Numerical simulations of two-dimensional magnetic domain patterns

Room-Temperature Intrinsic Ferromagnetic Chromium Tellurium Compounds with Thickness-Tunable Magnetic Texture

2D ferromagnetic chromium tellurides exhibit intriguing spin configurations and high-temperature intrinsic ferromagnetism, providing unprecedented opportunities to explore the fundamental spin physics and build spintronic devices. Here, a generic van der Waals epitaxial approach is developed to synthesize the 2D ternary chromium tellurium compounds with thicknesses down to mono-, bi-, tri-, and few-unit cells (UC). The Mn_0.14 Cr_0.86 Te evolves from intrinsic ferromagnetic behavior in bi-UC, tri-UC, and few-UC to temperature-induced ferrimagnetic behavior as the thickness increases, resulting in a sign reversal of the anomalous Hall resistance. Temperature- and thickness-tunable labyrinthine-domain ferromagnetic behaviors are derived from the dipolar interactions in Fe_0.26 Cr_0.74 Te and Co_0.40 Cr_0.60 Te. Furthermore, the dipolar-interaction-induced stripe domain and field-induced domain wall (DW) motion velocity are studied, and multibit data storage is realized through an abundant DW state. The magnetic storage can function in neuromorphic computing tasks, and the pattern recognition accuracy can reach up to 97.93%, which is similar to the recognition accuracy of ideal software-based training (98.28%). Room-temperature ferromagnetic chromium tellurium compounds with intriguing spin configurations can significantly promote the exploration of the processing, sensing, and storage based on 2D magnetic systems.

High-degeneracy Potts coarsening

I examine the fate of a kinetic Potts ferromagnet with a high ground-state degeneracy that undergoes a deep quench to zero temperature. I consider single spin-flip dynamics on triangular lattices of linear dimension 8≤L≤128 and set the number of spin states q equal to the number of lattice sites L×L. The ground state is the most abundant final state, and is reached with probability ≈0.71. Three-hexagon states occur with probability ≈0.26, and hexagonal tessellations with more than three clusters form with probabilities of O(10^{-3}) or less. Spanning stripe states-where the domain walls run along one of the three lattice directions-appear with probability ≈0.03. "Blinker" configurations, which contain perpetually flippable spins, also emerge, but with a probability that is vanishingly small with the system size.

Topology-controlled Potts coarsening

We uncover unusual topological features in the long-time relaxation of the q-state kinetic Potts ferromagnet on the triangular lattice that is instantaneously quenched to zero temperature from a zero-magnetization initial state. For q=3, the final state is either the ground state (frequency ≈0.75), a frozen three-hexagon state (frequency ≈0.16), a two-stripe state (frequency ≈0.09), or a three-stripe state (frequency <2×10^{-4}). Other final state topologies, such as states with more than three hexagons, occur with probability 10^{-5} or smaller, for q=3. The relaxation to the frozen three-hexagon state is governed by a time that scales as L^{2}lnL. We provide a heuristic argument for this anomalous scaling and present additional new features of Potts coarsening on the triangular lattice for q=3 and for q>3.

Coarsening dynamics in the Swift-Hohenberg equation with an external field

We study the Swift-Hohenberg equation (SHE) in the presence of an external field. The application of the field leads to a phase diagram with three phases, i.e., stripe, hexagon, and uniform. We focus on coarsening after a quench from the uniform to stripe or hexagon regions. For stripe patterns, we find that the length scale associated with the order-parameter structure factor has the same growth exponent (≃1/4) as for the SHE with zero field. The growth process is slower in the case of hexagonal patterns, with the effective growth exponent varying between 1/6 and 1/9, depending on the quench parameters. For deep quenches in the hexagonal phase, the growth process stops at late stages when defect boundaries become pinned.

