Dynamics of ferrimagnetic skyrmionium driven by spin-orbit torque

Magnetic skyrmionium is a skyrmionlike spin texture with nanoscale size and high mobility. It is a topologically trivial but dynamically stable structure, which can be used as a nonvolatile information carrier for next-generation spintronic storage and computing devices. Here we study the dynamics of a skyrmionium driven by the spin torque in a ferrimagnetic nanotrack. It is found that the direction of motion is jointly determined by the internal configuration of a skyrmionium and the spin polarization vector. Besides, the deformation of a skyrmionium induced by the intrinsic skyrmion Hall effect depends on both the magnitude of the driving force and the net angular momentum. The ferrimagnetic skyrmionium is most robust at the angular momentum compensation point, whose dynamics is quite similar to the skyrmionium in antiferromagnet. The skyrmion Hall effect is perfectly prohibited, where it is possible to observe the position of the skyrmionium by measuring the magnetization. Furthermore, the current-induced dynamics of a ferrimagnetic skyrmionium is compared with that of a ferromagnetic and antiferromagnetic skyrmionium. We also make a comparison between the motion of a ferrimagnetic skyrmionium and a skyrmion. Our results will open a new field of ferrimagnetic skyrmioniums for future development of ferrimagnetic spintronics devices.

Figure 1

Schematic diagram of an isolated ferrimagnetic skyrmionium in a two-dimensional film. (a) Illustration of a Néel-type ferrimagnetic skyrmionium spin structure with the two antiparallel magnetic sublattices. The magnetization orientation along any centerline through the skyrmionium core is sketched below. Red and blue colors indicate the sublattices 1 and 2, respectively. (b) The simulation snapshot of a relaxed ferrimagnetic skyrmionium, where the $z$ component of Néel vector $\mathit{n}$ is coded in the pixel color as shown in the color bar on the right.

Figure 2

Comparison between the motion of a Néel-type skyrmionium and a Bloch-type skyrmionium. The top views of the (a) Néel-type skyrmionium and the (b) Bloch-type skyrmionium with the in-plane component of the order parameter $\mathit{n}$ indicated by the black arrows (left), and the corresponding $y$ component of these skyrmioniums (right). (c) and (d) The motion of these two types of skyrmioniums driven by the spin current with different $\mathit{p}$, where the yellow dashed circle represents the initial position of the skyrmionium and green arrows denote the polarization direction.

Figure 3

Comparison between numerical results and analytical results. The ferrimagnetic skyrmionium velocity as a function of (a) the magnetization ratio $\eta =M_2/M_1$ and (b) the damping coefficient $\alpha$ for $\eta =0.9$. Here the driving current density $j$ is fixed at 10 $\mathrm{MA}{\mathrm{cm}}^{-2}$ to ensure that the skyrmionium keeps a circle shape without severe distortion. Symbols represent the results from numerical simulations, and the dashed lines indicate the analytical results based on Eq. (7).

Figure 4

Driving current density dependence of the ferrimagnetic skyrmionium velocity. (a) Ferrimagnetic skyrmionium velocity as a function of the driving current density $j$ for $\eta =0.9$, 1.0, 1.1, 1.2, and 1.3. The inset is a zoom-in of the current-velocity relationship when $j<30\mathrm{MA}{\mathrm{cm}}^{-2}$. The cross represents the skyrmionium with severe distortion under such a driving current density. (b) The critical driving current density that causes a severe distortion for a ferrimagnetic skyrmionium, as a function of the magnetization ratio $\eta$. The blue symbols indicate the spin density $\sigma$ corresponding to different $\eta$.

Figure 5

Analysis of the skyrmionium deformation. (a) The components of dissipative tensor $\mathcal{D}$ and tensor $\mathcal{I}$ as functions of the ratio $\eta$ when the skyrmionium moves along the nanotrack without severe distortion. Here $j=10\mathrm{MA}{\mathrm{cm}}^{-2}$. (b)–(d) The time evolution of $D_{ij}$ and $I_{ij}$ for $\eta =$ 0.9, 1.1, and 1.3 (corresponding to $\sigma =+0.58,0.0$, and $-0.58{\mathrm{A}}^2\mathrm{s}{\mathrm{m}}^{-2}$), where the driving current density is 40, 70, and 45 $\mathrm{MA}{\mathrm{cm}}^{-2}$, respectively. Insets show the top view of a ferrimagnetic skyrmionium marked with a deformation angle $\psi$.

Figure 6

Comparison between the current-induced dynamics of (a) FM, (b) FiM ($\eta =0.9$), and (c) AFM skyrmioniums. Insets show the top view of a skyrmionium driven by the selected current density $j=20\mathrm{MA}{\mathrm{cm}}^{-2}$ at $t=6$ ns, which is indicated by the vertical dashed line. Symbols represent the results from numerical simulations, and the dashed lines indicate the analytical results based on Eq. (7).

Figure 7

Snapshots of (a) and (b) a FM skyrmion, (c) a FiM skyrmion, and (d) a FiM skyrmionium during their motion driven by a spin current, where the yellow dashed line represents the initial position of spin structures, and the gray lines stand for their trajectories. The time evolution of velocities for these magnetic structures driven by a spin current with (e) $j=25\mathrm{MA}{\mathrm{cm}}^{-2}$ and (f)–(h) $j=30\mathrm{MA}{\mathrm{cm}}^{-2}$. Here $\eta =0.9$ for a typical uncompensated FiM system.