Bloch point structure in a magnetic nanosphere

A Bloch point singularity can form a metastable state in a magnetic nanosphere. We classify possible types of Bloch points and analytically derive the shape of the magnetization distribution for different Bloch points. We show that an external gradient field can stabilize the Bloch point: The shape of the Bloch point becomes radially dependent. We compute the magnetization structure of the nanosphere, which is in good agreement with performed spin-lattice simulations.

Figure 1

Schematic of different types of Bloch points. Magnetization distribution in azimuthal vortex Bloch points in a sphere, see (a) and (b), and both Bloch points in the axial part of the cylinder-shaped sample during the vortex polarity switching process, see (c). The same for the azimuthal antivortex Bloch points, see (d) and (e), and both singularities in the axial part of the astroid-shaped sample during the switching, see (f). During the switching process shown in (c) and (f), two Bloch points move along the disk axis in opposite directions and finally annihilate halfway.

Figure 2

The Bloch point energy vs the rotation angle for ${\mathrm{BP}}_1^1$: analytical result (12) (solid curve) and simulations (symbols). Simulations parameters: sphere diameter $2R=35a_0$, exchange length $\ell =3.95a_0$, and damping parameter $\eta =0.5$.

Figure 3

Reduced rotation angle $g_{\varepsilon }\left(r\right)$, see Eq. (20) for different field intensities and $\varepsilon =0.05$: analytical result (A1) (solid curve) and numerical solution of Eq. (18) (dashed curves).

Figure 4

Bloch point under the action of the gradient field. (a) Critical behavior where the rotation angle vs field intensity $b$ near the critical field $b_c=1.47$ from the numerical solution of Eq. (18) (blue curves) and theoretical estimation by Eq. (23) (red curves) with $\varepsilon =0.05$. Solid lines correspond to rotation angle $\gamma \left(0\right)$, and the dashed line corresponds to $\gamma \left(1\right)$. (b) Phase diagram for solutions of Eq. (18). The upper (hedgehog) phase corresponds to the solution $\gamma =0$, and the lower (nonhomogeneous) one corresponds to the radially dependent Bloch point with $\gamma \left(r\right)$. Dashed lines correspond to the analytical result for the critical field $b_c\approx 1.46$ for $\varepsilon =0.05$, see text.

Figure 5

Radial dependence of rotation angle $\gamma$ in a spherical particle. Line: numerical integration of Eq. (18). Symbols: slasi simulations for crystallographic directions [110] and [111]. Parameters are the same as in Fig. 2.

Figure 6

Dynamics of total spin along the $z$ axis of the sample. The Bloch point is initially shifted by $\Delta z=-2a_0$ from the center of the sample. The insets show magnetization distribution in $z=-0.5a_0$ and $y=-0.5a_0$ planes for different times. The color bar indicates $S_{z,\mathit{n}}$ for different lattice nodes. The applied field amplitude $b=1$ and the other parameters are the same as in Fig. 2.

Figure 7

Reduced rotation angle $g_{\varepsilon }$ vs reduced exchange length $\varepsilon$ at $r=0$ (solid curve) and $r=1$ (dashed curve).