Propelling ferrimagnetic domain walls by dynamical frustration

Model

We consider a 1D classical spin model, ∣S∣ = 1, with the following Hamiltonian

where J, Δ > 0 impose AFM couplings in the x– and y-directions, but FM interactions for the z-component. Our main results apply also to easy-cone ferromagnets obtained for J < 0, Δ > 0 as long as long-ranged dipolar interaction can be neglected. δ₂ and δ₄ > 0 introduce a uniaxial anisotropy. For δ₂ < 2J(Δ − 1) and B_z = 0, one obtains a ferrimagnetic ground state with a constant \({S}_{i}^{z}=\pm {m}_{0}^{z}\), where \({m}_{0}^{z}=\sqrt{\frac{-{\delta }_{2}-2J(1-\Delta )}{{\delta }_{4}}}\ge 0\). The g-factors g_i are chosen differently for even and odd sites. Such staggered g factors naturally occur in magnets with, e.g., a screw axis⁴⁴ and are used here to avoid that B_z(t) couples to the conserved magnetisation only. The dynamics of the system is described by the stochastic Landau-Lifshitz-Gilbert equation, which includes a phenomenological damping term α and a corresponding noise term proportional to the temperature T, see methods. We first study the noiseless case, T = 0.

To drive the system out of thermal equilibrium, we consider the effect of a rapidly oscillating magnetic field, \({B}_{z}(t)={B}_{0}\cos (\omega t)\), ω ~ J. In the ferrimagnetic phase, \(0 \, < \, | {m}_{0}^{z}| \, < \, 1\), this oscillation ‘activates’ the Goldstone mode. The staggered xy-order, described by the angle φ, begins to precess collectively,

where the formulas are valid for small B₀ and a small FM magnetisation. The fields m^z and φ are defined by \(\langle {S}_{i}^{z}\rangle={m}^{z}(x)\), \(\langle ({S}_{i}^{x},{S}_{i}^{y})\rangle={(-1)}^{i}\sqrt{1-{({m}^{z})}^{2}}\left(\cos (\varphi (x)),\sin (\varphi (x))\right.\). Here, x is the coarse-grained position variable, with x = ia at the position of spin S_i. 〈…〉 denotes an average over the oscillation period of the external field (in plots, however, for simplicity’s sake we show stroboscopic images instead, see methods). An analytical calculation of ω_rot using second order perturbation theory in B₀ for arbitrary m^z is given in Supplementary material, App. C. The precise way in which non-equilibrium is implemented is expected to be irrelevant as long as the xy-symmetry remains intact. In an experimental system one could, e.g., also use a laser to create electronic excitations. In this case, no staggered g-factors are needed and one expects that γ is proportional to the power of the laser.

Moving domain walls

In our system, the xy-magnetisation rotates in the clockwise or anticlockwise direction when the FM magnetisation points up or down, respectively. As argued in the introduction, this leads to dynamical frustration at the centre of the domain wall where m^z changes sign. Figure 1b, c shows the result of a numerical simulation of such a domain wall. Depending on the initial conditions, the domain wall moves either to the right or to the left with a constant velocity. Far from the domain wall boundary, the angle φ develops a finite slope that builds up behind the moving domain wall.

To develop an analytical theory for the velocity of the moving domain wall, we first consider the limit of vanishing damping, α → 0. The following calculation uses the interplay of spin (super) currents and domain wall motion as studied previously by Kim and Tserkovnyak³⁵, see also refs. ^36,38. For a domain wall moving with the velocity v, we make the ansatz

see Supplementary material App. D.5 for details. This ansatz captures the magnetisation profile shown in Fig. 1b, c, obtained in the long-time limit. The term \(\tilde{\omega }t\) is needed to capture the asymmetry between the left and the right side. Note that both the direction of v and the sign of the phase gradients arise from spontaneous symmetry breaking by tiny asymmetries in the initial condition. We need only one extra input to solve the steady-state problem: for vanishing damping, the magnetisation m^z is conserved, ∂_tm^z + ∂_x j^z = 0 and the spin (super) current is proportional to the gradient of the phase^35,36,45,46,

where \({\rho }_{s}\approx J{a}^{2}(1-{({m}_{0}^{z})}^{2})\) is the spin stiffness and a the lattice constant. Substituting in Eq. (4), and integrating the continuity equation over space from  − x₀ to x₀, one obtains

