ELECTRONIC TRANSPORTTRANSPORT PROPERTIES OF LIQUID NON-SIMPLE METALS

(1)

HAL Id: jpa-00220208

https://hal.archives-ouvertes.fr/jpa-00220208

Submitted on 1 Jan 1980

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ELECTRONIC TRANSPORTTRANSPORT

PROPERTIES OF LIQUID NON-SIMPLE METALS

F. Yonezawa

To cite this version:

F. Yonezawa. ELECTRONIC TRANSPORTTRANSPORT PROPERTIES OF LIQUID NON- SIMPLE METALS. Journal de Physique Colloques, 1980, 41 (C8), pp.C8-447-C8-457.

�10.1051/jphyscol:19808112�. �jpa-00220208�

(2)

JOURNAL DE PHYSIQUE CoZZoque C8, suppldment au n08, Tome 41, aou't 1980, page C8-447

ELECTRONIC TRANSPORT.

TRANS?ORT P R O P E R T I E S . OF L I Q U I D NON-SIMPLE METALS.

F. Yonezawa

Research I n s t i t u t e for FundamentaZ Physics, Kyoto University, Kyoto, Japan 606

A b s t r a c t . - We d i s c u s s t h e t h e o r e t i c a l a p p r o a c h e s t o t h e t r a n s p o r t p r o p e r t i e s o f l i q u i d non-simple t h e s c a t t e r i n g p o t e n t i a l s a r e s t r o n g . To t h i s c a t e g o r y b e l o n g l i q u i d t r a n s i t i o n m e t a l s and t h e i r a l l o y s , r a r e e a r t h m e t a l s , expanded m e t a l l i c f l u i d s , e t c . F o r t h e s e m e t a l s , t h e p i c t u r e o f n e a r l y f r e e e l e c t r o n s b e i n g weakly s c a t t e r e d by n e u t r a l pseudo-atoms i s n o t p l a u s i b l e and a c c o r d i n g l y t h e Ziman t h e o r y f o r t h e e l e c t r i c a l r e s i s t i v i t y i s no l o n g e r j u s t i f i e d . Although a n a i v e e x t e n s i o n of t h e o r i g i n a l Ziman t h e o r y t o t h e s t r o n g - s c a t t e r i n g s y s t e m s h a s b e e n p r o p o s e d , no s a t i s f a c t o r y g r o u n d s f o r t h e s o - c a l l e d e x t e n d e d Ziman f o r m u l a have been g i v e n .

I t i s r e q u i r e d t h e r e f o r e t o e s t a b l i s h a s y s t e m a t i c scheme of s t u d y i n g t h e t r a n s p o r t p r o p e r t i e s o f t h e s e m e t a l s from f i r s t p r i n c i p l e s . One p r o m i s i n g way t o t h i s end i s t o e x t e n d and a p p l y t o t h e problem t h e t h e o r e t i c a l methods s o f a r developed t o examine o n e - e l e c t r o n p r o p e r t i e s such a s t h e d e n s i t y o f s t a t e s and s p e c t r a l f u n c t i o n s . I n t h e c o u r s e o f e x t e n s i o n , we emphasize t h e i m p o r t a n c e o f t h e b a s i c c o n s e r v a t i o n laws s u c h a s embodied by t h e g e n e r a l i z e d o p t i c a l theorem. Among v a r i o u s methods, t h e e f f e c t i v e medium a p p r o x i m a t i o n (EMA), which h a s been shown t o b e t h e b e s t s i n g l e - s i t e

theory f o r o n e - e l e c t r o n p r o p e r t i e s , a l s o s e r v e s a s a good f i r s t a p p r o x i m a t i o n f o r t h e c o n d u c t i v i t y . The EMA and o t h e r t h e o r i e s a r e a p p l i e d t o t h e t i g h t - b i n d i n g model and t h e m u f f i n - t i n p o t e n t i a l model. Horeover, t h e s-d h i b r i d i z a t i o n model i s a p p r o p r i a t e t o d e s c r i b e t h e s i t u a t i o n a s i s found i n l i q u i d t r a n s i t i o n m e t a l s where t h e n e a r l y - f r e e s - e l e c t r o n s and t h e t i g h t l y - b o u n d d - e l e c t r o n s a r e b o t h p r e s e n t .

Our p h i l o s o p h y u n d e r l y i n g t h e s e t h e o r i e s and t h e f o r m u l q t i o n s a r e b r i e f l y g i v e n , and t h e a p p l i - c a r i o n s t o l i q u i d t r a n s i t i o n m e t a l s and expanded mercury a r e d e m o n s t r a t e d .

1. INTRODUCTION

The present paper deals w i t h some r e c e n t theo- r e t i c a l progress concerning t h e t r a n s p o r t p r o p e r t i e s o f l i q u i d non-simple metals. *

I t i s now e i g h t years since t h e Tokyo conference and four years s i n c e t h e B r i s t o l conference,*

and we can see t h a t t h e t h e o r e t i c a l researches i n t h i s f i e l d have been d e f i n i t e l y making n o t i c e a b l e progress. A t t h e Tokyo conference, i t was p o i n t e d o u t t h a t t h e r e was a gap between ' t h e o r e t i c a l ob- j e c t s ' u s u a l l y s t u d i e d by t h e o r e t i c i a n s and ' e x p e r i - mental o b j e c t s ' measured by e x p e r i m e n t a l i s t s , where t h e t r a n s p o r t p r o p e r t i e s such as c o n d u c t i v i t y were c l a s s i f i e d as t h e experimental o b j e c t s h u t , i t i s these t r a n s p o r t p r o p e r t i e s we a r e now d e a l i n g w i t h . A t t h e B r i s t o l conference, t h e e f f e c t i v e medium approximation ( h e r e a f t e r r e f e r r e d t o as t h e EM)was mentioned as one of the most s o p h i s t i c a t e d theories: b u t i t i s t h e EMA f o r c o n d u c t i v i t y which we are now t a l k i n g about.

I t i s w i d e l y accepted t h a t t h e problem concer- n i n g simple 1 iq u i d metals has been e s s e n t i a l l y s o l - ved from t h e t h e o r e t i c a l p o i n t o f view. I n these li-

q u i d metals, scatter'ings a r e weak and t h e e l e c t r o - n i c s t r u c t u r e and t r a n s p o r t p r o p e r t i e s a r e w e l l described by t h e n e a r l y f r e e - e l e c t r o n (NFE) t h e o r i e s due t o Ziman, Faber and Edwards!'' I n p a r t i c u l a r , t h e Ziman formula f o r r e s i s t i v i t y has a t t a i n e d an outstanding success i n e x p l a i n i n g the e s s e n t i a l features of t h e weak c o u p l i n g s i t u a t i o n . I n sim- p l e l i q u i d metals a r e c l a s s i f i e d a l k a l i metals and some ( n o n - t r a n s i t i o n ) p o l y v a l e n t metals.

