Microscopic Calculation of Spin Torques in Textured Antiferromagnets

Manipulation of spin textures using electric current forms an intriguing subfield of spintronics. The effect of currents on ferromagnetic (FM) textures is well-understood through the spin angular momentum transfer between the conduction electrons and magnetization Ralph2008 ; Bazaliy ; Brataas2012 . However, a similar picture is not feasible in antiferromagnets (AFs) MacDonald2011 ; Duine2011 ; Jungwirth2016 since the magnetic order parameter and conduction electrons do not carry macroscopic spin angular momenta Xu2008 ; Swaving2011 ; Hals2011 ; Tveten2013 ; Yamane2016 ; Park2020 ; Fujimoto2021 . This makes microscopic studies indispensable for understanding spin torques in AFs.

In FM, electrons moving in a spin texture with exchange coupling exhibit a spin polarization,

$\displaystyle\langle\hat{\bm{\sigma}}\rangle\ \propto\ {\bm{n}}\times({\bm{v}}\cdot\nabla){\bm{n}},$

(1)

where ${\bm{n}}$ is the magnetic order parameter (magnetization vector for FM) and ${\bm{v}}$ is a velocity that characterizes the electron flow (spin current for FM). The spin polarization arises as a reactive response Kohno_book and exerts a reaction torque, known as the spin-transfer torque (STT), on the FM spins. In AF, Xu et al. Xu2008 and Swaving and Duine Swaving2011 numerically obtained the same form of spin polarization as Eq. (1) with ${\bm{n}}$ now representing the Néel vector. Analogous to FM, this spin polarization emerges through a reactive process, and gives rise to a torque that conserves total angular momentum, which may thus be called the STT. However, in contrast to FM, the coefficient cannot be determined by a macroscopic argument based on the conservation law. Moreover, there is in general another type of torque, called the $\beta$ torque, that arises as a dissipative response due to spin relaxation Zhang2004 ; Parkin2008 , the analytic expression of which is yet to be determined for AFs.

In this Letter, we present a microscopic calculation of the STT, the $\beta$ torque, and the damping torques in AF metals. A careful treatment is given to the effects of spin relaxation, which we model by magnetic impurities. We find a STT proportional to the electric current but with a coefficient different from that in FM. The $\beta$ torque is proportional to the spin-relaxation rate. Interestingly, both torques in AFs drive the texture in the opposite direction compared to those in FMs. Using collective coordinates, it is shown that only the $\beta$ torque drives AF domain walls (DWs) Hals2011 ; Tveten2013 , because the effect of STT is nullified by an effect similar to the intrinsic pinning in FM. Finally, a recent experiment on the current-assisted DW motion in ferrimagnets at the angular-momentum compensation temperature Okuno2019 is discussed.

We consider the s-d model consisting of localized spins ($H_{S}$), conduction electrons ($H_{\rm el}$), and the s-d exchange interaction ($H_{\rm sd}$) between them,

$\displaystyle H$

$\displaystyle=H_{S}+H_{\rm el}+H_{\rm sd}.$

(2)

The space dimensionality, $d$, can be arbitrary in the general formulation, but explicit calculations will be done for a two-dimensional square lattice, $d=2$.

We first sketch the derivation of the equations that describe long-wavelength, low-frequency dynamics of AF spins coupled to conduction electrons. We start with the lattice model,

	$\displaystyle H_{S}=J\sum_{\langle i,j\rangle}{\bm{S}}_{i}\cdot{\bm{S}}_{j}-K\sum_{i}({S}_{i}^{z})^{2},$		(3)
	$\displaystyle H_{\rm sd}=-J_{\rm sd}\sum_{i}{\bm{S}}_{i}\cdot c_{i}^{\dagger}{\bm{\sigma}}\,c_{i},$		(4)

where ${\bm{S}}_{i}$ is a localized, classical spin at site $i$, $J>0$ is the AF exchange coupling constant between nearest-neighbor (n.n.) sites $\langle i,j\rangle$, and $K>0$ is the easy-axis magnetic anisotropy constant. In $H_{\rm sd}$, $c_{i}^{\dagger}=(c_{i\uparrow}^{\dagger},c_{i\downarrow}^{\dagger})$ is the electron creation operator at site $i$, ${\bm{\sigma}}$ is a vector of Pauli matrices, and $J_{\rm sd}$ is the s-d exchange coupling constant.

