' Hungari an ‘Academy o f‘Sciences

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' Hungar i an ‘Academy o f ‘S ciences C E N T R A L

R E S E A R C H

I N S T I T U T E F O R P H Y S I C S

B U D A P E S T

KFKI-1980-9 3

ELECTRON DENSITIES OF LIQUID AND AMORPHOUS METALS

B. Vasvári

Central Research Institute for Physics H-1525 Budapest 114, P.O.B, 49, Hungary

To appear in the Proceedings of the Conference on Metallic Glasses:

Science and Technology, Budapest, Hungary, June 30 - July 4, 1980;

Paper E-18

HU ISSN 0368 5330 ISBN 963 371 739 6

АННОТАЦИЯ

С использованием математических методов, применяемых для расчета струк­

туры полос кристаллических материалов, разработан метод определения локаль­

ных плотностей электронных состояний в структурно-неупорядоченных металлах.

Целью разработки являлась разработка простого, но реально описывающего алго­

ритма для определения плотностей состояний. Вследствие изотропности этих си­

стем имеется возможность существенного упрощения, если при расчетах до конца придерживаться представлению данных в форме координат. Приведены предваритель­

ные цифровые результаты для аморфного железа.

KI VONAT

Egy eljárást fejlesztettünk ki a szerkezetileg rendezetlen fémek lokális elektron-állapotsürüségeinek (DOS) meghatározására a kristályos anyagok sáv- szerkezetének számítására szokásos matematikai módszerek felhasználásával.

A célkitűzés egy egyszerű, mégis realisztikus algoritmus kifejlesztése az ál- lapotsürüség meghatározására. A szóbanforgó rendszerek izotrópiájának kö­

vetkeztében lényeges egyszerüsitési lehetőségek adódnak, ha a számítások so­

rán mindvégig a koordináta-reprezentációban maradunk. Előzetes numerikus eredményeket mutatunk be az amorf vasra vonatkozóan.

ABSTRACT

A f o r m a l i s m is d e v e l o p e d to c a l c u l a t e the local d e n s i t y of states (DOS) in a structurally d i s o r d e r e d metal u s i n g the m a t h e m a t i c a l techniques of band structure c a l c u l a t i o n s of c r y s t a l l i n e materials.

The aim is to develop a simple, b u t r e a l i s t i c a l g o r i t h m for the c a l c u l a t i o n of the DOS. R e m a i n i n g c o n s i s t e n t l y in the direct space t h r o u g h o u t the w h ole calculation, a consid e r a b l e s i m p l i f i c a t i o n occurs due to the isotropy of the systems involved. P r e l i m i n a r y results for amorphous iron are presented.

INTROD U C T I O N

R e a l i s t i c calcul a t i o n s for the e l e c t r o n i c d e n s i t y of states (DOS) of s t r u c t u r a l l y d i s o r d e r e d metals usu a l l y a p p l y the G reen's function f o r m a l i s m and the m u f f i n - t i n a p p r o x i m a t i o n [1]. S i m i larly to the case of c r y stalline metals, e x p a n s i o n s in the m o m e n t u m (k) space and ang u l a r m o m e n t u m (L = (£,m ) ) r e p r e s e n t a t i o n s lead to the K o r r i n g a - K o h n - R o s t o c k e r (KKR) type of formulas and an i n ­ d irect r e l a t i o n s h i p b e t w e e n the q u a s i - p a r t i c l e energy, E, a n d the wave n u m b e r vector, k,in the r e c i p r o c a l space. Due to the lack of

t r a n s l a t i o n a l invariance in a m o r p h o u s m etals, the w a v e number vector, k, is not a go o d q u a n t u m n umber for the one elec t r o n

states, therefore, c o m p l i c a t i o n s occur, like c o m p l e x wave n u m b e r vectors, finite lifetimes in the state k, etc. In t h i s -paper, we do also use the G reen's f u n c t i o n f o r m a l i s m and the m u f f i n - t i n approximation, but w i t h o u t i n t r o d u c i n g any w a v e - n u m b e r d e p e ndence into our formalism, in other w ords, we use c o n s i s t e n t l y the d i ­ rect, c o o r d i n a t e r e p r e s e n t a t i o n s of our quantities. As a c o n s e ­ quence we g e t rather simple r e s u l t s for the d e n s i t y of states in s t r u c t u r a l l y d i s o r d e r e d systems like liquid m etals or m e t a l l i c g l a s s e s .

