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Multiple-scattering theory of binary alloys in single-site approximation
Physica 113B (1982) 317-335 North-Holland Publishing Company
MULTIPLE-SCATTERING THEORY OF BINARY ALLOYS IN SINGLE-SITE APPROXIMATION J. d ' A L B U Q U E R Q U E e CASTRO* Department of Mathematics, Imperial College, London SW7 2BZ, UK Received 14 May 1981 Revised 23 November 1981 The application of the single-site approximation to investigate electronic properties of substitutional binary alloys is discussed within the framework of the muffin-tin model. On the basis of the multiple-scattering theory, new expressions for the complete and restricted averages of the site diagonal elements of the alloy scattering matrix are presented. These expressions are given in terms of an arbitrary crystalline reference medium and can be used in conjunction with either the coherent potential (CPA) or the average t-matrix (ATA) approximations. It is also shown that the previous expressions for the restricted averages of the alloy scattering matrix can only be used in conjunction with the CPA. The expressions in this paper reduce to the CPA results for a suitable choice of the reference medium.
1. Introduction T h e s t u d y of e l e c t r o n i c p r o p e r t i e s of n o n - c r y s t a l l i n e m a t e r i a l s has b e c o m e of g r e a t i n t e r e s t in t h e p a s t few years. D e s p i t e t h e e n o r m o u s difficulty in a p p r o a c h i n g t h e s u b j e c t t h e o r e t i c a l l y , c o n s i d e r a b l e p r o g r e s s has b e e n m a d e t o w a r d s t h e u n d e r s t a n d i n g of t h e e l e c t r o n i c s t r u c t u r e of d i s o r d e r e d subs t i t u t i o n a l b i n a r y alloys. S o p h i s t i c a t e d t e c h n i q u e s b a s e d on t h e m u l t i p l e - s c a t t e r i n g f o r m a l i s m i n t r o d u c e d b y L a x [1] a n d o t h e r s h a v e b e e n d e v e l o p e d to d e a l with t h e s e systems, a n d m a n y a p p r o x i m a t i o n s c h e m e s h a v e b e e n p r o p o s e d . A m o n g t h e m two of t h e m o s t c o m m o n l y u s e d h a v e b e e n t h e a v e r a g e t-matrix (ATA) and the coherent potential (CPA) approximations. T h e A T A has its origin in t h e w o r k s of K o r r i n g a [2] a n d B e e b y [3] while t h e C P A was first discussed b y S o v e n [4] a n d T a y l o r [5]. T h e s e two a p p r o x i m a t i o n s a n d r e l a t e d w o r k s u p to 1975 a r e r e v i e w e d by E h r e n r e i c h a n d S c h w a r t z [6] a n d E l l i o t t et al. [7]. B o t h A T A a n d C P A a r e b a s e d on t h e s o - c a l l e d single-site a p p r o x i m a t i o n , which d e c o u p l e s t h e m u l t i p l e - s c a t t e r i n g e q u a t i o n for t h e c o n f i g u r a t i o n a v e r a g e d alloy s c a t t e r i n g m a t r i x . This d e c o u p l i n g e n a b l e s o n e to w r i t e t h e c o n f i g u r a t i o n a v e r a g e d s c a t t e r i n g m a t r i x in t e r m s of t h e s c a t t e r i n g m a t r i x of a c r y s t a l l i n e effective m e d i u m , which can b e determined either self-consistently (CPA) or not (ATA). In this p a p e r w e a r e c o n c e r n e d with t h e a p p l i c a t i o n of t h e single-site a p p r o x i m a t i o n within the f r a m e w o r k of t h e muffin-tin m o d e l . T h e r e is n o w a c o n s i d e r a b l e b u l k of l i t e r a t u r e in this field a n d the n u m e r i c a l c a l c u l a t i o n s b a s e d on this m o d e l i n d i c a t e t h a t it p r o v i d e s a realistic d e s c r i p t i o n of the e l e c t r o n i c s t r u c t u r e of d i s o r d e r e d s u b s t i t u t i o n a l alloy. O u r discussion is r e s t r i c t e d to b i n a r y alloys AxBy a n d t h e o b j e c t s of c e n t r a l i n t e r e s t h e r e a r e t h e c o n f i g u r a t i o n a v e r a g e s of t h e s i t e - d i a g o n a l m a t r i x e l e m e n t s of t h e alloy t - m a t r i x . W e o r g a n i s e this p a p e r as follows. In s e c t i o n 2 we w r i t e d o w n t h e single-site e q u a t i o n for t h e * Permanent address: Departamento de Fisica, Universidade Federal de Silo Carlos, Via Washington Luiz, Km 235, Silo Carlos-SP-Brazil. 0378-4363/82/0000-4)000/$2.75 O 1982 N o r t h - H o l l a n d
318
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
c o m p l e t e c o n f i g u r a t i o n a v e r a g e of t h e alloy t - m a t r i x in t e r m s of an a r b i t r a r y c r y s t a l l i n e r e f e r e n c e m e d i u m . In section 3 w e c o n s i d e r t h e e x t e n s i o n of t h e u s u a l m u l t i p l e s c a t t e r i n g t h e o r y to d e a l with an a r b i t r a r y r e f e r e n c e m e d i u m , a n d solve t h e single-site e q u a t i o n for t h e c o m p l e t e c o n f i g u r a t i o n a v e r a g e of t h e o n - t h e - e n e r g y - s h e l l m a t r i x e l e m e n t s of t h e alloy t - m a t r i x . In section 4 we l o o k at t h e alloy e l e c t r o n i c d e n s i t y of states a n d w e see that it can b e w r i t t e n in t e r m s of t h e c o m p l e t e a n d r e s t r i c t e d c o n f i g u r a t i o n a v e r a g e s of t h e s i t e - d i a g o n a l o n - t h e - e n e r g y - s h e l l m a t r i x e l e m e n t s of t h e t - m a t r i x . T h e r e s t r i c t e d a v e r a g e is t h e a v e r a g e o v e r all t h e c o n f i g u r a t i o n s h a v i n g on A (or B) a t o m on a specified site. A l s o we d e r i v e a f u n d a m e n t a l e q u a t i o n r e l a t i n g t h e c o m p l e t e a n d r e s t r i c t e d c o n f i g u r a t i o n a v e r a g e s of t h e s i t e - d i a g o n a l m a t r i x e l e m e n t s of t h e t - m a t r i x a n d a n a l y s e t h e p r e v i o u s l y p r o p o s e d e x p r e s s i o n s for t h e s e r e s t r i c t e d a v e r a g e s . O n t h e basis of this f u n d a m e n t a l e q u a t i o n we p r o p o s e n e w e x p r e s s i o n s for t h e r e s t r i c t e d a v e r a g e s a n d s h o w in s e c t i o n 5 t h a t t h e y l e a d to t h e c o r r e c t l e a d i n g c o n t r i b u t i o n s to t h e d e n s i t y of states in t h e d i l u t e alloy limit. Finally, w e d r a w o u r c o n c l u s i o n in section 6. T w o a p p e n d i c e s a r e also p r e s e n t e d . T h e first c o n t a i n s e x p a n s i o n s of t h e muffin-tin G r e e n ' s f u n c t i o n at different p o i n t s in space. T h e s e c o n d d e a l s with t h e i n t e g r a t i o n of G r e e n ' s f u n c t i o n o v e r all s p a c e a n d o v e r t h e W i g n e r - S e i t z cell. In o r d e r to h e l p t h e r e a d e r , we p r e s e n t t a b l e I s u m m a r i z i n g t h e definition of t h e m a n y s c a t t e r i n g m a t r i c e s a n d c o m b i n a t i o n s of t h e m t h a t a r e u s e d in this w o r k . This t a b l e can b e r e f e r r e d to at any p o i n t in this p a p e r . Definitions t-matrices associated with the potentials VAand vB and relative to free space Alloy configuration-dependent t-matrix relative to free space t-matrix associated with the potential v (reference medium) and relative to free space Reference-medium t-matrix relative to free space t-matrices associated with the potentials VAand va and relative to the reference medium Alloy configuration-dependent t-matrix relative to the reference medium Single-potential t-matrix associated with the effective medium and relative to the reference medium Single-potential t-matrix associated with the effective medium and relative to free space
tA(k) and tB(k) T(k) i(k) 7"(k) ~'A(k)and "ra(k) .~(k) r~(k) t~(k )
~A = i-~(k ) - t-A~ and A¢ = [-l(k ) - t;l(k)
/iB(k) = i+l(k) - tz,l(k)
2. The single-site approximation L e t us a s s u m e t h e H a m i l t o n i a n for a given c o n f i g u r a t i o n of t h e b i n a r y alloy A x B r to h a v e t h e muffin-tin f o r m H(r) = Ho(r) + V(r) =
H0(r) +
v=(lr - R.I),
(1)
n
w h e r e H0 is t h e f r e e - e l e c t r o n H a m i l t o n i a n a n d vn a r e n o n - o v e r l a p p i n g s p h e r i c a l l y s y m m e t r i c p o t e n t i a l s c e n t r e d o n t h e lattice p o i n t s of a B r a v a i s lattice {R,}. T h e p o t e n t i a l s v, a r e e q u a l to VA o r vB d e p e n d i n g on t h e configuration. W e a s s u m e t h a t all t h e muffin-tin s p h e r e s h a v e t h e s a m e radius. T h e alloy e l e c t r o n i c p r o p e r t i e s can b e r e l a t e d to t h e c o n f i g u r a t i o n a v e r a g e d G r e e n ' s f u n c t i o n ( G ( z ) ) = ((z - H)-~), which has t h e full s y m m e t r y of the B r a v a i s lattice. It is c o n v e n i e n t to e x p r e s s G in t e r m s of t h e s c a t t e r i n g o p e r a t o r T d e f i n e d b y
J. d'Albuquerque e Castro / Multiple-scattering theory o[ binary alloys G = Go + GoTGo
319 (2a)
= Go + ~ GoT,~,Go,
(2b)
n,n '
where Go is the free-electron Green's function and T,., are the usual path operators. The nonoverlapping condition on the potentials v, implies that only the on-the-energy-shell matrix elements of T,~, enter the expression for G. They are given by [T(k)]~,' = [ ( t - ' ( k ) - B ( k ) ) - ' ] ~ / ,
(3)
where k = ~ / E , L = (l, m) are the angular m o m e n t u m indices, [B(k)]~,' are the structural functions and [t(k)],L,L,' = [t(k)],~6,,,8~, are the usual t-matrices associated with the isolated potentials v, (Ehrenreich and Schwartz [6]). We are interested in the alloy density of states per unit of volume, which can be written as (g(E)) = -~-~O
Im(f
d3rG(r, r, E + ) ) ,
(4)
where E += E + i0 + and g2 is the volume of the system. As we shall show in section 4, ( g ( E ) ) can be expressed in terms of the complete and restricted configuration averages of the site-diagonal matrix elements of [T(k)], namely <[T(k)]~'> and ([T(k)]~).=A and ([T(k)]~'),=B. T h e last two quantities involve all the alloy configurations having respectively an A or B atom on the site n. The complete average ([T(k)]) can be easily evaluated in the single-site approximation, but the restricted averages are more complicated. We return to this point in section 4. In order to calculate the configuration average of [T(k)] we introduce a crystalline reference medium described by ISI(r) = Ho(r) + ~'(r) = Ho(r) + ~
O~(]r -
R.I),
(5)
tt
where (,'(r) satisfies the muffin-tin conditions and may be energy dependent. The alloy configuration dependent scattering operator gr relative to this medium is defined by the Dyson equation G = (~ + t~3(~,
(6)
where ( ~ ( z ) = (z -/_~)-1 is Green's function of the reference medium. We can show that
X n
n,m m#n
+X X n,ra m#n
(7a)
p p#m
where T,, = [1 - (v,, - b)t~l-'(v. - tS).
(7b)
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
320
Eq. (7a) can be rearranged in the form 3 = Y...,~-..,, where the path operators 3.., satisfy the equation
~-.., = ~.a.., + ~-,,0 E er..,.
