Deterministic roughening in the dc-driven precessional regime of domain walls

We numerically study the dynamics of extended domain walls in homogeneous ferromagnets driven by a uniform magnetic field at zero temperature. Using both micromagnetic Landau-Lifshitz-Gilbert simulations and a collective-coordinate description, we show that flat chiral domain walls become linearly unstable above the Walker breakdown field and below a higher threshold, provided their length exceeds a characteristic scale. This instability is captured by a quasi-universal spectral stability diagram, parameterized solely by the Gilbert damping, which predicts the onset of deviations from rigid-wall behavior. Beyond the linear regime, large domain walls with bands of unstable modes develop spatiotemporal chaos, intricate Bloch-line dynamics, and deterministic roughening. At a critical field, the system undergoes a dynamical phase transition from a flat to a rough moving phase with universal features. Our results provide a framework for addressing domain-wall dynamics in the presence of thermal fluctuations and quenched disorder by disentangling their effects from intrinsic deterministic instabilities.

💡 Research Summary

This paper presents a comprehensive study of the dynamics of extended magnetic domain walls (DWs) in homogeneous ferromagnets driven by a uniform magnetic field at zero temperature. The authors combine full micromagnetic simulations based on the Landau‑Lifshitz‑Gilbert (LLG) equation with a reduced collective‑coordinate model, often referred to as the u‑ϕ model, which describes the wall position u(x,t) and its internal magnetization angle ϕ(x,t). By non‑dimensionalizing the governing equations, the Walker breakdown field H_W becomes the natural unit of magnetic field (h = H/H_W), and the Gilbert damping α emerges as the sole material parameter controlling the stability landscape.

In the stationary regime (h < 1) the flat wall solution is linearly stable. Above the Walker field (h ≥ 1) the internal angle precesses, rendering the linearized equations non‑autonomous. The authors re‑parameterize time by the precessional angle and apply Floquet theory to each Fourier mode κ. The monodromy matrix M_κ is obtained by integrating over one precessional period (π in the reduced angle), and its eigenvalues μ_κ (Floquet multipliers) determine stability: |μ_κ| > 1 signals exponential growth of that mode. The resulting stability diagram, plotted as |μ_κ| versus (κ, h), is quasi‑universal because it depends only on α. For a typical damping α = 0.27 the diagram displays “instability feathers” extending beyond the negative‑mobility interval (1 < h < h_S) predicted by rigid‑wall models. Characteristic scales follow simple power laws: the most unstable wave number κ_m ∝ α^{-1/2} and the critical field h_c ∝ α^{-2}.

Finite‑size analysis shows that if the system length L satisfies L/L_0 < 2π/κ_m, all non‑zero modes remain stable and the wall behaves as a rigid object. When L exceeds this threshold, the lowest mode becomes unstable, leading to the growth of spatial undulations.

Beyond the linear regime, the nonlinear term sin(2ϕ) in the u‑ϕ equations couples unstable modes, generating a cascade of energy toward higher wave numbers. Numerical integration of the full nonlinear equations reveals spatiotemporal chaos, intricate Bloch‑line nucleation and annihilation, and deterministic roughening of the wall profile. As the driving field is increased, the system undergoes a dynamical phase transition at a critical field h_c: the wall switches from a flat, chiral moving phase to a corrugated, achiral chaotic phase. This transition is accompanied by abrupt changes in average velocity, roughness exponent, and statistical properties of the wall, indicating universal behavior characteristic of nonequilibrium pattern‑forming systems.

The reduced model is validated against micromagnetic simulations performed with MuMax³ on a Co/Pt multilayer (α = 0.27, A_ex = 1.4×10⁻¹¹ J/m, M_s = 9.1×10⁵ A/m, K_u = 8.4×10⁵ J/m³). Because the effective domain‑wall width Δ′ in the full simulation differs slightly from the constant Δ assumed in the analytical model, the authors rescale the simulation length by the factor Δ′/Δ to ensure identical dimensionless system size. After this correction, the instability thresholds, growth rates, and nonlinear roughening patterns agree quantitatively between the two approaches.

Overall, the work establishes a quasi‑universal spectral stability diagram governed solely by Gilbert damping, predicts finite‑size criteria for the onset of instability, and elucidates the route from linear mode growth to fully developed deterministic roughening and chaos. The framework provides a solid baseline for future studies that incorporate thermal fluctuations or quenched disorder, allowing researchers to disentangle intrinsic deterministic instabilities from extrinsic noise effects in domain‑wall dynamics.