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The application of CPA and its extension to ferromagnetic transition-metal alloys
153 THE APPLICATION OF CPA AND ITS EXTENSION TO FERROMAGNETIC TRANSITIONMETAL ALLOYS
J. K A N A M O R I , H. A K A I , N. H A M A D A and H. M I W A Department of Physics, Osaka University, Toyonaka 560, Japan
A review is given of certain m e t h o d s for calculating the electronic and magnetic structure of ferromagnetic transition-metal alloys based on the coherent potential approximation and its extension. Three subjects are discussed in detail. (1) The response of the alloys to an external magnetic field is calculated on the basis of the single-site C P A combined with the H a r t r e e - F o c k approximation for fcc Ni-Fe, bcc F e - N i and F e - C o alloys. (2) A method of treating the effect of the nearest-neighbor composition on the basis of an extension of C P A is discussed. The d e p e n d e n c e of the Fe atomic magnetic m o m e n t and other quantities on the nearest neighbor composition in bcc F e - C o and F e - V alloys is calculated as an example of the applications of this method. (3) A method of carrying out the single-site C P A for the muffin-tin potential model is proposed. Some preliminary results are presented.
I. Introduction
The coherent potential approximation (CPA), which was proposed by Soven [I] in 1967, was subsequently applied with success to ferromagnetic alloys of transition metals by Hasegawa and K a n a m o r i [2] to explain the dependence on composition of the average magnetization, the atomic m o m e n t of each constituent and the electronic specific heat. Since then, discussions of ferromagnetic alloys have evolved towards a more detailed study of their electronic and magnetic structure. We intend in this paper to give a perspective view of the present state of the C P A theory of the ferromagnetic state in disordered alloys, by presenting our recent calculations on various subjects. We assume a tight binding model of d states in the discussions given in sections 2 and 3. The inter-atomic electron transfer integrals, or in other words the off-diagonal elements in the site representation, are assumed to be independent of atomic species, while the diagonal elements representing the effective atomic levels are random variables, depending on atomic species and also on the e n v i r o n m e n t surrounding the atom under consideration. H a s e g a w a and K a n a m o r i determine the diagonal elements in such a way that the average number of d electrons of each spin direction in a given atom satisfies the Hart r e e - F o c k condition for the atom e m b e d d e d in an average medium described by the coherent potential; the coherent potential is determined in turn by C P A for the random system defined by the diagonal elements satisfying the H a r t r e e Fock condition. This approach, which we call Physica 91B (1977) 153-161 © North-Holland
H F - C P A , is summarized in section 2, where we present also a calculation of the effect of a magnetic field on the ferromagnetic state in alloys. The latter subject has been discussed on the basis of H F - C P A by Inoue and Shimizu for the case of N i - C u alloy [3]. We discuss fcc N i - F e , bcc F e - C o and F e - N i alloys for which the tight binding model is better justified. It will be shown that the deformation of the density of states due to scattering plays an important role in the response to an external field. Section 3 will describe our efforts to take into account the fluctuation of atomic composition on the nearest-neighbor shell of a given atom, which is neglected in H F - C P A . The fact that the magnetic state of an atom can depend sensitively on its local environment in a given alloy was discussed long before the proposal of C P A . The most remarkable environment effects in ferromagnetic alloys are: (1) the environment stabilizes one of the two magnetic states with its magnetic m o m e n t either parallel or anti-parallel to the bulk magnetization; (2) the magnitude of the atomic m o m e n t changes sensitively with the environment; and (3) the atom b e c o m e s either magnetic (with a "localized m o m e n t " ) or nonmagnetic, depending on its environment. Typical examples for behaviour type (1) are Mn and Fe in fcc Ni alloys, which have been studied by Jo and Miwa [4] and Jo [5]. We discuss in this section the state of the Fe atom in bcc Fe alloys as an example of type (2) behaviour. We shall show also that we obtain essentially the same result as that of H F - C P A when, for example, the Fe m o m e n t is averaged o v e r the environments. Before presenting the calculation,
154 we discuss our m e t h o d of treating the local e n v i r o n m e n t in some detail. T h e type (3) beh a v i o u r can be r e p r e s e n t e d by the Ni a t o m cluster in N i - C u alloys. B r o u e r s et al. [6] have studied the p a r a m a g n e t i c susceptibility of such a s y s t e m by use of a m e t h o d similar to ours. H o w e v e r , our t h e o r y will not be appropriate for direct application to type (3) b e h a v i o u r where the cluster plays an i m p o r t a n t role, since certain statistical c o n s i d e r a t i o n s are required for dealing with a sort of percolation problem that arises in this case. in section 4 we return to the single-site ( ' P A to discuss e x t e n s i o n s of the tight binding model which we need for a more detailed description of the electronic properties of alloys. E x a m p l e s of the p r o b l e m s are: the asphericity of the a t o m i c magnetic m o m e n t density [7]: the electrical resistivity [8]; alloys b e t w e e n transition and nontransition metals [9]; etc. T h o u g h several p r o p o s a l s have been made so far. they involve a s s u m p t i o n s to simplify the C P A calculation. To appraise these a t t e m p t s we need a C P A calculation for the muffin-tin potential model, which can be m a d e the basis of e v e r y simplified model. We describe here a m e t h o d for c a r r y i n g out this muffin-tin C P A . The electrical resistivity of f c c N i - F e and N i - C r alloys is then calculated by use of the method. Since this last-mentioned w o r k has been reported elsew h e r e , we give here only a s u p p l e m e n t a r y discussion of it. W e present also a preliminary calculation for Fe-A1 alloys. We c a n n o t , h o w e v e r , go on to a general appraisal of the simplified t r e a t m e n t s at present, since to do so we would need more calculations. 2. A s u m m a r y of H F - C P A
We describe first the tight binding model ass u m e d in the following calculations. Refinem e n t s of the model are discussed in section 4. We a s s u m e a binary alloy b e t w e e n metals A and B. T h e model hamiltonian is given by
~ = ~ tiai,aj** + ~ Ei.~ai~ai~,t i,i,*
{1)
i,s
w h e r e a~ is the annihilation o p e r a t o r of an electron with spin s in a d orbital of the a t o m i: t# is the t r a n s f e r integral, w h i c h is i n d e p e n d e n t of a t o m i c species; E~.~ is the effective atomic level of the a t o m i; we use the subscript i in a
dual sense, d e n o t i n g the lattice site and the species of the atom at the site. In H F - C P A Ei, takes either wdue, EA~ or E~,, being d e t e r m i n e d self-consistently to satisfy Ei,
-
E i
+4(Ui
JiJtli,
+ 5~Jitl i ,
SI&I4H,
(2}
where E, ( i - - A or B) is the atomic level m the a b s e n c e of the e l e c t r o n - e l e c t r o n interaction: t/, and J~ are the c o u l o m b and e x c h a n g e integrals, respectively: nA is the n u m b e r of electrons with spin s per d orbital of the atom i. The last term on the r.h.s, of eq. (2) is the Z e e m a n e n e r g y of an external field H with s + 1. The calculation is carried out in the following way. We start with initial values of n , and hence Ei~ ( i = A and B, s + and or I" and ,1,) to determine a c o h e r e n t potential o-(E) in C P A . The local state densities obtained t h e r e b y yield a new set of n~; which are the input for the s e c o n d cycle. The iteration is c o n t i n u e d until the difference b e t w e e n input and output value of each n~, b e c o m e s smaller than a preset limit: the limit is set b e t w e e n 10 ~and 10 4 d e p e n d i n g on the situation. W e c h e c k that the final solution does not depend on the choice of initial values of n~ by changing the choice. The result of the calculation d e p e n d s of c o u r s e on the choice of the parameters, E~, U~ and J~, the total n u m b e r of electrons and the d e n s i t y - o f - s t a t e s function of pure metal which constitute the input informations. Generally we aim at elucidating the mechanism underlying various alloying effects rather than fitting the calculation to experimental data. D e p e n d e n c e on the input i n f o r m a t i o n s will be discussed in the e x a m p l e s presented in this section and the next. We present a calculation of the r e s p o n s e to external field of fcc Ni~ ~Fe~ with x ~> 0.5, where the saturation magnetization deviates f r o m the linear increase with x. Fig. I s h o w s an e x a m p l e of the calculation. We adopt the same model state-density f u n c t i o n as that used in the previous calculation [2]. The choice of other p a r a m e t e r s is given in fig. I. We adjust them, in particular U and J, such thai the deviation from the linear increase starts at a value of x a r o u n d 0.5. S o m e of the majority spin ( T ) states !ie a b o v e the Fermi level when x e x c e e d s this critical value. We can see f r o m fig. I that an external field tends to restore the saturation magnetization to the S l a t e r - P a u l i n g curve. We
155
X (lO-4emu/mol)
M (IJ.e/otom) 0
......... ... H= 3xlOSOe
I
19
2 /
,
[ I
'",,,
20 i
,
A,O'""
",,,,
AVE~
I0 i
-
"~'",
'
. ...................
~
at%Ni
24
A
""'Q'"', .
