Phenomenology of current-induced skyrmion motion in antiferromagnets

We consider a two-sublattice antiferromagnet within the exchange approximation with the sublattice magnetizations ${{\bf{M}}}_{1}\simeq -{{\bf{M}}}_{2}$ and $| {{\bf{M}}}_{1}| =| {{\bf{M}}}_{2}| ={M}_{s}$, where ${M}_{s}$ is the saturation magnetization. Further, we consider a noncentrosymmetric lattice of the crystallographic class ${C}_{{nv}}$ (an example of such an antiferromagnet is ${{\rm{K}}}_{2}{{\rm{V}}}_{3}{{\rm{O}}}_{8}$) [14]. This class encompasses also two-dimensional films which have structural inversion asymmetry along the $\hat{z}$-direction, for example, due to the presence of an interface.

It is most convenient to formulate the theory in terms of the antiferromagnetic order parameter (also called the Néel vector) ${\bf{n}}=({{\bf{M}}}_{1}-{{\bf{M}}}_{2})/2{M}_{s}$ and the total magnetization ${\bf{m}}=({{\bf{M}}}_{1}+{{\bf{M}}}_{2})/2{M}_{s}$. Our phenomenological approach is based on the exchange approximation [22], which requires rotational invariance of the magnetization vectors and invariance of the theory with respect to an exchange of the two sublattices, i.e., under the transformations ${\bf{n}}\to -{\bf{n}}$ and ${\bf{m}}\to {\bf{m}}$. The magnetic energy that follows from these considerations is⁸ [14, 22]

$$\begin{eqnarray}F & = & \displaystyle \int {\rm{d}}{\bf{r}}\left\{A\displaystyle \sum _{i=x,y}{({\partial }_{i}{\bf{n}})}^{2}+{H}_{\mathrm{exc}}{M}_{s}{{\bf{m}}}^{2}-{H}_{\mathrm{an}}{M}_{s}{({\bf{n}}\cdot \hat{z})}^{2}-2{M}_{s}{\bf{m}}\cdot {\bf{H}}\right.\\ & & +2{M}_{s}{\bf{d}}\cdot ({\bf{m}}\times {\bf{n}})+D[(\hat{z}\cdot {\bf{n}})({\rm{\nabla }}\cdot {\bf{n}})-({\bf{n}}\cdot {\rm{\nabla }})(\hat{z}\cdot {\bf{n}})]\phantom{\rule{0ex}{3.95ex}}\}.\end{eqnarray} \tag{ 2 }$$

The first and the second terms describe the inhomogeneous and homogenous exchange interaction with the constants $A$ and ${H}_{\mathrm{exc}}$, respectively. The uniaxial anisotropy is parameterized by the constant ${H}_{\mathrm{an}}\gt 0$ and the external magnetic field is denoted by ${\bf{H}}$. The remaining terms describe the DM interactions, where ${\bf{d}}\parallel \hat{z}$ and D represent the homogeneous and inhomogeneous parts, respectively. The latter part of the DM interactions, also called a Lifshitz invariant, is the main ingredient needed to stabilize a skyrmion in this system and inhomogeneous textures in general [14].

The antiferromagnetic vectors obey the constraints ${{\bf{n}}}^{2}=1$ and ${\bf{n}}\cdot {\bf{m}}=0$. Throughout this work we make the assumption that the homogeneous exchange interaction ${H}_{\mathrm{exc}}$ is the dominant energy scale, so that ${H}_{\mathrm{exc}}\gg d,{H}_{\mathrm{an}}$. For typical values of the external magnetic field that do not destroy the antiferromagnetic order the exchange constants dominates the field too, ${H}_{\mathrm{exc}}\gg H$ [6].