(8)
m
raCn
We now consider the configuration average of the above equation:
(e%) = (~)8.., + (~)d E (er.,.,) + (0". - (r))d E (erm.,- (~m.,))), m
m
mYn
m#n
(9a)
where (r) = x r . + Y~'B = x[1 - (VA-- tS)(~I-'(VA-- tS)+ y[1 -- (Vs -- ,~)dl-'(v,,- t~),
(gb) (9c)
and introduce the single-site approximation, which consists in neglecting the third term on the right-hand side of eq. (9a). This term is difficult to deal with and certainly not negligible if the reference medium is not properly chosen. This point is discussed in section 6. Leaving this problem aside for the moment, eq. (9a) is then replaced by
(3r,~,) = O')&,.' + (r)G ~'~ (~m.').
(10)
m rain
Although the general solution of eq. (10) is not readily available, it can be easily solved for ([~r(k)]~,'), where [ff(k)]~,' = f dap, dap2YL(Pl)dp,(k, p,)~..,(p,, Pz, k )~b,,(k, p2)YL,(P2) •
(11)
H e r e ~bt(k, p) is the component of the regular solution of the Schr6dinger equation for the isolated potential fi, YL(x) are the real spherical harmonics, Pl = r - R, and P2 = r ' - R , , . This solution is obtained in the next section, where we also use it to obtain an expression for ([T(k)]~;).
3. The solution of the single-site equation Before discussing the solution for ([3(k)]) in the single-site approximation we have to derive an expression for the configuration-dependent matrix [8-(k)] relative to an arbitrary reference medium. For this purpose we first consider the case of a single A atom on site n of a periodic reference medium described by the Hamiltonian in eq. (5). The Hamiltonian of the perturbed medium is then ~a(r) = where
I:I(r)+ 8v.(Ir- R~I),
(12a)
J. d'AIbuquerrluee Castro / Multiple.scatteringtheory of binary alloys BOA(It -- Rnl) = vA(lr - R,I) -
~(Ir- R,I).
321 (12b)
It is convenient to introduce the matrix [/(k)]~' = [i(k)ILS..'SLL' 8nn'8~' f d3p d3p'yLOg)jt(kp)[(k, p, p')jr(kp')YL,(IJ),
(13)
where the operator t" = ( 1 - 6G0)-1~ describes the scattering due to ~ in free space. Now we look at the operator rA = (1--8VAt~)-lSVA, which is related to the perturbed Green's function ~A(Z)= (Z --~A) -~ via the Dyson equation: ~A(r, r', k ) = (~(r, r', k ) + f darl d3r2G(r, rl, k)'rA(k, rl, r2)G(r2, r', k).
(14)
It follows from the definition of T A and 8VA that "rA(k, rl, r2) is only non-zero when rl and r2 lie inside the nth muffin-tin sphere. Eq. (14) enables us to derive an expression for the matrix elements [~'A(k)]LL' = f O3p d3p'YL(P)qb, (k, p)~'A(k, Pl, pE)~br(k, p') YL(P'),
(15)
where ~bl is defined at the end of the previous section and chosen to be normalized to j t ( k p ) ikil(k)ht(kp) at the surface of the nth muffin-tin sphere. Here jl(x) and hT(x)=jl(x)+ int(x) are the spherical Bessel and Hankel functions, respectively. Assuming in eq. (14) that r and r' lie in the interstitial region and expanding (0A and (~ according to eqs. (A.2), (A.4) and (A.5) we get
~, ~.. AnlL~(k , r)[TA(k)]L~L~And.2(k, r')
nlL1n2L2
= ~_~ ~_~ A,,~L~(k, r)[T(k)]~,~A~2L2(k, r ) -
L1L2
'
nlLl n2L2
7-1 LIL 7_ 1 L'L2} +~, ~'. A,~L~(k,r) { ~ , , [T(k)t (k)],~n[zA(k)]LL,[t (k)T(k)]~, 2 A,~L~(k,r'), rllLl
n2L2
(16)
where A~L(k, r) is defined by eq. (A,2),
- LIL2= [(i-'(k) [Ttk)l.,,~
-- B(k))
_,]~,.~. L,L2
(17a)
and
[(t-l(k)- B (k))-']'L~L~
L1L2 [ [Ta(k)ln,.2 = [t(k )]~'L' = [tA(k )]L', [t(k)]n,L, = [i(k)k,,
n'= n
(17b)
n'~n.
We should bear in mind that in all the cases of interest only a finite number of terms in the summation
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
322
over angular momentum indices will be considered, and, therefore, the order in which they are carried out in eq. (16) is immaterial. On the basis of eqs. (17) one can easily see that [TA(k)]
=
[7'(k)](1
-
[AAI[~(k)])
(lSa)
-~ ,
where matrix multiplication is implied and [ZiA(k)IL~L~= 6.~.26.1.6L,L2{[[-'(k )lL,- [t-Al(k )lL,}
=
6.,.26.,.aL,L:[A.(k)]L
(lSb)
, .
Thus, eq. (16) reduces to
(n~lL A~ILI(k, r)[T(k - )].~.) L1L {(1 - [AA(k )][ T(k )]..)-'[AA(k )] 1
_[i-l(k)][TA(k)l[i-l(k)]}LL,(Z [T(k)],,,, L'L2 2 A.2~a(k, r') ) = 0 n2L2
(19)
for any pair of points r and r' in the interstitial region. It follows immediately that [rh(k)] LL'= {[t(k )] (1 - [dh(k)][ 7"(k )]oo)-'[Zia(k )][/(k )]}LL',
(20)
where we have used that [~'(k)],, = [7"(k)]00. A similar result can be obtained for [rB(k)]. We point out that if the lattice has cubic symmetry and all the phase-shifts for !-> 3 are neglected (which is a good approximation for transition, noble and simple metals) then the matrices [~'A(k)] and [rB(k)] are diagonal in the angular momentum indices. Having derived expressions for ['rA(k)] and [rB(k)] we can now obtain an expression for [~(k)]~, defined in eq. (11). The starting point is eq. (8), which gives
Y.~,(k, r, r') = r.(k, r, r')6.., + ~. f d3rl d3r2"rn(k, r, r0(~(rl, r2, E)3-,..,(k, r2, r').
(21)
mall m;~n
It can be easily seen that ffn,,(k, r, r ' ) ~ 0 only when r and r' lie inside the nth and n'th muffin-tin spheres, respectively. Based on eqs. (11), (15) and (A.1) we immediately get
ff(k)~,' = [~'.(k)ILL'8..'+ ~ ~ [ r . ( k ) l L t , [ S ( k ) l t ~ t 2 I ~ ( k ) ] m L1L2
L2L' ,
(22)
where [~',(k)] is equal to [~'A(k)] or [ra(k)] depending on the alloy configuration and [ S ( k ) ] ~ , ' = (1 -
8~,)[i-'(k)~(k)i-'(k)]~,'.
(23)
The formal solution of eq. (22) is [~r(k)]~,' = [(~--'(k)- S(k))-']]~,'
(24a)
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
323
or equivalently [ff(k)]~,' = [i(k)(1 - zi (k)2P(k))-lzi (k)i(k)]~,',
(24b)
where [zi(k)]~,' = 6..,SLL~[[i-I(k)IL- [t~(k)]L}.
(24C)
One can easily verify that the matrix [if(k)] satisfies the relation [T(k)] = [T(k)] + [7"(k)l[i-'(k)l[3(k)l[i-l(k)l[7"(k)],
(25)
which can be derived from eqs. (2a) and (6). We note that ['rA(k)], [rB(k)], [S(k)] and [~r(k)] reduce to [/A(k)], [ta(k)], [B(k)] and [T(k)], respectively, when the reference medium is free space, as they should. This can be checked by taking the limit i(k)--> 0 for all l in eqs. (20), (23) and (24). We are now in a position to consider the calculation of ([if(k)]). In the single-site approximation the configuration average of eq. (22) is written as LIL 2
([~(k)]~,') = ([7.(k)]~,)6..,+ ~. ~] ([~'.(k)]~l)[S(k)].,. m
L2L'
([~-(k)],..,),
(26)
L1L 2
where ([r.(k)l) = [Tc(k)] = X[rA(k)] + y[~'B(k)] •
(27)
The solution of eq. (26) is given by ([~r(k)]~,') = [(~-2'(k)- S(k))
-1
LL' ]..,,
(28)
which is the scattering matrix of a new crystalline medium characterized by the single-potential scattering matrix [zc(k)] relative to the reference medium. This solution enables us to get ([T(k)]) immediately. Let us define the scattering matrix [t¢(k)] such that
x[rA(k )] + y[rB(k)] = [[(k )][,~¢(k )](1 - [7"(k )]oo[$c(k )])-'[i(k )] ,
(29a)
where [,~c(k)] =
[i-'(k)l-
[Gl(k)].
(29b)
On the basis of eq. (25) we finally get ([T(k)]~.) = [ ( t g ' ( k ) - B(k))-ll.U-.,' .
(30)
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
324
This is an important result, but it has to be interpreted carefully. In the single-site approximation the complete configuration average of [T(k)] is given by the scattering matrix of a crystalline effective medium, however restraint must be exercised in replacing the true alloy potential by the effective medium potential for the purpose of calculating other averaged quantities. This is particularly relevant for the calculation of the restricted averages ([T(k)]~')0=g and ([T(k)]~')0=~, as we will see in the next section. Before ending this section we want to comment briefly on the two schemes usually employed in the calculation of ([T(k)]), namely the coherent potential (CPA) and the average-t-matrix (ATA) approximations. In the CPA the reference medium is chosen such that [to(k)] = [tcp(k)] = [i(k)], which implies that
X[zA(k)] + y[~'B(k)l = 0.
(31)
From eq. (20) we can show that this condition is equivalent to the usual CPA equation (Gyorffy and Stocks [8])
[t~(k )1 = x [txl(k )1 + y [t~'(k)] + [A CP(k)1[ TCr(k)]oo[ACP(x)],
(32a)
where [,4 CAP(k)]= [t?~(k)] - [tX'(k)],
(32b)
[A aCP(k)] = [t~(k)] - [t~'(k)],
(32c)
[TCP(k)]oo = [(t~[(k)- B(k))-']oo = N - ' ~ [(t~l(k)-
Bq(k))-l].
(32d)
q
Here N is the number of unit cells in the system and Bq(k) are the well-known K K R structure functions (Kohn and Rostoker [9], Ham and Segall [10]). Eqs. (32) must be solved self-consistently for [to,(k)], which involves repeated Brillouin zone integrations. In the A T A a convenient choice of the reference medium is made and ([T(k)])= [TAT(k)] is calculated from eqs. (20), (29) and (30), with [T(k)]00 = N-1Eq [(i-l(k)-Bq(k)-l]. In the previous applications of this approximation free space was taken as the reference medium. The validity of this choice is doubtful since the results it gives in the dilute limit (x ~ 1 or y ~ 1) are in disagreement with the exact results (Bass [11]). This point is discussed in section 5. We emphasize, however, that the results derived in this section do enable one to make a different choice for the reference medium. Finally, we want to point out that eqs. (24) and (25) are also valid for non-crystalline reference media either with or without an underlying lattice.
4. Restricted averages and the density of states The alloy electronic density of states per unit of volume is given by eq. (4), which is equivalent to
(g(E))= --~-~ Im f d3r(G(r, r, E+)),
(33)
J. d'Albuquerque e Castro I Multiple-scatteringtheoryof binary alloys
325
where we have used the identity
~(f
d3rG(r,r'E+))=-~1 f d3r(O(r, r, E+)).
