Variational principle for transport phenomena

VARIATIONAL PRINCIPLE

FOR TRANSPORT

PHENOMENA

BY

G. E. TAUBER 1 SUMMARY

The role of the variational principle in transport theory and its application to the calculation of the transport coefficients are described. A generalized variational principle including the effect of an external magnetic field and lattice vibrations is discussed. The variational principle for inter-band scattering is derived, and is generalized by the introduction of an external magnetic field and lattice vibrations, leading to the most general variational principle including inter-band scattering, in addition to an external magnetic field and lattice vibrations, or any combination of these. The transport coefficients are expressed in terms of energy integrals, which for the various cases take the form of ratios of infinite determinants, the number and form of their matrix elements depending on the interactions. A formal solution is given by the Ritz method for the different cases of interest. I. INTRODUCTION

M a n y p h y s i c a l p r o b l e m s can be f o r m u l a t e d in t e r m s of a v a r i a t i o n a l principle, t h e s o l u t i o n of w h i c h is o f t e n s i m p l e r a n d m o r e a c c u r a t e t h a n t h e original f o r m u l a t i o n . T h i s is d u e t o t h e f a c t t h a t t h e v a r i a t i o n a l f u n c t i o n is n o t as s e n s i t i v e t o f l u c t u a t i o n s a n d e r r o r s as t h e d i f f e r e n t i a l (or i n t e g r a l ) e q u a t i o n o b t a i n e d u p o n v a r i a t i o n . F o r p r a c t i c a l calcul a t i o n s use is m a d e of t h e R i t z v a r i a t i o n a l m e t h o d in w h i c h t h e u n k n o w n f u n c t i o n is e x p a n d e d as a n infinite series w i t h coefficients d e t e r m i n e d b y t h e v a r i a t i o n a l m e t h o d (1). 2 In t h e case of t r a n s p o r t p h e n o m e n a t h e f u n d a m e n t a l e q u a t i o n determining the distribution function, and thus the various transport p h e n o m e n a , is a n i n t e g r o - d i f f e r e n t i a l e q u a t i o n k n o w n as t h e B o l t z m a n n e q u a t i o n (2). W e can w r i t e this e q u a t i o n as a n o p e r a t o r e q u a t i o n of the form L(F) = X (1) w h e r e L r e p r e s e n t s t h e i n t e g r a l o p e r a t o r d u e t o t h e c h a n g e of t h e distrib u t i o n f u n c t i o n b y collisions, a n d X t h e force d u e t o t h e e x t e r n a l field and temperature gradients. I t has b e e n s h o w n b y K o h l e r (3) t h a t t h i s e q u a t i o n is e q u i v a l e n t t o a v a r i a t i o n a l principle, for w h i c h t h e v a r i a t i o n a l f u n c t i o n is s i m p l y t h e i n n e r p r o d u c t (F;

F) = f FL(F)dk

(2)

x Department of Physics, Western Reserve University, Cleveland, Ohio; and The Franklin Institute Laboratories, Philadelphia, Pa. The boldface numbers in parentheses refer to the references appended to this paper. I75

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subject to the subsidiary condition (F, X ) = f

F X d k = (F; F) on account of (1).

(3)

It has been pointed out by Ziman (4) that the variational function (F; F) is not simply a mathematical device, but of physical interest as it is proportional to the rate of entropy production by collision processes, and thus affords a connection between microscopic statistical theory and irreversible thermodynamics. The positive definiteness of that function based on microscopic reversibility of the operator L assures us also that the Onsager reciprocity relations (S) among the various transport quantities are always satisfied (6). If we now expand the function F in a series of the form F =

E arer

(4)

r

where ~ is the energy, variation with respect to the coefficients ar will result in a system of linear equations for these quantities ar = Z d~.a~

(5)

where the coefficients a, and dr, can be evaluated from a knowledge of the form of L and X. The various transport coefficients, such as electrical and heat conductivity, can then be obtained in terms of energy integrals K which turn out to be just the subsidiary conditions (3). It has been shown by Enskog (7) that these energy integrals can be expressed in terms of infinite determinants, the elements of which can be evaluated for a given energy-momentum dependence. If deviations from equilibrium of the phonon distribution function are also taken into account this results in a system of coupled integral equations, which in operator form are given by X = L~(F) -k- L2(N) Y = L3(F) + L4(N).

(6)

Here N represents the phonon distribution function, in addition to the previously discussed carrier distribution function F. Furthermore, in addition to the operator L1 = L (of 1) there also appear the operators L2, L3 and L4 describing the interaction of the electrons with the phonons and phonons with phonons. A variational approach to this problem has been suggested by Ziman (4), and recently Dorn (8) obtained a variational principle and applied it to include the effect of longitudinal polarized vibrations for spherical energy surfaces. The

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V A R I A T I O N A L P R I N C I P L E IN TRANSPORT T H E O R Y

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variational function is given by the sum of terms of the form (2) (F; F)t + (F; N)2 + (N; F)3 + (N; N)4

(7)

where the subscripts i = 1 to 4 refer to the appropriate operator L~. The symmetry relations amongst the integral operators assure us of the positive definiteness of the variational function (7) and the existence of the Kelvin relations amongst the transport coefficients. If the lattice distribution function N is also expanded in terms of an infinite series analogous to the corresponding one for F (4) an application of the Ritz variational method will then again result in a system of linear equations for the unknown coefficients. The energy integrals are again given in terms of infinite determinants, the elements of which now depend on a knowledge of the operators L, (i = 1 to 4) and the functions X and Y. In the presence of an external magnetic field the above theory has to be modified. The Boltzmann equation for the carrier distribution function acquires an extra term, which represents the effect of the "magnetic scattering." Taking the symmetry property of the magnetic scattering operator under reversal of the field into account, it is still possible to obtain the Kelvin relations for that case (9). Although the magnetic field does not contribute to the entropy production it is possible to formulate a variational principle (10) which leads to the correct transport equation. Its solution is again achieved by an application of the Ritz method, except that two separate expansions for the conjugate sets F + and F- must be used, depending on the direction of the magnetic field H and - H . We have shown (11) how a generalized variational principle can be found, which is valid for an arbitrary direction of the electric field and polarization of the lattice vibrations in the presence of an external magnetic field, and does not depend on any special form of the energy surface. The various transport coefficients, both for thermoelectric and thermomagnetic phenomena, are then obtained by the Ritz method in terms of infinite determinants without requiring an explicit solution of the transport equation. In some cases of physical interest, notably in semiconductors, one has more than one energy surface, and it becomes necessary to consider the effect of transitions between bands. The appropriate operator equation is then given by Z L,.,~(F") = X , . n

(8)

where the subscripts m , n refer to the various bands. The variational function for this case turns out to be a sum of terms of the type given earlier (2) (F; F) ~- Z Z (F"; F-) (9) ?n n

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subject to the subsidiary conditions (F, X) = Z (F ~, Xm) = (F; F) on account of (8).

(10)

This variational principle can again be solved by an application of the Ritz m e t h o d where the coefficients ar (of 4) now acquire an extra subscript indicating the band m and become a ~ . Carrying out the variation with respect to these parameters one again obtains a set of linear simultaneous equations for these unknowns of the form (5), the difference being the appearance of extra indices in the quantities ar and &8 indicating the bands. The current density J and heat current density W m a y be considered as a sum over bands over the appropriate expressions for the individual bands J = EJm

and

W = EWm-

(11)

The transport coefficients are again expressed in terms of energy integrals K which can be obtained from infinite d e t e r m i n a n t s without the necessity of explicitly solving the system of equations for a~r, the only difference being t h a t now there appear more matrix elements including the contributions from all different bands m, n. A further generalization of the theory can be achieved by considering the effect of an external magnetic field o r / a n d the presence of lattice vibrations for this case. If the non-equilibrium phonon distribution function is taken into account a system of coupled equations of the form (6) is obtained X , . = Z L,,,,,'(F") + • L.~.~(N) n

Y=

n

(12)

~ L , . . 3 ( F -) + ~ ] L . . ? ( N ) m,n

m,n

and a variational principle can again be formulated, the variational function being of the type (7) where each term is now a sum over bands of the type (9). The effect of an external magnetic field again introduces an extra scattering operator L' into the formalism and produces a separation of the solution into two conjugate sets depending on the direction of the magnetic field H. The energy integrals, and hence the transport coefficients, can again be obtained by the Ritz m e t h o d and expressed in terms of infinite determinants, containing now m a n y more matrix elements describing the effect of the interaction of the electrons with the lattice, phonon-phonon interaction, and the various bands. The purpose of this paper is to give as complete a discussion as is possible of the variational principle in transport theory and its application to the calculations of the various transport coefficients. Starting

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VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

179

with the derivation of the variational principle developed by Kohler a we include the effect of an external magnetic field or lattice vibrations, and finally combine both these effects in a discussion of the generalized variational principle. The variational principle for inter-band scattering is set up next, and is afterwards generalized by the introduction of an external magnetic field and lattice vibrations, so t h a t we finally obtain a variational principle which includes inter-band scattering, in addition to an external magnetic field and lattice vibrations. In order to present a systematic presentation, Section II is devoted to the Boltzmann equation and contains a discussion of the various cases, viz. effect of magnetic field, lattice vibrations and inter-band scattering, and a combination of any or all of these. In Section IIl, the appropriate variational principles are derived and their solution indicated by an application of the Ritz method. Finally, in Section IV, starting from a general expression of the transport coefficients in terms of energy integrals, it is shown how these can be expressed in terms of infinite determinants, which differ from each other in the n u m b e r and nature of their matrix elements. In order to evaluate these matrix elements, the form of the integral operators must be known, together with the dependence of energy on m o m e n t u m . Work along these lines is in progress, and has been reported elsewhere (12), while here we have only been concerned with presenting a systematic discussion of the variational principle applied to transport p h e n o m e n a and in particular, expressing the various energy integrals necessary for the calculation of the transport coefficients, in terms of infinite d e t e r m i n a n t s for the various cases of interest. II. BOLTZMANTRANSPORTEQUATION

1. Nature of Transport EQuation For a good analysis of transport p h e n o m e n a it is necessary to determine the distribution of the particles both in regard to position and velocities. The fundamental equation determining the distribution function f(v, r) is an integro-differential equation known as the Boltzm a n n equation, which describes the probability of a particle being in a given state, and is therefore based on the one-electron model. It is, however, quite simple to generalize the distribution function, and hence its change with time to a n-dimensional phase space in which it is now a function of the co-ordinates and m o m e n t a of the n particles. Recently, several authors have adopted the approach of the density matrix in q u a n t u m statistics to derive the transport equation from general statistical consideration. However, for the present presentation, we will assume the validity of the standard theory and postpone a more rigorous derivation of the transport equation to a future discussion. 3 L o c . cit.

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If the number of particles which lie in an element of dr of configuration space and in dk of k-space at a time t is given by (4~r3)-~f(k, r, t)dkdr

(12)

then at time t + dt according to Liouville's theorem they will lie in an equal volume of six dimensional phase space about k + kdt, r + vdt. The difference, therefore, between f ( k + kdt, r + vdt, t + dt) and f ( k , r, t) must be due to the particles scattered into the new positions in the time dt. The total change of the distribution function f due to collisions with the lattice (and each other) and external fields must be zero, so that (Of/Ot)co~. + (Of/Ot),~e~a~ = 0 (1,3) where (Of/Ot)oon. represents the change due to collisions and (Of/Ot) ~ie~aB the change due to external fields, which is given by

df/dt = Of/Ot + k. Vkf + V" vrf.

