Control and manipulation of a magnetic skyrmionium in nanostructures

A magnetic skyrmionium is a nontopological soliton, which has a doughnut-like out-of-plane spin texture in thin films, and can be phenomenologically viewed as a coalition of two topological magnetic skyrmions with opposite topological numbers. Due to its zero topological number ($Q=0$) and doughnut-like structure, the skyrmionium has its distinctive characteristics as compared to the skyrmion with $Q=\pm 1$. Here we systematically study the generation, manipulation, and motion of a skyrmionium in ultrathin magnetic nanostructures by applying a magnetic field or a spin-polarized current. It is found that the skyrmionium moves faster than the skyrmion when they are driven by the out-of-plane current, and their velocity difference is proportional to the driving force. However, the skyrmionium and skyrmion exhibit an identical current-velocity relation when they are driven by the in-plane current. It is also found that a moving skyrmionium is less deformed in the current-in-plane geometry compared with the skyrmionium in the current-perpendicular-to-plane geometry. Furthermore we demonstrate the transformation of a skyrmionium with $Q=0$ into two skyrmions with $Q=+1$ in a nanotrack driven by a spin-polarized current, which can be seen as the unzipping process of a skyrmionium. We illustrate the energy and spin structure variations during the skyrmionium unzipping process, where linear relations between the spin structure and energies are found. These results could have technological implications in the emerging field of skyrmionics.

Figure 1

Illustrations of a magnetic skyrmionium with a topological number of $Q=0$, which can be seen as composed of a skyrmion with $Q=+1$ and a skyrmion with $Q=-1$. The arrows denote the spin directions. The dashed lines indicate the spin-to-spin mapping relation among the skyrmion with $Q=+1$, the skyrmion with $Q=-1$, and the skyrmionium with $Q=0$.

Figure 2

Total micromagnetic energy $E_{\mathrm{total}}$ for FM state, skyrmion, skyrmionium, and $3\pi$ state as a function of the DMI constant $D$ in the nanodisk with $r=75$ nm. $E_{\mathrm{total}}$ includes the Heisenberg exchange, PMA, demagnetization, and DMI energies. A zoomed view of the curves around $D=4$ mJ ${\mathrm{m}}^{-2}$ is given. Insets show the top views of the FM state, skyrmion, skyrmionium, and $3\pi$ state at $D=4$ mJ ${\mathrm{m}}^{-2}$, and that of the distorted skyrmion and skyrmionium at $D=5.6$ mJ ${\mathrm{m}}^{-2}$. The color scale, which has been used throughout the paper, represents the out-of-plane magnetization component $m_z$.

Figure 3

(a) Relaxed profiles $m_z\left(x\right)$ of the out-of-plane magnetization component along the radius for the FM state, skyrmion, skyrmionium, and $3\pi$ state in the nanodisk with $r=75$ nm and $D=4$ mJ ${\mathrm{m}}^{-2}$. $L=66$ nm and $L_0=47$ nm stand for the cycloid period of the skyrmionium and the anisotropy-free cycloid period, respectively. (b) Sketch of the four-well potential of the nanodisk with $r=75$ nm and $D=4$ mJ ${\mathrm{m}}^{-2}$, where the skyrmion and skyrmionium are favorable (stable) states, while the FM state and $3\pi$ state are unfavorable (metastable).

Figure 4

Generation of a skyrmionium by applying a spin-polarized current with the CPP geometry in the nanodisk with $r=75$ nm and $D=4$ mJ ${\mathrm{m}}^{-2}$. (a) Top views of the magnetization distribution at selected $t$. A 200-ps-long spin-polarized current is injected into the central region with a radius of $r_{\mathrm{i}}=r/2$, which is indicated by the black circle. (b) The spin-polarized current density $j$ and the topological number $Q$ as functions of $t$. (c) The normalized in-plane and out-of-plane magnetization components ($m_x,m_y,m_z$) as functions of $t$.