(34)
The evaluation of either the left or the right-hand sides of eq. (34) shows us that in order to calculate (g(E)) we have to obtain an expression for ([T(k)]00)0=A and ([T(k)]00)0=B. However, the important point here is the fact that if the single-site approximation is employed to calculate ([T(k)]) then eq. (34) reduces to a relation between ([T(k)]00), ([T(k)]00)0=A and ([T(k)]0o)0=B, which must hold in both CPA and ATA. Let us consider first the left-hand side of eq. (34). The integration of G(r, r, E +) over all space can be done using eqs. (A,1) and (A.2) and then we get (Appendix B)
1
d3'O(r,r))=
f Id' "l
tLclE
+
dt{31
~
dB ~
L~L
= Mo + ~1 L~ {x [dG,1][ dE JL([T]°° )0=A y [--d-~-]L([T]oo)o=B- ZnlLl[ ~ ] ° " i ([T]"I°)} ' (35) where Mo = 12-~f d3rGo(r, r) and g2c is the volume of the Wigner-Seitz cells. We have omitted the energy arguments in the matrix elements above. Eq. (35) is in agreement with Lloyd's formula [12]. On the right-hand side of eq. (34) we can make use of the fact that the configuration averaged Green's function has the full symmetry of the Bravais lattice and express the integration over all space in terms of the integration over one Wigner-Seitz cell
1
~ f d3r(G(r, r)) =-ffc f d3r(G(r, r)) (o)
= x_~f d3r(G(r,
r))o=A+-~ f d3r(G(r, r))o=s,
(o)
(36)
(o)
where the subscript (0) denotes the integration over the central Wigner-Seitz cell. Assuming that the cells are not too asymmetric (Morgan [13]) the Green's function inside the Wigner-Seitz cell is given by eq. (A.1), and therefore
( G(r, r))0=a = ~] YL(p)J~(p)[tx1]t.([ T]~')O=A[t~I]L,J~,(p) YL,(O)--~, YL(P)J~t(P)O~(P) YL(p) . LL'
L
(37)
A similar expression can be written for (G(o, p))0=n. From eqs. (36), (37) and (B.11) it follows that
f d'r(O(,, r)>=
1
dtX 1
~
+
>0-A
dtB 1
LL
)0=.
L1L _]_ } +x([ZAk, + ~ [WA]LL,([T]o0 )O=A y([ZB]~ + ~ [WB]LL,([T]6OOO=B , L|
LI
(38)
J. d'Albuquerque e Castro I Multiple-scattering theory of binary alloys
326 where
[ZA] = - [ a l [ t T J l - [/3],
(39a)
[WA] = [fl][tX11+ [t2][/31 + [tT~'l[al[tg 1] + [3'],
(39b)
and similarly for [ZB] and [WB]. The matrices [t~], [fl] and [3'] are defined in eqs. (B.12)-(B.14), respectively. So far no approximation has been made. However, if ([T(k)]~') is calculated in the single-site approximation we can make use of the expression for dB/dE (Jacobs and Zaman [14], Lodder and van Dijkum [15])
LdEJ.,,, = - ~ L1
{
}
+ LL1 LL1 L2L' t.,L' [B],,,,, [fl]L,L,+ Z [BI..2[a]L,L2[B].2.' + [Y]LL' n2L2
[B]u.,[Bl..,
(40)
and rewrite eq. (35) in the form 1
dG, l
LL
+
LIL }
+[z0]~ + Y~ [WclLL,([r]~ ) ,
(41)
L1
where we have used/~-1 EL [fllLz = M0. The definitions of [Zc] and [We] are similar to the ones of [ZA] and [WA] in eqs. (39). This result together with eqs. (34) and (38) give us the important relation
X~L {x([ZA]LL + ~[WA]LL,([T]oo£,L)0=A) + y ([ZBIL£+ ~ [W.l££,([Tloo )0=a Li
=
[Ze]LL+ ~, [W~]~,([T]oo ) ,
(42)
L1
valid in the single-site approximation. It is useful at this point to make some comments on the results we have derived so far. As we have seen above, within the muffin-tin framework the calculation of the alloy density of states requires the knowledge of the complete and restricted averages of the site-diagonal elements of the alloy scattering matrix. In the previous section we discussed the calculation of the complete average, which can be done either in the CPA or in the ATA. These two approximations (CPA and ATA) are based on the single-site approximation which decouples the multiple-scattering equations for the alloy t-matrix. For the purpose of calculating the density of states, it is necessary, however, to find expressions for the restricted averages as well. In view of that, the importance of eq. (42) becomes clear since this equation provides us with a relation between the complete and restricted averages of the site-diagonal elements of the alloy t-matrix. This relation has been derived from identity (34) only assuming that the complete average of the t-matrix is calculated in the single-site approximation. Therefore eq. (42) must hold in both CPA and ATA, and it can be used as a check for expressions for the restricted averages. We c a n easily verify that all the previously proposed expressions for the restricted averages ([T]nLnL)n=Aand ([T],,),=a LL (Schwartz and Bansil [16], Ehrenreich and Schwartz [6], Bansil [17], Faulkner
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
327
and Stocks [18]) do not satisfy (42) for a general choice of the reference medium. In fact, it is only when the reference medium is the self-consistent coherent potential medium (CPA) that the previous expressions for the restricted averages satisfy (42). However, self-consistent calculations based on the muffin-tin model are considerably time-consuming and for this reason it may be in some case desirable to do a non-self-consistent (ATA) calculation starting from a convenient choice of the reference medium. Thus, it is necessary to find expressions for the restricted averages which satisfy (42) for a general choice of the reference medium, and which, of course, reduce to the previously proposed expressions when the reference medium is the coherent potential medium. The derivation of general expressions for the restricted averages in the single-site approximation is not a straightforward problem. Argument based on mean-field theory (Schwartz and Bansil [16], Ehrenreich and Schwartz [6]) shows that in the CPA the restricted averages are given by the scattering matrix of a single A or B atom placed in the (self-consistent) coherent potential medium. The fact that those expressions cannot be used in a non-self-consistent (ATA) calculation indicates that the CPA is the only single-site approximation which is equivalent to a mean-field theory. We emphasize that the use of inappropriate expressions for the restricted averages in a non-selfconsistent calculation may lead to unsatisfactory results, as for example the conclusion that the two expressions (4) and (33) for the alloy density of states give different results when the reference medium is arbitrarily chosen. In other words, if one uses expressions for the restricted averages which are valid only in the CPA, eqs. (4) and (33) give the same result only if the reference medium is the self-consistent coherent potential medium. Finally, we remark that since eq. (35) is the same as one obtains from Lloyd's formula [12], the use of the latter as a basis for discussion of disordered systems is fully justified, provided adequate approximations for the restricted averages are used. Clearly the problem now is to find out expressions for the restricted averages ([T]nn)n=g and ([T],~)n=B which are adequate in the sense that they satisfy eq. (42). This equation can be simplified if the lattice has cubic symmetry and the phase-shifts for l -> 3 are neglected. In this situation, which will be assumed from now on, all the matrices in angular momentum indices which appear in that equation are diagonal and we can write
{x([Z.lu + [WAI~z([TlgL)O~A)+ y([ZBILL + [WBk,([TI~,L}o~O} L
= E {[Zok~ + [ W . ] u ( [ r l N ) } .
(43)
L
If in the above equation we make use of the identity ( [ T ] ~ ) = x ( [ T ] ~ ) 0 = . + y([T]~)0=B,
(44)
we immediately get the relations E {x([W*ILL -- [WBI~)([T]~)o=*} L
= E {([Zc]rr - [ 2 ] , , ) + ([Wc]tz - [WB]Ix)([T]~)},
(45a)
328
J. d'Albuquerque e Castro I Multiple-scattering theory of binary alloys
{y([ W , ] ~ - [ WAIu~X[TI~'-)o:.} L
= ~
{([Zol,./. [21.)+ ([wc],.. [WA]LL)(IT]~')}, -
-
(45b)
L
where (46)
[21 = x [ Z A + y [ Z . ] ,
which suggest that in the single-site approximation the restricted averages ([T]~LL)0= A and ([T]~oL)O=B are given by x([ T]~)0=A = ([ WA]LL -- [ WB]LL)-I{[Zc]LL -- [Z]LL -}- ([ Wc]LL -- [ WB]LL)([ T] ~)},
(47a)
y([T]~)0=B = ([ WB]~ - [WA]LL)-I{[Zc]LZ -- [Z]LL + ([Wc]LL -- [WA]~)([T]~)}.
(47b)
It can be easily shown that if [tc(k)] is set equal to [tcv(k)] given by eqs. (32) ([T]~)0=A = {1 -- ( [ t ~ ] t L - [txl]~)[TCe]~}-l[TCP]~oL,
(48)
and similarly for ([T]~)0=B, which are the expressions usually employed in the CPA calculations. Eq. (48) corresponds to an A atom on the centre of the otherwise effective medium and we emphasize that this result can only be obtained from (47) assuming that the reference medium is the coherent potential medium. /.z On the basis of the above results we conclude that if ([T]~)0=A and ([T]00)0=B are given by eqs. (47) then the alloy electronic density of states can be evaluated from any of the following equations: 0 1 (g(E))= g (E)---Im ~'/'2c
dt~,1 ~ + t T q t z dB__B_~ ~, { x [ ~ ] / ( [ T l o o ) o = A y[_~_~_] ([Tloo)o=B_dtfi ~ LL N-l~t Jq dE J' L q
(g(E)) = - rr~cIm ~ {x[LdtT'l] + y r<" ll + [Z,.:]~ + [W<:]~[TC]~} d E JL ([T]~L)°:A L d E JL ([T]~)°:B (g(E)) =
(49a)
(49b)
---Im~{x([dtXq [WA]LL)<[TIa>0-A+/rdtaq [WBI~)<[TN>0=.+[2I,,} I
~ac
(49c) where g°(E) = -(Trg2c)-1 Im f d3rG°(r, r, E+).
5. Dilute alloy limit In this section we investigate the behaviour in the dilute alloy limit of the results we derived in the previous section. This will provide a useful check o n them since in this limit exact results can be obtained.
J. d'Albuquerque e Castro / Multiple-scatteringtheory of binary alloys
329
Assuming that the positions of the A atoms are randomly distributed on the lattice we can show that in the limit x ~ 1, (g(E)) is given by (Ehrenreich and Schwartz [6]) (g(E)) = gB(E)- ~
Im t r { ( 1 - [ A l[ TB]oo)-I ~ E ([A l[TBloo)} + 6(x2),
(50)
where gB(E) is the density of states of the pure B crystal, [al = [t;'l - [U,1],
[TBI = [(t~'-- B ) - ' ] - '
and tr stands for the trace over the angular momentum indices only. We can also show that ([TIN') = [TBIN' + x
E
[T"lo., k , ( [zl l~L'
_ t--f'~s'L'L'r~ J.,., t" J',J a '-'f~mL'L't" J.,0 + ~7(X:)
(51)
nlLl
In order to derive this result we take the pure B crystal as the reference medium in eq. (22) and calculate the configuration average to the lowest order in x. Then making use of the relation (25) eq. (51) follows immediately. We now want to look at the behaviour of ([T]~}, ([T]~L)0=Aand ([T]~)0=B given by eqs. (30), (47a) and (47b), respectively. As Bass (1973) pointed out the reference medium must be chosen such that it reduces to that of the pure B (or A) crystal when x (or y) goes to zero. According to this we assume that [i] = [tB] + x[O] + ~(X2),
(52)
where [0] is an arbitrary diagonal matrix in the angular m o m e n t u m indices. From eqs. (17a), (20) and (29) we get [t~-q = [ta'l + x[zi l(1 - [TBloo[Zll)-' + ~7(x2) •
(53)
Taking this result into eq. (30), we immediately get eq. (51). On the basis of these results eqs. (47) give ([Tloo)o=a = [A l(1 -[TS]oo[ a ])-1 _~ (~(x)
(54a)
([Tloo)o=B = [T"loo + x E [TBlo.[Zl 1(1 -- [TBI,,,,IA I)-'[TB].o + 6(X2),
(54b)
and
n#0
where the first term on the right-hand side of eq. (54a) represents one A atom on the central site of the otherwise pure B crystal, and the first two terms on the right-hand side of eq. (54b) correspond to a pure B crystal with probability x of having an A atom on site n ~ 0. We should note that since ([T]00)0=A always appears multiplied by x we just have to consider the x independent term in eq. (54a). W e are now in a position to investigate the behaviour of the single-site expression for (g(E)) when x ,~ 1. The easiest way of getting the leading contributions to (g(E)) in this limit is to rewrite eq. (49a) as
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
330
(g(E)) = g°(E) - ~
d
Im ~ ~
ln{detlt~-1 -- Bq[}
q
1 Im tr{x ~-- ([t~:] - [t:l]X[T]00)0=A + y ~d ([ti t] _ [t~_l]X[T]00)0=a~ "rr/-/c J
(55)
and make use of eqs. (53) and (54). Then eq. (55) reduces to eq. (50). As we have seen, provided that the t-matrix of the reference medium satisfies eq. (52) the averaged quantities considered in this paper have the correct behaviour as x--* 0. However, it is interesting to notice that if the reference medium is free space then eq. (53) is replaced by ft,-l]''''= {f/B] + x([tA] -- [tB])}-1 .