(14)

Here hk is the change of the impulse and equal to the external force

hk = eE + e(v/c) X I-I

(15)

where E denotes the electric field, I-I the magnetic field, e the charge and c the velocity of light. The velocity of the electron v is expressed in terms of its wave vector k through ¢~v = Vk~(kx, kv, k,).

(16)

(In the steady state one also has Of/at = 0.)

a. Collision Operator Since binary collisions of electrons are neglected, the number of particles per unit volume with velocities in the range dv which have their velocities changed to lie in dv ~ in time dt must be proportional to f(v, r, t) and to the interaction W(v, v', r), which is independent of f and only depends on the interaction mechanism. Similarly, the number of particles which are scattered into the range dv are proportional to f(v', r, t) and the interaction W(v', v, r). The net difference between these two quantities, integrated over dr', determines (Of/Ot)col~., so that

(Of/or)con. -- f(v', r, t)W(v', v, r) - f(v, r, t)W(v, v', t)dv'.

(17)

The equilibrium distribution function f0 must be such that (Ofo/Ot) Corr.= 0, and thus W(v, v', r ) / W ( v ' , v, r) = fo(v', r)/f0(v, r).

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Now according to classical mechanics, f0 is a function of the energy only, and in fact, is the Maxwell-Boltzmann function. Also, the collisions are supposed to be elastic, so that energy is conserved and f0(v, r) - f0(v', r). Hence W(v, v', r) = W(v', v, r) and

(Of/Ot)ooU. = f

[-f(v', r, t) - - f ( v , r, t)]W(v, v', r)dv'.

(18)

In the case of electrons, Pauli's exclusion principle has to be taken into account. We describe the states of the electrons by the wave vector k (rather than the velocity v) and suppose the ~ (k, k') is the probability t h a t an electron makes a transition from the state k to the state k'. As a transition can only occur if the corresponding state is unoccupied, the number of electrons, per unit volume, gained by dk from dk' in time dt is W(k', k) f (k') [-1 - f(k)]dkdk'dt and the number of electrons lost from dk to dk' is W(k, k ' ) f ( k ) [-1 -- f(k')-]dkdk'dt so that the net change in f is given by

(Of/Ot) °0,. = f W(k', k)f(k') [-1 - f ( k ) ] - W(k, k')f(k) [-1 - f ( k ' ) ~dk'. (19) d

This expression must vanish in the equilibrium state, where there exists detailed balancing and therefore V(k, k') = W(k, k')f0(1 - f0') = W(k', k)f0'(1 - f0) = V(k, k')

(20)

in which we have simplified the notation by writing fo' to denote f0(k'), and which leads to the Fermi function f0 = [exp B(e -- f) + 1-]-1 ,

¢/ = 1/kT.

(21)

In problems of conductivity we are interested, primarily, in very small displacement from the equilibrium state. Thus we are justified in writing f = fo + fl = fo - OOfo/oe = fo + Of 0(1 - f0)5 (22) which follows from the form of the Fermi function f0 (22). Hence, if terms involving @ are neglected, which is equivalent to neglecting squares and products of the electric fields and temperature gradients, (19) becomes

(Of/Ot)co11. = fl f

V(k, k')[-Q(k) - Q(k')-]dk'.

(23)

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b. Operator Form of Transport Equation The transport equation is then obtained by equating (23) describing the effect of the collisions with the lattice to (14) due to the external field. From (21) one also obtains the relations Vkfo = vk~Ofo/O~ = hvOfo/O~ Vrf 0 =

afo/aT =

(241)

(Of#aT) v~T + (Ofo/O~,)v ~ ~ -1" ~ (af0/a

),

af0/a

= - af0/a

.

(25)

With the help of these relations, one can write (14) in the steady state up to first orders in f df/dt = - Ofo/Oe X viEeE, + TO/Ox,(~/T) + e/T(OT/axO~ i=1

e

+ (0f0/Oe) hc (v X tI) VkQ.

(26)

The form of this expression suggests the separation of the equation into two parts, one being proportional to ~ and one independent of it. Thus, if we write for Q = Z {[-eE,: + TO/Ox,.(~/T)-]f~. + (1/'T)(OT/Oxi)f~},

(27)

i

equations (26) (with H = 0) and (23) can be replaced by L(f~,) = v,Ofo/O~ = Xl,

(27a)

L(f~.O = v~Ofo/O~ = x2,

(27b)

where L is the integral operator defined by (23)

L(F) -- 13 f V(k, k ' ) [ F ( k ) - F(k')~dk'.

(28)

The symmetry property (20) leads to the symmetry relation ( F ; G ) = (G; F)

(29)

where we have defined the inner product (F; G) = fl f

d

F(k)L(G)dk.

(30)

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The truth of this result can be seen by writing (F;

G) = ~ f f

on account of (20). (F;G)

= ~

G(k')~dkdk'

V(k, k')F(k)[-G(k) -

Hence, by addition

ff

V(k, k')[-F(k)

-

F(k')][a(k)

-

G(k')]dkdk'

which is symmetrical in F and G and hence establishes the relation (29).

c. Effect of the Magnetic Field In the presence of an external magnetic field Eqs. (27) acquire an extra term arising from the last term in (26) and which can be taken into account by adding to the operator L the operator L'

L'(O) = (e/hc)(Ofo/Oe)(v ×

H) vkQ.

(31)

The operator L' is not symmetric, but satisfies the relation

f FL'(G)dk = - f GL'(F)dk

(32)

and, on reversal of the magnetic field, H,

f FL+'(G)dk = f GL_'(F)dk

(33)

where L+' and L_ p denote the operator L t with + H and - H respectively in the expression (31). These results follow from the fact that the integral f

v-[-(v X

H)FGOfo/O~dk

vanishes. It is also to be noted that both v and H are solenoidal. From (33) it is seen that the solutions of the transport equation depend on the direction of the magnetic field. On reversing the magnetic field the electrons will curve in the opposite direction and the condition of microscopic reversibility (20) then becomes V+(k, k') = V-(k', k).

(34)

It is thus possible to divide the set of solutionsfl~ and f ~ into conjugate

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sets fj,+ and f j , - (j = 1, 2) for which we have analogous to (27) the following system of equations x~,+ = L ( f , , +) + L + ' ( f , , +)

(35a)

x2,+ = L(f2, +) + L+'(f2, +)

(35b)

together with the corresponding ones for a magnetic field in the opposite direction, which are obtained from (35) by changing all + into quantities. The system of equations (27) or (35) can be obtained by means of a variational principle, which will be discussed in Section III. 2. The Lattice Distribution Function The transition probability, W(k', k), depends mainly on the interaction of the electrons with the lattice. If n,; denotes the distribution function of the lattice vibration, where q describes the state of the lattice vibrations, j the polarization (j = 1, 2 for transverse and j = 3 for longitudinal) the transition probability for a transition k --* k' = k + q with absorption of a lattice q u a n t u m is of the form W ( k , k') =

V ( k , k ' ) n q * ( , k -- *k' +

(36)

and for a transition k ~ k' --- k - q with emission of a lattice q u a n t u m W ( k , k') =

V(k, k')(nq +

-

*k' -- h q)

(37)

where the Dirac S-functions ensure t h a t energy is conserved in the transition. In other words, (36) vanishes unless ~k -- ~k' + huq = 0, and (37) vanishes unless ** - ~k' -- huq = 0. Here Vq is the frequency of the lattice wave in the state q. If the effect of lattice vibrations is neglected, nq m a y be replaced by its equilibrium value n0(q) = (e* - 1)-',

y = hv/kT.

(38)

However, in the general case considered here, both f and nqj depart from their equilibrium value, and in addition to (12), one has a transport equation for nq 3

(Onq/Ot) oo,l. = X U,(Ono/OT) (OT/Ox,) i=1

(39)

where U~ are the c o m p o n e n t s of the lattice wave velocity. Hence, the two functions f and nq m u s t be determined simultaneously as the solutions of the two Boltzmann equations (12) and (39).

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a. Collision Operators

For the two collision operators (Of/Ot)~oll. and (On/Ot)oo~., respectively, one obtains (taking into account the exclusion principle and the relations (36) and (37) and the reverse transitions k' --* k) : (of/ot)

P ~on.= J {E V(k', k)fk, (1 --fk) (nq+ 1) - V(k, k')fk (1 --fk')nq] X ~ (~k -- Ek'-~-hPq) + [ V ( k t) k)fk, (1 --fk)nq -- V(k, k')fk(1 - - f k , ) ( n q + l ) ~ 8 ( e k - - e k , - - h U q ) l a r k

(0n/0*)coll.

=/[V(k/) k)fk, (1 --fk) (nq-}- 1) --

V(k,

(40)

k')fk(1 --fk,)nq~ (41)

In the equilibrium state, these expressions must vanish. Hence, putting f = f0 and nq = no, and using the energy equation e' = e--b-hv we must have V(k', k)f0(~ + hp)[1 - fo(~)](no + 1) = V(k, k')f0(e)E1 - f0(~ + hv)-]

(42)

V(k', k)f0(~ - hv)[1 - fo(~)~no = V(k, k')f0(~)E1 - fo(~ - h~,)](no + 1).

(43)

Both these restrictions are satisfied if V ( k , k') satisfies the symmetry relation (20) which expresses the principle of microscopic reversibility for the present problem and forms the basis of the Kelvin relations to be discussed later. If we now expand nq; in the same manner as has been done f o r f n , i = no - Ni~Ono/Oy = no + flno(no + 1)Ni

the exact expressions (33) and (34) may be simplified. terms linear in Q and N i we obtain ( o f / o t ) oo,1. =

Keeping only

P /3J V(k, k') In0 (q)f0(k) (1 --f0 (k'))~8 (ek- ~k' "~-hPq) X [O (k') -- O (k) -- N (q) 3 + no (q) f0 (k') (1 - f 0 (k)) X a ( ~ , - ¢k,-hrq)[O(k') -O(k)-l-N(q)~]dk

(On/Ot) oo.. =

(44)

(45)

/3( V(k, k')-0 (q) [1 - f 0 (k')-]3 ({~k-- Ek'--~hVq) x[Q(k')--Q(k)--N(q)~tk

(46)

where the left-hand side of these expressions is given by (84) and (88), respectively. They are now a system of integro-differential equations for the two quantities Q and N;.

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b. Operator Form

The system of equations, (45) and (46) can be written in operator form xl~ = viOfo/OE = Ll(fx~) -J-L2(gl0 (47a) Yl, = 0 = L , ( f ~ ) + L4(g~0

(47b)

x2~ = Ev,Ofo/O~ = L,(f2i) + L2(g2i)

(47c)

y2~ = -

(47d)

U~TOno/OT = L3(f20 + L4(g~)

where the function N has been separated into two parts similar to Q (cf. 27) N = ~ {EeE~ + TO/Oxi(~/T)]g,~: + ( 1 / T O T / O x O g ~ } .

