High field magnetoresistance of inhomogeneous semiconductors and plasmas—The stratified medium

ANNALS

OF

High

26, 181-221

PHYSICS:

(1964)

Field Magnetoresistance conductors and Plasmas-The

of Inhomogeneous SemiStratified Medium

H. L. FRISCH AND J. A. MORRISON Bell

Telephone

Laboratories,

Murray

Hill,

Sew

Jersey

We describe the steady-state electron distribution of a high temperature, spatially inhomogeneous semiconductor subject to a large magnetic and an infinitesimal electric field, using classical mechanics (strictly valid if the range of the potential describing the inhomogeneities is considerably larger than the thermal electron wavelength), by a modified Boltzmann equation retaining both spatial and velocity variables. Such a classical transport equation can also describe the behavior of electrons in a plasma subject to given spatially dependent density fluctuations. In the case of the semiconductor the collision term refers to collisions with phonons which we treat in this paper in the relaxation time approximation with a constant relaxation time, this being our assumption (1). For mathematical convenience we restrict ourselves in this paper to a stratified medium in which the inhomogeneities are distribut,ed in sheets perpendicular to say the x-axis, this being our assumption (2). We obtain the asymptotic solution of the transport equation in reciprocal powers of the strength of the magnetic field X. The velocity moments corresponding to the first few terms of the solution are identical with the solution of the velocity moment (4, 10, etc.) equations obtained by ret,aining only 2. 3, etc. generalized Hermite polynomials in the electron velocity, in the representation of the solution by these polynomials. These moment solutions provide us, in principle, with a method of solving approximately a general linear transport equation for which assumption (1) is dropped. The leading term of t,he asymptotic solution corresponds to a macroscopic theory of the high field magnetoresistance obtained by Herring. The transverse magnetoresistance does not saturate but increases indefinitely with X2, while there is no longitudinal magnetoresistance in our stratified medium. The general first correction to the macroscopic theory results are explicitly obtained. The mathematical methods can in certain instances be extended to more realistic inhomogeneity distributions, i.e., not subject to assumpt,ion (2). These extensions will be t,he subject of a subsequent paper. The physical significance of these mathematical results is noted. I. INTRODUCTIOPi

An old problem in transport theory in semiconductors is the nonsaturation of the magnetoresistance under conditions which cannot apparently be related to the intervention of some unusual quantum effect. In this series of papers we 181

182

FRISCH

AND

MORRISON

examine the possibility that the high field nonsaturation of the magnetoresistante has a classical origin due to the presence in the sample of a more or less random distribution of spatial inhomogeneities of possibly varying sizes. The effect of random inhomogeneities in the classical limit on the high field magnetoresistance of a semiconductor has been recently treated theoretically among other topics by Herring (f ) . The scale of the inhomogeneities considered was to be small compared with the size of the specimen but large compared with dimensions such as a Debye length or a suitably defined mean free path. The latter restriction appeared to be enforced by the natural limitations of the macroscopic approach used by Herring (1) . This approach consisted of a straightforward, but elegant, Fourier solution of the equations of continuity for the local fluctuating electric current and field to first order in the fluctuations, subject to the boundary condition that the macroscopic (i.e., space averaged) field or macroscopic current is prescribed. The effective conductivity or reistivity tensor, connecting the macroscopic current and electric field, could then be computed to the second order in the fluctuations. A few special cases were also treated without assuming that the fluctuations were weak. One of the most important consequences implied by this theory (1) was that a material which, if uniform, would show a high field saturation of the transverse magnetoresistance (T.M.R.), would in the presence of appreciable inhomogeneities in the Hall constant exhibit a magnetoresistance which increases indefinitely with the magnetic field. Herring has ascribed this effect to the current distortions arising from the large and fluctuating Hall fields (1). An adequate extension of this macroscopic theory to inhomogeneities on an atomic scale, such as charged or neutral impurity or defect centers, requires an atomistic quantum mechanical theory. This is true even at sufficiently high temperatures for which the Landau quantization of the orbits of the electrical charge carriers (i.e., electrons) can be neglected, because the physical range of dimensions of the centers can easily be much smaller than the thermal electron wavelength. Unfortunately such a rigorous quantum mechanical formulation presents very formidable difficulties. We restrict ourselves to an approximate, classical, i.e., high temperature, atomistic theory based on a classical microscopic Boltzmann transport equation, in which position as well as momentum variables are retained, describing essentially uncorrelated electrons in the effective mass approximation. At least for charged impurity centers some justification of the classical approximation is suggested, by analogy, by the Brooks-HerringConwell-Weisskopf theory of ionized impurity scattering (2). It is well known from this theory that the important contributions to the effective scattering occur from the long wavelength fluctuations in the impurity potential. Thus a classical treatment may suffice, if the concentration of impurities is not too large so that the Debye length is large compared to the thermal electron wavelength. In the

HIGH

FIELD

MAGNETORESISTANCE

183

case of neutral impurity (or defect) centers we must assume that their range of interaction with the electrons is large compared to the thermal electron wavelength. Subject then to the restrictions imposed by the classical approximation, the interaction of the fixed “small” inhomogeneities, the impurity centers, distributed at random or periodically in a unit volume of our large sample, with an electron at x, will be described by a potential

v = g9i(x - Xi),

(1.1)

with @i the efiective potential contribution of a center located at xi . To describe approximately the shielding by other electrons of this potential V, we employ a correlation potential, am, an appropriate solution of the Poisson-Boltzmann equation, neglecting thus certain effects due to charge fluctuations. The electrons are described as classical point particles with an effective isotropic mass 111,whose collisions with phonons can be described by a stochastic (relaxation type) collision term on the right-hand side of the transport equation. Certain general features, as well as specific solutions, of the resulting spatially dependent transport equation have been previously obtained by Frisch and Lebowitz (3) in a different physical context, dealing with certain limiting cases of electron transport in the presence of impurities (cf. Eq. (1.2) of ref. 3). We propose to investigate the magnetoresistance, particularly that resulting from the inhomogeneity centers (whose range of interaction with the electrons can be large compared to their mean free path, resulting, say, from phonon scattering) employing a high magnetic field asymptotic solution of our transport equation. The classical, spatially dependent transport equation for the electrons in a semiconductor with which we are concerned can also describe the behavior of electrons in a gaseous plasma subject to spatially dependent density fluctuations. The latter can result, for example, in a model in which the heavy ions are completely localized to fixed positions. A number of studies dealing with the anomalous diffusion of a plasma across a magnetic field (4) or the electrical conductivity of a turbulent plasma (5) have noted and used this analogy to transcribe many results of the Herring theory to the plasma theory. Our results can thus also he immediately transcribed to extend these plasma investigations. In the next section we shall formulate the detailed, steady-state transport equation. The conceptual statistical mechanical details underlying the approach to a steady state in an open system, such as used by us in this study, have been discussed in some detail in Section 1 of ref. Y and need not be repeated here. We wish to stress the fact that to simplify the mathematical analysis we shall deal in this paper with a highly idealized model of an impure or inhomogeneous semiconductor in that (1) the relaxation time for the electron-phonon collisions

184

FRISCH AND MORRISON

? is taken to be a constant, and (2) we make the simplifying assumption that the impurities or inhomogeneities are distributed in planes perpendicular to the z-axis, so that the impurity potential is a function of x only. Some of the n&hematical techniques developed in this paper (e.g., the moment approximations) would apply equally well if assumption (1) were dropped. Assumption (2) presents us with a model system exhibiting a nonsaturating magnetoresistance and allows us to check the mutual consistency of a number of approximate mathematical approaches, some of which appear very promising in connection with more realistic problems in which assumption (2) can be dropped. These investigations, not subject to assumption (2)) will be reserved for a subsequent paper. Our aim here is threefold. First we shall show that the Herring macroscopic theory results are consequencesof the leading term of the high magnetic field asymptotic solution of the microscopic transport equation. The physical significance of this leading term is quite transparent and is discussed in the first few paragraphs of Section V of this paper. The result can be obtained in a few lines [subject to assumption (2)] from a straightforward Drude theory argument coupled with an ansatz asserting the local spatial independence of the correlation in velocity and spatial fluctuations. Second, we obtain, in full generality, the first correction to the macroscopic theory result. Third, we demonstrate that these terms correspond to successively more detailed “hydrodynamical,” spacedependent moment descriptions intermediate between the microscopic description in the phase space of an electron and a gross macroscopic theory on a scale on which spatial fluctuations can be completely neglected. For the sake of continuity we relegate the mathematical analysis to appendices. Only enough detail of this analysis is given to justify certain specific physical conclusions and to provide a sufficient framework for the extension in a subsequent paper of the mathematical development (where applicable) to more realistic inhomogeneity distributions. Throughout the body of this paper we shall deal with the T.M.R. Only in the concluding section shall we discuss the absence of a longitudinal magnetoresistante for our simplified model. Our specific results for the T.M.R. hold only for the model in which the impurity potential is a function of x only, as we have verified by solving the transport equation for somespecial casesin which assumption (2) does not apply. II. THE TRANSPORT

A. TRANSPORT

EQUATION

AND SOME OF ITS ELEMENTARY

PROPERTIES

EQUATION

We consider the stationary distribution of velocity 0 and position x of a representative electron in a unit of volume V in a very large sample of the semiconductor, f = j;(S, x) (we suppressthe explicit dependence of f on the xi through

HIGH

the impurity

potential

V).

