Spatiotemporal Patterns in Ultraslow Domain Wall Creep Dynamics

Dynamical heterogeneities of thermal creep in pinned interfaces

Disordered systems under applied loading display slow creep flows at finite temperature. Renormalization group arguments predicted that creep proceeds via thermal avalanches of activated events. Recently, thermal avalanches were argued to control the dynamics of liquids near their glass transition. Both theoretical approaches are markedly different. Here we provide a scaling description that seeks to unify dynamical heterogeneities in both phenomena, confirm it in simple models of pinned elastic interfaces, and discuss its experimental implications.

Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles

Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar-Parisi-Zhang (KPZ) terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random-field magnets. Emphasis is given to numerical and experimental tests of the theory.

Effect of thresholding on avalanches and their clustering for interfaces with long-range elasticity

Avalanches are often defined as signals higher than some detection level in bursty systems. The choice of the detection threshold affects the number of avalanches, but it can also affect their temporal correlations. We simulated the depinning of a long-range elastic interface and applied different thresholds including a zero one on the data to see how the sizes and durations of events change and how this affects temporal avalanche clustering. Higher thresholds result in steeper size and duration distributions and cause the avalanches to cluster temporally. Using methods from seismology, the frequency of the events in the clusters was found to decrease as a power-law of time, and the size of an event in a cluster was found to help predict how many events it is followed by. The results bring closer theoretical studies of this class of models to real experiments, but also highlight how different phenomena can be obtained from the same set of data.

Thermally activated flow in models of amorphous solids

Amorphous solids yield at a critical value Σ_{c} of the imposed stress Σ through a dynamical phase transition. While sharp in athermal systems, the presence of thermal fluctuations leads to the rounding of the transition and thermally activated flow even below Σ_{c}. Here we study the steady-state thermal flow of amorphous solids using a mesoscopic elastoplastic model. In the Hébraud-Lequex (HL) model we provide an analytical solution of the thermally activated flow at low temperature. We then propose a general scaling law that also describes the transition rounding. Finally, we find that the scaling law holds in numerical simulations of the HL model, a two-dimensional (2D) elastoplastic model, and previously published molecular dynamics simulations of 2D Lennard-Jones glass.

Universality classes of the domain-wall creep motion driven by spin-transfer torques

With the stochastic Landau-Lifshitz-Gilbert equation, we numerically simulate the creep motion of a magnetic domain wall driven by the adiabatic and nonadiabatic spin-transfer torques induced by the electric current. The creep exponent μ and the roughness exponent ζ are accurately determined from the scaling behaviors. The creep motions driven by the adiabatic and nonadiabatic spin-transfer torques belong to different universality classes. The scaling relation between μ and ζ based on certain simplified assumptions is valid for the nonadiabatic spin-transfer torque, while invalid for the adiabatic one. Our results are compatible with the experimental ones, but go beyond the existing theoretical prediction. Our investigation reveals that the disorder-induced pinning effect on the domain-wall rotation alters the universality class of the creep motion.

Local Probe Comparison of Ferroelectric Switching Event Statistics in the Creep and Depinning Regimes in Pb(Zr_{0.2}Ti_{0.8})O_{3} Thin Films

Ferroelectric materials provide a useful model system to explore the jerky, highly nonlinear dynamics of elastic interfaces in disordered media. The distribution of nanoscale switching event sizes is studied in two Pb(Zr_{0.2}Ti_{0.8})O_{3} thin films with different disorder landscapes using piezoresponse force microscopy. While the switching event statistics show the expected power-law scaling, significant variations in the value of the scaling exponent τ are seen, possibly as a consequence of the different intrinsic disorder landscapes in the samples and of further alterations under high tip bias applied during domain writing. Importantly, higher exponent values (1.98-2.87) are observed when crackling statistics are acquired only for events occurring in the creep regime. The exponents are systematically lowered when all events across both creep and depinning regimes are considered-the first time such a distinction is made in studies of ferroelectric materials. These results show that distinguishing the two regimes is of crucial importance, significantly affecting the exponent value and potentially leading to incorrect assignment of universality class.

Avalanche criticality during ferroelectric/ferroelastic switching

Field induced domain wall displacements define ferroelectric/ferroelastic hysteresis loops, which are at the core of piezoelectric, magnetoelectric and memristive devices. These collective displacements are scale invariant jumps with avalanche characteristics. Here, we analyse the spatial distribution of avalanches in ferroelectrics with different domain and transformation patterns: Pb(Mg_1/3Nb_2/3)O₃–PbTiO₃ contains complex domains with needles and junction patterns, while BaTiO₃ has parallel straight domains. Nevertheless, their avalanche characteristics are indistinguishable. The energies, areas and perimeters of the switched regions are power law distributed with exponents close to predicted mean field values. At the coercive field, the area exponent decreases, while the fractal dimension increases. This fine structure of the switching process has not been detected before and suggests that switching occurs via criticality at the coercive field with fundamentally different switching geometries at and near this critical point. We conjecture that the domain switching process in ferroelectrics is universal at the coercive field.

While classical approaches rely on the study of individual ferroelectric domain wall movement on long time scales, the authors consider collective movements of domain walls during short time scales, characterized by discrete jumps, as indicators of avalanches on a broad range of scales.

