THE COHERENT POTENTIAL APPROXIMATION AND FERROMAGNETISM

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THE COHERENT POTENTIAL APPROXIMATION AND FERROMAGNETISM

J. Kanamori

To cite this version:

J. Kanamori. THE COHERENT POTENTIAL APPROXIMATION AND FERROMAGNETISM.

Journal de Physique Colloques, 1974, 35 (C4), pp.C4-131-C4-139. �10.1051/jphyscol:1974423�. �jpa- 00215614�

JOURNAL DE PHYSIQUE CoZloque C4, supplkment au no 5 , Tome 35, Mai 1974, page C4-13 1

THE COHERENT POTENTIAL APPROXIMATION AND FERROMAGNETISM

J. KANAMORI

Department of Physics, Faculty of Science, Osaka University Toyonaka, Osaka 560, Japan

RBsumB. - Les applications de l'approximation du potentiel coherent dans les cas des alliages ferromagnetiques desordonnes des mktaux de transition sont passees en revue dans le contexte des developpements theoriques recents. L'approximation du potentiel cohkrent qui utilise I'approxima- tion de Hartree-Fock pour determiner le potentiel de chaque atome peut elucider la variation en fonction de la concentration du moment magnbtique atomique et de Ia chaleur spkcifique electronique dans les cas oh 1'6tat magnetique de chaque atome est bien defini. Dans les cas des alliages tels que Nil-,Mn, et N ~ I - ~ F ~ ~ on doit considher une extension de I'approximation pour tenir compte de I'existence de deux etats magnktiques de I'atome et aussi de I'effet de la configuration des atomes les plus proches a 1'6tat magnetique de l'atome central. Les extensions de la mkthode Hartree-Fock-CPA qui abordent ces problkmes sont discutees.

Abstract. - The applications of the coherent potential approximation to ferromagnetic disor- dered alloys of transition metals are reviewed in the light of recent theoretical developments. The CPA calculation which utilizes the Hartree-Fock approximation for determining the potential of each atom can elucidate the variation of the atomic magnetic moment and the electronic specific heat with concentration in the cases where the magnetic state of each atom is well defined. Alloys such as Nil-,Mnx and Nil-,Fez demand an extension of the approach to take into account the existence of two magnetic states of an atom and also the effect of the nearest neighbour configura- tion on the magnetic state of a given atom. Possible extensions of the Hartree-Fock-CPA method to meet such cases are discussed.

1. Introduction. - The ferromagnetic state in disordered transition metal alloys exhibits a variety of the alloying effects. Friedel [l] elucidated in 1958 the underlying mechanism in the case of dilute alloys ; his argument is a general one, being based on a deep insight into the effect of the impurity potential on the electronic structure of a system where a substitutional impurity atom is embedded in an otherwise pure ferromagnetic metal. Kanamori [2] formulated in 1964 Friedel's theory on a tight binding model, intro- ducing in the problem the concept of a spin dependent impurity potential which was derived by use of the Hartree-Fock approximation for the intra-atomic coulomb interaction between electrons of antiparallel spins. Making use of the coherent potential approxi- mation (CPA) proposed by Soven [3] and combinding it to the Hartree-Fock approximation (HF) with the tight binding model, Kanamori and Hasegawa [4]

extended the discussion to concentrated alloys in 1970.

Since then calculations based on this HF-CPA approach have been carried out for various ferro- magnetic binary and ternary alloys [5-71.

The purpose of the present paper is to discuss the capability of the HF-CPA calculation and to explore possible extensions. CPA is a single site approximation which replaces the atomic potentials of atoms sur- rounding a given atom in an alloy by the coherent potential independent of the species of neighbouring atoms. Thus the electronic structure of a given atom in

CPA represents the result of taking a self-consistent average over the configurations of neighbouring atoms. It has been discussed, however, by many people that the magnetic state of a given atom can be quite sensitive to the composition of its nearest neighbours [8]. Cable and Child [9] established an experimental evidence for the effect in ferromagnetic NiMn alloys. Apart from the general problem of the capability of CPA of describing electronic states in a disordered alloy, the near neighbour effect on the magnetic state poses a limit on the applicability of CPA to ferromagnetic alloys. This problem is the central theme of the present paper.

Since CPA corresponds to an extension of the impurity theory, we discuss in the next section the Hartree-Fock solution of the impurity problem as well as the HF-CPA approach. After appraising the practical applications of the latter approach in sec- tion 3, we discuss possible extensions in section 4.

The discussions given in section 2 and 4 are based largely on recent studies made by Miwa [lo]. Sup- plementary discussions are given in section 5.