(56)
NOW if we introduce the quantity [80 = [tA] -- [tBl
(57)
and expand both equations in powers of [rt] we get
[te l] ---- [tB 1]
- -
x[tal][St][ta ~] + (?(x[St] 2)
(58a)
and [t~-ll' s =
[taq -
x[t~ll[Stl[t{3 '] + 0(x216t]2).
(58b)
Therefore, at least in the dilute limit, eq. (56) is only correct to terms of the order [St]. The same result holds for the other averaged quantities considered before. The reason for this can only be the fact that when the reference medium is free space then the third term on the right-hand side of eq. (9a), which is neglected in the single-site approximation, gives a non-negligible contribution. Actually, for this choice of the reference medium, we can expect that term to give a contribution proportional to [St] 2, at least in the dilute limit. As we have said before, this result casts doubt on the validity of the choice of free space as the reference medium.
6. Conclusion
In this paper two important problems regarding the calculation of the density of states of a substitutional binary alloy in the single-site approximation h a v e been considered. The first is the calculation of the complete configuration average of the site-diagonal elements of the scattering matrix IT(k)] relative to free space in terms of an arbitrary crystalline reference medium. The second is the calculation of the restricted averages ([T(k)]..)n=A and ([T(k)],,.).=B in terms of the same reference medium. These points are discussed in sections 3 and 4, respectively, and on the basis of the results obtained there one can calculate the alloy density of states. We want to summarize here the steps involved in such a process, assuming that the lattice has cubic symmetry and that the phase-shifts for I -> 3 are neglected.
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
331
For a non-self-consistent calculation (ATA), the starting point is the choice of the t-matrix [/(k)] of the reference medium which can be chosen arbitrarily. The next step is to calculate the site-diagonal matrix elements [ T ( k ) ] ~ in terms of the Brillouin zone integration N -1Eq [(i-~(k)- Bq(k))-I]LL, and to make use of this result to obtain the t-matrix [to(k)] of the effective medium from eqs. (20) and (29). The last point involves just the solution of a simple algebraic equation for each value of k, Having determined [to(k)], the complete configuration average ( [ T ( k ) ] ~ ) is calculated in terms of a second Brillouin zone integration, N -1E s [(t~-l(k) - Bq(k))-I]LL, and this result should be used in eqs. (47) to obtain immediately the restricted averages ([T(k)]nt~n)n=A and ([T(k)]~),=B. Finally, (g(E)) can be calculated from any of eqs. (49). In a self-consistent (CPA) calculation, [i(k)] = [t~(k)] = [tcp(k)] is determined by solving selfconsistently eq. (32a), which involves repeated Brillouin zone integrations. For a discussion of the details involved in this process the reader is referred to the article by Stocks et al. [19]. Once [to,(k)] is known, the calculations of ( [ T ( k ) ] ~ ) , ([T(k)]n~n)~=A, ([T(k)]~)n=B and (g(E)) follow as in the previous case.
Acknowledgements I am grateful to Dr. R.L. Jacobs and Mr. R. Bechara Muniz for many important conversations. Financial support from the CNPq of Brazil is also gratefully acknowledged.
Appendix A: Expansions of the Green's function The Green's function G(r, r', E) for a muffin-tin Hamiltonian of the form given by eq. (I) can be written as (Lehmann [20], Hamazaki et al. [21])
G(p + Rn, p' + Rn,, E) = ~, YL(p)Jnt(k, p)[t-l(k )T(k )t-'(k )l~"Jn'r(k, P')YL'(P') LL'
-ann, ~, YL(p)J~,(k,p<)Q.t(k. P>)YL(P'),
(A.1)
L
when p and p ' lie inside the nth and n'th muffin-tin sphere, respectively. Here Jnt and Qnt are the l components of two solutions of the Schroedinger equation for the isolated potential vn, [T(k)] is defined by eq. (3), and p< and p> stand for the minimum and the maximum of Ipl and Ip'l. Jn, and Qnl are normalized at the surface of the nth muffin-tin sphere to jt(kp)-iktnt(k)h~(kp) and to t~l(k)jl(kp), respectively, and Jnt is regular at the origin. Since we will be always dealing with a finite number of terms in the summations over L and L' we need not worry about the order in which those summations are carried out (Morgan [13]). Except in situations of high asymmetry (Morgan [13]), eq. (A.1) can be extended to all the points inside the Wigner-Seitz cells. However, the following expressions are also useful, particularly in the cases where such extensions are not possible. (a) r and r' are both in the interstitial region:
G(r, r', E) = G°(r, r', E) + ~_~ A ~ ( k , r)[ T(k )l~,'A,,L,(k, r') , nL n'L'
(A.2)
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
332 where
A~L(k, r) = - i k h T(k lr - R,]) YL(r - R, ) .
(A.3)
(b) r is inside the nth sphere and r' is in the interstitial region:
G(r, r', E) = ~ ~, YL(p)J,a(k, p)[t-l(k)T(k)]~,'A.,L,(k, r'). L
(A .4)
n'L'
(c) r is in the interstitial region and r' is inside the n'th sphere:
G(r, r', E) = ~, ~, A~L(k, r)[ T(k )t-'(k )]~,'J.,r(k, P') YL'(P') . nL
(A.5)
L'
Eqs. (A.4) and (A.5) can be easily derived employing the same technique used by d'Albuquerque e Castro et al. [22] and they can be shown to apply, as well as eq. (A.1), even to amorphous systems. In order to prove that expansions (A.1), (A.2), (A.4) and (A.5) match at the surface of the muffin-tin spheres we need the relation
A~L(k. r) = - i k h T(klr = ~
jr(klr
R.l) Ydr - R.)
- R.,I)Y~,(~
(A.6)
- R.,),
L'
which is valid for Jr - R.,I < JR. - R.,I.
Appendix B: Integration of the Green's function The integration of the muffin-tin Green's function over the total volume of the system can be written as
(B.0
f d a r G ( r , r , E ) = ~" f d3rG(r, r,E+)+ f d3rG(r,r,E+), n ~0n
Oi
w h e r e / 2 , denotes the volume of the nth muffin-tin sphere and Oi the volume of the interstitial region. From eq. (A.1) we have
~n
:
rM
rM
0
0
÷ L
+
+c,}
{a,l,-ml., ÷ L
(B.2)
J. d'AIbuquerque e Castro / Multiple-scattering theory of binary alloys
333
where rM is the radius of the muffin-tin spheres and
at = ~
3
{[J'(x)] ~ - jl-l(X)jl+l(X)} ,
(B.3)
ik b, = ---~ r~r{2jt(x )h 7 (x ) - Yl-l(x )h l+,(x ) - jt+,(x )h ~--l(x)}- 21 + 1
(B.4)
ct =
(B.5)
4k 2 ,
rh{[h~[(x)l 2 - h M ( x ) h M ( x ) } ,
with x = krM. Also, from eq. (A.2), we have
f d3rG(r, r,E+) = f d3rG°(r,r,E+)+ ~ ~,L,[T]~,'f d3rA~(k, r)A.,L,(k,r) IJi
ni
=
-Qi
darG°(r,r,E+)+~, 1~i
[T]~(-Ebl[t-l].t-adt-2],t-c,)-~_,
nL
~
,~i L-j"t"
nlL 1
+ at[t-1].~ + 2bt}.
(B.6)
In order to derive this last equation we have to m a k e use of eq. (A.6) and of the relation
f
(t~g,,)
darA~(k, r)A.,L,(k, r) = - [dB]LL' _ xf _ ~j~(x) + j~(x)l [ B I ~ ,, [ d E J n,, 2 k 2 [ j ~ ( x ) - j r ( x ) J
+ik{~
h~'(x)at'~[B]~;
j~,(x) J
(B.7)
where the subscript (/~'.,) denotes the integration over all space except the interior of the nth and n'th (n ~ n') muffin-tin spheres, and j~(x) denotes the derivative of the Bessel function. W e remark that the presence of the small imaginary part in the energy, E + = E + i0 +, guarantees the convergence at infinity. Combining eqs. (B.1), (B.2) and (B.6) and noting that
bt = f d3rG°(r, r, E+), L
nn
we can write 1~1 -L1L) [dr-'1 [T]~- Z I-dB [~-~J,. [TI.I.j~ f d"r~r,r,E+)= f d~rO°(r,r,E+)+~{LdEj.~
(B.8)
nlL1
W e consider now the integration of the muffin-tin G r e e n ' s function over the central Wigner-Seitz cells. Assuming that expansion (A.1) can be extended to all the points inside the cell we have
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
334
f d3rG(r, r, E ÷) = ~,
tt-lTt'lo'-f dSoJot(k,p)JoL,(k,p)
(o)
(o)
--~L f dSrj°t(k'P)Q°L(k'P)'
(B.9)
(0)
where JoL(k,p)= Jo(k, P)YL(P) and similarly for QoL(k, p). These integrals can be evaluated with the help of the relation (Lodder [23])
f d3rpL(r)Ru(r) = f dA fdPL
dPL
OA
= WA[dPL/dE, RL.],
(B.10)
where PL and RL are two solutions at the same energy of the Schroedinger equation [-V 2+ V - E]PL = 0, and ~'~A is the volume delimited by the surface A. If RL is singular inside /2A then PL must be regular. On the basis of eq. (B.10) we finally have
rj d3r O(r, r, E +) = ~ [ d[ tdE - ' l [J0L[TI~°Z+ ~'~z,([Blu"[t-1]°L~ + [t-q0L[B]/~, (0) +[t-I]0L[~]LLI[t-I]0/., + [T]LL,)[T]~J L --
[alt~[t-lloL - [/3kt},
(B.11)
where [altt, = Wc[djL(r)/dE, jL,(r)],
(B.12)
[/3]t~, =-ikWc[djL(r)/dE, hb(r)],
(B.13)
[~'ltz, =-k2Wc[dh~.(r)/dE, hb(r)].
(B.14)
The subscript c denotes the surface of the Wigner-Seitz cell.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
M. Lax, Rev. Mod. Phys. 23 (1951) 237. J. Korringa, J. Phys. Chem. Solids 7 (1958) 252. J.L. Beeby, Proc. Phys. Soc. (London) A279 (1964) 82; Phys. Rev. 135 (1964) A130. P. Soven, Phys. Rev. 156 (1967) 809. D.W. Taylor, Phys. Rev. 156 (1967) 1017. H. Ehrenreich and L. Schwartz, Solid State Physics 31 (1976) 149. R.J. EUiott, J.A. Krumhansl and P.L. Leath, 'Rev. Mod. Phys. 46 (1974) 465. B. Gyortiy and G.M. Stocks, J. Phys. (Paris) 35 (Suppl. No. 5) (1974) C4-75. W. Kohn and N. Rostoker, Phys. Rev. 94 (1954) 111.
