Multiscale simulations of topological transformations in magnetic-skyrmion spin structures

Magnetic skyrmions belong to the most interesting spin structures for the development of future information technology as they have been predicted to be topologically protected. To quantify their stability, we use an innovative multiscale approach to simulating spin dynamics based on the Landau-Lifshitz-Gilbert equation. The multiscale approach overcomes the micromagnetic limitations that have hindered realistic studies using conventional techniques. We first demonstrate how the stability of a skyrmion is influenced by the refinement of the computational mesh and reveal that conventionally employed traditional micromagnetic simulations are inadequate for this task. Furthermore, we determine the stability quantitatively using our multiscale approach. As a key operation for devices, the process of annihilating a skyrmion by exciting it with a spin polarized current pulse is analyzed, showing that skyrmions can be reliably deleted by designing the pulse shape.

Figure 1

Dynamics of a skyrmion for different values of a constant out-of-plane field. The system shows an oscillatory behavior, where both (a) its size, expressed in terms of the skyrmion magnetic moment $S$, and (b) the angle $\theta$ between the in-plane magnetization components of the domain wall and the radial direction, reach a certain nonzero value before relaxing back to the equilibrium. The data corresponding to 250 kA/m shows the skyrmion annihilation. (c) The minimum magnetic field $H_{\mathrm{del}}$ necessary to adiabatically annihilate a skyrmion for different values of the DMI constant $D$ and linear spin density $a^{-1}$. $H_{\mathrm{del}}$ is shown to linearly increase as a function of the spin density. Data points corresponding to the lowest value of $a^{-1}$ were simulated in purely micromagnetic simulations.

Figure 2

(a) Path to annihilation of a skyrmion in 300 $\mathrm{kA}/\mathrm{m}$ external magnetic field. The internal energy $E_{\mathrm{int}}$ of the system and the total energy (internal plus Zeeman energies) are compared as functions of time. A potential barrier exists for the internal energy which has to be overcome by the application of an external field. All the quantities are presented in arbitrary units. (b) Dynamic snapshots at various stages of the annihilation process. The initial configuration of a Néel skyrmion is perturbed when the structure is shrinking. The scale on the axes is expressed in units of the micromagnetic computational cell (3 nm).

Figure 3

Spin structure of the skyrmion during the annihilation process in cases I and II. The color code shows the out-of-plane component, from red, to white, to blue. The initial state corresponds to a relaxed skyrmion centered on the cell with coordinates (50, 50). (a) Case I: A BL is formed in the domain wall of the skyrmion within a vortex-antivortex pair. The spins in the domain wall turn counter-clockwise starting from the position of the pair, meanwhile, the skyrmion increases in size and reaches a maximum then starts shrinking in size. As the skyrmion shrinks below the minimum size, it is finally annihilated. The system relaxes into the ferromagnetic ground state. (b) Case II: The vortex-antivortex pair annihilates, the skyrmion number immediately turns to zero, the system quickly relaxes back to the ferromagnetic state. (c) Different regimes depending on the peak height $J_0$ and half-width $\sigma$ of the Gaussian pulse. The error bars were evaluated by performing simulations with different values of $\sigma$.

Figure 4

Details of the case I annihilation process. (a) The topological charge density is accumulated in a vortex-antivortex pair. (b) The system relaxes. (c) The skyrmion size increases until the Néel character is recovered. (d) The skyrmion shrinks and then annihilates. For the topological charge density $q$ the relation min = $-\max$ holds.

Figure 5

Zoom in on the case II annihilation process. (a) The topological charge density is contained in a vortex-antivortex pair. (b) A Bloch line is formed. (c) The Bloch line annihilates, generating a topological charge density of the opposite sign. (d) The system relaxes into the ferromagnetic state. For the topological charge density $q$ the relation min = $-\max$ holds.