DC spinmotive force from microwave-active resonant dynamics of a skyrmion crystal under a tilted magnetic field

We theoretically show that a temporally oscillating spin-driven electromotive force or voltage with a large DC component can be generated by exciting microwave-active spin-wave modes of a skyrmion crystal confined in a quasi-two-dimensional magnet under a tilted magnetic field. The DC component and the AC amplitude of the oscillating electric voltage is significantly enhanced when the microwave frequency is tuned to an eigenfrequency of the peculiar spin-wave modes of the skyrmion crystal called “rotation modes” and the “breathing mode,” whereas the sign of the DC component depends on the microwave polarization and on the spin-wave mode. These results provide an efficient means to convert microwaves to a DC electric voltage by using skyrmion-hosting magnets and to switch the sign of the voltage via tuning the microwave frequency, which are important capabilities for spintronic devices.

Figure 1

(a) Schematics of the spin-transfer torque mechanism. Angular momentum transfer from conduction-electron spins of an injected spin-polarized current to magnetizations constituting a noncollinear magnetic texture drives the translational motion of the magnetic texture. (b) Schematics of spin-driven electromotive force, the so-called spinmotive force. Momentum transfer from a driven noncollinear magnetic texture to the conduction electrons via exchange coupling causes an effective electromotive force acting on the conduction electrons, resulting in the generation of spin-polarized electric currents. Here we consider a domain wall in a ferromagnetic nanowire as an example of the noncollinear magnetic texture.

Figure 2

Thin-plate specimen of a chiral-lattice magnet hosting a skyrmion crystal under (a) a perpendicular and (e) tilted external magnetic field ${\mathit{H}}_{\mathrm{ex}}=(H_{z}tan\theta ,0,H_z)$ with tilting angles of $\theta =0^{\circ }$ and $\theta \ne 0^{\circ }$, respectively. Skyrmion crystal under (b) the perpendicular and (f) tilted magnetic field ${\mathit{H}}_{\mathrm{ex}}$ with $\theta =0^{\circ }$ and $\theta ={30}^{\circ }$, respectively. Magnetization configuration of a skyrmion constituting the skyrmion crystal under (c) a perpendicular and (g) tilted magnetic field ${\mathit{H}}_{\mathrm{ex}}$. (d) Theoretical phase diagram of the spin model given by Eq. (2) at $T=0$ as a function of the out-of-plane component of ${\mathit{H}}_{\mathrm{ex}}$ when $\theta =0^{\circ }$.

Figure 3

Calculated imaginary parts of the dynamical magnetic susceptibilities $\mathrm{Im}{\chi }_{\mu }\left(\mu =x,y,z\right)$ of the skyrmion crystal confined in the two-dimensional system under perpendicular $\left(\theta =0^{\circ }\right)$ and tilted $\left(\theta ={30}^{\circ }\right)$ magnetic fields ${\mathit{H}}_{\mathrm{ex}}$ with $H_z=0.036$ as functions of the microwave frequency $\omega$. (a) $\mathrm{Im}{\chi }_x$ and $\mathrm{Im}{\chi }_y$ for the in-plane microwave polarization with ${\mathit{H}}^{\omega }\parallel \mathit{x},\mathit{y}$. (b) $\mathrm{Im}{\chi }_z$ for the out-of-plane microwave polarization with ${\mathit{H}}^{\omega }\parallel \mathit{z}$. The dominant mode component and the value of the eigenfrequency are shown at each peak position.

