Antiferromagnetic domain walls under spin-orbit torque

Domain walls in antiferromagnets under a spin-polarized current present dynamical behavior that is not observed in ferromagnets, and it is tunable by the current polarization. Precessional dynamics is obtained for perpendicular spin polarization. In-plane polarization gives propagating walls. We obtain the velocity as a function of current by a perturbation method for low velocities, and the wall profile is found to lack a definite parity. For high velocities, a power-law decay develops in the trailing tail of the wall. The main features of the wall profile are manifest in a direct solution of an equation that is valid in a limiting case. Oscillatory motion of domain walls is obtained for a spin polarization that has both perpendicular and in-plane components, and an analytical description is given. We discuss the modifications of the dynamics when a Dzyaloshinskii-Moriya interaction is present. Finally, we give the magnetization of the dynamical walls and find that this can become large, providing a potential method for observations.

💡 Research Summary

The paper presents a comprehensive theoretical and numerical study of antiferromagnetic (AFM) domain‑wall (DW) dynamics driven by spin‑orbit torque (SOT). Starting from a discrete spin‑chain model that includes antiferromagnetic exchange, Dzyaloshinskii‑Moriya (DM) interaction, and easy‑axis anisotropy, the authors derive a continuum nonlinear σ‑model for the Néel vector n(x,τ) with added damping (α) and spin‑torque (β) terms. The governing equation is
 n × (¨n – f + β n×p̂ + α ẋn) = 0,
where f = n″ – 2λ ê₂×n′ + n₃ ê₃ and λ∝D. Physical units are fixed by realistic parameters (J≈10¹³ s⁻¹, K≈0.0025 J, lattice spacing ≈0.5 nm), giving a velocity scale of ~1.4×10⁴ m s⁻¹.

The study distinguishes three main regimes depending on the spin‑polarization direction p̂:

1. In‑plane polarization (p̂ = ê₂) – Propagating walls.
   By assuming a traveling‑wave ansatz n(x−vτ) and setting Φ=0 (so n₂=0), the dynamics reduces to a single equation for the polar angle Θ(ξ):
   (1−v²)Θ″ + αvΘ′ = sinΘ cosΘ – β.
   This is mathematically equivalent to a damped particle in a tilted sine‑Gordon potential. For small β (β≪1) a perturbative expansion Θ = Θ₀ + βΘ₁ + …, v = βv₁ + … yields v₁ = π²/(2α), i.e. v ≈ (π²/α) β. This linear relation matches earlier collective‑coordinate results. Higher‑order corrections involve β³/α³ terms, whose coefficients depend on the detailed shape of Θ₁, Θ₂ but are not evaluated explicitly. An exact integral relation ∫Θ′² dξ = (πβ)/α holds for any velocity, allowing the total in‑plane magnetization to be expressed as M₂ = sπ√2 v. Numerical simulations confirm the linear law at low β and reveal a non‑trivial wall profile at higher currents: the trailing tail (ξ→−∞) decays as a power law (∝|ξ|⁻¹ᐟ²) while the leading edge decays exponentially. The wall therefore lacks a definite parity; the asymmetry grows with current.

2. Perpendicular polarization (p̂ = ê₁) – Precessional dynamics.
   When the spin polarization is orthogonal to the easy axis, the azimuthal angle Φ acquires a time‑dependent rotation while Θ remains essentially static. The reduced equations become Φ″ + αΦ′ = β cosΘ sinΦ, admitting a steady precessional solution. This regime reproduces the “precessional” domain‑wall motion previously reported for antiferromagnets.

3. Mixed polarization (p̂ = cosθ ê₁ + sinθ ê₂) – Oscillatory motion.
   With both components present, the wall oscillates between two equivalent Néel configurations. Analytically, the system behaves like a particle trapped in a double‑well potential under a periodic drive; the oscillation frequency scales as ω ∝ β sinθ, and the amplitude depends on damping. The authors provide an approximate analytical description that matches numerical trajectories.

The effect of a finite DM interaction (λ≠0) is examined by retaining the term −2λ ê₂×n′ in f. DM breaks the left‑right symmetry of the wall, biases its motion, and modifies the current‑velocity relation, especially at larger λ where the wall is tilted even in the absence of current.

A notable result concerns the dynamical magnetization of the wall. Using the tetramer coarse‑graining scheme, the magnetization density is m = ε²√2 n×ẋn, leading to a total moment M = sπ√2 v directed in‑plane. Because v can reach a sizable fraction of the characteristic velocity (~10⁴ m s⁻¹), the induced moment becomes large enough to be detected experimentally via techniques such as X‑ray magnetic circular dichroism or magneto‑optical Kerr effect, offering a practical route to visualize antiferromagnetic textures.

The paper is organized as follows: Section II introduces the discrete and continuum models; Section III treats propagating walls with in‑plane polarization, including perturbative low‑current analysis and numerical verification; Section IV discusses precessional motion for perpendicular polarization; Section V presents the oscillatory dynamics for mixed polarization; Section VI summarizes the findings and highlights experimental implications. All simulation data are made publicly available on Zenodo (doi:10.5281/zenodo.17898266).

In summary, the work elucidates how the direction and magnitude of spin‑orbit torque control antiferromagnetic domain‑wall motion, revealing regimes of steady translation, precession, and oscillation, and exposing novel features such as asymmetric wall profiles, power‑law tails, and sizable dynamical magnetization. These insights deepen the theoretical foundation of antiferromagnetic spintronics and suggest concrete experimental signatures for future studies.