Topological Spin Hall Effect in Antiferromagnets Driven by Vector Néel Chirality

Spin-charge interconversion has been extensively studied in spintronics with the aim of application to next-generation devices. It is typically achieved by the spin Hall effect (SHE) [1] originating from the relativistic spin-orbit coupling (SOC), mostly in nonmagnetic materials [2, 3, 4, 5, 6]. Ferromagnets (FMs) are another class of materials that enable spin-charge conversion, not just as a simple spin source, but also by emergent electromagnetism due to spatiotemporal magnetization dynamics. In particular, a magnetization texture forming a finite scalar spin chirality simulates a magnetic field that affects electrons’ orbital motion but in a spin-dependent way. The resulting Hall effect, often called the topological Hall effect (THE), is thus the SHE in essence [7].

Antiferromagnets (AF) are a material having both aspects, magnetic at the microscopic scale but nonmagnetic at the (semi)macroscopic scale, and offers a unique platform to generate pure spin currents. A large SHE was reported in ${\rm Ir_{20}Mn_{80}}$ [9], which originates from SOC. Recently, there are some proposals of SHE that arise from the antiferromagnetic spin texture, providing another means of pure spin-current generation without relying on relativistic SOC.

In this Letter, we explore theoretically the SHE in AF originating from AF spin textures. From the analogy with FMs, an AF with a textured Néel vector ${\bm{n}}$ is expected to generate a spin Hall current,

under an applied electric field $E_{j}$ ($\tilde{{\sigma}}_{\rm SH}$ is a coefficient, and $e>0$ is the elementary charge). Because of the scalar chirality, ${{\bm{n}}}\cdot({\partial}_{i}{{\bm{n}}}\times{\partial}_{j}{{\bm{n}}})$, this effect may be termed as a topological spin Hall (TSH) effect [10, 11, 12]. Such a texture has opposite spin chiralities on different sublattices and deflects spin-up and spin-down electrons in mutually opposite directions; the charge Hall currents then cancel out, but the spin Hall currents add up. However, Eq. (1) is not a macroscopically observable quantity since its sign depends on the definition of sublattice or ${\bm{n}}$; it changes sign under ${\bm{n}}\to-{\bm{n}}$. We define the physical spin current ${\bm{j}}_{{\rm s},i}$ through $\tilde{j}_{{\rm s},i}^{z}={\bm{n}}\cdot{\bm{j}}_{{\rm s},i}$, hence by

The factor $({\partial}_{i}{{\bm{n}}}\times{\partial}_{j}{{\bm{n}}})^{\alpha}$ may be identified as an emergent magnetic field in spin channel, and interestingly, it can be expressed as $(\partial_{i}a_{j}^{\alpha}-\partial_{j}a_{i}^{\alpha})/2$ with an emergent vector potential,

This is the vector chirality ($\sim{\bm{S}}_{1}\times{\bm{S}}_{2}$ for two spins) formed by the Néel vector, and we call it “vector Néel chirality” [13]. Spatially-averaged spin current $\langle j_{{\rm s},i}^{\alpha}\rangle$ is proportional to a winding number defined by the vector chirality, hence is “topological.” To date, the vector spin chirality is known to induce charge [14, 15] and (equilibrium) spin currents [16, 17, 18], but its AF counterpart in terms of the Néel vector has been less focused on.

In the following, we derive Eqs. (1) and (2) and demonstrate the topological character of Eq. (2). The effect is present in systems with AF merons [19] but not with AF skyrmions [20, 21, 22, 23, 24, 25, 26], and is enhanced in the weak-coupling regime. These results are confirmed numerically based on the Landauer-Büttiker formula.

We consider electrons on a square lattice and coupled to a static spin texture. The Hamiltonian

consists of nearest-neighbor hopping (first term), s-d exchange coupling to localized spins ${\bm{S}}_{i}$ (second term), and on-site impurity potential (third term), with electron operators $c_{i}={}^{t}(c_{i\uparrow},c_{i\downarrow})$ at site $i$ and Pauli matrices ${\bm{\sigma}}=(\sigma^{x},\sigma^{y},\sigma^{z})$. We assume a slowly-varying checkerboard type AF texture, ${\bm{S}}_{i}=S(-1)^{i}{\bm{n}}_{i}$, where ${\bm{n}}_{i}$ is the Néel vector varying slowly in space [Fig. 1 (a)].