On the mechanism behind the inverse melting in systems with competing interactions

The competition between a short range attractive interaction and a nonlocal repulsive interaction promote the appearance of modulated phases. In this work we present the microscopic mechanisms leading to the emergence of inverse transitions in such systems by considering a thorough mean-field analysis of a variety of minimal models with different competing interactions. We identify the specific connections between the characteristic energy of the homogeneous and modulated phases and the observed reentrant behaviors in the phase diagram. In particular, we find that reentrance is appreciable when the characteristic energy cost of the homogeneous and modulated phases are comparable to each other, and for systems in which the local order parameter is limited. In the asymptotic limit of high energy cost of the homogeneous phase we observe that the degree of reentrance decreases exponentially with the ratio of the characteristic energy cost of homogeneous and modulated phases. These mean-field results are confronted with Langevin simulations of an effective coarse grained model, confirming the expected extension of the reentrance in the phase diagram. These results shed new light on many systems undergoing inverse melting transitions by qualitatively improving the understanding of the interplay of entropy and energy around the inverse melting points.

Magnetic domain wall creep and depinning: A scalar field model approach

Magnetic domain wall motion is at the heart of new magnetoelectronic technologies and hence the need for a deeper understanding of domain wall dynamics in magnetic systems. In this context, numerical simulations using simple models can capture the main ingredients responsible for the complex observed domain wall behavior. We present a scalar field model for the magnetization dynamics of quasi-two-dimensional systems with a perpendicular easy axis of magnetization which allows a direct comparison with typical experimental protocols, used in polar magneto-optical Kerr effect microscopy experiments. We show that the thermally activated creep and depinning regimes of domain wall motion can be reached and the effect of different quenched disorder implementations can be assessed with the model. In particular, we show that the depinning field increases with the mean grain size of a Voronoi tessellation model for the disorder.

Coarsening of stripe patterns: variations with quench depth and scaling

The coarsening of stripe patterns when the system is evolved from random initial states is studied by varying the quench depth ε, which is a measure of distance from the transition point of the stripe phase. The dynamics of the growth of stripe order, which is characterized by two length scales, depends on the quench depth. The growth exponents of the two length scales vary continuously with ε. The decay exponents for free energy, stripe curvature, and densities of defects like grain boundaries and dislocations also show similar variation. This implies a breakdown of the standard picture of nonequilibrium dynamical scaling. In order to understand the variations with ε we propose an additional scaling with a length scale dependent on ε. The main contribution to this length scale comes from the "pinning potential," which is unique to systems where the order parameter is spatially periodic. The periodic order parameter gives rise to an ε-dependent potential, which can pin defects like grain boundaries, dislocations, etc. This additional scaling provides a compact description of variations of growth exponents with quench depth in terms of just one exponent for each of the length scales. The relaxation of free energy, stripe curvature, and the defect densities have also been related to these length scales. The study is done at zero temperature using Swift-Hohenberg equation in two dimensions.

Stripe patterns: role of initial state and boundary conditions

This paper presents results on stripe patterns by numerical solution of the Swift-Hohenberg equation. The focus is on the role of initial state and boundary conditions. We choose initial states which generate simple defect configurations and study their evolution. Various classes of defects are identified and their motion and relaxation is studied numerically. We first study the dynamics of a straight front and present a comparison of numerical results with some analytical results. We then study the domain-wall dynamics in configurations containing two and three domains and identify some mechanisms of their relaxation. Rates of domain-wall relaxation depend on several features like incommensuration, dislocations and orientations in neighboring domains, in addition to the curvature of the walls. For a generic class of domain walls the relaxation process has an intrinsic frustration which leads to generation of dislocations. This process also generates stripe curvature thereby making relaxation nonmonotonic. We have also generated some other topological defects and studied their evolution and the effect of boundary conditions on their stability.

Role of interface coupling inhomogeneity in domain evolution in exchange bias

Models of exchange-bias in thin films have been able to describe various aspects of this technologically relevant effect. Through appropriate choices of free parameters the modelled hysteresis loops adequately match experiment, and typical domain structures can be simulated. However, the use of these parameters, notably the coupling strength between the systems' ferromagnetic (F) and antiferromagnetic (AF) layers, obscures conclusions about their influence on the magnetization reversal processes. Here we develop a 2D phase-field model of the magnetization process in exchange-biased CoO/(Co/Pt)_×n that incorporates the 10 nm-resolved measured local biasing characteristics of the antiferromagnet. Just three interrelated parameters set to measured physical quantities of the ferromagnet and the measured density of uncompensated spins thus suffice to match the experiment in microscopic and macroscopic detail. We use the model to study changes in bias and coercivity caused by different distributions of pinned uncompensated spins of the antiferromagnet, in application-relevant situations where domain wall motion dominates the ferromagnetic reversal. We show the excess coercivity can arise solely from inhomogeneity in the density of biasing- and anti-biasing pinned uncompensated spins in the antiferromagnet. Counter to conventional wisdom, irreversible processes in the latter are not essential.