In this section we derive the equations of motion for the Néel order parameter for a time-independent magnetic field and electric current. The Landau–Lifshitz–Gilbert equations to leading order in ${H}_{\mathrm{exc}}$ are given by [18, 19, 23]

$$\begin{eqnarray}\dot{{\bf{n}}} & \simeq & -\displaystyle \frac{\gamma }{2{M}_{s}}\left[{\bf{n}}\times \displaystyle \frac{\delta f}{\delta {\bf{m}}}\right]+{\tau }_{{\bf{n}}},\\ \dot{{\bf{m}}} & \simeq & -\displaystyle \frac{\gamma }{2{M}_{s}}\left[{\bf{n}}\times \displaystyle \frac{\delta f}{\delta {\bf{n}}}+{\bf{m}}\times \displaystyle \frac{\delta f}{\delta {\bf{m}}}\right]+{\alpha }_{{\rm{G}}}{\bf{n}}\times \dot{{\bf{n}}}+{\tau }_{{\bf{m}}},\end{eqnarray} \tag{ 3 }$$

where γ is the gyromagnetic ratio⁹ , ${\alpha }_{{\rm{G}}}$ is a phenomenological Gilbert damping coefficient and ${\tau }_{{\bf{n}},{\bf{m}}}$ represent the respective current-induced torques to be discussed in section 2.3. The functional derivatives of the energy density f are

$$\begin{eqnarray}\displaystyle \frac{1}{2{M}_{s}}\displaystyle \frac{\delta f}{\delta {\bf{n}}} & = & -\displaystyle \frac{A}{{M}_{s}}{{\rm{\nabla }}}^{2}{\bf{n}}+{\bf{d}}\times {\bf{m}}-{H}_{\mathrm{an}}({\bf{n}}\cdot \hat{z})\hat{z}+\displaystyle \frac{D}{{M}_{s}}[({\rm{\nabla }}\cdot {\bf{n}})\hat{z}-{\rm{\nabla }}({\bf{n}}\cdot \hat{z})],\\ \displaystyle \frac{1}{2{M}_{s}}\displaystyle \frac{\delta f}{\delta {\bf{m}}} & = & {H}_{\mathrm{exc}}{\bf{m}}-{\bf{H}}+({\bf{n}}\times {\bf{d}}).\end{eqnarray} \tag{ 4 }$$

The static magnetization of this system without electric currents is found by taking the cross product of the first line of (3) with ${\bf{n}}$ and identifying the time-independent components under the constraints ${{\bf{n}}}^{2}=1$ and ${\bf{n}}\cdot {\bf{m}}=0$, which yields [14, 15]

$$\begin{eqnarray}&&{{\bf{m}}}_{0}=-\displaystyle \frac{1}{{H}_{\mathrm{exc}}}{\bf{n}}\times ({\bf{n}}\times {\bf{H}})-\displaystyle \frac{1}{{H}_{\mathrm{exc}}}{\bf{n}}\times {\bf{d}}.\end{eqnarray} \tag{ 5 }$$

The presence of the homogeneous DM interactions term d shows that even in the absence of an external magnetic field the total magnetization is nonzero. However, this term does not contribute to the motion of an antiferromagnetic texture within the approximations we consider and we neglect it for the remainder of the paper.

We follow a phenomenological approach to derive the current-induced torques. It is based on the Onsager reciprocity relations, which, in the case under consideration, relate the process of inducing charge currents by a time-varying magnetic texture to the effect that charge currents have on the magnetization dynamics.

Following [19], to lowest order in the spatial gradients and the magnetization ${\bf{m}}$ and zeroth order in spin–orbit coupling we find three contributions towards the magnetically pumped charge density ${{\bf{j}}}^{\mathrm{pump}}/\sigma$ (where σ is the electrical conductivity) that obey the symmetries of the system (rotation and ${\bf{n}}\to -{\bf{n}}$): $\eta /\gamma {\bf{n}}\cdot (\dot{{\bf{m}}}\times {\partial }_{i}{\bf{n}})$, $\beta /\gamma \;\dot{{\bf{n}}}\cdot {\partial }_{i}{\bf{n}}$ and $\zeta /\gamma \;{\bf{n}}\cdot (\dot{{\bf{n}}}\times {\partial }_{i}{\bf{m}})$ [19]. After applying the Onsager relations these are transformed and result in the torques

