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Electrical conductivity of an interacting electron gas
Physica 93A (1978) 171-190 © North-Holland Publishing Co.
E L E C T R I C A L C O N D U C T I V I T Y O F AN I N T E R A C T I N G E L E C T R O N GAS* David Y. KOJIMA$ r
C.P.R.D.-A.M.D., lnstituto De Energia Atdmica. Caixa Postal 11.049, Pinheiros, CEP 05508, Sao Paulo, SP. Brazil
Received 25 August 1977
The many-body theory by the propagator method developed by Montroll and Ward for equilibrium statistical mechanics, is reformulated to describe the electrical conductivity for an electron gas system containing impurity atoms. The theory includes electron-impurity interaction to infinite order and electron-electron interaction to the first order exchange effect. The propagator used by Montroll and Ward is separated into two propagators, each one of which satisfies either the Bloch or the Schr6dinger equation, to utilize the perturbation method. Correct countings of graphs are presented. The change in the relaxation time due to the electronic interaction and the temperature change is discussed and compared with recent works.
I. Introduction It has been a s s u m e d f o r a long time that the e l e c t r o n - e l e c t r o n ( e - e ) interaction did not have a large effect on the electrical c o n d u c t i v i t y and m a n y w o r k s w e r e r e p o r t e d with this assumption. K o h n ~) pointed out that the effect o f the e - e interaction only existed t h r o u g h the b r e a k d o w n of the Galilean translational s y m m e t r y as a result of the introduction of fixed impurities. L a n g e r has r e p o r t e d in his series of papers 2) the i m p o r t a n c e of the e - e interaction t h r o u g h the electronic screening effect on impurities, and the t e m p e r a t u r e d e p e n d e n c e . T h e f r e q u e n c y d e p e n d e n c e of the relaxation f u n c t i o n b y the r a n d o m p h a s e a p p r o x i m a t i o n has been calculated by G/)tze and W61fle 3) f o r a f r e q u e n c y range f r o m zero to values l a r g e r than twice the F e r m i e n e r g y EF, using the m e m o r y f u n c t i o n t e c h n i q u e d e v e l o p e d by Mori4). T h e t e c h n i q u e a p p e a r s to be p o w e r f u l , but the introduction o f the e - e interaction is not clear. O n e might have to take into c o n s i d e r a t i o n the locality o f the external electric field w h e n the e - e interaction is included b e c a u s e the C o u l o m b interaction is of long range, w h e r e a s the w a v e n u m b e r o f an electric field with a f r e q u e n c y to ~ 2EF is of the o r d e r of 102-103 ,~, Ting, Y a n g and Q u i n n 5) (T.Y.Q.) r e c e n t l y d e v e l o p e d a t h e o r y which closely r e s e m b l e d * Work supported by Conselno Nacional De Desenvolvmento Cientifico e Tecnologico Grant no. 2222.0072/76. Present address: Department of Applied Physics, Faculty of Engineering, Nagoya University, Chigusa-Ku, Nagoya, Japan. 171
172
D.Y. K O J I M A
Langer's. They introduced the center-of-mass coordinates of an electron gas system to show explicitly the points made by Kohn. They showed that the electron coupling was brought about only through the center-of-mass variable of the system. H o w e v e r , their theory, as well as Langer's, left the renorrealization factors of the Fermi level uncalculated. The propagator method developed for equilibrium statistical mechanics by Montroll and Ward6), and later by Isihara7), was first utilized by Montroll and Ward 8) to describe the electrical conductivity. H o w e v e r , they developed the theory only to describe a t w o - c o m p o n e n t Boltzmann gas system. The method of counting similar graphs presented by them is, therefore, good for the Boltzmann gas, of which c o m p o n e n t s are countable particles. The use of the distribution function of an equilibrium system in linear response theory is established by the K u b o formula. In section 2 we shall develop the propagator method to include the first order exchange effect explicitly, and in section 4 show that the effect only appears in the expression of the relaxation time. The time-correlated distribution function in the m o m e n t u m representation b(p~, t~,pl, t'~) is also developed in the process of the formalism in section 2. In section 3 we shall review the derivation of the conductivity of an electronic plasma. The discussions and the t e m p e r a t u r e d e p e n d e n c e of the electrical conductivity are given in section 5.
2. Formalism
The grand ensemble average of a m a c r o s c o p i c observable of a system is defined as: the sum over the n u m b e r of particles N of the products of the activity z to the p o w e r N, the partition function ZN and the partition function average of the observable, divided by the grand partition function _~, under the same volume and temperature. Thus, the grand ensemble average of the electrical conductivity is written as N IN) N=I Z Z N O r . v (o3),
Orgy(O) ) = I
(2.1)
where (r.~ (N (~o) is the partition function average of the electrical conductivity as given by the K u b o formula for N charge carriers:
'N"-.'= f dte '" f dXl
O"l.zv I I,rJ!
0
(2.2,
0
where the superscript ( N ) represents the total number of particles of the system in the volume V, O0 is the distribution function defined as po = J ,~N~(t) the current operator, J~"(t)= exp(-flH)/ZN, and exp(ih ~tH)J~,u) e x p ( - i h - ~ t H ) . One must take the limit in the a b o v e expression for the volume V tending to infinity. The conductivity is given as the response of the system to the external electric field: E = E0exp(iwt). The grand
C O N D U C T I V I T Y OF AN I N T E R A C T I N G E L E C T R O N GAS
173
ensemble average is no longer a function of the total number of the particles, but of the activity. It is thus n e c e s s a r y to introduce a s u p p l e m e n t a r y equation which relates the total n u m b e r with the activity, i.e. N - In_____~, 0 ~ ] 0 In z o.v"
(2.3)
Eq. (2.3) relates the Fermi m o m e n t u m PF, defined by z = e x p ( [ 3 p ~ / 2 m ) , where [3 = l / k T and m is the electron mass under consideration, with the absolute Fermi m o m e n t u m P0 defined by N = ~
pg
V.
(2.4)
The ratio PFIPo obtained from eqs. (2.3) and (2.4) serves as the renorrealization factor. Eqs. (2.1) and (2.3) together determine the conductivity as the function of the number density of the electron gas, usually by making use of the absolute Fermi m o m e n t u m , p0. The total hamiltonian ~ of the system in the absence of the external field can be described as follows: Y( -= H + O,
(2.5a)
N
• ~ ~ ~ 4~(ri - ri),
(2.5b)
H ~- ~ , H ( j ) =
+
V(r i -R~)
,
./=1
(2.5c)
where 4~(ri - ri) is the coulombic repulsive potential between the ith and jth electrons and V ( r i - R , ) is the potential between the jth electron at rj and the a t h impurity atom located at R~. The total hamiitonian in eq. (2.5a) is for N electrons, thus eq. (2.2) is described with this ~. The electrical conductivity cr~v(w) can be separated as
o-,~.(,.,) = o't,~(w) to~ + o - ~ ( ~ ) + o - ~ ( ~ ) + . • -,
(2.6)
in accordance with the number of e - e interactions involved. Thus, the first term on the right-hand side is that for the ideal gas, and the second one for the first order exchange graphs, and so on. Each term in eq. (2.6) is still a function of Pv, unless renormalized. The K u b o formula, eq. (2.2), is re-expressed when no magnetic field is present, as iN,
1
o'~,~ (w) = ~ N V
~
e~e~ [ d z e - i ~ f d
A
i.i mimj
o
o
× Tr~m[exp( - ([3 - A + ih-~r)~)pi~ e x p ( - ( A - ih-~r)~)pj~],
(2.7)
where ei and mi are the ith electronic charge and mass, and Pi~ is the uth
174
D.Y. KOJIMA
c o m p o n e n t of the ith m o m e n t u m in the operator form. The above expression is now rewritten by making use of propagators, as was done by Montroll and Ward: 0%. ~ ' (w) =
~Nvl~eie'ifei'°'ffi,i 'Himi /-- d r
dA
11
dplt,~( l " p(2) i " l f '11) i " 12)
{}
x K ( p 11~,18+ih %:pC-~.A)K(p~2~,A'.p~L',ifi Ir)"
12.8)
where p,i,=
(p~.. p'J'. p "'3 . ....
t~'")'~ .
p~" = ( p '",~. p ',','. P'/~).
ZN = Y~ e x p ( - / 3 E , ) = ( dp~i'K(p ~i~,~: p~i,. 0), n
J
and
K ( p ~]~, sl; pl2) $2) = Z Illn(P~I))IIJ+(P ~21)e x p ( - ( s l - s2)E,, ),
(2.9)
tl
where g,,(p~tt) is a Fourier transform of a wave function of an N-particle system with H, satisfying the antisymmetric relation of the exchange of any two variables. In this f o r m the current is no longer an operator and counting similar graphs b e c o m e s possible. One must notice here that eq. (2.9) is the propagator with which particles at p(2~ at a complex reciprocal temperature s~ propagate to pm at s~. The two propagators in eq. (2.8) can be rewritten by introducing two more propagators, it becomes
K ( p I'~, f i + i h tr: p~2~ a)K(p~2~.a;p,l~.ih
Lr)
= f d p I3~dpl4~K(p ~1~,i~ lr;p~31, O)K(pl31 ft. pl21 A ) x K(pl2~, A • p<41,O)K(pI41, O" pll~,ih Ir).
(2.10)
The second and third propagators are the G r e e n ' s functions of Bloch equations, and the first and fourth, those of Schr6dinger equations. The fourth one is the complex conjugate form and follows the c.c. form of the Schr6dinger equation. These two are identical to the F e y n m a n propagators1°), and follow the same condition of the time flow. The G r e e n ' s functions of the two equations satisfy the following relations:
O K ( f 1~, 181" r 12', 182) + H K = 6(~L - ¢12)6(r ~L~- rI21),
(2.11)
with the conditions:
K = 6 ( f l~-r~'-'), =0.
if18t+/3e+0, if /31
and
hOK(r il~, ih-[tl; i
Otl
r 12), ih It2)
H K = - h16=( tl
- t2)8(r 11' - r(:~).
(2.12)
CONDUCTIVITY OF AN INTERACTING ELECTRON GAS
175
with the conditions:
K = 6(r "~- r(2~), i f t l ~ t 2 + 0 , =0,
if t~
where 6(r "~- r ¢z,) consists of 6-functions of the Slater determinant with an appropriate weight factor. It is possible to combine these two types of propagators at any order of the perturbation series. Introducing eq. (2.8) into eq. (2.1), the grand ensemble average of the electrical conductivity yields
t3 O'~,~((O) = ~
e2f
e -i'°t dt
f
0
dA
A n ( l ) A~(2) ~(I)~(2)KI~(I) p]2)), f "Vl '-'Vl V t , , V ~ ' W , ,
(2.13a)
where
l N zNf
b(p~,), p~2)) = V--~
dPl') dp~2)
x K(p (t~, fl + ih-%; p(2~, X)K(p(2~, X ; p"), ih-%) = blOl(p~j, p~21)+ bin+ bI21+.. ".
