EMBEDDED MOLECULAR CLUSTER APPROACH TO THE ELECTRONIC STRUCTURE OF AMORPHOUS AND LIQUID METALS

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EMBEDDED MOLECULAR CLUSTER APPROACH TO THE ELECTRONIC STRUCTURE OF

AMORPHOUS AND LIQUID METALS

B. Delley, D. Ellis, A. Freeman

To cite this version:

B. Delley, D. Ellis, A. Freeman. EMBEDDED MOLECULAR CLUSTER APPROACH TO THE

ELECTRONIC STRUCTURE OF AMORPHOUS AND LIQUID METALS. Journal de Physique

Colloques, 1980, 41 (C8), pp.C8-437-C8-440. �10.1051/jphyscol:19808109�. �jpa-00220204�

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EMBEDDED MOLECULAR CLUSTER APPROACH TO THE ELECTRONIC STRUCTURE OF AMORPHOUS AND L I Q U I D METALS*

B. Delley, D.E. Ellis and A.J. Freeman

Department of Physics and Astronomy, N o r t h e s t e m University, Evmston, 11. 60201, U.S.A.

Abstract.- In this approach to the electronic structure of amorphous and liquid metals, we represent the system by molecular clusters which are embedded in an external potential chosen as a suitable representation of the rest of the system. We have determined the electronic structure of a number of Cu, Zr and Cu-Zr clusters using the self-consistent discrete variational-LCAO approach within local density functional theory. Effects due to deviations from perfect crystalline symmetry are analyzed. Total densities of states used to interpret observed photo-electron spectra are found to show good agreement. Core level shifts and the concentration of conduction electrons are determined. Results are compared with predictions of the Nagel and Tauc theory and with the pseudopotential theory of stability.

I. INTRODUCTION

Amorphous and l i q u i d metals share t h e common f e a t u r e o f a l a c k o f t r a n s l a t i o n a l symmetry which makes i n a p p l i c a b l e t r a d i t i o n a l energy band methods

f o r determining t h e i r e l e c t r o n i c s t r u c t u r e . By c o n t r a s t , molecular c l u s t e r methods a r e a t t r a c t i v e because they do n o t r e q u i r e t r a n s l a t i o n a l symmetry t o s i m p l i f y t h e l o c a l d e n s i t y Hamiltonian. I n order t o t r e a t b u l k matter, l a r g e c l u s t e r s should be used. However, i f instead of c a l c u l a t i n g the e l e c t r o n i c s t r u c t u r e of free molecules, s u i t a b l e boundary c o n d i t i o n s a r e a p p l i e d t o reduce surface e f f e c t s , much smaller c l u s t e r s can be used as

charge d e n s i t y . Ye have v e r i f i e d t h a t the overlap- p i n g s p h e r i c a l charge model imp1 i e d by the Mu1 1 i ken p o p u l a t i o n s i s near o p t i m a l , i n the l e a s t squares sense. The o u t p u t p o p u l a t i o n s a r e mixed w i t h the i n p u t p o p u l a t i o n s t o d e t e r n i n e new atomic c o n f i g u r - a t i o n s f o r b o t h c l u s t e r and e x t e r n a l atoms, and t h e i t e r a t i o n process i s continued t o convergence.

The embedding p o t e n t i a l which we use has t o

Supported by the United States National Science Foundation and the Swiss National Science Foundation.

models f o r t h e b u l k . T h i s paper describes our embedded m o l e c u l a r c l u s t e r l o c a l d e n s i t y theory approach t o determining the e l e c t r o n i c s t r u c t u r e of asorphous and l i q u i d metals and presents r e s u l t s obtained f o r a number o f Cu, Z r and Cu-Zr c l u s t e r s . 11. YETHOD

The s e l f -consi s t e n t one-el e c t r o n l o c a l d e n s i t y model used i n the present c a l c u l a t i o n s has been a p p l i e d t o a v a r i e t y o f atoms, molecules, and s o l - i d s . The Hamiltonian c o n s i s t s o f t h e usual k i n e t i c energy, Coulomb p o t e n t i a l , and exchange-correlation operators. The exchange-correlation p o t e n t i a l i s taken as a simple f u n c t i o n o f t h e e l e c t r o n i c charge d e n s i t y , w i t h t h e f o r m g i v e n by Gunnarsson and L u n d q v i s t [I].