With a unitary transformation, $c_{i}=U_{i}\tilde{c}_{i}$, which diagonalizes the s-d coupling, $U_{i}^{\dagger}({\bm{n}}_{i}\cdot{\bm{\sigma}})U_{i}=\sigma^{z}$, $H$ is transformed into $H=-t\sum_{(i,j)}\tilde{c}_{i}^{\dagger}{\rm e}^{iA_{ij}}\tilde{c}_{j}-J\sum_{i}(-)^{i}\tilde{c}_{i}^{\dagger}\sigma^{z}\tilde{c}_{i}+u_{\rm i}{\sum_{i}}^{\prime}\tilde{c}_{i}^{\dagger}\tilde{c}_{i}$, where $J=J_{\rm sd}S$, and $A_{ij}$ is the spin gauge field defined by $U_{i}^{\dagger}U_{j}={{\rm e}}^{iA_{ij}}$ [27, 28]. Because of slow variations of the texture, $A_{ij}$ is small and can be treated perturbatively. The unperturbed state (with $A_{ij}=0$) is a uniform AF, and the electron band splits into spin-degenerate upper and lower bands, $\pm E_{\bm{k}}$, with an AF gap $2|J|$ in between [Fig. 1 (b)]. Here, $E_{\bm{k}}\equiv\sqrt{\varepsilon_{\bm{k}}^{2}+J^{2}}$ with $\varepsilon_{\bm{k}}=-2t(\cos k_{x}+\cos k_{y})$. Also, $A_{ij}$ can be treated in the continuum approximation, $A_{ij}\to A_{\mu}$, where $\mu$ ($=x,y$) specifies the bond direction of $(i,j)$, and expanded as

where ${\alpha}=x,y,z$ and $\perp=x,y$. The spin-conserving component $A^{z}$ describes adiabatic processes, whereas the spin-flip component ${\bm{A}}^{\perp}$ induces nonadiabatic transitions. In FM, the latter can be important only in the weak-coupling regime [3], but in AF, both are important because of spin degeneracy of the AF bands. Both produce the same effective field, $(\nabla\times{\bm{A}}^{z})_{z}=({\bm{A}}_{x}^{\perp}\times{\bm{A}}_{y}^{\perp})^{z}={{\bm{n}}}\cdot({\partial}_{x}{{\bm{n}}}\times{\partial}_{y}{{\bm{n}}})$.

To calculate the spin Hall conductivity, $\sigma_{\rm SH}\equiv\frac{1}{2}(\sigma_{xy}^{z}-\sigma_{yx}^{z})$, we assume a good AF metal and focus on the Fermi-surface contribution [30],

where ${\cal J}_{{\rm s},i}^{z}$ and ${\cal J}_{j}$ are spin-current and number-current vertices, $\rm Tr$ means the trace in spin, sublattice, and ${{\bm{k}}}$ spaces (${{\bm{k}}}$, ${{\bm{k}}}^{\prime}$-integrals), and $\langle\cdots\rangle_{\rm i}$ represents impurity average. The Green’s function $G_{{{\bm{k}}},{{\bm{k}}}^{\prime}}^{\rm R(A)}=(\mu-H\pm i0)_{{{\bm{k}}},{{\bm{k}}}^{\prime}}^{-1}$ takes full account of impurities and the gauge field, and ${{\bm{k}}}_{\pm}={{\bm{k}}}\pm{{\bm{Q}}}/2$. We treat the impurity scattering in the Born approximation with ladder vertex corrections (VC) [31]. The superscript $z$ on $\sigma_{ij}^{z}$ and ${\cal J}_{{\rm s},i}^{z}$ indicates the spin component in the rotated frame, thus it is the component projected to the local Néel vector ${\bm{n}}$.