FIELD

185

MAGNETORESISTANCE

We normalize

this distribution

so that

n(x)=sJ(i+,x) w is the local number density by C->

e.g., the macroscopic

of electrons

(2.1)

at x in V. We denote a space average

(...) =+s,...dx,

number density

of electrons

(2.2)

is given by

ii = (n(x)).

(2.3)

The space-independent, Maxwellian (velocity) distribution will be denoted by .I,+, normalized to unity. A representative electron will be subject to collisions with phonons characterized by the constant relaxation time ? (assumption ( 1) of our model), as well as a Lorentz force due to the externally applied (constant) electric E and magnetic H fields. In the presence of impurities the electron is also subject to an extra local acceleration H(x), which we shall shortly see is essentially derivable from the impurity potential, I’, and the number density of electrons, n( x ) . We will be concerned not only with the distribution j(i+, xj but a.lso the velocity moments of f. In order not to carry extra factors of powers of fi?‘, k (Boltzmann’s constant), and T (constant temperature of the system), and still retain the advantages which can be incurred through checking the physicaldimensional consistency of our equations, we adopt physical dimensions, in which kT/m is unity. Thus we define the new electron velocity v, velocity of light c, electron distribution f = f(v, x), Maxwellian distribution fM , electron acceleration a(x), electron charge q, relaxation time 7, and dielectric constant of the system D in terms of the corresponding entity in the old dimensions ( represented by the corresponding letter with a tilde) by means of the identities: v = (m/kT)“%,

a(x)

= (rt~/k?‘)iii(x),

c = (m/kT)“‘~, q = (m/kT)ij,

7-l = (w/kT)““i-‘,

D = (m/kT)D, fM(v)

(~2.4)

f(v, x) = (r~tlkT)-~“f(B,

= jiM(v)(m/kT)

x),

= exp( -v”/2),/(27r)“‘*,

with ,n(x)

=

s

f( v, x) dv.

In these units, the reader should note that velocities

become dimensionless,

the

186

FRISCH

AND

MORRISON

acceleration a(x) has units of reciprocal distance, r has the units of an effective mean free path, D has units of time divided by distance, etc. The stationary distribution f(v, x) satisfies the modified Boltzmann equation (6) v.$

+ a- $

+ $ E. g + $

(v X H). g = i [n(x)fnn - f].

(2.5)

The left-hand side of (2.5) contains, besidesthe convection term, the effect of all the deterministic forces on the electron due to interaction with the spatially distributed impurity centers and the external fields. The stochastic, relaxationtype, collision term, resulting from interactions with phonons, randomizes only the velocity of a representative electron. Replacement of n(x)fM by the equilibrium electron distribution function to obtain a simplified but incorrect collision term, which randomizes both position and velocity, leads to the loss of the T.M.R. effect. The acceleration a(x) is derivable from an effective impurity potential #J(X) = *(x7 w 17

a(x) = - PC!!! m ax '

(2.6)

where IL(x) is the sum of the impurity potential V given by (1.1) and the correlation potential, cpC , accounting for shielding of the impurities by the electrons which, as is usual in the static approximation, can be taken to be an appropriately bounded solution of Poisson’sequation v2 cpc= 5 [?I0- n(x)],

(2.7)

where fro is the value of fi for vanishing 1E ( . While the use of a self-consistent # introduces limitations on our formulation, e.g., in that it disregards all correlation in local charge fluctuations, this is not, we believe, important for the main effect under consideration. The a priori justification of the use and the range of validity of our basic transport equation (2.5) is essentially the same as that of the modified Boltzmann equation in studying spatially inhomogeneous problems in the kinetic theory of gases(7). The highly restricted form of the collision term [assumption (1 )] leads to a transport theory which in the absence of the impurities (i.e., V = 0) does not exhibit any T.M.R. effect. Thus any T.M.R. exhibited by the current derived from a solution of (2.5) with V # 0 can be directly ascribed to the presence of the impurities. B. THE VELOCITY

MOMENTS

Theentities of physical interest will be space averages of the velocity moments of f(v, x) , since some of these entities correspond to the usual experimentally

HIGH

FIELD

187

MAGNETORESISTANCE

measurable properties of the inhomogeneous semiconductor sample. The simplest such velocity moment (of zeroth order) is the local number density n(x). The local particle current, J(x), is the principal moment we shall be concerned with. The tensor moments up to and including the fourth order ones are denoted by 4x>

=

Ji(X)

= / Vif(V, x> dv,

Pij(Xj

Hijkl(X)

s

f(v, xl dv,

= / vi vj f(v, x) dv,

=

1

2); Vj uk Vi f(v,

X>

( 2.8)

dv-

The second order moments Pij(x) are linearly related to the stress tensor components of a modified Boltzmann gas. Since the current in the semiconductor will vanish in the absence of an applied electric field, Pij(x) in the ohmic response approximation (i.e., in a solution retaining only first order terms in the electric field components) is related to the local stress tensor pi,i by the linear relation Pij(X)

= pij(X) + 6ijql.(X).

(2.9 )

The contraction of the heat flow tensor Xijk(x) yields a vector qi(x) twice the heat flow vector, =

q;(X)

which is (2.10;)

S;jj(Xj.

The spatial evolution of these velocity moments is easily obtained from the steady-state transport equation (2.5) by multiplying it by 1, 21;, v~j , . . , etc. and integrating over all v in accord with the definitions (2.8). After some integration by parts and some rearrangement one finds: dJi -= aXi dP.. 2

-

ai

aXj aSijk dXi

-

n

-

g E, '7?2

fi

o

-

(2.11’)

7

9

Ejrs J,

H,

+

J-l

1)2C

=

0,

(2.12’)

7

(aj Jk + ak Jj) - k ( Ej Jk -t Ek Jj)

(2.13) -5

H,(t,,,P,k+e,,,P,,!+~-lLSjk=O,

7

7

188

~aHiik1 -

axi

FltISCH AND MORRISON

(aj

pkl

+

ak

f’lj

+

az

Pjk)

-

5

(Ei

pkl

+

Ek

Plj

+

EE Pjk> (2.14)

etc., with Eijk the three-index, totally antisymmetric, Kronecker unit tensor. These linear equations possessa characteristic structure in that the equation for the nth order tensor moment contains the divergence of the (n + l)st order tensor moment arising from the (nondiagonal) operator vi a/axi . This infinite hierarchy of moment equations is directly related in a known fashion (7, 8) to the infinite set of coupled partial differential equations satisfied by the coefficients of a generalized Hermite polynomial (in v) representation of f(v, x) satisfying a transport equation such as (2.5). In the domain of convergence of such a representation the infinite set of Eqs. (2.11-2.14), etc. contains mathematical information equivalent to that contained in the original transport equation. In analogy to procedures in the kinetic theory of gases,we seek desirable, i.e., physically informative, approximations to the solution f(v, x) of the transport equation (2.5) by suitably truncating with some finite order moment the hierarchy (2.11-2.14), etc. to obtain a consistent set of self-determined equations. This requires an ansatz concerning some sufficiently high order moment. We shall seethat the high magnetic field asymptotic solution of the transport equation suggestsa natural sequenceof truncations, based essentially on the number of generalized Hermite polynomials retained in the representation of f(v, x) . The simplest such truncation scheme will yield for our model results identical with those obtained by applying Herring’s macroscopic treatment to this model. The physical significance of the first three moment equations is simple, since these are essentially the electron number, momentum, and energy conservation equations. Equation (2.11) is the electron density continuity equation and assertsthat there are no sources or sinks of electrons in the sample, so that the surface integral, over any simply connected region, of the local current Ji vanishes. Equation (2.12) asserts the balance of forces-the first term arising from the stress, the second term from the acceleration of the electrons by the impurities, the third and fourth terms giving the Lorentz force, and the last term being the contribution due to phonon-electron collisions. Similarly (2.13) can be decomposed. Upon truncation, the resulting set of finite moment equations provides a hydrodynamical description of our system similar to the usual hydrodynamical descriptions of various orders which approximate the behavior of a Boltzmann gas in the kinetic theory of gases (7). C. REDUCTION

OF THE TRANSPORT

AND R~OMENT

EQUATIONS

Clearly the solution of (2.5) or even somefinite set of truncated moment equations is a formidable mathematical problem. A considerable simplification of the

HIGH

FIELD

189

MAGNETORESISTAKCE

mathematics can be achieved by restricting the spatial dependence of the problem to a single coordinate, say x. In particular the moment equations then become ordinary differential equations rather than partial differential equations. It is for this reason that we introduce assumption (2) that V be a function of x only (and not y or z). The correlation potential (o, similarly can be taken to be a function of x only satisfying, instead of (2.7), the ordinary differential equation (2.15) so that G = (Do + V = #(x) is a function of x only, which in turn implies, by virtue of (2.6)) that the acceleration due to the impurities has only a single component parallel to the z-axis, a function of .r only given by (2.16) Thus we shall seek a solution of (2.5), subject to (2.16), which .T only, f(u, U, w, x), where v = (u, U, w), with n(x) a function

=

s

of x only. To investigate

is a function

f(u, u, w, s) dv

the T.RI.R.

it sufhces to take

E = (E,,E,,O); H = (0,0,x),

of

(2.17)

cl = fpc/111c.

