MODELLING THE (GALVANO-) MAGNETIC BEHAVIOUR OF PERMALLOY SENSORS

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MODELLING THE (GALVANO-) MAGNETIC BEHAVIOUR OF PERMALLOY SENSORS

R. de Ridder, J. Fluitman

To cite this version:

R. de Ridder, J. Fluitman. MODELLING THE (GALVANO-) MAGNETIC BEHAVIOUR OF PERMALLOY SENSORS. Journal de Physique Colloques, 1985, 46 (C6), pp.C6-287-C6-290.

�10.1051/jphyscol:1985650�. �jpa-00224905�

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JOURNAL DE PHYSIQUE

Colloque C6, supplément au n°9, Tome *6, septembre 1985 page Cé-287

MODELLING THE (GALVANO-) MAGNETIC BEHAVIOUR OF PERMALLOY SENSORS

R.M. de Ridder and J.H. Fluitman

Twente University of Technology, Dept. of Electrical Engineering, P.O. Box 217, 7500 AE Enschede, The Netherlands

Résumé - Les écarts entre les courbes magnétorésistives mesurées et les courbes théoriques peuvent être expliquées en majeure partie par

l'introduction de deux effets: l'épaisseur non-uniforme de la couche magnétique et la dispersion de l'intensité de champ d'anisotropie.

Abstract - The deviations from theory of measured magnetoresistance curves can be largely explained by introducing two effects: non-uniform thickness

(edge-tapering) and dispersion in the strength of the anisotropy field.

1 - INTRODUCTION

Galvano-magnetism in permalloy thin films is present because as a consequence of externally applied magnetic fields the magnetization vector in the film orients itself into a position of minimum (free) energy, while the material exhibits an anisotropy in the conductivity with respect to this orientation.

The direction of the magnetization vector is determined by the strength and orientation of the applied field and by the geometry of the film. The latter influence is caused by the occurence of self-fields as a consequence of gradients in the magnet-

ization distribution.

The configuration which has been investigated most thoroughly is depicted in fig.l, where it is assumed that the magnetization without an externally applied field is oriented along the strip axis, because of an intrinsic anisotropy, and that the magnetization vectors rotate under the influence of an externally applied field oriented in the plane of the film and perpendicular to the strip axis.

As a consequence of the self-fields the rotation of the magnetization is not homo- geneously distributed in the film plane. In the next section we give the equations and the solutions which govern this situation assuming a rectangular cross-section and homogeneous magnetic properties.

The experimental magnetoresistance curves mostly deviate somewhat from the theoretical ones. These deviations are generally small and need not be a problem in practice.

In fact they may even have the advantage of giving somewhat smoother curves compared to the theoretically predicted ones. In our experience the deviations, apart from incidental ones with sometimes exotic appearance, show general characteristics, which means that they must be explained by general principles. This is the first motivation for the underlying study.

Next, it is felt that for a proper design of sensors it is really needed to design models which explain the deviations mentioned. Especially in our work on magnetometers this need was felt, in order to explain some unexpected hysteresis effects /l/. From this research the suggestion that some of the deviations should be explained by a dispersion in the magnitude of the magnetic anisotropy was rather strong so we have worked this out in detail. (Earlier efforts to explain the deviations in terms of angular dispersion had failed.) It turned out that the deviations mentioned can be explained indeed with such a model, but it also proved that the strip edges play a significant role in the explanation. Therefore the effect of different edge profiles (e.g. tapered edges) has also been studied.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985650

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C6-288

2 -

^THEORY

JOURNAL D E PHYSIQUE

We

assume a striplike structure as depicted in fig.1 with a rectangular cross- section. The thickness is t, the width is

W. An

intrinsic magnetic anisotropy, if present, will

be

directed along the strip axis. The strength will

be

expressed in tens of an anisotropy field Hk.

If the cross-section has an elliptic profile it can easily be proven that this form gives rise to a (form) anisotropy of value Hkf=%/w, with MS the value of

the (saturation) magnetization.

?he total effective anisotropy in this case is: Hkeff=Hk+~kf.

For a rectangular profile such an effective anisotropy can not

be

defined because a homogeneous applied field does not lead to a homogeneous magnetization distribution.

Nevertheless we will use the factor Hk as a normalisation factor which proves

m

be

convenient in practice. We next S i n e the coefficient a =.Hk /Hkeff as a measure of the fraction of form anisotropy over the total anlsoEropy. For Hk>>%/w we have a-->@ and for Hk<<tMs/w we have a->l. If an applied field Hap 1 is present in the plane of the film and perpendicular to its axis the total fiefd H inside the strip is composed of this applied field and the generated self- field according to the formula:

Next we have for the resistance of the strip:

w/2

R = R

- E

0

W l/,

^(-pi)^S^:

^dye

As

long as there are no saturation effects, which means that the magnetization has nowhere reached its final direction perpendicular to the strip axis we have:

%/M, ⁼

sinp

=

H/Hk

with

My

the component of MS along the y-axis, and p the angle of rotation of the mgnetlzation vector from the strip axis. Since My is linearly dependent on H, and consequently on Hap 1 in this case, the resistance

R

will show

a

quadratic

behaviour whatever fhe spatial form of Haeel.

With the aid of a computerprogram we arrive at the results depicted in figure 2. In the unsaturated case (Happl<Hkeff)

m

find a purely quadratic behaviour which can be expressed as:

Figure 1. Sensor Figure 2. Theoretical mag- Figure 3. Relation between

configuration netoresistance curves

"y

and a.