$$\begin{eqnarray}{\tau }_{{\bf{n}},\mathrm{STT}} & = & \displaystyle \frac{\eta }{2}({\bf{j}}\cdot {\rm{\nabla }}){\bf{n}}+\displaystyle \frac{\beta }{2}({\bf{j}}\cdot {\rm{\nabla }}){\bf{n}}\times {\bf{m}}+\displaystyle \frac{\zeta }{2}{\bf{m}}\times [({\bf{j}}\cdot {\rm{\nabla }}){\bf{m}}\times {\bf{n}}],\\ {\tau }_{{\bf{m}},\mathrm{STT}} & = & \displaystyle \frac{\eta }{2}{\bf{m}}\times [({\bf{j}}\cdot {\rm{\nabla }}){\bf{n}}\times {\bf{n}}]+\displaystyle \frac{\beta }{2}({\bf{j}}\cdot {\rm{\nabla }}){\bf{n}}\times {\bf{n}}+\displaystyle \frac{\zeta }{2}{\bf{n}}\times [({\bf{j}}\cdot {\rm{\nabla }}){\bf{m}}\times {\bf{n}}].\end{eqnarray} \tag{ 6 }$$

Here, the terms parameterized by the coefficients $\eta ,\zeta$ describe reactive spin-transfer torques, whereas the term with β describes a dissipative spin-transfer torque.

Furthermore, the inversion symmetry breaking gives rise to another set of torques even in homogeneous systems, which do not involve gradients of the antiferromagnetic vectors. Following the same approach as above [24], we find the pumped charge currents that are lowest order in the spin–orbit coupling: ${C}_{1}/\gamma \;\hat{z}\times \dot{{\bf{m}}}$, ${C}_{2}/\gamma \;\hat{z}\times ({\bf{n}}\times \dot{{\bf{n}}})$ and ${C}_{3}/\gamma \;\hat{z}\times ({\bf{m}}\times \dot{{\bf{m}}})$, which after applying the Onsager relations lead to

$$\begin{eqnarray}{\tau }_{{\bf{n}},\mathrm{SOT}} & = & -\displaystyle \frac{{C}_{1}}{2}{\bf{n}}\times (\hat{z}\times {\bf{j}})-\displaystyle \frac{{C}_{2}}{2}{\bf{m}}\times [{\bf{n}}\times (\hat{z}\times {\bf{j}})]-\displaystyle \frac{{C}_{3}}{2}{\bf{n}}\times [{\bf{m}}\times (\hat{z}\times {\bf{j}})],\\ {\tau }_{{\bf{m}},\mathrm{SOT}} & = & -\displaystyle \frac{{C}_{1}}{2}{\bf{m}}\times (\hat{z}\times {\bf{j}})-\displaystyle \frac{{C}_{2}}{2}{\bf{n}}\times [{\bf{n}}\times (\hat{z}\times {\bf{j}})]-\displaystyle \frac{{C}_{3}}{2}{\bf{m}}\times [{\bf{m}}\times (\hat{z}\times {\bf{j}})].\end{eqnarray} \tag{ 7 }$$

Here, the terms proportional to ${C}_{1}$ are field-like (reactive) spin–orbit torques and the terms proportional to ${C}_{2},{C}_{3}$ are the anti-damping (dissipative) spin–orbit torques. In ferromagnetic systems similar torques have been discussed in [25, 26] and in antiferromagnets a microscopic analysis has been performed in [7].

While we cannot generically exclude the existence of spin–orbit torques beyond the exchange approximation, in microscopic models such as the Rashba model such torques do not occur. In that case, with the exception of higher harmonics, (7) captures the spin–orbit torques. The study of possible spin–orbit torques that break the exchange approximation falls outside the scope of the present work.

We emphasize that the spin–orbit torques and the spin-transfer torques differ in their nature. Whereas in the latter the free electrons are polarized by the local magnetization while moving through the texture and interact with it after being polarized, in the spin–orbit torques the polarization is due to the spin–orbit coupling in the system and not due to the magnetization. In that sense, the spin-transfer torques are a result of a non-local interaction between the electrons and the magnetic moments, while the spin–orbit torques are local.

In the later steps of the calculation, presented below, the form of both the spin-transfer and the spin–orbit torques will be simplified by retaining only the leading order terms in ${H}_{\mathrm{exc}}$.