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
335
[10] F.S. Ham and B. SegaU, Phys. Rev. 124 (1961) 1786. [11] R. Bass, Phys. Lett. A46 (1973) 189. [12] P. Lloyd, Proc. Phys. Soc. 90 (1967) 207; P. Lloyd and P.V. Smith, Adv. Phys. 21 (1972) 69. [13] J. van W. Morgan, J. Phys. C10 (1977) 1181. [14] R.L. Jacobs and N. Zaman, J. Phys. F7 (1977) 283. [15] A. Lodder and C.E. van Dijkum, Phys. Lett. 74A (1979) 110. [16] L. Schwartz and A. Bansil, Phys. Rev. B10 (1974) 3261. [17] A. Bansil, Phys. Rev. B20 (1979) 4025. [18] J.S. Fauikner and G.M. Stocks, Phys. Rev. B21 (1980) 3222. [19] G.M. Stocks, W.M. Temmerman and B.L. Gyortiy, in Electrons in Disordered Metals and Metallic Surfaces, P. Phariseau, B.L. Gyorffy and L. Scheire, eds. (Plenum), New York, 1979). [20] G. Lehmann, Phys. Stat. Sol. (b) 70 (1975) 737. [21] M. Hamazaki, S. Asano and J. Yamashita, J. Phys. Soc. Japan 41 (1976) 378. [22] J. d'Albuquerque e Castro, R. Bechara Muniz and R.L. Jacobs, Phys. Stat. Sol. (b) 99 (1980) 735. [23] A. Lodder, J. Phys. F10 (1980) 1117.
MULTIPLE-SCATTERING THEORY OF BINARY ALLOYS IN SINGLE-SITE APPROXIMATION J. d ' A L B U Q U E R Q U E e CASTRO* Department of Mathematics, Imperial College, London SW7 2BZ, UK Received 14 May 1981 Revised 23 November 1981 The application of the single-site approximation to investigate electronic properties of substitutional binary alloys is discussed within the framework of the muffin-tin model. On the basis of the multiple-scattering theory, new expressions for the complete and restricted averages of the site diagonal elements of the alloy scattering matrix are presented. These expressions are given in terms of an arbitrary crystalline reference medium and can be used in conjunction with either the coherent potential (CPA) or the average t-matrix (ATA) approximations. It is also shown that the previous expressions for the restricted averages of the alloy scattering matrix can only be used in conjunction with the CPA. The expressions in this paper reduce to the CPA results for a suitable choice of the reference medium.
1. Introduction T h e s t u d y of e l e c t r o n i c p r o p e r t i e s of n o n - c r y s t a l l i n e m a t e r i a l s has b e c o m e of g r e a t i n t e r e s t in t h e p a s t few years. D e s p i t e t h e e n o r m o u s difficulty in a p p r o a c h i n g t h e s u b j e c t t h e o r e t i c a l l y , c o n s i d e r a b l e p r o g r e s s has b e e n m a d e t o w a r d s t h e u n d e r s t a n d i n g of t h e e l e c t r o n i c s t r u c t u r e of d i s o r d e r e d subs t i t u t i o n a l b i n a r y alloys. S o p h i s t i c a t e d t e c h n i q u e s b a s e d on t h e m u l t i p l e - s c a t t e r i n g f o r m a l i s m i n t r o d u c e d b y L a x [1] a n d o t h e r s h a v e b e e n d e v e l o p e d to d e a l with t h e s e systems, a n d m a n y a p p r o x i m a t i o n s c h e m e s h a v e b e e n p r o p o s e d . A m o n g t h e m two of t h e m o s t c o m m o n l y u s e d h a v e b e e n t h e a v e r a g e t-matrix (ATA) and the coherent potential (CPA) approximations. T h e A T A has its origin in t h e w o r k s of K o r r i n g a [2] a n d B e e b y [3] while t h e C P A was first discussed b y S o v e n [4] a n d T a y l o r [5]. T h e s e two a p p r o x i m a t i o n s a n d r e l a t e d w o r k s u p to 1975 a r e r e v i e w e d by E h r e n r e i c h a n d S c h w a r t z [6] a n d E l l i o t t et al. [7]. B o t h A T A a n d C P A a r e b a s e d on t h e s o - c a l l e d single-site a p p r o x i m a t i o n , which d e c o u p l e s t h e m u l t i p l e - s c a t t e r i n g e q u a t i o n for t h e c o n f i g u r a t i o n a v e r a g e d alloy s c a t t e r i n g m a t r i x . This d e c o u p l i n g e n a b l e s o n e to w r i t e t h e c o n f i g u r a t i o n a v e r a g e d s c a t t e r i n g m a t r i x in t e r m s of t h e s c a t t e r i n g m a t r i x of a c r y s t a l l i n e effective m e d i u m , which can b e determined either self-consistently (CPA) or not (ATA). In this p a p e r w e a r e c o n c e r n e d with t h e a p p l i c a t i o n of t h e single-site a p p r o x i m a t i o n within the f r a m e w o r k of t h e muffin-tin m o d e l . T h e r e is n o w a c o n s i d e r a b l e b u l k of l i t e r a t u r e in this field a n d the n u m e r i c a l c a l c u l a t i o n s b a s e d on this m o d e l i n d i c a t e t h a t it p r o v i d e s a realistic d e s c r i p t i o n of the e l e c t r o n i c s t r u c t u r e of d i s o r d e r e d s u b s t i t u t i o n a l alloy. O u r discussion is r e s t r i c t e d to b i n a r y alloys AxBy a n d t h e o b j e c t s of c e n t r a l i n t e r e s t h e r e a r e t h e c o n f i g u r a t i o n a v e r a g e s of t h e s i t e - d i a g o n a l m a t r i x e l e m e n t s of t h e alloy t - m a t r i x . W e o r g a n i s e this p a p e r as follows. In s e c t i o n 2 we w r i t e d o w n t h e single-site e q u a t i o n for t h e * Permanent address: Departamento de Fisica, Universidade Federal de Silo Carlos, Via Washington Luiz, Km 235, Silo Carlos-SP-Brazil. 0378-4363/82/0000-4)000/$2.75 O 1982 N o r t h - H o l l a n d
318
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
c o m p l e t e c o n f i g u r a t i o n a v e r a g e of t h e alloy t - m a t r i x in t e r m s of an a r b i t r a r y c r y s t a l l i n e r e f e r e n c e m e d i u m . In section 3 w e c o n s i d e r t h e e x t e n s i o n of t h e u s u a l m u l t i p l e s c a t t e r i n g t h e o r y to d e a l with an a r b i t r a r y r e f e r e n c e m e d i u m , a n d solve t h e single-site e q u a t i o n for t h e c o m p l e t e c o n f i g u r a t i o n a v e r a g e of t h e o n - t h e - e n e r g y - s h e l l m a t r i x e l e m e n t s of t h e alloy t - m a t r i x . In section 4 we l o o k at t h e alloy e l e c t r o n i c d e n s i t y of states a n d w e see that it can b e w r i t t e n in t e r m s of t h e c o m p l e t e a n d r e s t r i c t e d c o n f i g u r a t i o n a v e r a g e s of t h e s i t e - d i a g o n a l o n - t h e - e n e r g y - s h e l l m a t r i x e l e m e n t s of t h e t - m a t r i x . T h e r e s t r i c t e d a v e r a g e is t h e a v e r a g e o v e r all t h e c o n f i g u r a t i o n s h a v i n g on A (or B) a t o m on a specified site. A l s o we d e r i v e a f u n d a m e n t a l e q u a t i o n r e l a t i n g t h e c o m p l e t e a n d r e s t r i c t e d c o n f i g u r a t i o n a v e r a g e s of t h e s i t e - d i a g o n a l m a t r i x e l e m e n t s of t h e t - m a t r i x a n d a n a l y s e t h e p r e v i o u s l y p r o p o s e d e x p r e s s i o n s for t h e s e r e s t r i c t e d a v e r a g e s . O n t h e basis of this f u n d a m e n t a l e q u a t i o n we p r o p o s e n e w e x p r e s s i o n s for t h e r e s t r i c t e d a v e r a g e s a n d s h o w in s e c t i o n 5 t h a t t h e y l e a d to t h e c o r r e c t l e a d i n g c o n t r i b u t i o n s to t h e d e n s i t y of states in t h e d i l u t e alloy limit. Finally, w e d r a w o u r c o n c l u s i o n in section 6. T w o a p p e n d i c e s a r e also p r e s e n t e d . T h e first c o n t a i n s e x p a n s i o n s of t h e muffin-tin G r e e n ' s f u n c t i o n at different p o i n t s in space. T h e s e c o n d d e a l s with t h e i n t e g r a t i o n of G r e e n ' s f u n c t i o n o v e r all s p a c e a n d o v e r t h e W i g n e r - S e i t z cell. In o r d e r to h e l p t h e r e a d e r , we p r e s e n t t a b l e I s u m m a r i z i n g t h e definition of t h e m a n y s c a t t e r i n g m a t r i c e s a n d c o m b i n a t i o n s of t h e m t h a t a r e u s e d in this w o r k . This t a b l e can b e r e f e r r e d to at any p o i n t in this p a p e r . Definitions t-matrices associated with the potentials VAand vB and relative to free space Alloy configuration-dependent t-matrix relative to free space t-matrix associated with the potential v (reference medium) and relative to free space Reference-medium t-matrix relative to free space t-matrices associated with the potentials VAand va and relative to the reference medium Alloy configuration-dependent t-matrix relative to the reference medium Single-potential t-matrix associated with the effective medium and relative to the reference medium Single-potential t-matrix associated with the effective medium and relative to free space
tA(k) and tB(k) T(k) i(k) 7"(k) ~'A(k)and "ra(k) .~(k) r~(k) t~(k )
~A = i-~(k ) - t-A~ and A¢ = [-l(k ) - t;l(k)
/iB(k) = i+l(k) - tz,l(k)
2. The single-site approximation L e t us a s s u m e t h e H a m i l t o n i a n for a given c o n f i g u r a t i o n of t h e b i n a r y alloy A x B r to h a v e t h e muffin-tin f o r m H(r) = Ho(r) + V(r) =
H0(r) +
v=(lr - R.I),
(1)
n
w h e r e H0 is t h e f r e e - e l e c t r o n H a m i l t o n i a n a n d vn a r e n o n - o v e r l a p p i n g s p h e r i c a l l y s y m m e t r i c p o t e n t i a l s c e n t r e d o n t h e lattice p o i n t s of a B r a v a i s lattice {R,}. T h e p o t e n t i a l s v, a r e e q u a l to VA o r vB d e p e n d i n g on t h e configuration. W e a s s u m e t h a t all t h e muffin-tin s p h e r e s h a v e t h e s a m e radius. T h e alloy e l e c t r o n i c p r o p e r t i e s can b e r e l a t e d to t h e c o n f i g u r a t i o n a v e r a g e d G r e e n ' s f u n c t i o n ( G ( z ) ) = ((z - H)-~), which has t h e full s y m m e t r y of the B r a v a i s lattice. It is c o n v e n i e n t to e x p r e s s G in t e r m s of t h e s c a t t e r i n g o p e r a t o r T d e f i n e d b y
J. d'Albuquerque e Castro / Multiple-scattering theory o[ binary alloys G = Go + GoTGo
319 (2a)
= Go + ~ GoT,~,Go,
(2b)
n,n '
where Go is the free-electron Green's function and T,., are the usual path operators. The nonoverlapping condition on the potentials v, implies that only the on-the-energy-shell matrix elements of T,~, enter the expression for G. They are given by [T(k)]~,' = [ ( t - ' ( k ) - B ( k ) ) - ' ] ~ / ,
(3)
where k = ~ / E , L = (l, m) are the angular m o m e n t u m indices, [B(k)]~,' are the structural functions and [t(k)],L,L,' = [t(k)],~6,,,8~, are the usual t-matrices associated with the isolated potentials v, (Ehrenreich and Schwartz [6]). We are interested in the alloy density of states per unit of volume, which can be written as (g(E)) = -~-~O
Im(f
d3rG(r, r, E + ) ) ,
(4)
where E += E + i0 + and g2 is the volume of the system. As we shall show in section 4, ( g ( E ) ) can be expressed in terms of the complete and restricted configuration averages of the site-diagonal matrix elements of [T(k)], namely <[T(k)]~'> and ([T(k)]~).=A and ([T(k)]~'),=B. T h e last two quantities involve all the alloy configurations having respectively an A or B atom on the site n. The complete average ([T(k)]) can be easily evaluated in the single-site approximation, but the restricted averages are more complicated. We return to this point in section 4. In order to calculate the configuration average of [T(k)] we introduce a crystalline reference medium described by ISI(r) = Ho(r) + ~'(r) = Ho(r) + ~
O~(]r -
R.I),
(5)
tt
where (,'(r) satisfies the muffin-tin conditions and may be energy dependent. The alloy configuration dependent scattering operator gr relative to this medium is defined by the Dyson equation G = (~ + t~3(~,
(6)
where ( ~ ( z ) = (z -/_~)-1 is Green's function of the reference medium. We can show that
X n
n,m m#n
+X X n,ra m#n
(7a)
p p#m
where T,, = [1 - (v,, - b)t~l-'(v. - tS).
