Asymmetric Motion of Magnetic Skyrmions in Ferromagnetic Nanotubes Induced by a Magnetic Field

Abstract

Magnetic skyrmions, featuring topological stability and low driving current density, are believed to be a promising candidate of information carriers. One of the obstacles to application is the skyrmion Hall effect, which can lead to the annihilation of moving skyrmions at the lateral boundary of thin-film tracks. In order to resolve this issue, it was recently proposed to exploit ferromagnetic nanotubes as alternative skyrmion guides. In this work, we investigate the field-effect of current-driven skyrmion motion in nanotubes using micromagnetic simulations. It is found that, in the presence of an axial field, the skyrmion motion becomes asymmetric in tubes. This is fundamentally different from the flat strip, in which a field has little influence on the skyrmion dynamics. Based on the dissipation tensor determined by the spin texture of the skyrmions, the solution of the Thiele equation is obtained, yielding a perfect match with simulations. We argue that the asymmetry of the skyrmion dynamics originates from the curvature of the nanotube.

1. Introduction

The magnetic skyrmion is a topological magnetic quasiparticle which can exist stably under room temperature [1,2,3,4,5]. The generation of Néel-type skyrmions are normally induced by the interfacial Dzyaloshinskii–Moriya interaction (DMI) contained in some multilayer films with broken interface asymmetry [6], whereas the Bloch-type ones originate from the bulk DMI existing in some spontaneous chiral magnets with no central symmetry [7]. Skyrmions can also be formed under strong enough easy-normal anisotropy, magnetostatic interaction or surface curvature, even without DMI [8,9,10,11]. Skyrmions feature the ability to surmount inclusion defects and to keep a steady spin texture while shifting on curved tracks, owing to their topological stability. In contrast to the magnetic domain walls, nanosized skyrmions also have a smaller threshold of driving current density [12,13]. All of the above properties support skyrmions to be a potential candidate of information carriers [14,15].

Skyrmions in flat strips, driven by the spin-polarized current, may annihilate at the lateral boundaries due to the skyrmion Hall effect [16,17]. A practicable solution is to utilize ferromagnetic nanotubes as the skyrmion guides [18,19]. As studied before, the curvature effect in nanotube geometry has significant impacts on spin dynamics [19,20,21,22,23,24,25,26,27,28]. One of the impacts is the broken dynamic chiral symmetry existing in cylindrical geometry [29,30]. For instance, the field-induced domain-wall motion in a nanotube, as well as the threshold magnitude of a magnetic field that encounters Walker breakdown [31], is asymmetric; the propagation of the spin-wave in the opposite direction is asymmetric, which derives from the asymmetric dispersion [32,33]. Despite extensive studies, no asymmetry in skyrmion dynamics was reported so far either in flat strips or tubes [17,18].

In this study, we demonstrate the asymmetric motion of skyrmions induced by the curvature of nanotubes via micromagnetic simulations. Particularly, we investigate the influence of the external field and the spin-polarized current on the symmetry of the motion and spin texture of skyrmions. Further, we theoretically calculate the specific value of the shifting velocity of skyrmions as well as the dissipative force tensor characterizing the spin texture. We conclude that the curvature effect can cause the asymmetric motion of skyrmions, which becomes more evident with the increase in the magnitude of the axial external magnetic field and current density. Our results provide another example of utilizing the shape of the nanostructure to manipulate sample properties besides other common approaches [34,35,36,37].

2. Materials and Methods

Employing the Mumax3 package [38,39], we performed finite-difference micromagnetic simulations to study the dynamics of skyrmions in magnetic nanotubes via solving the Landau–Lifshitz–Gilbert equation with the spin-transfer torque ${\mathsf{\Gamma }}_{STT}=-\left({\mathit{v}}_{\mathit{s}}·\nabla \right)\mathit{m}+\beta \mathit{m}\times \left({\mathit{v}}_{\mathit{s}}·\nabla \right)m$ [40,41,42],

$$\frac{d\mathit{M}}{dt}=-{\gamma }_0\left(\mathit{M}\times \mathit{H}\right)+\frac{\alpha }{M_s}\left(\mathit{M}\times \frac{d\mathit{M}}{dt}\right)+{\mathsf{\Gamma }}_{STT}.$$

(1)

