Magnetic isotherms of heterogeneous ferromagnets

Physica 112B (1982) l-5 North-Holland Publishing

MAGNETIC

Company

ISOTHERMS

OF HETEROGENEOUS

FERROMAGNETS

D. WAGNER Institit fib TheoretischePhysik 3, Ruhr-Uniuersitiit.

Bochum,

F.R. Germany

E.P. WOHLFARTH Department

of Mathematics,

Imperial

College, London SW7 2BZ,

U.K.

Received 23 March 1981 Revised 20 June 1981

The Arrott plots of heterogeneous (crystalline and amorphous) ferromagnets are discussed in terms of concentration fluctuations of the magnetic atoms, using a Heisenberg model. It is shown that the Arrott plots are curved, the curvature being dependent on the magnitude of the concentration fluctuations. The curvature is positive above Tc and negative below Tc. These results are quantitatively similar to those obtained earlier by Shtrikman and Wohlfarth [6] on the basis of a band model. Hence the present model is more generally useful. However, in contrast to [6], a downward shift of the Curie temperature due to the concentration fluctuations was also obtained. The present results are compared with recent experiments on crystalline and amorphous alloys.

1. Introduction

It is well known, that heterogeneous ferromagnets exhibit curved Arrott plots (M* versus H/M, where M is the magnetization in a field H) [l-5]. The curvature of the Arrott plots are a measure of the magnitude of the concentration fluctuations. This problem has been dealt with theoretically on the basis of a Landau-Ginzburg equation for inhomogeneous ferromagnetic systems [6-81. Recently some doubt has been cast on the validity of this approach [9]. Therefore it may be appropriate to reconsider this approach and its applicability to amorphous ferromagnetism, starting with a Landau-Ginzburg equation, which was derived on the basis of a localized model [lo]. As shown in the present paper, section 2, some results obtained on the present basis agree quantitatively with those obtained earlier (Shtrikman and Wohlfarth [6]), but others differ from these earlier results. We are thus planning to reconsider the same problem on the basis of the Hubbard Hamiltonian. The present 037%4363/82/OOOO-OOOO/$O2.75 @ 1982 North-Holland

calculations made it possible to deduce the concentration dependence of the Landau parameters; a knowledge of this dependence is essential if one wants to calculate the change of the ferromagnetic or paramagnetic Curie temperature on going, for example, from the crystalline to the amorphous state. Although these experiments are rarely carried out without changing the chemical composition, experiments on amorphous and on disordered crystalline systems of nearly equal chemical composition seem to be sufficiently well represented by this theory [lo-121.

2. Theory Starting with a Heisenberg model for a disordered spin system it is easy to see that the free energy of the system with a concentration c(r) of the magnetic atoms can be expanded in powers of the local magnetization M(r) as follows [lo]

2

D. Wagner and E.P. Wohlfarth

-/3F(H,

T) = 1

-

/ Magnetic

d3r 1 d3r’V(r - r’)M(r)M(r’)

I

d3r kg+:% 1

- PH(r)M(r))

(1)

.

Here a and 6 are positive constants; H(r) is the external magnetic field and V(r - r’) is proportional to the exchange interaction J(r - r’): V(r - r’) = pJ(r - r’);

+ B(r)M3(r) - CAM(r) = H(r),

state: In disordered systems one cannot obtain Tc hy considering the eigenvalues of the linearized equation of state for H = 0. To deduce the Arrott plots from eq. (2) it is convenient to follow Shtrikman and Wohlfarth [6] and to expand A and B only in powers of the concentration fluctuations

y(r) = Gc(rYc0, where c(r) = CO+ SC(r) and where co is the spatial mean value of the concentration. Then

fI = 1IkaT.

In principle V could depend on r and r’ separately but it is here taken as translationally invariant for simplicity. Minimising the free energy as a functional of M(r) gives the Landau-Ginzburg equation or the equation of state [lO] A(r)M(r)

isotherms of heterogeneous ferromagnets

(2)

A(r) = A0 + A’y(r) + A”y2(r) , B(r) = B. + B’y(r) + B”y2(r).

The expansion (4) differ from those in ref. [6] where, due to the implicit use of a simple band model, A” = 0, B’ = B” = 0. Introducing as in ref. [6] Fourier transformed quantities

y(r) = C

with

k

A(r)=s--2J,

B(r) =

‘ok exp(ik - r)

= M +

M(r)

j$$y

(4)

,

2 kfk exp(ik

-

r) ,

k#O

one finds from eq. (2) C = :Jp’,

P2 =

J=

I

J(r)d3r, x=Ao+y-@‘-6y)~b’k12

d3rJ(r)r2

I

(3) I

d3rJ(r)

It should be mentioned that it is important to take the concentration dependence of B into account, since this dependence is largely responsible for the decrease of the spontaneous magnetization with increasing concentration fluctuations, whereas the specific dependence of A on c(r) is mainly responsible for the shift of the ferromagnetic Curie temperature to lower values with increasing fluctuations of c(r) [lo]. The reason for not finding such a shift in a similar treatment of this problem by Ftinle [13] is due to an inconsistent treatment of the equation of

+tA’- 9Y)z k

3y 2 l%12~(k).

