Phase diagram of ultrathin ferromagnetic films with perpendicular anisotropy

Magnetic ordering in strong anisotropic ferromagnetic bilayers with dipolar and antiferromagnetic interlayer exchange interactions: Monte Carlo approach

In this work, multilayer films consisting of two strong-anisotropic ferromagnetic layers antiferromagnetically coupled by a nonmagnetic spacer are studied by Monte Carlo simulations. The system is modeled by an Ising-based Hamiltonian that depends on both the intralayer exchange and dipolar constants and on the interlayer exchange constant (IEC). The ground state of the monolayers (null IEC) corresponds to alternate stripe domains with width h defined by the ratio between the exchange and dipolar constants (δ). The results show that IEC alters the energy balance that controls the stripe domain formation, leading to a ground state characterized by in-plane stripes out-plane antiferromagnetically coupled. When temperature increases two regimes are identified: an IEC-dominated regime where the orientational and positional orders are simultaneously lost in both layers, driving the system to the tetragonal liquid (TL) phase, and a dipolar-dominated one where signs of layers decoupling and the onset of positional disorder are observed. The last could be related with an intermediate nematic phase (NM). From the study of the nonequilibrium dynamics, the phase transitions to TL phase are characterized as continuous and those to the NM one as Kosterlitz-Thouless type. Also, for both layers the critical temperatures are the same and increase with IEC magnitude. Furthermore, the obtained critical exponents depend on the IEC values, which is indicative of a weak universality. For the dipolar-dominated regime, the decoupling between layers is also evidenced by the difference between their critical exponents.

Turing-like patterns in an asymmetric dynamic Ising model

To investigate novel aspects of pattern formation in spin systems, we use a mapping between reactive concentrations in a reaction-diffusion system and spin orientations in a dynamic multiple-spin Ising model. While pattern formation in Ising models always relies on infinite-range interactions, this mapping allows us to design a finite-range-interactions Ising model that can produce patterns observed in reaction-diffusion systems including Turing patterns with a tunable typical length scale. This model has asymmetric interactions and several spin types coexisting at a site. While we use the example of genetic regulation during embryogenesis to build our model, it can be used to study the behavior of other complex systems of interacting agents.

Evidence of Kosterlitz-Thouless phase transitions in the Ising model with dipolar interactions

The ferromagnetic bidimensional Ising model with dipolar interactions has been proposed to model ultrathin films with strong out-of-plane anisotropy. The phase diagram presents a rich phenomenology that includes low-temperature phases characterized by stripes of width n (h_{n}) and a high-temperature phase with domains of stripes with mutually perpendicular orientations, named tetragonal liquid (TL). The latter phase can be reached by two possible ways. One of them is the direct transition h_{n} to TL, and the other one is through an intermediate phase with orientational order but short-range positional disorder, named nematic phase (NM). The regions of the phase diagram where these transitions occur, as well as their character, remain an open question and are the object of the present work. In order to clarify this topic, intensive Monte Carlo simulations were performed by employing short-time dynamics as the main tool for studying the phase transition behavior. The dynamic evolution of the orientational order parameter and its moments are measured for selected values of the ratio between the ferromagnetic exchange and dipolar constants, called δ. The obtained results indicate that the intermediate NM phase is present for δ≥2 in narrow ranges of temperatures. Also, the results suggest that both transitions, i.e., h_{n}-NM and NM-TL, have a Kosterlitz-Thouless character. This type of topological transition is observed in continuous bidimensional models and have been proposed for discrete ones, as in the case of the present work.

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.

Critical exponents and scaling invariance in the absence of a critical point

The paramagnetic-to-ferromagnetic phase transition is classified as a critical phenomenon due to the power-law behaviour shown by thermodynamic observables when the Curie point is approached. Here we report the observation of such a behaviour over extraordinarily many decades of suitable scaling variables in ultrathin Fe films, for certain ranges of temperature T and applied field B. This despite the fact that the underlying critical point is practically unreachable because protected by a phase with a modulated domain structure, induced by the dipole–dipole interaction. The modulated structure has a well-defined spatial period and is realized in a portion of the (T, B) plane that extends above the putative critical temperature, where thermodynamic quantities do not display any singularity. Our results imply that scaling behaviour of macroscopic observables is compatible with an avoided critical point.

Thermodynamic observables develop power laws and singularities when approaching the Curie point of a ferromagnetic phase transition. Here, Saratz et al. demonstrate that topological excitations (that is, magnetic domains in Fe/Cu(100) films that even persist above the Curie point) remove those singularities compatibly with an avoided critical point.