Subject to assumption (a), with (Pi given by (2.15), U(X) by (2.16), and the external fields by (2.17), the transport equation of our model reduces to

(2.18)

In the remainder of the paper, except in the concluding section, we shall work with this integro-differential equation. We shall be interested only in the ohmic response, and hence require only the perturbation solution of (2.18) valid for vanishingly small electric field. Thus retaining only terms of first order in the electric field components we can write 8% 11,w, s> = fo(u, u, w, xl + fi(U, 0, w, d, 42)

= no(x) + n(x),

$(x:) = V(x, i.4) 4x:)

+ (DC*(x)+ %1(X) = $0(x) + ih(x),

= so(x) + al(x),

(2.19)

190

FRISCH

AND

MORRISON

with fO(u, v, w, x) the solution of the transport equation in the absence of the electric field, and where similarly the subscript 0 denotes the absence of E in the other entities of (2.19). We note that d2 ---=$0 dx2

d2 V dx2

g ifi0 - n&>l;

so(x) = -3 35

(2.20)

m dx ’

d2 - +I = -g,,(x); dx2 al(x)

= - z 3!! m dx ’

(2.21)

by virtue of (2.15) and (2.16). The equation satisfied by fo

ug+ao(x)g+D

(

v$u$

>

+~[fo-?Lo(s)fMl=o,

(2.22)

possesses the obvious bounded solution which does not give rise to currents, fo(u, 8, w, xl = no(x)fdu, ao(x)no(x) no(x)

= no’(x) =

fi0

exp

v, WI, or

t-&0(x)lN(exp

(2.23) [-4~0(x)l4).

The same solution for fo holds, of course, if assumption (2) is dropped except that no and #o are functions of the vector x. We shall take in what follows tie(x) as a known function rather than V(x). This is certainly possible since V(x) is not known absolutely and in any casecan be found by a direct double quadrature of (2.20) [cf. (2.23)] d2 V -=dx2

d” fro dx2

(2.24)

Thus also so(x) = - (q/m)d$o/dx can be taken as an explicitly known function. The electric field perturbed part of the electron distribution, fi , satisfies, by virtue of (2.18), (2.22), and (2.23), the transport equation

- c&l(X) af,

1

au *

(2.25)

HIGH

The velocity

moments

FIELD

(2.8) can be written Ji

191

MAGNETORESISTANCE

=

Jj”’

pij = pi;’

as

+

J;“,

+ p$‘, (2.26)

Sijk = As:;: + As:,‘: )

Hijkl

= Hi;:1 + H:;:, ,

as a consequence of the break up of the electron distribution, with the superscript 0 entities arising from f. and the superscript 1 entities arising from fI . The former are given, generally, by J(O)

=

0.

Pty’ =

TLO(X)6ij

;

s$

= 0,

(2.27)

etc., with afO’(x)no(x)

= az. t

(2.28)

Thus the nonvanishing current and heat flow tensors arise only from fi . Since we will only be interested in the electric field perturbed parts of the velocity moments such as these, we henceforth drop the superscript 1 on Jj” and $3: , and simply write Ji and Sijk . Again generally, from (2.11-2.14),

L’Ji -=, axi aP$’ ax, asijk -

a.c-,

o

(2.29aj

[ujl’no + uj”‘nl] - ,; Ej no - 5 cjrs J, H,s + $ = 0,

12.2%)

-

-

The first four moment

equations

corresponding

to assumption

(.2) and (2.17j

192

FRTSCH

AND

MORRIS0

are, from (2.29a) and (2.29b),

dJzz o -= 7 dx

(2.30)

and dP::’ _- QJ, + 5 = ino( dx 7 dP’l’ 2

+ [al(x

+ QJ, + $ = f no(x)&,

+ ao(x)n~(x>l,

(2.31a)

,

(2.31b) (2.31~)

D. THE

FORM

OF THE ELECTRON

DISTRIBUTION

Before we consider the methods of approximate solution of the transport equation, at least for the local current from whose space average we can find the T.M.R., we shall make a formal substitution into (2.25) which will provide a number of interesting physical consequenceswithout requiring an explicit solution of the transport equation. An analog of this substitution exists when we drop assumption (2). We represent fi as a sum of four terms; the first two being suggested by the known solution of (2.25) in the absence of spatial inhomogeneity, nOfM(au + pv), with a! and 6 given by (2.33) below, and a remainder. Central in simplifying our subsequent calculations is the chosen break up of this remainder. Let fl

=

no(x)fna

[ au

+

(y = c/a-L + ~&4) m( 1 + U”) ’

flu

-

f

h(x)

+

G(u,

v,

p = dJ%/ - ~a) m(1 + w”)

w,

’

2)

1 ,

w = or.

(2.32) (2.33)

Substitution of (2.32) into (2.25), and use of (2.20) and (2.21), yields the following integro-differential equation for the determination of the reduced distritubion function G,

(2.34)

We have used

m(x) ~ + ; h(x) = no(x)

(2;)3,2 /jS_I exp [ - i (u” +

v2 +

wt)]

G dU dv dw, (2.35)

HIGH

FIELD

193

MAGNETORESISTANCE

which follows immediately from (2.32), in deriving (2.34). First we note that E, and E, enter (2.34) solely through the combination (Y given by (2.33). This aids considerably in computing the form of the T.M.R. Furthermore we note that (2.34) does not contain &(x) (or its derivatives) or nl(z),’ nor does w occur freely explicitly. Thus to obtain G we do not need to know #1(x); on the contrary, we can solve for til(x), once G is known, using (2.21) which now reads

(2.36) G du dv dw, by virtue of (2.35). In some instances we will be concerned with a periodic potential #o(x), and then spatial averages are taken over a period. However, we are not restricted to strictly periodic potentials, but can consider a wider class which we refer to as the random case, for an infinite sample, for which #o(x) is such that G remains bounded as x + f 00. Note that, since #o(x) corresponds to a physical I’, it cannot be entirely arbitrary but must be such that the solution V of (2.24) is properly behaved at infinity. We will make use of the fact that if F(x) is a bounded function, then (dF/dx) = 0 (cf. Eq. (1.5) or ref. 5). Now, from a theorem (9) on differential equations of the form (d2y/dx”) - f’(r)y = g(s), since no(z) is positive and bounded, both from above and away from zero, for bounded &(x), and G is bounded as a function of x, there is one and only one bounded solution &(x) of (2.36) for prescribed G. Actually, from (2.34), G is determined only to within an arbitrary constant, which reflects the fact that $i(r) is determined only to within an arbitrary constant, as is to be expected. The components of the local current in terms of G are

Since G does not depend on $1, neither does the current. Taking G as the solution of (2.34) which does not depend on w, i.e., G = G(u, u, x), we find by using the above and the moment equations the following three exact physical conse1 This appears astonishing at first sight since carriers on the impurities and hence the T.M.R.,

nl(.r) but

determines see (2.42)

the push and below.

of the electrical

194

FRISCH

AND

MORRISON

quences of (2.32) : (I) The current Ji and its spatial average (Ji), and thus the T.M.R., do not require knowledge of $i(z) but can be computed directly from knowledge of &(z) alone. (II) The current and its spatial average satisfy J, = 0, J, = (Jz)

= I,

(2.38) (a constant),

(2.39)

from (2.30), and (J,)

= [E fro E, - w(Jz)]

(2.40)

,

from (2.31b), i.e., the spatial average of the y component of the Lorentz force is just balanced by the spatial average of the force component resulting from the phonon scattering, as we would expect. (III) The current and components of the heat flow tensor are related by

smz = SW = sgvz = s,,, = 0:

(2.41)

Sws = J, .

Sm = Jz , Let

(2.42)

(Jz> = afiolx, with with

x a constant parameter, in general a function (2.40)) gives the resistivity tensor :2

El! m 0 -i &

of o. Then (2.42),

together

Mx - 1) + xl (2.43)

= qr?io

cd

Thus as long as (x - 1 )-’ = o( w”) as o + m our model will exhibit a nonsaturating T.M.R., as we shall see is the case. The fact that a single x defined via (2.42) suffices is a consequence of the fact that G depends only on (Y and not separately on E, and E, . III.

OUTLINE

A. METHODS

OF

DIRECTLY

THE

VARIOUS TRANSPORT

BASED

METHODS EQUATION

OF

ON THE TRANSPORT

SOLUTION

OF

THE

EQUATION

We consider first a direct solution of the transport equation satisfied by the 2 The particle

resistivity current,

the

tensor differs by electric current

a factor is clearly

q from qJ.

the

usual

definition.