2.5 "',,o
', O
A
E <
~
A
22
21
I
at % Fe
60
55
50
Fig. 1. The average magnetization per atom at H = 0 and H - 3 x 10~Oe (the dotted curves) and the "high field susceptibility" X in e.m.u./mol (the middle full curve) in Ni, ,Fe,. The full curves labelled Ni and Fe are the partial susceptibilities of Ni and Fe atoms, respectively, with the relation X-xx~:o+(l-x)xN,. The half band-width in pure metal is assumed to be 0.35 Ry. U,, = Uvo - 0.51 and JN, = JF~ = 0.19 are assumed, in units of the half band-width.
s h o w a l s o t h e s o - c a l l e d h i g h field s u s c e p t i b i l i t y , i.e. t h e d e r i v a t i v e w i t h r e s p e c t to t h e field s t r e n g t h H o f the a v e r a g e m a g n e t i z a t i o n at H = 0 a n d t h o s e o f the F e a n d N i a t o m i c m o m e n t s . W e find t h a t t h e field t e n d s to d e c r e a s e t h e n u m b e r o f e l e c t r o n s o n an F e a t o m a n d i n c r e a s e t h a t o n a N i a t o m . T h i s is d u e to a d e f o r m a t i o n of the local state densities of the minority-spin b a n d c a u s e d b y an i n c r e a s e o f t h e d i f f e r e n c e E r e ~ - E N ~ . In f a c t w e find an o p p o s i t e t e n d e n c y if w e k e e p t h e l o c a l s t a t e d e n s i t i e s rigid. Thus, we may conclude that the mechanism of t h e r e s p o n s e to an e x t e r n a l field is q u i t e diffe r e n t f r o m t h a t o b t a i n i n g in t h e rigid b a n d discussion. Fig. 2 s h o w s o u r c a l c u l a t i o n o f t h e field e f f e c t in t h e c a s e o f b c c F e - C o a n d F e - N i a l l o y s . T h e d i f f e r e n c e in t h e c o m p o s i t i o n d e p e n d e n c e b e t w e e n t h e s e t w o a l l o y s w a s d i s c u s s e d in d e t a i l p r e v i o u s l y [2]. W e find, h o w e v e r , t h a t t h e s t a t e d e n s i t y f u n c t i o n a d o p t e d t h e r e is i n a d e q u a t e f o r d i s c u s s i o n o f t h e field e f f e c t , b e i n g o v e r s i m p l i fied. W e a s s u m e h e r e a m o r e f a i t h f u l r e p r e s e n tation of Wakoh and Yamashita's calculation of t h e s t a t e d e n s i t y in b c c F e [10], s h o w n in fig. 3. There are two important features of the state d e n s i t y w h i c h d e t e r m i n e t h e a l l o y i n g e f f e c t in t h e s e a l l o y s . O n e is t h e v a l l e y s e p a r a t i n g t h e t w o m a i n p e a k s , a n d a n o t h e r is t h e s h o u l d e r o f
0
20
40
60
ot % Co
Fig. 2. The high field s u s c e p t i b i l i t y (full c u r v e s ) and a v e r a g e
magnetization at H = 0 (dotted curves) of bcc Fe-Co and Fe-Ni. The curves A and B are for Fe-Co and Fe-Ni. respectively. The open and filled triangles are the experimental X of Fe-Co and Fe-Ni [26], respectively; we have subtracted 1.19x 10 4 e.m.u./mol from the data to take account of the orbital and s band contributions. The open and filled circles are the data for the average magnetization of Fe-Co [14] and Fe-Ni [27], respectively. The half bandwidth of pure metal is assumed to be 0.4 Ry. This calculation is based on the single band model [21 with Uj,~ = 1.388, U . , = Uc,, 1.88. =
r 0,
I
,, O)
-----
....0
'
"
Fig. 3. (a) The model state density function for the d band of a bcc pure metal; (b) the local state densities of an Fe atom in bcc Fe09Co,,, with the nearest-neighbor shell specified by the number 0, 4 and 8 of Co atoms as indicated in the figure. The vertical line at E = 0 is the Fermi level. The parameters are chosen to be U,~o=0.6, JF~=0.19688, Uc,,=0.8, Jco= 0.27 and Evo E,.,, = 1.87718 in units of the half-width of the model state density.
t h e u p p e r p e a k at w h i c h t h e F e r m i l e v e l is s i t u a t e d in t h e m a j o r i t y - s p i n b a n d o f F e . T h e l a t t e r w a s n e g l e c t e d in t h e p r e v i o u s d i s c u s s i o n , w h i c h h a s to a d j u s t t h e d e p t h o f t h e v a l l e y to s o m e e x t e n t to o b t a i n a g r e e m e n t w i t h e x p e r i ment. Here we obtain good agreement with experiment for the derivative of the saturation m a g n e t i z a t i o n as w e l l t h o s e q u a n t i t i e s w h i c h were discussed previously. The explanation for the difference between the two alloys, however, is t h e s a m e as t h e p r e v i o u s o n e , i.e. it a r i s e s
156 f r o m the fact that the difference between the atomic e n e r g y levels of c o n s t i t u e n t a t o m s is larger in F e - N i than in F e - C o . In the latter alloy the shoulder persists b e c a u s e of the smallness of Ev~,~ -Ec<, T. Incidentally, the p r e s e n c e of this shoulder does not d e p e n d on the choice of the muffin-tin potential on which the band structure calculation is based, it arises from :+L difference of the peak position b e t w e e n de and d y states. Deviation from the charge neutrality of each c o n s t i t u e n t atom in alloy can be calculaled in C P A . It turns out that the quantity can be sensitive to the choice of the p a r a m e t e r s in some cases. We shall discuss an e x a m p l e in the next section. 3. Calculation of the local e n v i r o n m e n t effect
3. 1. M e t h o d o f c a l c u l a t i o n
The following discussion is based on the t h e o r y d e v e l o p e d by M i w a [ l l ] . C P A calculates the local electronic structure of a given atom e m b e d d e d in the m e d i u m described by the coherent potential. This idea has been e x t e n d e d to a (z + l)-atom cluster consisting a central a t o m and its z nearest neighbors e m b e d d e d in an a v e r a g e medium. It is a s s u m e d that the a v e r a g e m e d i u m is described by a site-diagonal c o h e r e n i potential o'(E) for each spin direction, e denoting the e n e r g y variable. The e n v i r o n m e n t of tin a t o m is c h a r a c t e r i z e d by the n u m b e r of A and B atoms, N A and Nu ( N a + Nu = =I on the nearestneighbor shell and their geometrical arrangement, and also by the a v e r a g e alloy concentration specifying the a v e r a g e medium. After verifying by calculation that the difference in the geometrical a r r a n g e m e n t has a relatively small effect, Miwa has p r o p o s e d a simplified t r e a t m e n t of the e n v i r o n m e n t in which the n e a r e s t - n e i g h b o r c o m p o s i t i o n is specified by N,x and NR only. An equivalent t h e o r y has been p r o p o s e d i n d e p e n d e n t l y by B r o u e r s et al. [121. With M i w a ' s simplification we easily obtain the G r e e n f u n c t i o n for the cluster of a given central a t o m labelled C and its nearest neighbors. The site-diagonal matrix element of the G r e e n function at the central site, for example, is given by
Gl~)(e" N.O = Li ,I
t'-
[ N.d(L~ ~
.V+#)I÷I '
N ~d(Lu i
where 1,, (i = A , B , C ) is the locater (~ t:'~,) J for lhe atom i, and wc o n l i l the spin subscript t'or simplicil,,: : / and .J, respectively, :+ire the diagonal :+ind off-diagonal elements of ihc indirecl lransfer through the nverLige Inediunl tween the ~ltOnlS on lhe n e a r e s t - n e i g h b o r shell, and are f u n c t i o n s of E ~rt~:l. t is the transfer
integral between a pair of nearest-neighboring atoms. Strictly speaking, we derive cq. ~3) for the lighl binding model in which the e l e c h o n transfer is confined io the nearest-neighbor site. H o w e v e r , expressing :i and .]~ in terms of lhc site-diagonal clement of the Green function in the :+tverage medium, and tlSillg a model ~,lale-
density ftinclion in the calculation of the (ireen function, we can show that (;[M~given by eq. (3! r e d u c e s to the Green function with the a s s u m e d model state-density function in the limit of a pure metal. In other words, eq. (3) can interpolate b e t w e e n two pure metals with tin arbitrary model state density, in any case eq. 13)is :+t reasonable interpohition, since the slate densily in a real transition metal does not deviate m u c h from thai of the n e a r e s t - n e i g h b o r transfer model. The vahie of t is d e t e r m i n e d by the condition that the s e c o n d m o m e n t of the adopted model state density coincides with that of the n e a r e s t - n e i g h b o r transfer model, i.e. zt ~. Miwa, and also Brouers el al., determine the c o h e r e n t potential erie) by the so-called central site a p p r o x i m a t i o n , which equates the configurational a v e r a g e of (7~x~(E:N, 0 o v e r the cluster c o m p o s i t i o n s to the site-diagonal element in the average medium. However, careful examinations, particularly for o n e - d i m e n s i o n a l cases, carried out by Nickel and Butler and by Butler 113l have revealed that the a p p r o x i m a t i o n often gives non-analytic solutions for o-(Et, especially in the case of split hands. A simihtr i n a d e q u a c y has also been f o u n d in our numerical calculations for three-dimensional cases. In the following calculation we take the condition that the configurational a v e r a g e of the trace of the Green function taken on the n e a r e s t - n e i g h b o r shell divided by = is equal io the site-diagonal element of the Green function in the a v e r a g e medium. We obtain essentially the same result if we extend the trace to the
.'i÷,~ll
t)
'J-
(3)
157
c l u s t e r i n c l u d i n g the c e n t r a l site. T h i s app r o x i m a t i o n is r e m i n i s c e n t of the " b o u n d a r y site a p p r o x i m a t i o n " d i s c u s s e d b y B u t l e r [13] f o r the o n e - d i m e n s i o n a l c a s e . O u r n u m e r i c a l calc u l a t i o n with this a p p r o x i m a t i o n has w o r k e d f o r all p a r a m e t e r s a n d c o n c e n t r a t i o n s we h a v e chosen. W e d e t e r m i n e the s p i n - d e p e n d e n t a t o m i c e n e r g y level E;~ b y eq. (2). S i n c e ni~ n o w v a r i e s a c c o r d i n g to the local e n v i r o n m e n t , E;s has a distribution even for a given species of atom. W e d e n o t e b y n~:(NA) the n u m b e r of e l e c t r o n s with spin s p e r d o r b i t a l of the c e n t r a l a t o m C ( = A o r B) h a v i n g NA n e a r e s t n e i g h b o r i n g A a t o m s ; c o r r e s p o n d i n g l y , the a t o m i c levels are d e n o t e d b y EC(NA). In the n u m e r i c a l c a l c u l a t i o n we a s s u m e initial v a l u e s for nC(NA) f o r C = A, B a n d N A = 0,1 . . . . . z a n d h e n c e Ec~'(NA). W e t a k e into a c c o u n t the d i s t r i b u t i o n of LA ~ on the n e a r e s t n e i g h b o r shell b y r e p l a c i n g I/(LA ~ - oW+ 5-) in eq. (3) b y its a v e r a g e with a p p r o p r i a t e conditional probabilities over those A atoms w h i c h h a v e the a t o m C at a given site on their n e a r e s t n e i g h b o r shells; the hAs of t h e s e A a t o m s is given b y the a s s u m e d Nff(NA). T h e s a m e a p p l i e s to 1 / ( L B ~ - 5 ~ + 5.). W e then d e t e r m i n e a c o h e r e n t p o t e n t i a l or(E) to m e e t the " b o u n d a r y s i t e " c o n d i t i o n m e n t i o n e d a b o v e . By i n t e g r a t i n g the local s t a t e d e n s i t y f o r e a c h C, NA a n d s to the F e r m i e n e r g y we o b t a i n a n e w set o f nC,'(NA) w h i c h s t a r t s the s e c o n d c y c l e . T h e i t e r a t i o n is c o n t i n u e d until the d i f f e r e n c e bet w e e n i n p u t a n d o u t p u t f o r e a c h nC(NA) bec o m e s s m a l l e r than 10 -3 a n d that f o r the a v e r a g e o f n~'(NA) s m a l l e r than 10 4. It s h o u l d be n o t e d that the use o f the a v e r a g e with the c o n d i t i o n a l p r o b a b i l i t y m e n t i o n e d a b o v e c o r r e s p o n d s in the dilute limit to the i m p o s i t i o n of the H a r t r e e - F o c k c o n d i t i o n , eq. (2), n o t o n l y f o r the c e n t r a l a t o m b u t also for the n e a r e s t n e i g h b o r s in the c l u s t e r . T h e m e t h o d c a n be a p p l i e d to a l l o y s with a t o m i c s h o r t r a n g e o r d e r , if o n e e m p l o y s the c o r r e c t p r o b a b i l i t y f o r e a c h c o n f i g u r a t i o n of the c l u s t e r . U n f o r t u n a t e l y , the a t o m i c s h o r t r a n g e o r d e r is not o f t e n c h e c k e d in the s a m p l e s of alloys used for magnetic measurement, and we h a v e little e x p e r i m e n t a l d a t a on its influence. A p r e l i m i n a r y c a l c u l a t i o n on F e - V a l l o y s s h o w s t h a t the a t o m i c s h o r t r a n g e o r d e r has an app r e c i a b l e effect on the a v e r a g e m a g n e t i z a t i o n as well as on the d i s t r i b u t i o n o f the m a g n e t i c m o -
ment of constituent atoms. We defer discussion of this p r o b l e m to a f u t u r e p u b l i c a t i o n .
