THEORY OF TRANSPORT PROPERTIES OF LIQUID NON-SIMPLE METALS IN THE EFFECTIVE MEDIUM APPROXIMATION

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THEORY OF TRANSPORT PROPERTIES OF

LIQUID NON-SIMPLE METALS IN THE EFFECTIVE MEDIUM APPROXIMATION

M. Itoh, K. Niizeki, M. Watabe

To cite this version:

M. Itoh, K. Niizeki, M. Watabe. THEORY OF TRANSPORT PROPERTIES OF LIQUID NON- SIMPLE METALS IN THE EFFECTIVE MEDIUM APPROXIMATION. Journal de Physique Col- loques, 1980, 41 (C8), pp.C8-508-C8-511. �10.1051/jphyscol:19808128�. �jpa-00220225�

JOURNAL DE PHYSIQUE CoZZoque C8, suppZ6ment au n08, Tome 41, aoCt 1980, page C8-508

THEORY OF TRANSPORT PROPERTIES OF LIQUID NON-SIMPLE METALS IN THE EFFECTIVE KEDIUM APPROXIMATION

M. Itoh, K. ~iizeki* and M. Watabe ++

Research I n s t i t u t e for FundmentaZ Physics, Kyoto University, Kyoto 606 + ~ e ~ a r t m e n t o f Physics, Tohoku University, Sendai, Japan

++ Faculty o f Integrated Arts and Sciences, Hiroshima University, Hiroshima, Japan.

Abstract.- Theory of transport properties of structurally disordered systems is formulated based on the tight binding model. The expressions for the conductivity tensors including nonorthogonality of atomic orbitals are given in quite general form with some numerical work. The extension of EMA to the electronic transport is made, and its application to liquid transition metals is discussed.

1. INTRODUCTION

Theoretical approaches to the electronic proper- ties of structurally dkordered systems have achieved considerable progress in recent years, some of which were already reported at the last Conference. Among others the effective medium approximation (EM) is now established as the best single site theory for the electronic density of states and successfully applied to liquid non- simple metals. 1 ~ 2

As for the transport properties of non-simple metals, however, we are still in a stage where fundamental theoretical studies are still required.

As a first step tward this, we extended the theory by Ishida and Yonezawa 3 , which is known as a suitable first order approximation to EHA, to the calculation of the electronic transport coefficient based on the tight binding m0de1.~'~ j 6

The present study is the continuation of our previous work. First we treat the problem of the nonorthogonality of atomic orbitals. Since we can not define the Wannier function for structurally disordered systems, the importance of this problem is not limited for practical use. The calculation of the conductivity tensor has lsiderable diffi- culty in this respect, while the nonorthogonality can be quite generally taken into account for the density of states. We give the general expressions for the dc and the Hall conductivity based on the non~rthogonalti~ht binding model, then apply it to the IY theory with some numerical work within single s-band model. Some general conclusions are derived from it. Next we consider the ex- tension of Eii to the calculation of the transport . coefficients, which will be of special interest

because for the present E M is considered to be the

most reliable theory for the electronic density of states. The formulation is performed in special care of the consistency with the calculation of the density of states. We also discuss the application of EIfA to liquid transition metals. In this the d- electrons are treated within tight binding scheme, and the treatment of s-electrons is analogous, but proir, to that of liquid simple metals due to

' Ziman.

2. TIGHT BINDING EXPRESSION OF COPJDUCTIVITY TENSOR WEEN NOI-IORTHOGONALITY EXISTS

Tie assume for simplicity a single s-orbital

<rln> for each n-th atom. Then the p-th eigen- function is expressed as N

whose eigen value equation is given by E

'

, St y' ^{= ffl} qu

Kere we have defined an N-dimensional column vector

$u whose n-th component is given by a'

.

matrix and % are defined by t

(W) =<mlGln> and <mi nz mn

respectively. The normalization and completeness conditions are given by

q,, i- St qv = 6 ~ v (2-3)

and

respectively, and the Green function matrix G ( z ) = [ z - S - M I-' *is expressed as

t

~ z = )n ( r

-

'=I

' '