We consider a two-sublattice unit cell $m$ with localized spins, ${\bm{S}}_{A,m}$ and ${\bm{S}}_{B,m}$, on each sublattice, and define the Néel component ${\bm{n}}_{m}$ and the uniform component ${\bm{l}}_{m}$ by Mikeska1991

$\displaystyle{\bm{n}}_{m}$

$\displaystyle=\frac{{\bm{S}}_{A,m}-{\bm{S}}_{B,m}}{2S},\ \ \ \ \ {\bm{l}}_{m}=\frac{{\bm{S}}_{A,m}+{\bm{S}}_{B,m}}{2S},$

(5)

where $S=|{\bm{S}}_{i}|$ is the (constant) magnitude of the localized spins. We assume the spatial variations are slow for ${\bm{n}}_{m}$ and ${\bm{l}}_{m}$, and adopt a continuum description, ${\bm{n}}_{m}\to{\bm{n}}({\bm{r}})$ and ${\bm{l}}_{m}\to{\bm{l}}({\bm{r}})$. We also exploit the exchange approximation, $|{\bm{l}}|\ll 1$, and neglect higher order terms in ${\bm{l}}$ Lifshitz_book . It then follows from $|\,{\bm{l}}\pm{\bm{n}}|=1$ that $|{\bm{n}}|=1$ and ${\bm{n}}\cdot{\bm{l}}=0$.

Since the definition in Eq. (5) violates sublattice-interchange symmetry Tveten2016 , it is convenient to work with the physical magnetization Nakane_AF_H ,

$\displaystyle\tilde{\bm{l}}$

$\displaystyle\equiv{\bm{l}}+\frac{a}{2}(\partial_{x}{\bm{n}}),$

(6)

where $a$ is the lattice constant, and the $x$ axis is chosen along the bond connecting two sites in the unit cell. This preserves the constraints, $\tilde{\bm{l}}\cdot{\bm{n}}=0$ and $|{\bm{n}}|=1$, within the exchange approximation, and simplifies the formalism. In terms of $\tilde{\bm{l}}$ and ${\bm{n}}$, the Lagrangian density is given by Nakane_AF_H

	$\displaystyle\mathcal{L}_{S}$	$\displaystyle=s_{n}\left\{\,\tilde{\bm{l}}\cdot({\bm{n}}\times\dot{\bm{n}})-{\cal H}_{S}-{\cal H}_{\rm sd}\right\},$		(7)
		$\displaystyle{\cal H}_{S}=\frac{zJS}{\hbar}\bigg{\{}\,\tilde{\bm{l}}^{2}+\frac{a^{2}}{4d}\sum_{i=1}^{d}(\partial_{i}{\bm{n}})^{2}-\frac{K}{zJ}n_{z}^{2}\bigg{\}},$		(8)
		$\displaystyle{\cal H}_{\rm sd}=-\frac{M}{s_{n}}\,(\,\tilde{\bm{l}}\cdot\hat{\bm{\sigma}}_{l}+{\bm{n}}\cdot\hat{\bm{\sigma}}_{n}),$		(9)

where $s_{n}=2\hbar S/(2a^{d})$ is the density of staggered angular momentum, $z$ is the number of n.n. sites of a given site, $\hat{\bm{\sigma}}_{l}$ and $\hat{\bm{\sigma}}_{n}$ are the uniform and staggered components of the electron spin density, and $M=J_{\rm sd}S$. The equations of motion are obtained as

$\displaystyle\left\{\begin{array}[]{cl}\dot{\bm{n}}&=\displaystyle{\bm{H}}_{l}\times{\bm{n}}+{\bm{t}}_{n},\\[8.0pt] \dot{\tilde{\bm{l}}}&=\displaystyle{\bm{H}}_{n}\times{\bm{n}}+{\bm{H}}_{l}\times\tilde{\bm{l}}+{\bm{t}}_{\,l},\end{array}\right.$

(12)

where ${\bm{H}}_{n}=\partial\mathcal{H}_{S}/\partial{\bm{n}}$ and ${\bm{H}}_{l}=\partial\mathcal{H}_{S}/\partial\tilde{\bm{l}}$ are the effective fields coming from the spin part (${\cal H}_{S}$), and

