SPIN SUSCEPTIBILITIES OF FERROMAGNETIC Pd-Ni and Pt-Ni Alloys

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SPIN SUSCEPTIBILITIES OF FERROMAGNETIC Pd-Ni and Pt-Ni Alloys

T. Kato, M. Shimizu

To cite this version:

T. Kato, M. Shimizu. SPIN SUSCEPTIBILITIES OF FERROMAGNETIC Pd-Ni and Pt-Ni Alloys.

Journal de Physique Colloques, 1974, 35 (C4), pp.C4-145-C4-147. �10.1051/jphyscol:1974425�. �jpa-

00215616�

JOURNAL DE PHYSIQUE

CoEEoque C4, supplkment au no 5, Tome 35, Mui 1974;page C4-145

SPIN SUSCEPTIBILITIES OF FERROMAGNETIC Pd-Ni and Pt-Ni Alloys

T. KATO and M. SHIMIZU

Department of Applied Physics, Nagoya University, Nagoya, Japan

RCsumC.

Nous dkterminons la susceptibilite g6n6ralisCe d'alliages ferrornagnktiques dans l'approximation RPA et dans l'hypothbse des potentiels localises (CPA). Dans ce modble nous calculons numtriquement la susceptibilite x(q

0) des alliages ferromagnktiques Pd-Ni et Pt-Ni.

Abstract. - Generalized spin susceptibility of ferromagnetic alloys is calculated within the random phase approximation under the assumption of perfect local screening. Using this result, uniform spin susceptibilities of ferromagnetic Pd-Ni and Pt-Ni alloys are numerically calculated.

1. Introduction. - Recently, Harris and Zucker- mann [l] and the present authors [2] have calculated the paramagnetic spin susceptibilities of Pd-Ni, Pt-Ni and Pd-Pt alloys by the Wolff-Moriya model in the random phase approximation (RPA) and in the coherent potential approximation (CPA), and the experimental results have been successfully explained.

However. it was difficult to extend this formulation to ferromagnetic alloys, for the wave vector dependent susceptibility in this model exhibits a discontinuity at zero wave vector, as the long-range part of the Coulomb interaction is disregarded [3]. In this paper this difficulty is avoided by the assumption of perfect local screening, and generalized spin susceptibility of a ferromagnetic alloy is calculated in the RPA and CPA and the appropriateness of this assumption is discussed in Sec. 2. In Sec. 3, uniform suscepti- bilities of ferromagnetic Pd-Ni and Pt-Ni alloys are numerically calculated by using the values of para- meters determined in the paramagnetic region. The calculated results are compared with the observed ones.

perfect local screening is assumed, that is, when H,' = 0 the excess charge of the impurity is completely screened within the atomic cell and the number of electrons at each site is a constant which depends only on the kind of atoms. Furthermore, it is assumed that this situation does not change when the magnetic field is applied. From these assumptions, the expecta- tion value of the number of a spin electrons at the i site can be written in the RPA as

where m, is < n, > when H,' = 0, and A i ^y, is the induced moment proportional to h, and is determined self-consistently. Then, (I) is reduced to

where

h; = h eiq.Ri +UiAi/2 ₍₂₎

2. Generalized spin susceptibility of ferromagnetic is an effective magnetic field applied to a system alloy. - Hamiltonian of electrons in a ferroma- described by & - ^{E at hi = 0.} The self-consistent gnetic transition metal alloy under the magnetic field equation to determine

is given by

H i exp [iq.r] in the Wannier representation is given A i = x 2 Ai,j h j ,

j

(3) by [21

H = 2 Tij a; uj, + 2 ^{l/i n ,} + ^{where 2 pg} Ai,j is the local susceptibility which gives

i,j,a i.o

the induced moment A i yB by hj in the system with go.

+ x ^{U z niT nLl - h,} x ^{(nir -} nil) eiq.Ri, (1) The local susceptibility of the system with Z i s defined

I i 2

by 2 ^yB xij with ^{= (f)} (aAi/ahj), where 12, = hqeiq'R'.

where Vi and U i are V and U + AU at impurity sites, From (2) and (3), we have and 0 and U at host sites, respectively, the form factor

is assumed to be unity and h, = ^p, H,". Now, the

⁼ Ai,j + C ^{Ai,l u, X,,}

. (4)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974425

C4-146 T. KATO AND M. SHIMIZU The A i , j is approximated as < AiTj >,, by taking

random average over configurations. Then, (4) can be solved within the CPA in the same way as before 121, and we have

with

where ~ ( k , q) and A(q) are the Fourier transforms of and < A,,, >,,, respectively, and c is the concen- tration of impurity atoms.

