ELECTRONIC STRUCTURE OF FERROMAGNETIC ALLOYS IN THE COHERENT POTENTIAL APPROXIMATION

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ELECTRONIC STRUCTURE OF FERROMAGNETIC ALLOYS IN THE COHERENT POTENTIAL

APPROXIMATION

J. Kanamori, H. Hasegawa

To cite this version:

J. Kanamori, H. Hasegawa. ELECTRONIC STRUCTURE OF FERROMAGNETIC ALLOYS IN

THE COHERENT POTENTIAL APPROXIMATION. Journal de Physique Colloques, 1971, 32 (C1),

pp.C1-280-C1-281. �10.1051/jphyscol:1971194�. �jpa-00214521�

STRUCTURE ELECTRONIQUE ET DENSITE DE SPIN

ELECTRONIC STRUCTURE OF FERROMAGNETIC ALLOYS IN THE COHERENT POTENTIAL APPROXIMATION

J. KANAMORI and H. HASEGAWA

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

Rhsumh. - L'approximation du potential coherent est appliquee dans les cas de Nil-,Fez et Nil-,Mnz en se basant sur un modele

&

une bande. I1 est tenu compte de I'energie d'echange dans le cadre de I'approximation de Hartree-Fock.

Le calcul pour Nil-Zez explique la variation en fonction de la concentration, des moments magnetiques du Ni et du Fe et aussi de la chaleur spkifique electronique.

Abstract. - The coherent potential approximation is applied to N i l - ~ F e ~ and Nil-,Mnz on the basis of a single band model. The exchange energy is taken into account within the framework of the Hartree-Fock approxin~ation. The calculation in the case of Ni

l-nFez

explains the concentration dependence of the magnetic moments of Ni and Fe and the electronic specific heat.

The coherent potential approximation which was proposed originally by Soven [I] and discussed by several authors through different approaches [2-51 gives an access to the calculation of the electronic structure of an alloy. We present in this paper an application to the ferromagnetic alloys Ni-Fe and Ni-Mn both in the fcc phase. Nil -, Fe, is characterized by the appea- rance of the invar alloy a t x

=

0.65, which is followed by the disappearance of the ferromagnetism a t x

%

0.8 [6]. I t has been shown by Shimizu and Hirooka [7]

and Mizoguchi [8] that the rigid band approximation is capable of locating the instability of the ferroma- gnetic state against the paramagnetic state at the observed value of x with reasonable choices of para- meters. Since in our opinion this is one of the most impressive successes of the rigid band approximation applied to ferromagnetic alloys, we feel that the problem should be reexamined in a more advanced approximation. In fact Velickjr, Kirkpatrick, and Ehrenreich [5] have shown that the density of states vs.

energy curve for an alloy deforms to a considerable extent even when the band shapes at x

=

0 and x

=

1 are the same. Moreover the rigid band approxi- mation which does not distinguish between Ni and Fe is not capable, for example, of discussing the average magnetic moments of Ni and Fe, which have been measured by neutron diffraction [9, 101.

Our calculation is based on the single band model proposed by Velickf, Kirkpatrick, and Ehrenreich [5].

The Hamiltonian is given by

where nio is the number operator of electron with spin o of the Wannier orbital a t the i-th lattice site ;

W represents the electron transfer terms between different sites which are assumed to be independent of the species of atoms occupying the sites. cia is equal to either cAU or

^EB,

depending on whether an atom A

(=

Fe or Mn) or B (= Ni) occupies the i-th site ; they depend also on the spin direction through

exchange energy. We calculate n: and : n self-consis- tently within the framework of the Hartree-Fock approximation. The coherent potentials 2, and Z-, which are spin-dependent through eq. (2) are deter- mined by the equations which are obtained from the standard equation given in [5] by replacing the spin- independent energies

EA

and

&,

by

EA,

and

,,&,

respec- tively. Altogether we have six simultaneous equations

A B

to determine six quantities, n+,, nr,, xu ^and xu, ^with

a given set of parameters EA, eB, UA, UB and x, and a given band shape of pure metal determined in principle by W.

The parameters

2,

and

^8,

are determined by the condition that the Fermi energies of pure metals A and B in the paramagnetic state coincide with each other for a given number of electrons for each metal and given values of UA and UB. We assume that pure Ni contains 9.415

=

1.88, pure Fe 1.44, and pure Mn 1.22 electrons per atom. In the calculation for Nil-,Fe, we assume UA

=

UB

=

U for simplicity. In figure 1 the assumed band shape for pure metals is

FIG.

1. -The dotted lines show the assumed state density for x = 0 with the

dotted

vertical line indicating the Fermi energy. The full lines represent the calculation

for

x =

0.6.

shown, which corresponds to a simplification of the density of states given by the standard calculation.

Figures 1-4 show the results of the calculation for Ni, -,Fe, in which U = 1.25 in the unit of a half of the total width of the band for pure metal is assumed.