On t h e o t h e r hand, we have n o t y e t been suc- c e s s f u l i n s y s t e m a t i c a l l y understanding 1 iq u i d non- simple metals, by which we mean those l i q u i d metals where t h e KFE model no l o n g e r holds, such as t r a n -

s i t i o n , noble and r a r e e a r t h metals and t h e i r a l l o y s , l i q u i d semiconductors and expanded metal- l i c f l u i d s . A t t h e B r i s t o l conference, a review was g i v e n by Eatabe7 o f t h e t h e o r e t i c a l approaches

...

* The numerical r e s u l t s and arguments presented i n t h i s a r t i c l e a r e based upon work c a r r i e d o u t i n c o l l a b o r a t i o n w i t h Prof.S.Asano and Dr.Y.Ishida.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19808112

(3)

C8-448 J O U R N A L DE PHYSIQUE

by 197G t o t h e e l e c t r o n i c p r o p e r t i e s o f these non- simple metals m a i n l y i n the t i g h t - b i n d i n g model.

I n t h e present a r t i c l e , we concern ourselves w i t h t h e t r a n s p o r t p r o p e r t i e s such as t h e e l e c t r i - c a l c o n d u c t i v i t y , and n a t u r a l l y we focus on t h e progress a f t e r B r i s t o l .

2. SOME CHARACTERISTIC PROPERTIES

Although some experimental work on compressed a1 k a l i metal s t 1 iq u i d t r a n s i t i o n and r a r e e a r t h me- t a l s and t h e i r a l l o y s : and l i q u i d Hg9 had been r e - p o r t e d even before t h e Tokyo conference, i t was a t the Tokyo conference when we saw t h e spectacular r e s u l t s o f experiments on t h e t r a n s p o r t and o t h e r p r o p e r t i e s o f these material^!^ Owing t o t h e tech- n i c a l developments o f temperature- and pressure- r e s i s t a n t c o n t a i n e r s and electrodes, a considerable amount o f experimental data has been accumulated by Among v a r i o u s features o f these m a t e r i a l s shown by experiments, t h e f o l l o w i n g p r o p e r t i e s a r e c h a r a c t e r i s t i c e s p e c i a l l y i n comparison w i t h t h e general p r o p e r t i e s o f l i q u i d simple metals.

[I] L i q u i d t r a n s i t i o n metals

(1) The p r o p o r t i o n a l i n c r e a s e i n t h e r e s i s t i v i t y , (pL -pS)/pS , observed a t the m e l t i n g p o i n t i s ab- normally small; values a r e 0.01, 0.09, and 0.33 f o r Fe, Co, and h i r e s p e c t i v e l y w h i l e t h e corresponding values f o r l i q u i d simple metals range from 0.6 t o 1.3.

( 2 ) The magnitude o f t h e r e s i s t i v i t y i s u s u a l l y of t h e o r d e r 100 p n cm and decreases from bin t o Cu i n order Mn -+ Fe + Co + N i -+ Cu.

(3) The r e s i s t i v i t y depends 1 i t t l e on temperature and sometimes t h e temperature c o e f f i c i e n t becomes negative, as i n a l l o y s w i t h Ge and as i n some li- q u i d r a r e e a r t h metals.

(4) The H a l l c o e f f i c i e n t does n o t s a t i s f y t h e f r e e e l e c t r o n formula and i s o f t e n p o s i t i v e , as i n Fe, Co and t4n (by e x t r a p o l a t i o n from r e s u l t s on a l l o y s w i t h Ge), and a l s o i n l i q u i d Ce and La.

(5) The s u s c e p t i b i l i t y h a r d l y changes on me1 t i n g , except i n Fe.

( 6 ) The s u s c e p t i b i l i t y increases i n order Mn + Fe + Co and t h e r e a f t e r decreases i n o r d e r Co -t F!i+ Cu.

This i s i n t e r e s t i n g e s p e c i a l l y i n r e l a t i o n t o t h e p r o p e r t y s t a t e d i n (2).

(7) The temperature c o e f f i c i e n t o f t h e suscepti b i -

l i t y suggests t h e e x i s t e n c e o f l o c a l i z e d magnetic

moments i n l i q u i d Fe, Co, N i and t h e i r a l l o y s w i t h Ge

.

[II] L i q u i d r a r e e a r t h metals

I n general, l i q u i d r a r e e a r t h metals show t h e beha- v i o u r more o r l e s s s i m i l a r t o t h a t o f t h e l i q u i d t r a n s i t i o n metals.

[111] Expanded m e t a l l i c f l u i d s and b i n a r y mix- t u r e s

( 1 ) A wide v a r i e t y o f e l e c t r o n i c p r o p e r t i e s o f expanded Hg (as w e l l as those o f l i q u i d semiconduc- t o r s such as kg-Bi a l o y s and some t e l l u r i u m a l l o y s ) a r e c o n s i s t e n t l y explained when t h e existence o f a pseudogap i s assumed.

(2) The t r a n s p o r t p r o p e r t i e s o f expanded Hg and some b i n a r y m i x t u r e s such as TI-Se, B i - B i B r 3 i n d i - c a t e t h e importance o f t h e d e n s i t y - and composition-

fluctuation^.'^ These p r o p e r t i e s a l s o suggest t h e c l o s e r e l a t i o n s h i p between t h e e l e c t r o n i c and t h e r - modynamic p r o p e r t i e s . ^14"

[ I V ] Other systems which a r e non-simple The study of i i q u i d semiriletals and si.lniconductors has a t t r a c t e d wide a t t e n t i o n , which i n c l u d e v a r i - ous metal-metal, metal-semiconductor, semiconductor- semiconductor, metal - s a l t and o t h e r a1 l o y s . l 8 The experimental r e s u l t s so f a r obtained a r e very f r u i t - f u l and suggestive.