Writing out the torques explicitly, the Landau–Lifshitz–Gilbert equations (3) become

$$\begin{eqnarray}\dot{{\bf{n}}} & = & -\displaystyle \frac{\gamma }{2{M}_{s}}\left[{\bf{n}}\times \displaystyle \frac{\delta f}{\delta {\bf{m}}}\right]+\displaystyle \frac{\eta }{2}({\bf{j}}\cdot {\rm{\nabla }}){\bf{n}}-\displaystyle \frac{{C}_{1}}{2}{\bf{n}}\times (\hat{z}\times {\bf{j}}),\\ \dot{{\bf{m}}} & = & -\displaystyle \frac{\gamma }{2{M}_{s}}\left[{\bf{n}}\times \displaystyle \frac{\delta f}{\delta {\bf{n}}}+{\bf{m}}\times \displaystyle \frac{\delta f}{\delta {\bf{m}}}\right]+{\alpha }_{{\rm{G}}}{\bf{n}}\times \dot{{\bf{n}}}+\displaystyle \frac{\eta }{2}{\bf{m}}\times [({\bf{j}}\cdot {\rm{\nabla }}){\bf{n}}\times {\bf{n}}]\\ & & +\displaystyle \frac{\beta }{2}({\bf{j}}\cdot {\rm{\nabla }}){\bf{n}}\times {\bf{n}}-\displaystyle \frac{{C}_{1}}{2}{\bf{m}}\times (\hat{z}\times {\bf{j}})-\displaystyle \frac{{C}_{2}}{2}{\bf{n}}\times [{\bf{n}}\times (\hat{z}\times {\bf{j}})].\end{eqnarray} \tag{ 8 }$$

Here, only terms to leading order in ${H}_{\mathrm{exc}}$ have been kept, apart from the η and ${C}_{1}$ ones in the second line, which we retained in order to keep the constraints ${{\bf{n}}}^{2}=1$ and ${\bf{n}}\cdot {\bf{m}}=0$ fulfilled. The term containing the functional derivative of the energy density with respect to ${\bf{m}}$ is of subleading order in ${H}_{\mathrm{exc}}$, however, it is of the same order as the left-hand side after substituting (9) below and needs to be kept as well.

An expression for the total magnetization can be obtained from the equation for $\dot{{\bf{n}}}$:

$$\begin{eqnarray}&&{\bf{m}}=\displaystyle \frac{1}{\gamma {H}_{\mathrm{exc}}}{\bf{n}}\times \dot{{\bf{n}}}+{{\bf{m}}}_{0}-\displaystyle \frac{\eta }{2\gamma {H}_{\mathrm{exc}}}{\bf{n}}\times ({\bf{j}}\cdot {\rm{\nabla }}){\bf{n}}+\displaystyle \frac{{C}_{1}}{2\gamma {H}_{\mathrm{exc}}}{\bf{n}}\times [{\bf{n}}\times (\hat{z}\times {\bf{j}})],\end{eqnarray} \tag{ 9 }$$

where ${\bf{m}}$₀ now does not contain the homogeneous DM interactions contribution ${\bf{d}}$, as discussed in section 2.2. With this, we are able to write a closed equation for the Néel order parameter

$$\begin{eqnarray}&&{\bf{n}}\times \ddot{{\bf{n}}}={{\bf{H}}}_{\mathrm{shape}}+{{\bf{H}}}_{\mathrm{forces}}.\end{eqnarray} \tag{ 10 }$$

Here, we have grouped the right-hand side into terms that determine the shape of the antiferromagnetic texture and terms that induce or affect its motion. The fields are given by

$$\begin{eqnarray}{{\bf{H}}}_{\mathrm{shape}} & = & \displaystyle \frac{2\gamma }{{M}_{s}}\dot{{\bf{n}}}({\bf{n}}\cdot {{\bf{H}}}_{\mathrm{eff}})-\displaystyle \frac{{\gamma }^{2}{H}_{\mathrm{exc}}}{2{M}_{s}^{2}}\left[{\bf{n}}\times \displaystyle \frac{\delta f}{\delta {\bf{n}}}\right]-\displaystyle \frac{{\gamma }^{2}}{{M}_{s}^{2}}({\bf{n}}\times {{\bf{H}}}_{\mathrm{eff}})({\bf{n}}\cdot {{\bf{H}}}_{\mathrm{eff}}),\\ {{\bf{H}}}_{\mathrm{forces}} & = & -\displaystyle \frac{\gamma {H}_{\mathrm{exc}}\eta }{2{M}_{s}}{\bf{n}}[({\bf{j}}\cdot {\rm{\nabla }}){\bf{n}}\cdot {\bf{m}}]+\displaystyle \frac{\gamma {H}_{\mathrm{exc}}\beta }{2{M}_{s}}({\bf{j}}\cdot {\rm{\nabla }}){\bf{n}}\times {\bf{n}}\\ & & -\displaystyle \frac{\gamma {H}_{\mathrm{exc}}{C}_{2}}{2{M}_{s}}{\bf{n}}\times [{\bf{n}}\times (\hat{z}\times {\bf{j}})]+\displaystyle \frac{\gamma {H}_{\mathrm{exc}}{\alpha }_{{\rm{G}}}}{{M}_{s}}{\bf{n}}\times \dot{{\bf{n}}},\end{eqnarray} \tag{ 11 }$$