2. The HF solutions. - In the following 'discus- sions we assume the tight binding model defined by the Hamiltonian,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974423

C4-132 3. KANAMORI where the first term represents the electron transfer

energy which is assumed to be independent of the species of atoms at the sites i and j ; ci of the second term is the spin independent atomic potential, being equal to either E, or E, in the case of a binary alloy A,B,-, ; Ui is the intra-atomic coulomb integral which may depend on the species of atom at the site i, being equal to either UA or U,. This single band Hamiltonian yields almost the same result as the degenerate band Hamiltonian in which the intra- atomic interband coulomb and exchange interactions are taken into account [ 5 ] . We discuss first the single impurity case which corresponds to x ⁼ 0 in the binary alloy A,B,-,. The Hartree-Fock solution of this problem was discussed by Kanamori [2], Gomes [11] and Campbell and Gomes [12]. The H F approxi- mation leads us to the spin dependent atomic potential given by

cka = E~ + Uknk-,(k = A o r B), (2) where nk-, is the average number of electrons with spin - o of atom A or B which is to be calculated self-consistently. As for the parameters ^E'S and U's three different approaches were attempted in lite- rature. The approach adopted by Campbell and Gomes [I21 is to impose the condition that the solution should satisfy the Friedel sum rule,

where Z is the nuclear charge difference between the impurity A and host atom B and y, is the phase shift of o electrons at the Fermi energy. Kanamori adopted the condition that the impurity atom should be electrically neutral in place of the condition (3). In either of these two approaches, we assume that E

and U of host atom are known and determine ^E and U of a given impurity atom so as to satisfy the imposed condition ^(I). As will be discussed below, however, the parameters thus determined vary with the type of the solution when two or more solutions are obtain- ed for a given value of Z. The third approach is to fix the values of ^E'S and U's for a given impurity atom ; of course we choose reasonable values of the para- meters on the basis of available experimental data.

We emphasize in this paper the fact that in any of the above three approaches there are two stable HF solutions in a wide range of Z . We denote the majority spin state of host ferromagnetic metal by up spin.

Assuming that the host metal is either Ni or Co, we suppose that the whole up spin band is below the Fermi energy in the ferromagnetic state of the host metal. As was discussed by Friedel [I], a localized level of up spin will appear above the Fermi level when ^CAI. (A corresponds to the impurity) exceeds a critical value E,,. In such a case the magnetic moment

(1) We need another condition to determine the parameters completely. Usually UA = UB or a similar relation between U's is assumed.

of the impurity atom will be antiparallel to bulk magnetization. On the other hand, if eAT is small enough, the whole up spin states remain below the Fermi level to result in a parallel impurity magnetic moment. Suppose we adopt the condition (3). It is shown by Campbell and Gomes [12] that with ^E,+ > E,,

the condition (3) can be satisfied only when j Z 1 is larger than a critical value Zm (we assume the negative Z case) ; if ^EAT < E,,, it is satisfied only when I Z I

is smaller than 2,. In the former case the impurity magnetic moment is antiparallel and in the latter it is parallel to bulk magnetization. We cannot conclude, however, from this argument that we have only the parallel solution for I Z I < Zm and only the anti- parallel solution for I Z I > 2,. It is shown by Miwa [lo] that we can obtain the HF solutions under the condition E A T = E,, ; in this situation the phase shift of up spin at the Fermi level can take intermediate values between - n and 0, which enables us to find the H F solutions. Figure 1 shows an example of Miwa's calculation which gives the H F solutions in the plane Z us. the impurity magnetic moment, ^p.

Similar curves are obtained for the H F solution even when we adopt the other alternative approaches for determining the parameters E'S and U's. The situation that we have three H F solutions for a wide range of Z remains the same also in the case of Fe as host where the Fermi level lies within the up spin band.

FIG. 1. - The Hartree-Fock solutions in the 2-,u plane, where Z is the atomic number difference between Ni and impurity, and ,u-0.6 is the difference between the impurity magnetic moment and that of Ni. The figure shows an example of results of the calculation in which the bell shape is assumed for the state density of Ni and the Friedel sum rule is assumed in determining

the impurity potential.

We may understand the reason why more than one H F solution are found in the following way. We assume Ni as host. When I Z I is small, the impurity potential rAT is almost equal to or even smaller than the corresponding host atom potential E,, in the solu- tion giving p parallel to bulk magnetization, because the U term in the expression (2) is smaller in cAT

THE COHERENT POTENTIAL APPR( )XIMATION AND FERROMAGNETISM C4-133

than in EBy with nAl < nBl. If we assume that p is antiparallel to bulk magnetization, however, nAL of this solution will be generally larger than that in the parallel solution, resulting in a larger r,+ which may produce a localized up spin level above or at the Fermi level. With more-than-one solutions only an energy comparison will determine which solution is realized.