Since

J is the

HIGH

reduced distribution

function

FIELD

195

MAGNETORESISTANCE

G defined by (2.32). From (2.34))

where (3.2) G being independent

of w. The subsidiary

equations

corresponding

to (3.1) are

ds u Two integrals

- (Da dh)

= ’

= [+cr)

/ ,;+

G]‘

(3*3)

of (3.3) are (V

+

CT)

=

colx3t.;

U2

+

z? +

-

II1

$0(x)

=

It might appear to be a straightforward matter to complete by a quadrature, assuming that (a(z) is known, and then, obtain an integral equation for @(z). However, a difficulty illustrated by considering a particular form of &(.r), such angular potential given by

const.

(3.4)

the solution to (3.3) by means of (3.2), arises which may be as the periodic tri-

(3.5) This we have done, but here omit the details for the sake of brevity. The remaining quadrature with respect to J: in (3.3)) using (3.4 j, is carried out in two regions, corresponding to u > 0 and u < 0. Then the conditions of periodicity, and of continuity at u = 0, lead to functional equations in two variables for the arbitrary functions a-hich arise in the integrations. Solution of these functional equations would then lead to a linear integral equation for a(~+‘), but even for this simple choice of $0(z) it has not been found possible to solve the functional equations, with a single exception. That is, in the absence of a magnetic field (Q = 0), the functional equations are in one variable only and a simple and explicit solution can be obtained, and a linear integral equation for (a(z) can be written down. Since the direct approach is blocked we restrict ourselves to obtaining asymptotic estimates to the solution of (3.1) valid for large W. This suffices for our purposes. For extremely large values of w the leading term of such an estimate will suffice. For lower values of the applied magnetic field X, the cyclotron radius will no longer be small compared to the range of the impurity forces; a correction

196

FRISCH

AND

MORRISON

to this result will be necessary which will involve the dimensionless ratio of these lengths. Another important dimensionless parameter of our problem will be denoted by IL, the difference between max(nJ/o(z)/m] and min(&O(x)/m) [a function of the Debye length in general; cf. (2.20)]. This parameter characterizes the maximum fluctuation (in our units) in the impurity potential energy divided by kT. We recognize the possibility of two different regimes of behavior depending on whether the dimensionless figure of merit PW is larger or much smaller than one. It will turn out, astonishingly, that for our (one-dimensional) model the leading terms of the asymptotic (in U) solution for P finite reduce to the leading terms of the perturbation solution of (2.34) in powers of p. This perturbation solution (for sufficiently small p) is certainly valid if PO << 1. The verification of this important property provides the motivation for the solution of the transport equation in the case of a small periodic impurity field. This solution, up to and including terms of order p2 can be found in Appendix 1. The results of this investigation will be summarized in the next section. The result obtained in Appendix 1 proves to be very useful3 since it suggests the general form (in a) which an asymptotic (as w + 00) solution of (3.1) must take. Setting G = CYH, x = T{ and (p/m) d$o/dx = dp(f)/df and introducing polar coordinates in the (u, ZJ) plane, u = r sin 8, v = -r cos 8, one finds (cf. Appendix 2) that (3.1) becomes aH 96

(H-F)

+rsintT’x--

& 4

al

l+--

cos 0 dH r de

,

(3.6)

where r exp (-r2/2)H

dr de.

(3.7)

Making use of the form [cf. (A1.15), (A1.21), (A1.28), (A1.29) and (A1.33)] suggested by the calculations in Appendix 1, we obtain an asymptotic solution of (3.6) in Appendix 2 by setting H = w2 Ho+:+:+ F = w2 a,+:+:+

..a -.a

>

, (3.8)

. >

This asymptotic solution applies equally well to randomly distributed as well as periodic distributions of the impurity sheets, provided that J/o(z) is such that 3 It implicitly equation, although

provides us with the moderate we are not overly concerned

and small o behavior of our transport with these regions in this paper.

HIGH

FIELD

MAGNETORESISTANCE

197

the reduced distribution function G is bounded as x -+ f 00 and also is sufficiently smoooth (i.e., possesses as many continuous derivatives as the number of terms in the series (3.8) required to adequately represent the solution of the transport equation). As can be seen more explicitly from the work in Appendix 2 it does not appear possible in this model, subject to assumption (2)) to write the asymptotic solution as directly parametrized by the ratio of the radius of the cyclotron orbit to the range of the impurity forces. Again our results are summarized in Section IV. B.

R~OMERTT

XIETHODS

We turn now to the solutions of the variously truncated moment equations. Our interest in these solutions is not only motivated in obtaining an independent derivation of the T.M.R. terms obtained from the asymptotic, in w, solutions of Appendix 2, but also in obtaining a method capable of dealing with a general Boltzmann transport equation whose collision term is an arbitrary continuous linear functional of the electron distribution [i.e., in particular not subject to assumption (1) of our model (7)]. Formally the three truncations with which we are concerned are obtained by representing G in (2.34), in decreasing order of accuracy, by four (the Oth, lst, 2nd and 3rd), three (Oth, 1st and 2nd), and two (0th and 1st) generalized Hermite polynomials in v. These truncations lead to 20 independent “hydrodynamical” equations for the components of Sljl; , PC?’ equations for the PI:‘, LJ, J. 1, and n 1 ; to 10 independent “hydrodynamical” Ji , and ~11; and 4 independent “hydrodynamical” equations for the Ji and nl , respectively. We henceforth refer to these as the 20-, lo-, and 4-moment equations. The termination of the infinite hierarchy of moment equations with the 20-moment equations, while suggestedby, is not based solely on a vague analogy with &ad’s 20.moment approximation (7) to the solution of the Boltzmann equation. Rather, our choice depends on the fact that the 20-, lo-, and 4-moment equations yield components of the heat flow tensor which satisfy the exact conditions (2.41) as well as Pii’ = nl . These conditions are not obtained, for example, by replacing the third order Hermite polynomial by the corresponding corltracted (over 2 indices) polynomial to obtain a 13-moment system of equations for the components qz , P$‘, Ji , nl , which we have investigated in some detail. To illustrate more explicitly we develop below the 4-moment equations. This will allow us also to make some remarks about the allowed forms of $0(~) for the “random” case. If G is not to contain generalized Hermite polynomials in v of higher order than the first, we must truncate by setting Pif)

= nlaij

(3.9)

in (2.31). Thus we need consider only (2.30) and (2.31). Using (3.9) we have

198

FRISCH

from (2.30),

(2.31b),

AND

MORRISON

and (2.31c), J, = I; J, =

J, = 0; $no(z)E,

- WI

the space average of the last equation setting [cf. (2.32)],

(3.10)

1 ,

being the exact (2.40).

m(z) = no(z) [Q(2) - ; h(r)] ]Q (x) is the 0th order velocity

moment

z = 7E, we obtain,

using (2.21),

(2.23),

&o(z)lm (2.33),

An equation for #i(z)

f.0”) $

,

= cp(F),

and (3.10), IV>

= [

1

ec(~) .

A0

(3.12)

may be obtained from (2.21). Averaging aA

(2.31a),

of jMG], and

(y (1:

From

(3.12) we obtain

= I(e-“)(e’),

(3.13)

which agrees with the leading term of the asymptotic solution to the transport equation given in Appendix 2. In averaging we have assumed that cp(t) is such that Q (hence G) is bounded, and hence (d&/do = 0.4 Alternatively, we can restrict ourselves to a very large, but still finite sample and require that Q and its derivatives, etc. satisfy periodic boundary conditions. 4 By definition,

We see that

if

2L is the length

as 5 -+ f

for & to be bounded

which places a restriction tion or if ~(6) is periodic.

onq(E). Actually,

[Lbounded as L -+ that G is bounded each stage of the

m. Similarly, as E ---f f asymptotic

dimension

of our volume

cc, we must

have,

V divided

in particular,

This condition is trivially satisfied for & to be bounded both

iie”L)d/]and[L(e’>-See.(L’di]

by 7,

if p(E) is an even

must

func-

remain

in Appendix 2 we have assumed that ~(6) is restricted m , and this leads to a restriction analogous to the above expansion.

so at

HIGH

FIELD

199

MAGNETORESISTANCE

The lo-moment equations are derived in Appendix 3. The asymptotic (in U) solution is obtained as is the exact solution for the periodic triangular potential (3.5). The 20-moment equations are also shortly discussed in Appendix 3. In Section IV we again summarize the results of the moment analysis as well as the various comparisons, shown diagrammatically in Fig. 1, between the perturbation (in P), asymptotic, moment, and Herring’s macroscopic analysis of the T.1I.R. problem. IV.

SUMMARY

OF

THE

T.M.R.

RENJLTS

We are essentially concerned in this paper only with the high magnetic field T.M.R., i.e., w >> 1. From (2.43) it follows that the T.M.R. will not saturate with increasing X, i.e., w, as long as (x - I)-’ = o(02) as w -+ CC, with x defined by (2.42). Since we are only dealing with an idealized model, as yet without a practical application, it suffices to state our results in terms of x rather than the more usually quoted magnetoresistance ratio

AP.ZZ_ PAX) PIdO)

- P,,(O) Pm(O)

(4.L)

*

To find ~~~(0) requires a solution of the transport equation in the absence of a magnetic field. We have already stated that we have reduced the solution of this

’ MODIFIED ; EQy+

BOLTZMANN

*

INFINITE HIERARCHY MOMENT EQUATIONS (2.29)~(2.3l),ETC.

