Skyrmion Dynamics at Finite Temperatures: Beyond Thiele’s Equation

Magnetic textures are often treated as quasiparticles following Thiele’s equation of motion. We demonstrate via spin model simulations of the current-driven and Brownian motion of ferromagnetic skyrmions that the existing theory based on Thiele’s equation is insufficient to describe the dynamics of skyrmions at finite temperatures. We propose an extended equation of motion that goes beyond Thiele’s equation by taking into account the coupling of the skyrmion to the magnonic heat bath leading to an additional dissipative term that is linear in temperature. Our results indicate that this so-far-neglected magnon-induced friction dominates for finite temperatures and Gilbert damping values typical for thin films and multilayers.

Figure 1

Diffusion coefficient $D$ versus $\alpha$ for various temperatures. Symbols are simulation results that are compared to the theoretical predictions based on the normal Thiele equation [Eq. (5)] (dotted lines) and our extended model [Eq. (7)] with $\eta =5.05\times {10}^{-17}\mathrm{kg}/(\mathrm{sK})$ (dashed lines).

Figure 2

Sketch of the transformation of a spin model to an effective rigid-body description: in the spin model (left part), the system is connected to the electronic or phononic heat bath via the Gilbert damping $\alpha$, which results in the $\alpha \mathfrak{D}$ term in the Thiele equation (right part). At finite temperatures, magnons are present in the spin model whose impact has to be accounted for by an additional $\eta T$ term in the effective rigid-body description.

Figure 3

Numerically obtained effective friction ${\mathrm{\Gamma }}^{\mathrm{eff}}$ versus $T$ for different types of skyrmionic spin structures. Dashed lines correspond to linear fits yielding $\eta \approx 5.05\times {10}^{-17}\mathrm{kg}/(\mathrm{sK})$ for skyrmionic spin structures with $Q=\pm 1$ and $\eta \approx 1.22\times {10}^{-16}\mathrm{kg}/(\mathrm{sK})$ for $Q=-2$ structures. The offset at $T=0$ corresponds to $\alpha \mathfrak{D}$.

Figure 4

Skyrmion Hall angle $\mathrm{\Theta }$, shifted by $\pi /2$, versus $\alpha$ for various values of $T$. Symbols are from simulations that are compared to the theoretical predictions based on Eq. (8) with $\eta =5.05\times {10}^{-17}\mathrm{kg}/(\mathrm{sK})$ (dashed lines).