Dynamic Manipulation in Piezoresponse Force Microscopy: Creating Nonequilibrium Phases with Large Electromechanical Response

Domain walls and topological defects in ferroelectric materials have emerged as a powerful tool for functional electronic devices including memory and logic. Similarly, wall interactions and dynamics underpin a broad range of mesoscale phenomena ranging from giant electromechanical responses to memory effects. Exploring the functionalities of individual domain walls, their interactions, and controlled modifications of the domain structures is crucial for applications and fundamental physical studies. However, the dynamic nature of these features severely limits studies of their local physics since application of local biases or pressures in piezoresponse force microscopy induce wall displacement as a primary response. Here, we introduce an approach for the control and modification of domain structures based on automated experimentation, whereby real-space image-based feedback is used to control the tip bias during ferroelectric switching, allowing for modification routes conditioned on domain states under the tip. This automated experiment approach is demonstrated for the exploration of domain wall dynamics and creation of metastable phases with large electromechanical response.

Correlations between avalanches in the depinning dynamics of elastic interfaces

We study the correlations between avalanches in the depinning dynamics of elastic interfaces driven on a random substrate. In the mean-field theory (the Brownian force model), it is known that the avalanches are uncorrelated. Here we obtain a simple field theory which describes the first deviations from this uncorrelated behavior in a ε=d_{c}-d expansion below the upper critical dimension d_{c} of the model. We apply it to calculate the correlations between (i) avalanche sizes (ii) avalanche dynamics in two successive avalanches, or more generally, in two avalanches separated by a uniform displacement W of the interface. For (i) we obtain the correlations of the total sizes, of the local sizes, and of the total sizes with given seeds (starting points). For (ii) we obtain the correlations of the velocities, of the durations, and of the avalanche shapes. In general we find that the avalanches are anticorrelated, the occurrence of a larger avalanche making more likely the occurrence of a smaller one, and vice versa. Examining the universality of our results leads us to conjecture several exact scaling relations for the critical exponents that characterize the different distributions of correlations. The avalanche size predictions are confronted to numerical simulations for a d=1 interface with short range elasticity. They are also compared to our recent related work on static avalanches (shocks). Finally we show that the naive extrapolation of our result into the thermally activated creep regime at finite temperature predicts strong positive correlations between the forward motion events, as recently observed in numerical simulations.

Seismiclike organization of avalanches in a driven long-range elastic string as a paradigm of brittle cracks

Crack growth in heterogeneous materials sometimes exhibits crackling dynamics, made of successive impulselike events with specific scale-invariant time and size organization reminiscent of earthquakes. Here, we examine this dynamics in a model which identifies the crack front with a long-range elastic line driven in a random potential. We demonstrate that, under some circumstances, fracture grows intermittently, via scale-free impulse organized into aftershock sequences obeying the fundamental laws of statistical seismology. We examine the effects of the driving rate and system overall stiffness (unloading factor) onto the scaling exponents and cutoffs associated with the time and size organization. We unravel the specific conditions required to observe a seismiclike organization in the crack propagation problem. Beyond failure problems, implications of these results to other crackling systems are finally discussed.

Localization of soft modes at the depinning transition

We characterize the soft modes of the dynamical matrix at the depinning transition, and compare the matrix with the properties of the Anderson model (and long-range generalizations). The density of states at the edge of the spectrum displays a universal linear tail, different from the Lifshitz tails. The eigenvectors are instead very similar in the two matrix ensembles. We focus on the ground state (soft mode), which represents the epicenter of avalanche instabilities. We expect it to be localized in all finite dimensions, and make a clear connection between its localization length and the Larkin length of the depinning model. In the fully connected model, we show that the weak-strong pinning transition coincides with a peculiar localization transition of the ground state.

Spatial shape of avalanches

In disordered elastic systems, driven by displacing a parabolic confining potential adiabatically slowly, all advance of the system is in bursts, termed avalanches. Avalanches have a finite extension in time, which is much smaller than the waiting time between them. Avalanches also have a finite extension ℓ in space, i.e., only a part of the interface of size ℓ moves during an avalanche. Here we study their spatial shape 〈S(x)〉_{ℓ} given ℓ, as well as its fluctuations encoded in the second cumulant 〈S^{2}(x)〉_{ℓ}^{c}. We establish scaling relations governing the behavior close to the boundary. We then give analytic results for the Brownian force model, in which the microscopic disorder for each degree of freedom is a random walk. Finally, we confirm these results with numerical simulations. To do this properly we elucidate the influence of discretization effects, which also confirms the assumptions entering into the scaling ansatz. This allows us to reach the scaling limit already for avalanches of moderate size. We find excellent agreement for the universal shape and its fluctuations, including all amplitudes.

Creep and thermal rounding close to the elastic depinning threshold

We study the slow stochastic dynamics near the depinning threshold of an elastic interface in a random medium by solving a particularly suited model of hopping interacting particles that belongs to the quenched-Edwards-Wilkinson depinning universality class. The model allows us to compare the cases of uniformly activated and Arrhenius activated hops. In the former case, the velocity accurately follows a standard scaling law of the force and noise intensity with the analog of the thermal rounding exponent satisfying a modified "hyperscaling" relation. For the Arrhenius activation, we find, both numerically and analytically, that the standard scaling form fails for any value of the thermal rounding exponent. We propose an alternative scaling incorporating logarithmic corrections that appropriately fits the numerical results. We argue that this anomalous scaling is related to the strong correlation between activated hops that, alternated with deterministic depinning-like avalanches, occur below the depinning threshold. We rationalize the spatiotemporal patterns by making an analogy of the present model in the near-threshold creep regime with some well-known models with extremal dynamics, particularly the Bak-Sneppen model.