(ASYMPTOTIC EXPANSION)

PERTURBATION SOLUTION FOR SMALL W (APPENDIX 1)

I (TRUNCATION)

OF

I

4+ .

10 MOMENT EQUATIONS

* (co=

0)

(APPENDIX

3) *

HERRING MACROSCOPIC. THEORY

FIG.

1. Flow

sheet

for

the

c

T.M.R.

problem

4 MOMENT EQUATIONS (SECTION 3)

comparisons

_

200

FRISCH

AND

MORRISON

rather formidable integro-differential equation to the solution of a single linear integral equation, at least for the case of a periodic triangular potential.5 We turn first, then, to the asymptotic in w solution of Appendix 2. We see that [cf. (A2.1) and (A2.11)]

G=awfHo(;)+o(;)],

(4.2)

with Ho(z/~) a bounded quadrature of (A2.18), so that, in general the electron distribution function diverges with w as w -+ 00. The magnitude of the leading term in (4.2) is directly proportional in this limit to only E, and not h’, as would be intuitively expected [cf. c11given by (2.33)J. In the absence of Ezl the electron distribution remains bounded. Subject to the restrictions on qO(zr) mentioned at the end of the last section our asymptotic results apply equally well to strictly periodic or “random” (in our sense) distributions of impurities. If &(x) is at least once differentiable (Jz) is given by (A2.29) and thus xdexp

[-a90(x>lml)(exp

X

[ 1 -

bJ~(~)hl)

&

((:

‘y>’

exp

M0(r)lml)

/

(exp

[q#~(+)/ml)]

,

(4’3)

where (exp

[-4#0(z)hl)(exp

[q~~(z>/ml)

=

(--J&j

2

(4.4)

1,

by virtue of (2.23). The T.M.R. thus diverges like w2 as w -+ 00 with the same coefficient as given by the macroscopic theory of Herring (1). If $o(z) possesses a cont.inuous second derivative the error in (4.3) is 0( l/w”). We are also interested in heat flow as w + 00. As expected, the exact relations connecting certain components of the heat flow tensor with the particle current, (2.41), are reproduced in the asymptotic solution, (A2.30) and (A2.31), which together with the asymptotic results (A2.37) yield qil Ji,5,

i = 2, y

(4.5)

+ ...

(4.6)

as found in the absence of any spatial inhomogeneity. Expanding (4.3) for small p we find that x = 1 + [([+)I”)

- ($))z]

- &([$*I;\

5 Readers interested in further details concerning this or other calculations, left out of this paper for the sake of brevity, can obtain information by sending us a request. 6 This corresponds to the usual value of the ratio of heat to particle current of 5 kT/2, remembering our choice of units and the definition of qi.

HIGH

FIELD

MAGNETORESISTANCE

201

in agreement with the exact solution to second order in p obtained in Appendix 1. To make the comparison with (A1.49) more transparent, consider q#~/?n = p cos kx, K = Icr in (4.6), which becomes (4.7 ) and note the definition of x in (2.42). The correction term to the T.RI.R. arising from K~/J*/SU~ is of importance only if p2w2 << 1. This smooth transition to the exact perturbation solution (to terms second order in 1) is consistent with the estimate that the perturbation solution is valid for w >> 1 if / p 1w << 1 and is possibly valid (cf. Appendix 1) if 1pK 1<<< 1. For sufficiently small p the w dependence of the particle current follows from (A1.43) to (A1.44), even for small w, although we are not interested in this behavior. Turning to the moment solutions we have already noted that the 4-moment result for (JI), (3.13), is identical with the first term in (JI) given by (A2.29). This is not astonishing since the 4-moment theory corresponds in our model to the Herring macroscopic theory. Furthermore the asymptotic (in w) solution of the lo-moment equations of Appendix 3 for (J,), Eq. (A3.31), is identical with (A2.29). Similarly, the first two terms of the asymptotic solution of the 20moment equation (J,) can be shown to correspond exactly to (A2.29)) the error being, for sufficiently smooth &,(z), 0( 1/w4). We have left open the question as to whether the term O(l/w4) of the asymptotic solution of Appendix 2 is identical with this error term of the asymptotic solution of the 20.moment equations, etc. The exact solution of the lo-moment equations for the periodic triangular #,I given by (A3.34 ) of Appendix 3 is not particularly physically suggestive except that the result implies that for a discontinuous d&/d.~ the error in (4.3) is 0( l// o I”). The ratio Q/J; is exactly equal to 5 in the lo-moment approximation by virtue of ( A3.1), the truncation condition. Finally, the moment equations give successively better estimates of (JJ) in the absence of a magnetic field, in powers of x2. We found this by considering the perturbation solution for the case of a small periodic potential. Rather than attempting to deduce the results for w = 0 as a limiting case of the results in Appendix 1, it is simpler to proceed directly from the pertinent equations. This wc have done, but again we omit the details. v.

D1SCU8S10N

We begin by demonstrating the T.M.R. result given by the 4-moment equations using an elementary argument similar to that first used by Drude (IO) to derive the electrical resistance of a metal. As we have stated in the introduction to this paper, we require the ansatz that the relaxation of velocity fluctuations occurs independently of the local spatial fluctuations in electron density. In

202

FRISCH

AND

MORRISON

lowest order for our modified Boltzmann electron fluid the relaxation of velocity fluctuations can be measured by the variance in the components of the local electron velocity, i.e., the stress tensor p;i . There are no stresses in a fully relaxed gas at equilibrium. To this order the independence ansatz then merely states that the spatial divergence of pij vanishes. This can be the case in our model, subject to assumption (2)) independently of the scale of the inhomogeneities, as long as classical mechanics applies and provided either (a) the magnetic field is very large, i.e., w >> 1 or (b) the magnetic field is very small, w << 1, and T sufficiently small so that K = lcr << 1. In the former case the radius of the cyclotron orbit is so small that the spatial variation of the impurity potential produces a negligible effect on the mean motion of an electron, while in the latter case the variation in the impurity force over a representative mean free path (determined by T) again produces a negligible effect on the mean velocity of the electron. Denoting averaging over velocities by a bar over the averaged quantity, we have, following Drude (10) (for a classical electron subject only to an electric field E, in the absence of any spatial inhomogeneities), the following statement of Newton’s Second Law

(5.1) Equation (5.1) immediately gives the Drude formula for the conductivity. This prescription of Drude for the velocity averaged electron acceleration can be immediately extended for our model with electric and magnetic fields given by (2.17) and stratified spatial inhomogeneity described by the effective potential $J(x) : First one replaces in (5.1) the acceleration due to E alone by the Lorentz acceleration, and second one adds the effective velocity averaged acceleration due to the inhomogeneity, A(x), not equal to a(x) and which vanishes with vanishing E, acting only parallel to the x-axis, so that instead of (5.1) we now have:

0 dU

= @$ + fig(x) + A(x) - u$

z

= 0,

(5.2)

and

=--

Q&x) - e(z> = 0. 7

(5.3)

The independence ansatz appears by the absence of terms due to the divergence of the stress tensor. In the absence of the spatial inhomogeneity (A (x) = 0) the solution of (5.2) and (5.3) is the well-known result that G = (a) = (Y and 6 = (0) = 0, with cx and /3 given by (2.33). Concerning A(x), when A (x) # 0, we need only note the necessary condition

HIGH

FIELD

MAGNETORESISTAXCE

203

that (A(x))

(5.4)

= 0,

since otherwise no spatially bounded steady-state current is possible.’ spatial averages of (5.2) and (5.3), and using (.5.4), we obtain

Taking

and

(5.6) i.e., (a(x)) = (Y and (e(x)) = 6 as in the absence of a spatial inhomogeneity. Since (G(Z)) and (6(x)) vanish as E vanishes, we can replace n(x) by no(x) in the ohmic local current Jz(z) Noting further

J,(.r)

= ~n”(z)ti(.x),

that electrons are conserved, Jz(x)

= I,

= no(zje(x).

i.e., div J(x)

= 0, we have that

a constant.

( 5.8)

Solving for G(E) from (5.7) and (5.8) and space averaging, I = (J,(z))

(5.7)

we obtain

= %I ci/<&j.

(5.9)

Equation (5.9) together with (3.10), obtained by multiplying (5.3 ) by nO(z j, again yields (2.43) on space averaging, with x given by (4.4) by virtue of the Boltzmann distribution which governs no(x) in equilibrium. Thus the appearance of a T.M.R. in the presence of inhomogeneities follows from the equilibrium spatial variation of the electron density no(x j , which in turn produces a fluctuating JU(x) and hence a locally fluctuating Hall field, which can no longer cancel uniformly the transverse force due to the magnetic field of an electron moving with the local mean velocity. Multiplying (5.2) by no(x) and using (.5.7), and comparing the result with that obtained by substituting in (2.31a) the I’::’ obtained by substituting (3.11) into (3.9), we see that A(z)

with

Q(x)

defined by (3.11).

7 Mathematically, we know from rivative of an essentially periodic

=

-

dQ(x)

We can verify the 4-moment function.