Figure 5

Dependence of the size and shape of the skyrmionium on the applied perpendicular magnetic field $H_z$ in the nanodisk with $r=75$ nm and $D=4$ mJ ${\mathrm{m}}^{-2}$. (a) Top views of the magnetization distribution at selected $H_z$. (b) Relaxed profile $m_z\left(x\right)$ along the nanodisk radius at selected $H_z$. (c) Relaxed profile $m_z\left(x\right)$ along the nanodisk diameter as a function of $H_z$. $H_z>0$ ($H_z<0$) denotes that the field is applied along the $+z$ direction ($-z$ direction). When $H_z>130$ mT or $H_z<-260$ mT, the degeneration of a skyrmionium with $Q=0$ to a skyrmion with $Q=\pm 1$ occurs.

Figure 6

The normalized out-of-plane magnetization component $m_z$ and the topological number $Q$ as a function of $H_z$ applied along (a) the $+z$ direction and (b) the $-z$ direction, corresponding to Fig. 5.

Figure 7

Velocity for the skyrmion $v_{\mathrm{sk}}$, skyrmionium $v_{\mathrm{skium}}$, and bilayer skyrmionium $v_{\mathrm{bi}-\mathrm{skium}}$ driven by the spin-polarized current with the CPP or CIP geometry as a function of the driving current density $j$ in the nanotrack with $l=1000$ nm, $w=150$ nm, and $D=3.5$ mJ ${\mathrm{m}}^{-2}$. The velocity at different $j$ is measured when steady motion is attained. The skyrmionium and skyrmion will be destroyed when the driving current density with the CPP geometry is larger than $15\mathrm{MA}{\mathrm{cm}}^{-2}$ and 17 $\mathrm{MA}{\mathrm{cm}}^{-2}$, respectively.

Figure 8

(a) A relaxed skyrmionium at $j=0\mathrm{MA}{\mathrm{cm}}^{-2}$. (b) A skyrmionium in the steady motion driven by the CPP geometry of $j=3\mathrm{MA}{\mathrm{cm}}^{-2}$, and (c) of $j=13\mathrm{MA}{\mathrm{cm}}^{-2}$. (d) A skyrmionium in the steady motion driven by the CIP geometry of $j=90\mathrm{MA}{\mathrm{cm}}^{-2}$, and (e) of $j=200\mathrm{MA}{\mathrm{cm}}^{-2}$. (f) A relaxed skyrmion at $j=0\mathrm{MA}{\mathrm{cm}}^{-2}$. (g) A skyrmion in the steady motion driven by the CPP geometry of $j=3\mathrm{MA}{\mathrm{cm}}^{-2}$, and (h) of $j=13\mathrm{MA}{\mathrm{cm}}^{-2}$. (i) A skyrmion in the steady motion driven by the CIP geometry of $j=90\mathrm{MA}{\mathrm{cm}}^{-2}$, and (j) of $j=200\mathrm{MA}{\mathrm{cm}}^{-2}$. Here, for the CIP geometry, $\beta =0.6$.

Figure 9

The components of the dissipative tensor $\mathcal{D}$, that is, ${\mathcal{D}}_{xx},{\mathcal{D}}_{yy},{\mathcal{D}}_{xy}$, and ${\mathcal{D}}_{yx}$, as functions of $j$ for the skyrmion (solid symbols) and skyrmionium (open symbols) driven by (a) CPP geometry, (b) CIP geometry with $\beta =0.15$, (c) CIP geometry with $\beta =0.3$, and (d) CIP geometry with $\beta =0.6$. Here, ${\mathcal{D}}_{xy}={\mathcal{D}}_{yx}$. The components of the $\mathcal{I}$ tensor, that is, ${\mathcal{I}}_{xx},{\mathcal{I}}_{yy},{\mathcal{I}}_{xy}$, and ${\mathcal{I}}_{yx}$, as functions of $j$ for the skyrmion (solid symbols) and skyrmionium (open symbols) driven by (e) CPP geometry, (f) CIP geometry with $\beta =0.15$, (g) CIP geometry with $\beta =0.3$, and (h) CIP geometry with $\beta =0.6$. Insets show the skyrmion and skyrmionium at selected $j$ and current injection geometries indicated by the vertical dashed lines.

