Topological quantization of the classical stochastic transport of a magnetic skyrmion driven by a ratchetlike spin-polarized electric current at finite temperature

We consider a magnetic skyrmion driven by a spin-polarized electrical current that is periodic in time, and is periodic and asymmetric in a direction different from that of the current itself. We study its classical stochastic transport in a finite temperature, by using the Fokker-Planck equation of the probability distribution, derived from the stochastic equation of motion, the Langevin equation. We also perform numerical simulation of the original Landau-Lifshitz-Gilbert equation describing the spins constituting the skrymion. The probabilistic average velocity of the skyrmion is along the direction of the periodicity. When the thermal energy is much lower than the potential energy, and their ratio is also much smaller than that between the time periodicity and the diffusion time, the time and probabilistic average velocity is the ratio between the spatial and temporal periodicities multiplied by topological integer called the Chern number. This result provides a practical way of realizing topological numbers in classical stochastic systems and suggests a convenient way of manipulating skyrmions at finite temperatures.

Figure 1

The skyrmion's average velocity along $x$ direction as a function of the amplitude $j_c$ of the polarized electric current. The unit of the velocity is $\frac{a}{\tau }$, while the unit of $j_c$ is $0.01\frac{2e}{a^2\tau }$. Different symbols and colors represent different values of $k_{B}T$ in unit of $J$, i.e., $k_{B}T/J$. The grey line represents the analytically predicted value of the velocity. The parameter values used in the simulation are $k=\frac{2\pi }{100}$, $\kappa =\frac{100}{864}$, $\mathcal{T}=5000\tau$. The amplitude $A$ in time oscillation is fixed to be 0.2. The damping parameter $\alpha$ is 0.1, the magnetic field along the $z$ direction $B_z$ is $0.015J$, the Dzyaloshinskii-Moriya interaction constant $D$ is $0.12J$; the anisotropic energy constant $K$ is $0.01J$.

Figure 2

The average velocity of the skyrmion divided by the basic unit $\frac{1}{\sqrt{1+{\kappa }^2}}\frac{L}{\mathcal{T}}$, which is denoted as $v_{\mathrm{standard}}$ here, as a function of $j_c$, for various values of the driving period $\mathcal{T}$. The temperature is fixed as $k_{B}T=0.01J$.

Figure 3

Comparison between functions $f_0\left(u\right)$ and $f\left(u\right)$.

Figure 4

A device that realizes the ratchetlike polarized electric current.