The matrix representation of the current operator defined by (v) =<m(l/ih[c,s] In> , is given by

mn

v = l / i h [ r V % ; ' . b - t % - ' r ] (2-6) t

where (~.)~~=<rn(?ln> , and (b)mn is the transfer integral between m-th and n-th aroms. By noting that for spherical orbital h- is written as

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19808128

ir = 1 / 2 ( IR St + St iR ) (2-7) where (IR) ⁼ * R m - s,,, , we c a n s e e a f t e r some manupula t i o n t h a t

t

< U ) V ~ V > = J u 5 , t - 1 / 2 ( ~ + ^{Ev), 9 ]} (2-8)

u

Here [A,E] = A.B - 03.8 and 8 = St - fi

.

The d c c o n d u c t i v i t y t e n s o r CXx i s g i v e n by t h e Kubo formula

x

Gv

I

'>

BY u s i n g (2-2)- (2-8), we c a n s e e t h a t eq. (2-9)

s(z1z2) = (Tr { Y ~ ( E ) G ( Z ~ ) V ~ ( E ) ~ ( Z ~ ) I> (2-lob) where

v(E) = [ IR , t - EaS ] , (2-11)

and we have t a k e n t h e a t o m i c l e v e l a s z e r o of e n e r g y . Eqs. (2-10) a r e q u i t e g e n e r a l e x p r e s s i o n of t h e c o n d u c t i v i t y t e n s o r , and i t i s c l e a r t h a t we c a n perform t h e c a l c u l a t i o n i n p a r a l l e l w i t h t h e o r t h o g o n a l c a s e by r e p l a c i n g t by t

-

.

When t h e magnetic f i e l d i s a p p l i e d t h e c a l c u - l a t i o n i s more c o m p l i c a t e d , b u t we c a n s t a r t w i t h t h e more g e n e r a l formula of t h e c o n d u c t i v i t y t e n s o r

Then t h e c a l c u l a t i o n goes a l m o s t i n t h e same way a s i n t h e c a s e of t h e d c c o n d u c t i v i t y , and we have

-a6(E:E')+Ea6(E-,E"> -4aB(Ef,E;)-zaB(E:EL) 1

x I r . where

-a6

z ( z l z 2 ) = < T r { G ( z l ) v a ( ~ ) ~ ( Z 2 ) V 6 ( ~ )

$

3 . APPLICATION TO I Y THEORY

The a p p l i c a t i o n of t h e formulae ,(2-10a,b) and (2-13a,b) t o any s p e c i f i c t h e o r y c a n b e performed i n p a r a l l e l w i t h t h e o r t h o g o n a l c a s e . Here we show t h e r e s u l t s of t h e I Y theorv.

d d 1

x d k [ - l / n ~ m ~ ( k ~ t ) 1 - ;ii;[l+S(k)-al [-l/aImG(kEf) 1 l

(3-2c) where S ( k ) i s t h e F o u r i e r t r a n s f o r m a t i o n of t h e m a t r i x element d i s t r i b i t i o n f u n c t i o n g(R (4;)

-

^-

W. n

a r e a l s o d e f i n e d i n a s i m i l a r way by ( t - E - s )

.f -, ^mn

; w i t h and w i t h o u t g(Rm-%) r e s p e c t i v e l y . G(k,z) i s t h e Green f u n c t i o n c a l c u l a t e d w i t h i n t h e I Y t h e o r y , C(z) i s t h e s e l f e n e r g y , and Gd(z) and a r e d e f i n e d by Gd(z)= [ z - ~ ( z ) ] - I and a = l i m z ~ ( z ) re s p e c t i v e l y . The H a l l conduc-

1213m

t i v i t y , otl = a x y / ~ , i s c a l c u l a t e d l i n e a r i n t h e m a g n e t i c f i e l d Ii based on t h e P e i e r l s a p p r o x i - m a t i o n . The c a l c u l a t i o n i s p r e t t y c o m p l i c a t e d

compared w i t h t h e o r t h o g o n a l c a s e , b u t t h e r e s u l t i s q u i t e s i m i l a r a p a r t from t h e s m a l l c o r r e c t i o n term o H ( l )

.