3. THE EXTENDED ZIMAN FORMULA

I n t h e weak s c a t t e r i n g case, t h e e l e c t r i c a l r e s i s t i v i t y pL o f l i q u i d metals i s e s s e n t i a l l y w e l l described by t h e Ziman f ~ r m u l a , ~ which i s obtained e i t h e r by s o l v i n g t h e weak-coupling Boltzmann equ- a t i o n o r by e v a l u a t i n g t h e Kubo-Greenwood formula f o r e l e c t r o n s c o l l i d i n g w i t h weak d i l u t e s c a t t e r e r s . A n a i v e extension of t h e Ziman formula t o t h e s t r o n g s c a t t e r i n g cases was proposed by Evans e t a1 . I 9

On t h e b a s i s o f p h y s i c a l arguments, they d e r i v e d a simple phenomenological formula - t h e s o - c a l l e d extended Ziman formula - by r e p l a c i n g t h e weak i o n pseudopotential i n the o r i g i n a l Ziman t h e o r y w i t h t h e s i n g l e - s i t e , on-shell t m a t r i x i n t h e scheme o f t h e m u f f i n - t i n p o t e n t i a l . Evans e t a l . 2 0 1 a t e r r e - d e r i v e d t h e formula using a s i m p l i f i e d v e r s i o n o f t h e theory based upon a f o r c e - f o r c e c o r r e l a t i o n f u n c t i o n . The t h e o r y r e q u i r e s t h e i n p u t parameters;

(1) a s u i t a b l e m u f f i n - t i n p o t e n t i a l f o r a l i q u i d ; (2) t h e Fermi energy EF;

(3) t h e e f f e c t i v e valence z*.

(4)

The choice o f these parameters a r e somewhat a r b i t - rary." However, t h e c a l c u l a t i o n s based on t h i s extended formula y i e l d good agreement w i t h e x p e r i - ment when these parameters a r e reasonably adjusted.

(See f o r instance r e f s 10,22,23). Besides, t h e model a s s e r t s t h a t t h e r e s i s t i v i t y and thermopower a r e s t r o n g l y governed by t h e p r o x i m i t y o f EF t o the resonance energy Eres,19 and e x p l a i n s t h e tendensy s t a t e d i n [ I - 2 1 o f $2.

I n s p i t e o f i t s q u a n t i t a t i v e successes, t h e extended Ziman formula i s n o t based upon t h e o r e t i c a l f i r s t p r i n c i p l e s and l i t t l e i s known o f t h e p h y s i c a l circunlstances under which i t m i g h t be v a l i d , nor of t h e c o r r e c t i o n s t h a t should be a p p l i e d when these c o n d i t i o n s a r e n o t we1 1 s a t i s f i e d . ^{2 4} I n p a r t i c u l a r , i t has been n o t i c e d f o r some time and r e c e n t l y poin- t e d o u t a f r e s h 2 ' t h a t t h e t h e o r e t i c a l value o f t h e r e s i s t i v i t y computed on t h e basis o f t h e formula i s extremely s e n s i t i v e t o small changes i n t h e i n p u t parameters. T h i s f a c t r e f l e c t s t h e v u l n e r a b i l i t y o f the extended Ziman formula. A c t u a l l y , Esposito e t a1 . 2 5 have made i t c l e a r t h a t , when z* and EF a r e r a t i o n a l l y chosen, t h e r e s i s t i v i t y values f o r Co and Fe a r e so bad t h a t t h e o m i s i o n o f m u l t i p l e - s c a t t e r i n g terms i s o b v i o u s l y serious. The mu1 t i p l e s c a t t e r i n g e f f e c t s a r e completely absent from t h e o r i g i n a l Ziman formula as w e l l . For l i q u i d simple metals, A s h c r o f t and S ~ h a i c h ~ ~ have i n c l u d e d some o f t h e m u l t i p l e - s c a t t e r i n g terms, b u t t h e i r numeri- c a l r e s u l t s s t r o n g l y suggest t h a t t h e e f f e c t s of these terms a r e n o t outstanding f o r these m a t e r i a l s . The m u l t i p l e - s c a t t e r i n g e f f e c t s a r e considered t o be more important i n s t r o n g s c a t t e r i n g system^.^

Attempts have been made t o r e l i e v e t h i s l a s t d i f f i - c u l t y by i n c l u d i n g some o f the mu1 t i p 1 e - s c a t t e r i n g terms and modifying t h e extended Ziman formula a l i t t l e f u r t h e r , 2 7 b u t t h e whole scheme of t h e extended Ziman theory s t i l l remains weakly based and t h e parameter-sensitive tendncy unaltered.

The philosophy u n d e r l y i n g t h e extended Ziman f ~ i r m u l a ~ ~ ' ~ ~ w i l l be discussed i n 56 i n connection w i t h t h e a s s e r t i o n due t o M ~ t t . ~ '

Before concluding t h i s section, l e t us quote from Ziman2" concerning t h e t h e o r y which uses t h e f o r c e - f o r c e c o r r e l a t i o n s . " I n t h e r e c e n t 1 i t e r a - t ~ r e , ~ ' t h e r e i s much d i s c u s s i o n o f an a l t e r n a t i v e f o r m u l a t i o n o f t h e l i n e a r response t h e o r y i n which i n v e r s e t r a n s p o r t c o e f f i c i e n t s (eg , t h e e l e c t r i c a l r e s i s t i v i t y ) a r e expressed i n terms o f t h e f o r c e - f o r c e c o r r e i a t i o n s seen by t h e e l e c t r o n s i n t h e

metal. Although t h i s approach seems a t f i r s t s i g h t t o g i v e some n i c e simple formulae, i t i s now c l e a r t h a t t h e exact c a l c u l a t i o n o f t r a n s p o r t c o e f f i c i e n t s by t h i s means i s no l e s s ardurous and s u b t l e than t h e d e r i v a t i o n s of t h e c o n d u c t i v i t y from t h e conventio- n a l Kubo formula."

4. ONE-ELECTRON PROPERTIES

Since a simple-minded extension o f t h e weak- s c a t t e r i n g t h e o r y t o the s t r o n g - s c a t t e r i n g systems has t u r n e d o u t t o be r a t h e r vulnerable, i t i s now r e q u i r e d t o e s t a b l i s h a systematic approach t o t h e t r a n s p o r t p r o p e r t i e s o f 1 iq u i d non-simp1 e metal s from f i r s t p r i n c i p l e s . A most standard approach t o t h i s end i s t o i n v e s t i g a t e a method of c a l c u l a t i n g t h e Kubo formula i n t h e case o f strong s c a t t e r i n g . Before we study t h e t r a n s p o r t , l e t us b r i e f l y r e - view t h e s i t u a t i o n f o r one-el e c t r o n p r o p e r t i e s .

When i t was made c l e a r t h a t t h e NFE p i c t u r e would n o t be p l a u s i b l e f o r 1 iq u i d non-simple metals, a f i r s t approach was t o b r i n g t r a d i t i o n a l l y suc- c e s s f u l methods o f energy band c a l c u l a t i o n i n crys- t a l l i n e s o l i d s t o bear on disordered systems. The t i g h t - b i n d i n g model i s more a p p r o p r i a t e t o d e s c r i b e t h e s t r o n g - s c a t t e r i n g s i t u a t i o n , where t h e s h o r t - range atomic c o r r e l a t i o n s a r e expected t o be more important. Another attempt i s t o employ t h e m u f f i n - t i n p o t e n t i a l model. The formal ism f o r a muffin- t i n p o t e n t i a l model i s a g e n e r a l i z a t i o n o f t h e KKR method t o a l i q u i d metal.