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A S U R V E Y OF THE M A T H E M A T I C A L T R E A T M E N T

In the following o n l y a short outline of the m a t h e m a t i c a l t r e a t m e n t can be given. The starting point is an e x p r e s s i o n for the local DOS, g(r,E), in the t e rms of i m m a g i n a r y p a r t of the e n s e m b l e a v e r a g e d G r e e n ' s function of the t o t a l system:

g(r,E) = - ^ lm < G(r,r,E)>. (1)

It is supposed that the p o t e n t i a l of the a m o r p h o u s s y s t e m can be w r i t t e n as s u p e r p o s i t i o n of n o n - o v e r l a p p i n g a t o m i c - l i k e p o t e n ­ tials :

V (r) = E v, (r-R ) .

— lR.

Here v^ir-F^) is the m u f f i n - t i n p o t e n t i a l a r o u n d the i-th a t o m c e n t e r e d at the p o s i t i o n R^.

It is w e l l known, the Green's function can be e x p a n d e d in the terms of the t-matrices, t., of the i n d i v i d u a l scatterers

l

G = G + E G t.G + E G t.* E G t.*G о i о г о . о r j + . о D о

+ E G t • E G t.* E G tn'G + ...

О i . . . о . О * О

i j+i J

(2)

w h e r e G^q is the f r e e - e l e c t r o n propagator. U s i n g a s imple change of the variables, the R^ d e p e n d e n c e can be t r a n s f o r m e d from t. = t (r-R. ,r.'-R. ) to the G 's: G >-G.. = G (r-r'- R .+ R .) . N e x t we suppose th a t we h a v e a simple, one c o m p o n e n t system, and assume, that the s c a t t e r i n g p r o p e r t i e s of the i n d i vidual atoms are all the same, i.e., the t^ m a t r i c e s are i n d e p e n d e n t of w h i c h a t o m is considered. T h i s a s s u m p t i o n is e q u i v a l e n t to the a verage t - m a t r i x a p p r o x i m a t i o n (ATA), w h e n the s c a t t e r i n g m a t r i x of the i n d i vidual atoms in a r a n d o m s y s t e m is r e p l a c e d by an average t-matrix.

A f ter a s t r a i g h t f o r w a r d a l g e b r a one g e t s for the density of e l e c t r o n s of E energy, inside a m u f f i n - t i n sphére

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g(r,E) = -j=- и S R„ (r)Rj (r) Y_ (r)Y_ (r) 4Tl LxL 2 ll %2 L 1 L 2

6T -lm - E j “ 1. M TT L 1L 2 И L L 1L L L 2

(3)

Here R^(r)YL (r) is the r egular s o l u t i o n of the radial Schrödinger e q u a t i o n with the m u f f i n - t i n p o t e n t i a l and for the

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energy E = и . The matr i c e s J and M are de f i n e d as follows:

J = [1-tD] ,

L1L2 L1L2

(4)

[tD]

L 1L 2

— tn • E В _ (R,x) , 2 R+o L 1 2

(5)

M L 1L 2

E t^, 2 By -rt (— R,x)B^ (RfX) t

L' R*o 1 2

(6)

where t^ = t^(x) a n d L are the e x p a n s i o n cofficients of the t-ma t r i x and the fre e - p a r t i c l e Green's functions, respectively, defined in the t e rms of phase shifts, л £ (E), and of the s p h e r i c a l Hankel functions, (xR) in the following way:

1 irU (E)

tÄ = - sin h£ (E)e , (7)

(£ - a')

B_ T (R,h ) = - 4nix E i 1 2 C T _ _ ,YT , ( R ) hl,(xR). (8)