Nematic phase in the J(1)-J(2) square-lattice Ising model in an external field

The J(1)-J(2) Ising model in the square lattice in the presence of an external field is studied by two approaches: the cluster variation method (CVM) and Monte Carlo simulations. The use of the CVM in the square approximation leads to the presence of a new equilibrium phase, not previously reported for this model: an Ising-nematic phase, which shows orientational order but not positional order, between the known stripes and disordered phases. Suitable order parameters are defined, and the phase diagram of the model is obtained. Monte Carlo simulations are in qualitative agreement with the CVM results, giving support to the presence of the new Ising-nematic phase. Phase diagrams in the temperature-external field plane are obtained for selected values of the parameter κ=J(2)/|J(1)| which measures the relative strength of the competing interactions. From the CVM in the square approximation we obtain a line of second order transitions between the disordered and nematic phases, while the nematic-stripes phase transitions are found to be of first order. The Monte Carlo results suggest a line of second order nematic-disordered phase transitions in agreement with the CVM results. Regarding the stripes-nematic transitions, the present Monte Carlo results are not precise enough to reach definite conclusions about the nature of the transitions.

Real-time observation of domain fluctuations in a two-dimensional magnetic model system

Domain patterns of perpendicularly magnetized ultra-thin ferromagnetic films are often determined by the competition of the short range but strong exchange interaction favouring ferromagnetic alignment of magnetic moments and the long range but weak antiferromagnetic dipolar interaction. Detailed phase diagrams of the resulting stripe domain patterns have been evaluated in recent years; however, the domain fluctuations in these pattern forming systems have not been studied in great detail so far. Here we show that domain fluctuations can be observed in ultra-thin two-dimensional ferromagnetic Fe/Ni/Cu(001) films with perpendicular magnetization in the stripe domain phase. Non-stroboscopic time-resolved threshold photoemission electron microscopy with high temporal resolution allows analysing the dynamic fingerprint of the topological excitations in the nematic domain phase. Furthermore, proliferation of domain ending defects in the vicinity of the spin reorientation transition is witnessed.

Nature of long-range order in stripe-forming systems with long-range repulsive interactions

We study two dimensional stripe forming systems with competing repulsive interactions decaying as r(-α). We derive an effective Hamiltonian with a short-range part and a generalized dipolar interaction which depends on the exponent α. An approximate map of this model to a known XY model with dipolar interactions allows us to conclude that, for α<2 long-range orientational order of stripes can exist in two dimensions, and establish the universality class of the models. When α≥2 no long-range order is possible, but a phase transition in the Kosterlitz-Thouless universality class is still present. These two different critical scenarios should be observed in experimentally relevant two dimensional systems like electronic liquids (α=1) and dipolar magnetic films (α=3). Results from Langevin simulations of Coulomb and dipolar systems give support to the theoretical results.

Nematic phase in stripe-forming systems within the self-consistent screening approximation

We show that in order to describe the isotropic-nematic transition in stripe-forming systems with isotropic competing interactions of the Brazovskii class it is necessary to consider the next to leading order in a 1/N approximation for the effective Hamiltonian. This can be conveniently accomplished within the self-consistent screening approximation. We solve the relevant equations and show that the self-energy in this approximation is able to generate the essential wave vector dependence to account for the anisotropic character of a two-point correlation function characteristic of a nematic phase.

Dipolar-energy-activated magnetic domain pattern transformation driven by thermal fluctuations

Two-dimensional ferromagnetic layers can serve as a playground for the study of basic physical properties of various pattern forming systems by virtue of their tuneable magnetic properties. Here we use threshold photoemission magnetic circular dichroism in combination with photoemission electron microscopy to investigate ultra-thin ferromagnetic Fe/Ni/Cu(001) films in the stripe domain phase near the spin reorientation transition as a function of film thickness, temperature and effective anisotropy. Here we report a metastable domain state with domain width larger than the thermodynamically stable one as a result of a rapid reduction of the anisotropy. The transformation into the equilibrium state takes place via the propagation of a transition front, which originates from defined steps in the film thickness.

Stripe-tetragonal phase transition in the two-dimensional Ising model with dipole interactions: partition function zeros approach

We have performed multicanonical simulations to study the critical behavior of the two-dimensional Ising model with dipole interactions. This study concerns the thermodynamic phase transitions in the range of the interaction δ where the phase characterized by striped configurations of width h = 1 is observed. Controversial results obtained from local update algorithms have been reported for this region, including the claimed existence of a second-order phase transition line that becomes first order above a tricritical point located somewhere between δ = 0.85 and 1. Our analysis relies on the complex partition function zeros obtained with high statistics from multicanonical simulations. Finite size scaling relations for the leading partition function zeros yield critical exponents ν that are clearly consistent with a single second-order phase transition line, thus excluding such a tricritical point in that region of the phase diagram. This conclusion is further supported by analysis of the specific heat and susceptibility of the orientational order parameter.