We use t h e a d i a b a t i c approximation and consider t h e motion o f e l e c t r o n s i n t h e s t a t i c f i e l d o f t h e ions which a r e regarded as being a t r e s t . The elec- tons a r e assumed t o be independent and t h e many- body e f f e c t s a r e taken i n t o account o n l y i n t h e con- s t r u c t i o n o f s e l f - c o n s i s t e n t l y screened p o t e n t i a l s.

A comnon understanding concerning disordered systems i s t h a t macroscopic physical q u a n t i t i e s do n o t de- pend upon the d e t a i l ed c o n f i g u r a t i o n s o f atoms b u t upon c e r t a i n s t a t i s t i c a l averages. Therefore, i t i s necessary t o t a k e t h e ensemble average of a r e - q u i r e d physical q u a n t i t y over a l 1 p o s s i b l e configu- r a t i o n o f ions. For s t r u c t u r a l l y disordered systems such as l i q u i d and amorphous metals, t h e ensemble average r e q u i r e s mu1 t i - i o t d i s t r i b u t i o n f u n c t i o n s , which a r e u s u a l l y approximated by t h e sums and products o f corresponding p a i r d i s t r i b u - t i o n f u n c t i o n s g(R). This i s because g(R) i s t h e o n l y d i s t r i b u t i o n f u n c t i o n a v a i l a b l e b o t h e x p e r i - m e n t a l l y and t h e o r e t i c a l l y . The choice o f how t o

(5)

c8-450 JOURNAL DE _PHYSIQUE

approximate higher d i s t r i b u t i o n f u n c t i o n s by means of g(R) c h a r a c t e r i z e s a theory. A reasonable choice i s e s p e c i a l l y r e q u i r e d i n systems w i t h stronq s c a t t e r i n g p o t e n t i a l s s i n c e short-range c o r r e l a - t i o n s o f ions a r e important i n these systems.

The Green f u n c t i o n method has been shown t o be u s e f u l f o r e v a l u a t i n g t h e e l e c t r o n i c p r o p e r t i e s o f disordered systems. For a given c o n f i g u r a t i o n o f atoms, t h e one-electron Green f u n c t i o n o p e r a t o r G(z) i s d e f i n e d i n terms o f t h e one-electron Hamil- t o n i a n o p e r a t o r H as

and t h e s p e c t r a l o p e r a t o r i s d e f i n e d a s 3 '

The average s p e c t r a l f u n c t i o n i s g i v e n i n the momentum r e p r e s e n t a t i o n as p ( k , ~ ) ~ { { k I d E ) lk)}

w h i l e t h e d e n s i t y o f s t a t e s (DOS) p e r atom s p i n i s described as

D(E) 5 N-'(tr p(E)> = ( ~ a ) - ' t r ~ m ( ~ ( ~ - ) ) (4.3) where N i s t h e number o f e l e c t r o n s . Here, t h e angu- l a r brackets denote t h e ensemble-average. Therefore, the c a l c u l a t i o n of t h e average DOS i s reduced t o t h e , e v a l u a t i o n of the average Green f u n c t i o n , ( ~ ( z ) ) , o r t h e self-energy ~ ( z ) d e f i n e d by (G(z))-[z-Z(Z)]-'.

5. SINGLE-SITE APPROXIMATIONS

For t h e l a s t decade, advances have been made i n c a l c u l a t i n g ( ~ ( z ) ) f o r l i q u i d and amorphous metals.

Since t h e exact average i s g e n e r a l l y d i f f i c u l t t o evaluate, most attempts made so f a r concern them- selves w i t h f i n d i n g a reasonable approximate average of t h e e n t i r e p e r t u r b a t i o n s e r i e s f o r the one-elec- t r o n Green f u n c t i o n . A number o f s i n g e - s i t e approximations (SSA) have been proposed, among which t h e Ishida-Yonezawa ( I Y ) theory3' and t h e E M A ~ ~ due t o Roth have been s t u d i e d most e x t e n s i v e l y . The com- p a r i s o n o f various SSAs has been made, and t h e v a l i d -

i t y o f t h e SSAs i s e ~ a m i n e d . ~ ' ~ ~ ' ~ " - ~ ~ (References of o t h e r SSAs a r e found i n Ref .7). The EM has been shown t o be t h e most s t r a i g h t f o r w a r d and s a t i s f a c t o - r y general i z a t i o n o f t h e coherent p o t e n t i a l approximation (CPA),37 o r i g i n a l l y developed f o r s u b s t i t u - t i o n a l l y disordered systems, t o 1 iq u i d metals The I Y t h e o r y i s 'the f i r s t approximation t o t h e EMA and advantageous i n t h e sense t h a t t h e I Y self-energy i s momentum-independent. This advantage o f t h e I Y theory i s g r e a t e s p e c i a l l y from numerical c a l c u l a -

t i o n p o i n t of view and i t i s u s e f u l t o know t h e range

o f v a l i d i t y o f t h e I Y t h e o r y compared t o t h e ~ ~ 4 4 . ~ ~ [ 1 5 . 1 ) !The CPA for SubstitutionaZ is order]

Since t h e SSAs have been described a t l e n g t h i n r e f . 7 , we do n o t go i n t o d e t a i l s . L e t us j u s t b r i e f l y o u t l i n e the s t r u c t u r e o f t h e approximations f o r l a t e r convenience. The idea o f t h e CPA i s il- l u s t r a t e d i n Fig.1 f o r t h e sake o f comparison. The l h s o f Fig.1 (a) represents a b i n a r y a l l o y composed o f A atoms denoted by open c i r c l e s and B atoms denoted by f i l l e d c i r c l e s , w i t h r e s p e c t i v e concentra- t i o n s cA and cB (cA+cg=l). The ensemble-average i s taken over a l l p o s s i b l e ways o f d i s t r i b u t i n g NcA-NA atoms on t h e N l a t t i c e s i t e s . When d i s o r d e r i s i n - cluded o n l y i n t h e s i t e - d i a g o n a l terms, then t h e system on t h e average has t h e same t r a n s l a t i o n a l symnetry as t h e o r i g i n a l l a t t i c e as expressed by t h e r h s o f F i g . l ( a ) . The average system i s character- i z e d by the e f f e c t i v e atoms denoted by hatched c i r c l e s . The p o t e n t i a l associated w i t h each effec- t i v e atom i s c a l l e d t h e coherent p o t e n t i a l . Since t h e average cannot be d e r i v e d exactly, i t must be s u i t a b l y approximated and one example i s given i n Fig.1 i b ) . The atoms i n a disordered c o n f i g u r a t i o n a r e replaced by t h e e f f e c t i v e atoms except f o r t h e atom a t , say,the i t h s i t e . Therefore, d i s o r d e r i s i n c l u d e d o n l y i n the atom a t s i t e i, and t h e ensemble-average reduces t o counting t h e r e s p e c t i v e p r o b a b i l i t i e s o f f i n d i n g A and B atoms a t s i t e i.