(5.10)

7)

that

equations

the analogous that,

-4(x)

Drude-like

is a complete

de-

204

FRISCH

AND

MORRISON

derivation, subject to the independence ansatz, applies when assumption (2) is dropped, i.e., we have a three-dimensional distribution of impurities, viz., 0 = $ replacing

- $ + &

(5.2) and (5.3))

(V X H) + A(x, y, z) =

0

$

,

(5.11)

with

J(z, Y,z> = nob, Y, ZF(T

Y, 21,

(5.12)

instead of (5.7), and div J(z, y, x) = 0, instead of (5.8). three-dimensional nob,

These equations are exactly the 4-moment distribution of impurities [# = $(x, y, z)]:

Y, 2) grad

&CT Y, 2)

(5.13) equations

= !$ ~o(x, Y, 2) + -& (J X H) - 1, 7

div J(z, y, z) = 0, ~I(~,Y,z)

= no(x:,~,z)

II

Q(x,Y,z)

- y~h(x,~,e)

for a

(5.14) (5.15)

1 ,

(5.16)

if we make the identification Nz, Y, z> = - grad Q(x, Y, z).

(5.17)

Unfortunately, the independence ansatz no longer applies independently of the scale of the inhomogeneity, for three-dimensional distributions of inhomogeneities, as long as the cyclotron radius is sufficiently small, i.e., (J >> 1. Still, the above results are gratifying in that they provide some physical insight into the meaning of the 4-moment equations. Returning to our model (one-dimensional distribution of impurities) we note the impurity concentration dependence of the leading asymptotic result given by (2.43) m

Pzz(X>, ---Ib2(x - 1) + xl. qrn0 Denoting by N/L the lineal density sense) distributed identical wells vanishes for 1x 1 > a and is constant lineal fraction of wells, C = 2&V/L,

(5.18)

of a set of periodic or “randomly” (in our each contributing potential a(z), which (E) for ] x 1 < a, and by C the dimensionless we have

(x - 1) = 4C(l - C) sinh2 % . 0

(5.19)

The concentration dependence of form (5.19) is of course a consequenceof the

HIGH

FIELD

205

MAGNETORESISTANCE

complimentarity between such a potential well in a specimen otherwise at zero potential and a hole well in a specimen otherwise at constant (nonzero) potential. We turn now to the correction term to the macroscopic theory result, which arises from the spatial variation of the stress. We note that the leading term [O(w’)] of the correction [cf. (4.3)] always acts to reduce the term x in the square bracket of (5.18), i.e., always decreases the T.M.R. to that order in W. The full physical significance of the dependence of the correction term on the impurity potential is not clear to us. Finally our model subject to assumption (2) camlot exhibit a longitudinal magnetoresistance since the current can always flow in tubes parallel to the magnetic field without hindrance from our stratified inhomogeneities. This ir shown explicitly in Appendix 4. APPENDIX

SOLUTIONOF IMPURITY

THE

TRANSPORT

EQCATION

1 IN

THE

CHASE

OF

A SMALL

PERIODIC

FIELD

We are here concerned with the solution of (3.1) in the case ; h(2)

= /.dr),

] /J [<< 1,

where P(X) is periodic. It is supposed that neigher cpnor T(&/&) panding in powers of p, G = /.LG~+ $Gz + . . . ;

(A1.1)

is large. F:.u-

a, = pa.1 + /a@? + . . . )

(Al .2)

we deduce from (3.1) and (3.2) that us+a

(

ug-ua;

>

+‘[G,-~~(.c)]=a~~, T

+ 1 [G* - @2(x)] = zg, 7 ,-

(Al.3 (Al.4

where exp]-$(u”

+ o’)]G,~ du d2~.

(A1.5)

From (2.37)) J, = ano(z)(l

+ A-),

J, = ~o(.~)(P+ 4,

C.41.6)

where

1

m-

CYY= u exp[-*(u” 27rss -m --m

(Al.71 + ?)]G rlu du,

206

FRISCH

AND

MORRISON

and, from (2.23) and (Al.l), no(x) The quantities x

of interest

= fi~e-~~(~)/ (e-“).

(A1.8)

are the spatial averages of J, and J, . Let

= /Lx1 + p2xz + *. * ;

Y = /.4Y1 + p2Y2 + . . . .

Without loss of generality we may take (q) = 0. From (A1.3) (D is periodic, it follows that (Xl) Hence, from (A1.6)

and (A1.7),

= 0 = (Y1).

since

(A1.lO)

and (A1.8),

- d-,>)I + . . . , - (~Yd)l + . +. .

(Jz) = afio[l + ~‘((x2 (J,> = fio[P + w2(W2 It is sufficient

(A1.9)

to consider the particular p(x)

(A1.11) (A1.12)

case

= coslcx.

(A1.13)

If a(x) is in general expressed as a Fourier series, it follows from (A1.3), (A1.4), (Al.ll), and (A1.12) that the spatial averages of J, and J, to order p2 may be obtained by superposition. So we will confine our attention to the impurity field corresponding to (A1.13). Let cl7 = l/6;

k = eD,

(A1.14)

and (A1.15)

G1 = 6i{iakreik”K}. Then, from (A1.3)

and (A1.5), iez&+

(

vg-u$

>

+6K=6(1+L),

(A1.16)

where

L=lxL* cc

To facilitate coordinates

exp[-+(u2

the solution of (A1.16) 9.4= r sin 0,

+

v2)]K

du

dv.

(A1.17)

00

we make the transformation

v = -rcose,

(0 4 8 5 27r).

of velocity (A1.18)

Then ia sin 0)K

= --6(1

+ L),

(A1.19)

HIGH

FIELD

207

MAGNETORESISTANCE

and L = & szr jm I- exp( -r”/2)K 0

Integration of (A1.19), implies single-valuedness, K(v, e) = ql

subject gives

+ L) exp

to the condition

-

(se

dr de.

(,41.20)

0

~UCOS

K(r,

Oj = K(r,

2x),

which

e)

2* e o exp(icr s

exp(k cos cp - 6~) dp (1 _ e-2,aj

(A1.21)

~0s cp - 6~) dp * I

Substitution of (A1.21) into (A1.20) leads, after considerable integration by parts with respect to 0, to

reduction

and au

where

N=sm

sinh(6e) sinh( &r)

? exp( -?/2)

0

cosh[6(?r - up)] sin[tv(cos

- secosh[6(8

- p)]sin[a.(cos

0e

-

X1

=

COS kX,

(A1.21),

YJ

and (A1.22),

=

-

$tCOS

cosh[S(?r - cp)lsin[er(cos

e-

(Al.23 j

cp)] dp de dr. i

cos

0

E’rom (A1.7), (A1.15), (A1.18), considerable reduction, that

CF)] dp

cos

it follows,

also after

(A1.24)

kX,

where

* *“f0

cos

cpjl cl9

(A1.25)

e

s0

sinh[6(0 - cp)lsin[er(cos

e

The expression for Xi can also be deduced from and the fact that J, is independent of x.

- cos cp)] dp de dr. 1 (Al&),

(A1.8),

and (A1.13),

208

FRISCH

NOW, from (A1.18)

AND

and (A1.21),

aK _= i&(1 + L)exp(&9 au

MORRISON

we find, after an integration

by parts, that

- ier cos 0) sin(8 - cp)exp(&r

cos cp - 6~) dp (A1.26)

(1 - eh2ra) e

-

sin(0 - cp)exp(ier cos cp - 6~) dq ’

Also, from (A1.15), (A1.27) Thus, from (A1.4),

(A1.5),

(A1.13), (A1.26), and (A1.27),

G2 = P sin 2kx + & cos 2kx + R,

(A1.28)

where P, Q, and R are independent of x. In order to determine the spatial averages of J, and J, it is sufficient, from (A1.7), (Al.ll), and (A1.12), to determine R. If we set R = -g&(1

+ L)S,

(A1.29)

then

$ - 6(X -

C) = -e*‘F(r,

e),

(A1.30)

F(r, @ ={(l-1e-2r8) c

sin (0 - cp) sin [cr(cos cp- cos e)] e-*’ dp (A1.31)

e s0

sin (0 - cp) sin [er(cos ‘p - cos e)] e? dp , !

and the constant C is given by r exp ( -r”/2)

S dr d0.

(A1.32)

We may take C = 0 without loss of generality. Then, integration of (A1.30), subject to the condition S(r, 0) = S(r, 2?r), which implies single-valuedness, gives F(r, tc) d$ - i’F(r,

#) d+] . (A1.33)

HIGH

FIELD

209

MAGNETORESISTANCE

Let

q = ;

(1 + L) I?’ irn r2

Then, from (Al.7),

(Al.lS),

(A1.28),

cos

(A1.22),

(A1.34), s

(A1.34)

e exp ( -?/2)

S & de.

iA41.35)

(Y2) = q.

(A1.36)

and (A1.35), T

OD

E2 r=-ao

S ok do,

and (A1.29),

(X2) = i-7 From

e exp( -?/2)

r2 sin

5=

r” exp( -r”/2)

s0

sin 8 [X(7*, 0) - S(r, 27~ - e)l dfl do; (Al.371 cos e [sb,

For convenience,

e) + S(T, 2~

- e)] de dr. (,41.38’)

we define

T(r, 9, p) = sin (+ - p) sin [~(cos

cp - cos

$11;

2a sinh [6(27r - cp)] Ur, $, cp) dp sinh (6~) 3 -2 cash [s(?r - cp)l T(j*, $, cp) s0

(A1.39)

(A1.40) d$.