	$\displaystyle{\bm{t}}_{n}$	$\displaystyle=\frac{M}{s_{n}}{\bm{n}}\times\langle\hat{\bm{\sigma}}_{l}\rangle,$		(13)
	$\displaystyle{\bm{t}}_{\,l}$	$\displaystyle=\frac{M}{s_{n}}\left\{\,{\bm{n}}\times\langle\hat{\bm{\sigma}}_{n}\rangle+\tilde{\bm{l}}\times\langle\hat{\bm{\sigma}}_{l}\rangle\right\},$		(14)

are the spin torques from electrons (${\cal H}_{\rm sd}$). We calculate $\langle\hat{\bm{\sigma}}_{l}\rangle$ and $\langle\hat{\bm{\sigma}}_{n}\rangle$ in response to an applied electric field ${\bm{E}}$ or to the time-dependent ${\bm{n}}$ and $\tilde{\bm{l}}$ using the Kubo formula Kohno2006 ; Kohno2007 .

To be explicit, we consider tight-binding electrons on a two-dimensional square lattice described by

$$H_{\rm el}=-t\sum_{\langle i,j\rangle}(c_{i}^{\dagger}c_{j}+{\rm h.c.})+V_{\rm imp},$$

(15)

where the first term expresses n.n. hopping, and

$\displaystyle V_{\rm imp}$

$\displaystyle=u_{\rm i}\sum_{l}c_{l}^{\dagger}c_{l}+u_{\rm s}\sum_{l^{\prime}}{\bm{S}}_{l^{\prime}}^{\rm imp}\cdot c_{l^{\prime}}^{\dagger}{\bm{\sigma}}\,c_{l^{\prime}},$

(16)

defines coupling to nonmagnetic and magnetic impurities. Combined with $H_{\rm sd}$, the hopping term gives upper and lower (spin-degenerate) bands, $\pm E_{\bm{k}}=\pm\sqrt{\varepsilon_{\bm{k}}^{2}+M^{2}}$, in a uniform AF state, where $\varepsilon_{\bm{k}}=-2t(\cos k_{x}+\cos k_{y})$. We take a directional average of ${\bm{S}}_{j}^{\rm imp}$ with second moment $\overline{S_{z}^{2}}$ ($\overline{S_{\perp}^{2}}$) for the component parallel (perpendicular) to ${\bm{n}}$. In the Born approximation, they appear through $\gamma_{\rm n}=\pi n_{\rm i}u_{\rm i}^{2}\nu$, $\gamma_{\perp}=\pi n_{\rm s}u_{\rm s}^{2}\,\overline{S_{\perp}^{2}}\,\nu$, and $\gamma_{z}=\pi n_{\rm s}u_{\rm s}^{2}\,\overline{S_{z}^{2}}\,\nu$, where $n_{\rm i}$ and $n_{\rm s}$ are the respective impurity concentrations, and $\nu=\frac{1}{N}\sum_{\bm{k}}\delta(|\mu|-E_{\bm{k}})$ is the density of states per spin ($N$ is the total number of sites) with the chemical potential $\mu$ measured from the AF gap center.

Vertex correction is necessary for a proper account of spin conservation, or its weak violation. Here, it is evaluated in the ladder approximation,

$\displaystyle\Pi_{\sigma\bar{\sigma}}$

$\displaystyle=\frac{2}{\pi\nu\tau^{2}}\frac{\mu^{2}}{\mu^{2}-M^{2}}\,\frac{1}{Dq^{2}-i\omega+\tau_{\varphi}^{-1}+\tau_{\rm s}^{-1}}.$

(17)

This describes diffusion, dephasing, and relaxation of transverse spin density, generalizing the result of Ref. Nakane2020 to include magnetic impurities. Here, $\tau^{-1}=2\,[\gamma_{+}+(M/\mu)^{2}\gamma_{-}]$, with $\gamma_{\pm}=\gamma_{\rm n}+\gamma_{z}\pm 2\gamma_{\perp}$, is the electron scattering rate, and

	$\displaystyle\frac{1}{\tau_{\varphi}}$	$\displaystyle=\frac{4M^{2}}{\mu^{2}}\left[\frac{\mu^{2}+M^{2}}{\mu^{2}-M^{2}}\gamma_{\rm n}+3\gamma_{\perp}+\frac{2(2\mu^{2}+M^{2})}{\mu^{2}-M^{2}}\gamma_{z}\right],$		(18)
	$\displaystyle\frac{1}{\tau_{\rm s}}$	$\displaystyle=4\,(\gamma_{\perp}+\gamma_{z}),$		(19)