,rhy;k)+

k

Here we used the abbreviations x = HIM,

f(k)=

y=B&i’

7

-$$&&s 0

where t2 is the correlation

length given by

f2= C/(A,,+3y). In deriving eq. (5) from the Landau-Ginzburg

D. Wagner and E.P. Wohlfarth I Magnetic isothermsof heterogeneousfewomagnets

equation

we used the fact that, due to eq. (3) A” = -A’(>O) and that B’ = -3Bo, B” = 6B0. The Arrott plots as given by eq. (5) now depend on the random function &z(r). Therefore, one has to perform an average over the possible distributions of the concentration The distribution function is fluctuations. generally unknown and depends on material constants as well as on the preparation parameters of the system under discussion. For simplicity we use the white noise spectrum* [7,81:

Here y2 is the variance of the relative concentration fluctuations. This choice of yk takes itIt0 account the fact that short wavelengths cannot contribute to the (unknown) distribution of the random function SC(~) since these have been taken into account already in deriving the free energy, eq. (1). Also k:’ is a very approximate measure of the inhomogeneity of the density of the magnetic atoms. The integrals can be easily evaluated. From eq. (5) we obtain x=AO+y-y2F,

(6)

where F contains the deviations from the linear Arrott plots. It is given by F =‘A’ _ 6y + 3 (A’ - 9y)(A’ - 3y) Ao+~Y

I-&arctan

k&)-zy

(H)

(7)

* Other distribution functions have also been considered; the results show that in the limits k,& 1 the formulae (8) and (9) tend to be independent of the details.

3

There are two regimes where this expression simplifies considerably: kc& * 1 and kc5 4 1. The region between these two extreme cases depends very markedly on k, or, stated somewhat more generally, on the details of the distribution function of the random function SC(~). For k& % 1 we obtain, independently of k,, x=A,,+y-(A’-6y)y’.

(8)

In this region, where the Curie temperature is approached, one thus obtains linear Arrott plots. The simplicity of these plots in this region is probably caused by the diverging correlation length 5 where inhomogeneity effects are smeared out. This region is only a small one in the x, y plane, as has already been discussed previously [6,8]. Incidentally, the more recent papers on this subject, mentioned above [9], are restricted to this region which is relatively unimportant for the discussion of the influence of heterogeneities on the Arrott plots. The region was, in fact, not considered in the previous work [6,8] which was concerned with the much larger area of the x, y plane outside this region corresponding to k,[ < 1. In this larger region, Kortekaas et al. [l] showed that reasonable results can be obtained for the concentration fluctuations on the basis of ref. [6]. Finally, we would like to point out that from eq. (8) we obtain the shift of the Curie temperature due to the concentration fluctuation y2. Putting x = 0, y = 0 we have A,(T,) = A’y2 or (Tc- T$‘)/T~‘= -y2, where T$? is the Curie temperature of the homogeneous system, exactly as given in ref. [lo]. However, it should be pointed out, that the validity of eq. (8) must be examined more carefully in this case, since H+ 0, T + Tc means A0 + 3y + 0 so that the prefactors in eq. (7) diverge. Nevertheless, writing t2 = &IT - T&T,)-’ one can show, that for kc&, S1 eq. (8) is still correct; for details we refer to ref. [lo]. This calculated shift of Tc is not obtained in the work of Shtrikman and Wohlfarth [6] since,

4

D. W. Wagner and E.P. Wohlfarth

I Magnetic

due to the use of a band model, A” = 0. However, the measurements of H&her et al. [12] on amorphous Fe-Ni-Si-B alloys seem to give such a shift as well as agreement with the calculated correlation of this shift [lo] with the Curie-Weiss constant and the saturation magnetization. Of greater importance is the region k,e 4 1, where, from eqs. (6) and (7) and again independently of k,, x=A,,+~+(A~+A’)~‘-A,,(~&$)~Y’.

(9)

To compare these Arrott plots with those found previously [6] it was decided to characterise them by obtaining their curvatured2yldx2. Then eq. (9) gives 3

= 54A,,

(10)

and from [6] it follows that

!EY=54A

’ (Ao%y)4

dx2

‘* ’

(104

These relations are very similar. Firstly, there is a change of sign of d2y/dx2 when going from T > TC to T < TC, with a negative curvature for T < TC(Ao< 0) and a positive curvature for T > TC(Ao>O). Secondly, the curvatures for two corresponding values A0 = +- lAoI are almost equal in magnitude, giving almost symmetrical Arrott plots about TC. This result follows from both (10) and (lOa). In (lo), using (3) and (4), it follows that Ao+A’=

-W,

(11)

so that (10) becomes 2

= 216A.