Far from the domain wall, the magnetisation takes its bulk value \({m}^{z}=\pm {m}_{0}^{z}\) and the spins rotate with \({\partial }_{t}\varphi=-v{\partial }_{x}\Phi+\tilde{\omega }=\mp | {\omega }_{{{\rm{rot}}}}|\), Eq. (3). Therefore, we obtain from Eq. (6) directly \(-2v{m}_{0}^{z}+2{\rho }_{s}\frac{| {\omega }_{{{\rm{rot}}}}| }{v}=0\) or, equivalently,

where the sign of the velocity is determined by the sign of the jump in the gradients, ∂_xΦ(x₀) − ∂_xΦ( − x₀). Eq. (7) is a noteworthy result for four reasons: (i) It has been derived without any knowledge of the shape of the domain wall. All microscopic parameters entering the equation are known analytically for small B₀. (ii) The direction of motion has to be chosen by spontaneous symmetry breaking and is, for example, determined by small fluctuations of φ in the initial state. (iii) As ω_rot is linear in \({m}_{0}^{z}\), the velocity of domain walls remains finite even for \({m}_{0}^{z}\to 0\), i.e., when the phase transition from an xy-ordered phase to the ferrimagnetic phase is approached. Indeed, in a previous analysis⁶, we found that this velocity characterises the field theory of the critical point in higher dimensions. Finally, (iv), the velocity is very fast. v is proportional to the square root of ω_rot, making it linear in the amplitude B₀ of the oscillating field or, equivalently, proportional to the square root of the power used to drive our system out of thermal equilibrium. For weak B₀, this makes v much larger than, e.g., ω_rotξ₀, where ξ₀ is the width of the domain wall. Actively moving phase boundaries have also been shown to exist in models containing mixtures of active and passive Brownian particles¹⁷.

Eq. (7) is valid for α → 0 for finite ω_rot. The calculation for finite α is more challenging. In Supplementary material, App. D, we derive an approximate formula for the velocity,

where ξ₀ is the width of the domain wall in equilibrium, \({\xi }_{0}=\frac{a\sqrt{2J\Delta /{\delta }_{4}}}{{m}_{0}^{z}}\), and η is a dimensionless numerical factor, with η = 3 close to the ferrimagnetic phase transition. For ω_rot → 0, the velocity of the domain wall is linear in ω_rot. In this limit, the domain wall moves by a large distance \(\frac{\pi \eta {\xi }_{0}}{\alpha {m}_{0}^{z}} \, \gg \, {\xi }_{0}\) during each rotation, inversely proportional to the (often very small) Gilbert damping α and the average magnetisation \({m}_{0}^{z} \, < \, 1\). In this regime, a similar formula has previously been obtained for a domain wall in a ferrimagnet driven by an external rotating field³⁷, but in this case the direction of motion is not chosen spontaneously but imprinted externally.

In Fig. 2, we show the velocity of the domain wall obtained from numerical simulations, together with Eq. (8). Taking into account that there is no fitting parameter and the approximate nature of the derivation of Eq. (8), the agreement for B₀ ≲ 0.2 is very good. The analytical formula Eq. (8) does not include the renormalisation of \({m}_{0}^{z}\) and ρ_s for large B₀, which may explain small deviations for larger B₀.

Across a wide parameter range, v is linear in B₀ and thus proportional to \(\sqrt{| {\omega }_{{{\rm{rot}}}}| }\), 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.

Above a critical strength of the driving field, B₀ > B_c, the moving domain wall solution ceases to exist. Instead, one obtains localised domain walls. In Supplementary material, App. A.1, we show that in this case, topology enforces the presence of singular configurations of φ(x, t) (phase slips or, equivalently, space-time vortices⁴⁷), which manifest as sudden bursts of spin excitations occurring periodically once per rotation period, T_rot = 2π/ω_rot.

Dynamics of ordering

To investigate the dynamics of ordering, we consider a quench starting from an initial state with long-ranged AFM order perturbed by a small random component of the magnetisation in the z-direction. An example for the dynamics after such a quench is shown in Fig. 3. The domain walls move and when two of them meet, they get annihilated. Thus, the correlation length, defined as the inverse of the domain wall density, ξ = 1/n_DW, grows rapidly, see Fig. 4. At intermediate times, the correlation length grows approximately with the speed of a single domain wall, ξ ≈ vt. In the long time limit, ξ also grows linearly in time,

where μ ≲ 1 is a numerical constant. μ vt, with μ ≈ 0.53 for the chosen parameters, is shown as a dashed black line in Fig. 4a. A linear growth of correlations has been reported before for active matter models with self-propelled particles^48,49.

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”.

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 \(C(x)=\langle {S}_{j}^{z}{S}_{j+x/a}^{z}\rangle\) (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.