(2.13b)
The function b(p °J, p~2~) is the singlet distribution function, which can be expanded also according to the order of the e - e interaction as is expressed in eq. (2.13b). 2.1. The zeroth order contribution in the e-e interaction (~ = O) When electrons do not interact among themselves it is well known that the N-particle trace in eq. (2.2) can be rewritten in terms of one-particle traces. This term was extensively investigated by Chester and Thellung ~1) in 1959 and by many others. We shall review this briefly by means of the propagator method with emphasis on the counting of the graphs. The singlet distribution function without the e - e interaction is given as
×Kt°l(p ~1~,/3 + ih-%; p~Z~,A)Kl°l(p~2~,A;p"),ih-~r),
(2.14)
and --~°= '~u zNZ[~1= ~ zN f dp")KI°J(P~' fl; p~l~, 0),
(2.15)
where the superscript [0] means the absence of the e - e interaction. Each kernel in eq. (2.14) consists of the Slater determinant carrying l/N! as the norm and it now becomes
~ (1) upi ,4~(21 b[Ol(p]l,, p]2,) _-- l_~g=o ~N [(N zN - 1)!] 2 f/__~ dpi × the two corresponding determinants of one-body propagators,
(2.16)
176
D.Y. KOJIMA
where the one-body propagator is defined with an eigenfunction hamiltonian H k ( j ) in eq. (2.5) and the eigenvalue E,, as + t2~) e x p ( - ( s l - s g E , ) . K , ( p ] " . sl • p jt2) , s2)=~]4~,(p,Iii )da,(Pi
dpn(r} k')
of the
(2.17/
n
The product of the two determinants consists of ( N ! ) 2 terms, and the variables in question, pt~ and p~2L are distributed in various ways as the parts of clusters. Obviously from the assumption of a h o m o g e n e o u s system, when the two variables are placed in separate clusters, the contribution is zero. It is enough to consider only cases with the two variables in the same cluster. A set of clusters, At, is defined in which the two variables are distributed in various unintegrated ways. The method of counting graphs is the same as the case without time variables• One cycle of 1-toron, however, consists of a time d e v e l o p m e n t from ih tr at P~I" to zero at an arbitrary coordinate p~'. a t e m p e r a t u r e d e v e l o p m e n t through a relay point (p~t2~, A) to a point (p)4,./3), and finally from zero to ih ~r at p~l~ with the reciprocal temperature /3. This process is shown in fig. la, in which the complex t e m p e r a t u r e flows from right to [eft. Summits of graphs always belong to the set p~'~, while planes belong to the set p(2) and A, the reciprocal t e m p e r a t u r e integral from zero r e m o v e to /3.
p1(11~
pl(11
l
IT
(a)
--e
P
~'~
I~
0i~
•
IB
pc2,
0
p,~
i I,]-I?:
(b)
i~-1'~
(C)
0
<
0 P
0
Fig. I. Graphs appearing in the ideal gas contribution. Graph (a) represents l-toron with complex reciprocal temperature. Graphs (b) and (c) represent two different 2-torons.
CONDUCTIVITY OF AN INTERACTING ELECTRON GAS
177
Figs. lb and lc stand for the following clusters (in this case, 2-torons)
b),,-, f dp~'K,(p~", 13 + ih-%; p~'~, ih-tr)
(1
× Kt(pl t~, fl + ih-'r; p~2~,A)Kt(p~2~, A; p~tt, i h - ' r ) = Kt(p~ 1~, 2fl + ih-%; p~:~ A)K~(p~Z~, A; p~t~, ih-'r),
(2.18)
(1 - c),~--~Kt(p~ '~, fl + ih %; p~2~ A)Kt(p~Zl, /3 + A; p~lj, ih-~r).
(2.19)
similarly
Extending this to the /-toron case, one can obtain for At that I-I
A, =- (-)t-lz' ~_, Kt(p~ '~, (I - s)fl + ih 'r; p~2~,A) s=0
x K~(p~ 2~, s~ + A; p~), ih-%'),
(2.20)
where (-)'-J appears because of the Slater determinant, since /-torons exchange (l - 1) times their positions, and z ~ comes from z u, leaving z N t for the rest of the configurations. First, the N particles are separated into l and N - I in such a way that I particles form an l-toron, with p~l~ and p~2j in it. Addressing the two variables in the I-toron, one finds l - 1 available spaces for each set of the variables. Those spaces filled with the rest of the variables in the sets p"l and pt2~, and are integrated so that characteristic differences no longer exist. The number of ways of filling the spaces are [(N-1)(N-2)...(N-I-(I-1))]
[(N - 1)!] 2 2= ~ /)!d"
The above argument only describes one of the elements in At in eq. (2.20); however, A, is the collection of l-torons, with different configurations of p ~ and p~2). Thus, the first part concludes l)!J At.
(2.21a)
The rest of the variables form an arbitrary number of both the same and different toron clusters, and the same configurations appear [ ( N - I)!]z times after integration of all variables. All the different configurations of forming the clusters out of N - l variables are given as [hOist
1-I '~'" (z') s', st!
with a condition
~
tst
=
N
-
1,
t
where b ° is defined as
(2.21b)
178
D.Y. KOJIMA b0 _ ( - ) ' - ' (t!t - 1)'" I Kl(p~ '~, t/3; p~,,, 0) dpt, ,.
(2.22)
The detailed derivation of eq. (2.22) is given in a textbook ~2) by Isihara. Collecting all the factors and introducing them into eq. (2.16), the singlet distribution function of the zeroth order is built up to give
b,O~(p,,,,, p~2,) = V~--~l
[(N -I 1)[l 2 ~
×[(N-t)!]2~H 1 V
~,~,(
~(N - 1)!] 2/)[j A;
(btz t )'~'
1
~,! -v,~,
A;
)~+'+'.... 'K,(p]" t/3+ih
IT:
p~2),A)
t= 1 s=0
× Kdp~l 2', s~ + A, p~', ih'c).
(2.23)
2.2. The first order contribution There are two types of graphs appearing as the first order of the e - e interaction as the partition function is constructed. One of the two is removed by the effect of the positive background, that is, the types of graphs characterized by u(q = 0) = 0, where u(q) is the Fourier transform of the coulombic potential between two electrons. When the perturbation method is applied to the right-hand side of eq. (2.10), it is easier to treat each kernel (propagator) in the real space and later transform it to the momentum space. Each kernel is expanded in terms of u(q) and the same kernel as used in eq. (2.14). The kernel involves the impurity-electron interaction as it is. Since both the time and reciprocal temperature kernels possess the same structure, these are recombined after the proper perturbation process at any order of perturbation. Of course, the directions of the times must be chosen correctly. After all the above processes, the singlet distribution function is expressed as
_ dp~tl b(p~t, p,l,,)= V,-,~o~,~l l • • • N=I ~ N2zN f ~__ i dpi12j x [Kl°l(p ~, 13 + ih ~': p~2~,A )Kl°l(p ~2~,A : p~, ih ~') 13+ih ir
A
x KE°J(p~2',A : pt4~ ~,: ii)KiO](p~4), ~,: p(1,, ih t-c) A
+ [ d~,KiOl(p,~,~+ih
i :p(4,,~,:q)
ih 17"
xKt°'(p'".fl':p'2',A)K'°'(p'2',A'p't',ih
%)}+C(u2(q))],
(2.24)
CONDUCTIVITY OF AN INTERACTING ELECTRON GAS
179
where
u(q)=f 4,(r,,)exp(-ir,,.~)dr,,, K(P " ) , s l ; p ( 2 ) , s g = ~
1f
(2.25)
K(r "),sl;
s2)
x exp ( - ~,ri t_(~ ) • p(l)_ r (2~. p(2))]"~dr,~ dr(2).
(2.26)
The summation in eq. (2.24) together with Kt°J(p (2~, A; p(4) fl,:q) represents N !/(2!. ( N - 2)!) Slater determinants, in each of which a pair of interaction momenta --_q is placed without repetition in two arbitrary columns of the set, p(4L Thus, in K(p (2', A; p(4) ~ , . q ) one finds .(4~ . . . . . Pj-I, .,") p~4),p~24). . . . . p!4)+q, l-'i+l
p }4, _
q. . . .
, P (~,.
The higher order terms with respect to u(q) are also important to form the grand partition function and the grand ensemble average, and cannot be neglected, although in eq. (2.24) only the first order is shown for the purpose of abbreviation. The first term in eq. (2.24) is that for the non-interacting electrons, and it produces 1
V ~ 0 ~ I . . . bI°l(p] t~, p~2))~0.