.T* (A0 + 3~)~ ‘* ’

and in (10a) A’ is a constant independent

(lob) of T.

isotherms of heterogeneous

ferromagners

Hence (lOa) and (lob) are identical in form. The almost symmetrical Arrott plots about TC (in the above sense) follow from the small influence of the term 3y in these two relations. A small asymmetry does occur, however, as shown in fig. 1 of [6]; the calculations gave slightly straighter Arrott plots at T < Tc than at T > TC. Such an effect has been observed [2,4] but here the observed effect of asymmetry is much more pronounced than calculated on the present basis. To explain this effect it is necessary to solve the Landau-Ginzburg equation more directly, for example, as in the work of Edwards et al. [14] for the dilute alloy problem (Pd-Ni). Here straighter Arrott plots were calculated, in agreement with experiment, for c > cctit than for c < ctit, where c,,+ is the critical concentration for ferromagnetism. This different behaviour according to T or c also follows generally from the present work, since A0 depends on TIC, see eq. (3). For some other alloy systems, such as some binary nickel alloys [15] a similar effect of the sign of A0 on Arrott plots has been observed. In yet others, see Schneider and Zaveta [2], the Arrott plots observed are, however, straighter and more symmetrical with respect to the sign of Ao. Hence, here the agreement with [6] and the present work is better. Where the present analysis is useful in this sense, it was possible (see [63 for ZrZn2 and [l] for Ni3Al and Ni-Pt) to estimate concentration fluctuations of the order of 1% from observed Arrott plots.

3. Conclusions We have obtained the following new results: (1) The magnetic isotherms of heterogeneous ferromagnets were obtained by using a white noise spectrum for the density fluctuations involving a cut-off wave vector k,. With 4 the correlation length, the two limiting cases k&+ C-J and k,[ + 0 were considered. (2) For k,,$ --, QI we get linear Arrott plots; the slope of the Arrott plots is increased by the

D. Wagner and E.P. Wohlfarth / Magnetic isotherms of heterogeneous ferromagnets

concentration fluctuations y2 = (6cIco)2. Moreover, under certain conditions, a shift of the Curie temperature is obtained, given by (Tc TT))/T$) = -y2; this shift was not obtained in the earlier work [6] which was based on a band model. (3) In the much larger region in the M, H plane defined by k&+0, Arrott plots were calculated and characterised by their curvature. This was found to be negative when T < Tc and positive when T < Tc and positive when T > Tc, exactly as in [6]. In both cases, the sign of the curvature is almost the same for A0 = +IAol. Some experimental results are discussed on this basis. (4) The agreement between the calculated An-Ott plots using the present Heisenberg model and the previous band model [6] makes the present model more generally useful. As shown in our previous papers [lo] the model leads to significant results on Curie temperature shifts as well as on shifts of the saturation magnetization. Some experimental results on amorphous alloys [12] are beginning to support this approach.

Acknowledgement

We wish to thank G. Hilscher and the anonymous referee for their invaluable help in correcting errors in an earlier version.

5

References [l] T.F.M. Kortekaas and J.J.M. Frame, J. Phys. F6 (1976) 1161. [2] J. Schneider, A. Handstein, J. Henke, K. Zaveta and T. Mydlasz, J. de Physique Suppl. au no. 8 (1980) 682 (Proc. 4th Conf. on Liquid and Amorphous Metals 1980). J. Schneider and K. Zaveta, Int. Symp. High Purity Mats., Dresden, 3 (1980) 18. [3] J. Schneider, K. Zaveta, A. Handstein, R. Hesske and Haubenreisser, Physica 91B + C (1977) 185. [4] A. LiCnard and J.P. Rebouillat, J. Appl. Phys. 49 (1978) 1680. [5] G. Hilscher and H. Kirchmayr, J. de Phys. 40 (1979) c5-1%. [6] E.P. Wohlfarth and S. Shtrikman, Physica 60 (1972) 427. [7] D. Shapero, J.R. Cullen and E. Callen, Phys. L&t. 50A (1974) 303. [S] H. Yamada and E.P. Wohlfarth, Phys. Lett. 51A (1975) 65. and M. Fahnle, Phys. Stat. Sol. b97 [91 H. Kronmiiller (1980) 513; M. FIhnle and H. Kronmiiller, Phys. Stat. Sol. b98 (1980) 219; G. Herzer, M. Finle, T. Egami and H. Kronmiiller, Phys. Stat. Sol. blO1 (1980) 713. [lOI D. Wagner and E.P. Wohlfarth, J. Phys. F9 (1979) 717; J. Magn. Magn. Mat. 15-18 (1980) 1345. WI M. Goto, Jap. J. Appl. Phys. 19 (1980) 51. WI G. H&her, R. Haferl, H. Kirchmayr, M. Miiller, and H.J. Giintherodt, to be published. P31 M. FiUmle, Phys. Stat. Sol. b99 (1980) 547. 1141 D.M. Edwards, J. Mathon and E.P. Wohlfarth, J. Phys. F5 (1975) 1619. J. Magn. Magn. Mat. 12 [151 F. Acker and R. Huguenin, (1979) 58.