Figure 10

The destruction process of a skyrmionium driven by the CPP geometry of $j=18\mathrm{MA}{\mathrm{cm}}^{-2}$ in the nanotrack with $l=1000$ nm, $w=150$ nm, and $D=3.5$ mJ ${\mathrm{m}}^{-2}$. The skyrmionium goes straight with deformation since the opposite Magnus forces act on the outer skyrmion with $Q=+1$ and the inner skyrmion with $Q=-1$, which results in the destruction of the skyrmionium into a skyrmion with $Q=+1$. After the destruction, the skyrmion with $Q=+1$ touches the upper edge of the nanotrack as the SkHE is active.

Figure 11

Top views of the motion of a magnetic bilayer skyrmionium driven by the CPP or CIP geometry in an antiferromagnetically exchange-coupled bilayer nanotrack with $l=750$ nm, $w=150$ nm, and $D=3.5$ mJ ${\mathrm{m}}^{-2}$. Top views of (a) the bottom and (b) the top FM layers at $t=0$ ps, where no driving current is applied. Top views of (c) the bottom and (d) the top FM layers when the CPP driving current of $j=30\mathrm{MA}{\mathrm{cm}}^{-2}$ is applied; the bilayer skyrmionium moves along the nanotrack without any distortion, reaching a steady velocity of $v_{\mathrm{bi}-\mathrm{skium}}=134\mathrm{m}{\mathrm{s}}^{-1}$. Top views of (e) the bottom and (f) the top FM layers when the driving current of $j=100\mathrm{MA}{\mathrm{cm}}^{-2}$ with the CIP geometry is applied with $\beta =0.6$; the bilayer skyrmionium moves along the nanotrack without any distortion, reaching a steady velocity of $v_{\mathrm{bi}-\mathrm{skium}}=78\mathrm{m}{\mathrm{s}}^{-1}$.

Figure 12

Unzipping of a skyrmionium with $Q=0$ into two skyrmions with $Q=+1$ in a nanotrack device with the junction geometry. (a)–(j) Top views of the magnetization configuration of the device at selected $t$.

Figure 13

(a) The topological number $Q$, (b) the normalized in-plane magnetization components $(m_x,m_y)$, (c) the normalized out-of-plane magnetization component $m_z$, (d) the total micromagnetic energy $E_{\mathrm{total}}$ and demagnetization energy $E_{\mathrm{demag}}$, (e) the exchange energy $E_{\mathrm{ex}}$ and PMA energy $E_{\mathrm{PMA}}$, and (f) the DMI energy $E_{\mathrm{DMI}}$ as functions of $t$. The topological transition of $Q=0\to Q=+2$ occurs within 1000 ps, indicating the success of the unzipping event. Vertical dashed lines indicate the selected times corresponding to the snapshots shown in Fig. 12. Light yellow background indicates the time span of the unzipping process.

Figure 14

(a) Domain wall spins and (b) core spins as functions of $t$. The domain wall spins are defined as the spins which have $-0.5<m_z<0.5$, while the core spins are defined as the spins which have $m_z<0$. Square, circle, and triangle symbols denote the data for the skyrmionium, unzipping process, and skyrmions, respectively.

Figure 15

(a) $m_z$ as a function of core spins. (b) $E_{\mathrm{ex}}$ and (c) $E_{\mathrm{DMI}}$ as functions of domain wall spins. Square, circle, and triangle symbols denote the data for the skyrmionium, unzipping process, and skyrmions, respectively.