The condition (3) or the alternative condition of a neutral impurity atom is inconvenient for that purpose, since the parameters vary with the type of the solu- tion. Miwa [lo] shows in the fixed parameter approach which was mentioned as the third alternative that we can derive an approximation expression of the energy as a functional of nAy and nAL ; the expression is such that the variation with respect to n,, gives rise to the Hartree-Fock self-consistent condition. Deferring fur- ther discussion of the energy expression to section 4, we note here that a H F solution then corresponds to either local energy minimum or a saddle point. When we have three solutions, two of them will give local minimums and the third one corresponds to a saddle point. The magnetic moment of the impurity atom is parallel to bulk magnetization in one local minimum solution and antiparallel (or parallel with a smaller magnitude) in another local minimum solution.

A similar situation is expected in the HF-CPA calculation of concentrated alloys. Here we solve essentially the impurity problem for an atom of each component metal ; the effective medium described by the coherent potential corresponds to the host metal in the real impurity problem. What is different from the latter problem is the fact that the electronic structure of each impurity atom reflects itself (or feeds back an effect) on the effective medium through the CPA condition for the t matrices determining the coherent potential. In our problem we introduce the spin dependent coherent potential for each spin direc- tion. Since the coherent potential has an imaginary part wherever the self-consistently determined density of states has a finite value, the localized level produced by the impurity potential is broadened into a band in this case. Nevertheless the basic mechanism of determining the H F solution is the same as in the single impurity problem. We may expect and have confirmed actually that in some cases we have two locally stable ferromagnetic solutions where the magnetic moment of a given species of atom is either parallel or antiparallel to that of another component atom.

In the single impurity problem the magnetic states of surrounding host atoms are perturbed to some extent by the presence of the impurity atom. This means that the impurity effect cannot be represented fully by the impurity potential which is assumed to be confined to the site occupied by the impurity atom.

It is known experimentally that the perturbing effect on surrounding atoms is relatively large when the impurity moment is antiparallel to bulk magnetization.

No effort has been made so far to take into account

an extended range impurity potential in the HF-CPA calculation because an extension of CPA is required to that purpose. We shall discuss this problem in section 4 when we discuss the effect of the nearest neighbour composition on the magnetic state of a given atom ^{( 2 ) .} Before going into more detailed discussions of general aspects of the HF-CPA approach we review briefly in section 3 the results of the HF-CPA calculation.

3. Application of the HF-CPA to ferromagnetic alloys. - The input informations of the HF-CPA calculation are the state density function of pure metal resulting from the electron transfer energy of the Hamiltonian (I), the total electron number of a given alloy and the parameters, c's and U's. The state density function can be taken from the band structure calcu- lation of pure metals. In the actual calculation we can simplify the band shape to some extent, since fine structures in an energy range small compared to the strength of the scattering potential, i. e., the difference between ^E,, and c,, are smeared out by the imaginary part of the coherent potential in concentrated alloys.

The total number of electrons can be estimated some- how as a function of concentration. As for the para- meters E'S and U's we determined them in the following way [5-71. We assume that the Fermi energy of pure metal A in the nonmagnetic state agrees with the corresponding Fermi energy of pure metal B. This means that we neglect the contact potential between them. This assumption which gives rise to a relation among the parameters does not affect the result seriously ; an adjustment of the parameters which is discussed below will offset the difference even when the contact potential is introduced. In the case of Fe where the Fermi level lies within the bands of both spin directions we can determine U,, from the knowledge of the saturation magnetization of the pure metal.

Also the impurity magnetic moment measured by neutron diffraction is used to determine the parameters for solute atom in such a case as Cr,Fe, -,. Otherwise we choose a set of the parameters satisfying the above- mentioned assumption to obtain the best fit to the experimental data. The choice of the parameters will be discussed again below.

The HF-CPA calculation explains generally well the concentration dependence of each atomic magnetic moment in a given alloy measured by neutron diffuse scattering and also the behavior of the electronic specific heat. The calculation has been carried out for Ni base fcc alloys such as NiCo, NiFe, NiMn and NiCr and ternary alloys NiCoFe and NiCoMn and also for Fe base bcc alloys such as FeNi, FeCo, FeMn and FeCr [5-71. In NiCo, NiFe, NiMn, FeCo, FeNi and FeMn we assume such solution that the magnetic moments are parallel to each other. In NiCr and FeCr

( 2 ) Note, however, that the latter effect is different from the

extended range impurity ^potential effect.