The constructure of nanotubes was assumed to be a rolled Co/Pt multilayer film, and the material parameters were as follows [43,44,45,46]: saturation magnetization $M_{sat}=580 \mathrm{kA}/\mathrm{m}$, exchange stiffness constant $A_{ex}=15 \mathrm{PJ}/\mathrm{m}$, easy-normal anisotropy parameter $K_u=800 \mathrm{kJ}/{\mathrm{m}}^3$, damping constant $\alpha =0.1$ and the interfacial DMI constant $D_{DMI}=2.3 \mathrm{mJ}/{\mathrm{m}}^2$. For spin-polarized current, electrical current polarization $P=0.5$ and nonadiabaticity of spin-transfer torque $\beta =0.5$. The nanotube in our simulations was defined by a 25 nm outer radius, 4 nm thickness and 1000 nm length. The initial magnetization was generally along the radial outward direction ($m_{\rho }=1$), whereas a small part at the center was magnetized oppositely ($m_{\rho }=-1$) to form a skyrmion, as shown in Figure 1.

3. Results

3.1. Velocity

In nanotubes, a Néel-type skyrmion with $-\rho$ polarization is stabilized and shifts in a left-handed helical trajectory driven by the spin-polarized current. The skyrmion velocity is defined by two components: one is parallel to the direction of electric current, $v_{∕∕}=\mathit{v}·\widehat{\mathit{z}}=v_z$, whereas the other is perpendicular, caused by the skyrmion Hall effect [43,47,48,49,50], $v_{\perp }=\mathit{v}·\widehat{\mathit{\phi }}=v_{\phi }$. Only the velocity at the outer radius is considered in our simulation results because of the different $v_{\phi }$ at different radius ($v_{\phi }=\omega r$).

Firstly, we measured the skyrmion velocity under three distinct field conditions: (1) $B=0T$, (2) $B_{\phi }=0.4T$ and (3) $B_z=0.4T$. Both directions of spin-polarized currents $j_{z+}$ and $j_{z-}$ were injected through the nanotubes one after the other. In the absence of an external field ($B=0T$), the magnitude of both velocity components $v_z$ and $v_{\phi }$ were identical, accounting for the complete symmetry of the skyrmion motion towards $z+$ and $z-$, as shown in Figure 2a. In the presence of $B_{\phi }=0.4T$, the disparity of $v_z$ remained indistinguishable (<$1\%$), whereas the disparity of $v_{\phi }$ varied slightly (≤$2.03\%$); these disparities manifest no positive correlation with the electric current density, as shown in Figure 2b. When an axial field was applied ($B_z=0.4T)$, it was obvious that the disparity of $v_{jz+}$ and $v_{jz-}$ was enlarged with the increasing current density, as shown in Figure 2c. Particularly, the difference of ${\mathit{v}}_{\perp }$ when skyrmions shifted towards the opposite direction reached up to 11% at $j=250\times {10}^{11}\mathrm{A}/{\mathrm{m}}^2$, as shown in Figure 2d.

Then, the electric current density was fixed to $150\times {10}^{11}\mathrm{A}/{\mathrm{m}}^2$ to make out the correlation between the field magnitude and the asymmetry of the skyrmion dynamics. Figure 3 indicates that if the direction of the electric current and the external field are parallel to each other, the velocity of the skyrmions in general remains the same with no changes. However, if the field and the current are antiparallel to each other, the skyrmion velocity decreases significantly with the increase of magnetic induction. This phenomenon also shows that the velocity disparity of skyrmions moving towards the two sides of the nanotube will gradually increase with the amplifying of the magnetic field.

Thus far, we demonstrated that the axial magnetic field breaks the symmetry of skyrmion motion in nanotubes, and this asymmetry is positively correlated to the current density and the magnetic field strength.

3.2. Shape

According to the Thiele equation of the skyrmion motion driven by in-plane spin-polarized current [51] with fixed material parameters, the skyrmion velocity is determined only by the current density $j_e$ and the dissipation tensor $D_{ij}$, where the latter in turn is determined by the configuration of the skyrmions in the system. We therefore investigated the effects of magnetic fields, electric currents and curvature effects on the spin texture of skyrmions.

In the steady state without external magnetic fields and current, the Néel-type skyrmions are stretched into elliptical structures along the axial direction due to the influence of the nanotube curvature [52]. Electric currents in different directions stretched the skyrmions along the Hall angle, but the skyrmions were basically symmetrical in configuration. When a tangential magnetic field was applied, the overall magnetic moments of the system were slightly deflected toward the magnetic field, but the symmetry of the skyrmions stretched by the electric current were basically the same as in the condition with no external field. When an axial external magnetic field was applied, the skyrmions were stretched into a configuration that was evidently asymmetrical and was distinct in size under the impact of current, as shown in Figure 4.