(2.27)
From the first order term two different categories of collections of graphs appear, other than the separation of the pair variables. The first category (i) is that the pair appears in a toron with the e - e interaction, and the second OiL is that the pair and the e - e interaction, u(q), are placed in different torons. For (i), it is seen that the independent clusters, which are separated from p ] ' , p~2) and u(q) yields the grand partition function of the ideal gas, -'-0, and it is multiplied by bm(p] ~, p]Z)). For (ii), the clusters minus the cluster with the pair make a group which does not involve u(q) and a toron which forms the first order exchange interaction so as to create finally E b~llz t, where b~u is an irreducible cluster of the first order. The first order term in eq. (2.24) will, therefore, result in the following form:
1 V-=0-~l .. •
btU(p~'),p~2')~-o+ bl°l-'-o ~ b~'lz'l. t=z
(2.28)
,
The second order term in the perturbation will also produce similar terms. Considering these higher order terms together, one obtains eq. (2.13b). To obtain the explicit form of bl"(p~ ~, p]2~), it is thus sufficient to observe the first appearance of the term in eq. (2.24) for the category (i). We now define Ci for the purpose given in eq. (2.29), which is expressed only by the one-body kernel and one e - e interaction, u(q). It contains all the possibilities of constructing l-torons with the momentum pair and u(q) placed in various locations, as was done to construct At:
180
D.Y. KOJIMA
CI = (--)lZl
f
/3+ih
dq
u(q) f dp';dp" z'+s+(t)+u=l Y~ t, s. I'll, t, u~o
I'r
A
× (p], R; (p~ + q),/3')(p", W ' , p ~ , i h
~')
+ (p~, X " (~'( - q ), ~')(p'(, s ~ , it'" + q ), 0) × ( p " , Y ' . p I , A)(p',, W ' . p , , i h '~-) + (p,, X : p',, ,~)(p'~, R : (p'~- q), ~') × (p~', t~" (p'g + q), O)(p", W : p,, ih 'r)}
~- i d][3'{(pI'X'(p]t-q,'~t)(p'('S~-}-/3tt:pI'*) ih lr × (p',, t[3 + X : (p" + q),13')(p", W : p~, ih Lr) + (p,, X : (p~'- q) , ¢3')(p'(, st3: (p" + q), O) × (p", t'[3 + ~i'; p',, A )(p'j, W " p,, ih Lr) +(p~,X;pi,
A)(p'~,t/3+Az(p'{
×(p'{,5;/3;(p~+q),O)(p~,
q),¢3') (2.29)
W : p l , i]i "r)~], JJ
with X = v/3 + i h
t~-:
R=s/3+A,
W = u/3 + / Y , W'= u/3+A: pl = p~l" and p; = p412',
Y=tfl+/3',
where (t) in the sum-restriction is either t or t', and each bracket represents the o n e - b o d y kernel defined in eq. (2.17). The first, second, f o u r t h and fifth terms are s c h e m a t i c a l l y s h o w n in fig. 2 a - d for the case of l = 2. The third and sixth start f r o m l = 3. The thick lines in fig. 2 represent the region where the /3' integral variable a s s o c i a t e d with u ( q ) m o v e s , and the w a v y horizontal lines, u ( q ) itself, respectively. In the first and fourth g r a p h s the pair is partitioned by the w a v y lines but the s e c o n d and third have no such structure. T h e s e four g r a p h s c a n n o t be c o m b i n e d in a simple m a n n e r in general. As an e x a m p l e of c o n s t r u c t i n g the general case of C~ we only s h o w schematically the graph c o r r e s p o n d i n g to the first term in C~ in fig. 3. Following the same t e c h n i q u e as in the derivation of bL°J(p~ I', p~2~), one can obtain the following result: b[,I(p~, p~2,) = V
=.
Ct.
(2.3(})
Here, besides the distribution of the m o m e n t u m pair, one must c o n s i d e r two more variables which belong to the set p~"~ in eq. (2.24), and which play roles of the e n t r a n c e and exit of the interaction variable q. The s e c o n d c a t e g o r y (ii)
CONDUCTIVITY
OF AN INTERACTING
ELECTRON GAS
181
cdl
Fig. 2. Graphs
for the first order
exchange
contributions
in the case of
I = 2.
Fig. 3. Graphical representation of the first term in eq. (2.29). where @ and @ represent 6 + iKiT-p’ and P’-A. respectively. The summand parameter t starts with zero; however (t - I)p appears because the boxes on both sides of the (I - l)p boxes can become identical.
is easily investigated following the ted here. By introducing eq. (2.23) into represent it by the one-body trace, electrical conductivity without the
same
technique,
but it will not be presen-
eq. (2.13a) and breaking one obtains the well-known e-e interaction:
the kernels to formula of the
182
D.Y. KOJIMA
c r ~ ( w ) = ~Vm
dr e ~" 0
dA
Tr'"{pu(r)f(H)p~(-ih
tr)(l
-f(H))},
(2.31)
0
where f(H) is the Fermi distribution function with the hamiltonian H(j). Similarly, by introducing eq. (2.30) and eq. (2.29) into eq. (2.13a) and performing the same procedure, the first order contribution to the conductivity. o-~(o)) yields t,I. O'~wtCO) =
S-'Vm 2 f d r e
'~*fda
o
f dp,"dp',"fdqh
3u(q)
o
/3+ih Ir
×[
d~'{a]P.(r+i~'h)lb)(clP,,(ih(~'-A))ld)
f A
+ (arf
p . ( i h ( I T - A ) ) P . ( r + i~'h)lb)(ca)
+ (a]P.(r + i~'h)p,,(ih(~' - A ))flb)(cd)} + i d~'{ (a]P"(r+i~'h)tb}(clfpAih(~'-At)fld) ih Ir
+ (a[P.(r + i¢~'h)p,,fih(~' - a ))f +(alfp.Iih(b'
Ib)(cd) (..3.)
a ) ) G ( r +i/~'h)lb)(cd)}],
where
I.> -= [pf'>:
f~f(H):
fb) -= Ip'," - q):
f ---l-f(HI:
f,') =-Ip',"): (cd)=(cIfld)
q), P.(tl:fp.(t)f.
fd) ~-Iv'?' +
and
The above expression is most general and cannot be expressed in terms of the one-body trace.
3. The conductivity of the electron plasma (no-impurity) When the impurity effect is not introduced, the whole theory must be reduced to that for the ideal gas even with the presence of the e - e interaction. Although the result is obvious it is worthwhile to show how it is derived in our theory. The hamiltonian H here consists of only the kinetic energy and commutes with the momentum operator. The kernels for eq. (2.23) are those for the ideal gas, then
K,(p~ '~, sl; p~t2~,s2) = 6~3'(pC~'~- p~2~)exp{-(s, - s2)(p~)2/2m}. As a result one obtains
(3. I)
CONDUCTIVITY OF AN INTERACTING ELECTRON GAS eZ P~ cr~(w) = 6 ~ iwm 2 x 3"n'2h 3"
183 (3.2)
T h e first o r d e r contribution is given as
mtw )= m----~-dw-~ e z/32 1 j( dq u(q) f dp cry,, x {p,(p~ + qv)f(p)[l - f ( p ) ] f ( p + q)[! - f ( p + q)] + p~,p~[1 -f(p)][1 - 2f(p)]f(p)f(p + q)} tOl 3me2
= tr~,,(to)---p-~h .
(3.3)
T h e grand partition function n e c e s s a r y to this order can be f o u n d elsewhere'3), and the o p e r a t i o n in eq. (2.3) gives N_ V
Po3 _ P~ ,, ~-g-~' + 3 s + . • .),
(3.4)
and then P 0 = pF(I + 3s + " • .)tn,
(3.5)
where
1 [4"n'11/3
me2
P0 _
Adding eq. (3.2) to eq. (3.3) and introducing eq. (3.5), one obtains
o-~(w) = e2n
iwm '
(3.6)
w h e r e n = p]/67rZh 3. T h r o u g h o u t the f o r m a l i s m the electronic spin f a c t o r w a s neglected.
4. Impurity effect on the conductivity T h e p r o c e d u r e of solving eq. (2.31) is c o m p l i c a t e d for finite f r e q u e n c i e s but it is possible in the case w h e n the f r e q u e n c i e s are m u c h smaller than the F e r m i e n e r g y as w a s done by T . Y . Q ) ) . To m a k e the situation simpler, let us a s s u m e the f r e q u e n c y to be zero here, and later c o m b i n e this result with the p l a s m a conductivity. C h e s t e r and Thellung s h o w e d the following description of the conductivity: T
cr, v = / 3 lim I dt Tr{½Po[J~(t)Jv(O) + J,,(0)J~(t)]} T-~e
. -T T
= ½/3 r-~lim f dt Tr{pj~(t)J~(0)}, -T
(4.1)
184
D.Y. KOJIMA
where O0 is either exp(/3[t~N-H])/~-o or exp(-/3H)/Z°u, and B ( t ) = exp(ih ~t~)B e x p ( - i h ~t~). Eq. (4.1) can be obtained with the assumption that o',v = o',~,. With this formula one notices some slight change in the complex reciprocal temperature in the propagator expression. In eqs. (2.23) and (2.29) the reciprocal temperatures,/3 + ih-%, v/3 + ih-% and t/3 + ih ~r are replaced by /3, v/3 and t/3, all the A values, by ih-tr, and ih-% in the last kernels, by zero, respectively. The A-integration must be replaced by /3. Because of the lengthy expression the explicit forms are omitted. The expression with operators are now given with the hamiltonian of one particle in eq. (2.5c) as 1
crt°' - e2/3 tim f dt f d p ( p l f ( H , ( l -
f(H))pu(Olp~(t)]p).
(4.2)
r 1
tr m -
e2/3 I~m f dt l a p f d p ' f dq u(q} T t3
x [ I d/3'{(P'lP"(ih/3'llP - q)(plP,.(t)lp + q) 0
+ (p'lf_p,(tlP.(i/3'h)lp - q)(plflp'+ q) (p'[Pg(ih/3')p~(t)flp - q)(p[flp'+ q)}
-
t
,f dr{(P']P"(-r)lP
+g
- q)(plfp~(t - r) - pat - ~')flp' + q)
0
+ (p'lP,(-r)p~(t - r) - pat - r)Pu(-r)lp - q)(p]f]p'+ q)}],
(4.3)
where f, f_, PAt) are defined in eq. (2.33). The first three terms can be combined by making use of the fact that ~4)
0
Ou. f =
f d/3' P.(ih/3') = f d/3' f(H)p.(-ih/3')(l 0
- f(H)),
(4.4a)
0
where f ' = [1 + exp(/3(H
- EF--p
• u))] -I,
with u being a c-number. One then obtains: /at m
e2/3 tim f d t f d p d p ' d q u ( q ) u~.O
x
-T
(4.4b)
C O N D U C T I V I T Y O F AN I N T E R A C T I N G E L E C T R O N G A S
185
t
same
,,.,,
as
0
To obtain the lowest order in impurity density, ni, it is only necessary to maintain the correct hamiltonian in the places of p~(t), and the rest of the hamiltonians are approximated by those for the free particles. The second part of the right-hand side of eq. (4.5) then disappears and eqs. (4.2) and (4.5) are re-expressed to give the following simple forms: r[°l ,.~ --
e2fl f dp(l-/o(p))fo(p)p~, m----~l~m T
×
[ (plp~(t)lp)dt,
(4.6a)
-T
~,~= ~e2fl _--l~m_f dp dq u(q ) O__.~_Ou~, [f;(P + q)[i
or[ll
-
f~(p)lf~(p)]
u-*0 T
X
[ (plp~(t)[p)dt.
(4.6b)
T
Assuming that the impurities are randomly distributed in the system with the density hi, and that only binary scattering is important, the time integral in eq. (4.6) can be given, following the same method as van Hove'7), as T
lim T~oo
[ (plp~(t)lp') dt = p~r(p)6p, p',
J -T
(4.7a)
where (2¢r)4ni
1
r(p)
(2~
f dk[V(k)126(Ep+k - Ep)
× [1 - c o s ( p , p + k)].