With this value of U the energy band of majority

A B

= EA +

uA

n - u

and

eBu = EB +

U, n - u , (2) spin (called up spin hereafter) remains below the Fermi energy for x smaller than 0.5. The deformation where n!,"', is the average number of electrons with of the up spin band is relatively by small because of spin ^-o on A or B atom ; the U terms represent the ^e cBi as is shown in figure 2, whereas the down

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971194

ELECTRONIC STRUCTURE OF FERROMAGNETIC ALLOYS IN THE COHERENT C l -281

0.6 0.4 0.2 O

X

FIG. 2. -

The concentration dependence of et's.

Ni-Fe t:

O 1 I 1 O

O 0.1 0.6 0.4 0.2 O

X X

FIG. 3. -

The magnetic moments of Ni and Fe and the ave- rage moments of N ~ I - ~ F ~ ~ and N ~ I - ~ M ~ ~ ; circles [9] and

triangles

[IO]

are the experimental data.

FIG.

4.

-

The calculated state density at the Fermi energy in the unit of states per atom per a half of the band width at x = O and the electronic specific heat y in 10-4 cal.(mole. deg2)-1.

spin band is deformed to a considerable extent (Fig. 1).

For x smaller than 0.5 the calculation is relatively

A B

easy since we can assume nt

=

nt

=

1. Above x

=

0.5 where the up spin band begins to pass the Fermi level, the computation becomes difficult. We determined nj and nb in any case within the accuracy of 0.015 by an iterative method. For

x

larger than 0.65, however, we find no ferromagnetic solution within this accuracy, though we can always find the paramagnetic solution.

This means perhaps that n,'s change rapidly with increasing x above x

=

0.65 to join the paramagnetic solution at a certain value of x. Such a limit of the ferromagnetic solution moves to a higher value of x with increasing the magnitude of U.

Discussion of the results for Ni, -, Fe,. - We plot the calculated magnetic moments of Ni and Fe and also the average magnetic moment per atom against x in figure 3. The average moment falls on the Slater- Pauling curve, showing a linear dependence on x for x below 0.5. The magnetic moment of Fe decreases from 3.15 p, a t x = O to 2.25 pB a t x

=

0.65, whereas the moment of Ni increases from 0.60

p,

to 0.70 ^p,.

The changes of the moments below x

=

0.5 reflect the changes of n i and n i which shift also and eBT through eq. (2), as is shown in figure 2. The increase of pushes up the up spin band across the Fermi energy at about x

=

0.5. Figure 3 shows also the experimental data of the moments measured by neu- tron diffraction. The moment of Fe for small x seems to be too high compared with the data. This is partly because Our Hamiltonian does not take into account the level shift caused by a deficit or an excess of elec- trons of the same spin. Anyhow the general tendency predicted by the coherent potential approximation is well supported by the experiment.

Figure 4 shows the calculatioil of the density of states per atom a t the Fermi level. The data of the electronic specific heat [ I l ] are also shown for compa- rison ; we fit the data to the calculation at a point on the horizontal part of the curves. The calculation does not depend very much on the choice of the band shape except for small x ; in fact another calculation assuming a flat state density around the Fermi level of down spin at x

=

O shows a similar decrease with increasing x above x

⁼

0.05. Since the state density of down spin deforms t o a considerable extent even for small values of x, we may conclude that the deduc- tion of the band shape from the specific heat data of alloys is hardly justified.

For x larger than 0.5 whether the up spin band crosses the Fermi level before the disappearance of ferromagnetism or not is a crucial point. Though we tried t o compare the energies of the para- and ferromagnetic states, we could not obtain any definite conclusion because of the limit of accuracy. We feel, however, that the present calculations explain more reasonably the increase of the electronic specific heat as well as the deviation of the magnetization from the Slater-Pauling curve than the coexistence of other phases, etc. postulated by the rigid band theory [7] at least for x smaller than 0.65.

Discussion of Ni, -, Mn,. - In figure 3 we show a preliminary calculation of the average magnetic moment of this system with UA

=-

1.10 and UB

=

1.25.

Details of the present calculation will be published elswhere.

Re

ferenees

[l] SOVEN (P.), Phys. Rev., 1967, 156, 809. [7] SHIMIZU (M.)

and

HIROOKA (S.), Phys. Letters, 1968, [2] TAYLOR (D. W.), Phys. Rev., 1967, 156, 1017. 27 A, 530.

[3] ONODERA (Y.)

and

TOYOZAWA (Y.), J. Phys. Soc. [8] MIZOGUCHI (T.), J. Phys. Soc. Japan, 1968, 25, 904.

Japan, 1968, 24, 341. [9] SHULL (C. G.) and WILKINSON (M. K.), Phys. Rev., [4] YONEZAWA (F.), Progr. Theor. Phys., 1968, 40, 734. 1955, 97, 304.

[SI VELICKY (B.), KIRKPATRICK (S.) and EHRENREICH (H.), [IO] COLLINS (M. F.), JONES (R. V.) and LOWDE (R. D.), Phys. Rev., 1968, 175, 747. J. Phys. Soc. Japan, 1962, 17B-III, 19.

[6] CHIKAZUMI (S.), MIZOGUCHI

(T.),

YAMAGUCHI (N.) [Il] GUPTA (K. P.), CHENG (C. H.)

and

BECK

(P.

A.),

and

BECKWITH

(P.),

J. Appl. Phys., 1968, 39, 939. J. Phys. Chem. Solids, 1964, 25, 73.