DENSITY OF ELECTRONIC STATES IN LIQUID METALS : AN EFFECTIVE MEDIUM APPROACH

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DENSITY OF ELECTRONIC STATES IN LIQUID METALS : AN EFFECTIVE MEDIUM APPROACH

W. John, W. Keller

To cite this version:

W. John, W. Keller. DENSITY OF ELECTRONIC STATES IN LIQUID METALS : AN EFFEC- TIVE MEDIUM APPROACH. Journal de Physique Colloques, 1980, 41 (C8), pp.C8-400-C8-402.

�10.1051/jphyscol:1980898�. �jpa-00220553�

JOURNAL DE PHYSIQUE CoZZoque C8, suppl6msnt au n08, Tome 4 1 , aoct 1980, page C8-400

DENSITY OF ELECTRONIC STATES I N L I Q U I D METALS

:

AN EFFECTIVE MEDIUM APPROACH W. J o h n and W. K e l l e r

Sektion Physik der Technischen Universitiit b e s d e n , DDfi

-

8027 Dresden, R.D.A.

A b s t r a c t . - We c o n s i d e r a c l u s t e r o f m u f f i n - t i n p o t e n t i a l s , embedded i n a homogeneous b u t e n e r g y d e p e n d e n t e f f e c t i v e medium. The e f f e c t i v e medium. The e f f e c t i v e medium i s d e t e r m i n e d from a g e n e r a - l i z e d Wigner-Seitz c o n d i t i o n . The m u l t i p l e s c a t t e r i n g w i t h i n t h e c l u s t e r i s t a k e n i n t o a c c o u n t i n t h e a t o m i c s p h e r e a p p r o x i m a t i o n . Numerical r e s u l t s f o r t h e d e n s i t y o f s t a t e s i n expanded l i q u i d mercury a r e p r e s e n t e d . The r e s u l t s show a t low d e n s i t i e s a gap between t h e 6 s and 6p s t a t e s i n good a g r e e - ment w i t h band s t r u c t u r e c a l c u l a t i o n s f o r h y p o t h e t i c a l c r y s t a l l i n e s t r u c t u r e s .

I t is w e l l c o n f i m e d t h a t t h e l o c a l den- t h e e f f e c t i v e medium e q u a t i o n g i v e n i n s i t y of e l e c t r o n s t a t e s i n a m e t a l is a /4/. Vle r e g a r d a t o p o l o g i c a l d i s o r d e r e d l o c a l p r o p e r t y which is m a i n l y d e t e r m i -

n a t e d by t h e s h o r t r a n g e o r d e r . T h i s c i r - cumstance s u g g e s t t o c a l c t l l a t e t h e d e n s i - t y o f s t a t e s (DOS) i n l i q u i d o r amor,?hous m e t a l s by a c l u s t e r approach. Following I.:cCill a a d Klima /I/ s e v e r a t a u t h o r s have c a l c u l a t e d t h e D3S i n t r a n s i t i o n m e t a l s from a m u f f i n - t i n c l u s t e r embedded i n a s e a o f f r e e e l e c t r o n s ( s e e e.g./2/).

The c l u s t e r approach works w e l l f o r t h e t i g h t l y bound d s t a t e s i n t r a n s i t i o n me- t a l s b e c a u s e t h e g e n e r a l f e a t u r e s of t h e

system o f i d e n t i c a l m u f f i n - t i n p o t e n t i a l s e n c l o s e d i n a box. The volume of t h e box can be d i v i d e d i n n o n r e g u l a r p o l y h e d r a i n s u c h a way t h a t e a c h polyhedron in- VOlvesthe same number of atoms. Next,

l e t u s c o n s i d e r a r e a l s o l u t i o n

v.' (r)

o f t h e S c h r o d i n g e r e q u a t i o n which f u l - f i l s t h e boundary c o n d i t i o n

Y/'