(4.7b)
Substituting eq. (4.7a) into eq. (4.6) and recognizing that a possible value of p is only at PF and that r(p) is a smooth function of p at least near PF, then t-(p) can be factored out of the integration, as r(pF). The final expression is identical to the case without impurities except that instead of 1/ioJ, 1"(pF) is introduced. Thus,
L~oJ e2n
= 6,~--m--¢(pF).
+3s+-..) (4.8)
Here, the first order exchange effect only appears through the value of PF in the expression of the relaxation time. The addition of relaxation times from
186
D.Y. K O J I M A
various sources is given as I
-
"/'tot
1 + --+--+ 1
TI
T2
1
. .
•
T~
and if one recognizes that 1/i~o is a part of the relaxation time mechanism. then eq. (3.6) may be added to eq. (4.8) to obtain -~ ~
o'~
e2n
r(pF) m 1 +i~T(pv)
.
(4.9)
It is important to notice that z(pv) also depends on the frequency ~o, in reality, but it may be neglected if ~o is much smaller than EF. Thus, the Drude formula for the low frequencies is obtained. The ideal gas part of the T.Y.Q.'s result has the opposite sign to our result. Our result is the consequence of the definition of the external electric field. It therefore does not contradict the classical result. G6tze and W61fle obtained the frequency dependence of the relaxation time quantitatively, and showed that the dependence is minor for small frequencies. The relaxation time at the Fermi momentum is given, by integrating the radial part of the momentum variable, by 1
"r(pv)
hireR0
(2rr)Z[pv]po]2[1 + 3s + • • "] f d/2 nimpv ( dO o'p~(O)(1 - cos 0),
= (2~.)2
o'pv(O)(I - cos 0) (4.10)
J
where
V(k)= f V ( r ) e x P ( h k . r ) d r
and
crpv(O)= l V(pF--
Here, the scattering amplitude, ~pF(0), depends on in front of the integral in eq. (4.10).
5.
pv
as well as the coefficient
Discussion
We have p e r f o r m e d explicit calculations of the e - e interaction for the electrical conductivity to the lowest order of impurity density by making use of quantum statistical mechanics, and also have investigated the pv-dependency of the electrical conductivity. The final expression is simple and only the relaxation time is purely pv-dependent. Its form is the same as that for the ideal gas relaxation time with impurity. The rest of the conductivity expression does not depend on PF but on the real n u m b e r density of the electron gas. It must be pointed out, however, that although the e - e interaction changes the magnitude of the relaxation time, this is not only because of the screening effect but also because of the decrease of the Fermi m o m e n t u m , which makes
CONDUCTIVITY OF AN INTERACTING ELECTRON GAS
187
electrons at the Fermi surface less active. To include the screening effect due to the electron cloud one has to involve the chain diagrammatic expression in between an impurity and an electron of interest; thus, V(k) will be modified approximately by
V(k) I + u(k)~ts(k)'
(5.1)
where Ai(k), the eigenvalues of the electron propagator, can be found in the same book by lsihara. The a p p e a r a n c e of this sort of screening effect is independent of the counting method of graphs which we have investigated. Assuming that only the j = 0 term with small m o m e n t u m is important, eq. (5.1) b e c o m e s
4~re2h:Z
r2
=
"r
2 FPF]
'P° [?-;0J'
(5.2)
where ~ = 412~:/3]2/3/rr 2, and Z is the number of excess valences of the impurity atom. Using eq. (5.2) the relaxation time has the form i
"r(pF)
-n'mp°r4e2Z]2(Pq3[a-A--1 +inL -] 71" [---~'---0 J
\PF] [1
+
(5.3)
A
with A = ~(r,/4)(po/PF). The ratio z(po) to r(pF) is approximately given as
• (00)
[p013
"/'(PF)
LPFJ
is.4)
The screening effect buffers the impurity effect so as to make the relaxation time in eq. (5.3) larger. On the other hand, the effects of the e - e interaction make PF decrease with an increase of rs so that, as can be seen from eq. (5.4), the real relaxation time decreases slightly from the ideal gas relaxation time. These two separate effects therefore play different roles on the relaxation time. Using eq. (4.10), the conductivity can be written as rl + 3s + ~?(s2)1 o',,,, = 8~ e2P~ t ni
x
If 0
sin 0 dOlv(p~--pF)12(1--COS(p~,pF))
}'
.
Comparing this with the result by Langer, one observes that the term (1 + 3s + ¢~(s2)) is not included in his expression. This c o m e s f r o m the fact that the Fermi m o m e n t u m kF in their theory, which differs from the absolute Fermi m o m e n t u m , is not identical to our PF. The Fermi m o m e n t u m which we use is directly connected with the activity or the chemical potential of an equilibrium state. The theory by T.Y.Q. is an extension of Langer's. T h e y have the final expression of the conductivity with an effective mass m* = po/ug, where ug is the group velocity at the Fermi level, and with the effective
188
D.Y. K O J I M A
relaxation time r*, which also contains u,. Re-expression of their result shows that the group velocity goes a w a y from the expression of conductivity. The result is given in their notation as
Ree2 T* ~
o"
nee2 m
1 _
nepom F,
,n o: (po, 0)3 2
r
~ [ '
po
ap0
J
× fdOlu(O)12(l - cos 0),
(5.5)
where P0 is the absolute Fermi m o m e n t u m . Comparing this expression with ours, eq. (4.10), their renormalization should be identified as
m O~(po, O)12 po Opo j "
pF]2[1 P00_l + 3 s + - - - l * + L 1
F
It is not clear in their notation how lu(0)] 2 depends on pv. When the classical equation of motion for the center-of-mass degree of freedom is used to obtain the relaxation time, i.e. the center-of-mass relaxation time, the m o m e n t u m appearing in the expression must be the Fermi momentum, p v , which involves the quantum effects and the e - e interaction effects. Our theory includes the t e m p e r a t u r e d e p e n d e n c e in a natural way. Using the facts that
2m f(p )[ 1 - f(p )] = d ~ f ( p ), (5.6) d
2m f(p)( 1 - f(p ))( 1 - 2f(p)) = -d-~{f(p )(1 - f(p ))}, and that the grand partition functions (~0, ~t) of the ideal gas, and the first order exchange contributions are given as In~o=~ V f dpln(l+z _
e
Op2/2m), (5.7)
In ~,x = ~2h,, f dp dq u(q)f(p)f(p + q), one finds that eqs. (2.32), (3.3) and (4.6) are e x p r e s s e d in terms of eq. (5.7) as O'~,
_[0l.
_[11
"4- •
•
•
0 In - % + In ,-~lx|1 + . . '. ,9 i n Z J
(5.8a)
Using eq. (4.9), it is also given as
Or
e2n
r(pv)
m I +itor(pF)'
(5.8b)
CONDUCTIVITY OF AN INTERACTING ELECTRON GAS
189
where n is the number density of the electron gas defined as
n-voInz
v.~"
Eqs. (5.8) imply that the ith order term in the electrical conductivity is directly related with the grand partition function of the same order in the same manner, but this is not an obvious fact. Our theory has no restriction on temperatures of the system. Ting, Ying and Quinn 5) shows that the electrical conductivity is related to the electronic interaction only through impurity potentials. Our result agrees with their emphasis even at finite temperatures. Langer discussed the temperature dependence of the conductivity in his paperZ), and showed that T z terms canceled within the lowest order in impurity density in a somewhat complicated way and that the possibility appeared from the impurity-independent term of the conductivity. In our theory not only T z terms but also all the orders in the temperature vanish in the expression of the conductivity, if the lowest order in the impurity density is considered. The expression of the conductivity only involves the real number density and the Fermi momentum PF in the expression of the relaxation time. Therefore, the Fermi momentum PF is the only source of the temperature dependence. This reflects the fact that the Fermi sphere changes its size due to the thermal agitation as well as interactions among themselves. In general when the temperature increases, the chemical potential Ev decreases, and so does PF- From the facts that the electrical conductivity is a function of PF and that PF is a function of the interaction strength and the thermal agitation, we conclude the importance of finding a correct expression of pF(rs,(kT)2). In the theory of Gell-Mann et al. ~6) and many others, the Fermi momentum PF was not evaluated explicitly, since it was not essential to express macroscopic quantities at absolute zero with respect to the density of the system. Before closing our discussion we would like to make a few points concerning our result. The dependence of the relaxation time and the conductivity on the Fermi momentum PF would not change even with the inclusion of the higher order e - e interactions, apart from the argument concerning screening effects. If so, the effective mass appearing in the conductivity is mainly due to the band effect. This may be true since the e - e interaction effect only appears through the impurity potential with PF. The Fermi momentum up to the third order ring diagram contribution (except the third order exchange) was recently calculated by the author and Isihara~5). In the metallic region the ratio, Pdpo, decreases from unity by 20-50%.
Acknowledgement The author would like to thank Prof. H. Tanaka in the Nagoya Institute of Technology for reading the manuscript and for many helpful discussions.
190
D.Y. KOJIMA
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) I1) 12) 13) 14) 15) 16) 17)
W. Kohn, Phys. Rev. 123 (1%1) 1242. J.S. Langer, Phys. Rev. 120 (1%0) 714; 124 (19611 1003:127 (1962) 5. W. G6tze and P. W/51fle, Phys. Rev, B6 (1972) 1226. H. Mori, Progr. Theor. Phys. (Kyoto) 33 (1965) 423; 34 (1965) 399. C.S. Ting, S.C. Ying and J.J. Quinn, Phys. Rev. B14 (19761 4439. E.W. Montroll and J.C. Ward, Phys. Fluids 1 (1958) 55. A. Isihara, Progr. Theor. Phys. Suppl. 44 (1969) I. E.W. Montroll and J.C. Ward, Physica 25 (1959) 423. R. Kubo, J. Phys, Soc. Jap. 12 (1957) 570. R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1%5). G.V. Chester and A. Thellung, Proc. Phys. Soc. 73 (1959) 745. A. lsihara, Statistical Physics (Academic Press, New York, 1971)oh. 11). A. Isihara and D.Y. Kojima, Z. Phys. B21 (19751 33. S. Fujita, Introduction to Non-equilibrium Quantum Statistical Mechanics IW.B. Saunders Co., Philadelphia and London, 1%6). D.Y. Kojima and A. lsihara, Z. Phys. B25 (1976) 167. M. GelI-Mann, Phys. Rev. 106 (1957) 369. M. GelI-Mann and K.A. Brueckner, Phys. Rev. 106 (1957) 364. L. Van Hove, Physica 21 (1955) 517:23 I1957) 411.