F i g . l ( a ) D e f i n i t i o n of t h e e f f e c t i v e medium f o r a disordered b i n a r y a l l o y ;

(b) The idea o f t h e CPA.

(6)

Obviously, t h e idea i s o f s i n g l e - s i t e nature. I n t h e language o f equations, t h e approximation i s s t a t e d as t a k i n g t h e s c a t t e r i n g t - m a t r i x f o r a s i n g l e i m p u r i t y t o be zero on t h e average; i . e .

where Gii i s t h e s i t e - d i a g o n a l Green f u n c t i o n f o r t h e average system.

C(5.21 The SSAs for StructuraZ is order]

On t h e o t h e r hand, t h e idea o f SSAs f o r l i q u i d and amorphous metals i s described by Fig.2 i n t h e t i g h t - b i n d i n g (TB) model. I t i s e a s i l y seen from (4.3) t h a t , i n t h e TB model, t h e DOS i s obtained by counting a l l t h e paths which s t a r t from, say, t h e i t h s i t e and r e t u r n t o the same s i t e . The ensemble- average i s performed over a l l p o s s i b l e d i s t r i b u - t i o n s o f atoms denoted by open c i r c l e s i n Fig.2.

The SSAs count o n l y those paths which a r e c a l l e d t h e extended chains o r i n o t h e r words which l o o k l i k e cactes as represented by t h i c k l i n e s i n Fig.2.

On t h e o t h e r hand, a l l m u l t i s i t e ( o r c l u s t e r ) terms d e s c r i b i n g r e p r a t e d s c a t t e r i n g s o f an e l e c t r o n by more than one atoms a r e completely absent from t h e SSAs. This means t h a t t h e k i n d o f p a t h as r e p r e - sented by t h i n l i n e s i n Fig.2 i s discarded. The average system o f t e n r e f e r r e d t o as t h e e f f e c t i v e medium r e s t o r e s t h e t r a n s l a t i o n a l i n v a r i a n c e o f f r e e e l e c t r o n s ; i.e. t h e system i s j e l l y - l i k e as i l l u s t r a t e d by Fig.2. I n t h e SSAs, a l l t h e e f f e c t s o f d i s o r d e r and average a r e included i n the e f f e c - t i v e s i t e energy Ed and t h e e f f e c t i v e t r a n s f e r energy M(R), u s u a l l y denoted as t h e diagonal and o f f - d i a g o n a l p a r t s o f t h e s e l f -energy, respective1 y.

The most e s s e n t i a l d i f f e r e n c e between t h e l i q u i d problem and t h e a l l o y problem c o n s i s t s i n t h e i n d i c a t i o n o f the ensemble-average; i n t h e a l l o y case, t h e average as i n d i c a t e d by F i g . l ( b ) u n i q u e l y

Fig. 2 D e f i n i t i o n of t h e e f f e c t i v e medium f o r a l i q u i d metal.

d e f i n e s t h e approximation w h i l e i n the l i q u i d case t h e average i n d i c a t e d by Fig.2 needs f u r t h e r speci- f i c a t i o n as t o t h e way how t o approximate t h e m u l t i - s i t e c o r r e l a t i o n s i n terms o f g(R). The I Y theory takes i n t o account t h e excluded volume e f f e c t o n l y between each p a i r o f two succeeding s i t e s on a p a t h w h i l e t h e EMA i n c l u d e s a l l t h e excluded volume e f f e c t w i t h i n each closed chain. A l l o t h e r SSAs a r e somewhere between t h e I Y and t h e EEN.

I n t h e EMA, t h e e f f e c t i v e s i t e energy Cd(z) and t h e F o u r i e r t r a n s f o r m o f t h e e f f e c t i v e t r a n s f e r energy, Kk(z), a r e d e f i n e d by

where G~(Z)~(G..(Z)}.=[Z-C~(Z)]-~, 11 1 n i s t h e number d e n s i t y and Gk, i s t h e F o u r i e r t r a n s f o r m o f

Y

[(Gii)i6(Rij) + ( G ~ ~ ) ~ ~ ] ; Hk, Hk and h(k) a r e t h e F o u r i e r transforms r e s p e c t i v e l y o f nH(R), nH(R)g(R) and h(R)=g(R)-1, where H(R. .)=H i s t h e o f f - d i a -

I J jj

gonal m a t r i x element o f t h e Hamiltonian i n t h e t i g h t - b i n d i n g r e p r e s e n t a t i o n . (See a l s o Fig.3 f o r t h e expressions by means o f diagrams.) Note t h a t Hii i s assumed t o be c o n f i g u r a t i o n independent and t h e o r i g i n o f energy i s chosen so t h a t Hii=O. The corresponding q u a n t i t i e s f o r t h e I Y t h e o r y a r e ob- t a i n e d by p u t t i n g h(k)=O i n (5.2) and (5.3). I n t h e above formulations, t h e TB bases a r e assumed t o be orthonormal and o n l y one o r b i t a l per s i t e i s taken i n t o account. The extension t o t h e case wherk t h e bases a r e n o t m u t u a l l y orthogonal has been shown t o

Fig. 3 P r e s c r i p t i o n s f o r diagram expression.

(7)

~ 8 - 4 5 2 JOURNAL DE PHYSIQUE

be r a t h e r easy. E s s e n t i a l l y , t h e extension i s achieved by r e p l a c i n g H(R) by [H(R)-ES(R)], S(R) being t h e o v e r l a p i n t e g r a l , and by t a k i n g proper care when we deal w i t h t h e o f f - d i a g o n a l p a r t o f t h e Green f u n c t i o n . The e x t e n s i o n t o m u l t i - o r b i t a l models i s a l s o ~ t r a i g h t f o r w a r d . ~ ' I n p r i n c i p l e , t h e m u l t i - o r b i t a l equations can be obtained by r e - p l a c i n g s c a l a r q u a n t i t i e s i n t h e equations f o r a s i n g l e - o r b i t a l model w i t h t h e corresponding matrices.