3.3. Role of the Curvature Effect

To further investigate the role of the curvature effect in the asymmetrical motion of skyrmions, we performed analogous numerical simulations in a planar structure in the absence of the curvature effect. The external magnetic fields corresponding to the simulation in nanotubes were applied: (1) $B=0T$, (2) $B_y=0.4T$ and (3) $B_x=0.4T$; notice that the easy-normal direction is along the z-axis. The geometric structure was set as a 1000 nm × 400 nm × 4 nm flat strip, with the interfacial DM interaction constant $D_{DMI}=2.7 \mathrm{mJ}/{\mathrm{m}}^2$. The effects of different external magnetic fields and spin-polarized currents on the motion and shape of skyrmions in planar structures were studied. The simulation results indicate that, as shown in

Table 1. Skyrmion velocity driven by a current under different external fields in a flat strip. The current density is fixed at $150\times {10}^{11}$ A/m².

Velocity(m/s)		$$\mathit{B}=0\mathit{T}$$	$${\mathit{B}}_{\mathit{y}}=0.4\mathit{T}$$	$${\mathit{B}}_{\mathit{x}}=0.4\mathit{T}$$
$${\mathit{j}}_{\mathit{x}}=150\times {10}^{11} A/m^2$$	v//	626	624	624
	v⊥	278	274	276
$${\mathit{j}}_{\mathit{x}}=150\times {10}^{11} A/m^2$$	v//	626	622	622
	v⊥	278	274	274

and Figure 5, the influence of external magnetic fields in any direction on the symmetry of the skyrmion velocity and shape is negligible in a planar structure.

4. Discussion

In this section, analytical calculations are performed to study the skyrmion dynamics in nanotube structures by solving the Thiele equation. In a planar structure, the Thiele equation for current-driven skyrmions can be expressed as [17,43,53,54]:

$$\mathit{G}\times \left(\mathit{v}-{\mathit{v}}_{\mathit{s}}\right)+\mathit{D}\left(\alpha \mathit{v}-\beta {\mathit{v}}_{\mathit{s}}\right)=0,$$

(2)

where ${\mathit{v}}_{\mathit{s}}=v_s\widehat{x}=\frac{-{\mu }_{B}P{j}_e}{e{M}_s\left(1+{\beta }^2\right)}\widehat{x}$ is the velocity of the conduction electrons, $j_e$ is the spin-polarized current density, ${\mu }_B$ is the Bohr magneton, P is the polarization rate of the current, $\mathit{G}=\mathcal{G}\widehat{g}$ is the gyrovector where $\mathcal{G}=\int \mathit{m}·\left(\frac{\partial \mathit{m}}{\partial x}\times \frac{\partial \mathit{m}}{\partial y}\right)dxdy=4\pi Q$, $Q=\pm 1$ is the skyrmion number, $\widehat{g}=\widehat{z}$ is the direction of the skyrmion core, $\mathit{v}$ is the drift velocity of the spin texture and $\mathit{D}$ is the dissipative force tensor:

$$D_{ij}=\int {\partial }_i\mathit{m}·{\partial }_j\mathit{m}{d}^{2}r.$$

(3)

For a geometrically perfectly symmetrical skyrmion, the components of $\mathit{D}$ own the following characteristics:

$$\left\{\begin{array}{c}{D}_{xx}=D_{yy}=D\\ D_{xy}=D_{yx}=0\end{array}\right..$$

(4)

In this case, the dissipation tensor $\mathit{D}$ is reduced to a scalar, and the components of the skyrmion velocity can be calculated as:

$$\left\{\begin{array}{c}{v}_{∕∕}=\frac{\alpha \beta D^2+{\mathcal{G}}^2}{{\alpha }^2{D}^2+{\mathcal{G}}^2}{v}_s\\ v_{\perp }=\frac{\mathcal{G}D\left(\alpha -\beta \right)}{{\alpha }^2{D}^2+{\mathcal{G}}^2}{v}_s\end{array}\right..$$

(5)