3.2. Magnetic moment of the Fe atom in bcc alloys It is well e s t a b l i s h e d that iron as a b c c m e t a l e x h i b i t s u n s a t u r a t e d f e r r o m a g n e t i s m ; that is to s a y it has h o l e s in b o t h the m a j o r i t y a n d the m i n o r i t y spin d b a n d s . T h i s a l l o w s the F e a t o m to c h a n g e the m a g n i t u d e of its m a g n e t i c m o m e n t e a s i l y w i t h o u t an a p p r e c i a b l e c h a n g e o f the n u m b e r of d e l e c t r o n s , in c o n t r a s t to the c a s e s of Co, F e a n d Mn a t o m s in f c c Ni a l l o y s . In fig. 3 we s h o w e x a m p l e s of the local s t a t e d e n s i t y of the F e a t o m with v a r i o u s e n v i r o n m e n t s . Fig. 4 s u m m a r i z e s o u r c a l c u l a t i o n on F e - C o a l l o y s . T h e c h a n g e o f the F e a t o m i c m o m e n t with the c o m p o s i t i o n o f the n e a r e s t - n e i g h b o r shell is larger in i r o n - r i c h a l l o y s a n d it is also l a r g e r with i r o n - r i c h n e a r e s t - n e i g h b o r c o m p o s i t i o n s at a given a l l o y c o n c e n t r a t i o n . T h e c o n f i g u r a t i o n a l a v e r a g e s of the F e a n d C o m o m e n t s a n d also the a v e r a g e m a g n e t i z a t i o n a g r e e well with the H F - C P A r e s u l t m e n t i o n e d in s e c t i o n 2. T h e
3.
2.
0
,2
.4
.6
Xco
Fig. 4. The Fe and Co atomic moments as function of x in Fe, xCo~. The upper group of dotted curves is the result of calculating the Fe moment for a given number of Co nearest neighbors which increases from zero for the bottom curve to 8 for the top curve; the lower group of dotted curves is the one for the Co moment with a number of Co nearest neighbors which increases from the bottom to the top. The full lines are the calculated average Fe and Co moments, with vertical bars representing the r.m.s, deviation of the moment from its average value. The curve labelled Av represents the calculated average magnetization per atom. Triangles show the neutron diffuse scattering data for Fe and Co moments [28]. Circles display the experimental average magnetization data of the disordered alloy; the crosses are those for the ordered alloys [14].
158
width of the d i s t r i b u t i o n of the Fe m o m e n l , i.e. the r o o t - m e a n - s q u a r e of the d e v i a t i o n from the a v e r a g e value, b e c o m e s s m a l l e r as the Co conc e n t r a t i o n i n c r e a s e s in spite of the i n c r e a s e of the f l u c t u a t i o n of the a t o m i c c o m p o s i t i o n in the n e a r e s t - n e i g h b o r shell. In the c o b a l t - r i c h region most of the slates in the m a j o r i t y spin b a n d are below the F e r m i level, so that the m a g n i t u d e of the m a g n e t i c m o m e n t c a n n o l c h a n g e w i t h o u t violating local charge n e u t r a l i t y . In the c a l c u l a t i o n s h o w n in fig. 4, the n u m b e r s of e l e c t r o n s per atom in pure iron and pure cobalt are a s s u m e d to be 7.1 and 8.2 respectively. T h e p a r a m e t e r Ek, Eu, is c h o s e n so lhat the F e r m i level of f e r r o m a g n e t i c pure iron coincides with that of n o n m a g n e t i c pure cobalt. The r e m a i n i n g p a r a m e t e r s , U and J for Fe and Co a t o m s , are c h o s e n to fit a p p r o x i m a t e l y our calc u l a t i o n to the e x p e r i m e n t a l data on the a v e r a g e m a g n e t i z a t i o n and a v e r a g e atomic m o m e n t s . T h e d e v i a t i o n s from these data, which can be seen from fig. 4, can be made s m a l l e r if we a d j u s t the p a r a m e t e r s more carefully. In particular, the i n c r e a s e of the a v e r a g e magn e t i z a t i o n o b s e r v e d in o r d e r e d F%~Co,~, 0.07 ~ per atom [14], which is m u c h larger than the c a l c u l a t e d value, can be e x p l a i n e d if the n u n > bets of e l e c t r o n s per a t o m in pure iron and pure coball are taken to be 7.0 and 8.15 r e s p e c t i v e l y , with a s m a l l e r value for Uc,,. Since we neglect v a r i o u s c o n t r i b u t i o n s such as that of the s b a n d . an e x a c t fit to the e x p e r i m e n t a l data would be withou! significance. Fig. 5 s h o w s the a v e r a g e n u m b e r of e l e c t r o n s at Fe and Co a t o m s , together with the width of the d i s t r i b u t i o n . The n u m b e r of e l e c t r o n s at a Fe a t o m is larger with iron-rich n e a r e s t n e i g h b o r c o m p o s i t i o n s , while its a v e r a g e i n c r e a s e s with the Co c o n c e n t r a t i o n . T h e latter b e h a v i o r is c o n s i s t e n t with the i s o m e r shift data of the M 6 s s b a u e r e x p e r i m e n t [15]. We note, h o w e v e r , that the d i r e c t i o n of the charge t r a n s f e r b e t w e e n
Fe and Co a t o m s is rather s e n s i t i v e to the choice of /£r~. K~.,,; if we a d o p t a value of E f t - E u , , larger by 0.05 in units of the halfwidth of the model state d e n s i t y , the d i r e c t i o n of the charge t r a n s f e r is r e v e r s e d . We show a similar c a l c u l a t i o n on F e - V allo~s in fig. 6. The Fe atomic m o m e n t c h a n g e s more s e n s i t i v e l y in this case than in F e - C o alloys. For e x a m p l e , the m a g n e t i c m o m e n t of an Fe atom in Fe~V<,~ varies from 2.5/J., (for Nr,. 8) to a b o u t zero ( N r + = 0 ) with the a v e r a g e value 1.15#E+ and with 0.5 >+ for the r.m.s, of the deviation+ In this c a l c u l a t i o n we used the model state d e n s i t y a s s u m e d by H a s e g a w a and K a n a m o r i : the fact that it is oversimplified will not affect the result a p p r e c i a b l y . We d e t e r m i n e E~. Er~. by the same p r o c e d u r e as in the case of F e - C o alloys. T h e a v e r a g e n u m b e r of e l e c t r o n s at an Fe atom t u r n s out to i n c r e a s e with the V c o n c e n t r a t i o n . which s e e m s to be i n c o n s i s t e n t with the isomer shift data [16]. H o w e v e r , we can r e v e r s e the d i r e c t i o n of the charge t r a n s f e r b e t w e e n Fe and V a t o m s by a d o p t i n g a slightly larger value of h,', E~.~. The n u m b e r of e l e c t r o n s at an Fe atom t u r n s out to be i n s e n s i t i v e to lhe n e a r e s t - n e i g h bor c o m p o s i t i o n in the p r e s e n t c a l c u l a t i o n . The a v e r a g e a t o m i c m o m e n t of V as an i m p u r i t y in iron is c a l c u l a t e d to be a b o u t 0.8/~u, which is a b o u t twice the value r e p o r t e d p r e v i o u s l y [17]. H o w e v e r , it is c o n s i s t e n t with a r e c e n t experim e n t [18]. The result that the V m o m e n t is ver} small for V c o n c e n t r a t i o n s more than 50% i~ c o n s i s t e n t also with this e x p e r i m e n t . -
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F'ig. 5. The calculated number of electrons of Fe and ('o atom,, in Fe, ,Co,. The bars represent the r.ln.s, deviation from the average value given by the full or,rye.
J F_TT'[--I
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Fig. 6. '['he Fe and V momenls vs. the number ol V ~itOlllXill the nearest-neighbor shell in Fe-V alloys. Each curve is fol a given alloy concenlration which is indicated in the figurc for the Fe moment. Fhe parameteP, are asstllned Io be U,.,. 0.5, .I~,. :: 0.17761: U , 0.32. J, :: I).07 in pure Fe and V metuls are 7.1 and 4.0 per alolll, renpcctivcl!,. The model stale density function is differcm from that in fig. ~ (see text).
