Some Soliton Properties of Colliding Vertical Bloch Lines

The collisions of two vertical Bloch lines are studied numerically. It is found that soliton like properties of the Bloch lines are obtained for a nonzero damping constant if the power input is large enough to balance the effect of dissipation. In the strict sense solitons are obtained in nondissipative nonlinear systems [I]. On the other hand, most solid state physical systems are dissipative. Interest in soliton-like excitations in dissipative media has increased recently 121. Solitary wave like properties have been found in a magnetic system the Bloch domain wall which is a damped system; it is driven by a uniform magnetic field a constant in time [3]. In domain wall dynamics [4], the solitary wave like excitations obtained in reference [3] are called horizontal Bloch lines where the term horizontal signifies that the kink-like rotation of the magnetic moment in the Bloch line occurs along the easy axis of the uniaxial anisotropy. In application oriented research on domain wall dynamics recent interest is centered on a different kind of Bloch line the vertical Bloch line in which the rotation occurs in a plane perpendicuIar to the anisotropy axis. Thus, it would seem to be interesting to see whether the vertical Bloch lines, which are described by a somewhat different set of equations of motion than horizontal Bloch lines, also have soliton-like properties. Below we show numerically that, when the parameters (the damping constant and the drive field) of the system are chosen judiciously, two vertical Bloch lines forces by the external drive field to collide pass through each other conserving their identity. The effect of dissipation is discussed. The calculations were done both with and without the long range magnetostatic interactions described in reference [5]. The only difference in the results was that the events calculated with the long range magnetostatic interactions included occurred with a slight phase shift with respect to those calculated without them. To describe the motion of the wall the equations of motion derived by Slonczewski [4] were solved numerically for a segment of a stripe domain wall with a one dimensional structure containing two vertical Bloch lines (see Fig. 1 and Eq. (1, 2) of Ref. [5]). The equations of motion used by us describe the motion of a one dimensional chain of magnetic moments corresponding to the center magnetic moments in the wall at the midplane of the film. Periodic boundary conditions at the end points of this chain were assumed and long range Fig. 1. The time evolution of the structure of the wall containing two vertical Bloch lines depicted every 5 ns. The collisions occur at about t = 20 ns and at about t = 45 ns. magnetostatic interactions following the results of [4] were included in the calculations. The following material parameters were used in the calculations: exchange constant A = 0.81 x lo-? erg/cm, saturation magnetization 4nM = 140 G, gyromagnetic ratio y = 1.73 x lo7 l/sOe, Bloch wall width A = 0.029 x lo-" cm, Gilbert damping constant cr = 0.156. The evolution with the time of a segment of the domain wall containing two vertical Bloch lines is shown in figure 1 for a moderate damping constant cr = 0.156 and the drive field magnitude Hz = 10.5 Oe. It can be seen that, due to the action of field applied to the wall and to the subsequent motion of the wall, the Bloch lines move towards the edges of the graph where the periodic boundary conditions are active. There they colide (t = 15 ns) , pass through each other and approach each other again. They next collide (t = 40 ns) and pass through each other. The repeated collisions would go on infinitely as long as the field were active. It is thus concluded that the vertical Bloch line may have soliton like properties. The motion of the vertical Bloch lines along the structure of the wall is accompanied by a distortion of the shape of the wall surface. This effect is the same as was discussed in reference [3] with respect to horizontal Bloch lines and will not be further described here. If the calculation of figure 1 was repeated and the applied field as well as the damping were set equal to zero the Bloch lines annihilated i.e. the soliton prop Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19888853 C8 1884 JOURNAL DE PHYSIQUE erties were not obtained. The collision of the Bloch lines results only in a vibration of the wall surface for which the energy of the Bloch lines is used. When only the damping is set equal to zero but the drive field is left on then, upon collision, the Bloch lines pass through each other as their energy has to be conserved. For a low, nonzero value of the damping constant (e.g. a = 0.01) the Bloch lines pass through each other even for small drive field magnitudes. However, when the moderate damping value a = 0.156 was used then, if the drive field was not large enough (i. e. less than 6 Oe), the Bloch lines annihilated upon collision leaving the wall in stationary motion with a unchanging structure. With higher magnituted of the drive the Bloch lines passed through each other. To conclude, the numerical experiments carried out show that in a nonlinear dissipative system such as the driven and damped Bloch wall soliton like properties of a kink excitation (a vertical Bloch line in the experiments) are obtained provided that the power input into the system (here proportional to the drive field) is large enough as compared to the rate of dissipation which is proportional to the damping constant. [I] Bishop, A. R., Phys. Scr. 20 (1979) 409. [2] Bishop, A. R., Fesser, K., Lombadhl, P. S. and Trullinger, S. E., Physica 7D (1983) 259. [3] ~ebrowski, J. J. and Sukiennicki, A. (Springer) Proc. Phys. 23 (1987) 130. [4] Malozemoff, A. P. and Slonczewski, J. C. , Magnetic domain walls (American Press, New York) 1979. [5] Kosibski, R. A. and Strzeszewski, J., Phys. Status Solidi B 133 (1986) 533.