6. THE I Y AND EMA CONDUCTIVITY [ ( ( ? . I ) The Vertex c o r r e c t i o n s j

The t r a n s p o r t c o e f f i c i e n t s a r e given by t h e Kubo formula as

a (E) = 2 n h t r ( j a 6 ( E - ~ ) j s 6 ( E - ~ ) ) ,

a s

where f ( E ) i s t h e Fermi-Dirac d i s t r i b u t i o n f u n c t i o n , and ja=eva stands f o r t h e a t h component o f t h e one- e l e c t r o n c u r r e n t operator. Therefore, t h e problem i s reduced t o the study o f t h e q u a n t i t y (jaG(zl)x jsG(z,$. Generally, i t i s n o t an easy t a s k t o evaluate t h i s q u a n t i t y . The s i t u a t i o n i s most t r a n s p a r e n t l y s t a t e d as follows f o r t h e case o f con- f iguration-independent c u r r e n t operators. That i s t o say, we cannot c a l c u l a t e t h e average o f a prod- u c t o f % Green f u n c t i o n s from knowledge o f t h e average o f a s i n g l e Green f u n c t i o n because (G 6) f (G) (G). A c t u a l l y , t h e d i f f e r e n c e i s u s u a l l y denoted as t h e v e r t e x - c o r r e c t i o n o p e r a t o r r defined by

When t h e operators a r e described i n t h e TB basis, the c u r r e n t operators a r e configuration-dependent and t h e s i t u a t i o n i s a l i t t l e more complicated a l - though no d i f f i c u l t y o f e s s e n t i a l nature i s i n t r o - duced thereby. I n any case, t h e v e r t e x c o r r e c t i o n s l' a r e d e f i n e d by some s e l f c o n s i s t e n t i n t e g r a l equa- t i o n o f t h e Bethe-Salpeter form, which i n most' cases must be solved numerically.

C(6.2) The I Y ~ o n d u c t i v i t y ]

The c o n d u c t i v i t y formula i n t h e approximation c o n s i s t e n t w i t h t h e I Y t h e o r y was f i r s t obtained by I t o h and ~ a t a b e , ~ ' and an a p p l i c a t i o n o f t h e formu- l a t o a simple model l i q u i d w i t h t h e s i n g l e s-band was r e p o r t e d a t t h e B r i s t o l conference; i n t h i s model, t h e c o n t r i b u t i o n s from t h e terms i n c l u d i n g t h e v e r t e x c o r r e c t i o n s simply vanish f o r more o r l e s s t h e same reason as t h a t by which t h e CPA v e r t e x c o r - r e c t i o n s a r e shown t o vanish f o r a s i n g l e s-band."

For mu1 t i - o r b i t a l s, t h e v e r t e x c o r r e c t i o n s g i v e f i n i t e c o n t r i b u t i o n s even i n t h e I Y theory.

However, a remarkable p o i n t about t h e I Y t h e o r y i s the f a c t t h a t t h e Bethe-Salpeter equation f o r t h e I Y v e r t e x c o r r e c t i o n s can be solved ana- l y t i c a l l y and therefore, once we o b t a i n (6) i n t h e I Y theory, t h e I Y c o n d u c t i v i t y i s c a l c u l a t e d w i t h - o u t any f u r t h e r trouble. I t i s i n t e r e s t i n g t o n o t e t h a t t h i s advantageous aspect o f t h e I Y c o n d u c t i v i t y i s analogous t o t h e advantage o f t h e mmentum-independent I Y self-energy which enormously s i m p l i f i e s t h e e v a l u a t i o n o f (G)Iy. Therefore, we emphasize here again t h a t i t i s important t o know t h e range o f v a l i d i t y o f t h e I Y theory.

The f o r m u l a t i o n o f t h e I Y c o n d u c t i v i t y w i t h nonorthogonal i t y terms has a1 so been obtained by I t o h e t a1

.

( p r i v a t e communication).

l ( 6 . 3 ) The GeneraZized Optical Theorem and t h e C o n d u c t i v i t y ~ o m m c k t i o n ]

Now, t h e n e x t s t e p we t a k e i s t o d e r i v e t h e EMA formula f o r t h e c o n d u c t i v i t y . T h i s o f course can be performed by summing o u t o f t h e p e r t u r b a t i o n expansion o f (6.3) o n l y those terms which a,re con- s i s t e n t w i t h t h e EMA for<G>. A p r e l i m i n a r y attempt i n t h i s l i n e was f i r s t made by I t o h 7 a n d i t s exten- s i o n has been s t u d i e d by I t o h e t a1 ( p r i v a t e commu- n i c a t i o n ) . The r e s u l t s a r e presented a t t h i s conference by I t o h e t a l 4 l and a l s o by Roth and Singh?2 The diagrams again f a c i l i t a t e t h e formulation g r e a t - l y . However, we would l i k e t o l o o k a t t h e problem from a d i f f e r e n t viewpoint.

I t i s w e l l known t h a t t h e general c o n s e r v a t i o n law y i e l d s t h e r e l a t i o n s h i p between t h e v e r t e x - c o r r e c t i o n o p e r a t o r and t h e self-energy operator, which f o r example can be expressed as

w h i l e r(z,z)=-dc(z)/dz f o r z,=z2=z although a ( 1 aB i s s i m p l i f i e d s t i l l f u r t h e r . The r e l a t i o n (6.5) can be expressed i n v a r i o u s forms, and i s o f t e n r e f e r r e d t o as t h e g e n e r a l i z e d o p t i c a l theoremY3

(8)

or the Ward identity." The optical theorem i s derived from the u n i t a r i t y of the S-matrix and related t o the probability conservation. The uni- t a r i t y of the S-matrix a l s o yields the energy and particle-number conservation. The importance of t h i s theorem should not be underestimated. I t i s a selfconsistency r e l a t i o n between the self-energy and the vertex corrections. I t t e l l s us how t o make approximations f o r these q u a n t i t i e s in a s e l f - consistent way; any approximation which does not comply w i t h (6.8) i s l i a b l e t o nonsensible r e s u l t s . For instance, i t has been shown t h a t a non-selfcon- s i s t e n t inclusion of c l u s t e r terms r e s u l t s in neg- a t i v e conductivity."+

The theorem was studied by ~ u b i o ~ ~ f o r disordered systems i n general and f o r 1 iquid metals a s an example while velicky4' examined the theorem in connection with the CPA and derived other conservation laws. In p a r t i c u l a r , although in Rubiols paper there a r e some parts which need more careful analy- ses, the paper is very suggestive.