(48)

i

The integral operators L~ (L = 1 to 4) are given by LI(F) = 13 f

V(k, k')[S(k, k') + S(k', k ) ] [ F ( k ) - F(k')-ldk'

(49a)

L2(F) = 13 f

V(k, k')gS(k, k')F(q) - S(k', k)F(q)]dk'

(49b)

L.~(F) = 13

f

L4(F) = 13f

v ( k , k')S(k, k')[-F(k)

F(k')-]dk

(49c) (49d)

V(k, k')S(k, k')F(q)dk

(It is important to note that q = k' - k is to be kept constant in (49c) and (49d) which are therefore functions of q.) S(k, k') is given by S(k, k') = n0(q)f0(k)~l -- f 0 ( k ' ) ~ 6 ( ~ k -- ek, + h~q)

(50)

corresponding to an absorption of a lattice quantum k --+ k' = k + q, (36), while S(k', k) corresponds to the emission of a lattice quantum k -+ k' = k - q and has the transition probability given by (37). For compactness of notation, let us now introduce the generalized inner product analogue to (30) (F; G)~ = f

F(k)L~(G)dk

i -- 1, 2, 3, 4

where the subscript i indicates the operator L~ defined by (49). symmetry property (20) leads to the symmetry relations

(51) The

(F; G), = (G; F),

(52a)

(F; a)2 = (G; F)3

(52b)

(F; G)~ = (G; F)~.

(S2c)

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18 7

The property (52a) follows from (29), while (52c) follows at once from the form of L4(F) (49d). To establish (52b) we write

(F;a)2=f F(k)L2(G)dk

f f V(k,k')F(k)ES(k,k')G(q)-S(k,k')G(q) dkdk' f f V(k,k')ES(k,k')F(k)G(q)-S(k, k')F(k')G(q)3dkdk' since V(k, k') = V(k', k).

= ff

Hence

V(k, k')S(k, k')l-F(k) -- F(k')~G(q)dkdk'

= ~ f G(q)L(F)dk' = (G; F)3 which completes the'proof.

c. Effect of a Magnetic Field As in the case without lattice vibrations taken into account, the presence of an external magnetic field introduces an extra term in the Boltzmann equations (47) which again can be represented by an operator L' defined by (31) and possessing the properties (32) and (33). T h e modified equations are then given by

xl, + = L , ( f l , +) + L2(g,,) + L+'(f,~ +) y,~ = L3(fl, +) + L4(gl,) x2,+ = Lx(f2, +) + L2(g2,) + L+'(f2, +) y2, = L3(f2, +) + L,(g2,)

(53a) (53b)

(53c) (53d)

with corresponding ones for a magnetic field in the opposite direction, which are again obtained from (53) by changing all + to - quantities. As L p does not operate on N, it is not necessary to separate g1¢ and g~.i into two conjugate sets, as m u s t be done for fie and f2~. T h e solution of this more complicated system of equations can also be obtained by a variational principle to be discussed in Section III.

3. Interband Scattering In some cases of physical interest, notably in semiconductors, one has more t h a n one energy surface, and it becomes necessary to consider the effect of transitions between bands. Before generalizing (27) to a

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multiple band model, we note t h a t the operator equations (27) can be written as X~ = E L ~ , J k , (54) k'

showing the connection between the forces Xk and currents Jk. From the properties of L(Q) we can also infer that the coefficients Lkk' form a positive definite symmetric matrix, with the s y m m e t r y relations Lkk, = Lk'k. Indeed, this last result forms the basis of the Kelvin relations amongst the thermoelectric coefficients to be discussed later. The s u m m a t i o n in (54) is to be carried out over all the states k. In order to take into account the other bands we simply extend the summation in (54) over the appropriate states. T h u s the system of equations (27) has to be replaced by Z Lm.(f,~") = v~Ofo/Oe,~ = x~ '~

(55a)

n

F. L,..(f2~") = v~e,.Ofo/OE,. = x.ox"

(55b)

n

where the operator Lm~ is defined by Lm.(F) = # f

V(k,~, k . ) [ F ( k , . ) -- F ( k . ) ] d k .

(56)

and m and n denote the possible energy bands. T h u s the effect of a scattering from band m to n is taken into account by the operator L~, acting on the various states F', = F(k,,) and producing a coupling between them. The s y m m e t r y property (20) in our present notation becomes

V(k.,, k.) = r(k.,, kin)

(57)

and again leads to the s y m m e t r y relation (F; G) = (G; F),

(29')

where., however, now the inner product (F; G) is a sum of terms of the type (F ~ ; G") such t h a t

(F; a) =

(58)

where we defined the product (F m; G ") by

(Fro; a-) = ]" F L .(a")dkm. ,]

(5%)

Sept., 1959.]

VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

18 9

The proof of the generalized s y m m e t r y relation is obtained by writing again

(F;C)=ex f f

V(k., k . ) F ( k ~ ) r G ( k ~ ) - G(k.)3dk,.dk.

(59b)

V(k~, k.)F(k~)EG(k~ ) - G(k.)~dk,.dk.

(59c)

m,n

m,rt

on account of (57).

Hence by addition V(k~, k~)[-F(k~) - F(k.)~ X [-G(k~) - G(k.)~dk,~dk.

(59d)

which is symmetrical in F and G, as for each term (F ~ ; G") in the sum we have a corresponding one (F"; G'), and thus establishes the t r u t h of (29').

a. Effect of the Magnetic Field In the presence of an external magnetic field, the equations (55) acquire an extra term, or rather a series of terms, of the type (31). T h u s the new modified operator equations are now given by x~,'~+ = ~ L=~(f,, n + ) + L m ' ( f J ~)

(60a)

n

x2,,~+ = Y:

L,~.(f2, "+) + L,,,'(f2, ~+)

(6oh)

n

where the + and - signs again denote the direction of the magnetic field, and where the sets of functions f l ? and f2~" are now divided into conjugate sets fl~± and f~"~ as before, and the operator L~' defined by

L.,'(F '~) = e/~c Ofo/O~.,(v., X I-I) Vk~F~. It should be noted t h a t the operator L~' describes the effect of the magnetic field on the band in question, and therefore only has a free index m, characteristic of the particular band.

b. Presence of Lattice Distribution Function We have seen t h a t the presence of a non-equilibrium lattice distribution function modifies the transport equation in two ways. First, the lattice distribution function satisfies a Boltzmann equation of the type (39) and secondly, the original Boltzmann equation for the carried distribution function acquires additional terms. T h e transition from band m to n is characterized by the emission (or absorption) of a lattice

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q u a n t u m according to k . = k~ -4- q, and hence the i n t e r b a n d scattering can be t a k e n into account formally by replacing k and k' in the expressions for the integral operators L¢ (49) by km and k., respectively. We note also t h a t in the case of La and L4 s u m m a t i o n m u s t be carried out over m and n, as the lattice function depends on q, and is therefore i n d e p e n d e n t of m and n. T h e system of equations (47) is then replaced analogous to (55) by the following system of operator equations X

~im =

y~.

Lm,~ P (f~i ~ ) 4- Y~ L~,~2(gli)

n

~- w~, J 4- Z L~n4(g~, ) mn

x "

(61a)

n

(61b)

mn

L

'{¢ '~

n

mn

yo./~ = E Lm. (f2~ ) 4- Y~ L.,,,4(g~O 3

*,tin

n

(61d)

ran

where the integral operators Lm~~ (i = 1 to 4) are now given by L~.~(F ,,) = 5

L.J(F-)

= B

f V(km, ko)ES(k~, k.) + S(k~, k~)] X [-F(k~) -- F(k.)3dk.

(62a)

V(k~, k.)[-S(k.,, k.)G(q) - S(k., k~)G(q)3dk.

(62b)

f V(k~, k.)S(k.,, k.)~F(km) -

Lm,,4(G) = 13 f

F(k.)-]dk..dk.

V(km, k.)S(k~, k.)G (q)dk~dk.

(62c) (62d)

with S(km, k.) = no(q)fo(kra)E1 - f0(k.)~6(em - ~. 4- hvq)

(63)

for an absorption of a lattice q u a n t u m k~ ~ k . = k~ 4- q, and S ( k . , k~) corresponding to the emission of a lattice q u a n t u m according to km --+ k . =kmq. We note t h a t the operators L ~ J and Lm. 4 have to be s u m m e d over both indices m and n, as the lattice distribution functions gl~ and g2~ are i n d e p e n d e n t of the interband interaction. This can also be seen by inspection of their operator forms, (59c) and (59d), which are integrated over both dk., and dk.. T h e appropriate generalized inner products analogous to (51) are now defined as (F;G)~ = ]E (F";G").,/ i = 1,3 (64a) (F;G)/

= Z (Fro; G~jm. ~' mn

/ = 2, 4

(64b)

Sept., I959.]

VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

I9I

with (F ~ ; G - ) ~ . ~ = f

(F';~).."

=f

F(k,.)L,..'(G")dk,.

i = 1, 3

(65a)

F(k,.)L,..'(G)dk.,

i = 2, 4.

(65b)

T h e need for differentiating between the two cases a and b arises from the fact t h a t in the second case the functions G represent the lattice distribution function and are independent of the band, as can be seen by inspection of the corresponding operators L 2 and L 4 (62b) and (62c). T h e superscript i gain indicates the operator L *, while the subscript mn shows the bands. T h e s y m m e t r y property (57) again leads to s y m m e t r y relations of the type (52). (F; G)I = (C; F)I (52~') (V; G)~ = (G; F)a

(52b')

(F; G), = (G; F),

(52c')

the first of which follows from (29') while the last can be reduced from the form of L 4. To establish (52b') we write again

(F; G)2 =

.Ef

F(k,,,)L,.,3(G)dk,.

= ~ ..z f f V(k~, k.)F(k~) X [-S(k.. k.)G(q) - S(k.,

k.~)G(q)3dk,.dk.

= ~ .. z f f V(k~, k.) X [-S(k,., k.) F(k,.)G (q) - S(k,., k.) F(k.)G (q)-]dk,,,dk. since V(k.., k.) = V(k,,, kin). (F;G)2 = ~

ff

=, f f

Hence

V(k,., k.)S(k,., k.)G(q)[F(k,.) - F(k.)-]dk.,dk. G(q)L,..a(F)dk = (G; F)a

which completes the proof. In the case of i = 2 and 4, the functions G do not have a superscript, as L~. operates on the lattice distribution function in these instances which follows from the definition of the appropriate operators (62b) and (62d). Finally, if in addition to non-equilibrium lattice vibrations we also have an external magnetic field as described in (a) of this section, the

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Boltzmann equations (61) will acquire extra terms of the type (57) and result in conjugate sets of solutionsf~ "~ and f2~"±, do not affect the functions gl~ and g2¢, as mentioned before. The modified system of equations is then given by x~,"+ --- E L,..~(f~# ~+) + 52 Lm~O'(g,¢) + L,,,'(f~ =+) n

(66a)

n

0 = E L,..3(f,, "+) + Z Lm.4(g~) $,n,n

(66b)

mr,

x~, "+ = Z L,~.*(f2, -'+) + E n

L

o .,,~-(g~) + Lm'(f~? "+)

(66c)

r~

y.o~ = ~ L,.,~3 ( f , ~n +)

+ y L,,~,,4(g2~)

(66d)

with corresponding ones for a magnetic field in the opposite direction, which are again obtained from (66) by changing all + to - quantities. The solution of the various systems of equations with increasing complexity, (55, 60, 61 and 66) can also be obtained by a variational principle which will be described in Section III. m. V*~TmN*L PR~NC~PL~

1. ~riational Principle for Distribution Function It has been shown by Kohler that the Boltzmann equation for the particle distribution function (cf. 27) is equivalent to a variational principle, which preserves the inherent symmetry properties. An application of the Ritz method enables one then to obtain a series solution for the distribution function and the transport coefficients in terms of infinite determinants, which can be evaluated to any required degree of accuracy. Consider an equation of the form

L(F) = X

(67)

where L is an integral operator of the type (28) with positive definite kernel V(k, k'), and X a known function, such as given by (27). It can be shown that (67) can be replaced by a variational principle. For compactness of notation, let us introduce in addition to the inner product (F; G) defined previously (30)

(F; O) = f FL(G)dk,

(68)

(F, X) = f FXdk.