Effect of magnetism on kinetics of γ-α transformation and pattern formation in iron

The kinetics of polymorphous γ-α transformation in Fe is studied numerically within a model taking into account both the lattice and the magnetic degrees of freedom, based on first-principle calculations of the total energy for different magnetic states. It is shown that a magnetoelastic phenomenon, namely the strong sensitivity of the potential relief along the Bain deformation path to the magnetic state, is crucial for a picture of the transformation. With increasing temperature, a scenario for the phase transformation evolves from a homogeneous lattice instability at T < M(s) (M(s) is the temperature of the beginning of the martensitic transformation) to the growth and nucleation of embryos of the new phase at T > M(s). In the latter case, the formation of a tweed-like structure occurs, with a strong short-range order and slow interphase fluctuations.

Finite-temperature phase diagram of ultrathin magnetic films without external fields

We analyze the finite-temperature phase diagram of ultrathin magnetic films by introducing a mean-field theory, valid in the low-anisotropy regime, i.e., close to the spin reorientation transition. The theoretical results are compared with Monte Carlo simulations carried out on a microscopic Heisenberg model. Connections between the finite-temperature behavior and the ground-state properties of the system are established. Several properties of the stripe pattern, such as the presence of canted states, the stripe width variation phenomenon, and the associated magnetization profiles, are also analyzed.

Geometrical properties of the Potts model during the coarsening regime

We study the dynamic evolution of geometric structures in a polydegenerate system represented by a q-state Potts model with nonconserved order parameter that is quenched from its disordered into its ordered phase. The numerical results obtained with Monte Carlo simulations show a strong relation between the statistical properties of hull perimeters in the initial state and during coarsening: The statistics and morphology of the structures that are larger than the averaged ones are those of the initial state, while the ones of small structures are determined by the curvature-driven dynamic process. We link the hull properties to the ones of the areas they enclose. We analyze the linear von Neumann-Mullins law, both for individual domains and on the average, concluding that its validity, for the later case, is limited to domains with number of sides around 6, while presenting stronger violations in the former case.

Curvature-driven coarsening in the two-dimensional Potts model

We study the geometric properties of polymixtures after a sudden quench in temperature. We mimic these systems with the q -states Potts model on a square lattice with and without weak quenched disorder, and their evolution with Monte Carlo simulations with nonconserved order parameter. We analyze the distribution of hull-enclosed areas for different initial conditions and compare our results with recent exact and numerical findings for q=2 (Ising) case. Our results demonstrate the memory of the presence or absence of long-range correlations in the initial state during the coarsening regime and exhibit superuniversality properties.

Surface patterning of low-dimensional systems: the chirality of charged fibres

Charged surfaces are interesting for their ability to have long-range correlations and their ability to be dynamically tuned. While the configurations of charged planar surfaces have been thoroughly mapped and studied, charged cylindrical surfaces show novel features. The surface patterning of cylindrically confined charges is discussed with emphasis on the role of chiral configurations. The origins of surface patterns due to competing interactions in charged monolayers are summarized along with their associated theoretical models. The electrostatically induced patterns described in this paper are important in many low-dimensional biological systems such as plasma membrane organization, filamentous virus capsid structure or microtubule interactions. A simple model effectively predicting some features of chiral patterns in biological systems is presented. We extend our model from helical lamellar patterns to elliptical patterns to consider asymmetrical patterns in assemblies of filamentous aggregates.

