Domain structures in ultrathin magnetic films

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.

Dielectric Modulation of Two-Dimensional Dipolar Materials

Spontaneous pattern formation plays an important role in a wide variety of natural phenomena and materials systems. A key ingredient for the occurrence of modulated phases is the presence of competing interactions, generally of different physical origins. We demonstrate that in dipolar films, a prototypical system for pattern formation, patterns can be induced by dielectric effects alone. A rich phase diagram arises, where striped and circular morphologies emerge with geometric properties that can be controlled through variation of particle shape and substrate permittivity or permeability. These effects are particularly enhanced by metamaterial substrates.

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.

Phase transitions and multicritical behavior in the Ising model with dipolar interactions

In this work, the phase diagram of the ferromagnetic Ising model with dipole interactions is revisited with the aim of determining the nature of the phase transition between stripe-ordered phases with width n (h_{n}) and tetragonal liquid (TL) phases. Extensive Monte Carlo simulations are performed in order to study the short-time dynamic behavior of the observables for selected values of the ratio between the ferromagnetic exchange and dipolar constants δ. The obtained results indicate that the h_{1}-TL phase transition line is continuous up to δ=1.2585, while for the h_{2}-TL line a weak first-order character is found in the interval 1.2585≤δ≤1.36 and becomes continuous for 1.37≤δ≤1.9. This result suggests the existence of a tricritical point close to δ=1.37. When it is appropriate, the complete set of critical exponents is obtained, and in all the studied cases they depend on δ but do not belong to the Ising universality class. Furthermore, short-time dynamic studies reveal that at the point where the mentioned lines meet the h_{1}-h_{2} line, i.e., at δ=1.2585, the critical phase corresponding to the h_{1}-TL transition coexists with the h_{2} phase.

Reconstructing interaction potentials in thin films from real-space images

We demonstrate that an inverse Monte Carlo approach allows one to reconstruct effective interaction potentials from real-space images. The method is exemplified on monomolecular ethanol-water films imaged with scanning force microscopy, which provides the spatial distribution of the molecules. Direct Monte Carlo simulations with the reconstructed potential allow for obtaining characteristics of the system which are unavailable in the experiment, such as the heat capacity of the monomolecularly thin film, and for a prediction of the critical temperature of the demixing transition.

Phase transitions and critical phenomena in the two-dimensional Ising model with dipole interactions: A short-time dynamics study

The ferromagnetic Ising model with antiferromagnetic dipole interactions is investigated by means of Monte Carlo simulations, focusing on the characterization of the phase transitions between the tetragonal liquid and stripe of width h phases. The dynamic evolution of the physical observables is analyzed within the short-time regime for 0.5≤δ≤1.3, where δ is the ratio between the short-range exchange and the long-range dipole interaction constants. The obtained results for the interval 0.5≤δ≤1.2 indicate that the phase transition line between the h=1 stripe and tetragonal liquid phases is continuous. This finding contributes to clarifying the controversy about the order of this transition. This controversy arises from the difficulties introduced in the simulations due to the presence of long-range dipole interactions, such as an important increase in the simulation times that limits the system size used, strong finite size effects, as well as to the existence of multiple metastable states at low temperatures. The study of the short-time dynamics of the model allows us to avoid these hindrances. Moreover, due to the fact that the finite-size effects do not significantly affect the power-law behavior exhibited in the observables within the short-time regime, the results could be attributed to those corresponding to the thermodynamic limit. As a consequence of this, a careful characterization of the critical behavior for the whole transition line is performed by giving the complete set of critical exponents.

Nanophase separation in monomolecularly thin water-ethanol films controlled by graphene

Control over nanoscale patterning of ultrathin molecular films plays an important role both in natural as well as artificial nanosystems. Here we report on nanophase separated patterns of water and ethanol within monomolecularly thin films confined between the cleavage plane of mica and single or a few layers of graphene. Employing scanning force microscopy of the graphene layers conforming to the molecular films we quantify the patterns using the ethanol-water cross correlation and the autocorrelation of domain wall directions. They reveal that lateral pattern dimensions grow and the domain walls stiffen upon increasing the thickness of the graphene multilayers. We attribute the control of the patterns through the graphene layers to the competition between the mechanical deformation energy of the graphene sheets and the electrostatic repulsion of dipoles normal to the interface. The latter results from charge transfer between graphene and the molecules confined between mica and graphene.

Vortex polarization dynamics in a square magnetic nanodot

Langevin simulations at finite temperature of two-dimensional magnetic nanodots were performed using the Landau-Lifshitz equation with exchange and dipolar interactions. In a wide range of temperatures, the dynamics of square samples with one central vortex was studied, focusing on the out-of-plane magnetic component at the vortex-core. This vortex-core undergoes polarization sign reversals in a thermally activated process. In the intervals between polarization flips, the out-of-plane spin components at the vortex-core show oscillations with identifiable frequencies connected with certain eigenfrequencies of the system associated with polarity active modes. The vortex-core positions were also monitored.

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.

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.