E L E C T R I C A L C O N D U C T I V I T Y O F AN I N T E R A C T I N G E L E C T R O N GAS* David Y. KOJIMA$ r
C.P.R.D.-A.M.D., lnstituto De Energia Atdmica. Caixa Postal 11.049, Pinheiros, CEP 05508, Sao Paulo, SP. Brazil
Received 25 August 1977
The many-body theory by the propagator method developed by Montroll and Ward for equilibrium statistical mechanics, is reformulated to describe the electrical conductivity for an electron gas system containing impurity atoms. The theory includes electron-impurity interaction to infinite order and electron-electron interaction to the first order exchange effect. The propagator used by Montroll and Ward is separated into two propagators, each one of which satisfies either the Bloch or the Schr6dinger equation, to utilize the perturbation method. Correct countings of graphs are presented. The change in the relaxation time due to the electronic interaction and the temperature change is discussed and compared with recent works.
I. Introduction It has been a s s u m e d f o r a long time that the e l e c t r o n - e l e c t r o n ( e - e ) interaction did not have a large effect on the electrical c o n d u c t i v i t y and m a n y w o r k s w e r e r e p o r t e d with this assumption. K o h n ~) pointed out that the effect o f the e - e interaction only existed t h r o u g h the b r e a k d o w n of the Galilean translational s y m m e t r y as a result of the introduction of fixed impurities. L a n g e r has r e p o r t e d in his series of papers 2) the i m p o r t a n c e of the e - e interaction t h r o u g h the electronic screening effect on impurities, and the t e m p e r a t u r e d e p e n d e n c e . T h e f r e q u e n c y d e p e n d e n c e of the relaxation f u n c t i o n b y the r a n d o m p h a s e a p p r o x i m a t i o n has been calculated by G/)tze and W61fle 3) f o r a f r e q u e n c y range f r o m zero to values l a r g e r than twice the F e r m i e n e r g y EF, using the m e m o r y f u n c t i o n t e c h n i q u e d e v e l o p e d by Mori4). T h e t e c h n i q u e a p p e a r s to be p o w e r f u l , but the introduction o f the e - e interaction is not clear. O n e might have to take into c o n s i d e r a t i o n the locality o f the external electric field w h e n the e - e interaction is included b e c a u s e the C o u l o m b interaction is of long range, w h e r e a s the w a v e n u m b e r o f an electric field with a f r e q u e n c y to ~ 2EF is of the o r d e r of 102-103 ,~, Ting, Y a n g and Q u i n n 5) (T.Y.Q.) r e c e n t l y d e v e l o p e d a t h e o r y which closely r e s e m b l e d * Work supported by Conselno Nacional De Desenvolvmento Cientifico e Tecnologico Grant no. 2222.0072/76. Present address: Department of Applied Physics, Faculty of Engineering, Nagoya University, Chigusa-Ku, Nagoya, Japan. 171
172
D.Y. K O J I M A
Langer's. They introduced the center-of-mass coordinates of an electron gas system to show explicitly the points made by Kohn. They showed that the electron coupling was brought about only through the center-of-mass variable of the system. H o w e v e r , their theory, as well as Langer's, left the renorrealization factors of the Fermi level uncalculated. The propagator method developed for equilibrium statistical mechanics by Montroll and Ward6), and later by Isihara7), was first utilized by Montroll and Ward 8) to describe the electrical conductivity. H o w e v e r , they developed the theory only to describe a t w o - c o m p o n e n t Boltzmann gas system. The method of counting similar graphs presented by them is, therefore, good for the Boltzmann gas, of which c o m p o n e n t s are countable particles. The use of the distribution function of an equilibrium system in linear response theory is established by the K u b o formula. In section 2 we shall develop the propagator method to include the first order exchange effect explicitly, and in section 4 show that the effect only appears in the expression of the relaxation time. The time-correlated distribution function in the m o m e n t u m representation b(p~, t~,pl, t'~) is also developed in the process of the formalism in section 2. In section 3 we shall review the derivation of the conductivity of an electronic plasma. The discussions and the t e m p e r a t u r e d e p e n d e n c e of the electrical conductivity are given in section 5.
2. Formalism
The grand ensemble average of a m a c r o s c o p i c observable of a system is defined as: the sum over the n u m b e r of particles N of the products of the activity z to the p o w e r N, the partition function ZN and the partition function average of the observable, divided by the grand partition function _~, under the same volume and temperature. Thus, the grand ensemble average of the electrical conductivity is written as N IN) N=I Z Z N O r . v (o3),
Orgy(O) ) = I
(2.1)
where (r.~ (N (~o) is the partition function average of the electrical conductivity as given by the K u b o formula for N charge carriers:
'N"-.'= f dte '" f dXl
O"l.zv I I,rJ!
0
(2.2,
0
where the superscript ( N ) represents the total number of particles of the system in the volume V, O0 is the distribution function defined as po = J ,~N~(t) the current operator, J~"(t)= exp(-flH)/ZN, and exp(ih ~tH)J~,u) e x p ( - i h - ~ t H ) . One must take the limit in the a b o v e expression for the volume V tending to infinity. The conductivity is given as the response of the system to the external electric field: E = E0exp(iwt). The grand
C O N D U C T I V I T Y OF AN I N T E R A C T I N G E L E C T R O N GAS
173
ensemble average is no longer a function of the total number of the particles, but of the activity. It is thus n e c e s s a r y to introduce a s u p p l e m e n t a r y equation which relates the total n u m b e r with the activity, i.e. N - In_____~, 0 ~ ] 0 In z o.v"
(2.3)
Eq. (2.3) relates the Fermi m o m e n t u m PF, defined by z = e x p ( [ 3 p ~ / 2 m ) , where [3 = l / k T and m is the electron mass under consideration, with the absolute Fermi m o m e n t u m P0 defined by N = ~
pg
V.
(2.4)
The ratio PFIPo obtained from eqs. (2.3) and (2.4) serves as the renorrealization factor. Eqs. (2.1) and (2.3) together determine the conductivity as the function of the number density of the electron gas, usually by making use of the absolute Fermi m o m e n t u m , p0. The total hamiltonian ~ of the system in the absence of the external field can be described as follows: Y( -= H + O,
(2.5a)
N
• ~ ~ ~ 4~(ri - ri),
(2.5b)
H ~- ~ , H ( j ) =
+
V(r i -R~)
,
./=1
(2.5c)
where 4~(ri - ri) is the coulombic repulsive potential between the ith and jth electrons and V ( r i - R , ) is the potential between the jth electron at rj and the a t h impurity atom located at R~. The total hamiitonian in eq. (2.5a) is for N electrons, thus eq. (2.2) is described with this ~. The electrical conductivity cr~v(w) can be separated as
o-,~.(,.,) = o't,~(w) to~ + o - ~ ( ~ ) + o - ~ ( ~ ) + . • -,
(2.6)
in accordance with the number of e - e interactions involved. Thus, the first term on the right-hand side is that for the ideal gas, and the second one for the first order exchange graphs, and so on. Each term in eq. (2.6) is still a function of Pv, unless renormalized. The K u b o formula, eq. (2.2), is re-expressed when no magnetic field is present, as iN,
1
o'~,~ (w) = ~ N V
~
e~e~ [ d z e - i ~ f d
A
i.i mimj
o
o
× Tr~m[exp( - ([3 - A + ih-~r)~)pi~ e x p ( - ( A - ih-~r)~)pj~],
(2.7)
where ei and mi are the ith electronic charge and mass, and Pi~ is the uth
174
D.Y. KOJIMA
c o m p o n e n t of the ith m o m e n t u m in the operator form. The above expression is now rewritten by making use of propagators, as was done by Montroll and Ward: 0%. ~ ' (w) =
~Nvl~eie'ifei'°'ffi,i 'Himi /-- d r
dA
11
dplt,~( l " p(2) i " l f '11) i " 12)
{}
x K ( p 11~,18+ih %:pC-~.A)K(p~2~,A'.p~L',ifi Ir)"
12.8)
where p,i,=
(p~.. p'J'. p "'3 . ....
t~'")'~ .
p~" = ( p '",~. p ',','. P'/~).
ZN = Y~ e x p ( - / 3 E , ) = ( dp~i'K(p ~i~,~: p~i,. 0), n
J
and
K ( p ~]~, sl; pl2) $2) = Z Illn(P~I))IIJ+(P ~21)e x p ( - ( s l - s2)E,, ),
(2.9)
tl
where g,,(p~tt) is a Fourier transform of a wave function of an N-particle system with H, satisfying the antisymmetric relation of the exchange of any two variables. In this f o r m the current is no longer an operator and counting similar graphs b e c o m e s possible. One must notice here that eq. (2.9) is the propagator with which particles at p(2~ at a complex reciprocal temperature s~ propagate to pm at s~. The two propagators in eq. (2.8) can be rewritten by introducing two more propagators, it becomes
K ( p I'~, f i + i h tr: p~2~ a)K(p~2~.a;p,l~.ih
Lr)
= f d p I3~dpl4~K(p ~1~,i~ lr;p~31, O)K(pl31 ft. pl21 A ) x K(pl2~, A • p<41,O)K(pI41, O" pll~,ih Ir).
(2.10)
The second and third propagators are the G r e e n ' s functions of Bloch equations, and the first and fourth, those of Schr6dinger equations. The fourth one is the complex conjugate form and follows the c.c. form of the Schr6dinger equation. These two are identical to the F e y n m a n propagators1°), and follow the same condition of the time flow. The G r e e n ' s functions of the two equations satisfy the following relations:
O K ( f 1~, 181" r 12', 182) + H K = 6(~L - ¢12)6(r ~L~- rI21),
(2.11)
with the conditions:
K = 6 ( f l~-r~'-'), =0.
if18t+/3e+0, if /31
and
hOK(r il~, ih-[tl; i
Otl
r 12), ih It2)
H K = - h16=( tl
- t2)8(r 11' - r(:~).
(2.12)
CONDUCTIVITY OF AN INTERACTING ELECTRON GAS
175
with the conditions:
K = 6(r "~- r(2~), i f t l ~ t 2 + 0 , =0,
if t~
where 6(r "~- r ¢z,) consists of 6-functions of the Slater determinant with an appropriate weight factor. It is possible to combine these two types of propagators at any order of the perturbation series. Introducing eq. (2.8) into eq. (2.1), the grand ensemble average of the electrical conductivity yields
t3 O'~,~((O) = ~
e2f
e -i'°t dt
f
0
dA
A n ( l ) A~(2) ~(I)~(2)KI~(I) p]2)), f "Vl '-'Vl V t , , V ~ ' W , ,
(2.13a)
where
l N zNf
b(p~,), p~2)) = V--~
dPl') dp~2)
x K(p (t~, fl + ih-%; p(2~, X)K(p(2~, X ; p"), ih-%) = blOl(p~j, p~21)+ bin+ bI21+.. ".
