Reversible magnetization processes in amorphous alloys

ELSEVIER

Journal of Magnetism and Magnetic Materials 133 (1994) 107-110

journal of magnetism ~ l ~ and magnetic materials

Reversible magnetization processes in amorphous alloys C. Appino, F. Fiorillo * Istituto Elettrotecnico Nazionale Galileo Ferraris and GNSM-1NFM, 10125 Torino, Italy

Abstract

The role of coherent moment rotations on the magnetization process of soft amorphous alloys has been investigated in theory and experiment. In particular, a theoretical formulation for the reversible magnetization curve has been worked out by modeling the spatial distribution of the magnetization in ribbon-shaped samples and its evolution under an applied field. Such a distribution is assumed to derive from the combined effects of frozen-in random anisotropies and the macroscopic in-plane anisotropy. Experiments are reported concerning Fe-based alloy ribbons, where a range of macroscopic anisotropy values are achieved through the application of a variable tensile/compressive stress. In order to determine the reversible component of the magnetization curve I~v(H) a method is devised that exploits the susceptibility after-effect, which permits one to discriminate between contributions from coherent spin rotations and domain wall displacements. The so obtained Ir~v(H) curve and its dependence on stress are found to agree with the theoretical prediction.

1. Introduction

The absence of magnetocrystalline anisotropy makes the magnetic behavior of amorphous alloys strongly susceptible to anisotropies induced by stress, field annealing and shape effects. These determine to a large extent the shape of the magnetization curve and the energy loss. It turns out in particular that, contrary to the case of conventional crystalline materials, reversible moment rotations can effectively compete with irreversible domain wall displacements during the magnetization process [1,2]. The ability to control the respective roles of these two mechanisms is of clear importance in applications, a circumstance that calls for quantitative methods of prediction of the rotational contribution. In this paper we outline a general theoretical approach to the reversible magnetization curve in amorphous ribbons. The theory takes into account the respective roles of: (1) random frozen-in

* Corresponding author. Tel: +39 (11) 3488933; fax +39 (11) 6507611; e-mail: [email protected].

anisotropies, basically deriving from internal stresses; (2) macroscopic anisotropies, contributed by applied stress, ordering and demagnetizing fields; and (3) the applied field. From the related minimum energy spatial distribution of the magnetization and its evolution with the applied field, the reversible recoil curve Ir~v(H) is calculated. It turns out in particular that Ir~v(H) can be accurately described by means of the incomplete function Ell~c, (H/Ho)C], with c = 1/x/2 and H 0 a suitable parameter, having the dimension of a field. The experimental reversible magnetization curve is determined in Fes]BI4Si3C 2 ribbons for various applied stress values ( - 8 0 MPa < o-_< 300 MPa). To extract such curve from the measured total I ( H ) behavior, a method is applied involving the study of the time decay of susceptibility under weak ac fields for various bias field values H. The method hinges on the assumption that while the domain walls are stabilized by relaxation, the coherent rotations are not, and their roles can consequently be distinguished. By a simple choice of the distribution functions for the orientation and the magnitude of the random anisotropies, a final good agreement between theory and experiments is obtained.

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C Appino, F. Fiorillo / Journal of Magnetism and Magnetic Materials 133 (1994) 107-110

108

2. Theory The model for calculating the reversible portion of the magnetization curve in amorphous alloy ribbons relies on the following assumptions: (1) the role of frozen-in stresses can be described by means of the correspondingly induced random in-plane uniaxial anisotropies k; (2) the direction and intensity of the randomly distributed anisotropies are defined over mesoscopic regions with minimum dimensions of the order of the ribbon thickness; (3) the contributions from such regions combine linearly to provide the magnetic state of the sample; (4) the applied stress ~r generates a macroscopic in-plane anisotropy k ~ = (3/2)A~o-. For Aso"> 0 (As~r < 0) the easy direction coincides with the LD (TD); and (5) an additional in-plane anisotropy k s exists, which is longitudinally directed in the as-quenched state. It is caused by magnetostatic effects a n d / o r structural ordering taking place during rapid quenching [3-5] and is modified by annealing, with or without an applied field. In order to simplify the problem, magnetostatic interactions between different regions, which are expected to influence the angular distribution of the magnetization I~, are not considered in the model. Although a full discussion cannot be given here, we will try to outline in the following the basic steps leading to the prediction of the I,.~v(H) curve. Let us therefore consider a specific region, as shown in Fig. 1, where the local anisotropy k makes an angle q5 with the LD. The macroscopic in-plane anisotropy k u = k,~ + ke is assumed to lie along the LD. This is the easy direction in positive As alloys, for applied tensile stresses and for compressive stress ~rc < 2ke/3A s. Higher values of % cause the easy direction to switch along the TD. The local magnetization vector I~, which is aligned with k when k u = 0 and H = 0, will rotate under the combined effects of k~, k and H towards a novel equilibrium direction making an angle y with the LD, where the energy