159 We note that the total cross-section for neutron magnetic diffuse scattering will be such that O" OC CACB(~L~ A -- ~,~B)2 -~- CA(A~/~A) 2 -~- CB(A~t£B) 2,
(4)
where the second and third terms represent the contribution of fluctuations in the magnitude of the magnetic moment due to the environment. For example, the fluctuation term for Fe atoms amounts to about one-quarter of the first term in Fe0.sV0.5. The width of the distribution of the Fe moment in F e - C o alloys seems to be much larger than that of the hyperfine field measured by the M6ssbauer experiment [19]. In the phenomenological analysis of the hyperfine field, one finds a contribution from the neighboring atoms which is proportional to their magnetic moment. Since an Fe atom with a larger magnetic m o m e n t is in a cobalt-rich environment with a smaller magnetic moment, the distribution of the hyperfine field might be narrower than that of the Fe moment. The distribution of the Fe moment in F e - V alloys seems to be consistent with a recent M6ssbauer measurement at least in the Fe-rich region [20]. Finally we note that the effect of second and more distant neighbors, which is often discussed in connexion with M6ssbauer experiments on dilute alloys, can be taken into account by a perturbation calculation in our approach. We defer further discussion to a future publication.
4. CPA for the muffin-tin potential model Several proposals have been made to apply the single-site C P A to more extended models which involve more than one kind of atomic states. Suppose the atomic states are the bases of the irreducible representations of the cubic point group. We may assume that the coherent potential is now a diagonal matrix with respect to the atomic states. As was discussed by Kirkpatrick et al. [21], we then obtain a set of simultaneous equations for determining the diagonal elements of the coherent potential. The fact that the diagonal elements are coupled with each other may be understood if one considers that one can make, for example, molecular orbitals of the de symmetry with respect to a lattice site from the d3, or s orbitals of surrounding atoms. An approximate decoupling of the equations by which one determines each c o m p o n e n t separately is assumed, for example,
in the calculation of the asphericity of the magnetic moment density in an alloy by Leoni and Sacchetti [7]. We do not intend here to deny the possibility that such a calculation is a reasonable interpolation between the two pure metals which constitute the alloy system. However, we need a check based on a C P A calculation carried out without such an approximation. For another example, an alloy between a transition metal and a nontransition metal such as FeAI, it is needless to say that random potentials must be introduced for s and p states as well as for d states. We have to consider the s band also in the discussion of the transport phenomena. In order to study these problems, we discuss the muffin-tin potential model of an alloy, which is a good starting point for a detailed description of the electronic structure. The C P A for the muffin-tin potential model, which assumes different muffin-tin potentials for atoms of different species, has been formulated by several people [22]. Practical calculation, however, has been thought to be difficult because of the amount of computation required. The calculation goes in parallel to that for the tight binding model. At each cycle of iterative determination of the coherent potential, or of a quantity equivalent to it such as the coherent t matrix, we have to calculate an integral over all states in the k space. Thus it seems that we have to carry out the band structure calculation of a pure metal described by the input trial coherent t matrix repeatedly until the final coherent t matrix is obtained. We propose here a new and more practicable method of computation. At the start, we replace the integral by a sum over several points in k space for which the Green function with an initial t matrix is calculated. Then we obtain an improved coherent t matrix from the C P A equation. We use this output to c~lculate the Green function at several additional points in k space, and include their contribution in the sum which approximates the integral. We then obtain an improved t matrix. The procedure is repeated to increase the number of k points to several thousand in ~ of the Brillouin zone. We find that the t matrix converges rapidly to its final value after 100 points. Thus the contribution of most points in the sum at the final stage is calculated with the final coherent t matrix; hence the in-
160
tegral is also calculated to good approximation with it. We may say, therefore, that the C P A equation is satisfied by the final coherent t matrix. In short, our idea is to carry out the calculation of the coherent t matrix in parallel to that of the k sum in the band structure calculation. In what follows we describe some preliminary applications of the above-mentioned method. We have applied the method to calculating the spin-dependent resistivity in fcc N i - F e and NiCr alloys [8]. In this calculation we reduce the muffin-tin potential model to an interpolation scheme model consisting of the tight binding hamiltonian of the d states and the free electron states. The tight binding electron transfer integrals are calculated in terms of the mixing of the d states with the free electron states of high energy. The s-d mixing elements for this tight binding part, as well as those for low-lying free electron states, are calculated by use of the pseudo-Greenian theory I23]. The muffin-tin potential was calculated with the Herman-Skillman atomic wave function. The 3d atomic levels, on the other hand, are taken from the H F - C P A calculation described in section 2: we fit the width of the model state density in H F C P A to the width of the d band in pure Ni metal calculated with the above-mentioned s-d mixing elements. With this choice of atomic energy levels the alloying effect on the total and local state densities in a given alloy in this calculation is quite similar to that in H F - C P A . We calculate the resistivity of electrons of each spin direction by use of the average Green function determined in the present C P A , following Velick}2"s treatment [24]. Fig. 7 shows the result. We note that the
resistivity of the minority-spin electrons in Ni~ +Fe+, P i , passes through a maximum around x - 0 . 3 . 1"he decrease of P i arises from the fact that the Fermi level slips off the high peak to go into the low state-density region in the minorityspin d band at around this value of x. We may say that our calculation of p+ represents the effect of the random atomic d levels at least semi-quantitatively. ()n the other hand. our calculation of P t is inadequate, remaining v e D small around x = 0.3 where the atomic d levcls of Fe and Ni become very close to each other. Since our theory neglects w~rious e f f e c t s - f o r example, the fluctuation of the levels duc It) the local environment e f f e c t - w e may expect that Or will increase with .v in a more improved calculation, though its wtlue is much smaller than p~. Thus the ratio p~/pl will begin to decrease around x~-0.25 where the bending over of the p~ vs. x curve starts, if Pt keeps a linear increase with x. This will be an exphmation for the observed rapid decrease of the spontaneous anisotropy of the resistivity for x > 0 . 2 5 which was ascribed to the transition from +'strong" ferromagnetism to the weak variety [25]. Our calculation of N i - C r alloys, on the other hand, seems to be in reasonable agreement with experiment. As another example in fig. 8 we show a preliminary calculation of the total state density of disordered Fe-AI alloys. Here we carry out the C P A in the K K R formalism. In this kind of problem, the choice or the self-consistent determination of a muffin-tin potential is a crucial I," Ry atom
FesAJe
2O 0 20C
5 I
. . . .
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'
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Gt % Cr --
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8
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Fig. 7. T h e r e s i s t i v i t i e s f o r t h e m a j o r i t y ( 1" ) a n d m i n o r i t y ,,pin e l e c t r o n s in N i , , F e , a n d N i , , C r , .
( 1, )
Fig. 8. T h e d e n s i t y of s t a t e s for e a c h s p i n d i r e c t i o n m Fe,,,Jxl,+:. T h e d o t t e d c u r v e is t h a l for p u r e b c c [:e metal+
161
p r o b l e m . T h e c a l c u l a t i o n s h o w n h e r e is o n l y p r e l i m i n a r y in this r e g a r d . W e u s e t h e m u f f i n - t i n p o t e n t i a l f o r p u r e AI c o n s t r u c t e d w i t h t h e H e r man-Skillman atomic wave function. We shift t h e A1 p o t e n t i a l b y a c o n s t a n t v a l u e to fit its c o n s t a n t p a r t (the m u f f i n - t i n p a n ) to t h a t o f the Fe potential taken from Wakoh and Yamashita's b a n d s t r u c t u r e c a l c u l a t i o n . O b v i o u s l y this r a i s e s t h e AI p o t e n t i a l t o o h i g h ; it r e s u l t s in t h e r e b e i n g o n l y 1.7 e l e c t r o n s w i t h i n t h e W i g n e r - S e i t z s p h e r e o f AI a t o m . In a n y e v e n t , o u r c a l c u l a t i o n s h o w s t h e n a r r o w i n g o f the d b a n d , a n d a d e c r e a s e o f the n u m b e r o f s t a t e s , w i t h i n c r e a s i n g AI c o n c e n t r a t i o n .
5. C o n c l u d i n g remarks W e h a v e p r e s e n t e d s e v e r a l e x a m p l e s o f the a p p l i c a t i o n o f C P A a n d its e x t e n s i o n to f e r r o m a g n e t i c t r a n s i t i o n - m e t a l a l l o y s . It is n o t i c e a b l e t h a t t h e m o r e w e p r e s s f o r fine d e t a i l o f t h e e l e c t r o n i c s t r u c t u r e , the m o r e c o m p l i c a t e d t h e required calculations inevitably become. Nevert h e l e s s w e b e l i e v e t h a t t h e C P A a p p r o a c h is a fruitful one for elucidating the mechanisms underlying a variety of alloying effects which the alloys exhibit.
References [I] P. Soven, Phys. Rev. 156 (1%7) 809. [2] H. Hasegawa and J. Kanamori, J. Phys. Soc. Jap. 31 (1971) 382; 33 (1972) 1599 and 1607. J. Kanamori, J. de Phys. 35 (1974) C4-131. [3] J. Inoue and M. Shimizu, J. Phys. Soc. Jap. 40 (1976) 1321. [4] T. Jo and H. Miwa, J. Phys. Soc. Jap. 40 (1976) 706.