Then, after some reduction,

-

2n cash [S(7i- + e - cp)l T(r It P) dP 7 , sinh (6~)

- I.* 2

[sCr, e) - s(r,

2* -

0

sinh

(A.141 )

He - cp)lT(r, A cp)& dJ/; >

e>l = shh (80) u(j.j sinh (6~)

(A1.-22)

210 Now,

FRISCH

from (Al.ll)-(A1.13),

AND

(A1.24),

MORRISON

and (A1.36), (A1.43)

(Jz) = afio[l + PQ- - $$;>I + * * * ,

(J,) = fia[P + w2 (q + $1

+ .. . ,

where M, N, I, and q are given by means of (A1.23), (A1.42). In view of (2.33) and (2.40) we should have 2N(c Partial checks of (Al .45) are N and M are 6M cancel for from (A1.14),

(A1.25),

and (A1.37)-

+ Sq) = (N - 6M).

(A1.45)

of this relationship have been made. In particular, the two sides odd functions of E. For small 6, the left-hand side is 0( c”), whereas both O(E). It has been verified that the terms of O(e) in N and arbitrary 6. Next, consider 1w 1>> 1 and kr = K = O(1). Then,

f

=

(A1.46)

K8,

and so E = O(S) . It is found that, under these conditions,

M

-

TK,

N-

K” 62

N

-

(A1.47)

?TK&

Also, ’ Thus, from (A1.43))

8’

2 q-5.

since wS = 1,

>I

+ ....

(J,) = aiio As a further

(A1.48)

4

(A1.49)

check on (Al .45), it has been verified that (N-6M)--K

37r 3 3 6. 4

Under these same conditions, let us examine the validity expansion in p. From (Al .26) and (Al .27), [I + O(S)] [I sin B I ( JI(KB~-)I +

(A1.50) of the perturbation

O(S)1,

(A1.51)

using (A1.22) and (Al.47). From (Al.Z)-(Al.4), we would expect the quantity on the left-hand side of (A1.51) to be small, if the perturbation expansion is to be valid. This is certainly true if I p/6 I << 1. However, there is an indication that the range of validity is somewhat larger. For, if we were to consider the Maxwellian distribution fM to be cut off and suitably normalized, so that 0 5 r 5 R

211

HIGH FIELD MAGNETORESISTANCE

where R >> 1 is the cutoff velocity, then, for j KS/ << l/R, the condition becomes 1/.iK1<< l/R << 1. APPENIIIX ASYMPTOTIC ~JIELD

SOLUTION

2

OF THE TRANSPORT

EQUATION

FOR HIGH

MAGNETIC

We are here concerned with the solution of (3.1) for 1w 1>> 1, w = L?T. bet, G = aH,

x = T‘$,

i #o(r) =cp (El.

(A2.1)

Then (,3.1) becomes, under the transformation of (Al.l8), I+-- cos 6 dH

I'

(A2.2)

a0

where

F=aizu l-

r’ exp (-?/2)

H dtxde.

(AM)

We are interested in the spatial averages of J, and J, , which, from (Al.(i)iA1.8), are given by -P(I) J, = tiu e J, = (A2.4) m (P + aY), noting that PL(~ has now been replaced by (o, where r2 sin @exp ( -?/2)

H dr df?, (A2.5)

r2 cos 0 exp ( -I-“/a)

H dr de.

Iqrom (2.33), (2.39)) (2.40), and (A2.4) it follows that

where C is a constant, and cd(e-‘X) + (ePY) = 0.

(~A2.7)

We seekan asymptotic solution of (A2.2) of the form, suggestedby the results of the perturbation analysis in Appendix 1, H = u2 Ho+z+$+ co

...

(A2.8 )

212 with

FRISCH

similar expansions

AND

MORRISON

for F, X, Y, and C. First, aH0 -= a0

o

we have

Ho = Ho&r>.

’

(A2.9)

Second, aH1

-

a0

(A2.10)

= (Ho - Fo) + r sin 0 &?! - sin L?* %!!’ . dg ar at

But, from single-valuedness, 2* aH1 --g de = 0, s0

and hence Ho = Fo which,

from (A2.3)) Ho = Ho&);

implies (A2.11)

$”

= 0.

I&((,

r) - r cos 13dHo dE

Thus, from (A2.10),

aH1 - rsin(jE!? d.$ ’ ae

H1 =

1

(A2.12)

.

Then, from (A2.2),

aH2 = ae

(El - Fl) + rsine-

a& - sine-- & dE1 dt ar at d2H - r cos 13dHo - r” sin 19cos e 0 dt2 4

From single-valuedness,

El = F1 which, EI = El(t);

from (A2.3), ‘$

(A2.13)

1 .

implies (A2.14)

= 0,

and, from (A2.13), 2

2

Hz=

Ez(&r)

dE

Now,

from (A2.5),

(A2.11),

(A2.12),

x0 = Xl = Yo = 0, Note that Yl = -X2

is consistent

with

1

. (A2.15)

-rco~B~-~sinB~~+>cos20~~

dt'

4

and (A2.15), y 1 2!5= 4 (A2.7).

co = 0 = Cl,

-x

2.

(A2.16)

From (A2.6)) (A2.17)

HIGH

FIELD

213

MAGNETORESISTARCE

and Cn

Averaging

p

=

(1+X2)

=(I-$$).

(A2.18)

this equation we obtain C&y

= 1.

(A2.19)

This gives the leading term in the quantity of interest, (J*). We remark that if, instead of using (A2.6) to obtain (A2.18), we go to the next term in the expansion of H, i.e., Ha , we are able to deduce that (A2.20) which leads to (A2.18). We need the correction to the leading term of (J,), and to obtain it we must carry the asymptotic expansion as far as H4 . For brevity we omit most of the details. It is first found that

whence C3(e7) = 0 *

(A2.22)

C3 = 0 = 5, .

Then,

where

1.

M&) =

( A2.25 >

Note that ((X4 + Y3) e-‘> = i ($

(e-“g)j

= 0,

( A2.26

214

FRISCH

which

is consistent

with

(A2.7).

AND

MORRISON

Also, from (A2.6)) (A2.27)

x4 = C,e”? From

(A2.18),

(A2.23),

and (A2.27), C4

it follows

that

1 ,‘dp d2Ho\ = - 3 1% p/

(0

(A2.28)

=- Cz 4

Hence, from

(A2.6),

(A2.17),

(A2.19),

(A2.22),

and (A2.28), (A2.29)

We are also interested in the heat flow terms. From (2.8) membering that G is independent of w, we find that, exactly, s,,,

= szrz = s,,

and (2.32))

re-

(A2.30)

= s,,, = 0,

and (A2.31)

Sz,, = J, , S,,, = J, . Also,

Szzz=dx)

3a +&~m~wu3exp[-~(uZ+v2)]Gdudv], m m

S maI = %b) S

[P + &SW

SW u’vexp[-4(u2+v2)]Gdudv

,

z~v=nob$Y+~~~ uv2 exp [-i(u”

+ v’)] G du dv

z?exp[-$(u2+v2)]Gdudv From

(2.23),

(A1.18),

S ZZZ=

1 1 1

(A2.32)

, .

and (A2.1),

-v(E) afio e (e-v> (3 + r>;

afio e-v(E) S WY = --(l+n);

fro e-vP(f’

SW =

(e-c>

& e-vP(f) s,,,=~(38+&

(p + a*);

(A233)

215

HIGH FIELD MAGNETORESISTANCE

where

1‘= & i2’ lrn r4 sin3Pexp ( -~‘12) *z-L n=l

H dr dtl;

2* m r* sin” 0 cos 13exp ( -~~/2) H;dr do, 2*0ss 0 2* m ,r4sin 0 COS’ e exp ( -I”/a) H dr do; 21ross 0 2n

1

xc-

2flos

(A2.34)

(0 s0

r4 cos3eexp( -?/2)H

dr de,

If we expand l’, A, II, and Z similarly to the expansion of H in (A2.8) then, from (A2.11), (A2.12) and (A2.15), we find that ro = ilo = II0 = 20 = 0 = r1 = II1 )

(AM.‘,)

and

A1 2!EL

-n

2,

dHo

&=3~=

-I?2.

(A2.36)

4

Hence, asymptotically,

from (A2.4) and (A2.16), x 152N 3Jz ; SZYY-

Jz ;

St,,, -

APPENDIX THE

lo-MOMENT

AND .%)-MOMENT

Sz,, -

Jv ;

(A2.37)

3J, .

3

APPROXIMATIONS

The lo-moment approximation corresponds to (2.29a)-( 2.29c), wherein we truncate by setting

Sijk = J$jk + Jj6ki + Jksij )

(A3.1)

in (2.29c). Thus,

Szzz = 3J,;

Szq,= Ju ;

Szw - J, ;

S,,, = 3J, ;

(A3.2)

and

&a = Jz ;

S,,, = J, ,

(A3.3)

and finaI.Iy, szz,, = 0;

Sm, = Jz = S,,, ;

S,,, = 3J, .

