THEORY OF MAGNETOSTRICTION IN AMORPHOUS FERROMAGNETS

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THEORY OF MAGNETOSTRICTION IN

AMORPHOUS FERROMAGNETS

M. Fähnle, J. Furthmüller, G. Herzer

To cite this version:

JOURNAL DE PHYSIQUE

Colloque C8, Supplement au no 12, Tome 49, decembre 1988

THEORY

OF MAGNETOSTRICTION IN AMORPHOUS FERROMAGNETS

M. Fahnle ( I ) , J. Furthmiiller ( I ) and G. Herzer (2)

( I ) Max-Planck-Institut fiir Metallforschung, Institut fir Physik, Heisenbergstr. 1 , 0-7000 Stuttgart 80, F.R.G.

(2) Vacuumschmelze GmbH, Postfach 2253, 6450 Hanau, F.R.G.

Abstract. - Based on the random anisotropy model we calculate within the linearized theory of elasticity the volume averages (&ij (I)) and ( E ; ~ (r)) of the total strain in magnetostrictively deformed amorphous and polycrystalline samples.

The mechanism of magnetostriction is clarified, and effects of annealing or application of stress are discussed.

In crystalline ferromagnets magnetostriction is de- termined by the strain-derivative of the macroscopic magnetic anisotropy tensor. In contrast, in macroscop- ically isotropic amorphous ferromagnets the macro- scopic magnetic anisotropy is zero. However, the mate- rial may be conceived [I-71 as consisting of very small structural units (basically an atom under considera- tion and its nearest neighbours) with strong uniaxial anisotropy and with easy axis varying randomly from site to site (Fig. 1). This so-called random anisotropy model has been justified by microscopic calculations based on the point-charge model by Cochrane et al. [6] for amorphous alloys containing rare-earth atoms, and by Elskser et al. [7] based on Hartree-Fock per- turbation theory for transition metal alloys.

The local magnetic anisotropy energy is a source for magnetostriction, because it may be lowered by the two deformation modes of the units:

(1) the local strain is responsible for: (a) a change of the local anisotropy strength; (b) a strain-induced rotation of the local easy axis (Fig. 2), yielding a better alignment with the magne- tization direction.

Fig. 1.

-

Schematic representation of local anisotropies (the symbols

1

denote the easy axis direction) in monocrystalline (left part) and amorphous (right part) materials.

Fig. 2.

-

Schematic illustration for the strain-induced re- orientation of the local easy axis (symbol f).

Both contributions, which constitute the so-called "conventionaI" magnetostriction mechanism [4, 51 are described by the magnetoelastic tensor Bklij = dKij/a&ki, where Kjj is the local anisotropy tensor;

(2) Rigid rotations of the local units also lead to a better alignment of the local easy axis with the magne- tization direction. This "reorientation" mechanism is not relevant for single crystals, because in this case all units rotate coherently, leading t o a rigid rotation of the whole sample. In contrast, due to the random easy axis orientation in amorphous ferromagnets (Fig. 1) the local rotations may yield a better alignment with the magnetization direction without rotation of the whole sample.

It is the first purpose to find out which of the above discussed mechanisms is responsible for magnetostric- tion of amorphous ferromagnets. The second pur- pose is to discuss the effect of the elastic coupling be- tween neighbouring units and to calculate the mag- netostriction constant from the magnetoelastic prop- erties of the local units (Kij, Bklij) and the elastic tensor Cijkl, which - for elastically anisotropic units - is also a random quantity due to the random orien- tation of the units. Neglecting the elastic coupling and the rigid rotations of the units for a moment, the structural units exhibit spontaneous magnetostric- tive strains, like small crystalline ferromagnets with the corresponding orientation of the easy axes. The effective magnetostriction tensor

BE:^

describing the macroscopic behaviour of the whole material then is given by a simple geometrical average over the local contributions Bklij of the differently oriented struc- tural units [8]. In reality, however, the elastic coupling prevents t h e single units from being strained in the same way as if they were elastically decoupled. As a result, there are deviations of

~ 2 : ~

from the simple average over the local contributions.

