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

TL;DR

The paper investigates deterministic instabilities of dc-driven extended domain walls in homogeneous ferromagnets at zero temperature. It develops a reduced $u$--$\phi$ model validated against zero-temperature micromagnetic simulations, and employs a Floquet stability analysis to derive a quasi-universal diagram controlled by the Gilbert damping $\alpha$, predicting when flat walls destabilize and how many modes participate. The main finding is a dynamical phase transition: below a finite-size threshold walls move rigidly, while above it they corrugate and enter a deterministic chaotic roughening regime with Edwards-Wilkinson-like scaling before transitioning back to a flat moving phase at higher fields. This framework isolates intrinsic deterministic instabilities from disorder and thermal effects and lays groundwork for incorporating additional interactions and higher-dimensional geometries.

Abstract

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.

Figures (12)

• Figure 1: Snapshot of a DW configuration highlighting its collective coordinates $u(x,t)$ and $\phi(x,t)$.
• Figure 2: (a) Linear stability diagram of DW modes for $\alpha = 0.27$. White regions are stable; colors indicate the magnitude of the largest Floquet multiplier for unstable modes with $|\mu_\kappa| > 1$. (b) Dependence of characteristic wavenumbers $\kappa_m$, $\kappa_c$, and critical field $h_c$ (as defined in inset of panel a) on Gilbert damping $\alpha$.
• Figure 3: (a) Absolute value of Floquet multipliers $|\mu_{\kappa}|$ as a function of driving field $h$, for $\alpha=0.27$, showing the only two conditionally unstable modes of a small system, for $\kappa_{1}=0.5$ and $\kappa_{2}=2\kappa_{1}=1.0$. (b) Normalized mean DW velocity and (c) roughness obtained from $u$--$\phi$ model and micromagnetic simulations for the same parameters and length scales. For comparison, panel (b) also shows the rigid wall velocity $\langle \dot u_{1}\rangle$ (blue line).
• Figure 4: Transient dynamics from a flat, perturbed initial condition. (a) Time evolution of the $S_\kappa$ [Eq. \ref{['eq:Skappa']}] at $t/\delta t=2^{0},2^{2},\dots,2^{18}$ with time step $\delta t=1.5$. Shaded area marks the band of unstable modes at $h=3$. (b) Steady-state power spectrum [Eq. \ref{['eq:powerspectrum']}] of the DW center-of-mass velocity. Vertical line indicates $\omega_0 \equiv 2\langle \dot \phi_1 \rangle$. Panels (c), (d) and (e) show DW snapshots at the initial condition, short and large times respectively (the color scale for $\phi$ corresponds to Fig. \ref{['fig:snap']})
• Figure 5: Snapshots of VBLs trajectories during the transient dynamics of Fig. \ref{['fig:transient']}, at increasing times. Color indicates the DW velocity $\dot u(\xi,t)$ relative to $\langle \dot u \rangle$, at each VBL position $\xi$ (See Appendix \ref{['app:vbl']}).