Here, the x0(q) is approximated by the susceptibility of a virtual crystal of alloys, and from the assumption of the perfect local screening, it is given by

with

where f (E) is the Fermi distribution function and

is the Fermi level of ^o spin electrons. The expression (7) can be easily obtained under the condition that the deviation of the electron density from the uniform density vanishes, that is, p(q) = 0 in eq. (2) of the ref. [3]. There is no discontinuity in xo(q) of (7) at q = 0. On the other hand, in the electron gas model, where bare Coulomb interaction exists, the discon- tinuity of x0(q) at q ⁼ 0 is also absent and there may be a deviation of the electron density from the uniform one at finite q, the x0(q) in the RPA is given by [3]

where

and the exchange interaction is approximated by U.

To see the validity of the assumption of perfect local screening, the xO(q) is numerically calculated for a ferromagnetic pure metal by (7) and (9) and also by the formula given by Penn [4] for the Hubbard model. All- of the three expressions are the same in the paramagnetic region. The energy spectrum is assumed to be free electron like, and the density of states at the paramagnetic Fermi level and the electron number are assumed to be 1.2 eV-I per atom per spin and 0.36 per atom, respectively, as in Pd metal. The calculated results of x0(q) are shown in figure 1 by curves 1, 2 and 3, which are read by the left-hand side ordinate, for the Bohr magneton number per atom n, = 0.24 and by curves 4 and 5, which are read by

FIG. 1.

Calculated results of xo(q) by (7) (curve

(9) (curves 2, 5), and by the formula of Penn [4] (curves 1, 4), for

nB =

0.24 (curves 1,

3) and

0.36 (curves 4, 5). The

is reduced by its value at the Brillouin zone boundary q~ and

is reduced by 2

N(EF).

the right-hand side ordinate, for n, = 0.36, where the corresponding values of U are 0.87 eV and 0.99 eV, respectively. Curves 1 and 4 are obtained by the formula of x0(q) given by Penn [4], curves 2 and 5 are obtained by (9) and curve 3 is obtained by (7) and the xo(q) given by (7) vanishes for n, = 0.36 in the case of strong ferromagnetism. It is found that the results of (7) are in good agreement with the results of (9) even at higher q. From the fact that the value of v(q) at the boundary of the Brillouin zone is about 5 eV and U is about 1 eV, it is satisfied that v(q) % U in (9) for all q's, so that we have the agreement between xo(q)'s given by (7) and (9). From these considerations, it is concluded that in the calculation of ~ ( q ) for the model where the long-range nature of the electron interaction is neglected as in (1) the assumption of perfect local screening gives a better result than that obtained without this assumption.

3. Numerical results of high-field spin susceptibility.

- The uniform spin susceptibility in the ferromagnetic region, that is, high-field spin susceptibility, ~ ( 0 ) = x

of Pd-Ni and Pt-Ni alloys are numerically calculated by (5) and (6). The function r,(q) defined by (8) is .-

approximated by (N(&g)/N(&,)) r ( q JeF/E;) both in Pd-Ni and Pt-Ni alloys, where hr(&) is the density of states of the virtual crystal,

is the Fermi level in the paramagnetic state and T(q) is the calculated result by Diamond [5] for Pd metal. The

determined so as to be consistent with the observed values of the magnetic moment for Pd-Ni alloys [6] and Pt-Ni alloys [7].

The paramagnetic spin susceptibility are calculated again for Pd-Ni and Pt-Ni alloys by using N(E) obtained by Mueller et al. [8, 91 as shown in figures 2 and 3.

The enhancement factors of Pd and Pt metals are

found as 10.0 and 3.5, respectively, by comparing

SPIN SUSCEPTIBILITIES O F FERROMAGNETIC Pd-Ni C4-147

FIG. 2. - Calculated results of

reduced by the observed susceptibility

of Pd metal for Pd-Ni alloys. Small circles [6]

and triangles (FAWCETT eet al., Phys. Rev. Lett. 21 (1968) 1183) are experimental data.