are, respectively, the spin-dephasing rate Manchon2017 ; Nakane2020 and the (transverse) spin-relaxation rate. In Eq. (17), $q^{-1}$ ($\omega^{-1}$) is the typical length (time) scale of the AF spin texture (dynamics), and $D$ is the diffusion constant. We assume $q\ell_{\varphi}\ll 1$ and $\omega\tau_{\varphi}\ll 1$, where $\ell_{\varphi}=\sqrt{D\tau_{\varphi}}$ is the spin-dephasing length, and let $q,\omega\to 0$ in the results. The constant terms in the denominator, $\tau_{\varphi}^{-1}+\tau_{\rm s}^{-1}$, reflect spin nonconservation in the electron system. The spin dephasing ($\tau_{\varphi}^{-1}$), characteristic of AF and absent in FM, is dominated by nonmagnetic impurities and vanishes at $M=0$ Manchon2017 , whereas $\tau_{\rm s}^{-1}$ comes solely from magnetic impurities and is essentially the same as that in FM Kohno2006 . It is convenient to decompose the former as $\tau_{\varphi}^{-1}=\tau_{\varphi 0}^{-1}+\tau_{\varphi 1}^{-1}$, where $\tau_{\varphi 0}^{-1}$ ($\propto\gamma_{\rm n}$) is the contribution from nonmagneic impurities and $\tau_{\varphi 1}^{-1}$ is from magneic impurities. The “dissipated” spin angular momentum via $\tau_{\varphi 0}^{-1}$ is actually transferred to the AF spin system.

We calculate electron spin density induced by an external electric field ${\bm{E}}$ in the presence of spin texture (for current-induced torques), or induced by time-dependent spins, $\dot{\bm{n}}$ and $\dot{\tilde{\bm{l}}}$ (for Gilbert damping). We assume weak spin relaxation, $\gamma_{z},\gamma_{\perp}\ll\gamma_{\rm n}$, and retain terms of lowest nontrivial order. The calculations are straightforward along the lines of Refs. Kohno2006 ; Kohno2007 ; Nakane2020 ; see Suppl for details.

Results.— We obtained

	$\displaystyle\langle\hat{\bm{\sigma}}_{n}\rangle$	$\displaystyle=-(s_{n}/M)\{\beta_{n}\,({\bm{v}}_{n}\!\cdot\!\nabla)\,{\bm{n}}+\alpha_{n}\dot{\bm{n}}\},$		(20)
	$\displaystyle\langle\hat{\bm{\sigma}}_{l}\rangle$	$\displaystyle=(s_{n}/M)\{{\bm{n}}\times({\bm{v}}_{n}\!\cdot\!\nabla)\,{\bm{n}}-\alpha_{l}\,\dot{\tilde{\bm{l}}}\},$		(21)

which are consistent with previous studies Xu2008 ; Swaving2011 ; Hals2011 ; Tveten2013 ; Yamane2016 ; Park2020 , and lead to the torques,

	$\displaystyle{\bm{t}}_{n}$	$\displaystyle=-({\bm{v}}_{n}\!\cdot\!\nabla){\bm{n}}-\alpha_{l}\,{\bm{n}}\times\dot{\tilde{\bm{l}}},$		(22)
	$\displaystyle{\bm{t}}_{\,l}$	$\displaystyle=-\beta_{n}\,{\bm{n}}\times({\bm{v}}_{n}\!\cdot\!\nabla){\bm{n}}-\alpha_{n}\,{\bm{n}}\times\dot{\bm{n}}.$		(23)

We retained dominant contributions, which come from $\langle\hat{\bm{\sigma}}_{l}\rangle$ for ${\bm{t}}_{n}$, and $\langle\hat{\bm{\sigma}}_{n}\rangle$ for ${\bm{t}}_{\,l}$.

Current-induced torques.— The first terms in ${\bm{t}}_{\,l}$ and ${\bm{t}}_{n}$ are current-induced torques, which are proportional to the charge current density, ${\bm{j}}=\sigma_{xx}{\bm{E}}=2e^{2}D\nu{\bm{E}}$, via

$\displaystyle{\bm{v}}_{n}=-\frac{\hbar}{2es_{n}}{\cal P}_{n}{\bm{j}}.$

(24)

The velocity ${\bm{v}}_{n}$ quantifies the STT, and we identify

$\displaystyle{\cal P}_{n}=\frac{\mu M}{\mu^{2}-M^{2}},$

(25)

to be the conversion factor from a charge current to STT. Note that $|{\cal P}_{n}|$ can be significantly larger than unity near the AF gap edge ($|\mu|\gtrsim|M|$). This contrasts to the case of FM, in which the corresponding factor $|P|$ is less than unity. The current-induced torque in ${\bm{t}}_{\,l}$ is characterized by a dimensionless parameter,

