2.~. S. V. TyabIikov, Methods of Quantum T h e o r y of Magnetism, Plenum P r e s s , New York (1967). 24. V. G. Vaks, A. I. Larkin, and S. A. Pikin, Zh. Eksp. T e o r . F i z . , 53, 1089 (1967). 25. N. M. Ptakida, P h y s . Lett. A, 45, 481 (1973). 26. K. Elk, Phys. Status Solidi B, 96, 251 (1979). 27. M. Peter et a l . , Heir. Phys. Acta, 47, 807 (1974). 28. P. B. Alien, "Phonons and the superconducting transition t e m p e r a t u r e , " in: Dynamical P r o p e r t i e s of Solids (eds. G. K. Horton and A. A. Maradudin), North-Holland Publ. Company (1980), Ch.2, pp.95-196.

INFLUENCE

OF

CONDUCTIVITY V.T.

s-d OF

HYBBIDIZATION LIQUID

ON T H E

TRANSITION

ELECTBICAL

METALS

Shvets

Betarded Green's functions are used to develop a theory of the electrical conductivity of liquid transition metals with systematic allowance for the hybridization of the s and d states of the conduction electrons. It is shown that the conductivity can be represented as a sum of three terms, one of which is due to the scattering of s electrons by ions, including resonance scattering by d states, the second is due to tunneling of d electrons through d states, and the third is due to transitions of almost localized d electrons to delocalized s states and reverse transitions from s to d states. Expressions for each of the contributions are obtained in the framework of perturbation theory with respect to the pseudopotential of the electron-ion interaction, the hybridization potential, and the resonance integral.

1.

Introduction

In r e c e n t y e a r s , there has been an appreciable i n c r e a s e in the interest shown toward study of t r a n s port phenomena in d i s o r d e r e d s y s t e m s . Among the most important s y s t e m s of this type are liquid transition m e t a l s , which occupy an i n t e r m e d i a t e position between simple liquid metals, whose p r o p e r t i e s can be well described in the approximation of almost free e l e c t r o n s , and amorphous s e m i c o n d u c t o r s , for which the tight-binding approximation is m o r e adequate. In transition metals, there are not only almost free s electrons but also a l m o s t bound d e l e c t r o n s . The approaches - based on a modified Ziman formula [1,2] used at the present time to interpret the experimental data on the static conductivity take into account the contribution to the conductivity due solely to the s e l e c t r o n s . The role of the d states is reduced m e r e l y to r e s o n a n c e s c a t t e r i n g of the s electrons by them. In the case of [1], the modification consists of replacing the pseudopotential of the e l e c t r o n - i o n interaction by the s i n g l e - p a r t i c l e t matrix; in the case of [2], it consists of replacing it by the potential of the s - d hybridization. Such an approach does not enable one to d e s c r i b e even qualitatively many p r o p e r t i e s of these metals, in p a r t i c u l a r , their e l e c t r i c a l conductivity in the optical frequency r a n g e and the Hall effect. M o r e o v e r , it has not yet been established to what extent the approach is applicable even for the description of the static conductivity of liquid transition metals. It is t h e r e f o r e n e c e s s a r y to take into account m o r e s y s t e m a t i c a l l y all the effects associated with the p r e s e n c e of the d e l e c t r o n s . In the p r e s e n t paper, the e l e c t r i c a l conductivity of liquid transition metals will be obtained in the f r a m e w o r k of a model that takes into account explicitly the p r e s e n c e of the d electrons and the hybridization of the s and d states. 2.

Formulation

of the

Problem

Apart f r o m the e l e c t r o n - e l e c t r o n interaction of the s e I e c t r o n s , which makes an appreciable c o n tribution to the r e s i s t i v i t y of both simple and transition metals only at v e r y low t e m p e r a t u r e s [3,4], it is n e c e s s a r y to take into account in transition metals the e l e c t r o n - e l e c t r o n interaction of the d electrons and the s and d e l e c t r o n s . If we proceed f r o m a model of d electrons localized at atoms, the interaction of the O d e s s a Technological Institute of the B e f r i g e r a t i o n Industry. Matematicheskaya Fizika, Vol. 53, N o . l , pp.146-155, October, 1982. 1981.

