Spin motive force driven by skyrmion dynamics in magnetic nanodisks

The spin motive force driven by the dynamics of the skyrmion structure formed in a nanomagnetic disk is numerically investigated. Due to the existence of the magnetic structure along the disk edge, the collective mode of the magnetization is modified from that of the bulk skyrmion lattice obtained by using the periodic boundary condition. For a single-skyrmion disk, the dynamics of the skyrmion core and the edge magnetization induce the spin motive force, and a measurable AC voltage is obtained by two probes on the disk. For a multi-skyrmions disk, the phase-locked collective mode of skyrmions is found in the lowest resonant frequency where the amplitude of the AC voltage is enhanced by the cascade effect of the spin motive force. We also investigate the effect of the Rashba spin-orbit coupling on the spin motive force.

Figure 1

(a) The schematic view of the calculated system. Two probes are set on the disk edge and the width is 5 nm. (b),(c) The ground state configuration of the magnetization at $B^{\mathrm{DC}}=26$ mT in real space. Each arrow represents the in-plane component of local magnetization. (b) and (c) indicates the single- and multi-skyrmion disk, respectively. (d),(e) The topological charge density of each disk.

Figure 2

The results of the single-skyrmion disk without Rashba spin-orbit interaction. (a) The magnitude of AC voltage signal as a function of the static field and the frequency. (b) The frequency dependence of the voltage and of the time averaged magnitude of the topological charge velocity in Eq. (4) (inset) with $B^{\mathrm{DC}}=26$ mT. (c) Each structure region for analyzing the velocity. The contours are comparable to Fig. 1. (d)–(g) The direction of velocity defined by $\theta \left(t\right)={tan}^{-1}\left(\frac{\langle v_y\rangle }{\langle v_x\rangle }\right)-\frac{\langle v_x\rangle }{|\langle v_x\rangle |}$ at each resonant mode, where $\langle \cdots \rangle$ is averaged over the region of each structure (c).

Figure 3

The results of multi-skyrmion disk without Rashba spin-orbit interaction. The plot conditions are the same as Fig. 2.

Figure 4

(Lower curves) The voltage as a function of the time. (Upper curves) The in-plane oscillating field. Each frequency of the field is $f=1.00$ GHz (single-skyrmion disk) and $0.95$ GHz (multi-skyrmion disk).

Figure 5

The snapshots of spin motive force with lowest resonant mode in multi-skyrmion disk. (a),(b) There are contours of topological charge. Arrows represent the direction of spin motive force and the color gradient shows the magnitude. (a) is an ordinary case without Rashba spin-orbit interaction (${\alpha }_{\mathrm{R}}=0\mathrm{eV}$ m). (b) With Rashba effect (${\alpha }_{\mathrm{R}}={10}^{-10}\mathrm{eV}$ m). (c) Comparing the voltage signal of both cases (a) and (b).