Key role of temperature in ferromagnetic Bloch point simulations

Bloch points in permalloy cylinders are investigated using a micromagnetic framework, where thermal effects are included via the Landau-Lifshitz-Bloch equation of motion. We show that this enables micromagnetic modeling of a Bloch point avoiding the problem of singularities, which have been reported in the literature so far. The details of the Bloch point which we reveal are compared with earlier analytic approximations describing its geometry and the magnetization drop in its center. The temperature dependence of characteristic parameters, like the Bloch point radius or the azimuthal inflow angle is given in the full temperature range.

Figure 1

Permalloy cylinder containing two antiparallel vortices and a Bloch point (BP) in between them. (Left) Schema of magnetization flow (thick arrows) around the BP. The antiparallel vortex cores are marked in blue (lower section) and red (upper section), where the magnetization has primarily a $z$ component. Close to the $xy$ plane the magnetization has primarily rotational components with some inflow tendency described by the angle $\gamma >{90}^{\circ }$, marked in gray. In other points in space it is a mixture of both these tendencies, approximated by us with model function ${\mathbf{M}}_{\mathrm{LH}}\left(\mathbf{r}\right)$ [see Eq. (5)]. (Right) Cross sections through the sample (results of simulations for $T=600\mathrm{K}$). Cross section (a) is in the $xy$ plane, while (b) and (c) are in the $xz$ plane. Color coding in part (b): red (upper) and blue (lower) according to $M_{\mathrm{z}}\left(\mathbf{r}\right)$ value. Color coding in parts (a) and (c): green (gray)-white according to magnetization length—here dark means $M\left(\mathbf{r}\right)\approx M_{\mathrm{e}}$, while white means $M\left(\mathbf{r}\right)\approx 0$. Parts (a)–(c) show only the central region of the sample as used for the fitting procedure. Note that the small asymmetry visible in parts (b) and (c) is due to the even number of simulations cells.

Figure 2

Theoretically predicted magnetization drop (normalized) close to the BP center as a function of the distance. Horizontal axis is scaled similarly as in Ref. 15. Exact function, as got by numerical solving of differential equation $u$, is shown together with its approximation $v$ [see Eq. (2)]. Difference $u-v$ is shown enlarged in the inset.

Figure 3

Results of our simulations as a function of the temperature in the range from 0 to $T_{\mathrm{C}}$. (Left axis) Quality parameter, $\Delta m_{\max }$. The gray region corresponds to the suggested condition $\Delta m_{\max }\lesssim 1/2$ (see text for more details). (Right axis) BP radius $a$, as returned by our fitting procedure (solid points; fitting errors are smaller than the points). The line shows the result of analytical theory; see Eq. (3).

Figure 4

Parameter determining the variation of the VC radius in the vicinity of the BP, $\kappa$ (points), as returned by our fitting procedure. Circles are for the cylinder sample, while triangles are for the disk. Fitting errors are smaller than the points. For comparison $\xi \left(T\right)$ from Eq. (A1) is shown.

Figure 5

Inflow angle $\gamma$ as returned by our fitting procedure. Circles are for the cylinder sample, while triangles are for the disk. Fitting errors are smaller than the points.

Figure 6

VC radius as a function of the film thickness. All values are normalized similarly to Ref. 22. The function got by Feldtkeller, $R^{\prime }\left(D\right)$,[22] is compared with our approximation, $R_{\mathrm{apr}}^{\prime }\left(D\right)$ (shown dashed).