E ( y ) = k~ sin2y + k sin2(~b - y ) - HI~ cos 3,

ku

-Zb- "

/7

(1)

it,, /

/

/

k H Fig. 1. Equilibrium direction of the magnetization I s under the effect of the applied field H in a region of local anisotropy k, making an angle ~b with LD. k~ represents the macroscopic anisotropy term.

is minimum. The distribution of magnitude k and angle ~b of the local anisotropy over the different regions leads to a corresponding distribution for the angle y. We define the density functions for k and ~b as f ( k ) and g(q~), respectively, with k and 4' independent random variables. We assume that f ( k ) and g(q5) are given by the expressions

f(k) = (2/(k))2k exp(-2k/(k)),

(2)

g(,b) = (1 - a)/'rr + a6(6 - ~r/2).

(3)

The Sawada function (2), where ( k ) = f ~ f ( k ) k dk, has been shown to describe correctly the distribution of the amplitude of internal stresses in amorphous ribbons [6]. gob) is assumed to be plane uniform in the range - ~ r / 2 < ,b -< -rr/2, but for an additional singular contribution accounting for those regions, of fractional volume a, where the easy axis is perpendicular to the ribbon plane [1]. p(y), the angular density function for I S in the presence of an applied field H, can be obtained from functions (2) and (3), introducing the relation between y and the random variables k and ~b, imposed by the minimum energy condition (1). First, we calculate according to standard statistical methods [7], the cumulative distribution function:

P(~,) = ffD~,,S(k)g(4~)

dk d~b,

(4)

where D(k, rb) is the suitable integration domain in the (k, ¢) plane. The domain boundary is defined, according to the solution of Eq. (1), by the function:

k(cb) = ( k , + H I J 2 cos 3') × (sin 23,/sin 2(¢b - 7))(5) The density function p ( y ) = d P ( y ) / d 7 can then be obtained and used to calculate the component of the magnetization along the field direction: I~ev(H) = I~[ ~/2 cos y p ( y ) dy, "

(6)

-rr/2

and the related susceptibility Xrev(H) = dlre,,(H)/dH. It should be stressed that fray(H), as given by Eq. (6), relates only to coherent spin rotations and does not aecount for a possible contribution from reversible wall motion. I f we are mainly interested in the reversible portion of the magnetization curve from saturation to remanence, we can neglect such a contribution. Consequently, Xrev(H)=xmt(H). Remarkably enough, the so calculated reversible recoil curve Ir~,,(H) turns out to be accurately described by the expression: Irev(/-/) = I s -

(1/c)

xlsz{r[1/cl-r[1/c, (H//40)~]}, (7)

C. Appino, F. Fiorillo/Journal of Magnetism and MagneticMaterials 133 (1994) 107-110 r

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eo

~

r

•

"

I

i"

I

'

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o.5 o e

r .,

I

I000

,,I

2000

,J

,,

,I

,

3000

,i

....

4000

Field (A/m) Fig. 2. Full points: experimental recoil curves I(H) in a Fe81B14Si3C2 alloy ribbon, determined for various applied compressive/tensile stress values (-80, 0, 30 and 100 MPa). Continuous lines: reversible curves Irev(H) predicted by Eq. (6), with fitting parameters (k) = 3000 J/m 3, k e = 350 J/m 3, a = 0.l (see text for an explanation of the symbols).

where

F[1/c] = f;exp(-t)g

109

(1/c)-1 dt

and

r[1/c, ( H/Ho) c] = f~ou/"°,c exp(-t)t (1/c,-1

at

are the complete and incomplete gamma functions, respectively, with c = 1 / v ~ . Z and H 0 are two parameters related to ( k ) and k u.

