Dynamics of ferromagnetic bimerons driven by spin currents and magnetic fields

A magnetic bimeron composed of two merons is a topological counterpart of a magnetic skyrmion in in-plane magnets, which can be used as the nonvolatile information carrier in spintronic devices. Here we analytically and numerically study the dynamics of ferromagnetic bimerons driven by spin currents and magnetic fields. Numerical simulations demonstrate that two bimerons with opposite signs of topological numbers can be created simultaneously in a ferromagnetic thin film via current-induced spin torques. The current-induced spin torques can also drive the bimeron and its speed is analytically derived, which agrees with the numerical results. Since the bimerons with opposite topological numbers can coexist and have opposite drift directions, two-lane racetracks can be built to accurately encode the data bits. In addition, the dynamics of bimerons induced by magnetic field gradients and alternating magnetic fields are investigated. It is found that the bimeron driven by alternating magnetic fields can propagate along a certain direction. Moreover, combining a suitable magnetic field gradient, the Magnus-force-induced transverse motion can be completely suppressed, which implies that there is no skyrmion Hall effect. Our results are useful for understanding the bimeron dynamics and may provide guidelines for building future bimeron-based spintronic devices.

Figure 1

(a)–(c) The geometry of the magnetic atom (orange), the DMI vector (green), and the magnetic anisotropy (red). When (a) the perpendicular magnetic anisotropy and isotropic DMI are adopted, (d) the skyrmion can be a stable solution. (b) Taking in-plane easy-axis anisotropy and isotropic DMI, the (e) asymmetrical bimeron is formed. (c) If we further rotate the DMI vectors by ${90}^{\circ }$ (anisotropic DMI), (f) the bimeron has symmetrical shape.

Figure 2

(a)–(h) The time evolution of the magnetization $m_z$ induced by a spin-polarized current with the polarization vector $\mathit{p}={\mathit{e}}_{\mathit{z}}$, where the damping-like spin torque is taken into account and the color represents the value of $m_z$. (i) The evolution of the topological number $Q$ and the injected current density $j$. In our simulations, the current of $j=1500\mathrm{MA}/{\mathrm{cm}}^2$ is injected in the central circular region with diameter of 30 nm [see green lines in (a)–(c)] and we adopt the following parameters [13]: $A=15$ pJ/m, $K=0.8$ MJ/${\mathrm{m}}^3$, $D=4$ mJ/${\mathrm{m}}^2$, $M_{\text{S}}=580$ kA/m, $\gamma =2.211\times {10}^5$ m/(A s), and ${\theta }_{\text{SH}}=0.2$. Here we take the damping $\alpha =0.5$, which is a realistic value in the ultra-thin FM layer grown on a heavy metal [54]. The mesh size of $0.3\times 0.3\times 0.5{\mathrm{nm}}^3$ is used to discretize the FM film with the size $120\times 120\times 0.5{\mathrm{nm}}^3$. Panels (a) to (h) only show the $m_z$ in the $96\times 96{\mathrm{nm}}^2$ plane.

Figure 3

The components (a) $m_x$ and (b) $m_y$ of the magnetization for this case of Fig. 2.

Figure 4

(a)–(f) The time evolution of the velocities ($v_x$, $v_y$) for bimerons with opposite signs of the topological number $Q$, where $j=5\mathrm{MA}/{\mathrm{cm}}^2$, ${\theta }_{\text{SH}}=0.2$, and $\alpha =0.5$. In addition, different polarization vectors $\mathit{p}={\mathit{e}}_{\mathit{x}},{\mathit{e}}_{\mathit{y}}$ and ${\mathit{e}}_{\mathit{z}}$ are adopted. The velocities (g) $v_x$ and (h) $v_y$ as functions of the damping $\alpha$ for the bimerons with positive sign of $Q$, where the numerical (symbols) and analytical (lines) results are obtained from our micromagnetic simulations and Eq. (5), respectively. In addition to $d_{xx}\sim 15.44$ and $d_{yy}\sim 12.93$, the numerical value of ($u_x,u_y$) was used in our calculations [Fig. S4 of Ref. [65] shows that it is equal to (0, $-13.18$ nm), (33.03 nm, 0), and (0, 25.91 nm) for the cases of $\mathit{p}={\mathit{e}}_{\mathit{x}},{\mathit{e}}_{\mathit{y}}$ and ${\mathit{e}}_{\mathit{z}}$, respectively.].