1040

0040-5779/82/5301-1040,r

Translated from Teoretieheskaya i Original a r t i c l e submitted July 2,

9 1983 Plenum Publishing C o r p o r a t i o n

s and d e l e c t r o n s has an exchange nature, with e n e r g y dependent on the orientation of the spin of the moving conduction electron with r e s p e c t to the spins of the localized electrons (the spins of the ions) [3,5]. In f e r r o magnetic transition metals below the Curie point, this m e c h a n i s m is one of the main conduction electron s c a t t e r i n g m e c h a n i s m s . When the t e m p e r a t u r e is r a i s e d above the Curie point, this contribution c e a s e s to depend on the t e m p e r a t u r e [5] and its importance d e e r e a s e s relatively, especially in the liquid phase,. For nonmagnetic transition metals, this contribution is absent. If we adopt a band r e p r e s e n t a t i o n of the d states, then the role of the s - d e l e c t r o n - e l e c t r o n interaction is the same as that of the e l e c t r o n - e l e c t r o n interaction of the s e l e c t r o n s [3]. The r o l e of the e l e c t r o n - e l e c t r o n interaction of localized eleetrons has been fairly fully investigated in the f r a m e w o r k of the Hubbard model [6, 7]. The most important eonsequenee of this interaction is its influence on the density of states of the localized electrons and, t h e r e f o r e , on the kinetic e h a r a e t e r i s t i c s of the s y s t e m [7]. B e s t r i c t i n g o u r s e l v e s to the ease of nonmagnetic liquid transition metals, we shall use a model of a metal that does not take into account explicitly the e l e c t r o n - e l e c t r o n interaction, and only at the end shall we c o n s i d e r how the obtained r e s u l t s a r e modified when allowance is made for the e l e c t r o n - e l e c t r o n i n t e r a c tion of the d e l e c t r o n s in the f r a m e w o r k of the Hubbard model [6J. As s i n g l e - p a r t i c l e basis states of the conduction e l e c t r o n s , we use fully orthogonalized plane waves [8]. We can r e p r e s e n t the field function in this c a s e as follows:

(1) k

n

Here, ]k> are fully orthogonalized plane waves d e s c r i b i n g the states of the s e l e c t r o n s , i ~ are o r t h o g o n a lized d states, these satisfying =0; ak, bn a r e electron annihilation o p e r a t o r s at the c o r r e s p o n d i n g states (n c h a r a c t e r i z e s both the number of the ion to which the eleetron belongs as well as an appropriate set of quantum numbers specifying the state of the d electron). An orthogonalized plane wave is related to a plane wave by the unitary t r a n s f o r m a t i o n : Ik>=f, lk> [8]. Then liquid t r a n s i t i o n metals can be d e s c r i b e d by means of the model HamiItonian

k

k,k"

n

n n"

ek6kk,+Wkk,=(klR.~[k')=
Ir n

e~=
J . . . . (~l//e~{n'),

(2}

h~k~ n

where ~ is the e n e r g y of the d levels of the n-th ion, d~,,, is the resonance integral r e s p o n s i b l e for the r t r a n s i t i o n s of the d e l e c t r o n s f r o m state n to state n , Ak~ is the hybridization potential leading to t r a n s i tions of the e l e c t r o n s between the s and d states, and Wkk, iS the F o u r i e r t r a n s f o r m of the pseudopotential of the e l e c t r o n - i o n interaction. Note that the pseudopotential W is weaker than the true potential witMn the ion c o r e s and may be r e g a r d e d as a small p a r a m e t e r of the theory, although such a t r e a t m e n t may not be applicable for the original potential. Also much less than unity for typical transition metals are the r a t i o s of Ak~ and :&~, to the F e r m i energy, which means that these r a t i o s can be r e g a r d e d as small p a r a m e t e r s of the theory. As in the case of liquid simple metals, we have ignored in (2) the dynamics of the ion s u b s y s t e m . In addition, we shall not a s s u m e that the states of the d electrons have a band nature. To find the conductivity % we use Kubo's linear r e s p o n s e theory, in a c c o r d a n c e with which [91 l ^^ t (~= --3V ((J, d))o = ~V-S ((i(t), d(O) )) dt,

((i(t), a (0) >)= - ~

(3)

0 (t) ([J(t), d(O) ]).