(2.13b)
The function b(p °J, p~2~) is the singlet distribution function, which can be expanded also according to the order of the e - e interaction as is expressed in eq. (2.13b). 2.1. The zeroth order contribution in the e-e interaction (~ = O) When electrons do not interact among themselves it is well known that the N-particle trace in eq. (2.2) can be rewritten in terms of one-particle traces. This term was extensively investigated by Chester and Thellung ~1) in 1959 and by many others. We shall review this briefly by means of the propagator method with emphasis on the counting of the graphs. The singlet distribution function without the e - e interaction is given as
×Kt°l(p ~1~,/3 + ih-%; p~Z~,A)Kl°l(p~2~,A;p"),ih-~r),
(2.14)
and --~°= '~u zNZ[~1= ~ zN f dp")KI°J(P~' fl; p~l~, 0),
(2.15)
where the superscript [0] means the absence of the e - e interaction. Each kernel in eq. (2.14) consists of the Slater determinant carrying l/N! as the norm and it now becomes
~ (1) upi ,4~(21 b[Ol(p]l,, p]2,) _-- l_~g=o ~N [(N zN - 1)!] 2 f/__~ dpi × the two corresponding determinants of one-body propagators,
(2.16)
176
D.Y. KOJIMA
where the one-body propagator is defined with an eigenfunction hamiltonian H k ( j ) in eq. (2.5) and the eigenvalue E,, as + t2~) e x p ( - ( s l - s g E , ) . K , ( p ] " . sl • p jt2) , s2)=~]4~,(p,Iii )da,(Pi
dpn(r} k')
of the
(2.17/
n
The product of the two determinants consists of ( N ! ) 2 terms, and the variables in question, pt~ and p~2L are distributed in various ways as the parts of clusters. Obviously from the assumption of a h o m o g e n e o u s system, when the two variables are placed in separate clusters, the contribution is zero. It is enough to consider only cases with the two variables in the same cluster. A set of clusters, At, is defined in which the two variables are distributed in various unintegrated ways. The method of counting graphs is the same as the case without time variables• One cycle of 1-toron, however, consists of a time d e v e l o p m e n t from ih tr at P~I" to zero at an arbitrary coordinate p~'. a t e m p e r a t u r e d e v e l o p m e n t through a relay point (p~t2~, A) to a point (p)4,./3), and finally from zero to ih ~r at p~l~ with the reciprocal temperature /3. This process is shown in fig. la, in which the complex t e m p e r a t u r e flows from right to [eft. Summits of graphs always belong to the set p~'~, while planes belong to the set p(2) and A, the reciprocal t e m p e r a t u r e integral from zero r e m o v e to /3.
p1(11~
pl(11
l
IT
(a)
--e
P
~'~
I~
0i~
•
IB
pc2,
0
p,~
i I,]-I?:
(b)
i~-1'~
(C)
0
<
0 P
0
Fig. I. Graphs appearing in the ideal gas contribution. Graph (a) represents l-toron with complex reciprocal temperature. Graphs (b) and (c) represent two different 2-torons.
CONDUCTIVITY OF AN INTERACTING ELECTRON GAS
177
Figs. lb and lc stand for the following clusters (in this case, 2-torons)
b),,-, f dp~'K,(p~", 13 + ih-%; p~'~, ih-tr)
(1
× Kt(pl t~, fl + ih-'r; p~2~,A)Kt(p~2~, A; p~tt, i h - ' r ) = Kt(p~ 1~, 2fl + ih-%; p~:~ A)K~(p~Z~, A; p~t~, ih-'r),
(2.18)
(1 - c),~--~Kt(p~ '~, fl + ih %; p~2~ A)Kt(p~Zl, /3 + A; p~lj, ih-~r).
(2.19)
similarly
Extending this to the /-toron case, one can obtain for At that I-I
A, =- (-)t-lz' ~_, Kt(p~ '~, (I - s)fl + ih 'r; p~2~,A) s=0
x K~(p~ 2~, s~ + A; p~), ih-%'),
(2.20)
where (-)'-J appears because of the Slater determinant, since /-torons exchange (l - 1) times their positions, and z ~ comes from z u, leaving z N t for the rest of the configurations. First, the N particles are separated into l and N - I in such a way that I particles form an l-toron, with p~l~ and p~2j in it. Addressing the two variables in the I-toron, one finds l - 1 available spaces for each set of the variables. Those spaces filled with the rest of the variables in the sets p"l and pt2~, and are integrated so that characteristic differences no longer exist. The number of ways of filling the spaces are [(N-1)(N-2)...(N-I-(I-1))]
[(N - 1)!] 2 2= ~ /)!d"
The above argument only describes one of the elements in At in eq. (2.20); however, A, is the collection of l-torons, with different configurations of p ~ and p~2). Thus, the first part concludes l)!J At.
(2.21a)
The rest of the variables form an arbitrary number of both the same and different toron clusters, and the same configurations appear [ ( N - I)!]z times after integration of all variables. All the different configurations of forming the clusters out of N - l variables are given as [hOist
1-I '~'" (z') s', st!
with a condition
~
tst
=
N
-
1,
t
where b ° is defined as
(2.21b)
178
D.Y. KOJIMA b0 _ ( - ) ' - ' (t!t - 1)'" I Kl(p~ '~, t/3; p~,,, 0) dpt, ,.
(2.22)
The detailed derivation of eq. (2.22) is given in a textbook ~2) by Isihara. Collecting all the factors and introducing them into eq. (2.16), the singlet distribution function of the zeroth order is built up to give
b,O~(p,,,,, p~2,) = V~--~l
[(N -I 1)[l 2 ~
×[(N-t)!]2~H 1 V
~,~,(
~(N - 1)!] 2/)[j A;
(btz t )'~'
1
~,! -v,~,
A;
)~+'+'.... 'K,(p]" t/3+ih
IT:
p~2),A)
t= 1 s=0
× Kdp~l 2', s~ + A, p~', ih'c).
(2.23)
2.2. The first order contribution There are two types of graphs appearing as the first order of the e - e interaction as the partition function is constructed. One of the two is removed by the effect of the positive background, that is, the types of graphs characterized by u(q = 0) = 0, where u(q) is the Fourier transform of the coulombic potential between two electrons. When the perturbation method is applied to the right-hand side of eq. (2.10), it is easier to treat each kernel (propagator) in the real space and later transform it to the momentum space. Each kernel is expanded in terms of u(q) and the same kernel as used in eq. (2.14). The kernel involves the impurity-electron interaction as it is. Since both the time and reciprocal temperature kernels possess the same structure, these are recombined after the proper perturbation process at any order of perturbation. Of course, the directions of the times must be chosen correctly. After all the above processes, the singlet distribution function is expressed as
_ dp~tl b(p~t, p,l,,)= V,-,~o~,~l l • • • N=I ~ N2zN f ~__ i dpi12j x [Kl°l(p ~, 13 + ih ~': p~2~,A )Kl°l(p ~2~,A : p~, ih ~') 13+ih ir
A
x KE°J(p~2',A : pt4~ ~,: ii)KiO](p~4), ~,: p(1,, ih t-c) A
+ [ d~,KiOl(p,~,~+ih
i :p(4,,~,:q)
ih 17"
xKt°'(p'".fl':p'2',A)K'°'(p'2',A'p't',ih
%)}+C(u2(q))],
(2.24)
CONDUCTIVITY OF AN INTERACTING ELECTRON GAS
179
where
u(q)=f 4,(r,,)exp(-ir,,.~)dr,,, K(P " ) , s l ; p ( 2 ) , s g = ~
1f
(2.25)
K(r "),sl;
s2)
x exp ( - ~,ri t_(~ ) • p(l)_ r (2~. p(2))]"~dr,~ dr(2).
(2.26)
The summation in eq. (2.24) together with Kt°J(p (2~, A; p(4) fl,:q) represents N !/(2!. ( N - 2)!) Slater determinants, in each of which a pair of interaction momenta --_q is placed without repetition in two arbitrary columns of the set, p(4L Thus, in K(p (2', A; p(4) ~ , . q ) one finds .(4~ . . . . . Pj-I, .,") p~4),p~24). . . . . p!4)+q, l-'i+l
p }4, _
q. . . .
, P (~,.
The higher order terms with respect to u(q) are also important to form the grand partition function and the grand ensemble average, and cannot be neglected, although in eq. (2.24) only the first order is shown for the purpose of abbreviation. The first term in eq. (2.24) is that for the non-interacting electrons, and it produces 1
V ~ 0 ~ I . . . bI°l(p] t~, p~2))~0.
