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CALCULATIONS OF ELECTRONIC STATES IN THE MUFFIN TIN MODEL OF A LIQUID METAL
L. Huisman, L. Schwartz, A. Bansil
To cite this version:
L. Huisman, L. Schwartz, A. Bansil. CALCULATIONS OF ELECTRONIC STATES IN THE MUF-
FIN TIN MODEL OF A LIQUID METAL. Journal de Physique Colloques, 1980, 41 (C8), pp.C8-441-
C8-442. �10.1051/jphyscol:19808110�. �jpa-00220206�
JOURNAL DE PHYSIQUE CG ZZoque C8, supplSment au n08, Tome 4 1 , aoCt 1980, page Cg-441
C A L C U L A T I O N S OF E L E C T R O N I C S T A T E S I N THE M U F F I N T I N MODEL O F A L I Q U I D METAL
L . Huisman, L . ~ c h w a r t z ' and A. B a n s i l *++
Department of Physics, Brandeis U n i v e r s i t y , WaZtharn, Massachusetts 02254, U. S. A.
*Department of Physics, Northeastern University, Boston, Massachusetts 02115, U.S.A.
A b s t r a c t . - We c o n s i d e r t h e a p p l i c a t i o n o f 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) t o t h e m u f f i n t i n model o f a l i q u i d m e t a l . The t h e o r y i s a p p l i e d t o a model i n which t h e atomic p o t e n t i a l s a r e r e p r e - s e n t e d by a s i n g l e r e s o n a n t
R
= 0 phase s h i f t and t h e EMA s p e c t r u m i s compared w i t h t h e r e s u l t s o f s e v e r a l o t h e r a p p r o x i m a t i o n s . Our c a l c u l a t i o n s i n d i c a t e t h a t t h e EMA d e n s i t y o f s t a t e s e x h i b i t s s u r - p r i s i n g l y s h a r p s t r u c t u r e a n d , a l s o , t h a t f o r c e r t a i n E and k, t h e EMA l e a d s t o n e g a t i v e s p e c t r a l d e n s i t i e s .I. I n t r o d u c t i o n . - Recent a u t h o r s have devoted con- s i d e r a b l e a t t e n t i o n 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 of d i s o r d e r e d t r a n s i t i o n m e t a l s y s t e m s . A l l such sys- tems have t h e common f e a t u r e t h a t t h e s c a t t e r i n g e x h i b i t s r e s o n a n t b e h a v i o r and t h u s c a n n o t be t r e a t e d by p e r t u r b a t i o n t h e o r y . I n t h e c a s e o f com- p o s i t i o n a l l y d i s o r d e r e d s y s t e m s ( i . e . , s u b s t i t u - t i o n a l a l l o y s ) i t i s w e l l e s t a b l i s h e d t h a t t h e a v e r a g e t - m a t r i x (ATA) and c o h e r e n t p o t e n t i a l 2-4
(CPA) a p p r o x i m a t i o n s y i e l d a r e a s o n a b l e mean f i e l d p i c t u r e o f t h e e l e c t r o n i c s p e c t r u m . V a r i o u s a u t h o r s have proposed e x t e n s i o n s o f t h e s e l f - c o n s i s t e n t CPA t o t h e problem o f s t r u c t u r a l d i s o r d e r . Among t h e s e , i t i s g e n e r a l l y a g r e e d t h a t t h e most p r o m i s i n g i s R o t h ' s e f f e c t i v e medium a p p r o x i m a t i o n (EMA)~. While t h e EMA h a s been s t u d i e d i n d e t a i l i n c o n n e c t i o n w i t h t h e t i g h t b i n d i n g model6, o n l y l i m i t e d r e s u l t s a r e a v a i l a b l e r e g a r d i n g i t s a p p l i c a t i o n t o t h e more r e a l i s t i c m u f f i n t i n model 7-9
.
11. S Phase S h i f t Model.- We c o n s i d e r a model i n which o n l y t h e R=O p h a s e s h i f t , 6 ( E ) , i s non-vani- s h i n g and i s t a k e n e q u a l t o t h e d-phase s h i f t o f Cu.
T h i s p h a s e s h i f t h a s a r e s o n a n c e o f w i d t h 0 . 0 3 Ry c e n t e r e d a t E = 0.385 Ry. I n t h e e n e r g y r a n g e between 0.375 Ry and 0.405 Ry we were u n a b l e t o f i n d s o l u t i o n s o f t h e EMA e q u a t i o n s . T h i s d o e s n o t imply t h a t s o l u t i o n s do n o t e x i s t , b u t r a t h e r , t h a t con- v e n t i o n a l i t e r a t i o n schemes9 a r e n o t c o n v e r g e n t i n t h e immediate v i c i n i t y o f t h e r e s o n a n c e . I n F i g . 1 t h e EMA s p e c t r u m i s compared w i t h t h e r e s u l t s o f t h r e e o t h e r a p p r o x i m a t i o n s : ( I ) t h e
i s o l a t e d atom : p(E) = po!E)+(2/n)d6,/dE. ( 2 ) t h e m o d i f i e d q u a s i c r y s t a l l i n e a p p r o x i m a t i o n (MQCA) 10
,
and ( 3 ) t h e l a t t i c e l i q u i d .
q he
l a t t i c e l i q u i d d e n s i t y o f s t a t e s was e v a l u a t e d w i t h i n t h e ATA] 1.