(69)

the product

Sept., I959.]

193

VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

Then, if F is a solution of (67) and G any function which satisfies the relation (G; G) = (G, X) (70) then of all functions G satisfying (70), F is the one which makes (G ; G) a m a x i m u m . The proof of this s t a t e m e n t depends on the relation (F; G) = (G ; F) and (G ; G) > 0 which are consequences of the symm e t r y property (20) and the fact t h a t V(k, k') is essentially positive definite. These relations have been derived in the last section, (29 and 29'). To prove the variational principle we now proceed as follows. If we multiply (67) by G and integrate over k-space, we find ( a ; F) = (G, X) = (G;C~) on account of (70).

F r o m (29') we have ( F - G ;

0 _< ( F - G ; F - - G )

(F-G;F-G)

(71)

= (F;F)--

(F;G) -

F-G)

>_0.

But

((7;F) + (G;G)

= ( F ; F ) - 2 ( G ; F ) -t- (G;G)

on account of (29)

= (F; F) -

on account of (29')

(G; G)

so t h a t finally (F; F) >_ (G;G)

(72)

which proves our assertion. T h u s our variational function is the inner product (F; F) subject to the subsidiary condition (F; F) -- (F, X).

(73)

Consider now the new variational function H = (F; F) -t- X(F, X) where ~ is a Lagrangian multiplier. results in

(74)

Variation of H with respect to ~F

6H = f dFE2L(F ) + XX-]dk = 0 J

which is equivalent to (67) provided we set ), = - 2. Applying these results to the system of equations (27) we introduce the variational functions Hz (l = 1, 2) (74) H1 = (fl~; f~j) + ~,1(f1,, xli)

(74a)

H2 = (f~,; f2j) + X2(f2,, x~j)

(74b)

which on variation with respect to fl~ and f2i respectively, lead to the system of equations (27) provided we set kl = X2 = - 2.

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a. Solution of the Variational P r i n c i p l e

T h e variational principle (74) is t h e n solved by the Ritz m e t h o d . In order to do this we expand f~¢ and f2~ in a power series in , (or ~c = , - ~- which m i g h t be more convenient) (75a)

fl~ = Z a.~"k~ s

(75b)

f2~ = Z b.gc"k~

where the coefficients a, and b~ are to be d e t e r m i n e d b y the Ritz m e t h o d . F r o m (68) we have ( f - ; f ~ J ) = Z a.-a~d~ ~j (76a) (f2~; f2~) = Z b~b.d~.'i

(76b)

r, s

where d~. ~' = (~:*k~; ~c"ki) = d . / i

(77)

while the subsidiary conditions (73) become Z d~iia, a. = r, ariiar r,s

(78a)

r

~_, d,~qb,b, = IF_,~ iia~ r,s

(78b)

r

where we have defined (79) T h e variational functions H (74a and 74b) then become H1 q = ~ a..a.d,.ii + Xt Y_, ariJa, r,s

H2 q = ~ brb.dr, q + X~ ~ 13riibr. r,s

(SOa)

r

(SOb)

r

Variation with respect to a, and b, t h e n results in 2 ~ aflr, q + Xlar ij = 0

(81a)

8

2 ~ b~d~q -k X2~ ~j = O. a

(81b)

Sept., I959.]

VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

~95

The Lagrangian multipliers X~ and ks are found to be )kl = ) k 2 = - - 2 by multiplying (81) by ar and 5. respectively and comparing with the subsidiary conditions (78). This, then, results in the following system of equations a,d,,~i = a l l (82a) $

(82b)

b,d,8 ~i = 3 / i 8

for the unknowns a, and br. The distribution function fl~ and f2~ can then be obtained in terms of infinite determinants. Let us denote by D the determinant whose elements are &8, thus D = [dr81 and D. ~and Do ~the determinants obtained from D by adding a column of ar and 3 , respectively, and a row consisting of ~crk~ the last diagonal element being zero. Then, it follows from (82) and (75) and the property of determinants that fl~ and f2~ are given by f l , = Z as~c'k, = D,,/D

(84a)

f2, = • a,~C'k, = Do/D.

(845)

b. Entropy Production

There exists a simple relationship between the variational function (F; F) and the rate of entropy production. The transport equation for the carrier distribution function can be written in the general form (cf. 13) Of/at]col,. = af/Ot q- v. vrf + e/~E. Vkf. (85) Multiplying this equation by ln f and adding to it the corresponding equation, where f has been replaced by (1 - f ) , (85) can be written after integrating over dk OS/Ot -k- div. s = - k f

,1

[-lnf - In (1 -f)]Of/Ot]co,,.dk,

(86)

where S and s are the entropy density and entropy current density, respectively, and defined by S = -- k f S

kf

[ - f l n f + (1 - f ) I n

(1 - f ) - ] d k

[ f l n f + (1 - f ) ln (1 - f)Jvdk.

(87a) (87b)

In obtaining (86) we have made use of the vector indentities div (Av) = v. VA + A div v.

(88)

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If we now use the expansion for f given by (22), then the right-hand side of (86) can be written up to first orders in Q as

-

1/Tf

EO(k) -

,/kT-]L(Q)dk.

(s9)

As the energy remains constant during the type of collisions considered here, the term proportional to the energy can be omitted, so t h a t we have finally

OS/Ot + div s = - 1 / r f Q(k)L(Q)dk = - 1/T(Q; Q)

(9(I)

which, apart from a factor ( - 1 / T ) , is just the variational function (Q ; Q) for the problem. Moreover, as we can prove from the H-theorem t h a t is always smaller than zero, it follows t h a t (Q;Q) is positive definite, as shown earlier. For the steady state, of course, OS/Ot + div s = 0 and hence the over-all change of f is also zero.

c. Effect of Magnetic Field In the presence of an external magnetic field, it has been shown that the Boltzmann transport equation acquires an extra term which we represented by the operator L' defined by (87), so that we can write the modified equation in operator form as X + = L I ( F +) + L+'(F+).

(91)

The entropy production due to magnetic scattering according to (90) is then given by

OS/Ot]mag. = -- 1 / r f FL'(F)dk = 0

(92)

which is seen to vanish on account of (32), so that there is no explicit contribution from magnetic scattering to the production of entropy. The only effect of the magnetic field is to bend the electrons without changing the order of the distribution. On account of the s y m m e t r y relation (33) on reversal of the magnetic field I-I, it is possible to obtain a variational principle for this case also, which will give rise to the correct transport equation (35), although the variational function does not contribute to the entropy production as shown above. Analogous to (74) let us define a variational function H by H = ( F - ; F ÷) + ( F - ; F + ) ' + X(F-, X) (93)

Sept., 1959.]

VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

197

where ( F - ; F +) is the inner product defined by (68) and the + and signs indicate the appropriate solution F ± depending on the direction of the field, and ( F - ; G+) ' denotes the inner product ( F - ; G+) ' = f

FL+'(G)dk = (G+; F - ) '

where the equality follows from (34).

(94)

Hence, on variation this results in

~H = ( 6 F - ; F + ) + ( F - ; 6 F +) + ( 6 F - ; F + ) ' + (F-;fiF+) ' + X(6F-, X) = 0.

(95)

Now, since L is independent of H, the first two terms in (95) are equal, while the next two are equal on account of (94), so t h a t (95) will reduce to (94) provided we choose again X = - 2. Applying this result to the system of equations (35) we now have for our new variational functions Hk (k = 1, 2) H1, = (fl,-;fa,+) + ( f , , - ; fl~+) ' - 2(/1~, Xll+)

(96a)

H=~ = (f2~-;f2, +) + (f2~-;f2j+) ' - 2(f=~-;x2~.+).

(96b)

T h e solution is again obtained by the Ritz method, where now the functions fl~ ± and f=~± are expanded according to f~,± = E a~± ~c'k,,

f=,± = E be± ~c'k,

S

(97)

s

and the new coefficients dr, are defined by d,, 'i = (~c'k,; ~c'k~) + (~c'k,; ~c'kj)'.

(98)

E v e r y t h i n g goes t h r o u g h as before, and we finally obtain two systems of equations of the t y p e (82a and 82b) depending whether the + or sign has been used for the u n k n o w n s a and b (and coefficients d, a and/3). 2. Generalized Variational Principle

We have seen t h a t the Boltzmann equation for the particle distribution function is equivalent to a variational principle. It can be shown t h a t in the presence of a lattice distribution function the resulting equations (47) can be complicated in this case due to the presence of additional integral operators and the lattice distribution function. Let Q and N be two functions satisfying the integral equations X = L~(Q) + L2(N)

Y = L3(Q) + L , ( N )

(99)

198

G . E . TAUBER

[J. F. I.

where Li (i = 1 to 4) are the integral operators defined by (49). thermore, let R and M be two functions satisfying the relations (R, X) = (R; R) I q- (R; M) 2 (M, Y) = (M; R)a -+- (M; M)4

Fur-

(lOO)

where the inner product (F; G)¢ has been defined by (51) and (F, X) by (69). As Q and N satisfy the equations (99) we have on multiplying by R and M, respectively, and integrating over k (R, X) = (R; 0) 1 --['- (R ; N) 2

(101)

(M, Y) = (M; @a + (M; N) 4

from which it follows that the right-hand sides of the system of equations (100) and (101) are equal. Consider now the linear combination of expressions of the type (F; G)~ which can be formed for these functions (F1F2; GIG2) = (F1; F2),-k (F1;G.o)2 + (G,; F2)a -k (G1;Ge)4

(102)

where Fj and Gj ( j = 1, 2) are any functions. From the symmetry relations (52) it follows that it is symmetric on interchange of the index 1 with 2, that is: (F2F1 ; (;~G1) = (F1F~ ; G1G2). (10,3) Moreover, on setting F1 = F2 = F and G1 = G2 = G, the resulting expression is positive definite, that is,

(F;G) = ( F ; F ) I + (F;G)2 q- (G; F)a + (G;G)4 >_0.

(104)

The truth of this statement can be established by the use of the symmetry relations and rearranging some of the terms. For example, due to the symmetry in k and k' the first term in (104) can be written following the argument leading to (29') as

OF; = f f

V(k,k')S(k,k')EF(k)

-

F(k')~2dkdk '.

As G is an odd function of its argument, the second term in (104) becomes, on making use of the form of the operator L2 (49n), =

ff

V(k, k ' ) S ( k ,

-

-

F(k')]G(q)dkdk'

and so finally, the whole expression (104) is given by ( F ; G ) = /3

ff

[-F(k) q- G(q)]2P°(k,k')dkdk '.

