Propelling ferrimagnetic domain walls by dynamical frustration

Abstract

Many-particle systems driven out of thermal equilibrium can show properties qualitatively different from any thermal state. Here, we study a ferrimagnet in a weak oscillating magnetic field. In this model, domain walls are not static, but are shown to move actively in a direction chosen by spontaneous symmetry breaking. Thus they act like self-propelling units. Their collective behaviour is reminiscent of other systems with actively moving units studied in the field of 'active matter', where, e.g., flocks of birds are investigated. The active motion of the domain walls emerges from 'dynamical frustration'. The antiferromagnetic xy-order rotates clockwise or anticlockwise, determined by the sign of the ferromagnetic component. This necessarily leads to frustration at a domain wall, which gets resolved by propelling the domain wall with a velocity proportional to the square root of the driving power across large parameter regimes. This motion and strong hydrodynamic interactions lead to a linear growth of the magnetic correlation length over time, much faster than in equilibrium. The dynamical frustration furthermore makes the system highly resilient to noise. The correlation length of the weakly driven one-dimensional system can be orders of magnitude larger than in the corresponding equilibrium system with the same noise level.

Fig. 1. Moving domain wall.

a Schematic picture of a domain wall in a driven ferrimagnet. The (staggered) in-plane magnetisation rotates in opposite directions on the left and right side, leading to dynamical frustration. This can lead to an active motion of the domain wall either to the left or right. Data from a numerical simulation of a right-moving domain wall are displayed in (b) and (c). b Magnetisation m^z(x, t), while c) shows the phase φ(x, t), describing the orientation of the staggered in-plane magnetisation. A substantial phase gradient ∂_xφ is built up behind the moving domain. Parameters: J = 1, Δ = 0.8, δ₂ = −0.6, δ₄ = 1, g₁ = 1, g₂ = 0.1, α = 0.1, B₀ = 0.15, ω = 3.6 resulting in a rotation period T_rot ≈ 612, used to scale the vertical axis for a system of 40,000 spins.

Fig. 2. Velocity v of a domain wall as function of the amplitude of the oscillating field B₀, for different values of anisotropy δ₂ and damping α.

Across a wide parameter range, v is linear in B₀ and thus proportional to $\sqrt{\mid {\omega }_{\mathrm{rot}}\mid }$, see Eq. (7). Lines: analytical calculation of the domain wall velocity, Eq. (8), without fitting parameters. Parameters: J = 1, Δ = 0.8, δ₂ = −0.6 and  −0.8, δ₄ = 1, g₁ = 1, g₂ = 0.1, α = 0.1 and 0.2, ω = 3.6 for a system of 40,000 spins. Numerical errors are smaller than the size of the symbols.

Fig. 3. Worldlines of domain walls after a quench from the AFM into the ferrimagnetic phase driven by a small oscillating field B₀ = 0.15 J.

Worldlines end either when a left-moving and a right-moving domain collide or when the distance between two domain walls shrinks to zero. Colour: amplitude of ∂_xφ in units of δ_φ, given by the difference of phase gradients δ_φ = ∂_xφ^left − ∂_xφ^right ≈ 0.176 across a single domain wall, see Fig. 1b, c. The motion of domain walls is subject to a long-range hydrodynamic interaction mediated by ∂_xφ. Parameters: same as in Fig. 1, initial state: ordered xy-AFM with a small random $S_i^z=\pm 0.1$, see section “Methods”.

Fig. 4. Buildup of FM correlations after a quench from an AFM-ordered phase into the ferrimagnetic phase in the absence of noise, T = 0, see also Fig. 3.

a Correlation length ξ = 1/n_DW as function of time after a quench from an AFM-ordered phase into the ferrimagnetic phase both for a driven and a non-driven system, marked by NEQ (non-equilibrium) and EQ (equilibrium) in the legend, respectively. The inset shows that the short-time dynamics of the driven and non-driven system is identical, but after a few rotation periods T_rot, the driven system shows a very fast increase of the correlation length. In the long-time limit, ξ grows linearly in time (~0.065t, black dashed line). For comparison, vt, where v is the velocity of a single domain wall, is also shown as a dashed red line. The linear growth of correlations with time can also be seen directly from a scaling plot of the equal-time correlation function $C\left(x\right)=⟨S_j^z{S}_{j+x/a}^z⟩$ (averaged over j), which is shown (for even x/a) in panel b as function of x/(vt). Here v is the velocity of a single domain wall. The plot shows that the maximal speed is 2v, arising from two domain walls moving in opposite directions. Parameters: as in Fig. 3, average over 20 initial states in simulations with 500,000 spins each (15 initial states for the equilibrium states). Error bars denote the corresponding standard deviation of the mean.

Fig. 5. Magnetisation dynamics after a quench into the ferrimagnetic phase.

a, b Magnetisation as function of x and t for two slightly different temperatures, and thus different noise levels, for systems of 250,000 spins (other parameters and colour scale as in Fig. 1). c $m^z\left(x\right)=S_i^z$ at the final timestep of the simulations shown in (a, b), for a small region with 2000 spins. The lower green curve shows that an equilibrium system has a much shorter correlation length at similar noise levels.

Fig. 6

Equal-time correlation function C(x) = 〈m^z(x₀ + x)m^z(x₀)〉 in the presence of thermal noise for systems of the size of 250,000 spins (parameters as in Fig. 1, only even x/a are shown, error bars denote the standard deviation of the mean). The data is measured at t ≈ 520,000/J ≈ 850 T_rot after a quench from an AFM state. The error bars are standard deviation of the mean obtained by averaging over four runs. With the exception of the curve for the lowest T (blue), the system has obtained a stationary state at that time, see Supplementary material, App. E.3. Note the logarithmic scale on the x-axis, needed due to a broad distribution of domain sizes, see Fig. 5. Compared to the non-driven system in thermal equilibrium (dashed lines), the correlation length becomes many orders of magnitude larger for T ≲ 0.025 J. Inset: The correlation length ${\xi }_{\max }$ grows faster than exponential with 1/T. For a discussion of error bars in the inset, see Supplementary material, App. E.3.