(4)

Here, a is the o p e r a t o r s of the dipole moment of the conduction electrons, J is the c u r r e n t o p e r a t o r , and the angular b r a c k e t s denote q u a n t u m - m e c h a n i c a l and statistical averaging. Using (1), we obtain for

~t= e s rkk,a~+a,,+ e s r~.,b~+b~,-k e s k,k'

ha'

[rk~ak+b~-~r~kb,,+ak], 1.kk,~(~klrlk'),

r . . . . (K[l'lg') ,

rk~=(k[rlg).

(5)

k,n

!041

S i n c e the fully o r t h o g o n a l i z e d p l a n e w a v e I~:) o s c i l l a t e s r a p i d l y w i t h i n the ion c o r e s (being c l o s e to an o r d i n a r y p l a n e w a v e o u t s i d e t h e m ) , and the o r t h o g o n a l i z e d w a v e f u n c t i o n s 1 ~ of the d s t a t e s o u t s i d e the ion c o r e s d e c r e a s e e x p o n e n t i a l l y , we c a n w i t h good a c c u r a c y i g n o r e r h c o m p a r e d with r~=, and rk~,. To the s a m e a c c u r a c y , we c a n i g n o r e r , . , f o r n ~ n c o m p a r e d w i t h r , and thus s e t r k k , = ( k l r l l ~ ' > ~ - ( k l r [ k ' ) = i08kk,/Ok. With a l l o w a n c e f o r t h e s e a p p r o x i m a t i o n s , the o p e r a t o r of the d i p o l e m o m e n t w i l l h a v e the f o r m

~ t = i e Z - -0~ 8i,k'ak+ak' -~ e Z kfl~'

r..b=+b. '

n

w h i c h a g r e e s with the e x p r e s s i o n u s e d in [10]. F o r the c u r r e n t o p e r a t o r , w e h a v e by d e f i n i t i o n ^

i

d]

(7)

g=L+L+L,,

(8)

a = T[Ro, o r , u s i n g (2) and (6), where

~,= ett s

ka +ak'

(9)

in k

L ~ ie

(10)

- - t - Z ] ~ ' ( r . . . , - - r ~ ) b,~+b,,,, n,n'

ie J~, = --~ s

OAk,,~ i OA,~k\ b,+a,.] " [ ( Ak.r.,,--i--~- ] ak+b,~ -- t A=kr.=q-i --~iT-)

(11)

.kn

In a c c o r d a n c e with (3), (8)-(11) (12)

o=(r.q-Oaq-od,, where

i

"

(14)

(;d== - ~ - I ((id=(t), [1(0) } dt.

(15)

H e r e . a s is the c o n t r i b u t i o n to the c o n d u c t i v i t y due to the s e l e c t r o n s , (rd is the c o n t r i b u t i o n due to the d e l e c t r o n s , and ads t h e c o n t r i b u t i o n due to t h e t r a n s i t i o n s of e l e c t r o n s b e t w e e n the s and d s t a t e s . 3.

Finding

of the

Conductivity

To find the c o n d u c t i v i t y , w e p r o c e e d a s f o l l o w s . F i r s t , we find it for a r a n d o m but fixed p o s i t i o n of t h e i o n s , and we then p e r f o r m s t a t i s t i c a l a v e r a g i n g of the o b t a i n e d e x p r e s s i o n with r e s p e c t to the ion coordinates. To find the t w o - t i m e r e t a r d e d G r e e n ' s f u n c t i o n s in the e x p r e s s i o n s (13)-(15), we u s e the m e t h o d of the e q u a t i o n s of m o t i o n [11]. By m e a n s of the H a m i l t o n i a n (2) f o r the G r e e n ' s function ,((ak+ak, d))0~ (the s u p e r s c r i p t e d e n o t e s a v e r a g i n g o v e r t h e c o o r d i n a t e s o f the e l e c t r o n s ) w e o b t a i n t h e h i e r a r c h y o f e q u a t i o n s ([ak+(0)ak(0) , d ( 0 ) ] ) e = 2

