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Submitted on 1 Jan 1988
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STRANGE BAND STATES OF A BLOCH WALL OF
FINITE LENGTH
J. Żebrowski
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Supplkment au no 12, Tome 49, d6cembre 1988
STRANGE BAND STATES OF A BLOCH WALL OF
FINITELENGTH
Institute of Physics, Warsaw Technical University, ul. Koszykowa 75, 00-662 Warszawa, Poland
Abstract.
-
Strange band chaotic states of a Bloch domain wall are found numerically between period-1 and period-2 att ractors.Recently, it has been shown [I] that the running- oscillatory state of the Bloch wall moving in a bulk material in drive fields larger than the Walker critical drive is unstable due to the extremely large sensitivity of the wall t o initial conditions. This feature of the wall causes a transition, in a properly chosen phase space of the domain wall, to occur under a small per- turbation directly from a fixed point state of the wall (which corresponds to the running-oscillatory motion) to a strange attractor.
If the size of the Bloch wall along the easy axis of the anisotropy is decreased below a certain value then, for drive fields only slightly larger than the Walker field, periodic states are obtained [2] and the strange attractor of reference [I] occurs only at higher drive fields. Depending on a combination of the drive field magnitude and the size of the system the periodic state obtained may be either period-1 or period-2 but no period doubling cascade has been obtained [2].
The transition from period-1 to period-2 states of the Bloch wall with a finite size moving in a uniaxial material with a large anisotropy is studied below. It is shown that instead of being sharp, the transition occurs through a strange band chaotic state which is caused by a coexistence of the period-1 and period-2 attractors for some combinations of the size of the wall and the magnitude of the drive.
to obtain the states of the wall. Force-free boundary conditions were used.
In reference [2] it was found that, depending on the distance between the end points of a Bloch wall of finite size for a fked magnitude of the drive field (12 Oe or
H,
+
1.08 Oe was used in Ref. [2]) several different periodic attractors are obtained having either a period of 1 or a period of 2.The behavior of the trajectories of the Bloch wall for a fixed value of the size of the wall h = 1.1 pm was followed through as the magnitude of the drive was changed during the numerical calculation during the motion of the wall. Several other values of the size h of the wall were used but, below, only the results for h = 1.1 pm are discussed in some detail.
From Hz = 11.0 Oe up to Hz = 11.28 Oe a period-
1 state was obtained. The shape of the wall surface and the corresponding evolution in time of the struc- ture of the wall for this and for the other attractors described below are similar to the ones given in refer- ence [2] and will not be discussed here for lack of space. The phase portrait for this case is depicted in figure 1: the image shown is a two dimensional~ojection ofie four d i m 2 i o n a l subsp-acce (q (t) - q (t), cp (t)
-
cp (t),d
(t)-
p (t), q (t) - q (t)) of the center point in the wall. On both axis of the phase portrait the variables are plotted relative t o their respective spatial averages over the length of the wall. By plotting this difference . - -Exactly the same equations of motion for the Bloch wall and numerical methods to solve them are used throughout this paper as were used in references [I, 21. These partial differential equations are the ones
originally derived by Slonczewski [3] for the bubble 4
garnet domain wall but with the surface stray field
term neglected. They describe the motion of the sur- \ face which is the locus of magnetic moments at the
;:::
center of the wall and the motion of these moments on-
u that surface. It has been shown elsewhere [3] that such-
2 - - a model for the motion of a domain wall is adequate-
4 for high anisotropy materials. In this model the wall is--
--
considered to be infinite in the direction perpendicular - ! : : : : : I
to the anisotropy .axis but fmite in size along it. The
-
1 0 - 1 material parameters used here are also those of refer-9'-
(P [ r a d l ences [I, 21 as well as the perturbation procedure used Fig. 1.-
Period-1 attractor obtained for Hz = 11.25 Oe.C8
-
1944 JOURNAL DE PHYSIQUEinstead of just the variables it is possible to get rid of the trivial components corresponding t o the aver- age motion of the wall. The center point of the wall was chosen here arbitrarily and the results obtained for other points in the wall are similar.
At Hz = 11.29 Oe a period-2 attractor was found. The phase portrait for this case is shown in figure 2.
Fig. 2.
-
Period-2 attractor obtained for Hz = 11.30 Oe.In a narrow range of the drive around Hz =
11.285 Oe a strange band state was obtained. The structure and wall shape (not shown here) were very similar t o those obtained for the periodic attractors described above. The phase portrait for Hz = 11.285 is shown in figure 3 for t = 750 ns-1 500 ns. It can be seen that the trajectories are similar t o each other and lie relatively close to each other. A repeller is sit- uated at (0, 0). It is the fixed point corresponding to the stationary motion of the Bloch wall at drive fields below the Walker critical field; at the range of drive fields of figure 1 the fixed point is unstable. Strange band behavior was obtained always for Hz = 11.283
-
11.287 Oe, if the strange band region was approached with a small enough increment (e.g. 0.001 Oe) either beginning at the period-2 attractor or at the period-1 attractor. On the other hand, by incrementing the drive field by 0.005 Oe the strange band could be ob- tained only for Hz = 11.278 Oe (starting within the period-1 region) and for Hz = 11.29 Oe (starting from the period-2 region).
By observing as the phase portrait in figure 3 is formed, one notices that the trajectory seems to trace the loop which is a special feaure of the period-2 attrac- tor only occasionally while the majority of the strange band trajectories seems to resemble the loop which is similar in both the period-1 and the period-2 attrac-
Fig. 3. - Strange band attractor obtained for Hz = 11.285 Oe.
tors. Thus the two periodic attractors - apart from residing in the same region of phase space
-
seem to overlap in parameter space.As the drive field was varied upwards from Hz = 11.29 up to Hz = 27 Oe several period-1 and period-2 states (three in all) were found for the size parameter h = 1.1 ,um - always separated by a strange band chaotic state. Above Hz = 21 Oe a strongly chaotic state was obtained of the type discussed in references [I, 21 for the case of a Bloch wall in a bulk material.
To conclude, in a Bloch domain wall of finite size, moving in a constant drive field larger than the Walker critical field, periodic states are separated by nar- row ranges of parameter values in which strange band chaotic attractors are obtained. The fixed point, which in this range of the drive field is unstable, acts as a repeller influencing the shape of the trajectories con- siderably. The periodic states seem to overlap to a certain extent.
Acknowledgements
The author wishes to thank prof. A. Sukiennicki for many fruitful discussions. Financial support was pro- vided by the University of Lodz under project CPBP
01.08.Bl.l.
[I] gebrowski, J. J., Sukiennicki, A., Springer Proc. Phys. 23 (1987) 130.
[2] gebrowski, J. J., to be published in Phys. Scr.
[3] Slonczewski, J. C., J . A p p l . Phys. 44 (1973) 1759. [4] Schryer, N. L., Walker, R. L., J. A p p l . Phys. 45