(2.27)
From the first order term two different categories of collections of graphs appear, other than the separation of the pair variables. The first category (i) is that the pair appears in a toron with the e - e interaction, and the second OiL is that the pair and the e - e interaction, u(q), are placed in different torons. For (i), it is seen that the independent clusters, which are separated from p ] ' , p~2) and u(q) yields the grand partition function of the ideal gas, -'-0, and it is multiplied by bm(p] ~, p]Z)). For (ii), the clusters minus the cluster with the pair make a group which does not involve u(q) and a toron which forms the first order exchange interaction so as to create finally E b~llz t, where b~u is an irreducible cluster of the first order. The first order term in eq. (2.24) will, therefore, result in the following form:
1 V-=0-~l .. •
btU(p~'),p~2')~-o+ bl°l-'-o ~ b~'lz'l. t=z
(2.28)
,
The second order term in the perturbation will also produce similar terms. Considering these higher order terms together, one obtains eq. (2.13b). To obtain the explicit form of bl"(p~ ~, p]2~), it is thus sufficient to observe the first appearance of the term in eq. (2.24) for the category (i). We now define Ci for the purpose given in eq. (2.29), which is expressed only by the one-body kernel and one e - e interaction, u(q). It contains all the possibilities of constructing l-torons with the momentum pair and u(q) placed in various locations, as was done to construct At:
180
D.Y. KOJIMA
CI = (--)lZl
f
/3+ih
dq
u(q) f dp';dp" z'+s+(t)+u=l Y~ t, s. I'll, t, u~o
I'r
A
× (p], R; (p~ + q),/3')(p", W ' , p ~ , i h
~')
+ (p~, X " (~'( - q ), ~')(p'(, s ~ , it'" + q ), 0) × ( p " , Y ' . p I , A)(p',, W ' . p , , i h '~-) + (p,, X : p',, ,~)(p'~, R : (p'~- q), ~') × (p~', t~" (p'g + q), O)(p", W : p,, ih 'r)}
~- i d][3'{(pI'X'(p]t-q,'~t)(p'('S~-}-/3tt:pI'*) ih lr × (p',, t[3 + X : (p" + q),13')(p", W : p~, ih Lr) + (p,, X : (p~'- q) , ¢3')(p'(, st3: (p" + q), O) × (p", t'[3 + ~i'; p',, A )(p'j, W " p,, ih Lr) +(p~,X;pi,
A)(p'~,t/3+Az(p'{
×(p'{,5;/3;(p~+q),O)(p~,
q),¢3') (2.29)
W : p l , i]i "r)~], JJ
with X = v/3 + i h
t~-:
R=s/3+A,
W = u/3 + / Y , W'= u/3+A: pl = p~l" and p; = p412',
Y=tfl+/3',
where (t) in the sum-restriction is either t or t', and each bracket represents the o n e - b o d y kernel defined in eq. (2.17). The first, second, f o u r t h and fifth terms are s c h e m a t i c a l l y s h o w n in fig. 2 a - d for the case of l = 2. The third and sixth start f r o m l = 3. The thick lines in fig. 2 represent the region where the /3' integral variable a s s o c i a t e d with u ( q ) m o v e s , and the w a v y horizontal lines, u ( q ) itself, respectively. In the first and fourth g r a p h s the pair is partitioned by the w a v y lines but the s e c o n d and third have no such structure. T h e s e four g r a p h s c a n n o t be c o m b i n e d in a simple m a n n e r in general. As an e x a m p l e of c o n s t r u c t i n g the general case of C~ we only s h o w schematically the graph c o r r e s p o n d i n g to the first term in C~ in fig. 3. Following the same t e c h n i q u e as in the derivation of bL°J(p~ I', p~2~), one can obtain the following result: b[,I(p~, p~2,) = V
=.
Ct.
(2.3(})
Here, besides the distribution of the m o m e n t u m pair, one must c o n s i d e r two more variables which belong to the set p~"~ in eq. (2.24), and which play roles of the e n t r a n c e and exit of the interaction variable q. The s e c o n d c a t e g o r y (ii)
CONDUCTIVITY
OF AN INTERACTING
ELECTRON GAS
181
cdl
Fig. 2. Graphs
for the first order
exchange
contributions
in the case of
I = 2.
Fig. 3. Graphical representation of the first term in eq. (2.29). where @ and @ represent 6 + iKiT-p’ and P’-A. respectively. The summand parameter t starts with zero; however (t - I)p appears because the boxes on both sides of the (I - l)p boxes can become identical.
is easily investigated following the ted here. By introducing eq. (2.23) into represent it by the one-body trace, electrical conductivity without the
same
technique,
but it will not be presen-
eq. (2.13a) and breaking one obtains the well-known e-e interaction:
the kernels to formula of the
182
D.Y. KOJIMA
c r ~ ( w ) = ~Vm
dr e ~" 0
dA
Tr'"{pu(r)f(H)p~(-ih
tr)(l
-f(H))},
(2.31)
0
where f(H) is the Fermi distribution function with the hamiltonian H(j). Similarly, by introducing eq. (2.30) and eq. (2.29) into eq. (2.13a) and performing the same procedure, the first order contribution to the conductivity. o-~(o)) yields t,I. O'~wtCO) =
S-'Vm 2 f d r e
'~*fda
o
f dp,"dp',"fdqh
3u(q)
o
/3+ih Ir
×[
d~'{a]P.(r+i~'h)lb)(clP,,(ih(~'-A))ld)
f A
+ (arf
p . ( i h ( I T - A ) ) P . ( r + i~'h)lb)(ca)
+ (a]P.(r + i~'h)p,,(ih(~' - A ))flb)(cd)} + i d~'{ (a]P"(r+i~'h)tb}(clfpAih(~'-At)fld) ih Ir
+ (a[P.(r + i¢~'h)p,,fih(~' - a ))f +(alfp.Iih(b'
Ib)(cd) (..3.)
a ) ) G ( r +i/~'h)lb)(cd)}],
where
I.> -= [pf'>:
f~f(H):
fb) -= Ip'," - q):
f ---l-f(HI:
f,') =-Ip',"): (cd)=(cIfld)
q), P.(tl:fp.(t)f.
fd) ~-Iv'?' +
and
The above expression is most general and cannot be expressed in terms of the one-body trace.
3. The conductivity of the electron plasma (no-impurity) When the impurity effect is not introduced, the whole theory must be reduced to that for the ideal gas even with the presence of the e - e interaction. Although the result is obvious it is worthwhile to show how it is derived in our theory. The hamiltonian H here consists of only the kinetic energy and commutes with the momentum operator. The kernels for eq. (2.23) are those for the ideal gas, then
K,(p~ '~, sl; p~t2~,s2) = 6~3'(pC~'~- p~2~)exp{-(s, - s2)(p~)2/2m}. As a result one obtains
(3. I)
CONDUCTIVITY OF AN INTERACTING ELECTRON GAS eZ P~ cr~(w) = 6 ~ iwm 2 x 3"n'2h 3"
183 (3.2)
T h e first o r d e r contribution is given as
mtw )= m----~-dw-~ e z/32 1 j( dq u(q) f dp cry,, x {p,(p~ + qv)f(p)[l - f ( p ) ] f ( p + q)[! - f ( p + q)] + p~,p~[1 -f(p)][1 - 2f(p)]f(p)f(p + q)} tOl 3me2
= tr~,,(to)---p-~h .
(3.3)
T h e grand partition function n e c e s s a r y to this order can be f o u n d elsewhere'3), and the o p e r a t i o n in eq. (2.3) gives N_ V
Po3 _ P~ ,, ~-g-~' + 3 s + . • .),
(3.4)
and then P 0 = pF(I + 3s + " • .)tn,
(3.5)
where
1 [4"n'11/3
me2
P0 _
Adding eq. (3.2) to eq. (3.3) and introducing eq. (3.5), one obtains
o-~(w) = e2n
iwm '
(3.6)
w h e r e n = p]/67rZh 3. T h r o u g h o u t the f o r m a l i s m the electronic spin f a c t o r w a s neglected.
4. Impurity effect on the conductivity T h e p r o c e d u r e of solving eq. (2.31) is c o m p l i c a t e d for finite f r e q u e n c i e s but it is possible in the case w h e n the f r e q u e n c i e s are m u c h smaller than the F e r m i e n e r g y as w a s done by T . Y . Q ) ) . To m a k e the situation simpler, let us a s s u m e the f r e q u e n c y to be zero here, and later c o m b i n e this result with the p l a s m a conductivity. C h e s t e r and Thellung s h o w e d the following description of the conductivity: T
cr, v = / 3 lim I dt Tr{½Po[J~(t)Jv(O) + J,,(0)J~(t)]} T-~e
. -T T
= ½/3 r-~lim f dt Tr{pj~(t)J~(0)}, -T
(4.1)
184
D.Y. KOJIMA
where O0 is either exp(/3[t~N-H])/~-o or exp(-/3H)/Z°u, and B ( t ) = exp(ih ~t~)B e x p ( - i h ~t~). Eq. (4.1) can be obtained with the assumption that o',v = o',~,. With this formula one notices some slight change in the complex reciprocal temperature in the propagator expression. In eqs. (2.23) and (2.29) the reciprocal temperatures,/3 + ih-%, v/3 + ih-% and t/3 + ih ~r are replaced by /3, v/3 and t/3, all the A values, by ih-tr, and ih-% in the last kernels, by zero, respectively. The A-integration must be replaced by /3. Because of the lengthy expression the explicit forms are omitted. The expression with operators are now given with the hamiltonian of one particle in eq. (2.5c) as 1
crt°' - e2/3 tim f dt f d p ( p l f ( H , ( l -
f(H))pu(Olp~(t)]p).
(4.2)
r 1
tr m -
e2/3 I~m f dt l a p f d p ' f dq u(q} T t3
x [ I d/3'{(P'lP"(ih/3'llP - q)(plP,.(t)lp + q) 0
+ (p'lf_p,(tlP.(i/3'h)lp - q)(plflp'+ q) (p'[Pg(ih/3')p~(t)flp - q)(p[flp'+ q)}
-
t
,f dr{(P']P"(-r)lP
+g
- q)(plfp~(t - r) - pat - ~')flp' + q)
0
+ (p'lP,(-r)p~(t - r) - pat - r)Pu(-r)lp - q)(p]f]p'+ q)}],
(4.3)
where f, f_, PAt) are defined in eq. (2.33). The first three terms can be combined by making use of the fact that ~4)
0
Ou. f =
f d/3' P.(ih/3') = f d/3' f(H)p.(-ih/3')(l 0
- f(H)),
(4.4a)
0
where f ' = [1 + exp(/3(H
- EF--p
• u))] -I,
with u being a c-number. One then obtains: /at m
e2/3 tim f d t f d p d p ' d q u ( q ) u~.O
x
-T
(4.4b)
C O N D U C T I V I T Y O F AN I N T E R A C T I N G E L E C T R O N G A S
185
t
same
,,.,,
as
0
To obtain the lowest order in impurity density, ni, it is only necessary to maintain the correct hamiltonian in the places of p~(t), and the rest of the hamiltonians are approximated by those for the free particles. The second part of the right-hand side of eq. (4.5) then disappears and eqs. (4.2) and (4.5) are re-expressed to give the following simple forms: r[°l ,.~ --
e2fl f dp(l-/o(p))fo(p)p~, m----~l~m T
×
[ (plp~(t)lp)dt,
(4.6a)
-T
~,~= ~e2fl _--l~m_f dp dq u(q ) O__.~_Ou~, [f;(P + q)[i
or[ll
-
f~(p)lf~(p)]
u-*0 T
X
[ (plp~(t)[p)dt.
(4.6b)
T
Assuming that the impurities are randomly distributed in the system with the density hi, and that only binary scattering is important, the time integral in eq. (4.6) can be given, following the same method as van Hove'7), as T
lim T~oo
[ (plp~(t)lp') dt = p~r(p)6p, p',
J -T
(4.7a)
where (2¢r)4ni
1
r(p)
(2~
f dk[V(k)126(Ep+k - Ep)
× [1 - c o s ( p , p + k)].