I n t h e f i g u r e 1 t h e n u m e r i c a l r e s u l t s of t h e dc and t h e H a l l c o n d u c t i v i t y a r e p l o t t e d . We have assumed t h e Is hydrogen-like a t o m i c o r b i t a l and t h e s t e p f u n c t i o n t y p e r a d i a l d i s t r i b u t i o n f u n c t i o n . For a s e t of p a r a m e t e r s chosen h e r e t h e e f f e c t of n o n o r t h o g o n a l i t y i s n o t s o l a r g e , b u t we s e e t h a t

t h e e f f e c t i s l a r g e r f o r l a r g e r v a l u e s of I E I . T h i s i s b e c a u s e t h e d e v i a t i o n of t h e " e f f e c t i v e "

t r a n s f e r , It-E-S, from t h e t r u e t r a n s f e r ti becones l a r g e r . T h e r f o r e t h i s tendency i s t h e g e n e r a l b e h a v i o u r i n d e p e n d e n t of

t h e a p p r o x i m a t i o n and w i l l b e i m p o r t a n t t o u n d e r s t a n d

t h e b e h a v i o u r of t h e

I /

I

.

F i g u r e 1. The d c and t h e I i a l l c o n d u c t i v i t y for o = 5 . 0 and p=0.25, where o i s t h e h a r d s p h e t e

d i a m e t e r i n u n i t s of t h e e f f e c t i v e B o h ~ , r a d i u s a*

and p=3211p23. The r e s u l t s of o r t h o g o n a l o r b i t a l n o d e l a r e a l s o p l o t t e d by t h e broken c u r v e s .

C8-510 JOURNAL DE PHYSIQUE

t r a n s p o r t c o e f f i c i e n t s o f , s a y , l i q u i d t r a n s i t i o n m e t a l s o r l i q u i d a l l o y s . The e f f e c t is a l s o v e r y i m p o r t a n t f o r t h e l i q u i d d i v a l e n t m e t a l s n e a r t h e c r i t i c a l r e g i o n s u c h a s expanded 'tig b e c a u s e t h e Fermi e n e r g y i s f a r from b o t h a t o m i c l e v e l s .

We a l s o n o t e t h a t t h e e f f e c t i s l a r g e r f o r t h e RaL1 e f f e c t . T h i s i s m a i n l y due t o t h e d i f f e r e n c e between t h e powers of t h e c u r r e n t d T ( k , ~ ) / d k and t h e s p e c t r a l f u n c t i o n - l / ~ I r n G ( k , E ) appeared i n eqs. (3-1) and (3-2b). T h e r e f o r e we c a n c o n c l u d e t h a t t h e n o n o r t h o g o n a l i t y may b e i m p o r t a n t f o r t h e H a l l e f f e c t even when i t h a s l i t t l e e f f e c t on t h e o t h e r e l e c t r o n i c p r o p e r t i e s .

4. CALCULATION OF CONDUCTIVITY TENSOR I N EMA For s i m p l i c i t y we a g a i n r e t u r n t o t h e o r t h o g o n a l o r b i t a l model. The e x t e n s i o n t o t h e n o n o r t h o g o n a l c a s e i s q u i t e s t r a i g h t f o r w a r d . For i l l u s t r a t i o n , t h e f o u r t h o r d e r term o f t h e p e r t u r b a t i o n e x p a n s i o n of eq. (2.10) i s r e p r e s e n t e d d i a g r a m m a t i c a l l y i n t h e f i g u r e 2. The t e r m s a r e summed up by t h e diagram e q u a t i o n i n t h e f i g u r e 3. These a r e w r i t t e n a s

+ +

[ G ( ~ z ~ ) G ( ~ z ~ ) - G ~ ( z ~ ) G ~ ( ~ ~ ) l A ( k ; z 1 ~ 2 ) ( 4 - 2 ) + +