[5] T. Jo, J. Phys. Soc. Jap. 40 (1976) 715; a paper presented at Int. Conf. Magnetism, Amsterdam, 1976. [6] F. Brouers, F. Gautier and J. Van der Rest, J. Phys. F5 (1975) 975. J. Van der Rest, F. Gautier and F. Brouers, J. Phys. F5 (1975) 995. [7] F. Leoni and F. Sacchetti, Nuova Cim. 21B (1974) 97. [8] H. Akai, a paper presented at Int. Conf. Magnetism, Amsterdam, 1976. [9] K. Terakura, this conference; J. Kanamori, Proc. Low Temp. Phys.-LTl3, Vol. 4 (Plenum, New York, 1972) p. 19. [10] S. Wakoh and J. Yamashita, J. Phys. Soc. Jap. 21 (1966) 1712. [11] H. Miwa, Progr. Theor. Phys. 52 (1974) 1. [12] F. Brouers, F. Ducastelle, F. Gautier and J. Van der Rest, J. Phys. F3 (1973) 2120. [13] B.G. Nickel and W.H. Butler, Phys. Rev. Len. 30 (1973) 373. W.H. Butler, Phys. Rev. 8 (1973) 4499. [14] D.I. Bardos, J. Appl. Phys. 40 (1969) 1371. [15] I. Vincze, I.A. Campbell and A.J. Meyer, Solid State Commun. 15 (1974) 1495. [16] R.S. Preston, D.J. Lam, M.V. Nevitt, D.O. Van Ostenburg and C.W. Kimball, Phys. Rev. 149 (1966) 440. [17] M.F. Collins and G.G. Low, Proc. Phys. Soc. 86 (1965) 535. [18] O. Yamashita, Y. Yamaguchi and H. Watanabe, J. Phys. Soc. Jap., to be published. [19] C.E. Johnson, M.S. Ridout and T.E. Cranshaw, Proc. Phys. Soc. 81 (1963) 1079. [20] M. Shiga, private communication. [21] S. Kirkpatrick, B. Velick~) and H. Ehrenreich, Phys. Rev. B1 (1970) 3250. [22] P. Soven, Phys. Rev. B2 (1970) 4715. H. Shiba, Progr. Theor. Phys. 46 (1971) 77. [23] J. Kanamori, K. Terakura and K. Yamada, Progr. Theor. Phys. 41 (1969) 1426; suppl, no. 46 (1970) 221. [24] B. Velick~,, Phys. Rev. 184 (1%9) 614. [25] I.A. Campbell, J. Phys. F4 (1974) LI81. [26] JH.M. Stoelinga and R. Gersdorf, Phys. Len. 19 (1966) 640. [27] J.C. Crangle and G.C. Hallam, Proc. Roy. Soc. A272 (1963) 119. [28] M.F. Collins and J.B. Forsyth, Phil. Mag. 8 (1963) 401.
J. K A N A M O R I , H. A K A I , N. H A M A D A and H. M I W A Department of Physics, Osaka University, Toyonaka 560, Japan
A review is given of certain m e t h o d s for calculating the electronic and magnetic structure of ferromagnetic transition-metal alloys based on the coherent potential approximation and its extension. Three subjects are discussed in detail. (1) The response of the alloys to an external magnetic field is calculated on the basis of the single-site C P A combined with the H a r t r e e - F o c k approximation for fcc Ni-Fe, bcc F e - N i and F e - C o alloys. (2) A method of treating the effect of the nearest-neighbor composition on the basis of an extension of C P A is discussed. The d e p e n d e n c e of the Fe atomic magnetic m o m e n t and other quantities on the nearest neighbor composition in bcc F e - C o and F e - V alloys is calculated as an example of the applications of this method. (3) A method of carrying out the single-site C P A for the muffin-tin potential model is proposed. Some preliminary results are presented.
I. Introduction
The coherent potential approximation (CPA), which was proposed by Soven [I] in 1967, was subsequently applied with success to ferromagnetic alloys of transition metals by Hasegawa and K a n a m o r i [2] to explain the dependence on composition of the average magnetization, the atomic m o m e n t of each constituent and the electronic specific heat. Since then, discussions of ferromagnetic alloys have evolved towards a more detailed study of their electronic and magnetic structure. We intend in this paper to give a perspective view of the present state of the C P A theory of the ferromagnetic state in disordered alloys, by presenting our recent calculations on various subjects. We assume a tight binding model of d states in the discussions given in sections 2 and 3. The inter-atomic electron transfer integrals, or in other words the off-diagonal elements in the site representation, are assumed to be independent of atomic species, while the diagonal elements representing the effective atomic levels are random variables, depending on atomic species and also on the e n v i r o n m e n t surrounding the atom under consideration. H a s e g a w a and K a n a m o r i determine the diagonal elements in such a way that the average number of d electrons of each spin direction in a given atom satisfies the Hart r e e - F o c k condition for the atom e m b e d d e d in an average medium described by the coherent potential; the coherent potential is determined in turn by C P A for the random system defined by the diagonal elements satisfying the H a r t r e e Fock condition. This approach, which we call Physica 91B (1977) 153-161 © North-Holland
H F - C P A , is summarized in section 2, where we present also a calculation of the effect of a magnetic field on the ferromagnetic state in alloys. The latter subject has been discussed on the basis of H F - C P A by Inoue and Shimizu for the case of N i - C u alloy [3]. We discuss fcc N i - F e , bcc F e - C o and F e - N i alloys for which the tight binding model is better justified. It will be shown that the deformation of the density of states due to scattering plays an important role in the response to an external field. Section 3 will describe our efforts to take into account the fluctuation of atomic composition on the nearest-neighbor shell of a given atom, which is neglected in H F - C P A . The fact that the magnetic state of an atom can depend sensitively on its local environment in a given alloy was discussed long before the proposal of C P A . The most remarkable environment effects in ferromagnetic alloys are: (1) the environment stabilizes one of the two magnetic states with its magnetic m o m e n t either parallel or anti-parallel to the bulk magnetization; (2) the magnitude of the atomic m o m e n t changes sensitively with the environment; and (3) the atom b e c o m e s either magnetic (with a "localized m o m e n t " ) or nonmagnetic, depending on its environment. Typical examples for behaviour type (1) are Mn and Fe in fcc Ni alloys, which have been studied by Jo and Miwa [4] and Jo [5]. We discuss in this section the state of the Fe atom in bcc Fe alloys as an example of type (2) behaviour. We shall show also that we obtain essentially the same result as that of H F - C P A when, for example, the Fe m o m e n t is averaged o v e r the environments. Before presenting the calculation,
154 we discuss our m e t h o d of treating the local e n v i r o n m e n t in some detail. T h e type (3) beh a v i o u r can be r e p r e s e n t e d by the Ni a t o m cluster in N i - C u alloys. B r o u e r s et al. [6] have studied the p a r a m a g n e t i c susceptibility of such a s y s t e m by use of a m e t h o d similar to ours. H o w e v e r , our t h e o r y will not be appropriate for direct application to type (3) b e h a v i o u r where the cluster plays an i m p o r t a n t role, since certain statistical c o n s i d e r a t i o n s are required for dealing with a sort of percolation problem that arises in this case. in section 4 we return to the single-site ( ' P A to discuss e x t e n s i o n s of the tight binding model which we need for a more detailed description of the electronic properties of alloys. E x a m p l e s of the p r o b l e m s are: the asphericity of the a t o m i c magnetic m o m e n t density [7]: the electrical resistivity [8]; alloys b e t w e e n transition and nontransition metals [9]; etc. T h o u g h several p r o p o s a l s have been made so far. they involve a s s u m p t i o n s to simplify the C P A calculation. To appraise these a t t e m p t s we need a C P A calculation for the muffin-tin potential model, which can be m a d e the basis of e v e r y simplified model. We describe here a m e t h o d for c a r r y i n g out this muffin-tin C P A . The electrical resistivity of f c c N i - F e and N i - C r alloys is then calculated by use of the method. Since this last-mentioned w o r k has been reported elsew h e r e , we give here only a s u p p l e m e n t a r y discussion of it. W e present also a preliminary calculation for Fe-A1 alloys. We c a n n o t , h o w e v e r , go on to a general appraisal of the simplified t r e a t m e n t s at present, since to do so we would need more calculations. 2. A s u m m a r y of H F - C P A
We describe first the tight binding model ass u m e d in the following calculations. Refinem e n t s of the model are discussed in section 4. We a s s u m e a binary alloy b e t w e e n metals A and B. T h e model hamiltonian is given by
~ = ~ tiai,aj** + ~ Ei.~ai~ai~,t i,i,*
{1)
i,s
w h e r e a~ is the annihilation o p e r a t o r of an electron with spin s in a d orbital of the a t o m i: t# is the t r a n s f e r integral, w h i c h is i n d e p e n d e n t of a t o m i c species; E~.~ is the effective atomic level of the a t o m i; we use the subscript i in a
dual sense, d e n o t i n g the lattice site and the species of the atom at the site. In H F - C P A Ei, takes either wdue, EA~ or E~,, being d e t e r m i n e d self-consistently to satisfy Ei,
-
E i
+4(Ui
JiJtli,
+ 5~Jitl i ,
SI&I4H,
(2}
where E, ( i - - A or B) is the atomic level m the a b s e n c e of the e l e c t r o n - e l e c t r o n interaction: t/, and J~ are the c o u l o m b and e x c h a n g e integrals, respectively: nA is the n u m b e r of electrons with spin s per d orbital of the atom i. The last term on the r.h.s, of eq. (2) is the Z e e m a n e n e r g y of an external field H with s + 1. The calculation is carried out in the following way. We start with initial values of n , and hence Ei~ ( i = A and B, s + and or I" and ,1,) to determine a c o h e r e n t potential o-(E) in C P A . The local state densities obtained t h e r e b y yield a new set of n~; which are the input for the s e c o n d cycle. The iteration is c o n t i n u e d until the difference b e t w e e n input and output value of each n~, b e c o m e s smaller than a preset limit: the limit is set b e t w e e n 10 ~and 10 4 d e p e n d i n g on the situation. W e c h e c k that the final solution does not depend on the choice of initial values of n~ by changing the choice. The result of the calculation d e p e n d s of c o u r s e on the choice of the parameters, E~, U~ and J~, the total n u m b e r of electrons and the d e n s i t y - o f - s t a t e s function of pure metal which constitute the input informations. Generally we aim at elucidating the mechanism underlying various alloying effects rather than fitting the calculation to experimental data. D e p e n d e n c e on the input i n f o r m a t i o n s will be discussed in the e x a m p l e s presented in this section and the next. We present a calculation of the r e s p o n s e to external field of fcc Ni~ ~Fe~ with x ~> 0.5, where the saturation magnetization deviates f r o m the linear increase with x. Fig. I s h o w s an e x a m p l e of the calculation. We adopt the same model state-density f u n c t i o n as that used in the previous calculation [2]. The choice of other p a r a m e t e r s is given in fig. I. We adjust them, in particular U and J, such thai the deviation from the linear increase starts at a value of x a r o u n d 0.5. S o m e of the majority spin ( T ) states !ie a b o v e the Fermi level when x e x c e e d s this critical value. We can see f r o m fig. I that an external field tends to restore the saturation magnetization to the S l a t e r - P a u l i n g curve. We
155
X (lO-4emu/mol)
M (IJ.e/otom) 0
......... ... H= 3xlOSOe
I
19
2 /
,
[ I
'",,,
20 i
,
A,O'""
",,,,
AVE~
I0 i
-
"~'",
'
. ...................