(103)

Sept., 1959.]

VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

199

This is clearly a positive definite function, as the probability P,(k, k') = V(k, k')S(k, k') is of necessity a positive definite function and the integrand in (105) consists of a square multiplied by it. This, then completes the proof of our assertion. It might be pointed out that (105) is simply related to the total entropy production for this process. If we now replace the function F by Q - R and G by N - M in (104) it can be shown that the following inequality holds (Q; Q)I -{- (Q; N)2 + (N; Q)a + (N; N)4 >_ ( R ; R ) I + (R;M)2 + ( M ; R ) a +

(M;M)4

(106)

where we have made use of the relations (101) and (102) and the symmetry properties (52). This enables us now to state the following theorem : If Q and N are two functions satisfying the integral equations (99), and if R and M are two arbitrary functions satisfying the relations (100), then of all these functions Q aiad N are the ones making (104) a maximum subject to the subsidiary conditions (101). Thus our variational function is the generalized inner product (Q;N) subject to the subsidiary conditions (Q, X) = (Q; Q) I + (Q; N) 2 (N, Y) = (N; Q)3 + (N; N),.

(lO7)

This theorem can also be applied to the system of equations (47) simply by replacing Q by f , (l = 1, 2). The expression to be maximized is then given by ( f . ; gzj) = (fu;fzs)l-}- ( f . ; g~i)2 + (gu;f~)a + (g.; g~s)4 l = 1,2 i , j = 1,2,3

(108)

and the subsidiary conditions (107) replaced by (fi,,Xlj) = ( f l i ; f l j ) l (gl,, Yli) = 0 =

+ (fa,;glj)2

(gli;fx~)a + (gl,; glj)4

(109a)

and (f~,, x2j) = (f2,;f2j)~ + (f2,; g2i)~

(g2,, x2j) = (g~,;f2;)a + (g2,; g2j)4

(109b)

From the form of these equations, it can be seen that we simply have two systems of equations (corresponding to the value of j = 1 or 2) for the functions fi~ and gj~. Consider now the new variational function H = (Q; N) + X(Q, x ) + u(N, Y)

(110)

200

G.E.

TAUBER

where X and g are two Lagrangian multipliers. with respect to O and N yields the equations

[J. F. I.

I n d e p e n d e n t variation

2LI(Q) -}- 2L2(N) q- XX = 0 2La(Q) q-2L4(N) q - g Y = 0 which are equivalent to (99) provided we set X -- # = - 2. Applying these results to the system of equations (47) we introduce the variational functions

Hi = (fli; gli) q- )t,(fli, Xlj) "q'- ],tl(gli, Ylj)

(llla)

H2 = (f2,; g2 ) -}- M(f2,, x2j) q- #2(g2,, y2y)

(111b)

which, on variation with respect to fl~ and gll, and f2¢ and go.i, respectively, lead to the system of equations (47) provided we set X~ = X,., =#1 =#2 = -2. a. Solution of the Generalized Variational Principle The generalized variational principle (108) is again solved by an application of the Ritz method. For this purpose we expand the functions gli and g2i in a power series in y = h v / k T , in addition to the expansion of the f ' s carried out previously (cf. 75). g l j = Y~ c~~y"uj

(112a)

s

g2j = E djsy'u~.

(l12b)

s

It should be noted t h a t there are three independent directions of the lattice waves in general, and it is therefore not permissible to omit the index j from the quantities c j8 and dj,, which are different for the various directions of the lattice polarization ( j = 1 for longitudinal and j = 2, 3 for transverse waves), uj are the components of velocity of the lattice waves. The coefficients a, b, c and d are then determined by the Ritz variational principle subject to the subsidiary conditions (109). With the help of the expansions (75) and (112) the functions to be maximized can then be written as (fl~; glj') = ~ (agz,d~ ~j + 2a~cj~g~ ~i + c~cj.h~, ~)

(l13a)

r,g

(f2~; g2;) = ~ (brb,d,, ~i q-2b,cj~g~,~i q-d,~ii,h,,~, ) r,s

(l13b)

Sept., 1959.]

VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

2OI

where the coefficients dr, ii, gr~ ii and h # i are given by (sc*k~; Sc*kj)~ = d./~

d~. ~i =

g # i = ( ocrk~; y.uj)2 = (y"u,;

(l14a)

(114b) (114c)

~c*ka-)3

h # i = (y~u~; y"uj)4 = h . / q

The subsidiary conditions (109) can also be transformed into linear equations in the unknowns a, b, c, and d and yield Y~ Otrliar ~_ y~ r

r,8

(115a) y" ,y,iicj~ = r

~_,

(ci,a,g,/i + ci~i,h# 0

r,8

E t3/ibr r

=

E r,8

(115b) (d~,b.g./i + d~gt~.h#O r

r,s

where, in addition to the a~ and /3~ defined previously (79), we also define = (u~, YliU), 6/i = (u~, y2jyO. (116) The variational functions H, (l = 1, 2) ( i l l ) then become H ~ i = ~ ( a , a , d # i + 2a~cj.~Sg,.ii + ci~j,ht, ii) r,8

+ Xl ~_, a/ia, -}- ]£1 ~ ~riicjr r

(117a)

r

H2~i = ~ (b,b,d,,'i + 2b,cj,g#i + d~ctj,h#O r,8

+ X2 ]E 3/+b~ + tz2 Z ~/sdi. r

(l17b)

r

Variation with respect to the quantities a,, br, c jr and di, then results again in a system of linear equations for these unknowns, where the Lagrangian multipliers X~ and m (l = 1, 2) are again found by multiplying with ar etc., respectively, and comparing with the subsidiary conditions (109). The final system of equations is thus given by

~_. (f~.i~a. + g~,~Jcj.) -- a / i = 0

(l18a)

s

(g~jia~ + h~.~c~) -- ,y/i = 0 a

(118b)

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(fl,~ib, + g#idj,) - jS/i = 0

(118c)

(g,,~ib, + h~,'idj,) - a / i = 0.

(l18d)

8

8

The distribution functions fz and gz can again be expressed in terms of infinite determinants. Let D denote 4 the d e t e r m i n a n t obtained from the matrix Dr.

=

,g~,(dt* h~,g~")

(119)

and D~,, and D~s be the d e t e r m i n a n t s obtained from D by adding a column of a'l or ~6 and a row consisting of ~c~k~ or y~u~ in the appropriate places. T h e other elements and the last diagonal element are zero. T h e n it follows from (118), (75) and (112) and the property of adding d e t e r m i n a n t s t h a t the distribution functions ft~ and g,j (l = 1, 2) are given by f ~ = a, sc~k~ = D~.~/D (120a)

f2~

br~rk< = D~o~/D

(120b)

gl~ = Cir~Cruj = D~,~/D

(120c)

g2~ = dir~cru~ = D~,~/D.

(120d)

=

In the absence of a lattice distribution function (and the corresponding interactions characterized by the operators L~ (i = 2, 3 and 4)) this, of course, reduces to the previous result (84).

b. Entropy Production The connection between the variational function (Q ; N) and entropy production can again easily be demonstrated. The transport equation for the phonon distribution function can be written in general form as (cf. 39) On/Ot]¢om = u. Vr + On~Or. (121) Multiplying this equation by I n n and subtracting from it the corresponding one for which n has been replaced by n + 1 (in accordance with the Bose-Einstein statistics) we obtain after integrating over dq

OS,/Ot + div s, = - k f

Eln n - In (n + 1)~(On/Ot)co~l.dq

(122)

where Sp and sv are the entropy density and e n t r o p y current density of the phonons, respectively, and defined by 4 To simplify the notation, the superscripts i and j are being omitted in (119) and similar expressions.

Sept., 1959.]

VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

203

l"

Sv = - k J [-n I n n -

(n -4- 1) In (n + 1)-]dq

(123a)

sv = - k f

(n + 1) In (n + 1)~udq.

(123b)

[-n In n -

For the corresponding e n t r o p y density and entropy current density of the carriers we again use the expressions (87), so t h a t the total entropy density and entropy current density are given by their sums S = Se + Sv

and

s = s~ + s~.

(124)

If we now use the expansions for f and n, (22) and (44) and limit ourselves again to terms linear in Q and N, both conservation laws can be written as

OS/Ot + div s = - 1/T f [-Q(k) - (e - t)]~L~(Q) + L2(N)-]dk

- lIT f

[N(q) - hv-][-L~(Q) + L,(N)-]dq

where we have made use of the operator equations for these functions (47) X = L I ( Q ) -4- L2(N)

(125a)

Y = Ls(Q) + L d N )

(125b)

to simplify the right-hand side. As the energy remains constant during the type of collision considered here, the terms proportional to ~ and hv give no contribution to the entropy production, so t h a t we have finally OS/Ot + div s = - 1/T(Q; F) (126) which is (apart from the factor 1/T) the variational function (Q; F> (105) and thus demonstrates independently t h a t (Q;F) is a positive definite function as shown earlier.

c. Effect of the Magnetic Field As has been shown previously, the presence of the magnetic field introduces an extra term into the transport equations characterized by the operator L' defined by (87) which obeys the s y m m e t r y relations (94). T h e appropriate transport equations in operator form (cf. 53) can be written as X + = LI(Q +) + L2(N) + L'(Q +) (127a)

Y = L3(Q +) + L4(N).

(127b)

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Analogous to (110) let us now define a variational function H by H = (O-, O+; N} + A(Q +, X) + u(N, Y)

(128)

where the inner product (Q-Q+; N) is given by

(Q--(?+; 57) = (Q-; Q+)I + (Q-; N)~ + (N; Q+)~

+ (N; N)4 q- ½(Q-; Q+)' -¢- ½(Q+; Q-)'.

(129)

The first four terms do not involve the magnetic field, and hence it is immaterial whether we use the -t- or - sign in the functions Q. Furthermore, on setting (2- = (2+, the last two terms in (129) disappear on account of (32) and (128), and thus reduce to (105) in the absence of the magnetic field. Let Q+ and N be two functions satisfying the integral equations (127) and Q- and N the ones obtained on reversing the direction of the field H. Then on forming the scalar products (Q, X) and (N, Y), one obtains the following subsidiary conditions

(Q-,X) = (Q-;Q+)~ + (Q-;N)2 + (Q-;Q+)' (N, Y) = (N; Q+)3 q- (N; N)4.

(130a) (130b)

Independent variation of H with respect to the functions /iQ- and aN then yields the equations (aQ-; Q+)I + (aQ-; N)2 + 1/2(aQ-; Q+)' + 1/2(Q+; aQ-)' + X(aQ-, x ) = 0

(131a)

(Q-; aN)2 + (aN, Q+)3 + (aN;N)4 + (N; aN)4 + u(aN, Y) = 0 (131b) which are satisfied by (127) on eliminating the Lagrangian multipliers X and/~ through the subsidiary conditions (120), which for this case are seen to be equal to X =/~ = - 1. (The difference from the previous result is due to the fact that we now consider (2- and Q+ to be distinct functions.) This result can now be applied to the system of equations (53) and their conjugate ones by replacing (2+ by fl~ + or f2~+, and N by gij or g,j. The appropriate variational functions are given by

HI

=

(fl,-,

fli+; glj} -- ( f l i - X l j +) -- (gll, y l j )

II2 = ( f 2 , - , f 2 , + ; g 2 j } -

(fo.~-x2i +) -- (g2~,y2j).