(4.7b)
Substituting eq. (4.7a) into eq. (4.6) and recognizing that a possible value of p is only at PF and that r(p) is a smooth function of p at least near PF, then t-(p) can be factored out of the integration, as r(pF). The final expression is identical to the case without impurities except that instead of 1/ioJ, 1"(pF) is introduced. Thus,
L~oJ e2n
= 6,~--m--¢(pF).
+3s+-..) (4.8)
Here, the first order exchange effect only appears through the value of PF in the expression of the relaxation time. The addition of relaxation times from
186
D.Y. K O J I M A
various sources is given as I
-
"/'tot
1 + --+--+ 1
TI
T2
1
. .
•
T~
and if one recognizes that 1/i~o is a part of the relaxation time mechanism. then eq. (3.6) may be added to eq. (4.8) to obtain -~ ~
o'~
e2n
r(pF) m 1 +i~T(pv)
.
(4.9)
It is important to notice that z(pv) also depends on the frequency ~o, in reality, but it may be neglected if ~o is much smaller than EF. Thus, the Drude formula for the low frequencies is obtained. The ideal gas part of the T.Y.Q.'s result has the opposite sign to our result. Our result is the consequence of the definition of the external electric field. It therefore does not contradict the classical result. G6tze and W61fle obtained the frequency dependence of the relaxation time quantitatively, and showed that the dependence is minor for small frequencies. The relaxation time at the Fermi momentum is given, by integrating the radial part of the momentum variable, by 1
"r(pv)
hireR0
(2rr)Z[pv]po]2[1 + 3s + • • "] f d/2 nimpv ( dO o'p~(O)(1 - cos 0),
= (2~.)2
o'pv(O)(I - cos 0) (4.10)
J
where
V(k)= f V ( r ) e x P ( h k . r ) d r
and
crpv(O)= l V(pF--
Here, the scattering amplitude, ~pF(0), depends on in front of the integral in eq. (4.10).
5.
pv
as well as the coefficient
Discussion
We have p e r f o r m e d explicit calculations of the e - e interaction for the electrical conductivity to the lowest order of impurity density by making use of quantum statistical mechanics, and also have investigated the pv-dependency of the electrical conductivity. The final expression is simple and only the relaxation time is purely pv-dependent. Its form is the same as that for the ideal gas relaxation time with impurity. The rest of the conductivity expression does not depend on PF but on the real n u m b e r density of the electron gas. It must be pointed out, however, that although the e - e interaction changes the magnitude of the relaxation time, this is not only because of the screening effect but also because of the decrease of the Fermi m o m e n t u m , which makes
CONDUCTIVITY OF AN INTERACTING ELECTRON GAS
187
electrons at the Fermi surface less active. To include the screening effect due to the electron cloud one has to involve the chain diagrammatic expression in between an impurity and an electron of interest; thus, V(k) will be modified approximately by
V(k) I + u(k)~ts(k)'
(5.1)
where Ai(k), the eigenvalues of the electron propagator, can be found in the same book by lsihara. The a p p e a r a n c e of this sort of screening effect is independent of the counting method of graphs which we have investigated. Assuming that only the j = 0 term with small m o m e n t u m is important, eq. (5.1) b e c o m e s
4~re2h:Z
r2
=
"r
2 FPF]
'P° [?-;0J'
(5.2)
where ~ = 412~:/3]2/3/rr 2, and Z is the number of excess valences of the impurity atom. Using eq. (5.2) the relaxation time has the form i
"r(pF)
-n'mp°r4e2Z]2(Pq3[a-A--1 +inL -] 71" [---~'---0 J
\PF] [1
+
(5.3)
A
with A = ~(r,/4)(po/PF). The ratio z(po) to r(pF) is approximately given as
• (00)
[p013
"/'(PF)
LPFJ
is.4)
The screening effect buffers the impurity effect so as to make the relaxation time in eq. (5.3) larger. On the other hand, the effects of the e - e interaction make PF decrease with an increase of rs so that, as can be seen from eq. (5.4), the real relaxation time decreases slightly from the ideal gas relaxation time. These two separate effects therefore play different roles on the relaxation time. Using eq. (4.10), the conductivity can be written as rl + 3s + ~?(s2)1 o',,,, = 8~ e2P~ t ni
x
If 0
sin 0 dOlv(p~--pF)12(1--COS(p~,pF))
}'
.
Comparing this with the result by Langer, one observes that the term (1 + 3s + ¢~(s2)) is not included in his expression. This c o m e s f r o m the fact that the Fermi m o m e n t u m kF in their theory, which differs from the absolute Fermi m o m e n t u m , is not identical to our PF. The Fermi m o m e n t u m which we use is directly connected with the activity or the chemical potential of an equilibrium state. The theory by T.Y.Q. is an extension of Langer's. T h e y have the final expression of the conductivity with an effective mass m* = po/ug, where ug is the group velocity at the Fermi level, and with the effective
188
D.Y. K O J I M A
relaxation time r*, which also contains u,. Re-expression of their result shows that the group velocity goes a w a y from the expression of conductivity. The result is given in their notation as
Ree2 T* ~
o"
nee2 m
1 _
nepom F,
,n o: (po, 0)3 2
r
~ [ '
po
ap0
J
× fdOlu(O)12(l - cos 0),
(5.5)
where P0 is the absolute Fermi m o m e n t u m . Comparing this expression with ours, eq. (4.10), their renormalization should be identified as
m O~(po, O)12 po Opo j "
pF]2[1 P00_l + 3 s + - - - l * + L 1
F
It is not clear in their notation how lu(0)] 2 depends on pv. When the classical equation of motion for the center-of-mass degree of freedom is used to obtain the relaxation time, i.e. the center-of-mass relaxation time, the m o m e n t u m appearing in the expression must be the Fermi momentum, p v , which involves the quantum effects and the e - e interaction effects. Our theory includes the t e m p e r a t u r e d e p e n d e n c e in a natural way. Using the facts that
2m f(p )[ 1 - f(p )] = d ~ f ( p ), (5.6) d
2m f(p)( 1 - f(p ))( 1 - 2f(p)) = -d-~{f(p )(1 - f(p ))}, and that the grand partition functions (~0, ~t) of the ideal gas, and the first order exchange contributions are given as In~o=~ V f dpln(l+z _
e
Op2/2m), (5.7)
In ~,x = ~2h,, f dp dq u(q)f(p)f(p + q), one finds that eqs. (2.32), (3.3) and (4.6) are e x p r e s s e d in terms of eq. (5.7) as O'~,
_[0l.
_[11
"4- •
•
•
0 In - % + In ,-~lx|1 + . . '. ,9 i n Z J
(5.8a)
Using eq. (4.9), it is also given as
Or
e2n
r(pv)
m I +itor(pF)'
(5.8b)
CONDUCTIVITY OF AN INTERACTING ELECTRON GAS
189
where n is the number density of the electron gas defined as
n-voInz
v.~"
Eqs. (5.8) imply that the ith order term in the electrical conductivity is directly related with the grand partition function of the same order in the same manner, but this is not an obvious fact. Our theory has no restriction on temperatures of the system. Ting, Ying and Quinn 5) shows that the electrical conductivity is related to the electronic interaction only through impurity potentials. Our result agrees with their emphasis even at finite temperatures. Langer discussed the temperature dependence of the conductivity in his paperZ), and showed that T z terms canceled within the lowest order in impurity density in a somewhat complicated way and that the possibility appeared from the impurity-independent term of the conductivity. In our theory not only T z terms but also all the orders in the temperature vanish in the expression of the conductivity, if the lowest order in the impurity density is considered. The expression of the conductivity only involves the real number density and the Fermi momentum PF in the expression of the relaxation time. Therefore, the Fermi momentum PF is the only source of the temperature dependence. This reflects the fact that the Fermi sphere changes its size due to the thermal agitation as well as interactions among themselves. In general when the temperature increases, the chemical potential Ev decreases, and so does PF- From the facts that the electrical conductivity is a function of PF and that PF is a function of the interaction strength and the thermal agitation, we conclude the importance of finding a correct expression of pF(rs,(kT)2). In the theory of Gell-Mann et al. ~6) and many others, the Fermi momentum PF was not evaluated explicitly, since it was not essential to express macroscopic quantities at absolute zero with respect to the density of the system. Before closing our discussion we would like to make a few points concerning our result. The dependence of the relaxation time and the conductivity on the Fermi momentum PF would not change even with the inclusion of the higher order e - e interactions, apart from the argument concerning screening effects. If so, the effective mass appearing in the conductivity is mainly due to the band effect. This may be true since the e - e interaction effect only appears through the impurity potential with PF. The Fermi momentum up to the third order ring diagram contribution (except the third order exchange) was recently calculated by the author and Isihara~5). In the metallic region the ratio, Pdpo, decreases from unity by 20-50%.
Acknowledgement The author would like to thank Prof. H. Tanaka in the Nagoya Institute of Technology for reading the manuscript and for many helpful discussions.
190
D.Y. KOJIMA
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) I1) 12) 13) 14) 15) 16) 17)
W. Kohn, Phys. Rev. 123 (1%1) 1242. J.S. Langer, Phys. Rev. 120 (1%0) 714; 124 (19611 1003:127 (1962) 5. W. G6tze and P. W/51fle, Phys. Rev, B6 (1972) 1226. H. Mori, Progr. Theor. Phys. (Kyoto) 33 (1965) 423; 34 (1965) 399. C.S. Ting, S.C. Ying and J.J. Quinn, Phys. Rev. B14 (19761 4439. E.W. Montroll and J.C. Ward, Phys. Fluids 1 (1958) 55. A. Isihara, Progr. Theor. Phys. Suppl. 44 (1969) I. E.W. Montroll and J.C. Ward, Physica 25 (1959) 423. R. Kubo, J. Phys, Soc. Jap. 12 (1957) 570. R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1%5). G.V. Chester and A. Thellung, Proc. Phys. Soc. 73 (1959) 745. A. lsihara, Statistical Physics (Academic Press, New York, 1971)oh. 11). A. Isihara and D.Y. Kojima, Z. Phys. B21 (19751 33. S. Fujita, Introduction to Non-equilibrium Quantum Statistical Mechanics IW.B. Saunders Co., Philadelphia and London, 1%6). D.Y. Kojima and A. lsihara, Z. Phys. B25 (1976) 167. M. GelI-Mann, Phys. Rev. 106 (1957) 369. M. GelI-Mann and K.A. Brueckner, Phys. Rev. 106 (1957) 364. L. Van Hove, Physica 21 (1955) 517:23 I1957) 411.