Within the linearized theory of elasticity we have de- rived and solved the magnetoelastic equations for the total strain in inhomogeneous magnetostrictive mate- rials. We have adopted a random anisotropy model by defining for every unit a local coordinate system for instance with ,+axis parallel to the local anisotropy axis in which the tensors

~:q"",

c&?

and B&' have

C8 - 1330 JOURNAL DE PHYSIQUE

hexagonal symmetry. For simplicity the local tensor components are assumed to be identical for all units. The only fluctuating quantities are the polar angles .9 and the azimuthal angles cp charaterizing the orien- tations of the local anisotropy axes with respect t o a global coordinate system. As a result of fluctuating angles the tensors Kij, Cijhl and Bklii in the global system are position dependent quantities with spatial fluctuations, e.g., 6Ciik1 (r)

.

The calculations [4, 51 have been performed by two different methods of elasticity theory, the balance-of- force method and the incompatibility method. The magnetoelastic equations have been solved iteratively, yielding a perturbation series with perturbation pa- rameter 6C(r). Because our continuum theory does not. contain any parameter characterizing the size of the units, it is valid both for amorphous and poly- crystalline ferromagnets. For all calculations we have inserted for the local material parameters the values of crystalline Co at 300 K and Gd at 4 K. (It should be noted that the Co values are not identical with those for near-zero magnetostrictive Co-based alloys, see Ref. [4] .)

The main results of the theory are:

(1) for isotropic distribution of the above defined angles 6 and p the volume average of the magne- tostrictive strain (&ij (r)) and hence the magnetostric- tion constant Xzff= 2/3 ((ell)

-

(EL))

(ell

and EL de- note the strain-components parallel and perpendicular to the magnetization) are exclusively determined by the conventional mechanism. There is a contribution of the reorientation mechanism only for volume aver- ages of the type (&& (1)) which characterize the local magnetostriction,'but this contribution is of some im- portance only for near-zero magnetostrictive Co-rich metallic glasses;

( 2 ) there is a contribution of both parts of the con- ventional mechanism, i.e. modifications of the local anisotropy strength (yielding A::, Tab. I) and strain- induced rotations of the easy axes (Xzff -A::)

.

The relative importance depends on the material parame- ters of the local units (Tab. I);

(3) the contribution from elastic coupling effects (which is approximately given by the difference be- tween the second-order approximations and the zeroth- order approximations in Tab. I) depends sensitively on the local material parameters;

(4) modifications of XZff due to annealing or ap- plication of external stress may be explained by an anisotropic distribution of 6 and cp and by modifica-

Table I. - Results for

x : ~

for local material parameters like i n crystalline Co or Gd ([4]), ,for different orders of perturbation theory and according to the balance-of- force method ( b f m ) or incompatibili~!y method (im). : A:

i s the contribution exclusively due to modifications of the local anisotropy strengths (see ]text).

tions of the local Btensor, respectively. For strong annealing we obtain [4] changes of' typically

AX:^

x

lov7,

and for an external stress of 1 MPA we find [4]

x some 10-lo. Both values agree qualitatively with experimental results [9, 101.

[I] O'Handley, R. C., J. Appl. Phys. 62 (1987) R15. [2] Szymczak, H., J. Magn. Magn. Mater. 67 (1987)

227.

[3] Fiihnle, M. and Egami, T., J'. Appl. Phys. 53

(1982) 2319.

[4] Furthmuller, J., Fiihnle, M. and Herzer, G., J.

Phys. F16 (1986) L255; J. Magn. Magn. Mater.

69 (1987) 79; J. Magn. Magn. Mater. 69 (1987) 89.

[5] Fiihnle, M. and Furthmuller, J., J. Magn. Magn. Mater. 72 (1988) 6.

[6] Cochrane, R. W., Harris, R. and Plischke, M., J. Non. Cryst. Solids 15 (1974) :239.

[7] Elsbser, C., F5hnle, M., Brandt, E. H. and Bohm, M. C., to be published.

[8] O'Handley, R. C. and Grant, N. J., Proc. Conf. on Rapidly Quenched Metals, Eds. S. Steeb and Warlimont, H. (North Holland, Amsterdam) 1985, p. 1125.

[9] Hernando, A., Vazquez, M., Madurga, V. and Kronmuller, H., J. Magn. d4agn. Mater. 37

(1983) 161.