$\displaystyle\beta_{n}$

$\displaystyle=\frac{2(\gamma_{\perp}+\gamma_{z})}{M}=\frac{\hbar}{2M\tau_{\rm s}},$

(26)

which originates from magnetic impurities, i.e., from spin relaxation, and is therefore a dissipative torque. The spin dephasing due to nonmagnetic impurities ($\tau_{\varphi\,0}^{-1}$) is microscopically a reactive process and does not contribute to $\beta_{n}$, whereas that from $\tau_{\varphi\,1}^{-1}$, combined with the self-energy terms, results in a contribution proportional to $\tau_{\rm s}^{-1}$. Along with the contribution originating from $\tau_{\rm s}^{-1}$ in Eq. (17), it leads to Eq. (26). The obtained two current-induced torques are related via $\beta_{n}{\bm{n}}\times$, which is reminiscent of the relation between the reactive and dissipative torques in FM; we call the former [$-({\bm{v}}_{n}\!\cdot\!\nabla){\bm{n}}$] the STT in AF, and the latter [$-\beta_{n}{\bm{n}}\times({\bm{v}}_{n}\!\cdot\!\nabla){\bm{n}}$] the $\beta_{n}$ torque. The expression of $\beta_{n}$ in terms of $\tau_{\rm s}$ and $M=J_{sd}S$ is also shared by FM Kohno2006 ; Zhang2004 .

The above two current-induced torques change their signs across the AF gap [see Eq. (25)], reflecting the fact that electrons in the upper and lower AF bands have mutually opposite spin directions. This feature of the STT was suggested in Ref. Swaving2011 . Interestingly, the driving direction is opposite to the naive expectation based on the two-FM picture of AF. Namely, for $\mu<0$, the spin polarization on the Fermi surface is positive (dominated by majority spin carriers) but the driving direction is opposite to the direction of electron flow. This is due to the intersublattice hopping in AF, namely, the electron spins exert torques on oppositely pointing neighboring spins, so the sign of the torques is reversed from that of FMs com_nn . The same is true for $\mu>0$.

Gilbert damping.— The second terms in Eqs. (22) and (23) describe damping. The damping parameters are calculated as

	$\displaystyle\alpha_{n}$	$\displaystyle=\left\{\gamma_{\perp}+\gamma_{z}+\frac{M^{2}}{\mu^{2}}(\gamma_{\perp}-\gamma_{z})\right\}\frac{2\hbar\nu}{s_{n}},$		(27)
	$\displaystyle\alpha_{l}$	$\displaystyle=\frac{(\mu^{2}-M^{2})(\mu^{2}+M^{2})}{\mu^{2}}\frac{\nu}{s_{n}}\tau.$		(28)

While $\alpha_{n}$ arises from spin relaxation (magnetic impurities), $\alpha_{l}$ does not necessitate it. Rather, $\alpha_{l}$ is proportional to $\tau$, like conductivity, hence can be very large in good metals. These features were pointed out in Refs. Liu2017 ; Simensen2020 based on the first-principles calculation.

It is interesting to compare $\alpha_{n}$ with the Gilbert damping in FM,

$\displaystyle\alpha_{\rm F}$

$\displaystyle=\sum_{\sigma}(\gamma_{z,\sigma}\nu_{\bar{\sigma}}+\gamma_{\perp,\sigma}\nu_{\sigma})\,\frac{\hbar}{s_{0}},$

(29)

obtained based on the same spin-relaxation model (magnetic impurities) Kohno2006 . Here, $\gamma_{\alpha,\sigma}=\pi n_{s}u_{s}^{2}\overline{S^{2}_{\alpha}}\,\nu_{\sigma}$ ($\alpha=\perp,z$), $\nu_{\sigma}$ ($\sigma=\uparrow,\downarrow$) is the density of states of electrons with spin $\sigma$, and $s_{0}=\hbar S/a^{d}$ is the angular-momentum density. We see that in the limit of spin-degenerate bands ($\nu_{\uparrow}=\nu_{\downarrow}$) and isotropic magnetic impurities ($\gamma_{\perp}=\gamma_{z}$), the above expressions of $\alpha_{n}$ (for AF) and $\alpha_{\rm F}$ (for FM) coincide. Therefore, in the current model of AF, the ratio $\beta_{n}/\alpha_{n}$ is of order unity, similar to FM Kohno_book .