~
at%Ni
24
A
""'Q'"', .
2.5 "',,o
', O
A
E <
~
A
22
21
I
at % Fe
60
55
50
Fig. 1. The average magnetization per atom at H = 0 and H - 3 x 10~Oe (the dotted curves) and the "high field susceptibility" X in e.m.u./mol (the middle full curve) in Ni, ,Fe,. The full curves labelled Ni and Fe are the partial susceptibilities of Ni and Fe atoms, respectively, with the relation X-xx~:o+(l-x)xN,. The half band-width in pure metal is assumed to be 0.35 Ry. U,, = Uvo - 0.51 and JN, = JF~ = 0.19 are assumed, in units of the half band-width.
s h o w a l s o t h e s o - c a l l e d h i g h field s u s c e p t i b i l i t y , i.e. t h e d e r i v a t i v e w i t h r e s p e c t to t h e field s t r e n g t h H o f the a v e r a g e m a g n e t i z a t i o n at H = 0 a n d t h o s e o f the F e a n d N i a t o m i c m o m e n t s . W e find t h a t t h e field t e n d s to d e c r e a s e t h e n u m b e r o f e l e c t r o n s o n an F e a t o m a n d i n c r e a s e t h a t o n a N i a t o m . T h i s is d u e to a d e f o r m a t i o n of the local state densities of the minority-spin b a n d c a u s e d b y an i n c r e a s e o f t h e d i f f e r e n c e E r e ~ - E N ~ . In f a c t w e find an o p p o s i t e t e n d e n c y if w e k e e p t h e l o c a l s t a t e d e n s i t i e s rigid. Thus, we may conclude that the mechanism of t h e r e s p o n s e to an e x t e r n a l field is q u i t e diffe r e n t f r o m t h a t o b t a i n i n g in t h e rigid b a n d discussion. Fig. 2 s h o w s o u r c a l c u l a t i o n o f t h e field e f f e c t in t h e c a s e o f b c c F e - C o a n d F e - N i a l l o y s . T h e d i f f e r e n c e in t h e c o m p o s i t i o n d e p e n d e n c e b e t w e e n t h e s e t w o a l l o y s w a s d i s c u s s e d in d e t a i l p r e v i o u s l y [2]. W e find, h o w e v e r , t h a t t h e s t a t e d e n s i t y f u n c t i o n a d o p t e d t h e r e is i n a d e q u a t e f o r d i s c u s s i o n o f t h e field e f f e c t , b e i n g o v e r s i m p l i fied. W e a s s u m e h e r e a m o r e f a i t h f u l r e p r e s e n tation of Wakoh and Yamashita's calculation of t h e s t a t e d e n s i t y in b c c F e [10], s h o w n in fig. 3. There are two important features of the state d e n s i t y w h i c h d e t e r m i n e t h e a l l o y i n g e f f e c t in t h e s e a l l o y s . O n e is t h e v a l l e y s e p a r a t i n g t h e t w o m a i n p e a k s , a n d a n o t h e r is t h e s h o u l d e r o f
0
20
40
60
ot % Co
Fig. 2. The high field s u s c e p t i b i l i t y (full c u r v e s ) and a v e r a g e
magnetization at H = 0 (dotted curves) of bcc Fe-Co and Fe-Ni. The curves A and B are for Fe-Co and Fe-Ni. respectively. The open and filled triangles are the experimental X of Fe-Co and Fe-Ni [26], respectively; we have subtracted 1.19x 10 4 e.m.u./mol from the data to take account of the orbital and s band contributions. The open and filled circles are the data for the average magnetization of Fe-Co [14] and Fe-Ni [27], respectively. The half bandwidth of pure metal is assumed to be 0.4 Ry. This calculation is based on the single band model [21 with Uj,~ = 1.388, U . , = Uc,, 1.88. =
r 0,
I
,, O)
-----
....0
'
"
Fig. 3. (a) The model state density function for the d band of a bcc pure metal; (b) the local state densities of an Fe atom in bcc Fe09Co,,, with the nearest-neighbor shell specified by the number 0, 4 and 8 of Co atoms as indicated in the figure. The vertical line at E = 0 is the Fermi level. The parameters are chosen to be U,~o=0.6, JF~=0.19688, Uc,,=0.8, Jco= 0.27 and Evo E,.,, = 1.87718 in units of the half-width of the model state density.
t h e u p p e r p e a k at w h i c h t h e F e r m i l e v e l is s i t u a t e d in t h e m a j o r i t y - s p i n b a n d o f F e . T h e l a t t e r w a s n e g l e c t e d in t h e p r e v i o u s d i s c u s s i o n , w h i c h h a s to a d j u s t t h e d e p t h o f t h e v a l l e y to s o m e e x t e n t to o b t a i n a g r e e m e n t w i t h e x p e r i ment. Here we obtain good agreement with experiment for the derivative of the saturation m a g n e t i z a t i o n as w e l l t h o s e q u a n t i t i e s w h i c h were discussed previously. The explanation for the difference between the two alloys, however, is t h e s a m e as t h e p r e v i o u s o n e , i.e. it a r i s e s
156 f r o m the fact that the difference between the atomic e n e r g y levels of c o n s t i t u e n t a t o m s is larger in F e - N i than in F e - C o . In the latter alloy the shoulder persists b e c a u s e of the smallness of Ev~,~ -Ec<, T. Incidentally, the p r e s e n c e of this shoulder does not d e p e n d on the choice of the muffin-tin potential on which the band structure calculation is based, it arises from :+L difference of the peak position b e t w e e n de and d y states. Deviation from the charge neutrality of each c o n s t i t u e n t atom in alloy can be calculaled in C P A . It turns out that the quantity can be sensitive to the choice of the p a r a m e t e r s in some cases. We shall discuss an e x a m p l e in the next section. 3. Calculation of the local e n v i r o n m e n t effect
3. 1. M e t h o d o f c a l c u l a t i o n
The following discussion is based on the t h e o r y d e v e l o p e d by M i w a [ l l ] . C P A calculates the local electronic structure of a given atom e m b e d d e d in the m e d i u m described by the coherent potential. This idea has been e x t e n d e d to a (z + l)-atom cluster consisting a central a t o m and its z nearest neighbors e m b e d d e d in an a v e r a g e medium. It is a s s u m e d that the a v e r a g e m e d i u m is described by a site-diagonal c o h e r e n i potential o'(E) for each spin direction, e denoting the e n e r g y variable. The e n v i r o n m e n t of tin a t o m is c h a r a c t e r i z e d by the n u m b e r of A and B atoms, N A and Nu ( N a + Nu = =I on the nearestneighbor shell and their geometrical arrangement, and also by the a v e r a g e alloy concentration specifying the a v e r a g e medium. After verifying by calculation that the difference in the geometrical a r r a n g e m e n t has a relatively small effect, Miwa has p r o p o s e d a simplified t r e a t m e n t of the e n v i r o n m e n t in which the n e a r e s t - n e i g h b o r c o m p o s i t i o n is specified by N,x and NR only. An equivalent t h e o r y has been p r o p o s e d i n d e p e n d e n t l y by B r o u e r s et al. [121. With M i w a ' s simplification we easily obtain the G r e e n f u n c t i o n for the cluster of a given central a t o m labelled C and its nearest neighbors. The site-diagonal matrix element of the G r e e n function at the central site, for example, is given by
Gl~)(e" N.O = Li ,I
t'-
[ N.d(L~ ~
.V+#)I÷I '
N ~d(Lu i
where 1,, (i = A , B , C ) is the locater (~ t:'~,) J for lhe atom i, and wc o n l i l the spin subscript t'or simplicil,,: : / and .J, respectively, :+ire the diagonal :+ind off-diagonal elements of ihc indirecl lransfer through the nverLige Inediunl tween the ~ltOnlS on lhe n e a r e s t - n e i g h b o r shell, and are f u n c t i o n s of E ~rt~:l. t is the transfer
integral between a pair of nearest-neighboring atoms. Strictly speaking, we derive cq. ~3) for the lighl binding model in which the e l e c h o n transfer is confined io the nearest-neighbor site. H o w e v e r , expressing :i and .]~ in terms of lhc site-diagonal clement of the Green function in the :+tverage medium, and tlSillg a model ~,lale-
density ftinclion in the calculation of the (ireen function, we can show that (;[M~given by eq. (3! r e d u c e s to the Green function with the a s s u m e d model state-density function in the limit of a pure metal. In other words, eq. (3) can interpolate b e t w e e n two pure metals with tin arbitrary model state density, in any case eq. 13)is :+t reasonable interpohition, since the slate densily in a real transition metal does not deviate m u c h from thai of the n e a r e s t - n e i g h b o r transfer model. The vahie of t is d e t e r m i n e d by the condition that the s e c o n d m o m e n t of the adopted model state density coincides with that of the n e a r e s t - n e i g h b o r transfer model, i.e. zt ~. Miwa, and also Brouers el al., determine the c o h e r e n t potential erie) by the so-called central site a p p r o x i m a t i o n , which equates the configurational a v e r a g e of (7~x~(E:N, 0 o v e r the cluster c o m p o s i t i o n s to the site-diagonal element in the average medium. However, careful examinations, particularly for o n e - d i m e n s i o n a l cases, carried out by Nickel and Butler and by Butler 113l have revealed that the a p p r o x i m a t i o n often gives non-analytic solutions for o-(Et, especially in the case of split hands. A simihtr i n a d e q u a c y has also been f o u n d in our numerical calculations for three-dimensional cases. In the following calculation we take the condition that the configurational a v e r a g e of the trace of the Green function taken on the n e a r e s t - n e i g h b o r shell divided by = is equal io the site-diagonal element of the Green function in the a v e r a g e medium. We obtain essentially the same result if we extend the trace to the
.'i÷,~ll
t)
'J-
(3)
157