This very efficient and fast growth of the correlation length is not what is expected from a theory of independently moving and pairwise annihilating domain walls. Consider a 1D model, where particles with velocity  ± v annihilate each other whenever two of them collide⁵⁰. In such a model, within time t roughly \({N}_{L}=N\pm \sqrt{N}\) left-moving and \({N}_{R}=N\pm \sqrt{N}\) right-moving particles interact with each other, where N ~ vt/d₀ if d₀ is the initial distance of the particles. After a time t, most of them will have annihilated, but this process leaves approximately \(| {N}_{R}-{N}_{L}| \sim \sqrt{N}\) particles in an area of size vt. Thus, in this model the correlation length grows relatively slowly with

for vt ≫ d₀^50,51, in contrast to our finding.

Therefore, we conclude that the rapid growth of ξ can only arise from a strong interaction of the domain walls, which is mediated by the gradients of the xy-order imprinted by dynamical frustration. This is shown by the following qualitative argument: within a FM domain, the angle φ grows linearly in time, φ ≈ ± ω_rott, with opposite signs for up and down domains. If domains have the typical size ξ(t), this implies that the typical gradient of φ is of the order of

If we now demand that gradients remain bounded for t → ∞, then it is plausible that ξ(t) has to grow linearly in t, which explains our numerical observation, Eq. (9). This derivation is not rigorous, as it is possible to construct solutions of, e.g., equally spaced domain walls with a time-independent distance. However, in our simulations such configurations are not stable due to their interactions with an inhomogeneous ∂_xφ-background.

Microscopically, the necessary interactions are mediated by φ. More precisely, the domain walls interact with \({\partial }_{x}^{2}\varphi\), as can be seen directly from the relevant Ginzburg-Landau theory, see methods. A negative \({\partial }_{x}^{2}\varphi\) leads to ∂_x j^z < 0, see Eq. (5), and thus to an increase of m^z, which acts as a repulsive force for an approaching domain with m^z < 0. Here, the main source of large \({\partial }_{x}^{2}\varphi\) are cases where two domain walls meet and annihilate. As can be seen in Fig. 3, each annihiliation event leaves behind a characteristic trace in \({\partial }_{x}^{2}\varphi\) where ∂_xφ changes sign. Close to x = 37,000 a in Fig. 3, a left-moving down domain wall changes its direction of motion when approaching a region with \({\partial }_{x}^{2} \, \varphi \, < 0\). After transforming into a right-moving domain wall, it finally annihilates with another left-moving domain wall. Importantly, the interactions mediated by \({\partial }_{x}^{2}\varphi\) are strongly retarded, as it takes a very long time proportional to ξ² for \({\partial }_{x}^{2}\varphi\) to decay by diffusion. This is much shorter than the time of order ξ/v needed for the approach of other domain walls, see Supplementary material App. A.3 for a hydrodynamic view of these effects.

Resilience to noise

Finally, we investigate the influence of thermal noise (described by a random magnetic field ξ_i(t), see methods) on our findings. In a 1D system in thermal equilibrium, such a noise term destroys long-ranged order and induces domain walls with a typical distance \({\xi }_{{{\rm{th}}}} \sim {e}^{{E}_{{{\rm{DW}}}}/{k}_{B}T}\), where E_DW is the domain wall energy at T = 0, and the temperature T controls the noise strength. For the parameters chosen in our figures, E_DW = 0.076 J. In the following, we will therefore always use T/E_DW to specify temperatures. Fig. 5a, b show the magnetisation of a driven ferrimagnet as function of space and time after a quench from the AFM phase for two different temperatures. As in the noiseless case studied above, domain walls actively move with a velocity approximately 10% larger than in the noiseless case, see Supplementary material, App. E.4. It is surprising that the velocity is affected so little, taking into account that the xy-correlation length ξ_AFM in the noisy and driven system becomes tiny, ξ_AFM ≲ 30, see Supplementary material, App. E.1.

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 \({m}^{z}(x)={S}_{i}^{z}\) 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.

Figure 5a shows the formation of giant domains with a size exceeding 10⁵ spins. This has to be compared to an equilibrium system, where for the same noise level the correlation length is less than 20 sites.

In Fig. 6 we show the steady-state correlation function for both the equilibrium and non-equilibrium systems. In the driven system, the correlation function decays very slow in space, roughly with a linear slope on a logarithmic scale, \(C(x)=C(0)\left(1-\ln [x/a]/\ln [{\xi }_{\max }(T)/a]\right.\), showing that the system hosts domains with very different sizes. \({\xi }_{\max }(T)\) can be interpreted as the largest correlation length in the system, see Supplementary material, App. E.3, for details. \({\xi }_{\max }(T)\) grows much faster than exponential with 1/T, see inset of Fig. 6 showing that the rapid growth does not arise from a thermal process. A direct comparison of the space-time correlations of the driven and non-driven systems, see Supplementary material, App. E.4, shows the qualitative change of dynamical properties of the driven system.