+

Wkk" ((ak+ak,, ~1));-- s k'

-

1~'

n"

(ek--ek,§

+ s ku(kr

1042

Wk'k"((ak ak,,,d))o--

'

Wkk' ((ak' ak, d)),

+

n

^ , {ak' +a,.,,d))o ^ . ]+ 9Wk'k[((ak +ak, d))0--

Z ~*t ( kJr ~#:k ' )

Wk"k

((ak,,

+a k ' , d^~ o " - t - s n'

((ak+b=,, d))o ^ ~-- Z A . , k n"

- o

((b~ a~, d))0,

((b., §ak,, d))0 ^ "

<(a~+b.,d)>o=-<[ak+(O)b.(O),'d(O)l>~+

A.~((a~ a~,d>>o

+

^ "

n'

Z

A~R"<
k'(k'~k)

(16)

n"

(e.--e~+g~) <(b. ak, d)>o : -- < [b" + (0)at (0), a (0) ] )* -- A ~.

<)o + h"

.~

+

^

n'

^

n"

9

^ " k'

k'(k'C'k)

e

(s.-e..+~5) ((b. b~,, d))o =-- <[b.+(0)b.,(0),~l(0)1> + Z],~,,,,, ~b.+b,~,,,d));--ZL,,,,, ((b+,,b,~,,~d)>'o+ nit

k

~t o

k

etc. The s y s t e m (16) can be solved for the G r e e n ' s function as in the case of liquid simple metals [12]. The solution has the f o r m ((ak+a~,d))o~=N~(k) ~ (k), (17) w h e r e -~(k) d e s c r i b e s the relaxation in the s u b s y s t e m of the s e i e c t r o n s and can be r e p r e s e n t e d in the f o r m of a s e r i e s in powers of W, A, and J: ^ 2~ ~-~(k), % - ' ( k ) = - ~ - E

,~-'(k)= n~2

2g I W k ~ , [ ' ~ 5 ( e k - - ~ k , ) [ t - - c o s ( k ' k ' ) ] + - ~ Z A,,k]25(e~--ek).

k'

(18)

n

i i 2~ (k) = - ~- < [ak + (0) ak (0): ~1(0) ] >~ -- -h--Z a'" (k), ~,(k)=

Wkk' <[ak+(O)ak'(O)'~l(o)]>~

E

e~--ek,+i5

k'

Z

Z W k , k <[ak'+-(O)ak(O)'~d(O)]>" ~8k'--ek+i5

k'

hk. ,< [ak+ (0) b. (0), d (0) ] >~ p , h.k <[ b"+ (0) ak (0), ~I(0) ] >~ sk--e.+i5 --~ e.--ek+i5 n

(19)

n

The second t e r m and the last t e r m s on the right-hand side of (19) take into account the s c a t t e r i n g of the s e l e c t r o n s by the ions induced by the e x t e r n a l e l e c t r i c field [13]. In a c c o r d a n c e with [12t, the t e r m s of a r b i t r a r y o r d e r in W, A, and J in the e x p r e s s i o n s (18) and (19) can also be r e a d i l y written down. Using (13) and (17), we obtain e i Z k (/V~(k),~ (k) >k ra 3V

o,

-

(20)

k

F r o m Eqs:(16), we can also find the G r e e n ' s functions ((ak+b~,~t>)0e and ((b,~+ak,~l))o". Then, using (11) and (14) we obtain for ~ds --e3~ Z