C4-134 J. KANAMORI

the magnetic moment of Cr is assumed to be anti- parallel to that of Ni and Fe. In NiCo and NiFe the up spin band lies below the Fermi energy in a wide range of the composition. and e,,~ turns out to be almost equal to ^{E N i +} in these alloys, because the difference between the spin independent terms is cancelled by the difference of the U terms in eq. (2).

This explains why the saturation magnetization of these alloys follows the Slater-Pauling curve in a wide range of the composition, since with a filled up spin band the bulk magnetization is determined by the total number of electrons. In Fe base bcc alloys the large gap between the two peaks in the d-band of a bcc metal which corresponds to the separation between the bonding states and antibonding states between the body-center and corner sites seems to play an impor- tant role in determining the concentration dependence of the ferromagnetic properties. A larger electron scattering in FeNi than in FeCo explains the well known difference of their magnetic properties. In FeCr where the magnetic moment of Cr is anti- parallel to that of Fe, the atomic potentials, E , , ~ and cCrl, turn out to be close to each other, making the deformation of the down spin band relatively small.

The calculated concentration dependence of the band shapes of both spin direction awaits experimental data, for example, those of the photoemission spectra of electrons. Since the results have been published elsewhere, we discuss here fcc NiFe and NiMn as the examples which demand further improvements of the approach.

With a choice of the parameters the whole up spin band in Nil-,Fe, lies below the Fermi level in the range of x ⁼ 0

-

The increase of down spin electrons at Fe atoms in Nil-,Fe, increases get through the U term to extend the up spin band towards higher energy. In the cal- culation shown in figure 2 the up spin band crosses the Fermi level at x = 0.5 at which the number of up spin electrons at Fe atoms, n,,+ and also nNit to less extent begins to decrease. The saturation magnetization deviates accordingly from the Slater-Pauling curve, i. e., the linear increase with increasing x after this concentration. The decrease of n,,? accelerates the increase of nFel, since it lowers eFeL. Thus the deviation of the saturation magnetization from the Slater-Pau- ling curve grows rapidly with increasing x after x = 0.5.

In the calculation we have chosen such U values that the calculated saturation magnetization starts deviat-

ing from the Slater-Pauling curve at the concentration which agrees approximately with the observed value, i. e., x r 0.5.

FIG. 2. - An example of the HF-CPA calculation on Nil-,Fez.

FIG. 2A shows the numbers of electrons of each spin direction at Ni and Fe as function of concentration. FIG. 2B shows the atomic magnetic moments. The circles, triangles, and square are the neutron diffuse scattering data. See t61 for the details of the

calculation and references for the experimental data.

Though the above-mentioned calculation seems to reproduce well the experimental data, it might not represent the real situation. In the discussion we have neglected the possibility of the antiparallel H F solu- tion discussed in the preceding section. Though we have not carried out numerical calculations yet, we may imagine that some Fe atoms acquire anti- parallel magnetic moments. Then we have some empty up spin states, which means some up spin states

THE COHERENT POTENTIAL APPROXIMATION AND FERROMAGNETISM C4-135 lying above the Fermi level. Thus the appearance of

antiparallel Fe magnetic moments may determine the concentration at which the deviation of the saturation magnetization from the Slater-Pauling curve starts. In fact the U values assumed in the calculation shown in figure 2 seems to be too small by factor 2 if we compare the resulting exchange splitting between the up and down spin bands in pure Ni to that calcu- lated by Connolly [14]. If we increase the U values, the deviation from the Slater-Pauling curve starts at larger x in the HF-CPA calculation. In order to take into account the antiparallel solution, however, we have to improve the approximation appropriately.

Before going into the discussion of this subject, we examine the case of Nil-,Mn, in the following.

In Nil -,Mn, the saturation magnetization deviates from a linear increase with increasing x at a very small value of x, x

-

The latter effect should be operating also in Ni, -,Fe,.

We discuss in the next section possible extensions of the HF-CPA approach to include the effects dis- cussed above.

4. Possible extensions of the HF-CPA. - Miwa [lo]

has proposed recently an extension of CPA which can take into account the nearest neighbour effect on the magnetic state of a given atom. We review in this section his theory and also his proposal of an energy expression which enables us to compare the energies of the HF solutions. Jo and Miwa ⁽³⁾ are applying these theories to the case of NiCo alloys containing a single atom of Mn an impurity. We review preli- minary results of this calculation also. After that, we discuss concentrated alloys whose example is Nil-,Fe, from the point of view obtained by these new approaches.