As e x p e c t e d , o u t s i d e o f t h e r e s o n a n c e r e g i o n , t h e f o u r c u r v e s shown i n F i g . 1 i n d i c a t e r o u g h l y t h e same d e n s i t y o f s t a t e s . The f o u r a p p r o x i m a t i o n s a l s o y i e l d s i m i l a r r e s u l t s f o r t h e n e t s p e c t r a l w e i g h t i n t h e r e s o n a n c e r e g i o n . F o r example, t h e change i n t h e i n t e g r a t e d d e n s i t y o f s t a t e s between E=0.30 and E=0.48 i s 2.28 ( f r e e e l e c t r o n plu: i s o l a t e d a t o m ) , 2.28 (MQCA), 2.32 ( l a t t i c e l i q u i d ) , and 2.26 (EMA).
W i t h i n t h e r e s o n a n c e r e g i o n t h e MQCA and l a t t i c e l i q u i d a p p r o x i m a t i o n l e a d t o q u a l i t a t i v e l y s i m i l a r r e s u l t s , b o t h showing a two peaked s t r u c t u r e i n t h e d e n s i t y o f s t a t e s . P h y s i c a l l y , t h e o r i g i n of t h i s s t r u c t u r e i s t h e h y b r i d i z a t i o n between t h e f r e e e l e c t r o n s t a t e s and t h e r e s o n a n t s - l e v e l . Observe, however, t h a t t h e minimum i n p(E) a t E = 0.4 By is c o n s i d e r a b l y d e e p e r i n t h e MQCQ t h a n i n t h e l a t t i c e l i q u i d . T h i s i n c r e a s e s t r u c t u r e may w e l l be unphy- s i c a l , s i n c e i t i s known t h a t , under c e r t a i n c i r - cumstances, t h e MQCA y i e l d s a n e g a t i v e s p e c t r u m i n t h e s t r o n g s c a t t e r i n g regime. T u r n i n g t o t h e EMA r e s u l t s , we s e e t h a t t h e d e n s i t y o f s t a t e s r i s e s a t l e a s t a s s h a r p l y a s t h e MQCA c u r v e on b o t h t h e h i g h and low e n e r g y s i d e s o f t h e r e s o n a n c e . S i n c e t h e EMA g i v e s e s s e n t i a l l y t h e same r e s u l t a s t h e MQCA and l a t t i c e l i q u i d f o r t h e a r e a under t h e r e s o n a n c e band, i t i s c l e a r t h a t , i n t h e r e g i o n 0.385 < E <
0.405, t h e EMA must a l s o e x h i b i t a two peaked s t r u c - t u r e w i t h a minimum a t l e a s t a s pronounced a s t h a t i n t h e MQCA. The p r e s e n c e of s u c h a s h a r p minimum i n t h e EMA s p e c t r u m i s q u i t e s u r p r i s i n g s i n c e one
+
Supported i n p a r t by t h e N a t i o n a l S c i e n c e Foun-d a t i o n u n d e r Grant No. DMR-77-27249. might r e a s o n a b l y have e x p e c t e d t h e d e g r e e o f s t r u c - t+ S u p p o r t e d i n p a r t by t h e N a t i o n a l S c i e n c e Faun- t u r e in t h e l i q u i d and t h e l a t t i c e l i q u i d t o be
d a t i o n u n d e r Grant No. DMR-79-02600.
r o u g h l y comparable.
A second r e a s o n f o r c o n c e r n r e g a r d i n g o u r s o l u t i o n s 33
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19808110
c8-442
JOURNAL DE PHYSIQUE. . . . . . .
Isolated atom--
MQC AREFERENCES
1 I I
1
0.3 0.35 0.4 0.45
Energy ( R y )
Fig. 1
-
Density of states (el per atom per Ry) in various approximations.of the EMA equations is that they lead to negative spectral densities at low k, for energies between 0 . 4 0 5 and 0 . 4 2 Ry. While these results do not lead to negative densities of states, they are certainly unphysical, and are furthur evidence that the EMA may not provide an entirely satisfactory generali- zation of the CPA to the liquid metal problem.
While our findings are to some extent negative, it may turn out that the difficulties we have encoun- tered are related to the artificial character of the s-phase shift model. Indeed, the EMA may well yield satisfactory results when applied to more realistic models. Calculations on such models are now in pro- gress and their results will be described in forth- coming publications.
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