After a standard procedure (see Supplemental Material [31]), we obtain Eq. (1) with $\tilde{\sigma}_{\rm SH}=\tilde{\sigma}_{\rm SH}^{(0)}+\tilde{\sigma}_{\rm SH}^{(1)}$,

where $\tilde{\sigma}_{\rm SH}^{(0)}$ is the contribution without VC, which comes from both adiabatic and nonadiabatic processes, and $\tilde{\sigma}_{\rm SH}^{(1)}$ is the contribution with VC, coming only from nonadiabatic processes. Here, $\nu=\nu(\mu)$ is the density of states (per spin) at chemical potential $\mu$, $C_{ij}=\langle 1-\cos k_{i}\cos k_{j}\rangle_{\rm FS}$ is the Fermi surface average [32], $\tau=[{\gamma}_{0}+(J/\mu){\gamma}_{3}]^{-1}/2$ is the scattering time (${\gamma}_{0}=\pi n_{\rm i}u_{\rm i}^{2}\nu$ and ${\gamma}_{3}=(J/\mu){\gamma}_{0}$ are the sublattice-independent and dependent parts, respectively, of the damping, and $n_{\rm i}$ is the impurity concentration), and

is the “spin dephasing” rate [2]. We introduced a finite spin relaxation rate $\tau_{\rm s}^{-1}$ by hand [34]; without $\tau_{\rm s}^{-1}$, we would have an unphysical result that $\tilde{\sigma}_{\rm SH}^{(1)}$ does not vanish in the limit $J\to 0$. Note that $\tau_{\varphi}^{-1}$ differs from $\tau_{\rm s}^{-1}$ in that it does not require spin-dependent scattering, randomizes only the transverse ($\perp{\bm{n}}$) components of the electron spin (see below), and vanishes as $J\to 0$. The results (7) and (8) are obtained at the leading order, i.e., second order in spatial gradient and second order in $\tau$.

The coefficients $\tilde{\sigma}_{\rm SH}^{(0)}$ and $\tilde{\sigma}_{\rm SH}^{(1)}$ are plotted in Fig. 2 (a). They are comparable in magnitude at large $J$ ($\sim 1.5t$), but as $J$ is reduced, $\tilde{\sigma}_{\rm SH}^{(1)}$ grows markedly whereas $\tilde{\sigma}_{\rm SH}^{(0)}$ decreases. The sum $\tilde{\sigma}_{\rm SH}=\tilde{\sigma}_{\rm SH}^{(0)}+\tilde{\sigma}_{\rm SH}^{(1)}$ is plotted in Fig. 2 (b) by solid lines, which grows as $J$ is reduced, especially near the AF gap edge, but finally vanishes at $J=0$. Since $\tilde{\sigma}_{\rm SH}^{(1)}$ comes solely from nonadiabatic processes, these results show that the combined effect of nonadiabaticity and the VC is important for the present SHE [35]. Physically, a nonadiabatic process produces a transverse spin polarization, and the VC describes its collective transport, which is however limited by spin dephasing [2, 28, 36]. The origin of the enhancement at small $J$ can be traced to the reduced dephasing at small $J$ [31]. As seen from Eq. (9), the spin dephasing arises through ${\gamma}_{3}$, a sublattice asymmetry in (nonmagnetic) scattering [28, 37], and its physical picture is illustrated in Fig. 3.

The obtained result, Eq. (1), needs to be interpreted with care. It arises with the scalar chirality formed by the Néel vector ${\bm{n}}$, and changes sign under ${\bm{n}}\to-{\bm{n}}$. This is not a pleasant situation since any physical quantity measurable by (semi)macroscopic means should not depend on the sign of ${\bm{n}}$, or on the definition of sublattice. This (apparent) puzzle is resolved if we note that the spin component of the calculated spin current $\tilde{j}_{x,{\rm s}}^{z}$ is the one projected to the Néel vector $\bm{n}$. Therefore, we write $\tilde{j}_{\rm s}^{z}={\bm{n}}\cdot{\bm{j}}_{\rm s}$ and identify ${\bm{j}}_{\rm s}$ as a physical spin current. The physical spin Hall current is thus given by Eq. (2).