(132a) (132b)

The solution of the variational principle is again achieved by the Ritz method, where the functions f,~ + and f~± are expanded according t<) (97), and coefficients d, g and h are defined by (114), except that the

Sept., 1959.]

VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

20 5

coefficients &,+ consist now of two terms according to (98) due to an additional term representing the magnetic field. The final system of equations is of the type (118), where, however, we have a different system of equations depending whether the q- or - sign has been used for the quantities a < and ba (and coefficients d, a ± and/3~), depending on the direction of the magnetic field.

3. Variational Principle for Interband Scattering In the case of interband scattering, we have seen t h a t the inner product (F; G) is defined as a sum of terms of the form (F=; G"), where m and n indicate the bands. If, in addition to t h a t inner product (cf. 58 and 59) (F;G) = Z (F';G") m,n

= ~_, ~ F"L,..(G")dR,.

(133)

m,n

we also define the product

(F, X) = ~., (F", X") = ~ . f F"X"dkm

(134)

it is easily seen t h a t the extremal function of the problem is given by (133) subject to the subsidiary condition

(F, x ) = Z

xm) = I2 (Fro; F-) = (F; V).

(135)

As before, we can again define a new variational function by H = Z (F'~; F-) q-X E (F'% X '~)

(136)

which, on variation with respect to ~F m leads to the system of operator equations E Lm,(F") = X m (137) n

where the Lagrangian multiplier is found to be equal to X -- - 2 by comparison with the subsidiary conditions (135). Applying this to our system of equations (55) we again obtain variational functions H1 and H2 of the form (74), the only difference being t h a t now each of the terms (f; f) and (f, x) appears as a sum over bands.

a. Solution of the Variational Principle The variational principle can again be solved by the Ritz method. For this purpose we expand the functions fi~ ~ and f ~ as an infinite

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power series in ~,. (or SC~) J"fl $. . . . .

f2,

TM

ki

E r

am,,(m)

(138a)

r

(138b)

= k, E b~.(~.,) ~ r

where now the u n k n o w n coefficients a ~ and bm~ and the energy e., also depend on the band under consideration. With the help of the expansion for f the inner products (f~,: ; f~i) and (f~i; f~j) become

(fli; f l j ) = Z Y~ a,.~a..d . . . . . ~i

(139a)

(f2i; f2j) = Z Y~,

(139b)

m,r n,s

1'.,~.. ; e

n,8

bm~b..d . . . . . ~

where we have introduced the notation d . . . . . ii = ( k #

~; k # . ' )

= d ...... Ji

(140)

while the subsidiary conditions become (141a) m,r

E m,r

n,8

mr

E b~b..d ..... q =

E b~JL,/~

n,s

mr

(141b)

where we have defined X ~ m)

(142a)

fl.,J~ = (k~emLX2/~).

(142b)

a~ji

= (ki~,

The variational functions H~ (l = 1, 2) are thus given by (bmrOlmr i]

(143a)

b.,rb.M . . . . . ~i + X2 E b ~ J 3 ~ j i .

(143b)

H~ = Y'. Y'. a ~ a . ~ d . . . . . ~i .Jr_ ~kI mr

n8

H 2 = F. E mr

n8

Z mr

mr

To find the values of the parameters we now vary these equations with respect to a=r and b=r. From the conditions for an e x t r e m u m we obtain the system of equations 2 E a . s d ..... ii + M a ~ j J ns

=

0

2 E a . s d ..... ii + h25.,/i = O. n~

(144a) (144b)

Sept., 1959.]

VARIATIONALPRINCIPLE IN TRANSPORT THEORY

20 7

The unknown Lagrangian multipliers X, and X2 are found on multiplying these equations by a~r and b,,~, respectively, summing over m and n 2 ~ ~ a.~a..d . . . . . ~i ~- )k 1 ~ amrOlmrlJ ~r~r

n~

mr

mr

n8

mr

-~-

0

and comparing with the subsidiary conditions (141). Thus from (144) we get two systems of linear equations for the unknowns a,, and b,, which satisfy the stationary condition and the subsidiary conditions (141) a ~ / i = E a . , d . . . . . ii (145a) ns

~/~

=

~

(145b)

b.,d ..... %

ns

We note that this result is analogous to our previous result (82) for the unknowns ar and b,, except t h a t now we also have to carry out summations over the various bands. The two distribution functions fl~ andf2~ can again be obtained in terms of infinite determinants. Let D denote the determinant obtained from the matrix d ..... where the elements are grouped according to increasing m (and n) for a given r (and s). Thus, for example, the determinant corresponding to m = n = r = s = 0, 1 is given by doo,oo doo,lo doo,ol doo,ll d,o,oo dlolO dlo,Ol dlo,ll t n = dol,oo dol',io do,,o~ doi,tl " (146) d,l,oo dl,,lo d11,Ol dll,n Furthermore, let Din, and Din0 be the determinants obtained from D by bordering it with a column of a,. and /~,8, respectively, and a row of k~(e,) r in the appropriate places, the others and the last diagonal element being zero. Thus, for example, D0o corresponding to (146) is given by doo,oo doo,lo doo,ol doo,ll

/~oo.

Do~=

dn,oo

d11,1o d11,ol d11,11

k~Eo°

0

k~o 1

/3ii. 1

(147)

0

Then, it follows t h a t the functions fli m and f~m are given by fl~" = f2~ =

-- D m i J / D is -- D ~ i i / D

ii.

(148a) (148b)

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b. Effect of a Magnetic Field In the presence of an external magnetic field the operator equations (55) have to be replaced by (60) which have an additional term L r / ( F m) describing the effect of the magnetic field. It has been noticed that this operator only has a free index describing the effect of the magnetic field on the band in question. Hence, let us define the product (F; F)' (F; F)' =

Z

(Fro; F'~) ' =

E

[FmL./(F")]

(149)

which, on account of the symmetry properties of L~', satisfies the relation (F-;G+) ' =

Z (F"-;G"+) ' =

F. ((;'"+;F'~-) ' =

(G+;F-) '.

(150)

m

In accordance with the procedure of Section 1, we take for our variational function H the combination H = ( F - ; F +) + (F-; F+) ' + X(F-, X)

(151)

where ( F - ; F +) is the inner product defined by (133) and ( F - ; F+) ' the one by (149). The q- and - signs again indicate the appropriate solutions depending on the direction of the field. Independent variation with respect to F = then results in the operator equation

X m+ = Y~ L,~,,(F ,,+) + Lm'(F ~-)

(152)

rt

for the distribution function, where the Lagrangian multiplier is found again to be X = - 2 in order to satisfy the subsidiary conditions (Fm-, X '~+) = ][] ( F m - ; F ~+) + ~ (F'~-;Fm+) '. m

rn n

(153)

rg~

Applying this result to the system of equations (60), we now have for our new variational function Hz (l = 1, 2) expressions of the form (96), the only difference being that now each of the inner products has to be summed over the various bands in accordance with the definitions (133,134 and 149). The solution is again obtained by the Ritz method, where now the functions fl~ m~ and f2~ are expanded according to TM

(154a)

f , , " ± = k, Y. r

(154b) r

and the new coefficients d ...... are defined by d ...... i, =

(kiE,nr; k s ~ # ) +

( k ~ . , ~ ; k,~, 8 ) ! .

(155)

Sept., x959.]

VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

20 9

The rest of the calculation proceeds as before, and we finally obtain two systems of equations of the type (145) for the unknown coefficients a , ~ and b~,±, respectively, depending whether the + or - sign has been used for the unknowns a and b and the coefficients d, a and ¢~. The distribution functions f can again be expressed in terms of infinite determinants, where again one has to differentiate between the -t- and sign characteristic of the two solutions.

c. Presence of Lattice Vibrations The carrier distribution function F m depends on the band in question, while the lattice distribution function G does not. In view of this, it is necessary to differentiate between operators which operate on F m and those which operate on G, and define the corresponding inner products accordingly, as can be seen from (64 and 65). Formally, however, we may expect that our previous results will still apply, provided that we take account of the appropriate definitions of the inner products. Let us take then, for our variational function H H = (Q; N) + X(Q, X) + ~(N, Y)

(156)

where the bracket (Q; N) denotes the linear combination

(Q;N) = (O;O)l + (O; N)2' + (N;Q)3 + ( N ; N ) , '

(157)

whose components have been defined by (64). Furthermore, the products (Q, X) and (N, Y) are given, respectively, by

(Q, X) = ~,, (Qm,Xm) = ~ f QmXr"dk,~

(158a)

(N, Y) = f N Ydq,

(158b)

only the first of which contains the dependence on the band. X and g are, of course, again Lagrangian multipliers. Independent variation of H with respect to Q~ and N yields the equations Z L~,I(Q ") + Z L~,2(N) = X m

(159a)

n

2 L~,3(Q ") + 2 Lm,4(N) = Y

(159b)

where the Lagrangian multipliers X and ~ are found to be equal to X = u = - 2 by comparison with the subsidiary conditions (Q; Q)I + (Q; N)2' = (Q, X)

(160a)

(N; @3 + (N; N)4' = (N, Y).

(160b)

G . E . TAUBER

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[J. F. I.

Applying our result to the system of equations (61), we use the variational functions of the form (111), where the various terms are to be interpreted in terms of our definitions for inner products, such as (64) and (158). Variation with respect to fl~ ~ and gli and f2~~ and g2~, respectively, then leads to the system of equations (61), where the Lagrangian multipliers have been eliminated with the help of the subsidiary conditions (109). In order to solve the variational principle we expand the functions f l C ' and f,2/~ according to (138) and gli and g2~ according to (54). With the help of these expansions the functions to be maximized can then be written as

(f,~; glj)----- E E mr

[am,bn~d . . . . . . . . ~"+2a,~rcj,g,n .... ~J+c~cj.h ...... ;,]

(161a)

]

(161b)

ns

(fo,; g2~)= E E [b.,.b..d . . . . . mr

ii

+2bm/t~g

.....

i]

+ d , d. . h .

......

i]

n~

where the coefficients d . . . . .

are given by

d ..... ~i = ( ~ k . i ; ~2kj),,,,? = d . . . . . ~'

(162a)

g . . . . . ~i = (~m~k~ ; y ' u i ) , , , j = (y'u~; ~ k ~ ) m J

(162b)

h . . . . . ~ = (y'u~; y*uj)mn 4.

(162C)

The subsidiary conditions can also be transformed into linear equations in the unknowns and are given by E am/"amr = E E (amra,,~d . . . . . i] + am~C~,g. . . . . ii)

(163a)

E %~"csr

(163b)

= E E (c~a..g ....... *i + c~rc.h . . . . . ~J)

r

mr

ns

mr

mr

ns

E &~ic~. r

= E E (d,.~b..g ...... " ~r

+ d ~ d i . h . . . . . ~i)

(163d)

ns

where the am, and fl=, are defined by (142) and % and ~r by (116) in accordance with (158). The variational functions H, (l = 1, 2) then become

H1

(fl~ ; glj) q- )tl E Olmrilgmr + jgl E "YriiCJr

(164a)

H.~ = (f2~; g2j) + X2 Z timJib,., + ta2 Z &~icj.