(\

~

N'a~(k,n)>

k,n

e

A.~r~H-g0A.k/0k/V~a(k,n) > } + s.--ek+i5

(21) k

In this expression, the t e r m containing l~ds^describes the transitions of the e l e c t r o n s f r o m the d to the s state, and the t e r m containing the function Nsd d e s c r i b e s those f r o m the s to the d states. The t e r m c o n taining the function Kds d e s c r i b e s transitions of e l e c t r o n s f r o m the d to the s state and s c a t t e r i n g of them by ions, this s c a t t e r i n g being s i m i l a r to that of the s e l e c t r o n s . The t e r m containing the function Ksd d e s c r i b e s the opposite p r o c e s s . The functions Ads and Nsd have a s t r u c t u r e s i m i l a r to the s t r u c t u r e of the

1043

function Ns,

Nd. (k, n) ~----- ~ - e - ~ - Z

~d,m(k,n),

(22)

i i " /'r (k, n) = -- -~- < [ b. + (0) ak (0), d (0) ] >~ -- - h - - 2 ~'~'* (k, n).

(23)

m~i

^

^

The functions K d s and K s d can also be expanded in p o w e r s of W, ~, and J: m ~d.~(k), ~d.~(k)= i T 2

/~d~( k ) = m~2

AkJnn--i OAkJOk ek--e~+i5 A~k,

(24)

n

~,

^

m

A~kr.~+i ~ h ~ / 0 k

m~2

n

As in the c a s e of ads , we can obtain an e x p r e s s i o n for rrd: ad = ~ - 2 e

l ~ (r.,~,--r~)e._e~,+i5 , ~]~(n,n')

~"

+ m 3V 2

n n"

(26)

k

The f i r s t t e r m in (26) d e s c r i b e s tunneling of the e l e c t r o n s through d s t a t e s , and the second tunneling with subsequent t r a n s i t i o n to the s state and s c a t t e r i n g by ions, and also the r e v e r s e p r o c e s s . We have i

fi/~(n,n.)=__~<[b

+(O)b~,(O),[l((})]> ~ -

~t- ~ =, ~^ ( ~ , n

. ),

(27)

rtt~i

/~a(k) =

~d~(k), ~ ( k ) = t--h * ~ --e.-e.,+i6 [ e . ' e k + i 6 7tt~3

t- ek--e~,+i8 ] ( r ' ' 2 r " ' ' ' ) "

(28)

rt,n"

The e x p r e s s i o n s we have obtained for as, rrd, and rrds a r e valid in a r b i t r a r y o r d e r in W, A, and J . Since g e n e r a l l y we must r e s t r i c t o u r s e l v e s to a finite o r d e r of p e r t u r b a t i o n t h e o r y , it is of i n t e r e s t to a n a l y z e in m o r e detail the e x p r e s s i o n s for as, crd, and ~rds in the lowest o r d e r of p e r t u r b a t i o n t h e o r y . Calculating the c o m m u t a t o r s on the r i g h t - h a n d sides of the e x p r e s s i o n s (19), (22), (23), and (27), we obtain < [ak+ (0) ak (0), ~1(0) l )~

(ek)/Ok,

(29)

< [ak+ (0)b= (0), d(0)] >o~=< [b+ (0) b=, (0), d(0)] >~

(30)

in the z e r o t h o r d e r and 9 <[at + (0) b. (0), d(O) ] >'" = er=~ ' + ie ~ ' ", Ok ([b.+(0)ak(0), r

9

]>.e = --er~. + i e - -

0

Ok

<[b.+(O)b~,(O), ff(O)_] >" ~ = e ( r ~ , . , - - r . . ) " ~

in the f i r s t o r d e r of p e r t u r b a t i o n t h e o r y .

,

",

(31)

(32) (33)

A c c o r d i n g l y , we have n (e~) - n (e~)

(ak+b~> , e = A~k ~

-,

(34:)

8k--l~n

,e

= Ak,~

n(e~) --n (s~)

",

(3 5)

8k--Sn

' ~=J~,~

n(EO--n(sn')

(36)

8n--F~n"

w h e r e n(e ) is the F e r m i - D i r a c

distribution function.