^{= 0}

a t t h e s u r f a c e of t h e box. Then t h e r e l a - t i o n

d bands depend mainly on t h e n e a r e s t i s v a l i d , where t h e s u r f a c e i n t e g r a l s a r e n e i g h b o u r s c a t t e r i n g . For s i m p l e m e t a l s

and f o r t h e s - l i k e s t a t e s a t t h e bottom o f t h e bands i n t r a n s i t i o n m e t a l s t h e c l u s t e r approach i s n o t s o e f f e c t i v e . On t h e o t h e r hand one c a n t r y t o c a l c u l a t e

t a k e n o v e r t h e boundary of a l l p o l y h e d r a . Using t h e d e f i n i t i o n of t h e Green f u n c - t i o n

t h e l o c a l DOS from a s i n g l e p o t e n t i a l we can f i n d t h e c o n d i t i o n embedded i n an e f f e c t i v e medium, which

s h o u l d s i m u l a t e t h e m u l t i p l e s c a t t e r i n g of the surrounding atoms 1 3 - 6 / . The e f f e c t i v e medium can be determined by a g e n e r a l i z e d Wlgner-Seitz c o n d i t i o n / 4 , 6 / . T h i s method g i v e s a r e a s o n a b l e d e s c r i p t i o n o f t h e s and p s t a t e s i n a metal and i s much s i m p l e r t o handle a s t h e m u l t i p l e s c a t t e r i n g approach developed by Roth, Yshida, Yonezawa and o t h e r a u t h o r s ( s e e e . g . /7/ and t h e r e f e - r e n c e s g i v e n t h e r e ) . However t h i s s i n g l e s i t e scheme does n o t g i v e a s a t i f a c t o r y account of t h e d bands i n a t r a n s i t i o n m e t a l / 4 / . T h e r e f o r e i t seems t o be r e a s o - n a b l e t o study t h e e l e c t r o n i c s t r u c t u r e o f a c l u s t e r embedded i n an e f f e c t i v e medium / 9 /

-

I n t h i s p a p e r we c o n s i d e r a s i n g l e clu- s t e r of m u f f i n - t i n p o t e n t i a l s embedded i n a homogeneous b u t energy-dependent p o t e n t i a l Vc(E). Our c o n d i t i o n f o r t h e p o t e n t i a l Vc(E) i s a c l u s t e r v e r s i o n o r

< f i t z

G(I-,<

^i} = 0 .

pf0

^{:r ( 3 )}

I n a c r y s t a l where t h e p o l y h e d r a form a r e g u l a r s t r u c t u r e , t h e c o n t r i b u t i o n o f a l l p o l y h e d r a are e q u a l due t o t h e Bloch theorem. Now we n e g l e c t f l u c t u a t i o n s i n t h e d i s o r d e r e d s t a t e and assume t h a t a l s o i n t h e d i s o r d e r e d system t h e c o n t r i b u - t i o n s o f a l l p o l y h e d r a a r e e q u a l . Repla- c i n g t h e s u r f a c e i n t e g r a l o v e r t h e sur- f a c e o f a c l u s t e r p o l y h e d r o n by a n i n t e - g r a l o v e r a s p h e r e w i t h t h e r a d i u s ro (Fig.1) we o b t a i n from ( 3 ) t h e c o n d i t i o n

' 3

~ \ ~ c ~ s ~ G c T - < E ) / -

^{t =I; =}

o

⁽⁴⁾

which must be f u l f i l l e d f o r each c l u s t e r . E q u a t i o n ( 4 ) can be u s e d i n o r d e r t o de- t e r m i n e t h e e f f e c t i v e medium s e l f - c o n - s i s t e n t l y . To do t h i s we c o n s i d e r a s i n g -

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980898

F i g . 1 C l u s t e r embedded i n a n e f f e c t i v e p o t e n t i a l Vc(E).