Effects of an oscillating field on magnetic domain patterns: Emergence of concentric-ring patterns surrounding a strong defect

Oscillating fields can make domain patterns change into various types of structures. Numerical simulations show that concentric-ring domain patterns centered at a strong defect are observed under a rapidly oscillating field in some cases. The concentric-ring pattern appears near the threshold of spatially uniform patterns in high-frequency cases. The threshold is theoretically estimated and the theoretical threshold is in good agreement with numerical one in a high-frequency region. The theoretical analysis gives also good estimations of several characteristics of domain patterns for high frequencies.

Strain localization driven by structural relaxation in sheared amorphous solids

A two dimensional amorphous material is modeled as an assembly of mesoscopic elemental pieces coupled together to form an elastically coherent structure. Plasticity is introduced as the existence of different minima in the energy landscape of the elemental constituents. Upon the application of an external strain rate, the material shears through the appearance of elemental slip events with quadrupolar symmetry. When the energy landscape of the elemental constituents is kept fixed, the slip events distribute uniformly throughout the sample, producing on average a uniform deformation. However, when the energy landscape at different spatial positions can be rearranged dynamically to account for structural relaxation, the system develops inhomogeneous deformation in the form of shear bands at low shear rates, and stick-slip-like motion at the shear bands for the lowest shear rates. The origin of strain localization is traced back to a region of negative correlation between strain rate and stress, which appears only if structural relaxation is present. The model also reproduces other well known effects in the rheology of amorphous materials, as a stress peak in a strain rate controlled experiment staring from rest, and the increase of the maximum of this peak with sample age.

Effects of an oscillating field on pattern formation in a ferromagnetic thin film: analysis of patterns traveling at a low velocity

Magnetic domain patterns under an oscillating field are studied theoretically by using a simple Ising-like model. We propose two ways to investigate the effects of the oscillating field. The first one leads to a model in which rapidly oscillating terms are averaged out and the model can explain the existence of the maximum amplitude of the field for the appearance of patterns. The second one leads to a model that includes the delay of the response to the field and the model suggests the existence of a traveling pattern which moves very slowly compared with the time scale of the driving field.

Morphologies of expansion ridges of elastic thin films onto a substrate

A model of a thin film elastically attached to a rigid substrate is considered. In the case in which the film expands relative to the substrate and assuming certain nonlinear elastic behavior of the film, expansion ridges may appear, in which the material collapses, and the density is higher on average. By studying numerically this process, the possible morphologies of these collapsed regions are presented. They range from circular spots and straight stripes, to wiggle polygonal patterns and ring-shaped domains. The similarity of some of these results with patterns observed in delamination of thin films and biphase epitaxial growth is emphasized.

Mechanism for nonequilibrium symmetry breaking and pattern formation in magnetic films

Magnetic thin films exhibit a strong variation in properties depending on their degree of disorder. Recent coherent x-ray speckle experiments on magnetic films have measured the loss of correlation between configurations at opposite fields and at the same field, upon repeated field cycling. We perform finite temperature numerical simulations on these systems that provide a comprehensive explanation for the experimental results. The simulations demonstrate, in accordance with experiments, that the memory of configurations increases with film disorder. We find that nontrivial microscopic differences exist between the zero field spin configuration obtained by starting from a large positive field and the zero field configuration starting at a large negative field. This seemingly paradoxical behavior is due to the nature of the vector spin dynamics and is also seen in the experiments. For low disorder, there is an instability which causes the spontaneous growth of linelike domains at a critical field, also in accord with experiments. It is this unstable growth, which is highly sensitive to thermal noise, that is responsible for the small correlation between patterns under repeated cycling. The domain patterns, hysteresis loops, and memory properties of our simulated systems match remarkably well with the real experimental systems.

Disorder-induced microscopic magnetic memory

Using coherent x-ray speckle metrology, we have measured the influence of disorder on major loop return point memory (RPM) and complementary point memory (CPM) for a series of perpendicular anisotropy Co/Pt multilayer films. In the low disorder limit, the domain structures show no memory with field cycling--no RPM and no CPM. With increasing disorder, we observe the onset and the saturation of both the RPM and the CPM. These results provide the first direct ensemble-sensitive experimental study of the effects of varying disorder on microscopic magnetic memory and are compared against the predictions of existing theories.