(7b)
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
320
Eq. (7a) can be rearranged in the form 3 = Y...,~-..,, where the path operators 3.., satisfy the equation
~-.., = ~.a.., + ~-,,0 E er..,.
(8)
m
raCn
We now consider the configuration average of the above equation:
(e%) = (~)8.., + (~)d E (er.,.,) + (0". - (r))d E (erm.,- (~m.,))), m
m
mYn
m#n
(9a)
where (r) = x r . + Y~'B = x[1 - (VA-- tS)(~I-'(VA-- tS)+ y[1 -- (Vs -- ,~)dl-'(v,,- t~),
(gb) (9c)
and introduce the single-site approximation, which consists in neglecting the third term on the right-hand side of eq. (9a). This term is difficult to deal with and certainly not negligible if the reference medium is not properly chosen. This point is discussed in section 6. Leaving this problem aside for the moment, eq. (9a) is then replaced by
(3r,~,) = O')&,.' + (r)G ~'~ (~m.').
(10)
m rain
Although the general solution of eq. (10) is not readily available, it can be easily solved for ([~r(k)]~,'), where [ff(k)]~,' = f dap, dap2YL(Pl)dp,(k, p,)~..,(p,, Pz, k )~b,,(k, p2)YL,(P2) •
(11)
H e r e ~bt(k, p) is the component of the regular solution of the Schr6dinger equation for the isolated potential fi, YL(x) are the real spherical harmonics, Pl = r - R, and P2 = r ' - R , , . This solution is obtained in the next section, where we also use it to obtain an expression for ([T(k)]~;).
3. The solution of the single-site equation Before discussing the solution for ([3(k)]) in the single-site approximation we have to derive an expression for the configuration-dependent matrix [8-(k)] relative to an arbitrary reference medium. For this purpose we first consider the case of a single A atom on site n of a periodic reference medium described by the Hamiltonian in eq. (5). The Hamiltonian of the perturbed medium is then ~a(r) = where
I:I(r)+ 8v.(Ir- R~I),
(12a)
J. d'AIbuquerrluee Castro / Multiple.scatteringtheory of binary alloys BOA(It -- Rnl) = vA(lr - R,I) -
~(Ir- R,I).
321 (12b)
It is convenient to introduce the matrix [/(k)]~' = [i(k)ILS..'SLL' 8nn'8~' f d3p d3p'yLOg)jt(kp)[(k, p, p')jr(kp')YL,(IJ),
(13)
where the operator t" = ( 1 - 6G0)-1~ describes the scattering due to ~ in free space. Now we look at the operator rA = (1--8VAt~)-lSVA, which is related to the perturbed Green's function ~A(Z)= (Z --~A) -~ via the Dyson equation: ~A(r, r', k ) = (~(r, r', k ) + f darl d3r2G(r, rl, k)'rA(k, rl, r2)G(r2, r', k).
(14)
It follows from the definition of T A and 8VA that "rA(k, rl, r2) is only non-zero when rl and r2 lie inside the nth muffin-tin sphere. Eq. (14) enables us to derive an expression for the matrix elements [~'A(k)]LL' = f O3p d3p'YL(P)qb, (k, p)~'A(k, Pl, pE)~br(k, p') YL(P'),
(15)
where ~bl is defined at the end of the previous section and chosen to be normalized to j t ( k p ) ikil(k)ht(kp) at the surface of the nth muffin-tin sphere. Here jl(x) and hT(x)=jl(x)+ int(x) are the spherical Bessel and Hankel functions, respectively. Assuming in eq. (14) that r and r' lie in the interstitial region and expanding (0A and (~ according to eqs. (A.2), (A.4) and (A.5) we get
~, ~.. AnlL~(k , r)[TA(k)]L~L~And.2(k, r')
nlL1n2L2
= ~_~ ~_~ A,,~L~(k, r)[T(k)]~,~A~2L2(k, r ) -
L1L2
'
nlLl n2L2
7-1 LIL 7_ 1 L'L2} +~, ~'. A,~L~(k,r) { ~ , , [T(k)t (k)],~n[zA(k)]LL,[t (k)T(k)]~, 2 A,~L~(k,r'), rllLl
n2L2
(16)
where A~L(k, r) is defined by eq. (A,2),
- LIL2= [(i-'(k) [Ttk)l.,,~
-- B(k))
_,]~,.~. L,L2
(17a)
and
[(t-l(k)- B (k))-']'L~L~
L1L2 [ [Ta(k)ln,.2 = [t(k )]~'L' = [tA(k )]L', [t(k)]n,L, = [i(k)k,,
n'= n
(17b)
n'~n.
We should bear in mind that in all the cases of interest only a finite number of terms in the summation
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
322
over angular momentum indices will be considered, and, therefore, the order in which they are carried out in eq. (16) is immaterial. On the basis of eqs. (17) one can easily see that [TA(k)]
=
[7'(k)](1
-
[AAI[~(k)])
(lSa)
-~ ,
where matrix multiplication is implied and [ZiA(k)IL~L~= 6.~.26.1.6L,L2{[[-'(k )lL,- [t-Al(k )lL,}
=
6.,.26.,.aL,L:[A.(k)]L
(lSb)
, .
Thus, eq. (16) reduces to
(n~lL A~ILI(k, r)[T(k - )].~.) L1L {(1 - [AA(k )][ T(k )]..)-'[AA(k )] 1
_[i-l(k)][TA(k)l[i-l(k)]}LL,(Z [T(k)],,,, L'L2 2 A.2~a(k, r') ) = 0 n2L2
(19)
for any pair of points r and r' in the interstitial region. It follows immediately that [rh(k)] LL'= {[t(k )] (1 - [dh(k)][ 7"(k )]oo)-'[Zia(k )][/(k )]}LL',
(20)
where we have used that [~'(k)],, = [7"(k)]00. A similar result can be obtained for [rB(k)]. We point out that if the lattice has cubic symmetry and all the phase-shifts for !-> 3 are neglected (which is a good approximation for transition, noble and simple metals) then the matrices [~'A(k)] and [rB(k)] are diagonal in the angular momentum indices. Having derived expressions for ['rA(k)] and [rB(k)] we can now obtain an expression for [~(k)]~, defined in eq. (11). The starting point is eq. (8), which gives
Y.~,(k, r, r') = r.(k, r, r')6.., + ~. f d3rl d3r2"rn(k, r, r0(~(rl, r2, E)3-,..,(k, r2, r').
(21)
mall m;~n
It can be easily seen that ffn,,(k, r, r ' ) ~ 0 only when r and r' lie inside the nth and n'th muffin-tin spheres, respectively. Based on eqs. (11), (15) and (A.1) we immediately get
ff(k)~,' = [~'.(k)ILL'8..'+ ~ ~ [ r . ( k ) l L t , [ S ( k ) l t ~ t 2 I ~ ( k ) ] m L1L2
L2L' ,
(22)
where [~',(k)] is equal to [~'A(k)] or [ra(k)] depending on the alloy configuration and [ S ( k ) ] ~ , ' = (1 -
8~,)[i-'(k)~(k)i-'(k)]~,'.
(23)
The formal solution of eq. (22) is [~r(k)]~,' = [(~--'(k)- S(k))-']]~,'
(24a)
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
323
or equivalently [ff(k)]~,' = [i(k)(1 - zi (k)2P(k))-lzi (k)i(k)]~,',
(24b)
where [zi(k)]~,' = 6..,SLL~[[i-I(k)IL- [t~(k)]L}.
(24C)
One can easily verify that the matrix [if(k)] satisfies the relation [T(k)] = [T(k)] + [7"(k)l[i-'(k)l[3(k)l[i-l(k)l[7"(k)],
(25)
which can be derived from eqs. (2a) and (6). We note that ['rA(k)], [rB(k)], [S(k)] and [~r(k)] reduce to [/A(k)], [ta(k)], [B(k)] and [T(k)], respectively, when the reference medium is free space, as they should. This can be checked by taking the limit i(k)--> 0 for all l in eqs. (20), (23) and (24). We are now in a position to consider the calculation of ([if(k)]). In the single-site approximation the configuration average of eq. (22) is written as LIL 2
([~(k)]~,') = ([7.(k)]~,)6..,+ ~. ~] ([~'.(k)]~l)[S(k)].,. m
L2L'
([~-(k)],..,),
(26)
L1L 2
where ([r.(k)l) = [Tc(k)] = X[rA(k)] + y[~'B(k)] •
(27)
The solution of eq. (26) is given by ([~r(k)]~,') = [(~-2'(k)- S(k))
-1
LL' ]..,,
(28)
which is the scattering matrix of a new crystalline medium characterized by the single-potential scattering matrix [zc(k)] relative to the reference medium. This solution enables us to get ([T(k)]) immediately. Let us define the scattering matrix [t¢(k)] such that
x[rA(k )] + y[rB(k)] = [[(k )][,~¢(k )](1 - [7"(k )]oo[$c(k )])-'[i(k )] ,
(29a)
where [,~c(k)] =
[i-'(k)l-
[Gl(k)].
(29b)
On the basis of eq. (25) we finally get ([T(k)]~.) = [ ( t g ' ( k ) - B(k))-ll.U-.,' .
(30)
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
324
This is an important result, but it has to be interpreted carefully. In the single-site approximation the complete configuration average of [T(k)] is given by the scattering matrix of a crystalline effective medium, however restraint must be exercised in replacing the true alloy potential by the effective medium potential for the purpose of calculating other averaged quantities. This is particularly relevant for the calculation of the restricted averages ([T(k)]~')0=g and ([T(k)]~')0=~, as we will see in the next section. Before ending this section we want to comment briefly on the two schemes usually employed in the calculation of ([T(k)]), namely the coherent potential (CPA) and the average-t-matrix (ATA) approximations. In the CPA the reference medium is chosen such that [to(k)] = [tcp(k)] = [i(k)], which implies that
X[zA(k)] + y[~'B(k)l = 0.
(31)
From eq. (20) we can show that this condition is equivalent to the usual CPA equation (Gyorffy and Stocks [8])
[t~(k )1 = x [txl(k )1 + y [t~'(k)] + [A CP(k)1[ TCr(k)]oo[ACP(x)],
(32a)
where [,4 CAP(k)]= [t?~(k)] - [tX'(k)],
(32b)
[A aCP(k)] = [t~(k)] - [t~'(k)],
(32c)
[TCP(k)]oo = [(t~[(k)- B(k))-']oo = N - ' ~ [(t~l(k)-
Bq(k))-l].
(32d)
q
Here N is the number of unit cells in the system and Bq(k) are the well-known K K R structure functions (Kohn and Rostoker [9], Ham and Segall [10]). Eqs. (32) must be solved self-consistently for [to,(k)], which involves repeated Brillouin zone integrations. In the A T A a convenient choice of the reference medium is made and ([T(k)])= [TAT(k)] is calculated from eqs. (20), (29) and (30), with [T(k)]00 = N-1Eq [(i-l(k)-Bq(k)-l]. In the previous applications of this approximation free space was taken as the reference medium. The validity of this choice is doubtful since the results it gives in the dilute limit (x ~ 1 or y ~ 1) are in disagreement with the exact results (Bass [11]). This point is discussed in section 5. We emphasize, however, that the results derived in this section do enable one to make a different choice for the reference medium. Finally, we want to point out that eqs. (24) and (25) are also valid for non-crystalline reference media either with or without an underlying lattice.
4. Restricted averages and the density of states The alloy electronic density of states per unit of volume is given by eq. (4), which is equivalent to
(g(E))= --~-~ Im f d3r(G(r, r, E+)),
(33)
J. d'Albuquerque e Castro I Multiple-scatteringtheoryof binary alloys
325
where we have used the identity
~(f
d3rG(r,r'E+))=-~1 f d3r(O(r, r, E+)).