l e c l u s t e r w i t h i n t h e r a d i u s ro, embedded i n t h e e f f e c t i v e p o t e n t i a l Vc(S) ( F i g . ? ) . The e f f e c t i v e medium Vc(E) is d e t e r m i n a - t e by t h e c o n d i t i o n t h a t t h e Green func- t i o n G(r,r9,E) of t h e s i n g l e c l u s t e r i n t h e e f f e c t i v e p o t e n t i a l Vc f u l f i l s equa- t i o n ( 4 ) . A s it was shown i n /4/ f o r a s i n g l e s c a t t e r e r from ( 4 ) t h e c o n d i t i o n

C ( . 2 C ~ f f i J i ~ / / ~ ~ -.&,I.= ⁰

⁽⁵⁾

I*

f o l l o w s , where L1 a r e t h e l o g a r i t h m i c d e r i v a t i v e s of t h e r e g u l a r s o l u t i o n s o f t h e S c h r o d i n g e r e q u a t i o n

yl

a r e t h e c o r r e s p o n d i n g q u a n t i t i e s f o r t h e s p h e r i c a l Hankel f u n c t i o n s w i t h t h e wave number kc = ( E - v ~ ) ' ' ~

S i m i l a r t o (5) one o b t a i n s f o r a c l u s t e r

where t h e m a t r i x e l e m e n t s DLL, a r e gene- r a l i z e d l o g a r i t h m i c d e r i v a t i v e s /9/ which can be c a l c u l a t e d from t h e r e g u l a r s o l u - t i o n s o f t h e S c h r o d i n g e r e q u a t i o n w i t h i n t h e c l u s t e r s p h e r e and f i s a d i a g o n a l m a t r i x w i t h m a t r i x e l e m e n t s

The s c a t t e r i n g s t a t e s w i t h i n t h e c l u s t e r are c a l c u l a t e d employing m u l t i p l e s c a t t e - r i n g t h e o r y and t h e a t o m i c s p h e r e appro- x i m a t i o n of O.K. Andersen

/lo/.

A f t e r d e t e r m i n a t i o n t h e p o t e n t i a l Vc(E) t h e l o c a l DOS w i t h i n t h e Wigner-Seitz s p h e r e ( r a d i u s r s ) f o r an atom of t h e c l u s t e r is g i v e n by

D e t a i l s of t h e method w i l l b e p u b l i s h e d e l s e w h e r e .

AS a n a p p l i c a t i o n of o u r method n u m e r i c a l c a l c u l a t i o n s a r e c a r r i e d o u t f o r expanded l i q u i d mercury. L i q u i d Xg u n d e r g o e s a c o n t i n u o u s m e t a l - i n s u l a t o r t r a n s i t i o n i n t h e d e n s i t y r a n g e between Q = 8.5 and l o g cm-3 / I ? / . T h i s t y p of metal-nonmetal t r a n s i t i o n i n a d i v a i e n t m e t a l l i k e mer- c u r y can be u n d e r s t o o d i n a s i m p l e c r y s t a l l i n e model. A t r e d u c e d d e n s i t i e s t h e o v e r l a p o f s and p bands d e c r e a s e s and a t a c r i t i c a l d e n s i t y a g a p o c c u r s . : c t u a l

,

t h e r e s u l t s o f v a r i o u s ban9 s t r u i - t u r e c a l c u l a t i o n s show t h a t a g a p is opened i n t h e r a n g e

P

⁼ 4 - 9.5 g/cm3 ( s e e /12/ and t h e r e f e r e n c e s g i v e n t h e r e ) . r'irst Yonezawa e t a 1 / l 3 / have c a l c u l a t e d

t h e DOS i n t h e l i q u i d s t a t e u s i n g a m u l t i - band t i g h t - b i n d i n g model. i ' r e l i m i n a r y c a l c u l a t i o n s u s i n g t h e e f f e c t i v e medium c o n d i t i o n ( 4 ) f o r a s i n g l e s c a t t e r e r have g i v e n a c r i t i c a l d e n s i t y i n good agree- ment w i t h n o n r e l a t i v i s t i c band s t r u c t u r e c a l c u l a t i o n s /6/.