3. Experiments

Amorphous ribbons of composition FesIB14Si3C 2 (A s = 30 x 10 -6) have been investigated. The complete static hysteresis curve, automatically corrected for the demagnetizing effect, was determined on single-strip specimens (length 200 mm, thickness 38 i~m, width 5 mm) up to a maximum applied field of 4 × 103 A / m . Tensile and compressive stresses could be applied by means of a spring loading device ( - 80 MPa _
and, according to domain observations, a = 0.1. There is a clear discrepancy at low fields between the experimental I(H) curve and the predicted reversible curve. This must be attributed to irreversible domain wall displacements, which contribute to the reversal of magnetization in association with coherent moment rotations and lead to a far lower remanence than expected from reversible processes alone. The problem therefore arises of separately evaluating the reversible and irreversible components of the I(H) curve. To understand how this can be achieved, let us define the weak field ac susceptibility measured around a generic point of the static I(H) curve as the sum of rotational and domain wall contributions X =Xrot +Xw. An experiment can be done where, after having magnetized the sample to saturation, the field is decreased, until a predefined point of the curve is reached. The weak field ac susceptibility X is then measured and its decay versus time suitably recorded. Such a decay must be attributed to the progressive stabilization of the domain walls induced by atom pair reorientations. On the other hand, small angle coherent spin rotations are negligibly influenced by relaxation after-effect. We are thus in a position to separately determine Xrot and Xw. We therefore define as t I and t2, with t 2 > t l , two generic times after the start of the experiment and suppose to determine the ratio /2 = X(tl)/X(t 2) under a condition characterized by exclusive domain wall motion (e.g. high tensile stresses in positive As alloys). Since in this case X = Xw, we achieve once and for all the quantity /2 =Xw(tl)/Xw(t2)> 1. /2 can in fact be taken to be substantially independent of the specific magnetic state and domain wall energy [10]. If X(tl, H) and X(t2, H) are the total ac susceptibilities measured at a generic point of the I(H) curve, we can therefore write:

x(t,, H)=Xrot(H)+Xw(tl, H),

(8)

X(t 2, n ) = X r o t ( n ) +Xw(t2, n ) .

(9)

Introducing the previously defined quantity/2, we get: Xrot(H) = X r e v ( n ) = dl~ev(H)/dH

= [~X(/4,

t2) - X ( / V ,

t~,)]/(~

- 1).

(10)

It is then a simple matter to obtain Irev(H) by integrating Eq. (10). We have measured X(tl, H) and X(t2, H) on the present samples through application of a smallamplitude ac field ( f = 1 kHz, induction swing= 10 -3 T), with measuring times t 1 = 10 s and t 2 = 1.8 × 103 s. ,(2 has been determined for o-= 300 MPa. The reversible magnetization curves Irev(H) obtained by means of the above procedure are shown by the full points in Fig. 3 and appear to be in satisfactory agreement with the previously shown theoretical curves (continuous lines, same as Fig. 2). As already stressed, the theoretical Irev(H) curve provided by Eq. (6) is

110

C. Appino, F. Fiorillo / Journal of Magnetism and Magnetic Materials 133 (1994) 107-110

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References

4000

[1] H. Kronmiiller, N. Moser and T. Reininger, Anales de Fisica B 86 (1990) 1. [2] J.M. Barandiaran, M. V~zquez, A. Hernando, J. Gonzalez and G. Rivero, IEEE Trans. Magn. 25 (1989) 3330. [3] H. Fujimori, in: Amorphous Metallic Alloys, ed. F.E. Luborsky (Butterworths, London, 1983) p.300. [4] P. Allia and F. Vinai, Europhys. Lett. 13 (1990) 367. [5] M. Tejedor, J.A. Garcia and J. Carrizo, J. Magn. Magn. Mater. 117 (1992) 141. [6] L. Malkinski, Z. Kaczkowski B. Augustyniak, J. Magn. Magn. Mater. 112 (1992) 323. [7] A. Papoulis, Probability Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965) p.187. [8] C. Appino and F. Fiorillo, J. Appl. Phys. 69 (1991) 5020. [9] C. Appino, F. Fiorillo and A. Maraner, IEEE Trans. Magn. 29 (1993). [10] H. Kronmiiller and N. Moser in: Amorphous Metallic Alloys, ed. F.E. Luborsky (Butterworths, London, 1983) p.341. [11] G. Bertotti, F. Fiorillo and M. Pasquale, IEEE Trans. Magn. 29 (1993).

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i

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i

2000

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3000

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Field (A/m) Fig. 3. Full points: experimental reversible curves l;ev(H), obtained from the I(H) curves shown in Fig. 2 through the measurement of the ac weak field susceptibility at various values of H. Continuous lines: as in Fig. 2,

accurately described by function (7). It has been shown that such function can be successfully introduced in the prediction of hysteresis loop shapes in amorphous alloys by means of improved Preisach modeling [11].