However, the geometric structure of skyrmions in nanotubes is not completely symmetrical under the action of an external magnetic field and the spin-polarized current; thus, Equation (4) is no longer applicable (but the accessible conclusion $D_{xy}=D_{yx}$ is still valid). At this time, the velocity components of the skyrmions (5) should be revised as:

$$\left\{\begin{array}{c}{v}_{∕∕}=\frac{\alpha \beta D_{xx}{D}_{yy}+{\mathcal{G}}^2-D_{xy}\mathcal{G}\left(\alpha -\beta \right)-\alpha \beta {D_{xy}}^2}{{\alpha }^2{D}_{xx}{D}_{yy}+{\mathcal{G}}^2-{\alpha }^2{D_{xy}}^2}{v}_s\\ v_{\perp }=\frac{D_{xx}\mathcal{G}\left(\alpha -\beta \right)}{{\alpha }^2{D}_{xx}{D}_{yy}+{\mathcal{G}}^2-{\alpha }^2{D_{xy}}^2}{v}_s\end{array}\right..$$

(6)

In order to calculate $D_{ij}=\left(\begin{array}{cc}{D}_{xx}& D_{xy}\\ D_{yx}& D_{yy}\end{array}\right)$, we unrolled the outermost layer of the nanotube into a planar film, as shown in Figure 6a, and then transformed the system magnetic moments $\left(\mathit{m}=m_x\widehat{x}+m_y\widehat{y}+m_z\widehat{z}\right)$ into ones expressed in a new Cartesian coordinate $\left(\mathit{m}=m_x^{\prime }{\widehat{x}}^{\prime }+m_y^{\prime }{\widehat{y}}^{\prime }+m_z^{\prime }{\widehat{z}}^{\prime }\right)$:

$$\left\{\begin{array}{c}{m}_x^{\prime }=m_z\\ m_y^{\prime }=m_{x}sin \theta -m_{y}cos \theta \\ m_z^{\prime }=m_{x}cos \theta +m_{y}sin \theta \end{array}\right.$$

(7)

$$\left\{\begin{array}{c}{\widehat{x}}^{\prime }=\widehat{z}\\ {\widehat{y}}^{\prime }=\widehat{x}\\ {\widehat{z}}^{\prime }=\widehat{y}\end{array}\right..$$

(8)

Using numerical results, we calculated the dissipation tensor from Equation (3) in the nanotube under different conditions.

We first investigated the influence of spin-polarized current density on $D_{ij}$ under different external fields. Figure 6b indicates that $D_{xx}$ was very slightly affected by current density, whereas $D_{yy}$ was significantly affected. Once the applied current density was greater than the threshold (pinning current $j_p\approx {10}^9 \mathrm{A}/{\mathrm{m}}^2$), $D_{yy}$ increased significantly regardless of whether the direction of the current was z+ or z−. In addition, when the external magnetic field was along the axial direction, the degree of the asymmetry of $D_{yy}$ in opposite current directions was the highest (the imparity between the maximum value and the minimum value was around 20%), whereas the one in the other two cases was less than 8%. In order to further study the influence of the axial external magnetic field on dissipation tensor $D_{ij}$, we fixed the current density at $j=150\times {10}^{11} \mathrm{A}/{\mathrm{m}}^2$, and changed the intensity of the axial magnetic field. The results (Figure 6c) are very similar to the effect of the magnetic field strength on the skyrmion velocity (Figure 3). When the direction of the driving current was antiparallel to the magnetic field, $D_{ij}$, especially $D_{yy}$, decreased obviously with the increase in the magnetic induction; when the direction of the driving current was the same as the magnetic field, $D_{xx}$ remained basically unchanged whereas $D_{yy}$ increased slightly with the magnetic induction. As for $D_{xy}$, the results of all numerical calculations were in the interval of (−1, 1), and there was no obvious regularity. Finally, we substituted the calculated dissipation tensor in each case into Equation (6) to obtain the theoretical value of skyrmion velocity (plotted with solid line in Figure 2) solved by the Thiele function, which was highly consistent with the micromagnetic simulation results shown in the previous section.

5. Conclusions

In this paper, we reported the impact of the magnetic field on skyrmion dynamics in ferromagnetic nanotubes. In contrast to a planar structure, in which the shape and moving velocity of skyrmions are completely symmetric under external fields, an axial field applied in nanotubes is found to be able to break the symmetry of the skyrmion motion. Numerically, we were able to extract the skyrmion profiles and moving velocities under various currents and fields. At the same time, the skyrmion velocities were also calculated by solving the Thiele equation. A perfect agreement is achieved between the analytical solution of the Thiele equation and micromagnetic simulations. Our results show that the curvature of the tubes also leads to a chiral effect in skyrmion dynamics with the help of an external field, which in principle provides new aspects for the manipulation of skyrmions.