Experimental phase diagram of perpendicularly magnetized ultrathin ferromagnetic films

We image the domain patterns in perpendicularly magnetized ultrathin Fe films on Cu(100) as a function of the temperature T and the applied magnetic field H. Between the low-field stripe phase and the high-field uniform phase we find a bubble phase, consisting of reversed circular domains in a homogeneous background. The curvature of the transition lines in the H-T parameter space is in contrast to the general expectations. The pattern transformations show yet undetected scaling properties.

Interplay between coarsening and nucleation in an Ising model with dipolar interactions

We study the dynamical behavior of a square lattice Ising model with exchange and dipolar interactions by means of Monte Carlo simulations. After a sudden quench to low temperatures, we find that the system may undergo a coarsening process where stripe phases with different orientations compete, or alternatively it can relax initially to a metastable nematic phase and then decay to the equilibrium stripe phase through nucleation. We measure the distribution of equilibration times for both processes and compute their relative probability of occurrence as a function of temperature and system size. This peculiar relaxation mechanism is due to the strong metastability of the nematic phase, which goes deep into the low-temperature stripe phase. We also measure quasiequilibrium autocorrelations in a wide range of temperatures. They show a distinct decay to a plateau that we identify as due to a finite fraction of frozen spins in the nematic phase. We find indications that the plateau is a finite-size effect. Relaxation times as a function of temperature in the metastable region show super-Arrhenius behavior, suggesting a possible glassy behavior of the system at low temperatures.

Competing interactions, the renormalization group, and the isotropic-nematic phase transition

We discuss 2D systems with Ising symmetry and competing interactions at different scales. In the framework of the renormalization group, we study the effect of relevant quartic interactions. In addition to the usual constant interaction term, we analyze the effect of quadrupole interactions in the self-consistent Hartree approximation. We show that in the case of a repulsive quadrupole interaction, there is a first-order phase transition to a stripe phase in agreement with the well-known Brazovskii result. However, in the case of attractive quadrupole interactions there is an isotropic-nematic second-order transition with higher critical temperature.

Magnetic bubble domain phase at the spin reorientation transition of ultrathin Fe/Ni/Cu(001) film

Magnetic domain phases of ultrathin Fe/Ni/Cu(001) are studied using photoemission electron microscopy at the spin reorientation transition (SRT). We observe a new magnetic phase of bubble domains within a narrow SRT region after applying a nearly in-plane magnetic field pulse to the sample. By applying the magnetic field pulse along different directions, we find that the bubble domain phase exists only if the magnetic field direction is less than approximately 10 degrees relative to the sample surface. A temperature dependent measurement shows that the bubble domain phase becomes unstable above 370 K.

Asymmetric domain nucleation and unusual magnetization reversal in ultrathin Co films with perpendicular anisotropy

We report unexpected phenomena during magnetization reversal in ultrathin Co films and Co/Pt multilayers with perpendicular anisotropy. Using magneto-optical Kerr microscopy and magnetic force microscopy we have observed asymmetrical nucleation centers where the reversal begins for one direction of the field only and is characterized by an acute asymmetry of domain-wall mobility. We have also observed magnetic domains with a continuously varying average magnetization, which can be explained in terms of the coexistence of three magnetic phases: up, down, and striped.

Hysteresis and noise from electronic nematicity in high-temperature superconductors

An electron nematic is a translationally invariant state which spontaneously breaks the discrete rotational symmetry of a host crystal. In a clean square lattice, the electron nematic has two preferred orientations, while dopant disorder favors one or the other orientations locally. In this way, the electron nematic in a host crystal maps to the random field Ising model. Since the electron nematic has anisotropic conductivity, we associate each Ising configuration with a resistor network and use what is known about the random field Ising model to predict new ways to test for local electronic nematic order (nematicity) using noise and hysteresis. In particular, we have uncovered a remarkably robust linear relation between the orientational order and the resistance anisotropy which holds over a wide range of circumstances.

Observation of stripe mobility in a dipolar frustrated ferromagnet

We have discovered two novel aspects of the stripe-domain to paramagnetic transition in perpendicularly magnetized Fe films on Cu(100). First, the width of the stripes carrying oppositely oriented spins decreases, close to the transition temperature, with a power law. Second, in a small temperature interval close to the transition temperature, the stripes--which form stationary patterns at low temperatures--become mobile. Various theoretical works have predicted stripe mobility in similar frustrated systems but no direct proof of this phenomenon has been reported so far.

