NÉEL LINES STRUCTURES IN UNIAXIAL FERROMAGNETS WITH QUALITY FACTOR Q > 1

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NÉEL LINES STRUCTURES IN UNIAXIAL

FERROMAGNETS WITH QUALITY FACTOR Q > 1

J. Miltat, P. Trouilloud

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C8, Suppl6ment au no 12, Tome 49, d6cembre 1988

NEEL

LINES STRUCTURES IN UNLAXIAL FERROMAGNETS WITH QUALITY FACTOR Q >1

J. Miltat and P. Trouilloudl

LabO~atOi~e de Physique des Solides, Bit. 510, Universite' de Paris-Sud, 91405 Orsay Cedex, France

Abstract. - Numerical calculations indicate that splay lines in uniaxial magnetic materials satisfying Q

>

1 induce wall buckling. Both splay and half-circular lines are found to be wider than predicted whereas wall contraction at line location is smaller than anticipated. Buckling amplitude, line width and wall contraction increase with decreasing Q.

An approximate analytical determination of NBel (Bloch) line structures is possible only in the limit Q

>

1, where Q is the usual quality factor Q =

K/2 TM: (K uniaxial anisotropy constant, M, sat- uration magnetization) [I, 21. On the other hand, line structures, numerically derived without constraining assumptions, are still scarce [3-51. The purpose of the present communication is t o extend those calculations and establish the behaviour vs. Q of relevant parame- ters such as line width, wall contraction and buckling at line location. Both splay and half-circular lines (the vertical and horizontal lines in bubble garnets, respectively) are considered. The two dimentional numerical procedure utilized has been described in [3].

1. Splay lines (line direction parallel to the easy axis)

The usual splay line structure model [I, 21 (Winter approximation [6]) implies wall contraction a t line location. Contraction is maximum a t line core (a: = 0)

where the wall width ratio amounts to: -1 -112

& / A o = ( ~ + ~ Q

)

.

(1) A. is the classical wall width parameter: Ao = (A/K)'/~ ( A exchange constant).

A splay line gives rise t o n charges, antisymmetric with respect t o the wall mid-plane and cr charges, sym- n- metric with respect to the line symmetry plane =

-

2 [Z]. Hubert [7] has suggested that the interaction be- tween T and cr charges may lead to wall buckling at line location. Buckling will be limited by the associated increase in wall length, hence wall energy. A

buckling effect may be inserted into the usual analytical model [2]. Assuming a classical line profile, width

Ano

(A0 is the classical line width parameter: A0 = ( A / ~ T M ~ )

,

X is a coefficient), a simple calculation shows that the wall displacement q (Fig. l a , b) and maximum reduced buckling amplitude obey the fol- lowing relations:

aq/83: = (2 Q X ~ ) sin 2 d:

-

d

Fig. 1. - (a, b) Splay line geometry; (c, d) Half-circular line geometry.

Equation (2) implies the buckling symmetries indi- cated in figure 2. Since the improved model assumes a sole reduction of the T charges, it is not surprising that the symmetries are independent of the a charge density.

All calculations have been performed on a 120

x

40

grid points rectangular lattice, with lattice parame- ters ~ A 0 / 1 5 and ah0/7.5. As was the case in previous calculations [4, 51, wall buckling is also found in the

(~/Ao),, N X - ~ Q - ~ / ~ . ₍₂₎ Fig. 2. - Expected splay lines buckling symmetries. 'NOW at: IBM T. J . Watson Research Center, Yorktown Heights, N.Y. 10598, U.S.A.

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C8 - 1948 JOURNAL DE PHYSIQUE

present calculations. Computed reduced buckling amplitude (q,,/Ao, 8 = a/2) us. Q is shown in figure 3, curve a, and compared to the plot of equation (2) in the limit X = 1 (curve labelled 2). It may be seen that the buckling amplitude is smaller (larger) than expected at small (large) Q'S. Figure 3, curve b, indi- cates the behaviour of the reduced line width (AlaAo) us. Q. Line width is determined by the slope of the line profile ($ (z, 0 = r/2)) at point x = 0

($

=

i)

.

For all Q values, it appears that line width is substantially larger than nAo and increases with decreasing Q. Fi- nally, line width ratio 6/Ao, deduced from the slope of the wall profile at

x

= 0 and 0 = a/2, is shown in figure 3, curve d, together with the plot of equation (1) (curve labelled 1). Wall contraction is observed but is smaller than anticipated according to the approximate analytical model. Further, the line profile is, for

a

$ =

5,

slightly asymmetric (not shown). This effect is amplified when Q decreases. Line width, buckling am- plitude and wall contraction increase with decreasing

Q.

Fig. 3.

-

(a) Computed reduced buckling amplitude q/Ao vs. Q for a splay line, (2) ibid, analytical; (b) computed reduced line width A / d o vs. Q for a splay line; (c) ibid, half-circular line; (d) computed wall width ratio 6/Ao vs.

Q for a splay line; (e) ibid, half-circular line, (1) ibid, ana- lytical.

2. Half-circular lines (line direction perpen- dicular to the easy axis)

The usual equations still describe the magnetization distribution in a Bloch wall carrying half-circular lines provided the polar and azimuthal angles 8 and 1(1 are assigned new-definitions depicted in figures lc, d. Wall contraction still obeys equation (1). However, in o p position to splay lines, half circular lines do not, in the limit Q 4 oo, bear a charges. Buckling is not

anticipated t o occur because, due to the orientation of the magnetization in adjacent domains, it would imply additional charges. Further, the wall mid-plane is expected t o be the line symmetry plane: buckling therefore proves undeterminate. The computed reduced line width and wall width ratio are shown in figure 3, curves c and el respectively. Although line width proves larger than aAo for all Q values, the agreement with the analytical model appears better than for splay lines. Wall contraction is essentially similar to wall contraction observed in splay lines. However, it is found that contraction is maximum in the circular portion of the line rather than at

x

= 0 (not shown).

Conclusion

Wall contraction being similar for splay and half circular lines, the larger splay line widths may be pre- dominantly attributed to the existence of a charges. In the case of splay lines, buckling symmetries agree with those shown in figure 2 (not shown). Wall buckling is therefore mainly due to the associated decrease in a charges. Finally, both larger than predicted line widths and smaller than anticipated wall contraction at line location should affect the expected line dynam- ical properties.

[I] Slonczewski, J. C., J. Appl. Phys. 45 (1974) 2705.

[2] Malozemoff, A. P. and Slonczewski, J. C., Mag- netic domain walls in bubble materials (Academic Press, New York) 1979.

[3] Trouilloud, P. and Miltat, J., J. Magn. Magn. Mater. 66 (1987) 194.

[4] Asselin, P. (unpublished).

[5] Nakatani, Y. and Hayashi, N., IEEE Trans. Magn. MAG-23 (1987) 2179.