L 1L 2 ~ L' L 1L 2L L

Equa t i o n (3) gives the density of e l e c t r o n s only for a g i v e n

configuration of t h e atoms, r e p r e s e n t e d b y their coordinates, R ^ . To calculate the e n s e m b l e a verage of g(r,E) one has to m a k e some a s s u mption for the d i s t r i b u t i o n of atoms in the system, the n to define a decoupling scheme, h o w to c a l c u l a t e the m a n y - p a r t i c l e averages in e q u a t i o n (3). A p a r t i c u l a r l y simple formula is r e ­ sulted if one s u p p o s e s that the atomic d i s t r i b u t i o n s can be r e p ­ rese n t e d by a s p h e r i c a l l y symmetric p a i r d i s t r i b u t i o n function, g(R) and if, for the higher o r d e r terms, one applies Ki r k w o o d ' s d e c o u p l i n g schemes. In that case the s u m in the (tD) and (M) matrices, over the atomic coordinates, c a n be repl a c e d by i n ­ tegrals containig the function g ( R ) . Due to the isotropy of this d i s t r i b u t i o n the a b o v e m e n t i o n e d m a t r i c e s are diag o n a l ones, and the inverse of J c a n easily be calculated. C a r r y i n g out the in-

4

tegrations w i t h respect to the a n g u l a r coordinates one gets the very simple final formulas for this case as follows:

, _. к „ 2H+1 n 2 . . 0 ( r ' E) * i

I

, _ 1 - 1

1 - I m —

к

J « ■ Ч - « ” *' - - 4npo t )tI (10)

M, = - ( 4 п н ) 2 E H,„

** Z r Z " ^ ^

( I D

where p is the average density of the material, the simple n u m ­ bers D (£,£{£") can e a s i l y be c a l c u l a t e d from the C l e b s c h - G o r d o n coefficients, and the I and are integrals:

I = e R g(R)RdR, о

(12)

H £ = I h * 2 (nR)g(R)R2dR. (13)

о

The integral of p(r,E) inside the atomic v olume with respect to the c o o rdinates r leads to the local density of states, p(E).

P r e l i m i n a r y numerical calcul a t i o n s wer e p e r f o r m e d for liquid or amorphous iron. The g(R) pair d i s t r i b u t i o n function was c a l c u ­ lated from the analitical structure factor of the h a r d sphere solution of the Per c u s - Y e v i c k equations. The m u f f i n - t i n potential was given by a simple analytical expression, the onl y p a r a meter of it was ch o o s e n to reproduce the resonance in the p hase- shift. U sing these ingredients a s i n g l e - p e a k e d local density of states curve was resulted (Fig. 1.), which, in our model, cor­

responds to a single scattering c e n t e r e m b e d d e d into a medium.

The effect of this m e d i u m is e x p r e s s e d by the diag o n a l matrices and , and resulted in the b r o a d e n i n g of the r e s o n a n c e level c h a r a c t e r i s t i c for transitional metals.

To improve the m o d e l d e s c r i b e d we intend to take into a ccount the local e n v i r o n m e n t of the i n d i vidual atoms by d i v i d i n g the sums over R^ in formulas (5) and (6) into two parts. In the first one the R^ takes the values of the few neighbours a r o u n d the original atom, creating a cluster, in the second p a r t s R^ runs over the a t o m i c coordinates outside of this cluster. Also, the

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m o d e l m u f f in-tin p o t e n t i a l w i l l be r e p l a c e d by a mor e realistic one, calculated from the w a v e functions of the n e i g h b o u r i n g atoms.

This sort of c a l culations are in progr e s s and w i l l be reported in later publications.

Fig. 1. Local density of states for amorphous Fe

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R E F E RENCE

[1] L. Schwartz, H. Ehrenreich: Annals of P h y s i c s 6£, 100-148 (1971);

L. Schwartz, H.K. Peterson, A. Bansil: Phys. Rev. B 1 2 , 3113- 3123 (1975);

J. Bethell, J.L. Beeby: J. Phys. F: Metal Phys. 1_, 1193-1205 (1977);

S. Asano, F. Yonezawa: J. Phys. F: Metal Phys. 10, 75-79 (1980).

61 -OQ 3

Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Tompa Kálmán

Szakmai lektor: Hargitai Csaba Nyelvi lektor: Hargitai Csaba Gépelte: Végvári Istvánná

Példányszám: 220 Törzsszám: 80-633 Készült a KFKI sokszorosító üzemében Felelős vezető: Nagy Károly

Budapest, 1980. október hó