(1648)

=

Variation with respect to the quantities a~r, bin,, c jr and d~-rthen results again in a system of linear equations for these unknowns, which after

Sept., 1959.]

VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

211

elimination of the Lagrangian multipliers finally results in the system of equations (d . . . . . iia~. + g . . . . . iJcs~) - a,.~ ii = 0 (165a) nil

5-'. Z (g ..... ~ia,~ + h ..... ~ici~) - q,~i = 0 ~rL

Z

ns

(d . . . . . ~ib., + g . . . . . ~ids~) -

Z Z (g ..... b.. tn

(165b)

ns

+ h ..... d¢,)

3,,,,.~i = 0

-- ~/;

= 0.

(165c) (165d)

n$

The distribution function fz ~ and gt (l = 1, 2) can again be expressed in terms of infinite determinants. For this purpose, we define D to be the d e t e r m i n a n t obtained from the matrix D . . . . . = ,r|d.....

g ..... \|

-,,g . . . . .

(166")

h ......

where the elements have been ordered according to the convention leading to (146). Furthermore, we define D,m,, and D , ~ to be the d e t e r m i n a n t s obtained from D by adding a column of a~,~r or 3m,~, and a row consisting of e~'k~ in the appropriate places, the others and the last diagonal element being zero. Also let D~,, and D,o, be again the det e r m i n a n t s obtained from D by bordering it with a column of a~,~, or 3~8~ and a row of y'u~, the last diagonal element being zero. T h e n it follows from the properties of d e t e r m i n a n t s t h a t the distribution functions f " and g are given by r

f27" = E b,.~E,.rk~ = D,,.o5/D

(167b)

r

glll"----

~ r

Cjryruj

g2: = ~ d j r y ' u j

D~.~/D

(167c)

= D~o,/D.

(167d)

=

r

If, in addition to the lattice distribution function g, we also have an external magnetic field, the appropriate variational principle is given by one of the form (132), where the various terms have to be interpreted in accordance with previous results. For example, the products ( f l ~ - , x~i +) and (f2~-, x~i +) are to be considered as sums over the various bands as in (134) with the + and - signs indicating the direction of the field. The inner product ( f - f + ; g) is of the form (129) where, in addition to

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the terms defined by (122), we also have to include those characteristic of the magnetic field and given by (149). I n d e p e n d e n t variation with respect to fz m± and g,i then leads to the appropriate equations. The solution of this more complicated variational principle is again achieved by the Ritz method, where the functions fl mi and f2 m± are expanded according to (154) and gl and g2 according to (112). The coefficients d, g, are defined as in (162) except t h a t the coefficients d ...... consist now of two terms and are given by (155). T h e final system of equations is of the type (165), where, however, we now have two sets of equations depending on whether the + or - sign has been used for the quantities a,n~± and bm~± (and coefficients) depending on the direction of the magnetic field. This, then, concludes our discussion of the variational principle as applied to transport theory, where we have tried to discuss the various cases arising whether (i) an external magnetic field, (ii) non-equilibrium lattice distributions or (iii) interband scattering are present, singly or in combination. In the next section we shall apply our result to the evaluation of the various transport coefficients. IV. TRANSPORT PHENOMENA

1. General Formulation of Transport Coefficients T h e various transport coefficients can be obtained from the general expressions for the electric current density J and energy current density W. T h e former density is given by

J, = - (e/Srr 3) f v,fdk

(i = 1, 2, 3)

(168)

while the latter consists of two contributions, one due to the electronic energy density, and one due to the lattice current density, and is given by W~ = (1/8rr a)

f v~fdk + (1/8rr a) f u~hv,,ndq

(i = 1, 2, ,3). (169)

Here f and n denote again the carrier and lattice distribution functions for which we use their respective expansions (22) and (44). The integrals containing fo or no will give no contributions to the current and energy current densities, while the terms containing Q and N can be separated into two parts according to (27) and (48). Of course, in the simple case where we neglect the deviation of the lattice distribution function from equilibrium, the latter will yield no contribution. Using these expansions we obtain for J~ and W~ (where i = 1 to 3 denote the c o m p o n e n t s of J and W)

J, = Y'. {K~/')[e2Ei --]-eTO/Oxi(~;/T)-] -t- K~/2~(e/T)OT/Ox j} i

(170)

Sept., I959.]

VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

2I 3

and

W~ = -- ~ {K~/3~[eEi + TO/Ox~(f/T)~ + K~/4)(i/T)OT/Ox i,

(171)

1"

respectively, where we have defined the energy integrals

f v~fljOfo/O,dk = (Ylj, Xl~)

(172a)

8~r3Kij.(2) = f v,f~jOfo/Oedk = (f2i, Xli)

(172b)

87r3Kij(I)

:

87r3Kij(~) = f viefliOfo/Oedk -

f u~TgliOno/O Tdq = (flj, x2~) + (gl~, y2~) (172c)

87r 3KI/(4) =-

f v,¢f2jOfo/Oedk -- f uiTg2jOno/O Tdq = (f2;, x2,) + (g2~, y2,)

(172d)

and where we have also made use of the definition (69). Hence it is seen that the energy integrals (172) are nothing else than the subsidiary conditions (109). The K's defined above are not independent, but satisfy the so-called Kelvin relations ..(1)

K~.(2) = K~¢3~,

r(..(4)

(173)

The proof of (173) can be carried out with the help of the symmetry relations (52) and the operator equations (47). From the fact that the electric current density J~ can be written as J~ = Y~ a~jE~.

(174)

i

we deduce from (168) t h a t the conductivity tensor ~ is given by

a~j = e2K# 1). If we denote the reciprocal tensor of Ki/1) by resistivity o~ is given by p~j = (l/e2) ~..(1)....

(175)

Kij(1)

then the electrical (176)

Equations (170) and (171) can be solved for the electric field and the heat current density

E, = E (o~jJ~- (1/e)~jOT/OxO -- (1/e)O~/Ox ~

(177)

i

W, = E (II,~.Jj - K,sOT/Ox,) -- (~/T)J~ i

(178)

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G . E . TAUBER

[J. F. I.

where S~i is the tensor of the Seebeck coefficient, H~j that of the Peltier coefficient, and K~j.the heat conductivity tensor. Comparisons of (177) and (178) with (170) and (171) yields for these tensors TSij = + ( ~ / e ) ~ i + E R i m ( ' ) K , . S 2)

~t

TK~j = K # ~) -

E K ~ , . ( 3 ) R m J ~ ) K . i (2~

m,n

(179)

(18o) (181)

where 6~j denotes the Kroenicker delta, &j = 1 if i = j, 6~j = 0 if i ¢ j. Moreover, from the Kelvin relations (173), it also follows that Flij = - TSjl

(182)

in accordance with known results. The expressions for the various transport coefficients (175, 176, 179-182) are quite general and do not depend on any particular model used. Their value depends only on the energy integrals which we shall evaluate for the various cases of interest by the Ritz variational principle. 2. Effect of Magnetic Field

We have seen that in the presence of an external magnetic field the transport equations acquire an extra term due to the interaction with the magnetic field, and that the distribution functions fl and f2 have to be divided into two conjugate sets fz + and f , - (l = 1, 2) corresponding to the direction of the field. The effect of this separation will be felt in the energy integrals which are now a function of the magnetic field H. The electric current and heat current densities are still given by their respective expressions (170) and (171), but the energy integrals Kij are now defined by 87~3Kij (1) ( H )

=

--

( f l j - , Xli +)

(183a)

x,,+)

(183b)

8~r3Ki/3)(H) = (f~y-, x2¢+) q- (gly, y2~)

(183c)

8rraK#~)(H) = (f~F, x=~+) + (g~, y2,)

(183d)

where the products (f-, x +) are of the type (69) with the appropriate direction of the magnetic field H. From (183) we can obtain the conjugate set corresponding to - H by changing all + to -- signs and vice

Sept., I959.]

VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

21 5

versa. It can be seen that the energy integrals (183) are again the subsidiary conditions (130) which differ from the previous ones (109) by having an extra term of the form (fz-; f~+)' and two sets of the distribution function f, (l = 1, 2). With the help of the s y m m e t r y relations (108) and (94) for inner products involving the operator L' (31) it can be shown that they satisfy the Kelvin relations (173), where now, however, the change of direction of the magnetic field must be taken into account. We thus obtain K,j(1)(H) = K j i ( 1 ) ( - H ) ; Kij(2)(H) = K ~ ( 3 ) ( - H ) K # 4) (H) = Ki,(4) ( - - H ) .

(184)

The various transport coefficients are still given by their appropriate expressions, (175, 176, 179-181), but where now the energy integrals defined by (183) must be used. From their symmetry relations under reversal of the magnetic field (184), one sees readily that the following relations hold among the various transport quantities o',s(H) = o ' j , ( - H ) ; TS,~(H)

K,i(H) =

-

= K~,(-H)

Ib,(-H).

(185)

If one separates the above tensors into their symmetric and antisymmetric parts, one finds that the symmetric parts of these tensors are even functions of the magnetic field, while their antisymmetric parts are odd functions of the field, that is, they change sign on reversal of the field. 3. Calculation of Energy Integrals

We have seen in the previous sections that all thermoelectric coefficients can be expressed in terms of the energy integrals K~.. Thus, te obtain expressions for these integrals, it is only necessary to calculato the various thermoelectric and magnetic coefficients. By applying an elegant method due to Enskog to the various cases, these can be obtained in terms of infinite determinants without the explicit solution of the transport equations. a. Carrier Distribution Function

Let us first consider the simple case for which the deviation of the lattice distribution function from equilibrium has been neglected, that is, n - no. The appropriate energy integrals are then given by 87r3K,i '

= f Vlfli(Ofo/O~)dk =

(flj', Xl,)

(186a)

8="K,~' = f vlf2~(Ofo/O~)dk (f~i, x,,)

(186b)

=

216

G.E.

TAUBER

[J. F. [.

8 7 r a K i j 3 =--

f v,~fl~(Ofo/OE)dk=

(fl~', x 2 0

(186c)

87raKcj 4 =

f v,~f2j(Ofo/OE)dk=

( f ~ j , x2,)

186d)

where the distribution functions f~i and f2~ are given by (184) and x~ and x2~ are defined by (27). F r o m these definitions and the ones for a, and 5~ (79), it is seen t h a t the various energy integrals are just sums of products of the coefficients a~ and b, and the integrals a~ and ~,. Thus, if we denote by D,o the d e t e r m i n a n t obtained from D = [d~.~[ 1)y adding a column of 5r and a row of a~ (with similar additions for the other types), we obtain for the energy integrals (186) 87raKi/ ---

-

D=a/D

187a)

=

-

D~e/D

187b)

=

-

D~/D

187c)

=

-

De~/D.