(A3.4)

216

FRISCH

In the case of x-dependence

only, the moment equations

AND

MORRISON

become, using (2.17), (A3.5)

tip::’ CiX

- QJ, + :

= ; no(x)E,

T

+ h(x)no(x)

+ dx)n~(x)],

(A3.6)

+ a~, + +’ = +(x)E,,

(A3.7)

dP6:’ J, -yjy+;=o,

(A3.8)

p(l)

21: - 2OP,,(‘I = n* 7

2

+ a[P::’

- P;;‘]

f%’ 7

dJ, dr-

+ 2ao(x) J, ,

7

+ F

QP;;’ + c

= ao(x>J, ,

= m(x) 7

) 2Qp;;’

(A3.9) (A3.10)

)

(A3.11)

= ao(x> J, ,

7

(A3.12) (A3.13) (A3.14)

P:t’ = n,(x). These equations (A3.13))

are supplemented

J, = 0;

by (2.21).

Pi:’

From

(A3.8),

(A3.12),

and

(A3.15)

= 0 = Pi:‘.

Thus, from (A3.4)) s,,,

= s,,,

= s,,,

(A3.16)

= s,,, = 0.

Equations (A3.3) and (A3.14)-(A3.16) are consistent with the exact solution to (2.34), since G is independent of 20. Note also that (A3.2) agrees with the high magnetic field asymptotic solution of (2.34)) as is seen from (A2.37). We proceed to reduce the remaining equations, and accordingly let

P$

= P;

J, = I;

Jv =

Pj;’

= [R% + nl(x)];

[

gndx)E,

+ K Pii’

1 ,

= [R, + nl(x)],

(A3.17) (A3.18)

HIGH

FIELD

217

MAGNETORESISTAiYCE

and 2

Then (A3.7)

=

r(;

w

and (A3.9)-(A3.11),

=

Qr;

g,o(x)

=

(A3.19)

dt).

using (2.20) and (2.23),

R,=

,,=,(,,-I$);

lead to

-2wP,

(A3.20)

and

$+d+K=O; $+$K+

(A3.21)

(1 +-Iw’)P

= 2~1%.

(A3.22)

1

(A3.23)

Also, if we set

pii’ = no(x) Q(x) - ; h(x) so that (A3.18) defines nl(z), and (A3.20))

dQ - = a(l + $) + (2 dt;

,

then (A3.6) gives, using (2.21), (2.23), (2.33),

e"(I) [ (UK - I) + 2$(coP

- I$)].

(A3.24)

An equation for #1(x) may be obtained from (2.21). We confine our attention to either the periodic case,or to the caseof an infinite sample where P(.$) is such that P, Q, and K are bounded. Then, averaging (A3.24), a%“(1 + ~0”) = I(e-‘> ([ 1 + 2 ($)J

e’> (A3.25) - w(e-‘)

From (A3.21) we also obtain (A3.26) Let

P = IR.

(A3.27)

218

FRISCH

Then, from (A3.25)

AND

MORRISON

and (A3.26))

aA = I(eC”)

(e’) + ~ ’ ‘\ (l+L) [ 2 (( f ) “/ { where, from (A3.21) and (A3.22), d$+g

f-

- 3~ ($$ Re9)

,

(1+4wz)11+3w~=o.

(A3.28)

(A3.29)

For sufficiently smooth p(C;) we deduce that, in the high magnetic field case Iw/>>L R 2*

(A3.30)

4w d[ *

Then, from (A3.28), afio - I(e-‘)

[ (e9) - 4~2((gcv)1

; (A3.31)

I~~[l+&‘9~~]~ in agreement with the asymptotic solution to the transport equation, as is seen from (A2.29). In the caseof the periodic triangular potential, as given by (3.5)) we can solve (A3.29) exactly. Let 72,

x = p!! c *

7

(A3.32)

From considerations of symmetry it follows that R is odd and hence R(0) = 0 = R(r). In the interval 0 < .$ < y we have &/di subject to (A3.33), is then R = (I y:w2)

(A3.33)

= X, and the solution of (A3.29))

11 + [(em’ - l)e” + (1 - e’y>e”El/(e’y - emY)), (A3.34)

where 1and m are the roots of y2 + xy -

(1 + 4w2) = 0.

(A3.35)

We have le-Cj = (1 - e-7 ; P

(e9) = (e” - 1) ; c1

(A3.36)

HIGH

FIELD

219

MAGNETORESISTANCE

and /,($f)'

e')

=

(A3.37)

X'(e').

Also, (1 cw2j .

(0

=

(em? -

it follows

(A3.28))

Jwy$ -

11 +

(1

emy) -

ey[e(m+A)y 072 +

A>

-

l]

(A3.38)

xj

that, for 1w 1>> 1, & p\ 3X2 R (IE e ,’ = 4 (e’) + 0

w Hence, from

+

l)[e'l+QY (I +

From (A3.35)

il

(A3.37))

(A3.39)

and (A3.39)) (A3.40)

The O( l/ ) w 1‘) term arises, rather than 0( 1/w4), because of the discontinuity in dp/d.$. The 20-moment approximation corresponds to (2.29a)-( 2.29d), wherein we truncate by setting HIij.z = [Pljl)8kl + Pj:‘6lj

+

Pz(i’6jk

+

+ P1(3)8& + Pj:‘&jl]

P::‘6ij

- nI(X)

(6ijskl

+

6idZj

+

(A3.41) 6iZ6,fk),

in (2.29d). We have carried out a reduction of the truncated 20 ordinary differential equations in an analogous manner to that used above for the IO-moment equations. This time we are finally left with six instead of two equations. We have obtained the asymptotic solution of these six equations for ) w / >> 1. APPENDIS LONGITUDIXAL

4

MAGNETORESISTAPXE

We here show that our one-dimensional model does not exhibit a longitudinal magnetoresistance. We have a = (U(X), 0,O j and there are two cases to consider. Case I. Here E = (E, 0, 0) and H = (X, 0, 0). Then, from (2.51) with 12 = q~/wrc, the transport equation is

220

FRISCH

f to first order in

We perturb

AND

MORRISON

E as in (2.19), which leads to

3fo+ so(x) 2 + D w g u-y& with solution

>

+ ; [fo - no(x>fnnl= 0,

(A4.2)

f. = no(z)fM as in (2.23), and

ml(z) fz - @ 9 - al(x) T nl au

af,

1

(A4.3)

au *

If we set fi = no(x)fM

e Eu - f #I(X) m

1

(A4.4)

+ G ,

then (A4.3) becomes, using (2.21) and (2.23),

+cf 7 exp[-$(u2

+ I? + w’)lG du dv dw

1

(A4.5)

.

We have also used (2.35) which, from (A4.4), is seen to hold. Thus $1(r) is again determined by (2.36). We merely observe that the solution G = G(x, u) for D = 0 is still a solution when Q # 0, and hence there is no magnetoresistance. Case II. Here E = (0, E, 0) and H = (0, X, 0). Then, from (2.5), the transport equation is u$+u(x)~+o

Perturbing

u&

w$

+;[f

-n(z)f.+,]

(A4.6)

= -q;$.

f to first order in E leads to

u g + so(x) $; + Q (u 2

- w

+ ; [fo - no(x>fd

= 0,

(A4.7)

with solution fo = no(x)fM as before, and

1

(A4.8)

nl(z) & - SE af, - (Q(x) 36 7 m av au * From (2.21) and (2.23)) A = no(x)fM [I;

Eu - ; h(x)]

(A4.9)

HIGH

FIELD

221

MAGNETORESISTANCE

is a solution of (A4.8)) independently is no magnetoresistance.

of the value of L?.Hence, once again, there

ACKNOWLEDGMENT

this

The authors work.

RECEIVED:

wish

to thank

Dr.

C. Herring

for

his helpful

discussions

and

interest

in

June 13, 1963 REFERENCES

J. Appl. Phys. 31,1939 (1960). This basic paper also contains an adequate review of the previous literature on these problems. H. BROOKS, ddvan. Electron. Electron Phys. 7, 85 (1955). H. L. FRISCH AND J. L. LEBOWITZ, Phys. Rev. 123,1542 (1961). S. YOSHIKAWA AND D. J. ROSE, Phys. Fluids 6,334 (1962), and a subsequent paper b) the same authors of the same title to appear shortly. S. YoSHIKtaW.4, Phys. Fluids 6, 1272 (1962). P. L. BHATNAGER, E. P. GROSS, AND M. KROOK, Phys. Rev. 94, 511 (1954). H. GRAD, “Handbuch der Physik,” vol. 12, p. 205 Springer, Berlin, 1958. J. L. LEB~WITZ, H. L. FRISCH, AND E. HELFAND, Phys. Fluids 3,325 (1960). R. BEIZVIAN, “Stability Theory of Differential Equations,” p. 138, Ex. 6. McGraw-Hill, New York, 1953. P. I)RI-DE, -inn. Physik (4th Ser.) 1, 575 (1900).

1. C. HERRINQ,

2. 3.

4. 5. 6.

7. 8. 9.

/O.