The i t e r a t i o n process s t a r t s from some assumed i n p u t charge d e n s i t y , t y p i c a l l y taken t o be t h e s u p e r p o s i t i o n o f n e u t r a l atoms on l a t t i c e s i t e s . The c l u s t e r SchGdinger equation i s solved, and 1 eve1 s a r e popul ated according t o Fermi -Dirac s t a t i s t i c s . A f t e r each i t e r a t i o n atomic o r b i t a l occupation numbers are obtained by performing a M u l l i k e n - l i k e p o p u l a t i o n a n a l y s i s o f t h e c l u s t e r

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19808109

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c8-438 JOURNAL DE PHYSIQUE

s i n u l a t e t h e a t t r a c t i v e c h a r a c t e r o f e x t e r i o r atom w e l l s , s u b j e c t t o the P a u l i p r i n c i p l e c o n s t r a i n t s which prevent the m i g r a t i o n o f c l u s t e r e l e c t r o n s t o l o w - l y i n g s t a t e s on e x t e r i o r atoms. The simple model p o t e n t i a l used here can be w r i t t e n as V=VLD+

Vpp, where VLD i s t h e usual l o c a l d e n s i t y p o t e n t i a l and Vpp i s a parameterized r e p u l s i v e core pseudo- p o t e n t i a l which t r u n c a t e s e x t e r n a l p o t e n t i a l w e l l s t o a value Vo. The r e s u l t i n g p o t e n t i a l d i s p l a y s the f u l l a n i s t r o p i c bonding c h a r a c t e r o f t h e e n v i - ronment, i n c o n t r a s t t o s p h e r i c a l l y averaged models

The d i s c r e t e variational-LCAO method used i n the present work r e l i e s on an expansion basis o f sym- m e t r i z e d numerical atomic o r i o n i c wavefunctions.

T h i s b a s i s p r o v i d e s an e f f i c i e n t r e p r e s e n t a t i o n o f c l u s t e r states, whose p o p u l a t i o n a n a l y s i s provides c s e f u l physical i n t e r ~ r e t a t i o n s . The expansion c o e f f i c i e n t s and energies are obtained by s o l v i n g the f a m i l i a r s e c u l a r equation ($-i$)C=O

-

w i t h m a t r i x elements determined by numerical i n t e g r a t i o n . For 13 atom cubic c l u s t e r s we f i n d t h a t 3000 i n t e g r a - t i o n p o i n t s a r e s u f f i c i e n t t o converge valence and shallow core l e v e l s t o b e t t e r than 0.1 eV.

The s e c u l a r equation i s reduced t o b l o c k form f o r symmetrical c l u s t e r s and i n o r d e r t o s i m p l i f y t h e c a l c u l a t i o n s we have considered systems o f Oh, DQh, and D3d symmetry (see Fig. 1). The Oh cubic c l u s t e r s f o r a s i n g l e metal, say Cu13, can be compared w i t h DQh ( f o u r - f o l d synunetry) and D3d ( t h r e e - f o l d ) s u b s t i t u t i o n s l i k e Cu7Zr6. By m a n i p u l a t i n g c l u s t e r s i z e and geometry one can approach v a r i o u s i n t e r e s t i n g a1 l o y compositions.

111. RESULTS

A. Results f o r Pure metal c l u s t e r s

The physics contained i n t h e several hundred eigenvalues obtained i n a c l u s t e r c a l c u l a t i o n can best be displayed as a d e n s i t y o f s t a t e s (DOS) curve. I n o r d e r t o draw a smooth curve we have

chosen a L o r e n t z i a n broadening w i t h a damping o f 0.1 eV. F i g 2 , compares DOS curves w i t h band s t r u c t u r e r e s u l t s obtained f o r ( i ) b u l k copper C21 and ( i i ) a copper monolayer t h i n f i l m C31. I t i s obvious from the Figure, t h a t b u l k r e s u l t s a r e ap- proached w i t h i n c r e a s i n g c l u s t e r size. Already f o r 13 atom c l u s t e r t h e d - s p l i t t i n g t u r n s o u t t o be g r e a t e r than t h a t found f o r t h e monolayer. As t h e c l u s t e r s i z e increases t h e d - s p l i t t i n g i s seen t o increase and t o approach the b u l k band s t r u c t u r e r e s u l t s .

