Spin eigenexcitations of an antiferromagnetic skyrmion

We study spin eigenexcitation of a skyrmion in a collinear uniaxial antiferromagnet by means of analytical and numerical methods. We found a discrete spectrum of modes which are localized on the skyrmion. Based on a qualitatively different dependence of the mode eigenfrequencies on the skyrmion radius $R_0$, we divided all localized modes into two branches. Modes of the low-frequency branch are analogous to the localized magnon modes of a ferromagnetic skyrmion, their frequencies scale as $R_0^{-2}$ for the large radius skyrmions, while the modes of the high-frequency branch have no direct ferromagnetic counterpart and do not demonstrate the significant radius dependence and are compactly situated at the magnon continuum. All the modes, except the radially symmetrical one, are doubly degenerated with respect to the sense of rotation around the skyrmion center: clockwise or counterclockwise. An out-of-plane magnetic field removes the degeneracy (for all modes except translational), resulting in a frequency splitting, which for the small fields is linear in field. The possibility of excitation of the modes by means of the external ac magnetic fields is discussed. To explain our numerical results for the low-frequency modes, we introduce a string model for an antiferromagnetic domain wall representing boundary of the large radius skyrmion.

Figure 1

Sketch of setup a thin film uniaxial collinear AFM with two magnetic sublattices ${\mathit{M}}_1$ and ${\mathit{M}}_2$ on top of a heavy metal substrate. Inset: Parametrization of the Néel vector in terms of spherical coordinates $\theta$ and $\varphi$.

Figure 2

Structure of the AFM skyrmion. (a) Distribution of magnetic moments, black and white arrows correspond to the different sublattices and the color scheme codes the perpendicular component $n_z$. (${\mathrm{a}}^{\prime }$) Distribution of the magnetization for $h=0.3$ (disk radius ${\rho }_{\max }=8$). (b) Skyrmion profiles determined by Eqs. (6) and (c) the corresponding static magnetization determined by Eq. (3) for various magnetic fields $h<h_c$. The vertical lines are located at the radius $R_0$ of the respective skyrmion. The DMI constant $\delta =0.9$ is the same for all field values.

Figure 3

Field dependence of the skyrmion radius for different values of DMI constants. The solid and dashed lines correspond to the numerical solution of Eqs. (6) and the analytical approximation $R_0(h,\delta )$ determined in Eqs. (8), respectively. The vertical asymptotes are $h_c=\sqrt{1-{\delta }^2}$.

Figure 4

Schematics of the perturbation (orange line) around the circular skyrmion configuration (black line) for $\left|\mu \right|=3$. For positive (negative) $\mu$, the rotation is clockwise (counterclockwise).

Figure 5

Schematics of the spectrum. Classification of the modes with frequency ${\omega }_{\mu }$ according to their quantum number $\mu$. Gray dashed lines connect the symmetry related resonances. The modes with zero frequency and $\mu =\pm 1$ are doubly degenerate (DD). The field-induced narrowing of the gap (between the blue zones) is not shown.

Figure 6

Spectrum of localized eigenmodes of an AFM skyrmion. (a) spectrum for $h=0$ and $h=0.5$, (b) high-frequency modes for $h=0$ and $h=0.01$ obtained by numerically solving the EVP Eqs. (14) with $\delta$ running the interval $[{\delta }_{\mathrm{ini}}\left(h\right),0.998]$ where ${\delta }_{\mathrm{ini}}\left(0\right)=0.08,{\delta }_{\mathrm{ini}}\left(0.01\right)=0.59,{\delta }_{\mathrm{ini}}\left(0.5\right)=0.45$. The modes with $\left|\mu \right|>4$ are not shown. Solid lines show the eigenfrequencies for $h=0$. Dashed and dotted lines show the eigenfrequencies of the modes $\mu \ge 0$ and $\mu <0$, respectively, in the presence of a magnetic field. Thin black lines show the asymptotic approximation Eq. (23) of the LFB modes.

Figure 7

Influence of the magnetic field on spectrum of the localized modes. Field dependence of the eigenfrequencies is shown in panel (a), and HFB is detailed in panel (b). Panel (c) shows the value of splitting for the low-frequency modes $\mu =\pm 2$ and $\mu =\pm 3$, where $\mathrm{\Delta }{\omega }_{\mu }^{\mathrm{LFB}}={\omega }_{\left|\mu \right|}-{\omega }_{-\left|\mu \right|}$. Here $\delta =0.9$ is fixed and $0\le h<h_c=\sqrt{1-{\delta }^2}$. The spectrum is obtained by means of the numerical solution of EVP Eqs. (14). Note that $\mu \to -\mu$ when $h\to -h$.

Figure 8

Examples of eigenfunctions of localized eigenmodes for $\left|\mu \right|\le 2$. The upper and lower rows correspond to HFB an LFB, respectively. The presented eigenfunctions and eigenfrequencies are obtained by means of numerical solution of EVP Eqs. (14). The vertical lines indicate the skyrmion radius $\rho =R_0$. The thin yellow lines shown for modes $\mu =\pm 1$ of LFB correspond to the eigenfunctions of the translational modes: $f_{\pm 1}=-{\mathrm{\Theta }}^{\prime }$ and $g_{\pm 1}=\pm sin\mathrm{\Theta }/\rho$, when the normalization is applied. Note that in the absence of the field, $f_0\equiv 0$ for the HFB (the in-plane component is only excited) and $g_0\equiv 0$ for the LFB (the out-of-plane component is only excited). However, both in-plane and out-of-plane components are excited for modes with $\left|\mu \right|>0$ or in presence of a field.

Figure 9

Cartoons of localized eigenmodes for (a) $\left|\mu \right|=1$ LFB showing the gyration mode with a minor effect on the domain wall thickness, (b) $\left|\mu \right|=1$ HFB higher order gyration mode, where the skyrmion wall profile changes, (c) $\left|\mu \right|=2$ LFB shape deformation, and (d) $\left|\mu \right|=2$ HFB elliptical deformation of the wall width. The color code corresponds to the out-of-plane component of the Néel vector. For better visibility, the amplitudes of the corrections have been enhanced.