Suppose we have an atom C whose magnetic state we are interested in at a given site. We assume that the atoms at the nearest neighbouring sites consist of atoms A and B. Depending upon the numbers of atoms A and B and their geometrical distribution, we have many (but a finite number of) configurations of the nearest neighbours. In Miwa's theory we do not specify farther neighbours, replacing them by effective atoms whose atomic potential is given by a self- consistently determined potential. The latter potential is called coherent potential in the following. Note that in CPA even the nearest neighbours are replaced

(3) Jo, T. and MIWA, H., unpublished work.

by the effective atoms. For simplicity we shall not specify the spin direction in the following discussion.

The atoms A and B are characterized by the locators Lk = ( E - 8J-l ( k = A o r B ) , (4) where E is the energy. Correspondingly the effective atom replacing a farther neighbour is characterized by the coherent locator,

z

La, ⁼ La hap , (6) where La is the locator of the atom occupying the site a,

being equal to either LA or L. The matrix L specifies a configuration of the nearest neighbouring atoms. We assume that the electron transfer given by the first term of the Hamiltonian (1) takes place only between nearest neighbouring atoms. We can obtain easily the following equation for the expectation value of the Green function at the central atom, Goo :

Goo = L C + ^{L C} t(C, L,) tGoo +

+ LC t[Za,(L.T. ')a,] GOO (7) where LC is the locator of the central atom ; t is the transfer integral ; T is a matrix whose columns and rows refer to the nearest neighbouring sites, a, P, ...

The ap component of T is the sum of those processes in the locator expansion of the Green function in which an electron starting at the site a is allowed to visit lattice sites except the central site before reaching the site p. The lattice sites including a and P can be visited many times in the sum. It is more convenient to introduce another matrix 7 which may be called irreducible transfer matrix ; we sum in 7 those pro- cesses in which the nearest neighbour sites as well as the central sites are forbidden to come in. 7 is related to T by

T = [I - ? . ~ ] - l ? . (8)

By use of eq. (7) and (8) we obtain an expression of Goo given by

Goo = [LG' - t 2 Cap { L - T),g1]-l . (9) By definition ? is independent of the configuration of the nearest neighbours. We can calculate 7, once the coherent locator describing the effective medium is determined. L, on the other hand, represents the nearest neighbour effect on Goo.

The self-consistent condition for

z

x.

The self-consistent equation reduces to that of CPA if we replace each component of L by

z.

C4-136 J. KANAMORI

shown that this method yields a state density almost identical with that of CPA in the case of a nonmagne- tic alloy ^(4). Once is determined, we can calculate Goo for each configuration of the nearest neighbours.

Note also that we can include in principle a potential produced by the central atom at the nearest neighbour- ing sites in the calculation by taking account of it in the locator matrix L.

Though various extensions of CPA have been proposed so far 1151, Miwa's formalism seems to be most promising at least in the problem of the ferro- magnetic state. Jo and Miwa have applied the theory to the case of Ni, -,Cox containing an atom of Mn as impurity by combining it to the H F approxi- mation. Here we are interested in the magnetic state of Mn which is known to have a parallel magnetic moment in pure Ni and an antiparallel magnetic moment in pure Co. We expect that the nearest neigh- bour configuration is decisive on the question of the direction (and also the magnitude) of Mn magnetic moment in NiCo alloys. Figure 3 shows some examples of the result of a preliminary calculation. In the calculation they simplify the irreducible transfer matrix ? such that !its inondiagonal elements are assumed to be equal to each other. With this simpli- fication each configuration of the nearest neighbours is characterized by the numbers of Ni and Co only, in other words, Go, becomes independent of the geometry of the distribution. Figure 3 shows that we

(4) This applies to the case where the potential difference is small. In the split band case the result differs from that of CPA to some extent.

FIG. 3. - A preliminary calculation of the nearest neighbour effect on the magnetic state of a Mn atom as substitutional impurity in NiCo alloys. Nco is the number of Co atoms at the nearest neighbouring sites. FIG. 3 A shows the magnetic moment of Mn in the parallel and antiparallel solutions and the difference between the energy of the parallel solution, Ep, and that of the antiparallel solution, EA, in Nio.99Coo.ol ; the unit of energy is the half-width of the band of pure metal. FIG. 3B shows the numbers of electrons of each spin direction of Mn atom in Nio.99Coo.ol. Fm. 3C shows the calculation in the case of Ni0.7Co0.3. The parallel solution ceases to exist for Nco > 6 in the case of figures 3A and 3B and for NCO > 9 in the case of figure 3C, whereas the antiparallel solution exists always. In the present calculation the coherent locator (see text) is calculated by the HF-CPA method and the calculation of the Green func- tion of pure metal is carried out with the same state density function as that used in the HF-CPA calculation given in [6].