and ${{\bf{H}}}_{\mathrm{eff}}={\bf{H}}-{C}_{1}/2\gamma (\hat{z}\times {\bf{j}})$.

In deriving the equation for the Néel order parameter only terms up to linear order in $\dot{{\bf{n}}}$ and ${\bf{j}}$ have been kept. An example of a term of higher order that has been omitted is ${\bf{n}}\times ({\bf{j}}\cdot {\rm{\nabla }})\dot{{\bf{n}}}$.

Equation (10) is an important result of this work and describes the magnetization dynamics of a uniaxial antiferromagnet with inversion symmetry broken along the $\hat{z}$-direction under the influence of an external time-independent magnetic field and DC electric current.

Previous work has shown that the dynamics of magnetic textures in ferromagnets and antiferromagnets can often be described by only a few variables [15, 20]. The approach necessitates the choice of a finite set of collective coordinates ${\xi }_{i}(t)$ which are used to specify the time evolution of the Néel order parameter ${\bf{n}}({\bf{r}},t)={\bf{n}}({\bf{r}},\{{\xi }_{i}(t)\})$. In particular, we use

$$\begin{eqnarray}\dot{{\bf{n}}} & = & \displaystyle \sum _{i}\displaystyle \frac{\partial {\bf{n}}}{\partial {\xi }_{i}}{\dot{\xi }}_{i},\\ \ddot{{\bf{n}}} & = & \displaystyle \sum _{i}\displaystyle \frac{\partial {\bf{n}}}{\partial {\xi }_{i}}{\ddot{\xi }}_{i}+{ \mathcal O }({\dot{\xi }}_{i}^{2}),\end{eqnarray} \tag{ 12 }$$

where the second term in the last equation is neglected because it is quadratic in the velocities, which are assumed to be small [20].

Here, we apply the approach to the translational motion of an antiferromagnetic skyrmion to analyze its current-induced dynamics. We assume that the skyrmion profile is composed of a static, cylindrical and rigid component ${{\bf{n}}}_{\mathrm{sk}}$ and motion- and current-induced corrections $\delta {\bf{n}}$ that break the cylindrical symmetry (see section 3.2). For the time evolution we use the ansatz ${\bf{n}}({\bf{r}},t)={\bf{n}}({\bf{r}}-{\bf{R}}(t))$, where ${\bf{n}}={{\bf{n}}}_{\mathrm{sk}}+\delta {\bf{n}}$ and ${\bf{R}}(t)$ is the skyrmion position. As collective coordinates we take $\{{\xi }_{i}\}=\{{R}_{x},{R}_{y}\}$. After multiplying (10) by¹⁰ ${\bf{n}}\times \partial {\bf{n}}/\partial {x}_{\alpha }$ for $\alpha =x,y$ and integrating over space, the equation of motion for the skyrmion position to leading order in the electric currents becomes

$$\begin{eqnarray}&&{m}_{\mathrm{eff}}\ddot{{\bf{R}}}=-{\rm{\Gamma }}\dot{{\bf{R}}}+{\rm{\Delta }}{\bf{j}}.\end{eqnarray} \tag{ 13 }$$

