Current-driven nucleation and propagation of antiferromagnetic skyrmionium

We present a theoretical study on nucleation and propagation of antiferromagnetic skyrmionium induced by spin current injection. A skyrmionium, also known as $2\pi$ skyrmion, is a vortexlike magnetic structure characterized by a topological charge, the so-called skyrmion number, being zero. We find that an antiferromagnetic skyrmionium can be generated via a local injection of spin current with toroidal distribution. We examine the threshold current density for successful skyrmionium nucleation and the robustness of the proposed mechanism against thermal fluctuation. A spatially uniform spin current is then demonstrated to induce propagation of the skyrmionium. We derive an expression for the skyrmionium velocity based on a collective-coordinate model, which agrees well with our numerical results.

Figure 1

Schematic of the system for skyrmionium nucleation. A 9 ps of radial electric current (light blue) is induced by illuminating the antenna with a fs-laser pulse (purple) due to the applied bias voltage. The current flowing in the NM gives rise to a spin current with circulating spin polarization (white), penetrating into the AFM.

Figure 2

Snapshots of the magnetization profile at selected times during the skyrmionium nucleation. The electric current pulse is applied in the toroidal region (see Fig. 1) from $t=0$ to 9 ps. After the current is turned off, the excited structure relaxes into a skyrmionium state, passing through the stage with increased skyrmion-number [see also Fig. 3]. The scale bar corresponding to 50 nm is shown at the upper left ($t=0$ for $n_3$).

Figure 3

Temporal evolution of (a) the skyrmion number and (b) the spatially averaged magnetic energy density during the skyrmionium nucleation shown in Fig. 2. The duration of the electric current pulse is indicated by the gray area.

Figure 4

The dependence of the threshold current density $j_{\mathrm{c}}^{\mathrm{thr}}$ on the sizes of the electrodes. The square and triangle symbols are numerical results, with the solid lines being guides to the eye. The blue (red) symbols indicate $j_{\mathrm{c}}^{\mathrm{thr}}$ as a function of the disk-electrode radius (ring-electrode inner radius), with the radius of the other electrode fixed to 100 (10) nm. For the material parameters, the same values as used in the previous calculation for Figs. 2 and 3 are employed.

Figure 5

The dependence of the threshold current density $j_{\mathrm{c}}^{\mathrm{thr}}$ on the Gilbert damping constant $\alpha$. The other material parameters are the same as before. The radius of the disk electrode and the inner radius of the ring electrode are 20 and 100 nm, respectively.

Figure 6

Snapshots of the magnetization profile at selected times during the skyrmionium nucleation. The calculation condition is the same as that in Fig. 2, except that the Gaussian random field due to thermal fluctuation is now included in the effective magnetic fields. The initial state is prepared at 300 K. During the pulse application (from $t=0$ to 9 ps), the temperature is increased to 600 K all over the sample. After the current pulse is turned off ($t>9$ ps), the temperature goes back to 300 K. The scale bar corresponding to 50 nm is shown at the upper left ($t=0$ for $n_3$).

Figure 7

Temporal evolution of (a) the skyrmion number and (b) the spatially averaged magnetic energy density during the skyrmionium nucleation shown in Fig. 6 with the thermal fluctuation included. The duration of the electric current pulse is indicated by the gray area.

Figure 8

Skyrmionium velocities $v_1$ and $v_2$, in the $x_1$ and $x_2$ directions, respectively, as functions of electric current density $j_{\mathrm{c}}$. For the skyrmionium propagation, the spin-current injection is spatially uniform with its polarization along the $-x_2$ direction. The inset is the $n_3$ profile at $t=0.1$ ns in the case of $j_{\mathrm{c}}={10}^{12}$ A/${\mathrm{m}}^2$.