Figure 5

(a) The time evolution of the velocities ($v_x$, $v_y$) for the bimeron with positive $Q$, where the magnetic field gradient ($\mathit{H}=-dH/dx·x{\mathit{e}}_{\mathit{x}}$ with ${\mu }_{0}dH/dx=1$ mT/nm) is adopted as the driving source, and the damping $\alpha =0.5$. (b) The velocities ($v_x$, $v_y$) at $t=0.2$ ns as functions of the magnetic field gradient ${\mu }_{0}dH/dx$ for $\alpha =0.5$, where the symbols are the results of numerical simulations and the lines are given by Eqs. (5) and (6).

Figure 6

(a)–(c) The trajectories of bimerons induced by an alternating magnetic field $\mathit{H}=H_0\sin \left(2\pi ft\right){\mathit{e}}_{\mathit{x}}$ [(a) the asymmetrical bimeron with positive $Q$; (b) the asymmetrical bimeron with negative $Q$; (c) the symmetrical bimeron with positive $Q$], where ${\mu }_0{H}_0=50$ mT, $f=10$ GHz and $\alpha =0.5$. To form the symmetrical bimeron, the anisotropic DMI is used in our simulation. (d)–(f) The time evolution of the guiding center ($r_x$, $r_y$) for the cases of panels (a) to (c), respectively.

Figure 7

(a), (b) The bimeron velocities ($v_x$, $v_y$) as functions of the frequency $f$ of the alternating magnetic field $\mathit{H}=H_0\sin \left(2\pi ft\right){\mathit{e}}_{\mathit{x}}$, where ${\mu }_0{H}_0=10$ mT and $\alpha =0.1$, 0.3, and 0.5. The inset in Fig. 7 is the absorption spectrum. Applying a magnetic field ${\mathit{B}}_{\mathit{s}}=0.5\mathrm{mT}·\sin (2\pi ft$)/($2\pi ft){\mathit{e}}_{\mathit{x}}$ with $f=100$ GHz yields the average magnetization $〈m_x\left(t\right)〉$ of the system, and then the Fourier transform of $〈m_x\left(t\right)〉$ gives the absorption spectrum, where the damping $\alpha =$ 0.008 is used in our calculation. (c), (d) The bimeron velocities ($v_x$, $v_y$) as functions of the damping $\alpha$, where the alternating magnetic field has the amplitude of 10 mT and frequency of 20 GHz. The inset in panel (d) shows the ratio of $v_y/v_x$ with different damping constants.

Figure 8

(a) The schematic diagram of the forces acting on the bimerons, where ${\mathit{F}}_{\text{net}}$ and ${\mathit{F}}_{\text{grad}}$ are the forces induced by the alternating magnetic field and magnetic field gradient, respectively, and ${\mathit{F}}_{\alpha }$ and $\mathit{G}\times \mathit{v}$ denote the dissipative force and Magnus force. The color of the background represents the value of the space-dependent magnetic field. (b) The trajectories of bimerons induced by an alternating magnetic field $\mathit{H}=H_0\sin \left(2\pi ft\right){\mathit{e}}_{\mathit{x}}$ with ${\mu }_0{H}_0=10$ mT and $f=20$ GHz, where we set the damping $\alpha =0.1$ and the total time $t=1$ ns in our simulations. In addition, the magnetic field gradient ($\mathit{H}=-dH/dx·x{\mathit{e}}_{\mathit{x}}$ with ${\mu }_{0}dH/dx=0$, 0.2, 0.28 and 0.6 mT/nm) is applied. (c, d) The bimeron velocities ($v_x$, $v_y$) as functions of the magnetic field gradient ${\mu }_{0}dH/dx$, where $\alpha =0.1$, the frequency $f$ of alternating magnetic fields is 20 GHz, and the amplitudes ${\mu }_0{H}_0$ of 10 and 20 mT are adopted. The symbols and lines denote the results of numerical simulations and Eq. (5), respectively.