The coefficients read

$$\begin{eqnarray}{m}_{\mathrm{eff}} & = & \displaystyle \frac{{M}_{s}^{2}}{{\gamma }^{2}}\displaystyle \frac{{x}_{0}}{{H}_{\mathrm{exc}}},\\ {\rm{\Gamma }} & \simeq & -\displaystyle \frac{{M}_{s}}{\gamma }{x}_{0}\;{\alpha }_{{\rm{G}}},\\ {\rm{\Delta }} & \simeq & \displaystyle \frac{{M}_{s}}{2\gamma }[\beta {x}_{0}-{C}_{2}{x}_{0}^{2}I],\end{eqnarray} \tag{ 14 }$$

where Γ represents a friction term and Δ characterizes the effect of the dissipative current-induced torques. The dimensionless constant I is determined by the skyrmion profile, which we discuss later. The characteristic lengthscale ${x}_{0}$ is the domain wall width of the system and is given in the following section. The dependence of the effective mass ${m}_{\mathrm{eff}}$ on the exchange constant ${H}_{\mathrm{exc}}$ is the main difference compared to ferromagnetic skyrmion motion and results from the different nature of the magnetization dynamics in antiferromagnets. A relation for the effective mass similar to (14) has also been obtained for domain walls in antiferromagnets [27].

In deriving (13) we have considered both homogeneous and inhomogeneous current-induced torques, as well as an external magnetic field applied in the -direction. Our results thus paint a richer picture of the current-induced antiferromagnetic skyrmion motion than discussed recently. References [16, 17] predicted longitudinal current-induced forces on the skyrmion position, as opposed to the ferromagnetic skyrmion motion, where the nonzero magnetization always leads to a transverse force. Reference [16] dealt with homogeneous torques only, whereas in [17] only gradient torques have been considered. In both references no magnetic field has been included. We find, however, that even in the presence of an applied field the skyrmion motion remains longitudinal. This is further substantiated by the findings of [15, 28], where it is shown that gyroscopic forces (which are linear in the magnetic field) are not present in antiferromagnets for objects that exhibit the topology of skyrmions.

For an electric current applied in the $\hat{x}$-direction, an expression for the longitudinal velocity ${v}_{\mathrm{sk}}$ can be readily obtained by the steady-state solution of (13), which yields

$$\begin{eqnarray}&&{v}_{\mathrm{sk}}=-\left[\displaystyle \frac{\beta }{2{\alpha }_{{\rm{G}}}}-\displaystyle \frac{{C}_{2}{x}_{0}}{2{\alpha }_{{\rm{G}}}}I\right]\;{j}_{x}\equiv {v}_{\mathrm{sk},\beta }+{v}_{\mathrm{sk},{C}_{2}}.\end{eqnarray} \tag{ 15 }$$

This is the velocity corresponding to the zero-field scenario considered in [16, 17]. Before we proceed with its estimate, we need to analyze the skyrmion shape in more detail.

To calculate the constant I, we need to determine the profile of the antiferromagnetic skyrmion. The model in (2) allows for skyrmion solutions, as long as the external field is applied along the -direction or is zero [14, 15]. We assume the external magnetic field to be the dominant contribution towards ${{\bf{H}}}_{\mathrm{eff}}$, so that the current-induced contribution towards the effective field does not destroy the skyrmion (that is, $H\gg {C}_{1}j/2\gamma$). The skyrmion profile ${\bf{n}}={{\bf{n}}}_{\mathrm{sk}}+\delta {\bf{n}}$ is determined from the steady state of (10) in the absence of dissipative and damping terms

$$\begin{eqnarray}&&{\bf{n}}\times \displaystyle \frac{\delta f}{\delta {\bf{n}}}+\displaystyle \frac{2}{{H}_{\mathrm{exc}}}({\bf{n}}\times {{\bf{H}}}_{\mathrm{eff}})({\bf{n}}\cdot {{\bf{H}}}_{\mathrm{eff}})=\displaystyle \frac{4{M}_{s}}{\gamma {H}_{\mathrm{exc}}\;}\dot{{\bf{n}}}({\bf{n}}\cdot {{\bf{H}}}_{\mathrm{eff}}).\end{eqnarray} \tag{ 16 }$$

Here, the static component ${{\bf{n}}}_{\mathrm{sk}}$ solves the equation with both $j,\dot{{\bf{n}}}=0$, whereas the corrections $\delta {\bf{n}}$ originate from a nonzero dynamic term $\dot{{\bf{n}}}$ and the current-induced effective field ${C}_{1}/2\gamma (\hat{z}\times {\bf{j}})$. In [27] it has been discussed that a deformation of an antiferromagnetic texture occurs at velocities close to the limiting velocity. This is different in the case of ferromagnets. We are considering slow skyrmion dynamics, therefore, the corrections can be assumed small, $| \delta {\bf{n}}| \ll | {{\bf{n}}}_{\mathrm{sk}}|$.