Equations of AF spin dynamics.— With the obtained torques and ${\cal H}_{S}$ [Eq. (8)], the equations of motion are explicitly written as

	$\displaystyle\dot{\bm{n}}$	$\displaystyle=\tilde{J}\,\tilde{\bm{l}}\times{\bm{n}}-({\bm{v}}_{n}\!\cdot\!\nabla)\,{\bm{n}},$		(30)
	$\displaystyle\dot{\tilde{\bm{l}}}$	$\displaystyle=-(c^{2}\tilde{J}^{-1}\nabla^{2}{\bm{n}}+\tilde{K}n^{z}\hat{z})\times{\bm{n}}$
		$\displaystyle\quad+(\alpha_{n}\dot{\bm{n}}+\beta_{n}({\bm{v}}_{n}\!\cdot\!\nabla)\,{\bm{n}})\times{\bm{n}}$
		$\displaystyle\quad+{\bm{n}}\,[\,\tilde{\bm{l}}\!\cdot\!({\bm{v}}_{n}\!\cdot\!\nabla){\bm{n}}],$		(31)

with $c=(zJSa)/(\hbar\sqrt{d})$, $\tilde{J}=2zJS/\hbar$, and $\tilde{K}=2SK/\hbar$. Damping terms in the first equation are dropped as they are higher order in $\tilde{\bm{l}}$. Solving Eq. (30) for $\tilde{\bm{l}}$ as $\tilde{\bm{l}}=\tilde{J}^{-1}{\bm{n}}\times[\dot{\bm{n}}+({\bm{v}}_{n}\!\cdot\!\nabla)\,{\bm{n}}]$, and substituting it in Eq. (31), one can obtain a closed equation for ${\bm{n}}$,

	$\displaystyle\ddot{\bm{n}}\times{\bm{n}}$	$\displaystyle=(c^{2}\,\nabla^{2}{\bm{n}}+\tilde{J}\tilde{K}n_{z}\hat{z})\times{\bm{n}}$
		$\displaystyle\ \ \ -\tilde{J}(\alpha_{n}\dot{\bm{n}}+\beta_{n}({\bm{v}}_{n}\!\cdot\!\nabla)\,{\bm{n}})\times{\bm{n}}$
		$\displaystyle\ \ \ -[({\bm{v}}_{n}\!\cdot\!\nabla)\,\dot{\bm{n}}]\times{\bm{n}}.$		(32)

This differs slightly from Ref. Swaving2011 due to the difference in $H_{\rm sd}$ (i.e., ${\bm{l}}$ vs. $\tilde{\bm{l}}$), and leads to the magnon dispersion,

	$\displaystyle\omega$	$\displaystyle=\sqrt{c^{2}{\bm{q}}^{2}+\tilde{J}\tilde{K}+({\bm{v}}_{n}\!\cdot\!{\bm{q}}-i\tilde{J}\alpha_{n})^{2}/4+i\tilde{J}\beta_{n}{\bm{v}}_{n}\!\cdot\!{\bm{q}}}$
		$\displaystyle\ \ \ \pm({\bm{v}}_{n}\!\cdot\!{\bm{q}}-i\tilde{J}\alpha_{n})/2,$		(33)

where damping enters only through $\alpha_{n}$ and $\beta_{n}$.

DW motion.— Here we study the AF DW motion using collective coordinates. Since $\mathcal{L}_{S}$ [Eq. (7)] is written with ${\bm{n}}$ and $\tilde{\bm{l}}$, we consider collective coordinates for both ${\bm{n}}$ and $\tilde{\bm{l}}$ Nakane_AF_H . Assuming for ${\bm{n}}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ a DW form, $\cos\theta(x,t)=\pm\tanh\left(\frac{x-X(t)}{\lambda}\right)$ and $\phi(x,t)=\phi_{0}(t)$, where $\lambda=a\sqrt{zJ/4Kd}$ is the DW width, we treat the DW position $X(t)$ and the angle $\phi_{0}(t)$ as dynamical variables Tatara2008 . As for $\tilde{\bm{l}}$, we expand it as Nakane_AF_H

