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MAGNETIC ANISOTROPY OF Ni-Cu ALLOYS

G. A U B E R T and B. M I C H E L U T T I Laboratoire de Magn~tisme, C.N.R.S., BP 166 X CT, 38042-Grenoble-C~dex, France

The temperature and concentrat ion dependences of the anisotropy of energy and magnetizat ion of N i - C u alloys (up to 3% Cu) are interpreted by a model in which the anisotropic properties result f rom a perturbat ion of the electronic densi ty of states by the rotation of the magnetization.

I. Introduction

The anisotropic properties of a ferromagnetic crystal at constant pressure and temperature are macroscopically introduced by means of the free energy of the unit volume of a spherical sample written in the form

- - 1 4 r e x 2 F = E~ - H • M --r ~ ~Trlvl . (1)

E a is the magnetocrystalline anisotropy energy and can be described by various expansions as a function of the direction of the magnetization M. Together with this anisotropy of energy, fer- romagnets also exhibit an anisotropy of the mag- netization [1] which, despite its smallness, can be measured and gives relevant information for the understanding of the fundamental mechanisms.

2. Experiment

Using the experimental technique already de- scribed [l], we have studied the anisotropy of energy and magnetization of Ni -Cu alloys (up to 3% Cu) in the temperature range 4.2-300K. Detailed results will be reported elsewhere and here we shall limit ourselves to the discussion of the magnetocrystalline anisotropy energy and magnetization differences AE~= E ~ ( l l l ) - Ea(100) and AM = M ( 1 1 1 ) - M(100) between the two directions of highest symmetry in a cubic crystal.

As for pure Ni, A E a and AM are found to be field independent up to 20 kOe and their temper- ature dependences are very similar to that of Ni. The energy curves of the alloys can be related to that of pure Ni by means of a concentrat ion dependent scale factor but the exponential form proposed for it by Franse [2] appears to be inadequate. Although the thermal variations of AM look roughly the same for all alloys, the concentrat ion dependence of AM is more com- plicated than that of AEa: at room temperature

AM decreases with increasing Cu concentrat ion but at low temperature AM begins to increase before decreasing as it is shown in table I which gives some values at 4.2 K. This oscillation of AM has been confirmed by repeated experiments on different samples.

Table I - Ae~ in 10 _2 K/Ni at. and Am in I0 "/~B/Ni at. at 4.2 K for pure Ni and two N i - C u alloys (% Cu in mass) .

% Cu 0 1.52 3.02

- Ae~ 3.126 2.700 2.359 Am 1.201 1.316 1.051

3. A tentative model

Many attempts have been made in order to understand the order of magnitude and the ther- mal behaviour of the magnetocrystalline anisot- ropy of 3d metals. The models based on an introduction of spin-orbi t coupling effects in band calculations are rather promising [3-4] but the difficulties encountered make a complete interpretation from the first principles very ques- tionable. This is the reason why we tried to interpret our experiments by the following semi- phenomenological model. Let g (E) be the density of states for spin up or spin down electrons of a Ni atom. This density of states is supposed not to be modified by alloying at these concentrations. The effect of S.O. coupling is to introduce a dependence in the direction of M that is Ag (e) = g(E)( l l l ) - -g(E)(100) . We consider that fer- romagnetism does not affect the shape of the up and down spin bands and we simply use an effective field approximation with a splitting of the bands 2Win where m is the magnetization in Bohr magneton per Ni atom (C.P.A. calculations for Ni rich alloys [5] show that the d holes are located primarily on Ni sites and we suppose here that Cu atoms do not participate in the

Physica 86.-88B (1977) 295-296 C~) North-Holland

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anisotropy and magnetization and anyway, this is numerically unimportant at low concentrations). Le t /x be the chemical potential of the electrons at a given temperature (/3 = 1/kT) , we call /x ~' = iz + Wm, lx ], = lz - Wm, g ~ o r g ~ g ( / x t or

/.t ~ ) and the same notation for other functions of E or/x. Our purpose is to determine Ag(e) from the experimental data.

By an almost straightforward calculation and some obvious approximations we get

- A e . = ( A H I' + A H $ ),

A m = l / g , [ + l / g ] , - 4 W \ g ~ g ~ - '

(2)

where Aea is AEa by Ni atom, Am is AM in Bohr magneton per Ni atom,

f +~ 1 de, Ah (/3,/x) = -~ Ag (e) exp [/3 (e - / . t )] + 1

AH(/3,/x) is the sim;lar expression obtained by replacing Ag (E) by AG (E) = f'~ Ag (e) de with AG(+ oc) = 0 and, because S.O. coupling does not change the mean energy of the band, f+_2AG(e)de =0 . We then introduce A~d(E)= f_'~AG(~)de with Aq$(+oo)=0 and make a Fourier transform of A% that is

,f0 A~(e) = ~ [M(a) cos ae + ~ ( a ) s i n a e ] d a

(3)

with obvious relations between ~¢ (a), ~ (a) and A~(e) or Ag(e). By means of some manipula- tions of the Fermi-Dirac integrals, we get:

_ ~ f o ~ a rr k T A H = sg'(a ) - - sh a rrk T

d a ,

~ , ~ fo ~ aTrkT Ah = A '(a ) - - da, sh arrkT

with

(4)

M ' ( a ) = M(a) cos a/.t + ~ ( a ) sin a/x,

A '(a ) = a ~ ( a ) cos alx - a M ( a ) sin a/x. (5)

It is thus possible through relations (2), (4) and (5) to analyse the experimental data and by (3) to get Ag(e). This inversion of the data involves some reasonable assumptions about the fer- romagnetic splitting and the variation o f / s with alloying (these assumptions must be consistent with the experimental magnetizations and sus- ceptibilities). It is not possible to give here a complete discussion which will be published elsewhere, so we mention only the most striking results:

(a) The variations of the anisotropy energy with temperature can be fitted within 10 3 be- tween 0 and 300 K by a single function Ae~ = A e ° ( a T / s h a T ) with a = 0.018 K -~ for all alloys and A e ~ = - 3 . 1 3 x 10-"K/at. for pure Ni. This means that Ag(E) is essentially, in the region of e = tz $ and (or) ~ t , an oscillating function with a "per iod" in energy 27r-'/a = 7 6 2 c m ~ which is precisely the order of magnitude that can be expected from S.O. coupling effects. From Aea ° we get an amplitude of the oscillations of about 10 -2 eV-I/at.

(b) It is impossible to account completely for Am by a spin contribution only and the orbital part must be of the same order of magnitude as the spin one, which confirms the Kondorskii evaluations at 0 K [3].

4. C o n c l u s i o n

The approach that has been briefly presented can constitute an interesting intermediate step between the experiments and the ab initio calcu- lations. The order of magnitude of the changes in the electronic density of states which are neces- sary to account for the experimental results confirm that these calculations will not be so easily completed.

R e f e r e n c e s

[1] G. Aubert and P. Escudier, Proc. Inter. Conf. Magn. Moscow 1 (1973) 215.

[2] J.J.M. Franse, G. de Vries and T.F.M. Kortekaas, Proc. Inter. Conf. Magn. Moscow 3 (1973) 84.

[3] E.I. Kondorskii and E. Straube, Zh ETF Pis. Red. 17 (1973) 41.

[4] N. Mori, Y. Fukuda and T. Ukai, J. Phys. Soc. Japan 37 (1974) 1263.

[5] S. Kirkpatrick, B. Velicky and H. Ehrenreich, Phys. Rev. B8 (1970) 3250.