FIG. 3. - Calculated results of x reduced by the observed susceptibility

of Pt metal for Pt-Ni alloys. Small circles [7]

and triangles (BESNUS et al., Phys. Lett., 39A (1972) 83) are experimental data.

N ( E ~ ) with the observed values of the susceptibilities of Pd and Pt metals. The values of AUNi/Upa and AUNi/Upt are determined as 1.26 and 0.75 by fitting to the observed critical concentrations of 2.3 at. % Ni in Pd and of 44 at. % Ni in Pt, respectively. Although the values of AU7s are changed a little from those given in [2] corresponding to the different values of N(zF), the present calculated results of x in the para-

[I] HARRIS, R. and ZUCKERMANN, M. J., Phys. Rev. B 5 (1972) 101.

[2] KATO, T. and SHIMIZU, M., J. Phys. Soc. Japan 33 (1972) 363.

[3] TAKAHASHI, I. and SHIMIZU, M., Prog. Theor. Phys. 43 (1970) 249.

[4] PENN, D. R., Phys. Rev. 142 (1966) 350.

[5] DIAMOND, J. B., J. Appl. Phys., 42 (1971) 1543.

[6] CHOUTEAU, G., FOURNEAUX, R., GOBRECHT, K. and TOURNIER, R., Phys. Rev. Lett. 20 (1968) 193.

magnetic region are almost the same as the previous results [2].

Using the values of parameters determined in the paramagnetic region, the x in the ferromagnetic region is self-consistently calculated by (5) and (6). Near the critical concentration c,, however, the numerical results of x are very sensitive to the shape of N(E) in the neighbourhood of E,. As a mater of fact, phy- sically reasonable results of x are obtained only above 7 at. % Ni for Pd-Ni alloys and only above 48 at. % Ni for Pt-Ni alloys. This is related to the fact that the N(E) curve of Pd and Pt metals concave downward at z, and the condition of ferromagnetism is barely satisfied near c,. To obtain the physically reasonable results of x near c,, the shape of N(z) in the neighbourhood of

is modified downward by less than about 3 % for Pd and by less than 0.5 %

for Pt, without changing the value of N(E~). The calculated results of x are shown in figure 2 for Pd-Ni alloys and in figure 3 for Pt-Ni alloys with the observed data. It is noticed that the results at higher concen- trations of Ni, which are not so affected by the small modification of N(E), agree well with the observed values. On the other hand, the numerical values of x near c, should not be taken seriously, as the shape of N(E) is modified a little. But this modification is qualitatively reasonable, as the values of N(&) in alloys are expected to be smaller than those of the virtual crystal, as explained in the following way.

The difference between the scattering potentials of o spin electrons at the host and impurity sites in the

N

system H , is given by - a(UNi mNi - Up, mpd)/2 for Pd-Ni alloys, where mNi pB and ~z,, pB are magnetic moments at Ni and Pd sites. From neutron diffraction experiments [lo], the potential difference is estimated as about 0.9 eV. On the other hand, the width of the peak of N(E) of Pd metal is about 0.4 eV. Therefore, the alloying of Ni atoms to Pd metal has the effect to lower the values of N(E;) of both spin bands.

The circumstance that the numerical results of x

near c, are extremely sensitive to N(E) will disappear in the theory which takes into account the existence of a localized magnetic moment on the Ni atoms near c,. According to the experiments 16, 101, it is considered that the localized magnetic moment of Ni atoms really exists just before the bulk ferromagnetism appears in Pd-Ni alloys. This problem is beyond the scope of this paper.

[7] GILLESPIE, D. J., MACKLIET, C. A. and SHINDLER, A.

Proc. Intern. Symp. Amorphous Magnetism, Detroit (1972) 343.

[8] MUELLER, F. M., FREEMAN, A. J., DIMMOCK, J. 0. and FURDYNA, A. M., Phys. Rev. B 1 (1970) 4617.

191 MUELLER, F. M., GARLAND, J. W., COHEN, M. H. and BENNEMANN, K. H., Ann. Phys. 67 (1971) 19.

[lo] CABLE, J. W. and CHILD, H. R., Phys. Rev. B 1 (1970)

3809.