$\displaystyle\tilde{\bm{l}}(x,t)$

$\displaystyle=[\,l_{\theta}(t)\,{\bm{e}}_{\theta}+l_{\phi}(t)\,{\bm{e}}_{\phi}\,]\,\varphi_{0}(x)+\cdots,$

(34)

where ${\bm{e}}_{\theta}\equiv\partial_{\theta}{\bm{n}}$ and ${\bm{e}}_{\phi}\equiv{\bm{n}}\times{\bm{e}}_{\theta}$ are orthonormal vectors normal to ${\bm{n}}$. The function $\varphi_{0}(x)=\Bigl{[}\cosh\frac{x-X}{\lambda}\Bigr{]}^{-1}$ reflects the spatial profile of ${\bm{n}}\times\dot{\bm{n}}$, and naturally extracts $l_{\theta}$ and $l_{\phi}$ in the first term of $\mathcal{L}_{S}$ [Eq. (7)]. The obtained Lagrangian, $L_{\rm DW}=2s_{n}(\pm\dot{X}l_{\phi}-\lambda\dot{\phi}_{0}l_{\theta})-H_{S}$, shows that $l_{\phi}$ and $l_{\theta}$ are canonical conjugate to $X$ and $\phi_{0}$, respectively. The equations of motion are given by

	$\displaystyle\pm\lambda\,\dot{l}_{\phi}$	$\displaystyle=\beta_{n}v_{n}-\alpha_{n}\dot{X},$		(35)
	$\displaystyle\pm\dot{X}$	$\displaystyle=\pm v_{n}+v_{J}\,l_{\phi}+\alpha_{l}\lambda{\dot{l}}_{\phi},$		(36)
	$\displaystyle\dot{l}_{\theta}$	$\displaystyle=\alpha_{n}\,\dot{\phi}_{0},$		(37)
	$\displaystyle\lambda\,\dot{\phi}_{0}$	$\displaystyle=-v_{J}l_{\theta}-\alpha_{l}\lambda\dot{l}_{\theta},$		(38)

where $v_{J}=4JS\lambda/\hbar$, and $\pm$ is the topological charge of the AF DW. The first two equations describe the translational motion, and the remaining two describe the rotational motion of the DW plane (defined by the Néel vector). Unlike in FM Tatara2004 , these two motions are decoupled in AF. The term $\pm v_{n}$ in Eq. (36) describes the spin-transfer effect, and $\beta_{n}v_{n}$ in Eq. (35) describes the momentum-transfer effect (a force on the DW). The terms with $\alpha_{l}$ are negligible in effect, but retained here for the sake of comparison with FM (see below).

Let us overview the translational motion of AF DW under a stationary $v_{n}$ Hals2011 . When $\alpha_{n}=\beta_{n}=0$, $l_{\phi}$ is a constant of motion. With an initial condition $l_{\phi}=0$ (no canting), the DW moves at a constant velocity $\dot{X}=v_{n}$ by the spin-transfer effect Swaving2011 . If the DW is initially canted, $l_{\phi}=l_{\phi}^{0}$, the constant velocity is modified to $\dot{X}=v_{n}\pm v_{J}\,l_{\phi}^{0}$. For finite $\alpha_{n}$, $l_{\phi}$ is no longer conserved, and approaches a terminal value, $l_{\phi}\to\mp(1-\beta_{n}/\alpha_{n})(v_{n}/v_{J})$. Then, from Eq. (36), the DW velocity approaches

$\displaystyle\dot{X}\ \to\ v_{n}-\left(1-\frac{\beta_{n}}{\alpha_{n}}\right)v_{n}=\frac{\beta_{n}}{\alpha_{n}}\,v_{n},$

(39)

which is solely determined by the $\beta_{n}$ torque. If $\beta_{n}=0$, the spin-transfer effect is completely nullified by the canting $l_{\phi}=v_{n}/v_{J}$, and the aforementioned steady movement eventually ceases Hals2011 . This is quite similar to the intrinsic pinning in FM. For finite $\beta_{n}$, canting $l_{\phi}$ is reduced, and the cancellation of the spin-transfer effect is incomplete. Finally, the case $\beta_{n}=\alpha_{n}$ is special in that there is no canting, and the spin-transfer effect is undisturbed.