I n t h e f o l l o w i n g we c o n s i d e r a c l u s t e r of 9 atoms i n t h e bcc c o n f i g u r a t i o n . The p o t e n t i a l w i t h i n t h e Yiigner-Seitz s p h e r e i s c o n s t r u c t e d from a t o m i c d e n s i t i e s u s i n g t h e i d a t t h e i s s p r e s c r i p t i o n . F o r t h e exchange p o t e n t i a l t h e S l a t e r approxima- t i o n w i t h cC = I i s used. The r a d i u s ro o f t h e c l c s t e r s p h e r e is chosen s o t h a t t h e a t o m i c d e n s i t y w i t h i n t h e c l u s t e r e q u a l s t o t h e d e n s i t y i n t h e l i q u i d . I n Fig. 2 t h e DOS of t h e c e n t r a l atom i n t h e c l u s t e r and t h e s i n g l e s i t e r e s u l t s a r e p l o t t e d f o r v a r i o u s d e n s i t i e s . The e n e r g y i s measured i n e a c h c a s e from t h e m u f f i n - t i n z e r o . The bottom o f t h e bands S b i s g i v e n by t h e c o n C i t i o n L0(Zb) = 0.

The e f f e c t i v e p o t e n t i a l s t a r t s a t t h e band bottom w i t h Vc(Zb) = Sb and depends o n l y weakly on t h e energy. At normal d e n s i t i e s (13,55 g/cm5) t h e 3 0 s o f t h e s i n g l e s c a t t e r e r model is f r e e - e l e c t r o n - l i k e whereas t h e r e s u l t f o r t h e c l v . s t e r

~ h o w s a s t r u c t u r e i n t h e 3 0 s . D e c r e a s i n g

JOURNAL DE PHYSIQUE

Fig. 2 D e n s i t y o f s t a t e s n(E) of a c l u s t e r ( s o l i d l i n e ) and a s i n g l e p o t e n t i a l ( b r o k e n l i n e ) a t a )

y

= 13,55 b ) 8.0 and C ) 5 , 6 g/cm3.

t h e d e n s i t y t h e o v e r l a p of s and p s t a t e s i s r e d u c e d and a band gap is 3pened a t a c r i t i c a l d e n s i t y

C;, 5

5 , 6 g/cm3 i n t h e s i n g l e s i t e c a l c u l a t i o n . U n f o r t u n a t e - l y t h e s i n g l e s c a t t e r e r model g i v e s a l e r m i e n e r g y which is s i t u a t e d a t t h e bottom o f t h e p band. T h i s i s due t o t h e f a c t t h a t i n c o n t r a d i c t i o n t o t h e c r i - s t a l l i n e c a s e t h e number of s t a t e s i n t h e s band is s l i g h t l y s m a l l e r t h a n two. The r e s u l t s o f t h e c l u s t e r i n d i c a t e t h a t t h e g a p o c c u r s a t l o w e r d e n s i t i e s .

I n o r d e r t o compare o u r r e s u l t s w i t h e x p e r i r r e n t a l d a t a r e l a t i v i s t i c e f f e c t s must b e t a k e n i n t o a c c o u n t . Due t o r e l a - t i v i s t i c e f f e c t s t h e 6 s s t a t e s a r e s h i f - t e d t o lower e n e r g i e s r e l a t i v t o t h e 6 p s t a t e s . T h e r e f o r e , t h e g a p o c c u r s at h i g h e r d e n s i t i e s a s o b t a i n e d by non- r e l a t i v i s t i c c a l c u l a t i o n s . R e l a t i v i s t i c s i n g l e s i t e c a l c u l a t i o n s

/9/

g i v e a c r i t i c a l d e n s i t y o f

9

⁼ 9 , 7 g/cm3 i n good agreement w i t h r e l a t i v i s t i c band s t r u c t u r e c a l c u l a t i o n s /13/.

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