Now, our p h i l ~ s o p h y ' ~ ' ' ~ i s t o regard the i d e n t i t y (5.5) as t e l l i n g us not only how t o make approximations s e l f c o n s i s t e n t but a l s o how t o derive t h e vertex c o r r e c t i o n s direct1 y from the know1 edge of the self-energy. Then, why should we not use the i d e n t i t y as a tool w i t h which t o evaluate the vertex corrections rather than calculating the vertex corrections through some other methods, proving there- a f t e r t h a t the r e s u l t s thus obtained f u l f i l l the i d e n t i t y and asserting t h a t the v a l i d i t y of the re- s u l t s a r e therefore ascertained? I t may be too quick t o conclude a t t h i s point t h a t t h i s i s "the Almighty" on a l l occasions, but we believe t h a t t h i s

idea i s of practical use in many cases when proper care i s taken of each case. A t l e a s t , t h i s i s the way how we have obtained the EMA conductivity.

I ( 6 . 4 ) The EMA ~ o n d u c t i v i t ~ ]

The l h s of (6.5) f o r the EMA i s in the momen-

t u m representation divided into two parts a s

X k ( z l ) - ' k ( ~ 2 )

Z l - ^{2 2}

- Cd(z2) Mk(zl) - Mk(z2)

- - -

z1 - ^z2 ^z1- ^{2 2} ^(6.6)

Using (5.2), we can rearrange the f i r s t term of (6.6)' a s

C d ( ~ L ) - ^zd(z2)

21

-

^'3

where

and

On the other hand, (5.3) applied t o the second term of (6.6) leads to;

-3

can be obta~ned by puttlng h ( R - ) = o.

2 11

=

-a6 ( 2 1 , ~ ~ ) (jCLG(zl ) j s ~ ( z ~ ) >

Fig.4. The EMA conductivity.

Fig. 5. Definition of the e f f e c t i v e current, e t c .

(9)

C8-454 ^JOURNALDE PHYSIQUE

where

-..

Xkl (zl,z2) = Gk(~L)Gk(z2)/Gd(~l)Gd(z2) - 1

-

^(6.11)

Equations (6.7) and (6.10) a r e t h e coupled Bethe- Salpeter equations which can be converted i n t o a more compact form.

By t a k i n g proper c o n s i d e r a t i o n f o r two average o f t h e c u r r e n t operators and u s i n g (6.7) and (6.10), we can d e r i v e t h e EMA c o n d u c t i v i t y . I n order t o save t h e space, l e t us describe our f i n a l form f o r t h e c o n d u c t i v i t y tensor i n t h e language o f diagrams, which i s i l l u s t r a t e d by Fig.4 w h i l e the e f f e c t i v e c u r r e n t s jeff (Ri j) e t c . a r e d e f i n e d by t h e diagram equations shown i n Fig.5.

From Figs.4 and 5, we can summarize t h e steps t o c a l c u l a t e the EMA c o n d u c t i v i t y as;

( 1 ) To e v a l u a t e t h e average Green f u n c t i o n (G) i n t h e EMA:

(2) To solve t h e Bethe-Salpeter equation f o r t h e v e r t e x c o r r e c t i o n s , o r e q u i v a l e n t l y f o r t h e e f f e c t i v e c u r r e n t s where the i n p u t data Gd, Gk, Mk a r e provided by (G)EMA-

I n most cases, t h e Bethe-Salpeter i n t e g r a l equation cannot be solved a n a l y t i c a l l y , t h e v e r t e x c o r r e c t i o n s ( o r t h e e f f e c t i v e c u r r e n t s ) must be c a l c u l a t e d numerically, and the s o l u t i o n s a r e usual 1 y evaluated by a s e l f c o n s i s t e n c y i t e r a t i o n procedure. Although t h i s sounds awful, t h e prac- t i c a l c a l c u l a t i o n s a r e n o t so s u b t l e because t h e i n t e g r a l equation i s l i n e a r i n t h e e f f e c t i v e c u r r e n t . This means t h a t , once t h e average Green f u n c t i o n (G) f o r t h e EMA i s obtained, t h e c a l c u l a - t i o n o f t h e EMA c o n d u c t i v i t y i s comparatively easy.

I t i s easy t o see t h a t t h e diagrams i n c l u d e d i n Fig.4 a r e t h e sum o f those terms which a r e con- s i s t e n t w i t h t h e EMA f o r (G). I t i s a l s o s t r a i g h t - forward t o show t h a t t h e I Y c o n d u c t i v i t y i s ob- t a i n e d by p u t t i n g h(R)=h(k)=O i n t h e f o r m u l a t i o n f o r t h e EMA c o n d u c t i v i t y . The extension t o m u l t i - o r b i t a l cases can again be achieved by r e -

p l a c i n g s c a l a r q u a n t i t i e s i n t h e above f o r m u l a t i o n w i t h t h e corresponding m a t r i c e s and by arranging

T

^I^-,'---

.

^{I Y} Fig.6. The DOS by t h e I Y t h e o r y and EMA f o r

D(E) a model described i n

I I t h e t e x t .

2 0

I

, /

E E

Fig.7. The I Y & EMA Fig.8. The EMA c o n d u c t i v i - c o n d u c t i v i t i e s . t i e s w i t h and w i t h o u t t h e

v e r t e x c o r r e c t i o n s . t h e m a t r i c e s i n proper order.

Before we conclude t h i s subsection, l e t us mention t h a t we can g i v e p h y s i c a l l y more transpar- e n t i n t e r p r e t a t i o n o f t h e s i n g l e - s i t e approxima- t i o n s f o r t h e c o n d u c t i v i t y . I t i s n o t d i f f i c u l t t o see from Fig.4 t h a t t h e essence o f t h e SSAs f o r the c o n d u c t i v i t y i s e q u i v a l e n t t o r e q u i r i n g t h e v e r t e x c o r r e c t i o n s f o r t h e d e v i a t i o n o f t h e c u r r e n t from t h e e f f e c t i v e v a l u e be zero on t h e average;

i.e.

where t h e e f f e c t i v e c u r r e n t j:ff(~; z1 ¶ z 2 ) i s de- f i n e d by Fig.5(a). On t h e o t h e r hand, we can use (6.11) t o define jEff. T h i s i s analogous t o t h e idea used by ~ i i z e k i " ~ t o d e r i v e t h e CPA c o n d u c t i v i t y . Here again, our s e l f c o n s i s t e n t requirement (6.11 ) i s d i f f e r e n t from t h e corresponding equation f o r t h e CPA i n t h e sense t h a t (6.11) i n c l u d i n g t h e ensemble-average over s t r u c t u a l l y disordered c o n f i g - u r a t i o n s does n o t d e f i n e jeff uniquely. When t h e average i s taken by t r e a t i n g t h e m u l t i - s i t e c o r r e - l a t i o n s on t h e basis o f t h e 1Y theory, then (6.11) d e f i n e s the I Y e f f e c t i v e c u r r e n t jeff. On t h e o t h e r hand, (6.11) d e f i n e s t h e EMA e f f e c t i v e c u r - r e n t when the m u l t i - s i t e c o r r e l a t i o n s a r e approximated by the EMA.