It is instructive to make a more detailed comparison with FM. In FM, the current-driven DW motion is described by

	$\displaystyle\pm\lambda\,\dot{\phi}_{0}$	$\displaystyle=\beta v_{\rm s}-\alpha\dot{X},$		(40)
	$\displaystyle\pm\dot{X}$	$\displaystyle=\pm v_{\rm s}+v_{K}\sin 2\phi_{0}+\alpha\lambda\dot{\phi}_{0},$		(41)

where, now, $X$ and $\phi_{0}$ are coupled. ($\phi_{0}$ here is defined by the uniform magnetization, and $\pm$ is the topological charge of the FM DW.) A close similarity to Eqs. (35) and (36) is evident, and here $\phi_{0}$ plays the role of $\l_{\phi}$. The effect of current appears in $v_{\rm s}=-(\hbar/2es_{0})Pj$, where $P$ is the current polarization factor, and the velocity $v_{K}=K_{\perp}S\lambda/2\hbar$ is defined with the hard-axis anisotropy constant $K_{\perp}$. At low current, $v_{\rm s}<v_{K}$, and with $\beta=0$, the DW plane tilts by $\phi_{0}=(1/2)\sin^{-1}(v_{\rm s}/v_{K})$ and the DW ceases to move, $\dot{X}=0$. This is the intrinsic pinning in FM Tatara2004 ; Koyama2011 . If $v_{\rm s}$ exceeds $v_{K}$, $v_{K}$ can not nullify the spin-transfer effect $v_{\rm s}$ and the DW is released from intrinsic pinning. The corresponding term in Eq. (36) has the linearised form, $v_{J}\,l_{\phi}$, which is justified since $v_{J}$ of AF is much larger than $v_{K}$ of FM (by 2-3 orders of magnitude), and the intrinsic pinning is robust in AF. It is interesting to note the contrasting origins of intrinsic pinning; in FM it is the explicit breaking of spin rotation symmetry, $K_{\perp}$, whereas in AF it is the AF order itself, i.e., spontaneous breaking. Finally, the case $\alpha=\beta$ provides a special solution $\phi_{0}=0$ and $\dot{X}=v_{s}$, similar to the case $\alpha_{n}=\beta_{n}$ for AF.

Recently, current-assisted field-driven DW motion was experimentally studied in a ferrimagnetic GdFeCo near its angular-momentum compensation temperature Okuno2019 . They analyzed the data by the Landau-Lifshitz-Gilbert equation for the uniform moment ${\bm{m}}$ (parallel to ${\bm{n}}$), and obtained a very large, negative value of $\beta/\alpha\simeq-100$. They assumed $\sim\beta P{\bm{j}}$ for the $\beta$-torque coefficient (that acts on ${\bm{m}}$), with a small factor $P$ $(\simeq 0.1)$ included. If, however, the main driving is the $\beta_{n}$ torque that acts on the Néel vector ${\bm{n}}$, as studied in the present work, we would conclude ${\cal P}_{n}\beta_{n}/\alpha_{n}\simeq-10$. While $\beta_{n}/\alpha_{n}\simeq 1$ as in FM (for positive $J_{\rm sd}$ Hoshi2020 ; com2 ), $|{\cal P}_{n}|$ can be significantly larger than unity near the AF gap edge. Therefore, the large value of $|{\cal P}_{n}|\sim 10$ may lie within the scope of the present results. The negative sign can be explained likewise from ${\cal P}_{n}$ with a negative $\mu$, which reflects intersublattice hopping in AF. Such “antiferromagnetic transport” in GdFeCo is supported by a recent magnetoresistance measurement Park2021 .

In conclusion, we have presented a microscopic model calculation of current-induced torques and damping torques in AF metals, paying attention to the effects of spin relaxation (and spin dephasing). A formulation in terms of the Néel vector and physical magnetization is given to study the AF spin dynamics in metallic AFs with s-d exchange interaction. The current-induced torques are found to be opposite in direction to those of FMs, reflecting the AF transport character, and the current-to-STT conversion factor can be significantly larger than that in FM. These results seem to be relevant to the recent experiment on GdFeCo.

We would like to thank T. Okuno, T. Ono and T. Taniyama for valuable discussions. We also thank K. Nakazawa, T. Funato, Y. Imai, T. Yamaguchi, A. Yamakage, and Y. Yamazaki for helpful discussion. This work was partly supported by JSPS KAKENHI Grant Numbers JP15H05702, JP17H02929 and JP19K03744, and the Center of Spintronics Research Network of Japan. JJN is supported by a Program for Leading Graduate Schools “Integrative Graduate Education and Research in Green Natural Sciences” and Grant-in-Aid for JSPS Research Fellow Grant Number 19J23587.