The U values are assumed to be U M ~ = 1.1, Uco = 1.35 and U N ~ = 2.7 in the unit of the half-width of the band of pure metal. The assumed state density function is not consistent with the assumption that the electron transfer takes place only between the nearest neighbouring pairs (see text). Another calcu- lation with the state density which is consistent with the assump- tion, however, yields qualitatively similar results of the nearest neighbour effect. We do not intend to claim in the present prelimi- nary calculation that the choice of the parameters is well justified

quantitatively.

P A R A L L E L

P A R A L L E L UP

5 r 0 1-

P A R A L L E L DOWN

w

P A R A L L E L

2 , 5

THE COHERENT POTENTIAL APPROXIMATION AND FERROMAGNETISM C4-137 have both parallel and antiparallel H F solution in a

wide range of the number of nearest neighboring Co.

The magnetic moment of Mn in the antiparallel solution is more dependent on the number of nearest neighbouring Co than that in the parallel solution.

This might explain the absence of the NMR signal arising from antiparallel Mn atoms in the experiment done by Yasuoka, Hoshinouch and Nakamura [16]

on NiCo alloys containing a small amount of Mn.

We proceed to the discussion of the calculation of the energy of the H F solutions. It is desirable to derive an expression of the energy which is a functional of the numbers of electrons, nk,'s. If it yields the Hartree-Fock equations as the Euler equations of the variation with respect to nk,'s, it will elucidate the physical meaning of the solutions. In fact we adopt in numerical calculation an iterative method which starts with a trial set of nk,'s for E~,'s given by eq. (2) and continues the calculation to reach an approximate agreement between the assumed and the calculated nkO7s ; this assumes implicitly that the solution corres- ponds to local minimum or maximum of a functional.

Miwa proposes the following approach to derive such an expression for the impurity problem. We assume that we know the electronic structure of the medium surrounding a given atom. The medium can be a pure metal or the CPA medium described by the coherent potential or the above discussed medium consisting of a given configuration of the nearest neighbows and the effective medium described by the coherent locator. We first place at the central site an atom whose atomic potential is known. We can calculate the Green function of the system. This system will be used as the reference system for the calculation of the energy. When we replace the central atom by the atom C which we are interested in, the change of the electronic structure is caused by the potential difference at the central site, V,, given by

where E,, is the known atomic potential of the reference atom. Noting that the relative phase shift to the reference system is given by

- (1l.n) Im log (1 ^- V, GG)

with G& representing the expectation value of the Green function of the reference system at the central site, we obtain an energy expression given by

EF

E = E, - (1171) Z, ~ r n (E ^- ~d (dId.1 x - m

x log (1 - V, G;P,) ds - U , n,, nc-, (11) where E, is the energy of the reference system, ^E, is the Fermi energy. yF is a functional of n,, determined by

EF

nc, = - (11.) 1m

1

It can be easily shown that the expression (1 1) yields the H F solutions as those satisfying the Euler equations in the variational problem with respect of n,,'s.

We can derive other expressions which yield the H F solutions as those giving extremum also. Deferring the discussion of the choice of the energy expression to the next section, we note here that the expression (1 1) might allow us to discuss the energy of unstable states as well.

Using the expression (1 1) in the problem of Mn impurity in NiCo alloys, Jo and Miwa have found that the energy difference between the parallel and antiparallel solutions actually changes its sign when the number of Co at the nearest neighbouring sites exceeds a critical value (see Fig. 3) ; the parallel solution is favored when the number of Co atoms at the n. n. sites is small. Though the existence of the critical number of Co depends on the choice of para- meters, we find it in wide ranges of the parameters covering the region where reasonable estimates of them fall in. The critical number may depend in principle on the composition of the host alloy, though the example shown in figure 3 does not show that effect. The energy difference, however, becomes small when the average concentration of Co is increased.

As was discussed before, we anticipate the presence of Fe atoms having antiparallel magnetic moments in Nil-,Fe, in the region where the saturation magnetization deviates from the Slater-Pauling curve.