(10)

[ (6.5) Numerical R e s u l t s for the EMA ~ o n d u c t i v i t y ] The c a l c u l a t e d EMA d e n s i t y o f s t a t e s f o r a model l i q u i d i s compared i n Fig.6 w i t h the I Y den- s i t y o f states. The model i s c h a r a c t e r i z e d by t h e Percus-Yevic s o l u t i o n f o r g(R) and by hydrogen-1 i ke 1s atomic o r b i t a l s o f e f f e c t i v e Bohr r a d i u s a*.

The hard-core parameter i s taken t o be 7.0 a* which has been estimated as t h e a p p r o p r i a t e value f o r d e l e c t r o n s i n t r a n s i t i o n metals. ^{3 2}The dimensionless d e n s i t y P=32nn(a*) 3 i s taken t o be 0.15. The non- o r t h o g o n a l i t y e f f e c t s a r e discarded. The who1 e be- haviour o f t h e b o t h DOS i s v e r y s i m i l a r t o t h e I Y and EMA DOS c a l c u l a t e d by R ~ t h ~ ~ f o r a l i t t l e d i f - f e r e n t p o t e n t i a l . The EMA c o n d u c t i v i t y w i t h t h e v e r t e x c o r r e c t i o n s f o r t h e afore-mentioned model i s shown i n Fig.7 i n comparison w i t h t h e I Y con- d u c t i v i t y . The d i f f e r e n c e between t h e I Y t h e o r y and t h e EM i s l a r g e r f o r t h e c o n d u c t i v i t y than f o r t h e DOS, which i s more o r l e s s expected from t h e f o r m u l a t i o n i t s e l f . I n order t o show t h e e f f e c t s o f t h e v e r t e x corrections, t h e c o n d u c t i v i t i e s w i t h and w i t h o u t t h e v e r t e x c o r r e c t i o n s a r e i l l u s t r a t e d i n Fig.8, from which t h e importance o f i n c l u d i n g t h e v e r t e x c o r r e c t i o n s i s s e l f - e v i d e n t . The e f f e c t s o f t h e v e r t e x c o r r e c t i o n s a r e most dominant near t h e c e n t r e o f t h e band.

7. DISCUSSION

It seems t h a t t h e problem o f e v a l u a t i n g the e l e c t r i - c a l c o n d u c t i v i t y o r r e s i s t i v i t y i n l i q u i d non-simple metals i s now w i t h i n our reach, and t h e importance o f t h e r o l e played by d e l e c t r o n s i n l i q u i d t r a n s i - t i o n metals seems undeniable " 9 ' 5 3 . A c a r e f u l ana- l y s i s of t h e r e s u l t s from f i r s t p r i n c i p l e s i n d i - cates t h a t t h e random phase approximation (RPA) which leads t o the formula aa[D(EF)12 becomes l e s s r e 1 ia b l e when t h e short-range c o r r e l a t i o n s o f t h e atomic d i s t r i b u t i o n s a r e i m p ~ r t a n t . ' ~ This i s physi- c a l l y p l a u s i b l e because, under s t r o n g atomic c o r r e - l a t i o n s , t h e assumption o f 'random phase' f o r elec- t r o n wavefunctions i s expected t o be weakly based.

Other problems l e f t open i n c l u d e (c.f. the charac- t e r i s t i c p r o p e r t i e s o f these m a t e r i a l s i n 52):

(1) The s i g n and t h e magnitude o f the H a l l c o e f f i - c i e n t s ;

( 2 ) A b e t t e r understanding o f t h e pseudogap; ^{l 6} 5 4

( 3 ) The e f f e c t s o f d e n s i t y - and composition f l u c t u - a t i o n s ;

( 4 ) The i n t e r r e l a t i o n s h i p between the e l e c t r o n i c and thermodynamic p r o p e r t i e s .

( 5 ) The study o f o t h e r t r a n s p o r t p r o p e r t i e s such as thermopower, AC c o n d u c t i v i t y and termal conduc- t i v i t y .

( 6 ) Problems concerning amorphous metals.

As f o r ( I ) , several attempts 5 5 5 4 have been made i n the l i n e o f t h e Boltzmann equation, which so f a r do n o t seem t o be v e r y successful. Another promising approach i s t o c a l c u l a t e t h e H a l l c o e f f i c i e n t s from f i r s t p r i n c i p l e s by means o f the t h e o r e t i c a l scheme described i n t h e preceding sections. Some i n v e s t i g a - t i o n s i n t h i s d i r e c t i o n have been made o f a model l i q u i d o f s i n g l e s-band w i t h i n t h e I Y theory.56 3 9

A study o f a more r e a l i s t i c system i n the EMA i s one o f our f u t u r e plans. The second problem o f the pseudogap r e q u i r e s t h e d e t a i 1 ed study of t h e Anderson l o c a l i z a t i o n i n t h e l i q u i d . The t h i r d problem of f l u c t u a t i o n s i s o u t s i d e t h e scope of t h e SSAs, and a d i f f e r e n t viewpoint would be necessary; the concept such as the,inhomogeneous media i s now required.

The f o u r t h problem i s d e f i n i t e l y important. The more experimental data i n connection w i t h t h i s problem a r e accumulated, t h e more urgent i t becomes f o r us t o i n v e s t i g a t e t h i s problem. This f o u r t h problem seems t o be r a t h e r t r a c t a b l e compared t o t h e s u b t l e problems s t a t e d i n ( 2 ) and (3). The e v a l u a t i o n of o t h e r t r a n s p o r t p r o p e r t i e s i s undoubt- e d l y more d i f f i c u l t and f a l l s i n the f u t u r e problem.

Before conclusion, l e t us mention t h a t , since most o f t h e amorphous metals a r e non-simple, t h e methods developed f o r l i q u i d non-simple metals can as w e l l be a p p l i c a b l e t o amorphous metals.

ACKNOWLEDGEMENTS

The author i s deeply indebted t o Dr.1shida and Prof. S.Asano f o r c o l l a b o r a t i o n . She i s a l s o g r a t e f u l t o P r o f . H.Endo, P r o f . T.Matsubara, Dr. T.

Ogawa and Dr. M.Yao f o r v a l u a b l e discussions; t o Dr. M.Itoh f o r f r u i t f u l discussions and f o r h i s h e l p i n t h e course o f preparing t h e manuscript; t o P r o f . I.Ono f o r h i s u s e f u l suggestions about t h e computer work; t o Prof.Watabe and t o P r o f . Cusack

f o r t h e i r c r i t i c a l reading o f t h e manuscript.