The r e n o r m a l i z e d c u r r e n t A ( k , z l , z 2 ) i s o b t a i n e d from (4-2) by i t e r a t i o n ; t h e l a b o r r e q u i r e d i s n o t beyond t h a t f o r t h e s e l f e n e r g y . The importance of t h e s e e q u a t i o n s i s t h a t t h e y a r e c o n s i s t e n t w i t h t h e c a l c u l a t i o n of o n e - p a r t i c l e Green f u n c t i o n and do n o t v i o l a t e t h e Ward i d e n t i t y . The most n o t a b l e

F i g u r e 2. The i l l u s t r a t i o n of t h e c a l c u l a t i o n of E ( z 1 , z 2 )

F i g u r e 3 . Diagrammatic r e p r e s e n t a t i o n of t h e i n t e g r a l e q u a t i o n s (4-1) and (4-2).

d i f f e r e n c e between t h e I Y t h e o r y and EMA i s t h a t i n t h e l a t t e r t h e o r y t h e v e r t e x c o r r e c t i o n s d o n o t v a n i s h even i n t h e c a s e of s i n g l e s-band s o t h e r e l a x a t i o n t i m e f o r t h e t r a n s p o r t h a s d i f f e r e n t v a l u e from t h e o r d i n a r y r e l a x a t i o n time.

We have f o r m u l a t e d t h e t h e o r y w i t h i n s i n g l e s- band f o r s i m p l i c i t y , b u t t h e e x t e n s i o n t o t h e m u l t i - band c a s e i s s t r a i g h t f o r w a r d .

5. APPLICATION TO LIQUID TRANSITIOIJ IiETALS We d e a l w i t h t h e f o l l o w i n g model H a m i l t o n i a n t o d e s c r i b e l i q u i d t r a n s i t i o n m e t a l s i n which n e a r l y f r e e s - e l e c t r o n s and t i g h t l y bound d - e l e c t r o n s a r e i n t e r a c t i n g t h r o u g h mixing e f f e c t ;

For s i m p l i c i t y we a s s h e o n l y s i n g l e non-degenerate t i g h t b i n d i n g band which we c a l l h e r e a f t e r a s "d- band" f o r convenience. We f u r t h e r assume t h a t

< k l k ' > = 6 k k , , < k l m > = O , < m l n > = 6 mn' '

and t h e "d" o r b i t a l In> h a s s p h e r i c a l symmetry. The e x t e n s i o n t o more r e a l i s t i c model based on t h e d e z e n e r e t e n o n - s p h e r i c a l d o r b i t a l s i s q u i t e s t r a i g h t f o r w a r d . A s e t of i n t e g r a l e q u a t i o n s of t h e m a t r i x e l e m e n t s of t h e Green f u n c t i o n

< k l e ( z ) lk'> = Ykk.(z) , <nlG(z) ( k > = $nk(z) ,

and < n l e ( z ) I m > = j m ( z ) D i s g i v e n by

~ l ( ~+ )G: ~( k ) n ~Z Y ( k ) / ~ e x ~ ( ~ ~ ~ ~ . n ) 4$nk'

+

gmk

-

gkk8,

gnk

qnn

which can b e s o l v e d f o r m a l l y . The r e s u l t i s e x p r e s s e d compactly i n t h e m a t r i x form a s

e

-

@ ( z )

d

where we have i n t r o d u c e d t h e f o l l o w i n g n o t a t i o n s (d)mn=tmn , ( u s ) = P 1 " ^{2 S}

mn N k e x p ( i k R n m ) ~ (k) ' G ~ (k)

e,

(

9

ynk

Eqs. (5-3) a r e t h e e x a c t r e s u l t s , and any s p e c i f i c t h e o r y can b e a p p l i e d t o c a l c u l a t e

9

which g ( k , r ) and g k ( z ) a r e o b t a i n e d e a s i l y . The c a l c u l a t i o n of .

9

fi

The e s s e n t i a l p o i n t i s t h e i n t r o d u c t i o n of t h e

" e f f e c t i v e t r a n s f e r " tS ; t h e t r a n s f e r p r o c e s s d + s + d i s t r e a t e d a s r e n o r m a l i z e d t r a n s f e r p r o c e s s d - d s o t h a t we can d e l e t e d s - e l e c t r o n s a p p a r e n t l y .