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.

How can even a relatively weak oscillating field increase the correlation length by many orders of magnitude and why is the driven system so resilient to noise? In the regime investigated here, fluctuations are not small and even within an ordered domain, they often induce a sign change, see the upper curve in Fig. 5c. Locally, fluctuations are very similar to the thermal system. The difference is, however, that the driven system is very efficient in suppressing the generation of defects and also in healing some of the most frequently occurring defects. The main origin of this healing effect is the dynamical frustration of domains rotating in opposite direction. In Supplementary material, App. B, we discuss and investigate the detailed microscopic mechanisms behind this observation.

One may ask whether one can (or indeed should) use the term ‘active magnetic matter’ to describe the system studied here. As discussed in the introduction, active matter is usually defined as being composed of large numbers of ‘active agents’, each breaking detailed balance on a local level and giving rise to collective non-equilibrium phenomena. In our case, one can either view each spin or each domain wall as one of the active agents. Each spin in our system is rotating either clockwise or counter-clockwise, thus breaking detailed balance on a local level. Similarly, a domain wall moves actively in space, either to the right or to the left side. Furthermore, we have shown that the collective dynamics of domain walls is dominated by long-range interactions and cannot be understood from their individual motion. In our case of an active magnetic system, the external drive is implemented in a continuous way by an oscillating external field. This is also the case in more traditional active systems, for example of biological origin. There, some type of feeding is required, but continuous drive is not needed in most biological systems, where energy can be stored. Other examples of continuously driven active matter systems use light or ultrasound as a stimulus^9,16,52,53.

In conclusion, we show that a simple oscillating field transforms a ferrimagnet into an ‘active magnet’. This is a state of matter with properties very different to those of equilibrium systems. Ferrimagnets recently attracted considerable interest in the field of spintronics⁵, as they combine the advantages of FM and AFM systems. Here, we mainly need the rotation of an xy-order parameter in a driven system to be controlled by the orientation of a FM component perpendicular to it. The rotation of the xy-order parameter leads to an effect which we call ‘dynamical frustration’, resulting in an efficient propulsion of the domain wall, which now moves actively either to the left or to the right by spontaneous symmetry breaking. The net effect is a very fast buildup of FM correlations on huge length scales, indicating a high degree of robustness of the system to thermal noise due to an efficient non-equilibrium self-healing mechanism. While our study has focused on one-dimensional systems, we show in Supplementary material App. F that key results such as the active motion of domain walls and the rapid buildup of correlations also apply in two dimensions. The most promising route for an experimental realisation of our model is to use widely studied magnetic multilayer systems, whose properties (including magnetic anisotropies) can largely be tuned by layer thickness and the combination of different materials^54,55. For example, Ślȩzak et al.⁵⁶ recently demonstrated direct coupling of an easy-plane antiferromagnetic NiO layer to an adjacent ferromagnet Fe layer, providing a direct route towards a room-temperature realisation of our model. Experimentally, it should be rather straightforward to observe the motion of domain walls using standard imaging methods based on the Faraday or Kerr effect, which are routinely used to observe the motion of domain walls⁵⁷.

A surprising aspect of our results is that very weak perturbations have a profound effect on the qualitative and quantitative behaviour of the system. For the chosen parameters, the speed of rotation of the magnetisation was tiny, ω_rot J ≈ ±0.01. This tiny perturbation changes the dynamics of the system, affects the correlations both on small and large length scales, see Fig. 6, and can increase the correlation length by orders of magnitude. In equilibrium systems, tiny perturbations (if they are ‘relevant’ in the renormalisation-group sense) can also sometimes strongly affect properties of correlation functions, but only at very long distances. We are not aware of any equilibrium systems in which small perturbations cause such a profound change of magnetic correlations on all length scales. Our driven system is different, as it induces active motion of domain walls.

In our study, we have investigated a minimal model of a driven magnet. We anticipate that it is possible to activate (approximate) Goldstone modes in a large class of systems and to use the principle of dynamical frustration to realise actively moving defects. For example, basic features of our models can be realised directly by coupling a superconducting layer to a ferroelectric or a polar metal in the presence of external radiation using, e.g., oxide heterostructures⁵⁸. Given the enormous variety of complex symmetries and associated topological defects available in solid-state systems, we anticipate numerous opportunities to realise systems which may be identified as active magnetic matter or, more broadly, active quantum matter. An important aspect of this challenge will be finding experimental realisations of such systems.