c l u s t e r i n c l u d i n g the c e n t r a l site. T h i s app r o x i m a t i o n is r e m i n i s c e n t of the " b o u n d a r y site a p p r o x i m a t i o n " d i s c u s s e d b y B u t l e r [13] f o r the o n e - d i m e n s i o n a l c a s e . O u r n u m e r i c a l calc u l a t i o n with this a p p r o x i m a t i o n has w o r k e d f o r all p a r a m e t e r s a n d c o n c e n t r a t i o n s we h a v e chosen. W e d e t e r m i n e the s p i n - d e p e n d e n t a t o m i c e n e r g y level E;~ b y eq. (2). S i n c e ni~ n o w v a r i e s a c c o r d i n g to the local e n v i r o n m e n t , E;s has a distribution even for a given species of atom. W e d e n o t e b y n~:(NA) the n u m b e r of e l e c t r o n s with spin s p e r d o r b i t a l of the c e n t r a l a t o m C ( = A o r B) h a v i n g NA n e a r e s t n e i g h b o r i n g A a t o m s ; c o r r e s p o n d i n g l y , the a t o m i c levels are d e n o t e d b y EC(NA). In the n u m e r i c a l c a l c u l a t i o n we a s s u m e initial v a l u e s for nC(NA) f o r C = A, B a n d N A = 0,1 . . . . . z a n d h e n c e Ec~'(NA). W e t a k e into a c c o u n t the d i s t r i b u t i o n of LA ~ on the n e a r e s t n e i g h b o r shell b y r e p l a c i n g I/(LA ~ - oW+ 5-) in eq. (3) b y its a v e r a g e with a p p r o p r i a t e conditional probabilities over those A atoms w h i c h h a v e the a t o m C at a given site on their n e a r e s t n e i g h b o r shells; the hAs of t h e s e A a t o m s is given b y the a s s u m e d Nff(NA). T h e s a m e a p p l i e s to 1 / ( L B ~ - 5 ~ + 5.). W e then d e t e r m i n e a c o h e r e n t p o t e n t i a l or(E) to m e e t the " b o u n d a r y s i t e " c o n d i t i o n m e n t i o n e d a b o v e . By i n t e g r a t i n g the local s t a t e d e n s i t y f o r e a c h C, NA a n d s to the F e r m i e n e r g y we o b t a i n a n e w set o f nC,'(NA) w h i c h s t a r t s the s e c o n d c y c l e . T h e i t e r a t i o n is c o n t i n u e d until the d i f f e r e n c e bet w e e n i n p u t a n d o u t p u t f o r e a c h nC(NA) bec o m e s s m a l l e r than 10 -3 a n d that f o r the a v e r a g e o f n~'(NA) s m a l l e r than 10 4. It s h o u l d be n o t e d that the use o f the a v e r a g e with the c o n d i t i o n a l p r o b a b i l i t y m e n t i o n e d a b o v e c o r r e s p o n d s in the dilute limit to the i m p o s i t i o n of the H a r t r e e - F o c k c o n d i t i o n , eq. (2), n o t o n l y f o r the c e n t r a l a t o m b u t also for the n e a r e s t n e i g h b o r s in the c l u s t e r . T h e m e t h o d c a n be a p p l i e d to a l l o y s with a t o m i c s h o r t r a n g e o r d e r , if o n e e m p l o y s the c o r r e c t p r o b a b i l i t y f o r e a c h c o n f i g u r a t i o n of the c l u s t e r . U n f o r t u n a t e l y , the a t o m i c s h o r t r a n g e o r d e r is not o f t e n c h e c k e d in the s a m p l e s of alloys used for magnetic measurement, and we h a v e little e x p e r i m e n t a l d a t a on its influence. A p r e l i m i n a r y c a l c u l a t i o n on F e - V a l l o y s s h o w s t h a t the a t o m i c s h o r t r a n g e o r d e r has an app r e c i a b l e effect on the a v e r a g e m a g n e t i z a t i o n as well as on the d i s t r i b u t i o n o f the m a g n e t i c m o -
ment of constituent atoms. We defer discussion of this p r o b l e m to a f u t u r e p u b l i c a t i o n .
3.2. Magnetic moment of the Fe atom in bcc alloys It is well e s t a b l i s h e d that iron as a b c c m e t a l e x h i b i t s u n s a t u r a t e d f e r r o m a g n e t i s m ; that is to s a y it has h o l e s in b o t h the m a j o r i t y a n d the m i n o r i t y spin d b a n d s . T h i s a l l o w s the F e a t o m to c h a n g e the m a g n i t u d e of its m a g n e t i c m o m e n t e a s i l y w i t h o u t an a p p r e c i a b l e c h a n g e o f the n u m b e r of d e l e c t r o n s , in c o n t r a s t to the c a s e s of Co, F e a n d Mn a t o m s in f c c Ni a l l o y s . In fig. 3 we s h o w e x a m p l e s of the local s t a t e d e n s i t y of the F e a t o m with v a r i o u s e n v i r o n m e n t s . Fig. 4 s u m m a r i z e s o u r c a l c u l a t i o n on F e - C o a l l o y s . T h e c h a n g e o f the F e a t o m i c m o m e n t with the c o m p o s i t i o n o f the n e a r e s t - n e i g h b o r shell is larger in i r o n - r i c h a l l o y s a n d it is also l a r g e r with i r o n - r i c h n e a r e s t - n e i g h b o r c o m p o s i t i o n s at a given a l l o y c o n c e n t r a t i o n . T h e c o n f i g u r a t i o n a l a v e r a g e s of the F e a n d C o m o m e n t s a n d also the a v e r a g e m a g n e t i z a t i o n a g r e e well with the H F - C P A r e s u l t m e n t i o n e d in s e c t i o n 2. T h e
3.
2.
0
,2
.4
.6
Xco
Fig. 4. The Fe and Co atomic moments as function of x in Fe, xCo~. The upper group of dotted curves is the result of calculating the Fe moment for a given number of Co nearest neighbors which increases from zero for the bottom curve to 8 for the top curve; the lower group of dotted curves is the one for the Co moment with a number of Co nearest neighbors which increases from the bottom to the top. The full lines are the calculated average Fe and Co moments, with vertical bars representing the r.m.s, deviation of the moment from its average value. The curve labelled Av represents the calculated average magnetization per atom. Triangles show the neutron diffuse scattering data for Fe and Co moments [28]. Circles display the experimental average magnetization data of the disordered alloy; the crosses are those for the ordered alloys [14].
158
width of the d i s t r i b u t i o n of the Fe m o m e n l , i.e. the r o o t - m e a n - s q u a r e of the d e v i a t i o n from the a v e r a g e value, b e c o m e s s m a l l e r as the Co conc e n t r a t i o n i n c r e a s e s in spite of the i n c r e a s e of the f l u c t u a t i o n of the a t o m i c c o m p o s i t i o n in the n e a r e s t - n e i g h b o r shell. In the c o b a l t - r i c h region most of the slates in the m a j o r i t y spin b a n d are below the F e r m i level, so that the m a g n i t u d e of the m a g n e t i c m o m e n t c a n n o l c h a n g e w i t h o u t violating local charge n e u t r a l i t y . In the c a l c u l a t i o n s h o w n in fig. 4, the n u m b e r s of e l e c t r o n s per atom in pure iron and pure cobalt are a s s u m e d to be 7.1 and 8.2 respectively. T h e p a r a m e t e r Ek, Eu, is c h o s e n so lhat the F e r m i level of f e r r o m a g n e t i c pure iron coincides with that of n o n m a g n e t i c pure cobalt. The r e m a i n i n g p a r a m e t e r s , U and J for Fe and Co a t o m s , are c h o s e n to fit a p p r o x i m a t e l y our calc u l a t i o n to the e x p e r i m e n t a l data on the a v e r a g e m a g n e t i z a t i o n and a v e r a g e atomic m o m e n t s . T h e d e v i a t i o n s from these data, which can be seen from fig. 4, can be made s m a l l e r if we a d j u s t the p a r a m e t e r s more carefully. In particular, the i n c r e a s e of the a v e r a g e magn e t i z a t i o n o b s e r v e d in o r d e r e d F%~Co,~, 0.07 ~ per atom [14], which is m u c h larger than the c a l c u l a t e d value, can be e x p l a i n e d if the n u n > bets of e l e c t r o n s per a t o m in pure iron and pure coball are taken to be 7.0 and 8.15 r e s p e c t i v e l y , with a s m a l l e r value for Uc,,. Since we neglect v a r i o u s c o n t r i b u t i o n s such as that of the s b a n d . an e x a c t fit to the e x p e r i m e n t a l data would be withou! significance. Fig. 5 s h o w s the a v e r a g e n u m b e r of e l e c t r o n s at Fe and Co a t o m s , together with the width of the d i s t r i b u t i o n . The n u m b e r of e l e c t r o n s at a Fe a t o m is larger with iron-rich n e a r e s t n e i g h b o r c o m p o s i t i o n s , while its a v e r a g e i n c r e a s e s with the Co c o n c e n t r a t i o n . T h e latter b e h a v i o r is c o n s i s t e n t with the i s o m e r shift data of the M 6 s s b a u e r e x p e r i m e n t [15]. We note, h o w e v e r , that the d i r e c t i o n of the charge t r a n s f e r b e t w e e n
Fe and Co a t o m s is rather s e n s i t i v e to the choice of /£r~. K~.,,; if we a d o p t a value of E f t - E u , , larger by 0.05 in units of the halfwidth of the model state d e n s i t y , the d i r e c t i o n of the charge t r a n s f e r is r e v e r s e d . We show a similar c a l c u l a t i o n on F e - V allo~s in fig. 6. The Fe atomic m o m e n t c h a n g e s more s e n s i t i v e l y in this case than in F e - C o alloys. For e x a m p l e , the m a g n e t i c m o m e n t of an Fe atom in Fe~V<,~ varies from 2.5/J., (for Nr,. 8) to a b o u t zero ( N r + = 0 ) with the a v e r a g e value 1.15#E+ and with 0.5 >+ for the r.m.s, of the deviation+ In this c a l c u l a t i o n we used the model state d e n s i t y a s s u m e d by H a s e g a w a and K a n a m o r i : the fact that it is oversimplified will not affect the result a p p r e c i a b l y . We d e t e r m i n e E~. Er~. by the same p r o c e d u r e as in the case of F e - C o alloys. T h e a v e r a g e n u m b e r of e l e c t r o n s at an Fe atom t u r n s out to i n c r e a s e with the V c o n c e n t r a t i o n . which s e e m s to be i n c o n s i s t e n t with the isomer shift data [16]. H o w e v e r , we can r e v e r s e the d i r e c t i o n of the charge t r a n s f e r b e t w e e n Fe and V a t o m s by a d o p t i n g a slightly larger value of h,', E~.~. The n u m b e r of e l e c t r o n s at an Fe atom t u r n s out to be i n s e n s i t i v e to lhe n e a r e s t - n e i g h bor c o m p o s i t i o n in the p r e s e n t c a l c u l a t i o n . The a v e r a g e a t o m i c m o m e n t of V as an i m p u r i t y in iron is c a l c u l a t e d to be a b o u t 0.8/~u, which is a b o u t twice the value r e p o r t e d p r e v i o u s l y [17]. H o w e v e r , it is c o n s i s t e n t with a r e c e n t experim e n t [18]. The result that the V m o m e n t is ver} small for V c o n c e n t r a t i o n s more than 50% i~ c o n s i s t e n t also with this e x p e r i m e n t . -
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F'ig. 5. The calculated number of electrons of Fe and ('o atom,, in Fe, ,Co,. The bars represent the r.ln.s, deviation from the average value given by the full or,rye.