Figure 4

Calculated time profiles of spin voltages induced by the microwave-active spin-wave modes of a skyrmion crystal confined in a thin-plate magnet under (a)–(c) a perpendicular and (f)–(h) tilted magnetic field ${\mathit{H}}_{\mathrm{ex}}$ with $\theta =0^{\circ }$ and $\theta ={30}^{\circ }$, respectively, and $H_z=0.036$ for various microwave polarizations and frequencies. (a) and (f) ${\mathit{H}}^{\omega }\parallel \mathit{x}$ with a frequency fixed at the eigenfrequency of the counterclockwise rotation mode, (b) and (g) ${\mathit{H}}^{\omega }\parallel \mathit{y}$ with a frequency fixed at the eigenfrequency of the counterclockwise rotation mode, and (c) and (h) ${\mathit{H}}^{\omega }\parallel \mathit{z}$ with a frequency fixed at the eigenfrequency of the breathing mode. The amplitude of the applied microwave is fixed at $H^{\omega }=3\times {10}^{-3}$ for the in-plane polarized microwave ${\mathit{H}}^{\omega }\parallel \mathit{x},\mathit{y}$ in (a), (b), (f), and (g), whereas it is fixed at $H^{\omega }=2\times {10}^{-3}$ for the out-of-plane polarized microwave ${\mathit{H}}^{\omega }\parallel \mathit{z}$ in (c) and (h). Here the values of the AC amplitude $V_x^{\mathrm{AC}}$, the DC component $V_x^{\mathrm{DC}}$, the angular frequency $\omega (=2\pi f)$, and the decay rate $\tau$ of the generated time-dependent spin voltage are shown for each case, as obtained by fitting the simulation results (see text). The results show that the DC voltage $V_x^{\mathrm{DC}}$ is always zero under the perpendicular magnetic field ${\mathit{H}}_{\mathrm{ex}}$, whereas it becomes finite when the magnetic field ${\mathit{H}}_{\mathrm{ex}}$ is tilted. (d), (e), (i), and (j) Snapshots of the spatial distribution of the spinmotive force (arrows) at selected times. The out-of-plane magnetization components $m_z$ are also shown in color to facilitate the visualization of the temporally varying skyrmion shape.

Figure 5

Calculated microwave-frequency dependence of the AC amplitude $V_{\mu }^{\mathrm{AC}}$ and the DC component $V_{\mu }^{\mathrm{DC}}\left(\mu =x,y\right)$ of the spin voltage under application of a tilted magnetic field ${\mathit{H}}_{\mathrm{ex}}$ with $\theta ={30}^{\circ }$ and $H_z=0.036$ for the (a)–(c) in-plane and (d)–(f) out-of-plane microwave polarizations ${\mathit{H}}^{\omega }\parallel \mathit{x},\mathit{y}$ and ${\mathit{H}}^{\omega }\parallel \mathit{z}$, respectively, i.e., (a) and (d) $V_x^{\mathrm{AC}}$ and $V_y^{\mathrm{AC}}$, (b) and (e) $V_x^{\mathrm{DC}}$, and (c) and (f) $V_y^{\mathrm{DC}}$. Here the amplitude of the applied microwave is fixed at $H^{\omega }=0.6\times {10}^{-3}$ for all simulations.

Figure 6

Calculated microwave-amplitude dependence of (a) the AC amplitudes $V_x^{\mathrm{AC}}$ and $V_y^{\mathrm{AC}}$ and (c) the DC components $V_x^{\mathrm{DC}}$ and $V_y^{\mathrm{DC}}$ of the spin voltages, respectively, for the counterclockwise rotation mode and the breathing mode excited by the in-plane microwave field ${\mathit{H}}^{\omega }\parallel \mathit{x}$ under application of a tilted magnetic field ${\mathit{H}}_{\mathrm{ex}}$ with $\theta ={30}^{\circ }$ and $H_z=0.036$. (b) and (d) Magnified view of the area indicated by the dashed rectangle in (a) and (c), respectively. The frequency of the applied microwave field is fixed at $\omega =0.04938$ for the counterclockwise rotation mode and at $\omega =0.0664$ for the breathing mode.

Figure 7

Time profiles of the spin voltage and the average of six Bragg peaks. Snapshots of the skyrmion-crystal configurations and the Bragg peaks in momentum space at selected moments are also displayed. In the simulations, the skyrmion crystal is continuously excited by an intense in-plane polarized microwave ${\mathit{H}}^{\omega }\parallel \mathit{x}$ with $H^{\omega }=3\times {10}^{-3}$ and $\omega =0.04938$ under application of a tilted magnetic field ${\mathit{H}}_{\mathrm{ex}}$ with $\theta ={30}^{\circ }$ and $H_z=0.036$.