Resonance modes and microwave-driven translational motion of a skyrmion crystal under an inclined magnetic field

We theoretically investigate the microwave-active resonance modes of a skyrmion crystal on a thin-plate specimen under application of an external magnetic field that is inclined from the perpendicular direction to the skyrmion plane. In addition to the well-known breathing mode and two rotation modes, we find novel resonance modes that can be regarded as combinations of the breathing and rotation modes. Motivated by the previous theoretical work of [Wang et al., Phys. Rev. B 92, 020403(R) (2015)], which demonstrated skyrmion propagation driven by breathing-mode excitation under an inclined magnetic field, we investigate the propagation of a skyrmion crystal driven by these resonance modes using micromagnetic simulations. We find that the direction and velocity of the propagation vary depending on the excited mode. In addition, it is found that a mode with a dominant counterclockwise-rotation component drives much faster propagation of the skyrmion crystal than the previously studied breathing mode. Our findings enable us to perform efficient manipulation of skyrmions in nanometer-scale devices or in magnetic materials with strong uniaxial magnetic anisotropy such as ${\mathrm{GaV}}_4{\mathrm{S}}_4$ and ${\mathrm{GaV}}_4{\mathrm{Se}}_4$, using microwave irradiation.

Figure 1

(a) Three types of magnetic skyrmion structures. Distributions of the magnetizations (top panels) and those of the scalar spin chiralities $C_i=({\mathit{m}}_{i+\widehat{\mathit{x}}}\times {\mathit{m}}_{i+\widehat{\mathit{x}}+\widehat{\mathit{y}}})·{\mathit{m}}_i$ (bottom panels) are shown. (b) Top view of the skyrmion crystal under a perpendicular magnetic field. (c) Thin-plate specimen hosting a skyrmion crystal under a magnetic field ${\mathit{H}}_{\mathrm{ex}}=(H_x,0,H_z)$ that is inclined with a finite in-plane component $H_x=H_{z}tan\theta$. (d) Skyrmion structures under the inclined magnetic field. (e) Bloch-type skyrmion crystal under the inclined magnetic field. (f) Theoretical phase diagram of the spin model given by Eq. (1) at $T=0$ as a function of $H_z$ for a perpendicular magnetic field ${\mathit{H}}_{\mathrm{ex}}=(0,0,H_z)$ with $\theta =0^{\circ }$.

Figure 2

(a) and (b) Imaginary parts of the calculated dynamical magnetic susceptibilities of skyrmion crystal confined in a two-dimensional system under perpendicular ($\theta =0^{\circ }$) and inclined ($\theta \ne 0^{\circ }$) magnetic fields as functions of the angular frequency $\omega$. (a) Those for the in-plane microwave polarization with ${\mathit{H}}^{\omega }\parallel \mathit{x},\mathit{y}$. (b) Those for the out-of-plane microwave polarization with ${\mathit{H}}^{\omega }\parallel \mathit{z}$. Note that these spectra are calculated for the Bloch-type skyrmion crystal, but it was confirmed that the Néel-type and the antivortex-type skyrmion crystals have perfectly equivalent spectra. Here the inclined magnetic field is given by ${\mathit{H}}_{\mathrm{ex}}=(H_{z}tan\theta ,0,H_z)$, with $H_z=0.036$. A dominant component of the oscillation is indicated below the name of each mode, where CW and CCW indicate the clockwise and the counterclockwise rotations, respectively. The spectra for mode 4 are magnified in the insets. (c) and (d) Spectral intensities as functions of $\theta$ for modes 1–4 for (c) ${\mathit{H}}^{\omega }\parallel \mathit{x}$, $\mathit{y}$ and ($\mathrm{d}){\mathit{H}}^{\omega }\parallel \mathit{z}$. (e) Resonant frequencies ${\omega }_{\mathrm{R}}$ as functions of $\theta$ for modes 1–4.

Figure 3

(a) Snapshots of the four resonance modes (modes 1–4) of skyrmion crystal activated by the in-plane microwave field ${\mathit{H}}^{\omega }\parallel \mathit{y}$ (left panels) and those of the three resonance modes (modes 1–3) activated by the out-of-plane microwave field ${\mathit{H}}^{\omega }\parallel \mathit{z}$ (right panels) under an inclined magnetic field ${\mathit{H}}_{\mathrm{ex}}=(H_{z}tan\theta ,0,H_z)$, with $H_z=0.036$ and $\theta ={30}^{\circ }$. One skyrmion in the skyrmion crystal is focused on because all the skyrmions in the skyrmion crystal show identical motions. We show here the results for the Bloch-type skyrmion crystal, but the other two skyrmion types show equivalent behaviors, with the exception of the rotation sense of the antivortex-type (see text and Fig. 4). (b) and (c) Imaginary parts of the dynamical magnetic susceptibilities calculated by tracing the time profiles of net magnetization through applying a sinusoidal ac magnetic field $\mathit{H}\left(t\right)$ for (b) ${\mathit{H}}^{\omega }\parallel \mathit{y}$ and (c) ${\mathit{H}}^{\omega }\parallel \mathit{z}$. Both spectra exhibit peaks at frequencies equivalent to those of the spectra for $\theta ={30}^{\circ }$ shown in Fig. 2, which were calculated by applying a short rectangular pulse. This indicates that both the short pulse and the ac field excite identical modes and validates our method to study the modes.