I n o r d e r t o c a l c u l a t e t h e c o n d u c t i v i t y , we must c a l c u l a t e i n s t e a d of (2-10)

3 ( z 1 z 2 ) =

<

-ss -dd

=

-

where (5-4)

ESS(z1z2)

Z ; k ( < ( k > ~ k k , ( ~ 1 ) < k * ( ~ l k 2 > ~ k , k ( ~ 2 )

=

Ckk.

~ ~ ~ ( z ~ z ~ )

>

( 5 ~ 5 a )

=

<

k mn

>

= < i _ijmn I < i l G l j > ~ j m ( z l ) < m ~ G ~ n > ~ i ( z 2 ~ (5-5c) W e can a p p l y t h e r e s u l t s o f t h e l a s t s e c t i o n t o t h e c a l c u l a t i o n of (5-5a,b,c). The r e s u l t s a r e

~ ~ ~ (= zp-'j($3<kl~lk>Gs(k~1)Gs(kz2)<kl~~k> ~ z ~ )

(5-6c) Here t (k) i s t h e F o u r i e r t r a n s f o r m a t i o n of t

d d mn

m u l t i p l i e d by g(ib;gn), GS(k,z) and G ( k , z ) a r e t h e Green f u n c t i o n of s- and d - e l e c t r o n s c a l c u l a t e d

.b r

by t h e EMA. f i d ( z ; z l z 2 ) and As(k;z1z2) a r e t h e r e -

n o r m a l i z e d c u r r e n t s c o r r e s p o n d i n g t o ( ~ ~ ) ~ ~ = < m l G l n >

and v S ( k ; z l z 2 ) = ?(k){(k~~)<kl~lk>~~(kz~)T(k)~

r e s p e c t i v e l y . AZtk; z l z 2 ) and Az(k;zlzZ) a r e d e f i n e d by

and (0

AS(k;z1z2) = As(k;zlz2)

-

W e n o t e h e r e t h a t t h e s a t i s f a c t o r y r e s u l t s a r e n o t o b t a i n e d e x c e p t by E1.U f o r two r e a s o n s . F i r s t t h e d i v e r g e n c e of t h e c o n d u c t i v i t y a r i s i n g from t h e u n p e r t u r b e d t e r m G;(k,z) i n $? k k ( z ) c a n n o t b e t r e a t e d r e a s o n a b l y u n l e s s t h e c u r r e n t p a r t i s p r o p e r l y r e n o r m a l i z e d . Secondly t h e v e r t e x cor- r e c t i o n s p l a y a n i m p o r t a n t r o l e f o r s - e l e c t r o n s , a s w e l l known from Ziman f o r m u l a . His t r e a t m e n t i s e q u i v a l l e n t t o t a k e t h e l a d d e r of d o t t e d h - l i n e i n t h e c a l c u l a t i o n of t h e v e r t e x c o r r e c t i o n , t h e r e f o r e t h e t r e a t m e n t by ~ 1 4 A ' i s a d e q u a t e f o r s - e l e c t r o n s .

REFERENCES

1. Yonezawa F, I s h i d a Y , M a r t i n o F and Asano S Proc. 3 r d I n t . Conf. on L i q u i d H e t a l s , B r i s t o l

(ed. r.. Evans and D.E. Greenwood 1977) p.385 2. Asano S and Yonezawa F J.Phys.F:i.letal Phys.

1 0 , 75(1980)

3. I s h i d a Y and Yonezawa F Prog. Theor. Phys.

49 (1973) 731-53

4. Watabe 11 P r o c . 3 r d I n t . Conf. on L i q u i d M e t a l s , B r i s t o l 288-304

5. I t o h 14 and Watabe 11 J . P h y s . F : ? i e t a l Phys. 8 (1978) 1725-49

6. I t o h 14 PhD T h e s i s Tohoku U n i v e r s i t y 1979