It is in fact possible to obtain Eq. (2) directly. By assuming $J$ is small and treating it perturbatively, we found the spin current arises at second order in $J$ [31],

This contrasts with the THE in FM caused by scalar spin chirality, which starts at third order ($\sim J^{3}$) [38], and demonstrates that the essential quantity for the present SHE is the vector (not scalar) chirality. That Eq. (10) is an even function of $J$ (or $J{\bm{n}}$) is consistent with the fact that the spin current is even under time reversal.

The expression Eq. (2) holds locally in space (as far as the variation of ${\bm{n}}$ is sufficiently slow). As a spin current measured experimentally, we consider a spatially-averaged one, $\langle{\bm{j}}_{\rm s}^{\alpha}\rangle=\Omega^{-1}\int{\bm{j}}_{\rm s}^{\alpha}dxdy$ (in two dimensions), where $\Omega$ is the sample area. It can be written as

where

and $a_{i}^{\alpha}=({\bm{n}}\times\partial_{i}{\bm{n}})^{\alpha}$ [Eq. (3)] is an emergent vector potential in spin channel. The line integral is taken along the sample perimeter. If the system has easy-plane magnetic anisotropy, and the Néel vector on the sample edge is constrained to lie in-plane, e.g., $x$-$y$ plane, the line integral of the vector chirality defines a topological winding number $m^{z}\in\mathbb{Z}$ in $\pi_{1}(S^{1})$. The spin Hall conductivity is thus proportional to the topological number density $m^{z}/\Omega$, and this fact resurrects the naming “topological” spin Hall effect. We emphasize that it is characterized by the vector chirality of Néel vector along the sample edge. Therefore, the present TSHE is absent for AF skyrmions, in which the Néel vector at the edge is uniaxial. On the other hand, it is finite for AF merons, which have finite in-plane winding of the Néel vector along the edge.

To verify these results, we have conducted numerical works based on the four-terminal Landauer-Büttiker formula [39]. We consider ballistic systems with $L\times L$ sites without disorder, and containing a single AF skyrmion or a single AF meron. For both textures, the spin Hall conductance $G_{\rm SHC}^{z}$ shows a strong peak just below the AF gap [Fig. 4 (a) and (b)], which, however, behave oppositely as $L$ is increased (with the skyrmion/meron size fixed); for the AF skyrmion the peak decreases with $L$ and seems to vanish in the thermodynamic limit. In contrast, for the AF meron it increases with $L$ [Fig. 4 (c)]. This is consistent with the analytical result, which is valid for infinite-size systems. Plots for several $J/t$ values are shown in Fig. 4 (d) for the AF meron system, showing that it is indeed enhanced at small $J/t$. All these agree with the analytic results, except for the detailed shape of $\mu$-dependence.

The discrepancy in shape ($\mu$-dependence) between the numerical [Fig. 4 (d)] and analytic results [Fig. 2 (b)] may be understood as due to the nonlocality effect in the former. To illustrate this, let us first consider the diffusive regime. As the typical wave number $q$ of the Néel texture (i.e., inverse of meron/skyrmion size) is increased, Eq. (8) is modified as

in the denominator, where $D$ is the diffusion constant. When electron spin diffusion ($Dq^{2}$) occurs faster than spin dephasing ($\tau_{\varphi}^{-1}$), the effective field becomes “nonlocal”. Similar feature has been noted for FMs, in which $Dq^{2}$ is compared to the exchange splitting [3]. Here in AF, it is compared to the (much smaller) spin dephasing rate, $\tau_{\varphi}^{-1}$, hence the present SHE enters the nonlocal regime rather easily compared to the THE in FM. More explicitly, the nonlocality appears if

where $l$ is the mean free path. In Fig. 2 (b), the analytic results with $q^{-2}=6000$ (with lattice constant taken unity) are plotted by dotted lines. The suppression due to nonlocality is more significant at larger $|\mu|$ (away from the AF gap), leaving a sharp peak in the vicinity of the AF gap edge. Since cleaner systems enter the nonlocal regime more easily [see Eq. (14) and a red dotted line in Fig. 2 (c)], this feature is expected to persist into the ballistic regime with a wider nonlocality region. The shape of Fig. 4 (d) may thus be understood as due to the nonlocality effect.