B. Results f o r Cu-Zr C l u s t e r s

U n l i k e the case o f c r y s t a l 1 in e s o l i d s , each s i t e i n a d i s o r d e r e d a l l o y has i t s own d i f f e r e n t p o t e n t i a l . I n o u r c l u s t e r c a l c u l a t i o n s , we f i n d t h a t t h e c l u s t e r r e s u l t s do n o t depend s t r o n g l y on t h e embedding p o t e n t i a l . For t h i s reason an averaged embedding p o t e n t i a l can be used. Here we ern- p l o y a weighted average of t h e c l u s t e r s i t e poten- t i a l s w i t h weights chosen p r o p o r t i o n a l t o the number o f corresponding atoms i n t h e c l u s t e r . This y i e l d s an embedding p o t e n t i a l which has no n e t o v e r a l l charge. The i n c l u s i o n o f two neighbor s h e l l s p r o v e s t o be s u f f i c i e n t f o r producing con- verged r e s u l t s , i.e., when a t h i r d neighbor s h e l l i s added, t h e r e s u l t s a r e e s s e n t i a l l y unchanged.

T y p i c a l c a l c u l a t e d DOS curves f o r CuZr a l l o y s a r e d i s p l a y e d i n F i g . 3. A l l curves have been s c a l e d ' t o t h e same peak h e i g h t . A broadening o f 0.3 eV has been a p p l i e d i n t h i s f i g u r e . The upper DOS curve corresponds t o an embedded 13 Copper

atoms c l u s t e r . The s i t e s have been chosen accord- i n g t o an f c c l a t t i c e f o r Cu a t room temperature.

The lower two DOS curves correspond t o embedded Copper-Zirconium c l u s t e r s w i t h D3d symmetry.

The f u l l curve corresponds t o a c l u s t e r c o n t a i n i n g 6 Cu atoms and 7 Z r atoms. The broken curve c o r - responds t o a c l u s t e r w i t h 7 Cu and 6 Z r atoms.

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l e v e l i s m a i n l y due t o Z r d - s t a t e s . T h i s can be i n f e r r e d c o n s i d e r i n g t h e d i f f e r e n t Z r c o n c e n t r a - t i o n s i n t h e two c l u s t e r s . T h i s has a l s o been v e r i f i e d by p r o j e c t i n g t h e p a r t i a l DOS. The main peak below t h e Fermi l e v e l i s due t o Cu d - s t a t e s . I n c a n p a r i s o n w i t h p u r e Cu a s h i f t t o h i g h e r b i n d - i n g e n e r g i e s i s e v i d e n t . As can be seen f r o m t h e f i g u r e , t h e c l u s t e r c a l c u l a t i o n s a l s o y i e l d a n a r - r o w i n g o f t h e Cu d-bandwidth and t h e shape o f t h e Cu d-band becomes more symmetrical i n t h e a l l o y . B o t h t h e s e f e a t u r e s have a l s o been observed e x p e r i - m e n t a l l y .

I n a d d i t i o n t o t o t a l and p a r t i a l ( b y a n g u l a r momentum and atom t y p e ) DOS, we have c a l c u l a t e d e l e c t r o n c h a r g e d e n s i t i e s ( t o s t u d y charge t r a n s f e r ) and c o r e and v a l e n c e e l e c t r o n energy l e v e l s ( t o s t u d y c o r e l e v e l s h i f t s ) f o r comparison w i t h experiment. Ke have a l s o d e f i n e d a p r o j e c t i o n o p e r a t o r t o d e t e r m i n e t h e number o f c o n d u c t i o n e l e c t r o n s , nc, which i s r e l e v a n t f o r a s s e s s i n g thermal s t a - b i l i t y a g a i n s t c r y s t a l l i s a t i o n as w e l l as f o r t h e Ziman t h e o r y o f r e s i s t i v i t y . T a b l e I shows a comparison o f t h e s e r e s u l t s w i t h experiment. B o t h t h e o r y and e x p e r i m e n t show t h a t t h e c o r e l e v e l s o f b o t h a l l o y i n g components t e n d t o s h i f t t o h i g h e r b i n d i n g e n e r g i e s . T h i s c a n n o t be e x p l a i n e d b y a s i m p l e charge t r a n s f e r model, where t h e 1 eve1 s o f one a t o m i c component s h o u l d s h i f t t o l o w e r b i n d i n g e n e r g i e s . F u r t h e r , a s s e e n f r o m T a b l e I,

t h e c l u s t e r r o d e l a l s o o i v e s a p o o d p r e - d i c t i o n f o r t h e s h i f t o f t h e Cu d - l e v e l s t o h i g h e r b i n d i n g e n e r g i e s . T h e n u m b e r n c o f c o n d u c t i o n e l e c t r o n s g i v e n i n T a b l e cannot be compared d i r e c t 1 y t o experiment. However, nc% 1.7