In such a case we had better introduced a ternary alloy picture in which we distinguish between Fe atoms having parallel moments and Fe atoms having antiparallel moments. Forgetting about the nearest neighbour effect for a while, we adopt the HF-CPA approach. Suppose we somehow determined the coherent potential. Then we solve the impurity problem of Fe embedded in the medium described by the coherent potential. By a preliminary calculation we have confirmed that we may have two stable H F solutions in this case ; one solution corresponds to the state with a magnetic moment parallel to bulk magnetization and the other to that with an anti- parallel magnetic moment whose magnitude is different from that of the parallel solution. The energies of these solutions depend on the coherent potential.

When we give the concentration ratio of the parallel Fe atoms to the antiparallel Fe atoms, we can calculate the coherent potential and the H F solutions through the HF-CPA approach. A way to determine the concentration ratio is to adjust it such that the energy of the antiparallel solution agrees with that of the parallel solution. Of course if we find also two stable solutions for Ni, we have to introduce the concept of a quaternary alloy consisting of both kinds of Ni and Fe. We might be able to pursue the ferromagnetic state in Nil-,Fe, from the concentration where the deviation from the Slater-Pauling curve starts to its disappearance at x E 0.74 with this approach. Without carrying out the calculation, however, it is difficult

C4-138 J. KANAMORI

to predict the behavior of the solution. With increasing x the number of the antiparallel Fe atoms would increase ; at the same time the antiparallel magnetic moment might tend to have a smaller magnitude or even to change its sign. Since some of the up spin states will be emptied, the magnitude of the parallel magnetic moment will decrease also. Whether we reach the nonmagnetic state smoothly or not is unpredic- table. Because a complicated self-consistent mecha- nism is built in, it may be possible to have several ferromagnetic solutions. Also the ferromagnetic solution may cease to exist at a certain x, as is observed in the simple HF-CPA based on the binary alloy concept. In such a case, a transition of the first kind to either the nonmagnetic phase or another phase might be concluded theoretical!^ by an energy comparison.

Then the nearest neighbour effect, i. e., the possibility of finding of various kinds of Fe atoms (and also Ni atoms) depending upon the nearest neighbour compo- sition might become important.

We can of course extend Miwa's theory of the nearest neighbour effect to the present case. We start again with the ternary or quaternary alloy picture.

We have to specify not only the atomic configuration but also the magnetic state of each atom of the nearest neighbours. The locators of the nearest neighbours could be prefixed somehow by introducing the concept of the average parallel and antiparallel Fe atoms.

The concentration ratio of the parallel atom to the antiparallel atom could be determined self-consis- tently by determining energetically which solution the central atom will take with a given configuration of the nearest neighbours and counting the probability of obtaining each solution. We might encounter a conceptual difficulty, however, when we could not determine which H F solution would correspond to which average magnetic state of a given atom.

An alternative approach would be to treat the cluster consisting of the central atom and its nearest neighbours as a unit in a self-consistent calculation.

Anyway we hope that future experiments will show us the limit and applicability of approximate concepts such as the ternary or quaternary alloy picture dis- cussed here.

5. Concluding remarks. - We presented in this paper a review of the application of CPA to ferro- magnetic alloys. We combine the Hartree Fock approximation to CPA or its extension to determine the magnetic state of a given atom. In the simplest approach we assume that atoms of each component metal are in the same magnetic state. The calculation based on this approach was able to elucidate the alloying effect on the ferromagnetic and electronic properties of various alloys. There are, however, the cases which demand further extension of the approach. The effect of the nearest neighbour composi- tion on the magnetic state of a given atom which is established experimentally in NiMn alloys is an

example. Also the deviation of the saturation magneti- zation from the Slater-Pauling curve in NiFe alloys requires a more detailed treatment. The Hartree- Fock approximation in the single impurity problem yields several solutions in a wide range of the impurity potential. A theoretical discussion on this fact and an extension of CPA to take into account the nearest neighbour effect both studied by Miwa are reviewed.

A preliminary calculation on Mn as impurity in NiCo alloys already reveals interesting features of the nearest neighbour effect. Possible extensions of the theory to concentrated alloys such as NiFe alloys are discussed subsequently. Finally we emphasize that our aim in these theoretical discussions is to establish useful approximate concepts of the ferromagnetic state in disordered alloys. Undoubtedly the HF-CPA and its extension are a promising tool for that purpose.

Before concluding this paper, we discuss the impli- cation of the energy expression given by eq. (11).

Miwa has shown that we can draw an energy contour map in the n,, n,-, plane by use of the expression (1 I).

