Multiscale and multimodel simulation of Bloch-point dynamics

We present simulation results on the structure and dynamics of micromagnetic point singularities with atomistic resolution. This is achieved by embedding an atomistic computational region into a standard micromagnetic algorithm. Several length scales are bridged by means of an adaptive mesh refinement and a seamless coupling between the continuum theory and a Heisenberg formulation for the atomistic region. The code operates on graphical processing units and is able to detect and track the position of strongly inhomogeneous magnetic regions. This enables us to reliably simulate the dynamics of Bloch points, which means that a fundamental class of micromagnetic switching processes can be analyzed with unprecedented accuracy. We test the code by comparing it with established results and present its functionality with the example of a simulated field-driven Bloch-point motion in a soft-magnetic cylinder.

Figure 1

Error estimate for the intrinsic deviation of the micromagnetic exchange energy formulation from a Heisenberg calculation. The error is displayed for the test case of a spin spiral of wavelength $\lambda$, where $\Delta$ is the distance between neighboring magnetic moments along the direction of the spin spiral.

Figure 2

Two different models are used to calculate the exchange interaction in the MS: Heisenberg (blue) and micromagnetic (green and orange). The green to orange shading illustrates a transition region from the outer sample to the MS, inside of which the MS stamps its properties onto the sample (green), and vice versa (orange).

Figure 3

Set of calotte structures acting as “refinement patches” for different distances of the MS to the boundary of the cylindrical sample. As the MS reaches the surface, the transition is achieved with calotte meshes of appropriate discretization density, which are chosen dynamically from a predefined set of meshes according to the penetration depth of the MS into the surface.

Figure 4

As a test case for the multimodel-multiscale code, we use a permalloy disk of 100 nm diameter and 35 nm height with a vortex structure. A spatially homogeneous in-plane field pulse with Gaussian time profile is applied to the relaxed vortex structure. The field pulse is applied along the $x$ axis; it has an amplitude of 50 mT, width $\sigma =100$ ps, and a peak delay of $t_{\mathrm{Max}}=300$ ps. This perturbation of the system generates a well-known micromagnetic low-frequency excitation (gyrotropic motion) which represents a spiraling motion of the vortex core around the equilibrium position. The background mesh has a cell size of 1.75 nm. On the upper left, the evolution of the total energy of the system is displayed as a function of time, while on the upper right, the spatially averaged and normalized $y$ component of the magnetization $\langle m_y\rangle$ is shown. The results obtained from the purely micromagnetic code and the multiscale-multimodel code are identical. The main frame on the bottom shows a snapshot of the magnetic structure at the moment indicated by the yellow dots in the upper frames. The color code represents the out-of-plane component $m_z$ of the magnetization and the semitransparent representation shows the position of the MS sphere at the core of the vortex. In the core of that sphere, the magnetic structure is calculated on an atomistic level. The crossing ribbons are the $m_x=0$ and $m_y=0$ isosurfaces, which we use routinely to track the position of the vortex (cf. Ref. [30]).

Figure 5

(a) Total energy and (b) average $y$ component of the magnetization $\langle m_y\rangle$ for the previously discussed case of a gyrating vortex. The various lines correspond to the results obtained with transition regions of different thickness ${\delta }_T$. (c) The error of the total energy connected with the multiscale-multimodel method as a function the thickness of the transition region ${\delta }_T$. Each of the lines refers to a moment in time (as indicated) after the beginning of the simulation at $t=0$. Here, the exact value of the energy is assumed to be the one obtained from the micromagnetic simulation, and the deviation from this value is displayed in percentage points.

Figure 6

Bloch-point velocity as a function of the applied field in a wire with a diameter of 60 nm. Different propagation behaviors are found, depending on the relative orientation of the cylinder axis and the atomic lattice. The last data point with zero velocity represents the depinning field. For both orientations, two metastable dynamic domain-wall configurations can be distinguished. The first is a propagation mode with almost field-independent velocity, as indicated by empty symbols. The system can also display a second propagation mode, with a different domain-wall configuration and for which the velocity increases significantly with the external field.

Figure 7

Evolution of the Bloch-point velocity vs times in the low field (regime below the magnonic limit) for a wire with a diameter of 60 nm and a cylinder axis oriented in the (100) direction.

Figure 8

Snapshots of the simulated vortex domain-wall (DW) propagation in a ferromagnetic cylinder [35] with a Bloch point in its core. The two magnified images show the magnetic configuration around the Bloch point in the equilibrium configuration and after applying an external field of 2 mT for four nanoseconds. The crossing of the three $m_{\{x,y,z\}}=0$ isosurfaces indicates the position of the BP in the center of the multimodel sphere, and the coloring represents the axial magnetization component.