f i e s t h e c r i t e r i o n 2k %k where k i s t h e wave

F P' P

number o f t h e f i r s t peak o f t h e s t r u c t u r e f a c t o r . T h i s i s , a c c o r d i n g t o t h e 7iman t h e o r y o f r e s i s - t i v i t y , a c r i t e r i o n f o r t h e occurrence o f a n e g a t i v e t e m p e r a t u r e c o e f f i c i e n t o f r e s i s t i v i t y i n m e t a l s . P e g a t i v e t e m p e r a t u r e c o e f f i c i e n t s have been f o u n d e x p e r i m e n t a l l y i n Cu-Zr C71. Furthermore, v a r i o u s a u t h o r s s u p p o r t t h e i d e a t h a t 2kF ^?. k p r o v i d e s

P

a l s o a c r i t e r i o n f o r s t a b i l i t y a g a i n s t c r y s t a l l i s a - t i o n C8,91.

F i n a l l y , i n o r d e r t o i n v e s t i g a t e e f f e c t s o f p o s i t i o n a l d i s o r d e r i n l i q u i d s and amorphous a l l o y s , c a l c u l a t i o n s f o r a number o f l o w symmetry c l u s t e r s have a1 s o been performed. Comparison w i t h r e s u l t s f o r h i g h symmetry O h ( f c c ) c l u s t e r s shows t h a t a t d e v i a t i o n s of u p t o 10% n e a r e s t n e i g h b o r d i s t a n c e

f r o m t h e r e g u l a r l a t t i c e p o s i t i o n s , o n l y s m a l l changes i n t h e e l e c t r o n i c s t r u c t u r e o c c u r .

TABLE I. R e s u l t s o f embedded c l u s t e r c a l c u l a t i o n s f o r Cu-Zr compared w i t h experiment.

b ) C l u s t e r w i t h Dgd s y m e t r y

Core l e v e l s h i f t s ( e V l Cu d-band (eV1 s m e t r y

'4 h '3d Experiment refs. C4.5.63

. . . FIGURE 1: a ) C l u s t e r w i t h D,h symmetry,

Z r ( 4 s l Center Lig.

0.0 t 0 . 9 0.0 0.1-0.5

( 4 p ) i O . S (34) 20.

C U O S I Center L l g . t 0 . 4 '0.4-4.0 +0.7-b1.0 0.3-.0.8

( 3 ~ ) t1.3 (ZP) 0.6

nc

1.6-.1.8 1 . 6 1 . 7 s h i f t

i 0 . 4 4 . 7 t O . W . 9 t O . W . 8

n a r r w r n g

0.+1.0 0 . 4 9 . 7

0.1

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c8-440 JOURNAL PHYSIQUE

I

I I

I ]

-4

^-L.ⁿ ^{E F} ^[eVl

FIGURE 2. DOS f o r Cu' : a ) monolayer ref. C33,

FIGURE 3 : DOS f o r Cu and Cu-Zr c l u s t e r s , broaden- ing 0.02 Ry: a ) 13 Cu atoms Oh symmetry. b) 6 Cu and 7 Zr atoms D3d symmetry, C) 7Cu and 6 Zr atoms Dgd symnetry.

REFERENCES:

C11 Gunnarsson e t a l . , PRB 13, 4274, 1976.

C21 Moruzzi e t a l . , Calculated E l e c t r o n i c Proper- t i e s of Metal s , Pergamon, N Y 1978, p. 96.

C31 Wang and Freeman, PPB l a , 1714, 1978.

C43 Amamou and K r i l l , s s t c 78, 957, 1978.

C51 Oel haven e t a l . , PRB 43, 1134, 1979.

C61 C a r t i e r e t a l . , J . Phys. F. 10, L21, 1980.

C71 Laubscher e t a l . , t h i s conference.

C83 Nagel and Tauc, PRL 35, 380, 1975.

C91 Beck and Oberle, Proceedfngs of 3 I n t e r n a l Conference on Rapid1 y Cuenched Metals I I I , Vol. 1, 416, 1978.

b) 13 atom c l u s t e r Oh synrmetry, c ) 19 atom c l u s t e r Oh symmetry, d ) 28 atom c l u s t e r Td symmetry, e ) 43 atom c l u s t e r Oh symmetry, f ) bulk r e f C21.