The parallel and antiparallel solutions in the single impurity problem of Ni actually correspond to energy minimums in such a map. A similar situation is expect- ed in the HF-CPA treatment of more concentrated alloys and also Miwa's extension of it. Then the solution whose energy lies higher than that of the other may represent an excited state associated with a given atom. The possibility of two atomic states with diffe- rent magnitudes of the magnetic moment have been suggested by several authors. The argument given by Weiss [17] on NiFe alloys of compositions around the invar alloy (Ni,.,, Fe,.,,) is a typical example in which he postulates the low and high spin states of Fe atom ^{( 5 )} Campbell [18] has pointed out that a similar idea can explain anomalous temperature dependence of the effective field acting on a nucleus observed in various alloys. These ideas might be substantiated in future along the line discussed above.

In order to establish the concept of two atomic states, we have to discuss the energy of intermediate states which are off the minimums. Miwa has shown that various modifications of the energy expression given by eq. (1 1) are possible ; for example, we may use

Va9s given by eq. (10) as variables in place of n,,'s.

Though these energy expressions agree with each other in the point that they yield the H F solutions as extremums, they differ from each other with respect to the energy of intermediate states. It is a future problem to find a suitable energy expression which can describe the longitudinal fluctuations of the magnetic moment at finite temperature. We may note that the free energy of a magnetic atom embedded in a nonmagnetic medium derived by Evenson, Schrieffer and Wang [19] and Hamann [20] by use

(5) As for the origin of the invar effect itself, we have a different opinion from Weiss' one (see 1211).

THE COHERENT POTENTIAL APPROXIMATION AND FERROMAGNETISM C4-139

of the functional integral method in the static approxi- Acknowledgments. - The author would like to mation is closely related to the anticipated energy express his sincere thanks to H. Miwa for valuable expression in our discussions ^{( 6 ) .} discussions on his unpublished work on which the present paper is largely based. Thanks are also due to T. Jo a i d bther members of the theoretical solid state

( 6 ) In their case the energy as a function of a variable corres-

ponding to an effective field has two symmetric minimums of physics group at the department of physics of Osaka

the same depth. In a ferromagnetic medium one of them may University for discussions and helps in the course

have a higher energy than the other or even disappear. of preparing this manuscript.

References

[I] FRIEDEL, J., NUOVO Cimento 2 (1958) Suppl. 287.

[2] KANAMORI, J., J. Appl. Phys. 36 (1965) 929.

[3] SOVEN, P., Phys. Rev. 156 (1967) 809.

[4] KANAMORI, J. and HASEGAWA, H., J. Physique 32 (1971) C1-195.

[5] HASEGAWA, H. and KANAMORI, J., J. Phys. SOC. Japan 31 (1971) 382.

[6] HASEGAWA, H. and KANAMORI, J., J. Phys. SOC. Japan 33 (1972) 1599 and 1607.

[7] Jo, T., HASEGAWA, H. and KANAMORI, J., J. Phys. Sac.

Japan 35 (1973) 57.

[8] JACCARINO, V. and WALKER, L. R., Phys. Rev. Lett. 15 (1965) 258.

[9] CABLE, J. W. and CHILD, H. R., J. Physique, 32 (1971) C1-67.

[lo] MIWA, H., to be published in Prog. Theoret. Phys.

Kyoto, Japan.

I l l ] GOMES, A. A., J. Phys. & Chenz. Solids, 27 (1966) 451.

[12] CAMPBELL, I. A. and GOMES, A. A., PYOC. R. SOC. A. 91 (1967) 319.

[I31 KUNITOMI, N., NAKAI, Y., YAMASAKI, K. and SCHIBUYA, N., J. Physique, 35 (1974) C4-149.

[14] CONNOLLY, J. W. D., Phys. Rev. 159 (1967) 415.

[15] For example, Foo, E-Ni, AMAR, H. and Aus~oos, M., Phys. Rev. B 4 (1971) 3350.

[16] YASUOKA, H., HOSHINOUCHI, S. and NAKAMURA, Y., J.

Phys. Soc. Japan 34 (1973) 1192.

[17] WEISS, R. J., Proc. R. SOC. A 82 (1963) 281.

[I81 CAMPBELL, I. A., J. Phys. C (Solid State) 3 (1970) 2151.

[19] EVENSON, W. E., SCHRIEFFER, J. R. and WANG, S. Q., J. Appl. Phys. 41 (1970) 1199 ;

WANG, S. Q., EVENSON, W. E. and SCHRIEFFER, J. R., Phys.

Rev. Lett. 23 (1969) 92.

[20] HAMANN, D. R., Phys. Rev. Lett. 23 (1969) 95.

[21] KANAMORI, J., Proc. of 1973 International Conference on Magnetism (Moskow).