Figure 4

Snapshots of mode 1 for Bloch-type, Néel-type, and antivortex-type skyrmion crystals under an inclined magnetic field ${\mathit{H}}_{\mathrm{ex}}=(H_{z}tan\theta ,0,H_z)$, with $H_z=0.036$ and $\theta ={30}^{\circ }$ activated by the in-plane microwave field ${\mathit{H}}^{\omega }\parallel \mathit{y}$. Note that the rotation sense of the antivortex type is opposite those of the Bloch type and the Néel type.

Figure 5

Translational motion of Bloch-type skyrmion crystal in a thin-plate specimen driven by microwave irradiation through activation of the resonance modes under an inclined magnetic field ${\mathit{H}}_{\mathrm{ex}}=(H_{z}tan\theta ,0,H_z)$, with $H_z=0.036$ and $\theta ={30}^{\circ }$. Here, the microwave field is given by $H_{\alpha }^{\omega }sin\omega t$ ($\alpha =x,y$), with $H_{\alpha }^{\omega }=0.0006$. The skyrmion crystal moves approximately towards the positive (negative) $y$ direction when a microwave field ${\mathit{H}}^{\omega }\parallel \mathit{x}$ (${\mathit{H}}^{\omega }\parallel \mathit{z}$) with $\omega =0.0494$ ($\omega =0.0666$) activates mode 1 (mode 2) with a dominant counterclockwise-rotation (breathing) component. Displacement vectors connecting the original position (dashed circles) and the position after microwave irradiation for 400 ns (solid circles) are indicated by the thick arrows in the right panels. The motion driven by mode 1 is much faster than that driven by mode 2.

Figure 6

Calculated $\omega$ dependence of the velocity $\mathit{v}=(v_x,v_y)$ for translational motion of the Bloch-type skyrmion crystal induced by a microwave field ${\mathit{H}}^{\omega }$ under an inclined magnetic field ${\mathit{H}}_{\mathrm{ex}}=(H_{z}tan\theta ,0,H_z)$, with $H_z=0.036$ and $\theta ={30}^{\circ }$. Here, the microwave field is given by $H_{\alpha }^{\omega }sin\omega t$ ($\alpha =x,y$), with $H_{\alpha }^{\omega }=0.0006$, and $\omega =2\pi f$ is its angular frequency. The velocities show peaks at the resonant frequencies of the modes, while their signs vary depending on the mode.

Figure 7

Calculated $\theta$ dependence of the velocity $v=\sqrt{{v_x^2+v_y^2}^{\phantom{0}}}$ of the skyrmion crystal driven by a microwave field ${\mathit{H}}^{\omega }$ through activation of (a) mode 1 with the dominant counterclockwise-rotation component and (c) mode 2 with breathing oscillations for different microwave polarizations. Here, $\theta$ is the inclination angle of the external magnetic field ${\mathit{H}}_{\mathrm{ex}}=(H_{z}tan\theta ,0,H_z)$, with $H_z=0.036$, and the microwave field is given by $H_{\alpha }^{\omega }sin\omega t$ ($\alpha =x,y$), with $H_{\alpha }^{\omega }=0.0006$. All velocity data are measured at the resonant frequencies of the modes. Calculated $\theta$ dependence of the propagation direction of the skyrmion crystal driven by (b) mode 1 and (d) mode 2 for different microwave polarizations.

Figure 8

Definitions of the angle $\varphi$ that describes the direction of propagation of the (a) Néel-type and (b) antivortex-type skyrmion crystal when driven by modes 1 and 2. The $\theta$ dependence of the direction of propagation for the Bloch-type skyrmion crystal in Fig. 7 also holds for the Néel-type and antivortex-type skyrmion crystals if we replace the definitions of $\varphi$ with them. Note that they are related to each other through ${90}^{\circ }$ rotation around the $z$ axis and mirror operation with respect to the $zx$ plane.

Figure 9

Calculated $H_z$ dependence of the velocity $v=\sqrt{{v_x^2+v_y^2}^{\phantom{0}}}$ of skyrmion crystal when driven by a microwave field ${\mathit{H}}^{\omega }$ through activation of (a) mode 1 with the dominant counterclockwise-rotation component and (b) mode 2 with breathing oscillations for different microwave polarizations. Here, $H_z$ is the perpendicular component of the external magnetic field ${\mathit{H}}_{\mathrm{ex}}=(H_{z}tan\theta ,0,H_z)$, with $\theta ={30}^{\circ }$, while the microwave field is given by $H_{\alpha }^{\omega }sin\omega t$ ($\alpha =x,y$), with $H_{\alpha }^{\omega }=0.0006$.