(34)
The evaluation of either the left or the right-hand sides of eq. (34) shows us that in order to calculate (g(E)) we have to obtain an expression for ([T(k)]00)0=A and ([T(k)]00)0=B. However, the important point here is the fact that if the single-site approximation is employed to calculate ([T(k)]) then eq. (34) reduces to a relation between ([T(k)]00), ([T(k)]00)0=A and ([T(k)]0o)0=B, which must hold in both CPA and ATA. Let us consider first the left-hand side of eq. (34). The integration of G(r, r, E +) over all space can be done using eqs. (A,1) and (A.2) and then we get (Appendix B)
1
d3'O(r,r))=
f Id' "l
tLclE
+
dt{31
~
dB ~
L~L
= Mo + ~1 L~ {x [dG,1][ dE JL([T]°° )0=A y [--d-~-]L([T]oo)o=B- ZnlLl[ ~ ] ° " i ([T]"I°)} ' (35) where Mo = 12-~f d3rGo(r, r) and g2c is the volume of the Wigner-Seitz cells. We have omitted the energy arguments in the matrix elements above. Eq. (35) is in agreement with Lloyd's formula [12]. On the right-hand side of eq. (34) we can make use of the fact that the configuration averaged Green's function has the full symmetry of the Bravais lattice and express the integration over all space in terms of the integration over one Wigner-Seitz cell
1
~ f d3r(G(r, r)) =-ffc f d3r(G(r, r)) (o)
= x_~f d3r(G(r,
r))o=A+-~ f d3r(G(r, r))o=s,
(o)
(36)
(o)
where the subscript (0) denotes the integration over the central Wigner-Seitz cell. Assuming that the cells are not too asymmetric (Morgan [13]) the Green's function inside the Wigner-Seitz cell is given by eq. (A.1), and therefore
( G(r, r))0=a = ~] YL(p)J~(p)[tx1]t.([ T]~')O=A[t~I]L,J~,(p) YL,(O)--~, YL(P)J~t(P)O~(P) YL(p) . LL'
L
(37)
A similar expression can be written for (G(o, p))0=n. From eqs. (36), (37) and (B.11) it follows that
f d'r(O(,, r)>=
1
dtX 1
~
+
>0-A
dtB 1
LL
)0=.
L1L _]_ } +x([ZAk, + ~ [WA]LL,([T]o0 )O=A y([ZB]~ + ~ [WB]LL,([T]6OOO=B , L|
LI
(38)
J. d'Albuquerque e Castro I Multiple-scattering theory of binary alloys
326 where
[ZA] = - [ a l [ t T J l - [/3],
(39a)
[WA] = [fl][tX11+ [t2][/31 + [tT~'l[al[tg 1] + [3'],
(39b)
and similarly for [ZB] and [WB]. The matrices [t~], [fl] and [3'] are defined in eqs. (B.12)-(B.14), respectively. So far no approximation has been made. However, if ([T(k)]~') is calculated in the single-site approximation we can make use of the expression for dB/dE (Jacobs and Zaman [14], Lodder and van Dijkum [15])
LdEJ.,,, = - ~ L1
{
}
+ LL1 LL1 L2L' t.,L' [B],,,,, [fl]L,L,+ Z [BI..2[a]L,L2[B].2.' + [Y]LL' n2L2
[B]u.,[Bl..,
(40)
and rewrite eq. (35) in the form 1
dG, l
LL
+
LIL }
+[z0]~ + Y~ [WclLL,([r]~ ) ,
(41)
L1
where we have used/~-1 EL [fllLz = M0. The definitions of [Zc] and [We] are similar to the ones of [ZA] and [WA] in eqs. (39). This result together with eqs. (34) and (38) give us the important relation
X~L {x([ZA]LL + ~[WA]LL,([T]oo£,L)0=A) + y ([ZBIL£+ ~ [W.l££,([Tloo )0=a Li
=
[Ze]LL+ ~, [W~]~,([T]oo ) ,
(42)
L1
valid in the single-site approximation. It is useful at this point to make some comments on the results we have derived so far. As we have seen above, within the muffin-tin framework the calculation of the alloy density of states requires the knowledge of the complete and restricted averages of the site-diagonal elements of the alloy scattering matrix. In the previous section we discussed the calculation of the complete average, which can be done either in the CPA or in the ATA. These two approximations (CPA and ATA) are based on the single-site approximation which decouples the multiple-scattering equations for the alloy t-matrix. For the purpose of calculating the density of states, it is necessary, however, to find expressions for the restricted averages as well. In view of that, the importance of eq. (42) becomes clear since this equation provides us with a relation between the complete and restricted averages of the site-diagonal elements of the alloy t-matrix. This relation has been derived from identity (34) only assuming that the complete average of the t-matrix is calculated in the single-site approximation. Therefore eq. (42) must hold in both CPA and ATA, and it can be used as a check for expressions for the restricted averages. We c a n easily verify that all the previously proposed expressions for the restricted averages ([T]nLnL)n=Aand ([T],,),=a LL (Schwartz and Bansil [16], Ehrenreich and Schwartz [6], Bansil [17], Faulkner
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
327
and Stocks [18]) do not satisfy (42) for a general choice of the reference medium. In fact, it is only when the reference medium is the self-consistent coherent potential medium (CPA) that the previous expressions for the restricted averages satisfy (42). However, self-consistent calculations based on the muffin-tin model are considerably time-consuming and for this reason it may be in some case desirable to do a non-self-consistent (ATA) calculation starting from a convenient choice of the reference medium. Thus, it is necessary to find expressions for the restricted averages which satisfy (42) for a general choice of the reference medium, and which, of course, reduce to the previously proposed expressions when the reference medium is the coherent potential medium. The derivation of general expressions for the restricted averages in the single-site approximation is not a straightforward problem. Argument based on mean-field theory (Schwartz and Bansil [16], Ehrenreich and Schwartz [6]) shows that in the CPA the restricted averages are given by the scattering matrix of a single A or B atom placed in the (self-consistent) coherent potential medium. The fact that those expressions cannot be used in a non-self-consistent (ATA) calculation indicates that the CPA is the only single-site approximation which is equivalent to a mean-field theory. We emphasize that the use of inappropriate expressions for the restricted averages in a non-selfconsistent calculation may lead to unsatisfactory results, as for example the conclusion that the two expressions (4) and (33) for the alloy density of states give different results when the reference medium is arbitrarily chosen. In other words, if one uses expressions for the restricted averages which are valid only in the CPA, eqs. (4) and (33) give the same result only if the reference medium is the self-consistent coherent potential medium. Finally, we remark that since eq. (35) is the same as one obtains from Lloyd's formula [12], the use of the latter as a basis for discussion of disordered systems is fully justified, provided adequate approximations for the restricted averages are used. Clearly the problem now is to find out expressions for the restricted averages ([T]nn)n=g and ([T],~)n=B which are adequate in the sense that they satisfy eq. (42). This equation can be simplified if the lattice has cubic symmetry and the phase-shifts for l -> 3 are neglected. In this situation, which will be assumed from now on, all the matrices in angular momentum indices which appear in that equation are diagonal and we can write
{x([Z.lu + [WAI~z([TlgL)O~A)+ y([ZBILL + [WBk,([TI~,L}o~O} L
= E {[Zok~ + [ W . ] u ( [ r l N ) } .
(43)
L
If in the above equation we make use of the identity ( [ T ] ~ ) = x ( [ T ] ~ ) 0 = . + y([T]~)0=B,
(44)
we immediately get the relations E {x([W*ILL -- [WBI~)([T]~)o=*} L
= E {([Zc]rr - [ 2 ] , , ) + ([Wc]tz - [WB]Ix)([T]~)},
(45a)
328
J. d'Albuquerque e Castro I Multiple-scattering theory of binary alloys
{y([ W , ] ~ - [ WAIu~X[TI~'-)o:.} L
= ~
{([Zol,./. [21.)+ ([wc],.. [WA]LL)(IT]~')}, -
-
(45b)
L
where (46)
[21 = x [ Z A + y [ Z . ] ,
which suggest that in the single-site approximation the restricted averages ([T]~LL)0= A and ([T]~oL)O=B are given by x([ T]~)0=A = ([ WA]LL -- [ WB]LL)-I{[Zc]LL -- [Z]LL -}- ([ Wc]LL -- [ WB]LL)([ T] ~)},
(47a)
y([T]~)0=B = ([ WB]~ - [WA]LL)-I{[Zc]LZ -- [Z]LL + ([Wc]LL -- [WA]~)([T]~)}.
(47b)
It can be easily shown that if [tc(k)] is set equal to [tcv(k)] given by eqs. (32) ([T]~)0=A = {1 -- ( [ t ~ ] t L - [txl]~)[TCe]~}-l[TCP]~oL,
(48)
and similarly for ([T]~)0=B, which are the expressions usually employed in the CPA calculations. Eq. (48) corresponds to an A atom on the centre of the otherwise effective medium and we emphasize that this result can only be obtained from (47) assuming that the reference medium is the coherent potential medium. /.z On the basis of the above results we conclude that if ([T]~)0=A and ([T]00)0=B are given by eqs. (47) then the alloy electronic density of states can be evaluated from any of the following equations: 0 1 (g(E))= g (E)---Im ~'/'2c
dt~,1 ~ + t T q t z dB__B_~ ~, { x [ ~ ] / ( [ T l o o ) o = A y[_~_~_] ([Tloo)o=B_dtfi ~ LL N-l~t Jq dE J' L q
(g(E)) = - rr~cIm ~ {x[LdtT'l] + y r<" ll + [Z,.:]~ + [W<:]~[TC]~} d E JL ([T]~L)°:A L d E JL ([T]~)°:B (g(E)) =
(49a)
(49b)
---Im~{x([dtXq [WA]LL)<[TIa>0-A+/rdtaq [WBI~)<[TN>0=.+[2I,,} I
~ac
(49c) where g°(E) = -(Trg2c)-1 Im f d3rG°(r, r, E+).
5. Dilute alloy limit In this section we investigate the behaviour in the dilute alloy limit of the results we derived in the previous section. This will provide a useful check o n them since in this limit exact results can be obtained.
J. d'Albuquerque e Castro / Multiple-scatteringtheory of binary alloys
329
Assuming that the positions of the A atoms are randomly distributed on the lattice we can show that in the limit x ~ 1, (g(E)) is given by (Ehrenreich and Schwartz [6]) (g(E)) = gB(E)- ~
Im t r { ( 1 - [ A l[ TB]oo)-I ~ E ([A l[TBloo)} + 6(x2),
(50)
where gB(E) is the density of states of the pure B crystal, [al = [t;'l - [U,1],
[TBI = [(t~'-- B ) - ' ] - '
and tr stands for the trace over the angular momentum indices only. We can also show that ([TIN') = [TBIN' + x
E
[T"lo., k , ( [zl l~L'
_ t--f'~s'L'L'r~ J.,., t" J',J a '-'f~mL'L't" J.,0 + ~7(X:)
(51)
nlLl
In order to derive this result we take the pure B crystal as the reference medium in eq. (22) and calculate the configuration average to the lowest order in x. Then making use of the relation (25) eq. (51) follows immediately. We now want to look at the behaviour of ([T]~}, ([T]~L)0=Aand ([T]~)0=B given by eqs. (30), (47a) and (47b), respectively. As Bass (1973) pointed out the reference medium must be chosen such that it reduces to that of the pure B (or A) crystal when x (or y) goes to zero. According to this we assume that [i] = [tB] + x[O] + ~(X2),
(52)
where [0] is an arbitrary diagonal matrix in the angular m o m e n t u m indices. From eqs. (17a), (20) and (29) we get [t~-q = [ta'l + x[zi l(1 - [TBloo[Zll)-' + ~7(x2) •
(53)
Taking this result into eq. (30), we immediately get eq. (51). On the basis of these results eqs. (47) give ([Tloo)o=a = [A l(1 -[TS]oo[ a ])-1 _~ (~(x)
(54a)
([Tloo)o=B = [T"loo + x E [TBlo.[Zl 1(1 -- [TBI,,,,IA I)-'[TB].o + 6(X2),
(54b)
and
n#0
where the first term on the right-hand side of eq. (54a) represents one A atom on the central site of the otherwise pure B crystal, and the first two terms on the right-hand side of eq. (54b) correspond to a pure B crystal with probability x of having an A atom on site n ~ 0. We should note that since ([T]00)0=A always appears multiplied by x we just have to consider the x independent term in eq. (54a). W e are now in a position to investigate the behaviour of the single-site expression for (g(E)) when x ,~ 1. The easiest way of getting the leading contributions to (g(E)) in this limit is to rewrite eq. (49a) as
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
330
(g(E)) = g°(E) - ~
d
Im ~ ~
ln{detlt~-1 -- Bq[}
q
1 Im tr{x ~-- ([t~:] - [t:l]X[T]00)0=A + y ~d ([ti t] _ [t~_l]X[T]00)0=a~ "rr/-/c J
(55)
and make use of eqs. (53) and (54). Then eq. (55) reduces to eq. (50). As we have seen, provided that the t-matrix of the reference medium satisfies eq. (52) the averaged quantities considered in this paper have the correct behaviour as x--* 0. However, it is interesting to notice that if the reference medium is free space then eq. (53) is replaced by ft,-l]''''= {f/B] + x([tA] -- [tB])}-1 .