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These results surmrize the magnetoresistance behaviour of a strip under the given assumptions.

The deviations which can be found in practice mostly show a slower approach to satu- ration and/or a rather short range of pure quadratic behaviour. Two characteristic cases are shown in figures 4a and 4b, where the crosses mark the measurement data and the drawn curves (1) are obtained from the model described above.

In order to study the possible causes of these deviations the computerprogram was extended with two facilities, one to include the magnitude dispersion mentioned above and one to include variations in thickness over the width of the strip (e.g. edge tapers).

3 - COMPUTER MODEL

A computer program for calculating the magnetization pattern in a permalloy strip having rectangular cross-section has been described earlyer

/3/.

The influence of anisotropy dispersion on the behaviour of orthogonal thin film magnetometers is described by an analytical model /4/ which however needs the assumption that magni- tude dispersion is "small" with respect to the mean anisotropy field. A computer model accounting for anisotropy dispersion /5/ uses position-dependent demagnetizing factors, which is computationally efficient but can lead to large errors when saturation of the film occurs.

Therefore, we have chosen to extend our previous model

/3/,

which uses an iterative method to calculate the demagnetizing field accurately, with the capability to deal with discrete anisotropy distributions and non-uniform film thickness. First, an extra term accounting for thickness-variation was added to the expression calculating the demagnetizing field. Second, the strip is considered as a collection of

"numerical domains", each described by 4 parameters: position in the strip, direction of the easy axis, strength of the anisotropy field and a weight factor giving the relative area of this domain. All domains individually obey the Stoner-Wohlfarth single domain model under the influence of the local magnetic field. The only coupling between domains occurs via the (homogeneous) applied field and the

(position-dependent) demagnetizing field.

4 - EXPERIMENTS

Samples have been produced in three different ways. One series (A) was produced using shadow masks, a second series (B) using photolithography and (wet) chemical etching and a third series

(C)

was produced using the lift-off technique. The masked samples where produced in a single run by vacuum evaporation (film thickness 150

m ) , the

other ones were sputter deposited (thickness 50

m);

both series were produced within the same run. Strip widths ranged from 10 to 4000pm for B and C and from 350 to 2100

p

for A. The composition of the material was choosen to be the one that is free from

magnetostriction.

The results of the series were all different. Especially the curves for the narrowest samples differed greatly. It turned out, however, that the curves for these small devices could be described assuming edge tapers. If the same tapers were introduced in the simulation of the results for the wider samples a good fit was absent and a mgnitude dispersion of the anisotropy was needed to fit theory with experiment.

It is reasonable to suppose that edge taper effects decrease the wider the samples

are because of a relatively decreasing edge area. On the other hand it is reasonable

to suppose that the effects of magnitude dispersion of the anisotropy, visible in the

widest samples, disappear in smaller samples because of the increasing dominance of

form anisotropy over intrinsic anisotropy. Therefore we derived a taper form from the

results of the smallest samples and a magnitude distribution from the results of the

widest samples. For the intermediate samples both effects were introduced in the

simulation of the curves, a necessary and sufficient measure to fit theory to experi-

ment, and this worked without further adjustment of taper form or dispersion dis-

tribution. Figure 4 gives

a

characteristic example of this exercise. During measure-

ments, we applied a small bias-field parallel to the easy axis. This leads to some

extra deviations from the theoretical curves in fig. 2. The computer model accounts

for this.

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JOURNAL DE PHYSIQUE

The (discrete) anisotropy distribution we need for fitting depends somewhat on the permalloy deposition method and is far from being a normal one: typically 5..7% of the film must have a local Hk>2Hk0 and 1.5..3% has Hk>2@Hk~, where Hk0 is

the mean value of the anisotropy field.

Each of the series A,B,C needed different tapering parameters. Series C (lift-off) needs a slightly concave taper of 3.5pm length. The etched devices (B) have only a very short taper (in the order of film thickness), which could not be determined more accurately because of the discretization of our model. The shadow-masked strips (A) show an "inverse" taper i.e. the edges are thicker than the rest of the film.

a b) c )

Figure 4. Fitting of model parameters to experimental data of permalloy strips produced using the lift-off technique. a) v4000 p, b) ~ 1 p, 0 c) ~ 1 0 0

p.

Curves: (1): no correction,

(2):

anisotropy dispersion, (3):edge taper, (4): (2)+(3) 5 - CONCLUSION

It can be concluded that all results can

be

described by the simple model of section 2 corrected for edge tapers and magnitude dispersion. The first is an extrinsic effect which is dominant in samples where the anisotropy is determined by the aspect ratio t/w. It must

be

said that in this case the magnetoresistance curves may be strongly affected by relatively small deviations from the rectangular profile form and thus will

be

strongly dependent on the method of preparation.

The second effect has an intrinsic nature and dominates the description in relatively wide samples where the intrinsic anisotropy is significant and the edge region is small compared to the total sample surface.

An explanation of the results in terms of only one of the model corrections is not possible. The significance of magnitude dispersion, which showed up in our research on magnetometers, where relatively wide samples are used, seems to be confirmed by the investigations described here.

We like to thank P. de Haan, J.P.J. Groenland, H.J.M. Geraedts, H.W. Krabbe and

C.