The profile equation needs to be solved numerically [14]. For the purposes of the present work, we restrict the further analysis to the static component ${{\bf{n}}}_{\mathrm{sk}}$ (the corrections $\delta {\bf{n}}$ will not lead to a qualitative change in the skyrmion velocity estimate). It is convenient to rewrite (16) into spherical coordinates for the Néel vector, ${{\bf{n}}}_{\mathrm{sk}}=(\mathrm{sin}\theta \mathrm{cos}\psi ,\mathrm{sin}\theta \mathrm{sin}\psi ,\mathrm{cos}\theta )$, and cylindrical coordinates for the spatial variables, ${\bf{r}}=\rho (\mathrm{cos}\phi ,\mathrm{sin}\phi )$. For the crystallographic class under consideration, a skyrmion solution exists when the Néel order parameter has the same azimuthal direction as the cylindrical spatial vector (that is $\psi =\phi$, see figure 2) [14]. The corresponding equation becomes

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}\theta }{\partial {\tilde{\rho }}^{2}}+\displaystyle \frac{1}{\tilde{\rho }}\displaystyle \frac{\partial \theta }{\partial \tilde{\rho }}-\displaystyle \frac{\mathrm{sin}\theta \mathrm{cos}\theta }{{\tilde{\rho }}^{2}}+\displaystyle \frac{4D}{\pi {D}_{0}}\displaystyle \frac{{\mathrm{sin}}^{2}\theta }{\tilde{\rho }}-\left(1-\displaystyle \frac{{H}^{2}}{{H}_{0}^{2}}\right)\mathrm{sin}\theta \mathrm{cos}\theta =0.\end{eqnarray} \tag{ 17 }$$

Here ${x}_{0}=\sqrt{A/| {H}_{\mathrm{an}}| }$ is the characteristic lengthscale (domain wall width) of the system, ${H}_{0}=\sqrt{| {H}_{\mathrm{exc}}{H}_{\mathrm{an}}| }$ the spin-flop field, ${D}_{0}=4/\pi \sqrt{A| {H}_{\mathrm{an}}| }$ the threshold value of the inhomogeneous DM interactions constant that stabilizes modulated magnetic structures at zero magnetic field and $\tilde{\rho }=\rho /{x}_{0}$ the rescaled radial coordinate.

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Figure 3 shows the skyrmion profile for the particular choice of $D/{D}_{0}=0.9$ and $H/{H}_{0}=0.3$. The boundary conditions used are $\theta (\tilde{\rho }=0)=\pi$ and $\theta (\tilde{\rho }\to \infty )=0$, where the latter condition is implemented by a shooting method. With this, we are in a position to calculate numerically the integral I. The result is given in appendix B.

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Now, we are in a position to give an estimate for the magnitude of the longitudinal skyrmion velocity (15). We estimate the spin–orbit torque coefficient to be ${C}_{2}\simeq 3.4\times {10}^{-3}\;{{\rm{m}}}^{2}\;{{\rm{A}}}^{-1}\;{{\rm{s}}}^{-1}$ (see appendix C and [7]). The Gilbert damping parameter is ${\alpha }_{{\rm{G}}}\simeq 0.01$ [19]. The characteristic length of the system is of the order of the skyrmion size, which is typically ${x}_{0}\simeq {10}^{-8}\;{\rm{m}}$ [14, 19, 20]. Typical experimentally used current densities in ferromagnets are of the order of $j\simeq {10}^{11}{\rm{A}}\;{{\rm{m}}}^{-2}$ [25]. With this, we estimate

$$\begin{eqnarray}&&{v}_{\mathrm{sk},{C}_{2}}\simeq 255\;{\rm{m}}\;{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 18 }$$

In contrast, [16] predicts a skyrmion velocity of $\sim 1700\;{\rm{m}}\;{{\rm{s}}}^{-1}$. In that work, homogeneous torques of a similar form, but of different origin have been considered. The homogeneous torques there arise from a spin-polarized current injected vertically into the system, whereas the homogeneous torques in the present work are due to the spin–orbit coupling in the antiferromagnet. Using the values that they provide (see table 1), we arrive at ${v}_{\mathrm{sk},{C}_{2}}\simeq 500\;{{\rm{m}}{\rm{s}}}^{-1}$, which has the same order of magnitude as the result of [16].

