Naukova Dumka, Kiev, 1981) (in Russian). [58]lB. Levinson and V.F. Gantmakher, Charge Carrier Scattering in Metals and Semiconductors (Nauka, Moscow, 1984) (in Russian). [59]lB. Levinson, Th. Eksp. Teor. Fiz. 65 (1973) 331. [60] J. Shah, Solid-State Electronics 21(1978) 43.
[61]Yu.G. Gurevich and O.L. Mashkevich, Fiz. Tverd. Tela 26 (1984)120. [62]O.L. Mashkevich, Fiz. Tekh. Poluprovodn. 20 (1986) 1853. [63]AN. Gurevich and A.B. Shvartsburg, A Nonlinear Theory of Radio Propagation in the Ionosphere (Fizmatgiz, Moscow, 1973) (in Russian). [64]Yu.G. Gurevich and O.L. Mashkevich, Fiz. Tekh. Poluprovodn. 15 (1981) 659. [65]Yu.G. Gurevich and O.L. Mashkevich, Fiz. Tverd. Tela 26 (1984) 3154. [66]M. Abramovitz and IA. Stegun, eds, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Natl. Bur. Stand., Washington, D.C., 1964). [67]J. Arfken, Mathematical Methods for Physicists (Academic Press, New York, 1972). [68]Yu.G. Gurevich and O.L. Mashkevich, Fiz. Tekh. Poluprovodn. 15 (1981) 1780. [69]O.L. Mashkevich, Fiz. Tekh. Poluprovodn. 15 (1981) 1951. [70]I. Yamashita and M. Watanabe, J. Phys. Soc. Jpn 7 (1952) 334. [71] O.L. Mashkevich, Ukr. Fiz. Zh. 27 (1982) 533.
394
Yu. G. Gurevich and 0.L. Mashkevich, The electron-phonon drag and transport phenomena in semiconductors
[72]H. Fröhlich, Proc. R. Soc. Ser. A 188 (1947) 532. [73]lB. Levinson, Fiz. Tverd. Tela 8 (1966) 2077. [74] Yu.G. Gurevich and O.L. Mashkevich, Izv. VUZ Fiz. 27 (1984) No. 2, 32. [75]Yu.G. Gurevich, Ukr. Fiz. Zh. 24 (1978)1601. [76]L.D. Landau and E.M. Lifshitz, Statistical Physics, Part I (Nauka, Moscow, 1976) (in Russian). [77]I. Gyarmati, Nonequilibrium Thennodynamics (Springer, Berlin, 1970). [78]L. Onsager, Phys. Rev. 37 (1930) 405. [79]L. Onsager, Phys. Rev. 38 (1931) 2265. [80]L.D. Landau and EM. Lifshitz, Electrodynamics of Continuous Media (Nauka, Moscow, 1982) (in Russian). [81]E.H. Sondheimer, Proc. R. Soc. A 234 (1956) 391. [82]A.V. Bochkov, Yu.G. Gurevich and O.L. Mashkevich, Fiz. Tekh. Poluprovodn. 20 (1985) 572, 2130. [83]L.M. Tsidilkovski, Thermomagnetic Phenomena in Semiconductors (Fizmatgiz, Moscow, 1960) (in Russian). [84]V.L. Bonch.Bruevich and S.G. Kalashnikov, Physics of Semiconductors (Nauka, Moscow, 1977) (in Russian). [85]B.N. Mogilevski and A.F. Chudnovski, Heat Conduction in Semiconductors (Nauka, Moscow, 1972) (in Russian). [86]Yu.G. Gurevich and A.M. Konin, Litov. Fiz. Sb. 20 (1980) No. 3, 57, 64. [87]AN. Bochkov, Yu.G. Gurevich and O.L. Mashkevich, Pis’ma Zh. Eksp. Teor. Fiz. 42(1985) 281. [88]M.Ya. Granovski and Yu.G. Gurevich, Fiz. Tekh. Poluprovodn. 9 (1975)1552. [89]A.I. Klimovskaya and O.V. Snitko, Pis’ma Zh. Eksp. Teor. Fiz. 7 (1968) 194. [90]E.I. Rashba, Z.S. Gribnikov and V.Ya. Kravchenko, Usp. Fiz. Nauk 119 (1976) 3. [91]A.M. Konin, Fiz. Tekh. Poluprovodn. 16 (1982) 1877. [92]A.V. Bochkov and O.L. Mashkevich, lzv. VUZ Fiz. 31(1988) No. 2, 50. [93]A.V. Bochkov and O.L. Mashkevich, Fiz. Tekh. Poluprovodn. 22 (1988) 710. [94]VS. Bochkov, Yu.G. Gurevich and L.A. Shekhtman, Fiz. Tekh. Poluprovodn. 19 (1985)1796. [95]VS. Bochkov, Yu.G. Gurevich and L.A. Shekhtman, Izv. VUZ Fiz. 30 (1987) No. 8, 73. [96]V.S. Bochkov, Fiz. Tverd. Tela 13 (1971) 2535. [97]N.A. Prima, Fiz. Tekh. Poluprovodn. 7 (1973) 338. [98]F.G. Bass, V.S. Bochkov and Yu.G. Gurevich, Fiz. Tekh. Poluprovodn. 7 (1973)3. [99]VS. Bochkov and Yu.G. Gurevich, Fiz. Tekh. Poluprovodn. 17 (1983) 728. [100]L.S. Stilbans, Physics of Semiconductors (Soviet Radio Publishers, Moscow, 1967) (in Russian). [101]Z.S. Gribnikov and NA. Prima, Viz. Tekh. Poluprovodn. 5 (1971)1274. [102]VS. Bochkov and Yu.G. Gurevich, Fiz. Tverd. Tela 11(1969) 714. [103]F.G. Bass, VS. Bochkov and Yu.G. Gurevich, Viz. Tverd. Tela 9 (1967) 3479. [104]I.V. Klyatsina et al., Fiz. Tekh. Poluprovodn. 13(1979) 1089. [105]A.!. Vakser and Yu.G. Gurevich, Ukr. Fix. Zh. 24 (1979) 1208. [106]A.L. Vakser and Yu.G. Gurevich, Fix. Tekh. Poluprovodn. 12 (1978) 82. [107]A.S. Bugaev, Yu.V. Gulyaev and O.L. Mashkevich, Fiz. Tverd. Tela 26 (1984) 1056. [1081M.Ya. Granovski and Yu.G. Gurevich, Fiz. Tverd. Tela 17 (1975) 421. [109]F.G. Bass and M.Ya. Granovski, Fix. Tverd, Tela 16 (1974) 1882. [110]T,S. Gredeskul, Yu.G. Gurevich and O.L Mashkevich, Fix. Tekh. Poluprovodn. 23 (1989) 905.