(56)
NOW if we introduce the quantity [80 = [tA] -- [tBl
(57)
and expand both equations in powers of [rt] we get
[te l] ---- [tB 1]
- -
x[tal][St][ta ~] + (?(x[St] 2)
(58a)
and [t~-ll' s =
[taq -
x[t~ll[Stl[t{3 '] + 0(x216t]2).
(58b)
Therefore, at least in the dilute limit, eq. (56) is only correct to terms of the order [St]. The same result holds for the other averaged quantities considered before. The reason for this can only be the fact that when the reference medium is free space then the third term on the right-hand side of eq. (9a), which is neglected in the single-site approximation, gives a non-negligible contribution. Actually, for this choice of the reference medium, we can expect that term to give a contribution proportional to [St] 2, at least in the dilute limit. As we have said before, this result casts doubt on the validity of the choice of free space as the reference medium.
6. Conclusion
In this paper two important problems regarding the calculation of the density of states of a substitutional binary alloy in the single-site approximation h a v e been considered. The first is the calculation of the complete configuration average of the site-diagonal elements of the scattering matrix IT(k)] relative to free space in terms of an arbitrary crystalline reference medium. The second is the calculation of the restricted averages ([T(k)]..)n=A and ([T(k)],,.).=B in terms of the same reference medium. These points are discussed in sections 3 and 4, respectively, and on the basis of the results obtained there one can calculate the alloy density of states. We want to summarize here the steps involved in such a process, assuming that the lattice has cubic symmetry and that the phase-shifts for I -> 3 are neglected.
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
331
For a non-self-consistent calculation (ATA), the starting point is the choice of the t-matrix [/(k)] of the reference medium which can be chosen arbitrarily. The next step is to calculate the site-diagonal matrix elements [ T ( k ) ] ~ in terms of the Brillouin zone integration N -1Eq [(i-~(k)- Bq(k))-I]LL, and to make use of this result to obtain the t-matrix [to(k)] of the effective medium from eqs. (20) and (29). The last point involves just the solution of a simple algebraic equation for each value of k, Having determined [to(k)], the complete configuration average ( [ T ( k ) ] ~ ) is calculated in terms of a second Brillouin zone integration, N -1E s [(t~-l(k) - Bq(k))-I]LL, and this result should be used in eqs. (47) to obtain immediately the restricted averages ([T(k)]nt~n)n=A and ([T(k)]~),=B. Finally, (g(E)) can be calculated from any of eqs. (49). In a self-consistent (CPA) calculation, [i(k)] = [t~(k)] = [tcp(k)] is determined by solving selfconsistently eq. (32a), which involves repeated Brillouin zone integrations. For a discussion of the details involved in this process the reader is referred to the article by Stocks et al. [19]. Once [to,(k)] is known, the calculations of ( [ T ( k ) ] ~ ) , ([T(k)]n~n)~=A, ([T(k)]~)n=B and (g(E)) follow as in the previous case.
Acknowledgements I am grateful to Dr. R.L. Jacobs and Mr. R. Bechara Muniz for many important conversations. Financial support from the CNPq of Brazil is also gratefully acknowledged.
Appendix A: Expansions of the Green's function The Green's function G(r, r', E) for a muffin-tin Hamiltonian of the form given by eq. (I) can be written as (Lehmann [20], Hamazaki et al. [21])
G(p + Rn, p' + Rn,, E) = ~, YL(p)Jnt(k, p)[t-l(k )T(k )t-'(k )l~"Jn'r(k, P')YL'(P') LL'
-ann, ~, YL(p)J~,(k,p<)Q.t(k. P>)YL(P'),
(A.1)
L
when p and p ' lie inside the nth and n'th muffin-tin sphere, respectively. Here Jnt and Qnt are the l components of two solutions of the Schroedinger equation for the isolated potential vn, [T(k)] is defined by eq. (3), and p< and p> stand for the minimum and the maximum of Ipl and Ip'l. Jn, and Qnl are normalized at the surface of the nth muffin-tin sphere to jt(kp)-iktnt(k)h~(kp) and to t~l(k)jl(kp), respectively, and Jnt is regular at the origin. Since we will be always dealing with a finite number of terms in the summations over L and L' we need not worry about the order in which those summations are carried out (Morgan [13]). Except in situations of high asymmetry (Morgan [13]), eq. (A.1) can be extended to all the points inside the Wigner-Seitz cells. However, the following expressions are also useful, particularly in the cases where such extensions are not possible. (a) r and r' are both in the interstitial region:
G(r, r', E) = G°(r, r', E) + ~_~ A ~ ( k , r)[ T(k )l~,'A,,L,(k, r') , nL n'L'
(A.2)
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
332 where
A~L(k, r) = - i k h T(k lr - R,]) YL(r - R, ) .
(A.3)
(b) r is inside the nth sphere and r' is in the interstitial region:
G(r, r', E) = ~ ~, YL(p)J,a(k, p)[t-l(k)T(k)]~,'A.,L,(k, r'). L
(A .4)
n'L'
(c) r is in the interstitial region and r' is inside the n'th sphere:
G(r, r', E) = ~, ~, A~L(k, r)[ T(k )t-'(k )]~,'J.,r(k, P') YL'(P') . nL
(A.5)
L'
Eqs. (A.4) and (A.5) can be easily derived employing the same technique used by d'Albuquerque e Castro et al. [22] and they can be shown to apply, as well as eq. (A.1), even to amorphous systems. In order to prove that expansions (A.1), (A.2), (A.4) and (A.5) match at the surface of the muffin-tin spheres we need the relation
A~L(k. r) = - i k h T(klr = ~
jr(klr
R.l) Ydr - R.)
- R.,I)Y~,(~
(A.6)
- R.,),
L'
which is valid for Jr - R.,I < JR. - R.,I.
Appendix B: Integration of the Green's function The integration of the muffin-tin Green's function over the total volume of the system can be written as
(B.0
f d a r G ( r , r , E ) = ~" f d3rG(r, r,E+)+ f d3rG(r,r,E+), n ~0n
Oi
w h e r e / 2 , denotes the volume of the nth muffin-tin sphere and Oi the volume of the interstitial region. From eq. (A.1) we have
~n
:
rM
rM
0
0
÷ L
+
+c,}
{a,l,-ml., ÷ L
(B.2)
J. d'AIbuquerque e Castro / Multiple-scattering theory of binary alloys
333
where rM is the radius of the muffin-tin spheres and
at = ~
3
{[J'(x)] ~ - jl-l(X)jl+l(X)} ,
(B.3)
ik b, = ---~ r~r{2jt(x )h 7 (x ) - Yl-l(x )h l+,(x ) - jt+,(x )h ~--l(x)}- 21 + 1
(B.4)
ct =
(B.5)
4k 2 ,
rh{[h~[(x)l 2 - h M ( x ) h M ( x ) } ,
with x = krM. Also, from eq. (A.2), we have
f d3rG(r, r,E+) = f d3rG°(r,r,E+)+ ~ ~,L,[T]~,'f d3rA~(k, r)A.,L,(k,r) IJi
ni
=
-Qi
darG°(r,r,E+)+~, 1~i
[T]~(-Ebl[t-l].t-adt-2],t-c,)-~_,
nL
~
,~i L-j"t"
nlL 1
+ at[t-1].~ + 2bt}.
(B.6)
In order to derive this last equation we have to m a k e use of eq. (A.6) and of the relation
f
(t~g,,)
darA~(k, r)A.,L,(k, r) = - [dB]LL' _ xf _ ~j~(x) + j~(x)l [ B I ~ ,, [ d E J n,, 2 k 2 [ j ~ ( x ) - j r ( x ) J
+ik{~
h~'(x)at'~[B]~;
j~,(x) J
(B.7)
where the subscript (/~'.,) denotes the integration over all space except the interior of the nth and n'th (n ~ n') muffin-tin spheres, and j~(x) denotes the derivative of the Bessel function. W e remark that the presence of the small imaginary part in the energy, E + = E + i0 +, guarantees the convergence at infinity. Combining eqs. (B.1), (B.2) and (B.6) and noting that
bt = f d3rG°(r, r, E+), L
nn
we can write 1~1 -L1L) [dr-'1 [T]~- Z I-dB [~-~J,. [TI.I.j~ f d"r~r,r,E+)= f d~rO°(r,r,E+)+~{LdEj.~
(B.8)
nlL1
W e consider now the integration of the muffin-tin G r e e n ' s function over the central Wigner-Seitz cells. Assuming that expansion (A.1) can be extended to all the points inside the cell we have
J. d'Albuquerque e Castro / Multiple-scattering theory of binary alloys
334
f d3rG(r, r, E ÷) = ~,
tt-lTt'lo'-f dSoJot(k,p)JoL,(k,p)
(o)
(o)
--~L f dSrj°t(k'P)Q°L(k'P)'
(B.9)
(0)
where JoL(k,p)= Jo(k, P)YL(P) and similarly for QoL(k, p). These integrals can be evaluated with the help of the relation (Lodder [23])
f d3rpL(r)Ru(r) = f dA fdPL
dPL
OA
= WA[dPL/dE, RL.],
(B.10)
where PL and RL are two solutions at the same energy of the Schroedinger equation [-V 2+ V - E]PL = 0, and ~'~A is the volume delimited by the surface A. If RL is singular inside /2A then PL must be regular. On the basis of eq. (B.10) we finally have
rj d3r O(r, r, E +) = ~ [ d[ tdE - ' l [J0L[TI~°Z+ ~'~z,([Blu"[t-1]°L~ + [t-q0L[B]/~, (0) +[t-I]0L[~]LLI[t-I]0/., + [T]LL,)[T]~J L --
[alt~[t-lloL - [/3kt},
(B.11)
where [altt, = Wc[djL(r)/dE, jL,(r)],
(B.12)
[/3]t~, =-ikWc[djL(r)/dE, hb(r)],
(B.13)
[~'ltz, =-k2Wc[dh~.(r)/dE, hb(r)].
(B.14)
The subscript c denotes the surface of the Wigner-Seitz cell.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
M. Lax, Rev. Mod. Phys. 23 (1951) 237. J. Korringa, J. Phys. Chem. Solids 7 (1958) 252. J.L. Beeby, Proc. Phys. Soc. (London) A279 (1964) 82; Phys. Rev. 135 (1964) A130. P. Soven, Phys. Rev. 156 (1967) 809. D.W. Taylor, Phys. Rev. 156 (1967) 1017. H. Ehrenreich and L. Schwartz, Solid State Physics 31 (1976) 149. R.J. EUiott, J.A. Krumhansl and P.L. Leath, 'Rev. Mod. Phys. 46 (1974) 465. B. Gyortiy and G.M. Stocks, J. Phys. (Paris) 35 (Suppl. No. 5) (1974) C4-75. W. Kohn and N. Rostoker, Phys. Rev. 94 (1954) 111.
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