Thus, as in the case of THE in FM [3], the present TSHE in AF exhibits various characteristic regimes [Fig. 2 (c)]. These are summarized as follows. First, for a ballistic and local regime, the effect is truly topological. As $q$ is increased and the nonlocal effects become important, the SHC deviates from the topological expression. In the diffusive case, it is difficult to have the topological expression because of dephasing (and nonlocality), but the effect is enhanced for weak-coupling AF with small AF gap. An interesting possibility may be found in mesoscopic systems, for which the effect can be topological even if the system is in a diffusive regime.

The emergent vector potential ${\bm{a}}^{\alpha}$ in the spin channel, identified here through TSHE, has more generality. In a study on THE in canted AF [28], an emergent vector potential in charge channel was identified as $l^{\alpha}{\bm{a}}^{\alpha}$, where $l^{\alpha}$ is the canting (uniform) moment. Also, ${\bm{a}}^{\alpha}$ can be expressed as $a_{i}^{\alpha}=-({\cal R}{\bm{A}}^{\perp}_{i})^{\alpha}$ [40], where ${\cal R}$ is an SO(3) matrix that connects the rotated and the original frames (e.g., ${\bm{n}}={\cal R}\hat{z}$), showing its conformity with the spin gauge field ${\bm{A}}^{\perp}_{i}$. These facts reinforce our interpretation of ${\bm{a}}^{\alpha}$ as an effective vector potential in spin channel.

To realize the present TSHE experimentally, a prime candidate texture is ${\bm{n}}$-meron. Such texture was found very recently in $\alpha$-Fe₂O₃ [19], which is however an insulator; search for metallic systems is desired. Another candidate is a canted AF; if the ferromagnetic moment ${\bm{l}}$ (due to canting) forms a skyrmion (called “${\bm{l}}$-skyrmion” in [28]), topological consideration shows that the Néel vector winds at least twice around the skyrmion, i.e., $m^{z}=2$ per skyrmion [28, 41]. A recent experiment on thin films of Ce-doped CaMnO₃, a canted AF, observed skyrmion bubbles formed by the (weak) ferromagnetic moment [42]. Therefore, this system can also be a candidate for the present TSHE.

Finally, we discuss the relationship with previous theoretical studies. In Ref. [11], the TSH conductivity was investigated in an AF skyrmion lattice with a focus on the intrinsic (Berry curvature) contribution. It is a future issue to investigate such intrinsic contribution in our framework. For example, one may consider a “meron-antimeron lattice” which contains a vortex with $m=\pm 2$ per unit cell. In Ref. [12], the Landauer-Büttiker method was used to study the TSHE in AF skyrmion systems, and a finite TSHE was found for finite-size systems, which does not contradict with our result because of finite size. More importantly, an increase of the spin Hall conductivity was pointed out for special impurity configurations, and it is also interesting to investigate how the impurity configuration affect the spin dephasing and TSHE.

To summarize, we have studied a spin Hall effect due to magnetic textures in AF metals. By analytic calculations, we found a topological contribution proportional to the winding number defined by vector chirality. This is finite for AF merons but not for AF skyrmions, and is enhanced at weak coupling. These results may provide hints to enhance spin currents and give directions for experimental investigations and device fabrications. The results are confirmed by numerical calculations based on the Landauer-Büttiker formula. Important roles played by nonadiabatic processes and spin dephasing are pointed out.

We would like to thank A. Yamakage, J. Fujimoto, T. Yamaguchi, Y. Imai, A. Matsui, and T. Nomoto for valuable discussions. This work was partly supported by JSPS KAKENHI Grant Numbers JP15H05702, JP17H02929, JP19K03744 and No. JP21H01799, and the Center of Spintronics Research Network of Japan. KN is supported by JSTCREST (JP-MJCR18T2) and JSPS KAKENHI Grant Number JP21K13875. JJN is supported by a Program for Leading Graduate Schools “Integrative Graduate Education and Researchin Green Natural Sciences” and Grant-in-Aid for JSPS Research Fellow Grant Number JP19J23587.

Supplemental material
Topological Spin Hall Effect in Antiferromagnets Driven by Vector Néel Chirality