Coarsening dynamics of topological defects in thin permalloy films

We study the dynamics of topological defects in the magnetic texture of rectangular permalloy thin-film elements during relaxation from random magnetization initial states. Our full micromagnetic simulations reveal complex defect dynamics during relaxation towards the stable Landau closure domain pattern, manifested as temporal power-law decay, with a system-size-dependent cutoff time, of various quantities. These include the energy density of the system and the number densities of the different kinds of topological defects present in the system. The related power-law exponents assume nontrivial values and are found to be different for the different defect types. The exponents are robust against a moderate increase in the Gilbert damping constant and introduction of quenched structural disorder. We discuss details of the processes allowed by conservation of the winding number of the defects, underlying their complex coarsening dynamics.

Figure 1

The elementary topological defects: (a) vortex, (b) antivortex, (c) positive edge defect, and (d) negative edge defect. The film is viewed from the direction of the $z$ axis, with the arrows showing the magnetization direction in the $xy$ plane. The vortex-antivortex cores pointing out of plane are denoted with $\otimes$. In the case of edge defects, the cores (not necessarily out of plane) are marked with $◯$ and the black bar represents the edge of the sample.

Figure 2

The relaxation process in a square thin film with lateral size $L=1024$ nm. The color wheel in the center shows the direction of the magnetization corresponding to each color. In this case, the final vortex (black dot at the center of the bottom right picture) is a $-z$-polarized clockwise vortex.

Figure 3

A schematic view of the different defects and the (ideal) surrounding magnetization in the nearest neighbors in the $xy$ plane, as in Fig. 1. The numbers in the corners of the cells show the cell traversal order when determining the defect type.

Figure 4

The various kinds elementary defects present in the relaxation of the largest film: clockwise vortices (squares), counterclockwise vortices (circles), antivortices $(+$ signs), and edge defects (triangles). The color of a bulk defect shows its polarization (black for $-z$ and white for $+z)$. For the edge defects, the color indicates whether the winding number of the defect is positive (black) or negative (white).

Figure 5

The total average winding numbers for the four film sizes used. The inset shows that the winding number for the largest sizes takes more time to converge into 1.

Figure 6

The two most commonly encountered metastable states, shown for the system with $L=1024$ nm. (a) A two-edge-defect, two-vortex state which also displays significant bending of domain walls. (b) A more complex state, showing an isolated vortex and an arc of other defects.

Figure 7

The time evolution of the total energy of the films, consisting of an initial fluctuation phase independent of the system size and a defect annihilation-coarsening phase displaying power-law relaxation terminated by a cutoff time increasing with the film size. The linear dependence of energy density and defect density during the coarsening phase is shown in the inset.

Figure 8

The time evolution of the densities of (a) all defects, (b) vortices, (c) antivortices, and (d) edge defects, with solid lines corresponding to power law fits as guides to the eye. The power laws can be seen most clearly in the largest film sizes. In the case of edge defects, the positive edge defects are noted with dashed symbols.

Figure 9

The annihilation of a vortex and an antivortex with (a) antiparallel and (b) parallel polarizations.

Figure 10

Though somewhat difficult to see, during this edge defect-vortex annihilation, the lower edge defect emits an antivortex with which the vortex actually annihilates.

Figure 11

In this antiparallel annihilation, the negatively polarized antivortex (black dot) generates a dip particle, which then splits into a positively polarized vortex-antivortex pair. Thus two annihilations occur: an antiparallel annihilation of the original antivortex and the generated vortex $\left({\mathrm{AV}}_{\mathrm{O}}\text{−}{\mathrm{V}}_{\mathrm{G}}\right)$, and a parallel annihilation of the generated antivortex and the original vortex $\left({\mathrm{AV}}_{\mathrm{G}}\text{−}{V}_{\mathrm{O}}\right)$.

Figure 12

The core switching of a vortex due to a momentary absorption into an edge defect. Usually before and after the absorption and emission of the bouncing defect, the edge defect cores gain short-lived out-of-plane magnetization components.

Figure 13

Main figure shows the average time evolution of the total number density of defects ${\rho }_{\text{d}}$ for $L=4096$ nm with four different $\alpha$ values. For larger $\alpha$, the power-law character of the relaxation (black lines indicate the the power-law fits used) starts earlier due to the strongly damped initial magnetization fluctuations. The inset shows the resulting ${\eta }_{\text{d}}$ exponent as a function of $\alpha$.