J F_TT'[--I
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~.
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Fig. 6. '['he Fe and V momenls vs. the number ol V ~itOlllXill the nearest-neighbor shell in Fe-V alloys. Each curve is fol a given alloy concenlration which is indicated in the figurc for the Fe moment. Fhe parameteP, are asstllned Io be U,.,. 0.5, .I~,. :: 0.17761: U , 0.32. J, :: I).07 in pure Fe and V metuls are 7.1 and 4.0 per alolll, renpcctivcl!,. The model stale density function is differcm from that in fig. ~ (see text).
159 We note that the total cross-section for neutron magnetic diffuse scattering will be such that O" OC CACB(~L~ A -- ~,~B)2 -~- CA(A~/~A) 2 -~- CB(A~t£B) 2,
(4)
where the second and third terms represent the contribution of fluctuations in the magnitude of the magnetic moment due to the environment. For example, the fluctuation term for Fe atoms amounts to about one-quarter of the first term in Fe0.sV0.5. The width of the distribution of the Fe moment in F e - C o alloys seems to be much larger than that of the hyperfine field measured by the M6ssbauer experiment [19]. In the phenomenological analysis of the hyperfine field, one finds a contribution from the neighboring atoms which is proportional to their magnetic moment. Since an Fe atom with a larger magnetic m o m e n t is in a cobalt-rich environment with a smaller magnetic moment, the distribution of the hyperfine field might be narrower than that of the Fe moment. The distribution of the Fe moment in F e - V alloys seems to be consistent with a recent M6ssbauer measurement at least in the Fe-rich region [20]. Finally we note that the effect of second and more distant neighbors, which is often discussed in connexion with M6ssbauer experiments on dilute alloys, can be taken into account by a perturbation calculation in our approach. We defer further discussion to a future publication.
4. CPA for the muffin-tin potential model Several proposals have been made to apply the single-site C P A to more extended models which involve more than one kind of atomic states. Suppose the atomic states are the bases of the irreducible representations of the cubic point group. We may assume that the coherent potential is now a diagonal matrix with respect to the atomic states. As was discussed by Kirkpatrick et al. [21], we then obtain a set of simultaneous equations for determining the diagonal elements of the coherent potential. The fact that the diagonal elements are coupled with each other may be understood if one considers that one can make, for example, molecular orbitals of the de symmetry with respect to a lattice site from the d3, or s orbitals of surrounding atoms. An approximate decoupling of the equations by which one determines each c o m p o n e n t separately is assumed, for example,
in the calculation of the asphericity of the magnetic moment density in an alloy by Leoni and Sacchetti [7]. We do not intend here to deny the possibility that such a calculation is a reasonable interpolation between the two pure metals which constitute the alloy system. However, we need a check based on a C P A calculation carried out without such an approximation. For another example, an alloy between a transition metal and a nontransition metal such as FeAI, it is needless to say that random potentials must be introduced for s and p states as well as for d states. We have to consider the s band also in the discussion of the transport phenomena. In order to study these problems, we discuss the muffin-tin potential model of an alloy, which is a good starting point for a detailed description of the electronic structure. The C P A for the muffin-tin potential model, which assumes different muffin-tin potentials for atoms of different species, has been formulated by several people [22]. Practical calculation, however, has been thought to be difficult because of the amount of computation required. The calculation goes in parallel to that for the tight binding model. At each cycle of iterative determination of the coherent potential, or of a quantity equivalent to it such as the coherent t matrix, we have to calculate an integral over all states in the k space. Thus it seems that we have to carry out the band structure calculation of a pure metal described by the input trial coherent t matrix repeatedly until the final coherent t matrix is obtained. We propose here a new and more practicable method of computation. At the start, we replace the integral by a sum over several points in k space for which the Green function with an initial t matrix is calculated. Then we obtain an improved coherent t matrix from the C P A equation. We use this output to c~lculate the Green function at several additional points in k space, and include their contribution in the sum which approximates the integral. We then obtain an improved t matrix. The procedure is repeated to increase the number of k points to several thousand in ~ of the Brillouin zone. We find that the t matrix converges rapidly to its final value after 100 points. Thus the contribution of most points in the sum at the final stage is calculated with the final coherent t matrix; hence the in-
160
tegral is also calculated to good approximation with it. We may say, therefore, that the C P A equation is satisfied by the final coherent t matrix. In short, our idea is to carry out the calculation of the coherent t matrix in parallel to that of the k sum in the band structure calculation. In what follows we describe some preliminary applications of the above-mentioned method. We have applied the method to calculating the spin-dependent resistivity in fcc N i - F e and NiCr alloys [8]. In this calculation we reduce the muffin-tin potential model to an interpolation scheme model consisting of the tight binding hamiltonian of the d states and the free electron states. The tight binding electron transfer integrals are calculated in terms of the mixing of the d states with the free electron states of high energy. The s-d mixing elements for this tight binding part, as well as those for low-lying free electron states, are calculated by use of the pseudo-Greenian theory I23]. The muffin-tin potential was calculated with the Herman-Skillman atomic wave function. The 3d atomic levels, on the other hand, are taken from the H F - C P A calculation described in section 2: we fit the width of the model state density in H F C P A to the width of the d band in pure Ni metal calculated with the above-mentioned s-d mixing elements. With this choice of atomic energy levels the alloying effect on the total and local state densities in a given alloy in this calculation is quite similar to that in H F - C P A . We calculate the resistivity of electrons of each spin direction by use of the average Green function determined in the present C P A , following Velick}2"s treatment [24]. Fig. 7 shows the result. We note that the
resistivity of the minority-spin electrons in Ni~ +Fe+, P i , passes through a maximum around x - 0 . 3 . 1"he decrease of P i arises from the fact that the Fermi level slips off the high peak to go into the low state-density region in the minorityspin d band at around this value of x. We may say that our calculation of p+ represents the effect of the random atomic d levels at least semi-quantitatively. ()n the other hand. our calculation of P t is inadequate, remaining v e D small around x = 0.3 where the atomic d levcls of Fe and Ni become very close to each other. Since our theory neglects w~rious e f f e c t s - f o r example, the fluctuation of the levels duc It) the local environment e f f e c t - w e may expect that Or will increase with .v in a more improved calculation, though its wtlue is much smaller than p~. Thus the ratio p~/pl will begin to decrease around x~-0.25 where the bending over of the p~ vs. x curve starts, if Pt keeps a linear increase with x. This will be an exphmation for the observed rapid decrease of the spontaneous anisotropy of the resistivity for x > 0 . 2 5 which was ascribed to the transition from +'strong" ferromagnetism to the weak variety [25]. Our calculation of N i - C r alloys, on the other hand, seems to be in reasonable agreement with experiment. As another example in fig. 8 we show a preliminary calculation of the total state density of disordered Fe-AI alloys. Here we carry out the C P A in the K K R formalism. In this kind of problem, the choice or the self-consistent determination of a muffin-tin potential is a crucial I," Ry atom
FesAJe
2O 0 20C
5 I
. . . .
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'
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Gt % Cr --
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Fig. 7. T h e r e s i s t i v i t i e s f o r t h e m a j o r i t y ( 1" ) a n d m i n o r i t y ,,pin e l e c t r o n s in N i , , F e , a n d N i , , C r , .
( 1, )
Fig. 8. T h e d e n s i t y of s t a t e s for e a c h s p i n d i r e c t i o n m Fe,,,Jxl,+:. T h e d o t t e d c u r v e is t h a l for p u r e b c c [:e metal+
161
p r o b l e m . T h e c a l c u l a t i o n s h o w n h e r e is o n l y p r e l i m i n a r y in this r e g a r d . W e u s e t h e m u f f i n - t i n p o t e n t i a l f o r p u r e AI c o n s t r u c t e d w i t h t h e H e r man-Skillman atomic wave function. We shift t h e A1 p o t e n t i a l b y a c o n s t a n t v a l u e to fit its c o n s t a n t p a r t (the m u f f i n - t i n p a n ) to t h a t o f the Fe potential taken from Wakoh and Yamashita's b a n d s t r u c t u r e c a l c u l a t i o n . O b v i o u s l y this r a i s e s t h e AI p o t e n t i a l t o o h i g h ; it r e s u l t s in t h e r e b e i n g o n l y 1.7 e l e c t r o n s w i t h i n t h e W i g n e r - S e i t z s p h e r e o f AI a t o m . In a n y e v e n t , o u r c a l c u l a t i o n s h o w s t h e n a r r o w i n g o f the d b a n d , a n d a d e c r e a s e o f the n u m b e r o f s t a t e s , w i t h i n c r e a s i n g AI c o n c e n t r a t i o n .
5. C o n c l u d i n g remarks W e h a v e p r e s e n t e d s e v e r a l e x a m p l e s o f the a p p l i c a t i o n o f C P A a n d its e x t e n s i o n to f e r r o m a g n e t i c t r a n s i t i o n - m e t a l a l l o y s . It is n o t i c e a b l e t h a t t h e m o r e w e p r e s s f o r fine d e t a i l o f t h e e l e c t r o n i c s t r u c t u r e , the m o r e c o m p l i c a t e d t h e required calculations inevitably become. Nevert h e l e s s w e b e l i e v e t h a t t h e C P A a p p r o a c h is a fruitful one for elucidating the mechanisms underlying a variety of alloying effects which the alloys exhibit.
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