9.3.5 Surface Charge Density

Another important aspect in comparison with the rigid vortex model is the magnetostatic energy and the surface charges, which generate the magnetostatic field. On the top and bottom circular surface the surface charges are proportional to $M_z$, because the normal vector $\boldsymbol{n}$ of the top and bottom is simply $e_z$ and $-e_z$, respectively. However, on the circumference the normal vector is, of course, $e_r$.

Fig. 9.16 shows a contour plot of the surface charge density on the nanodot in zero field. As expected, the equilibrium position of the vortex core is in the center of the nanodot and the circular magnetization distribution avoids any surface charges on the circumference (Fig. 9.17).

Figure 9.16: Contour plot of the surface charge density on the nanodot in zero field. Green indicates no surface charges, blue indicates negative surface charges.

Figure 9.17: Contour plot of the surface charge density on the circumference of the nanodot in zero field.

If an in-plane external field is applied, the vortex core is shifted perpendicular to the direction of the field (Fig. 9.18). As a result, surface charges appear on the circumference (Fig. 9.19).

Figure 9.18: Contour plot of the surface charge density on the nanodot in an external field. Blue indicates negative charges, green zero charges, and red positive surface charges.

Figure 9.19: Contour plot of the surface charge density on the circumference of the nanodot in an external field.

Fig. 9.20 shows the surface charge distributions for different applied fields. The ``jitter'' in this plot arises from the slightly different values across the height of the nanodot. The nanodot has been discretized by an unstructured mesh with four ``layers'' of finite elements. Thus, there are typically five nodes of the finite element mesh across the height of the dot and the values of the surface charge density of all five nodes has been plotted in Fig. 9.20.

Figure 9.20: Surface charge density on the circumference of the nanodot for different applied fields.

The surface charge densities for different vortex core displacements, as calculated with the FE model, is given in Fig. 9.21.

The surface charge distributions have been calculated, as they are predicted by the rigid vortex model. A comparison of the results with the finite element model can be found in Fig. 9.22.

For small external fields and therefore small vortex displacements there is very good agreement between the analytical rigid vortex model and the finite element simulation. As the external field increases more surface charges appear on the circumference of the nanodot. However, the rigid vortex model overestimates these surface charges. The values for the average magnetization is in good agreement, but the surface charge distribution is not. The reason is, that the magnetization distribution close to the circumference is disturbed by the strong demagnetizing fields. As we further increase the external field and the vortex displacement this deviation becomes more and more pronounced. In addition, we also find some deviation in the center of the nanodot, which arises from a more ``elliptical'' shape of the magnetization distribution as the vortex is pushed towards the boundary. Contour plots of the difference between the magnetization distribution calculated by the FE simulation and the rigid vortex model $d=\vert\boldsymbol{M}_\mathrm{FE}-\boldsymbol{M}_\mathrm{rv}\vert$ are shown in Fig. 9.23 for $H_\mathrm{ext}=66.0 \mathrm{kA/m}=830 \mathrm{Oe}$, $\langle M_x \rangle/M_\mathrm{s}=-0.72$, and $b/R=-0.76$. The red areas at the circumference and in the center of the nanodot indicate differences between the rigid vortex model and the FE simulation.

Figure: Surface charge distribution as a function of the polar angle on the circumference of the nanodot for different vortex shifts (indicated by different $\langle M_x \rangle$).

Figure 9.22: Surface charge distribution on the circumference of the nanodot as a function of the polar angle. An in-plane external field shifts the vortex (cf. Fig. 9.23) and leads to surface charges on the circumference. The rigid vortex model (gray lines) overestimates the charge density as compared to the FE simulation (black lines).

[
$H_\mathrm{ext}=0.8 \mathrm{kA/m} \hat= 10 \mathrm{Oe}$
$\langle M_x \rangle/M_\mathrm{s}=-0.02$
$b/R=-0.03$] [
$H_\mathrm{ext}=8.8 \mathrm{kA/m} \hat= 110 \mathrm{Oe}$
$\langle M_x \rangle/M_\mathrm{s}=-0.12$
$b/R=-0.13$]

[
$H_\mathrm{ext}=16.7 \mathrm{kA/m} \hat= 210 \mathrm{Oe}$
$\langle M_x \rangle/M_\mathrm{s}=-0.23$
$b/R=-0.25$] [
$H_\mathrm{ext}=25.5 \mathrm{kA/m} \hat= 320 \mathrm{Oe}$
$\langle M_x \rangle/M_\mathrm{s}=-0.34$
$b/R=-0.37$]

(Fig. cont. on next page)

Figure: Contour plot of $d=\vert\boldsymbol{M}_\mathrm{FE}-\boldsymbol{M}_\mathrm{rv}\vert$. Blue areas indicate good agreement of the magnetization distribution between the rigid vortex model (black cones) and the FE model (gray cones), red areas indicate larger differences.

[
$H_\mathrm{ext}=34.2 \mathrm{kA/m} \hat= 430 \mathrm{Oe}$
$\langle M_x \rangle/M_\mathrm{s}=-0.44$
$b/R=-0.48$]

[
$H_\mathrm{ext}=42.2 \mathrm{kA/m} \hat= 530 \mathrm{Oe}$
$\langle M_x \rangle/M_\mathrm{s}=-0.52$
$b/R=-0.57$]

[
$H_\mathrm{ext}=54.1 \mathrm{kA/m} \hat= 680 \mathrm{Oe}$
$\langle M_x \rangle/M_\mathrm{s}=-0.62$
$b/R=-0.67$] [
$H_\mathrm{ext}=66.0 \mathrm{kA/m} \hat= 830 \mathrm{Oe}$
$\langle M_x \rangle/M_\mathrm{s}=-0.72$
$b/R=-0.76$]

In remanence, the demagnetizing field arising from the vortex structure is mainly concentrated in the vortex core (Fig. 9.24). It has a dominating $z$-component and a smaller radial component.

Figure: $H^\mathrm{dem}_z$ and $H^\mathrm{dem}_r$ across the nanodot.