Breathing skyrmions in chiral antiferromagnets

Breathing oscillations of skyrmions in chiral antiferromagnets can be excited by a brief modification of the Dzyaloshinskii-Moriya interaction or magnetocrystalline anisotropy strength. We employ an adiabatic approximation and derive a formula for the potential energy that directly implies breathing oscillations. We study the nonlinear regime and the features of larger-amplitude oscillations, and we verify the validity of the adiabatic approximation. We show that there is a maximum amplitude supported by the potential. As a consequence, we predict theoretically and observe numerically skyrmion collapse and subsequent annihilation events due to excitation of large-amplitude breathing oscillations. The process is efficient when the skyrmion is mildly excited so that its radius initially grows, while the annihilation event is eventually induced by the internal breathing dynamics. We reveal the counterintuitive property that the skyrmion possesses a nonzero kinetic energy at the instance of its annihilation. Finally, the frequency of small-amplitude breathing oscillations is determined.

Figure 1

The blue dotted line shows the potential energy $V_{\lambda }$ for the static skyrmion versus its radius $R=R\left(\lambda \right)$. The red solid lines show the potential energy $V_{{\lambda }_0}\left({\mathrm{\Theta }}_{\lambda }\right)$ given in Eq. (23) with ${\lambda }_0=0.40,0.47,0.57$ and corresponding equilibrium radii $R_0=0.71,1.14,2.39$. The radius $R$ is in scaled units and the actual length is given, as in Eq. (8), by $aR/\epsilon$ [it is $\epsilon =0.1$ for parameter values in Eq. (2)].

Figure 2

Skyrmion annihilation via breathing. (a) The parameter $D$ is shown vs time. A square pulse with a duration of 7 ps is modulating the DM parameter to the value $D=0.057J_n$ ($\lambda =0.57$). After the pulse is switched off, it is $D=0.047J_n$ ($\lambda =0.47$). (b) The total energy from Eq. (1) vs time is shown by a blue solid line. The skyrmion radius is shown by a red dashed line. It is given in units of lattice spacing $a$. (c) Spatial distribution of $n_3$ at time points $t_i$ (indicated on the energy graph). Snapshot $t_0$ shows the initial skyrmion [static profile for parameter values of Eq. (2)], snapshot $t_1$ shows the skyrmion when the pulse is switched off, snapshot $t_2$ shows the skyrmion before annihilation, and snapshot $t_3$ shows the generated waves after the skyrmion has been annihilated. Only part of the simulation space is shown.

Figure 3

Kinetic energy ($T$) shown by a solid line for the annihilation event of Fig. 2. The skyrmion radius ($R$) is also shown by a grey dashed line for comparison. The results are plotted up to the time point where $V\approx 4\pi$, after which the continuum theory ceases to be valid. At this point, the skyrmion collapses while it can be clearly seen that $T>0$, in agreement with the theoretical prediction.

Figure 4

We apply a square pulse that modifies the DM parameter, as in Fig. 2, but now with a duration of 13 ps. We start the simulation with $D=0.057$ ($\lambda =0.57$) and we restore to $D=0.047$ ($\lambda =0.47$) at the time marked by the dotted red line. The damping parameter is $\alpha =0.001$. (a) Skyrmion radius vs time shows damped breathing oscillations. (b) Kinetic energy (16). The kinetic energy is close to zero at both turning points of the oscillation.

Figure 5

The Néel field $\mathit{n}$ and the magnetization field $\mathit{m}$ of the skyrmion during breathing. (a) The Néel field for the initial skyrmion. (b) The magnetization field during skyrmion expansion, at time $t_1$ indicated in Fig. 4. (c) The magnetization field during skyrmion shrinking, at time $t_2$. The magnetization points azimuthally in both cases as anticipated by Eq. (28).

Figure 6

For ${\lambda }_0=0.47$ ($D=0.047$), we show the maximum and minimum radii of skyrmion breathing oscillations, when the initial skyrmion profile is ${\mathrm{\Theta }}_{\lambda }$. Lines show theoretical results based on Eq. (22) and Fig. 1. Circles show results of numerical simulations. The simulations are initiated with skyrmion profiles at the blue circles (they give the one extremum of the oscillation). The other extremum of the oscillation is shown by red circles. [The radius $R$ is in scaled units and the actual length is given, as in Eq. (8), by $aR/\epsilon$, with $\epsilon =0.1$ for parameter values (2)].

Figure 7

The angular frequency for small-amplitude breathing oscillations as a function of the skyrmion radius $R$ (corresponding to the parameter ${\lambda }_0$). The angular frequency goes to zero for large $R$ (as ${\lambda }_0$ approaches $2/\pi$). The rate of convergence is consistent with Eq. (37) within numerical accuracy. Physical units for frequency ${\omega }_b/\left(2\pi \right)$ are restored by multiplying by 2.68 THz.