The r e s u l t s take on a p a r t i c u l a r l y s i m p l e f o r m if for the h y b r i d i z a t i o n potential we use the a p p r o x i mation Ak.~Akexp (ikr~). Then, p e r f o r m i n g s t a t i s t i c a l a v e r a g i n g in the e x p r e s s i o n (20) as in the c a s e of s i m p l e m e t a l s [12], and introducing a smooth distribution g(e ) of the e n e r g y of the d s t a t e s such that g(e )de

1044

is the probability of finding the e n e r g y of a d state in the interval v, e x p r e s s i o n for a s in the lowest o r d e r of perturbation theory:

+ d~, we obtain the following o

(~8~e21"tT/Wt~

(37)

i w~(x)S2(x)x3dX+_~AkFg(eF), 2n T-'= ('}2-'(kF)> i2~mz ~

(38)

where S~(q)~N-~* is the t w o - p a r t i c l e s t r u c t u r e function of the ion subsystem, and

Wkk,~w(k--k ~)

~(k--k'). If we a s s u m e that OA~./ak and A ~ a r e of the s a m e o r d e r of s m a l l n e s s , then the t e r m s that take into account in the e x p r e s s i o n s (2t) and (26) the p r o c e s s e s of tunneling of d electrons with subsequent s c a t t e r i n g by ions, and also the r e v e r s e p r o c e s s e s , which have a s t r u c t u r e s i m i l a r to that of the expression for ~rs, have at the s a m e time a higher o r d e r than the expression (37) for as, and in the lowest approximation of perturbation t h e o r y they can be ignored. The r e m a i n i n g t e r m s have the form [

( ~ = - - "3---V-h ne~ ~'~ Z..J\-d'k /dAk dkd h~ n(ek)--n(en)ek_ea

]

~ ~ dh~ ~ [6(ek--Sn21-0)21-5(ek--En--0)]).-~--~----h-~eF [ [ ~ j ~ ] g(eF), e n

gin

(39)

k, r~

=

n e2 ~ - l

(Y~* ~ - - ~

"

2~e~

(IJ~,.12(r~,.-r~)25(e~--er)5(s~-e~,))~,.~-~a12g

2

(oF). ,

(40)

n,n'

In obtaining the last e x p r e s s i o n , we have r e s t r i c t e d o u r s e l v e s to the n e a r e s t - n e i g h b o r approximation, ].~,~], and we have set r.,~,--r..'za. 4.

Discussion

of the

Results

Equation (20), which gives the contribution to the conductivity due to the s electrons has the f o r m c h a r a c t e r i s t i c of liquid simple metals except that in addition to the o r d i n a r y scattering by ions ~-' and N s also contain a contribution c o r r e s p o n d i n g to r e s o n a n c e s c a t t e r i n g of s electrons by d states. If we r e s t r i c t o u r s e l v e s to the lowest o r d e r of perturbation t h e o r y - the expression. (37) - and in it set A = 0~ then we obtain the well-known f o r m u l a of Ziman, which gives a good description of the static conductivity of liquid simple m e t a l s . In c o n t r a s t , for W = 0 we obtain the Mott formula [2]. The p r e s e n c e of the contribution a d is c h a r a c t e r i s t i c of any d i s o r d e r e d s y s t e m with electron conduction. The expression (40) is equal to the e x p r e s s i o n for the no-phonon contribution to the conductivity of doped and compensated or amorphous s e m i conductors [14]. In addition, the e x p r e s s i o n s (20), (21), and (26) a g r e e weI1 with the c o r r e s p o n d i n g e x p r e s sions for the conductivity of a p l a s m a [10] despite certain differences in the original models. The approximations for the hybridization potential and the r e s o n a n c e integral used to obtain Eqs. (36)(40) r e d u c e to a m i n i m u m the difference between the liquid and solid phases. T h e r e f o r e , these approximate e x p r e s s i o n s can be used only for the t r a n s i t i o n metals whose conductivity changes little on melting, which in r e a l i t y is the c a s e for a number of typical transition metals [2]. In the general c a s e , however, it is n e c e s s a r y to use Eqs. (20), (21), and (26). A c h a r a c t e r i s t i c feature of the e x p r e s s i o n s (36)-(40) is their strong dependence on the density of states of the d e l e c t r o n s at the F e r m i level. As this density i n c r e a s e s , the contribution of ~s d e c r e a s e s , the contribution of ads i n c r e a s e s linearly, and the contribution of ~d i n c r e a s e s as the square of the density. In this connection, the contributions ads and a d for typical transition metals may be v e r y appreciable c o m pared with the contribution a s. As we have a l r e a d y said, the e l e c t r o n - e l e c t r o n interaction of the d electrons can have a significant influence on the density of states of the d e l e c t r o n s . T h e r e f o r e , to find this density, we must add to the Hamiltonian (2)a t e r m taking into account the c o r r e s p o n d i n g interaction. For the Hubbard model, it has the f o r m