187d)

~2 a ~ a / , i = r

8~raK~i2

=

Y" b ~ a / i r

8rraKiJ a

=

E a~5/i r

8rraK~/

=

F. b t S / ' r

The addition of an external magnetic field leaves that result unchanged in principle, except t h a t the matrix elements d~ (77) have to be replaced by the corresponding ones including the effect of the field (98), and + or - signs have to be used for the coefficients a~+ and ~ ± depending on the direction of the field. In other words, we will now have two sets of energy integrals K o ( H ) and K~i(-I-I) of the type 8rraKo2(H) = -

D~-e+/D;

87raKo2(-H) = -

D~+a-/D

(188)

and similar ones for the other cases in (187).

b. Effectof Lattice Vibrations In the general case where the deviation of the lattice distribution function from equilibrium is also taken into account, the energy integrals Ko. are given by (172), the distribution functions f , and g , (l = 1, 2) by (120), and the coefficients at, /3r, % and ~r by (79) and (116), respectively. It can be seen again t h a t the energy integrals are just sums of products of the coefficients at, br, cr and integrals a , ~ , %. and 6r, and therefore can be expressed again in terms of infinite determinants. With the help of the defining equations and the properties

Sept., 1959.] VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

2I 7

of adding of d e t e r m i n a n t s we obtain for the energy integrals K~j in this case 8~raKis ~ = ~_, (a,a/~ + csCY/O = -- D , , , ~ / D (189a) =

-

D~.,/D

(189b)

=

-

D~,e~/D

(189c)

8~r3Ko * = 2~ (b~13/i + dj~a/O = - D~,a~/D

(189d)

8rraK,J 2 = E ( b r a / i + d J r / O r

8rr3K*i a = E ( a , 5 / i + c i , ~ / O r

r

where D~,0s is the d e t e r m i n a n t obtained from D by a line consisting of a'y and a column of fl~ with similar definitions for the other determinants. (It should be noted t h a t in this case D is the d e t e r m i n a n t obtained from the matrix Dr, defined by (119).) As in the simple case discussed above, the addition of an external magnetic field provides only minor modifications of t h a t result. For the matrix elements dr, we have to use the ones which include the effect of the field (98), and + and - signs have to be used for the quantities a, • and /L±, depending on the direction of the field. Of course, the integrals %± and 8r± are not affected by the field and remain unchanged. T h e final expressions for the energy integrals are still of the form (189), with the modification introduced by (88) resulting in two sets of integrals K~j(H) and K ~ i ( - H ) as defined by (183), from which the various transport quantities are to be calculated in accordance with (175, 176, 179-181). 4. Transport Coefficients for Interband Scattering a. Without Taking Account of Lattice Vibrations

In the case of interband scattering the electric and energy current densities m a y be considered to be given as sums of the appropriate expressions for each band. T h u s

J = E J"

(19o)

tn

where {K~i m l [e 2 Es + e T O / O x i ( ~ / T ) ] J?,, = _ (e/8~-3) f v¢fmdk = Y'. i + KC"2(e/T)OT/Ox~}

(191)

and W = XW,~ ,~

(192)

218

G.E.

TAUBER

[J. F. I.

with W~,,~ = (1/8~") / v , 4 " & = - Z {K,/~"~eEi + d

i

+ K,£"4(1/T)OT/Oxj}.

(193)

The K ' s are the energy integrals for the various bands and analogous to (186) defined by 87r3K;i TM = (flj ~, x~,~) (194a)

87r3Kij 2,n

(f2/", x~,")

(194b)

87r3Kifl ~ = (fli ~, x2i~)

(194c)

87r3K~/~ = (f:F, x2/").

(194d)

-~

As before, the indices m and n refer to the bands and i and j to the direction. From (190-193) it can be seen t h a t the current and energy current densities can be written as J, = Z {K,~aEe2Ej + eT(O/Oxj)(~/T)-] + K , / ( e / T ) O T / O x j }

(195)

i

and W, = -

Z { K , f f e E i + T ( O / O x i ) ( ~ / T ) ] + K~i(1/T)OT/Ox~},

(196)

i

respectively, where we introduced the notation K~? = ~ K , ....

(u = 1 , 2 , 3 o r 4 ) .

(197)

The expressions (195) and (196) are, therefore, of the same form as the corresponding ones in the standard theory (170) and (171), the only difference being the definition (197) for the energy integrals. From (195) and (196), all transport coefficients can now be obtained in the usual m a n n e r (see Sections 1 and 2). T h e energy integrals can be obtained again in terms of infinite d e t e r m i n a n t s by applying the results of Section I I I to this case. It has been shown there t h a t the distribution functions fx~m and f2~m are given by (148), and the various K ' s are then obtained by forming the inner products of the f ' s with Xl or x2 in accordance with (197). Thus, denote by D=~o~i the d e t e r m i n a n t obtained from Din0~i by replacing the row of k~= r by amr in the appropriate places, the others and the last diagonal element being again zero. In other words, D ~ o ij is of the same form as D~0 except t h a t it has more rows and columns and some zeros in the last row. For example, Do~ ~i obtained from Doo~i (147) is given by

Sept., I959.] VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

doo, oo doo,lo doo,ol doo,ll

219

/3oo

(198)

D o,,~ =

dll, oo dll, lO dll, Ol dll, 11 fill 0~00

0

0/01

0

0

We t h e n obtain for the various K¢?m's Ki/"~

=

-- D,.,,,di/D

(199a)

b,.~am~ =

-

D,..o/D

(199b)

a,..l~,..

-

D,.¢~/D

(199c)

E a,.,.a.,r = r

K ~ J TM =

Z r

K ~ j 3. . . .

~

=

r

K ~ j 4"~ =

~

b,.,13,.r =

-- D,.~e/D.

(199d)

r

S u m m a t i o n over m t h e n yields the Kij u (u = 1, 2, 3 or 4) (197). Inspection of the d e t e r m i n a n t s D ~ such as (198) shows t h a t the s u m m a tion over m can be carried out directly in the d e t e r m i n a n t s . Hence, if we define D,0 to be a d e t e r m i n a n t obtained from D by bordering it with a row of amr and a column of flmr (defined by (142)), where the last diagonal element is again zero, the K~j ~ are t h e n given b y K~j 1 =

-

D~=/D

(200a)

Kij ~ =

-

D~/D

(200b)

K~i 3 =

-

D~/D

(200c)

Kij 4 =

-- D~/D

(200d)

in accordance with our previous results (187), the only difference being t h a t now we have m o r e rows and columns corresponding to the different values of ( m r , n s ) . T h e various t r a n s p o r t coefficients are t h e n obtained from (200) and the appropriate formulae. T h e effect of an external m a g n e t i c field is t a k e n into account by replacing the m a t r i x elements d ..... by the ones including the effect of the field (155), and supplying + or - signs to the integrals amr~ and /~mr± depending on the direction of the field. As before, we will again have two sets of e n e r g y integrals K~ju(H) and K ~ i ~ ( - H ) of the t y p e (188), the only difference being t h a t each set is of the form (200). b. P r e s e n c e

of Lattice

Vibrations

In the case for which deviations from equilibrium of the lattice distribution function are also t a k e n into account, the energy integrals

220

G.n.

TA~:I~ER

[J. F. I.

K,~j..... for the individual bands are given analogous to (172) by 87raKe/m = ( f t F ~, Xli m)

(2(11a)

87raKe7 " = (f2/% x ~ " )

(201b)

8rraK;9 m = ( f , / " , x2, TM) + (glj, Y2i)

(201c)

8rraK~j 4" = (f..,/~, x ~ " ) -4- (g~,, y ~ ) .

(201d)

The distribution functions f~m and gzj ~ (l = 1, 2) are given by (167), where the appropriate integrals and matrix elements have been defined by (116) and (142). T h e various K's are then obtained from (167) by forming the inner products of sums of f ' s and g's with x and y. From (201), the various defining equations and the properties of addition of determinants, we then obtain for the energy integrals in this case 8rcaK¢/" = ~ (a.,~a,./~ + cj~'~ ~i) = - D,,,.~.~/D

(202a)

r

87raKe7 = = ~ ( b ~ a = / i + d s ~ y / O = - D . , o , . ~ / D

(202b)

r

87raK¢i TM = Z (a~13~/i -}- cj,&,:i) = -- D~.~ea/D

(202c)

r

87raK~j 4~ = Y'. (bm~3,~/s + ds~& ~i) = -- Dmo~e~/D

(202d)

r

where the d e t e r m i n a n t Dmo,.v is the sum of the d e t e r m i n a n t s D .... v and D~.v replacing, however, the row of emrk¢ by ~mr or amr and the row of yru~ by % or ~ in the appropriate places, the others and the last diagonal element being zero. S u m m a t i o n over m then yields the K¢i" for this case. It can be seen t h a t this s u m m a t i o n can be carried out again directly in the determinants. Thus, if we denote by D~o~ the det e r m i n a n t obtained from D (as given in (166)) by bordering it with a row of Olmr~{r and a column of time%, where the last diagonal element is again zero, the K ¢ j u are given by the previous expressions (189), the difference being t h a t we now have more rows and columns corresponding to the different values of (mr, ns) and t h a t the various matrix elements, d ...... g ..... are given by (162) and the integrals a~,/3m~, %, ~ by (116) and (142). T h e modification introduced by an external magnetic field results again in two sets of energy integrals K~i"(H) and K~j"(--H) with the matrix elements d ..... given by (155) and + or - signs introduced in the quantities ~ and /3~± depending on the direction of the field. From these expressions for the energy integrals, the various transport quantities can again be calculated in accordance with their defining equations (175, 176, 179-181) as before.

Sept., I959.]

VARIATIONAL PRINCIPLE IN TRANSPORT THEORY

221

We have thus been able to demonstrate how a variational principle can be formulated which is equivalent to the transport equations for carrier distribution functions, lattice vibrations, external magnetic field, interband scattering and combinations of all or any of these. General expressions for the various transport coefficients have been given in terms of energy integrals, which have been evaluated in terms of infinite determinants for any of the above cases. To evaluate the different determinants the coefficients d, etc. must be calculated for which purpose it is necessary to make definite assumptions about the {orm of the interaction and energy dependence. Work along these lines for the calculation of the transport coefficients for warped energy surfaces is in progress and has been reported elsewhere (12). In this paper we have only been interested to give a systematic discussion of the variational principle as applied to transport theory and its application to the calculation of the transport coefficients. REFERENCES

(I) P. M. MORSEANDH. FESCHBACH,"Methods of Theoretical Physics," New York, McGrawHill Book Co., Inc., 1953, Part II, Chapter 9. (2) A. H. WILSON., "The Theory of Metals," New York, Cambridge University Press, 1953. (3) M. KOHLER, Z. Physik, Vol. 124, p. 772 (1948)'; ibid., Vol. 125, p. 679 (1949). (4) J. M. ZIMAN, Can. J'. Phys., Vol. 34, p. 1256 (1956). (5) L. ON'SAGER,Phys. Rev., Vol. 37, p. 405 (1931) ; ibid., Vol. 38, p. 2265 (1931). (6) E. H. SON'DHEIMER,Proc. Roy. Soc. A, Vol. 234, p. 391 (1956). (7) D. EN.SKOG,dissertation, University of Utrecht, 1917. (8) D. DORN., Zeits. f. Naturf., Vol. 12a, p. 739 (1957). (9) M. KOHLER, Ann. Physik, Vol. 40, p. 165 (1941). (10) F. G. MOLINER AND S. SIMONS, Proc. Camb. Phil. Soc., Vol. 53, p. 848 (1957). (11) G. E. TAUBER, Can. J. Phys., Vol. 36, p. 1308 (1958). (12) G. E. TAUBER, AND L. SOFFER, f . Phys. Chem. Solids, Vol. 8, 138 (1959).