Magnetic stripe domains in coupled magnetic sandwiches

Magnetic stripe domains in the spin reorientation transition region are investigated in (Fe/Ni)/Cu(001) and Co/Cu/(Fe/Ni)/Cu(001) using photoemission electron microscopy. For (Fe/Ni)/Cu(001), the stripe domain width decreases exponentially as the Fe/Ni film approaches the spin reorientation transition point. For Co/Cu/(Fe/Ni)/Cu(001), the Fe/Ni stripe orientation is aligned with the Co in-plane magnetization, and the stripe domain width decreases exponentially with increasing the interlayer coupling between the Fe/Ni and Co films. By considering magnetic stripes within an in-plane magnetic field, we reveal a universal dependence of the stripe domain width on the magnetic anisotropy and on the interlayer coupling.

An inverse transition of magnetic domain patterns in ultrathin films

Inverse freezing and inverse melting are processes where a more symmetric phase is found at lower temperatures than at higher temperatures. Such inverse transitions are very rare. Here we report the existence of an inverse transition effect in ultrathin Fe films that are magnetized perpendicular to the film plane. The magnetization of these films is not uniform, but instead manifests itself as stripe domains with opposite perpendicular magnetization. Predictions relating to the disordering of this striped ground state in the limit of monolayer film thicknesses are controversial. Mean-field arguments predict a continuous reduction of the stripe width when the temperature is increased; other studies suggest that topological defects, such as dislocations and disclinations, might penetrate the system and induce geometrical phase transitions. We find, from scanning electron microscopy imaging, that when the temperature is increased, the low-temperature stripe domain structure transforms into a more symmetric, labyrinthine structure. However, at even higher temperatures and before the loss of magnetic order, a re-occurrence of the less symmetric stripe phase is found. Despite the widespread theoretical and experimental work on striped systems, this phase sequence and the microscopic instabilities driving it have not been observed before.

Computer simulations of a two-dimensional system with competing interactions

The results and methodology of large scale computer simulations of the two-dimensional dipolar Ising model with long-range interactions are reported. Systems as large as 117,649 particles were studied to elucidate the elementary excitations and phase diagram of two-dimensional systems, such as Langmuir monolayers, thin garnet films, and adsorbed films on solid surfaces, which spontaneously form patterns of stripes, bubbles, and intermediately shaped domains. The challenging numerical investigations of large scale systems with long-range interactions at low temperatures were made possible by combining the fast multipole method and a non-Metropolis Monte Carlo sampling technique. Our simulations provide evidence that, at sufficiently high ratios of the repulsive to the attractive coupling constant for the model, twofold stripe order in the systems of interest is lost through a defect-mediated mechanism. Heat capacity data and the excitations observed in our simulations as the system disorders indicate that it is most likely an instance of a Kosterlitz-Thouless phase transition. The results from simulations with and without external field are in excellent agreement with the predictions of an analytic scaling theory [A. D. Stoycheva and S. J. Singer, Phys. Rev. E 64, 016118 (2001)], confirming the phase diagram furnished by the analytic model. The scaling theory suggests that, under certain conditions, defect-mediated stripe melting may be supplanted by Ising like disordering within stripes for small repulsion strength. A qualitative discussion of a model that supports both disordering mechanisms is presented.

Topological textures in a ferromagnet-superconductor bilayer

The homogeneous state of a ferromagnet-superconductor bilayer (FSB) with the magnetization perpendicular to the layer can be unstable with respect to the formation of vortices in the superconducting layer. The developing topological instability in the FSB leads to formation of domains in which the direction of the magnetization in the magnetic film and the direction of vorticity in the superconducting film alternate together. This is a new, combined topological structure, which does not appear in isolated layers. Equilibrium domains can appear in the FSB even if the single magnetic film is in a monodomain state.

Stripe melting in a two-dimensional system with competing interactions

A model with competing short-ranged attractions and long-ranged repulsions that describes self-organized patterns in systems like Langmuir monolayers, magnetic films, and adsorbed monolayers is studied using numerical simulations and analytic theory. Simulations provide strong evidence confirming that the stripe phase order is destroyed in a defect unbinding transition. Large scale computer simulations are in agreement with an analytic scaling theory, which also predicts an eventual crossover from defect-mediated stripe melting to a spin-disordering (or particle-mixing) mechanism with decreasing repulsion strength.

Two-step disordering of perpendicularly magnetized ultrathin films

We have imaged the stripe domain structure of perpendicularly magnetized fcc ultrathin Fe films grown on Cu(100). The stripe phase has a strong local orientational order and sustains the two kinds of fluctuations predicted by Abanov et al. [Phys. Rev. B 51, 1023 (1995)]: meandering and dislocations. Before reaching the Curie temperature, the stripes transform into a new and so far unobserved domain structure, characterized by domains with predominantly square corners. We argue that this phase is the tetragonal liquid phase proposed by Abanov et al. to separate the stripe phase from the paramagnetic phase. This two-step disordering is reminiscent of a two-dimensional melting process.