9.3.4 Average Magnetization

Fig. 9.14 shows profiles of $M_z$ along the $y$-axis for different external fields. As a result, the vortex is shifted and the profile ``moves'' towards the circumference of the dot ($\vert y/R\vert=1$). From this plot the position of the vortex core for a given external field has been extracted. The corresponding average magnetization $\langle M_x \rangle$ is plotted in Fig. 9.15 (open circles). For symmetry reasons $M_y$ is zero (the vortex is shifted along the $y$-axis, since we applied a field in $x$-direction).

Figure 9.14: Profiles of $M_z$ along the $y$-axis through the center of the nanodot for a vortex moving in $-y$ direction due to an external field increasing in $x$ direction.

By integrating the magnetization distribution of the rigid vortex model over the surface of the nanodot the average magnetization $\langle M_x \rangle$ has been calculated. The result is given in Fig. 9.15. We find very good agreement between the rigid vortex model and the finite element simulation. The small difference can be understood by considering small deviations of the magnetization distribution due to surface charges on the circumference (cf. Sec. 9.3.5).

Figure: Comparison of $\langle M_x \rangle$ as a function of the vortex displacement between the FE simulation and the rigid vortex model. $y=0$ corresponds to a centered vortex, $y=-0.78$ is the maximum shift before vortex annihilation occurs in the FE simulation.