The p r e s e n c e of this t e r m in the Hamiltonian does not change the c u r r e n t o p e r a t o r but leads to the appearance in the leading o r d e r s of perturbation t h e o r y of additional t e r m s for (~,, ~d~, a~ . If it is assumed that I 0 has the same o r d e r as W, 4, J, then in the expansions of T-' and Ns in powers of these p a r a m e t e r s these t e r m s

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will have third o r d e r . Accordingly, for ads and a d they will make a contribution beginning with the t e r m s of third o r d e r . Thus, in the lowest o r d e r s of perturbation t h e o r y for the conductivity the e l e c t r o n - e l e c t r o n interaction of the d electrons need not be taken into account explicitly. It follows f r o m the e x p r e s s i o n s (20), (21), and (26) that s - d hybridization introduces a contribution in each of the t e r m s as, ads , and a d making up the conductivity a. The t e r m ads is entirely due to the hybridization of the s and d s t a t e s . In c o n t r a s t , the role of the d states r e d u c e s not only to r e s o n a n c e s c a t t e r i n g by them of s e l e c t r o n s but also to a direct contribution of the d electrons to the conductivity, this being due both to their tunneling through d states as well as their transitions to s states caused by the s - d hybridization. The r e s u l t s obtained in the p r e s e n t paper can be used to find the frequency dependence of the c o n ductivity and also the conductivity in a constant and homogeneous magnetic field. I a m v e r y grateful to Z. A. Gurskii and V. N. Bondarev for discussing the r e s u l t s and for valuable comments. LITEBATURE 1. 2. 3. 4. 5. 6. 7.

8. 9o 10. 11. 12. 13. 14.

CITED

P. Evans,: D. A. Greenwood, and P. Lloyd, Phys. Lett. A, 35, 57 (1971). J. M. Mott, Phitos. Mag., 26, 1249 (1972). J. M. Ziman, E l e c t r o n s and Phonons, Clarendon P r e s s , Oxford (1960). R. N. Gurzhi, Zh. Eksp. T e o r . F i z . , 35, 965 (1958). K. N. 1~. T a y l o r and M. I. Darby, P h y s i c s of B a r e Earth Solids, London (1972). I. Hubbard, P r o e . P. Soc. London, 281, 401 (1964). V. L. Bonch-Bruevich, I. P. Zvyagin, B. Kaiper, A. G. Mironov, P. ~nderlain, and B. ~sser,

Electron Theory of Disordered Semiconductors [in Pussian], Nauka, Moscow (1981). Z. A. Gurskii and B. A. Gurskii, Fiz. Met. Metalloved., 50, 928 (1980). D. N. Zubarev, NonequilibriumStatistical Thermodynamics, Plenum, New York (1974). G. l~Spke, Teor. Mat. Fiz., 46, 279 (1981). D. N. Zubarev, Usp. Fiz. Nauk, 71, 71 (1960). V. T. Shvets, Teor. Mat. Fiz., 42, 271 (1980). N. M. Plakida, Zh. Eksp. Teor. Fiz., 53, 2041 (1967). I. I. Fishehuk, Ukr. Fiz. Zh., 22, 1477 (1977).

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