Table 1.  Estimated skyrmion velocities using the parameters in the present work and in [16, 17]. In all cases the velocities ${v}_{\mathrm{sk},{C}_{2}}$ and ${v}_{\mathrm{sk},\beta }$ are estimated according to the expressions in (15). The numerical results for v are taken from [16, 17], respectively.

Parameters	Units	This work	[16]	[17] b
${\alpha }_{{\rm{G}}}$	−	0.01	0.3	0.01
${C}_{2}$	${{\rm{m}}}^{2}\;{{\rm{A}}}^{-1}\;{{\rm{s}}}^{-1}$	0.0034	0.2a	−
β	−	0.01	−	0.1
${x}_{0}$	${\rm{m}}$	10−8	10−8	−
j	${\rm{A}}\ {{\rm{m}}}^{-2}$	1011	1011	$3.2\times {10}^{12}\text{}$ c
Estimated
${v}_{\mathrm{sk},{C}_{2}}$	${{\rm{m}}{\rm{s}}}^{-1}$	255	500	−
${v}_{\mathrm{sk},\beta }$	${{\rm{m}}{\rm{s}}}^{-1}$	5	−	1000
Numerical results
v	${{\rm{m}}{\rm{s}}}^{-1}$	−	∼1700	∼2000

^aThis value has been calculated from the expression that the authors provide in [16] for their Slonczewski-like spin-transfer torque coefficient $\beta =| {\hslash }/({\mu }_{0}e)| P/(2{{dM}}_{s})$, multiplied by the gyromagnetic ratio $| \gamma | =2.211\times {10}^{5}\;{\rm{m}}\ {{\rm{A}}}^{-1}\;{{\rm{s}}}^{-1}$. Here, ${\hslash }$ is the reduced Planck constant, ${\mu }_{0}$ the vacuum permeability, e the electron charge, P = 0.4 is the polarization rate of the spin-polarized current, $d=0.4\;\mathrm{nm}$ the film thickness and ${M}_{s}=290\;\mathrm{kA}\;{{\rm{m}}}^{-1}$ is the saturation magnetization. ^bFor comparison, we focus only on one set of values for ${\alpha }_{{\rm{G}}},\beta$ and v of the range provided in [17]. ^cThe authors use a value of $j=200\;{\rm{m}}\;{{\rm{s}}}^{-1}$ for the drift velocity of the electrons. We calculate the corresponding current density by taking the electron density to be $n\simeq {10}^{29}\;{{\rm{m}}}^{-3}$ and the electron charge $e=1.6\times {10}^{-19}{\rm{A}}\;{\rm{s}}$.

In (8) the dissipative spin-transfer torque coefficient β has dimensions of ${{\rm{m}}}^{3}\;{{\rm{A}}}^{-1}\;{{\rm{s}}}^{-1}$. Its dimensionless counterpart $\tilde{\beta }$ is obtained by $\tilde{\beta }=\beta {ne}$, where n is the electron density and e the electron charge. Typically, in ferromagnetic systems this value is taken to be of the order of the Gilbert damping, $\tilde{\beta }\simeq {\alpha }_{{\rm{G}}}$ [29]. Typical metallic electron densities are of the order of $n\simeq {10}^{29}\;{{\rm{m}}}^{-3}$ and $e=1.6\times {10}^{-19}{\rm{A}}\;{\rm{s}}$, so that for these parameters

$$\begin{eqnarray}&&{v}_{\mathrm{sk},\beta }\simeq 5\;{\rm{m}}\;{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 19 }$$

This component of the velocity corresponds to the skyrmion velocity in [17]. Our estimate agrees with the findings of that work for the same choice of parameters (see table 1).

Typical magnon velocities in antiferromagnets are of the order of $c\simeq 30\;\mathrm{km}\ {{\rm{s}}}^{-1}$ [30]. The largest estimate of the skyrmion velocity that we obtained (using the parameters of [17], see table 1) is still much smaller than the magnon velocity. This justifies our assumption of slow skyrmion dynamics.