Stress induced stripe formation in Pd/W(110)

A stress-induced stripe phase of submonolayer Pd on W(110) is observed by low-energy electron microscopy. The temperature dependence of the pattern is explained by the change both in the boundary free energy and elastic relaxation energy due to the increasing boundary width. The stripes are shown to disorder when the correlation length of the condensed phase becomes comparable to its period, while the condensate to lattice-gas transition takes place at a higher temperature, as revealed by low-energy electron diffraction.

Microphase separation in two-dimensional systems with competing interactions

The formation of clusters in condition of thermodynamic equilibrium can be easily observed both in two and three dimensions. In two dimensions relevant cases include pattern formation in Langmuir monolayers and ferrofluids, while in three dimensions cluster phases have been observed in colloids and in protein solutions. We have analyzed the problem within the scenario of competing interactions: typically, a short-range attractive interaction against a long-range repulsive one. This simplified approach is suggested by the fact that the forces, governing self-organization, act on a length scale which is larger than the molecular size; as a consequence many specific details of the molecules of interest are not necessary for studying the general features of microphases. We have tackled the microphase formation by simulations in bidimensional fluids, exploiting the parallel tempering scheme. In particular, we have analyzed the density range in which the particles arrange in circular domains (droplets), and the temperature range in which the system goes from microphases to the homogeneous fluid phase. As the density increases, the droplet size increases as well, but above a certain density the morphology changes and stripes are formed. Moreover at low density, we observe the formation of a liquidlike phase of disordered droplets; at higher densities, instead, the droplets tend to arrange onto a triangular superlattice. Such a change affects the features of the static structure factor, which gives well defined signatures of the microphase morphology. In each case, the specific heat exhibits a peak close to the transition from microphases to the homogeneous fluid phase, which is due to the breaking up of the clusters. The saturation of the height of the specific heat peak, with the increasing of the system size, suggests the possibility of a Kosterlitz-Thouless transition.

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.

Molecular ordering and phase transitions in alkanol monolayers at the water–hexane interface

The interface between bulk water and bulk hexane solutions of n-alkanols (H(CH(2))(m)OH, where m=20, 22, 24, or 30) is studied with x-ray reflectivity, x-ray off-specular diffuse scattering, and interfacial tension measurements. The alkanols adsorb to the interface to form a monolayer. The highest density, lowest temperature monolayers contain alkanol molecules with progressive disordering of the chain from the -CH(2)OH to the -CH(3) group. In the terminal half of the chain that includes the -CH(3) group the chain density is similar to that observed in bulk liquid alkanes just above their freezing temperature. The density in the alkanol headgroup region is 10% greater than either bulk water or the ordered headgroup region found in alkanol monolayers at the water-vapor interface. We conjecture that this higher density is a result of water penetration into the headgroup region of the disordered monolayer. A ratio of 1:3 water to alkanol molecules is consistent with our data. We also place an upper limit of one hexane to five or six alkanol molecules mixed into the alkyl chain region of the monolayer. In contrast, H(CH(2))(30)OH at the water-vapor interface forms a close-packed, ordered phase of nearly rigid rods. Interfacial tension measurements as a function of temperature reveal a phase transition at the water-hexane interface with a significant change in interfacial excess entropy. This transition is between a low temperature interface that is nearly fully covered with alkanols to a higher temperature interface with a much lower density of alkanols. The transition for the shorter alkanols appears to be first order whereas the transition for the longer alkanols appears to be weakly first order or second order. The x-ray data are consistent with the presence of monolayer domains at the interface and determine the domain coverage (fraction of interface covered by alkanol domains) as a function of temperature. This temperature dependence is consistent with a theoretical model for a second order phase transition that accounts for the domain stabilization as a balance between line tension and long range dipole forces. Several aspects of our measurements indicate that the presence of domains represents the appearance of a spatially inhomogeneous phase rather than the coexistence of two homogeneous phases.

Magnetic phase transition in Co/Cu/Ni/Cu(100) and Co/Fe/Ni/Cu(100)

Magnetic phase transitions in coupled magnetic sandwiches of Cu/Co/Cu/Ni/Cu(100) and Cu/Co/Fe/Ni/Cu(100) are investigated by photoemission electron microscopy. Element-specific magnetic domains are taken at room temperature to reveal the critical thickness at which the magnetic phase transition occurs. The results show that a coupled magnetic sandwich undergoes three types of magnetic phase transitions depending on the two ferromagnetic films' thickness. A phase diagram is constructed and explained in the process of constructing Monte Carlo simulations, which corroborate the experimental results.

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.

Monte Carlo study of the three-dimensional Coulomb frustrated Ising ferromagnet

We have investigated, by Monte Carlo simulation, the phase diagram of a three-dimensional Ising model with nearest-neighbor ferromagnetic interactions and small, but long-range (Coulombic) antiferromagnetic interactions. We have developed an efficient cluster algorithm and used different lattice sizes and geometries, which allows us to obtain the main characteristics of the temperature-frustration phase diagram. Our finite-size scaling analysis confirms that the melting of the lamellar phases into the paramagnetic phase is driven first order by the fluctuations. Transitions between ordered phases with different modulation patterns are observed in some regions of the diagram, in agreement with a recent mean-field analysis.

Scaling theory for two-dimensional systems with competing interactions

We derive an analytic scaling theory for a two-dimensional system in which spontaneous patterns of stripes, bubbles, and intermediately shaped domains arise due to the competition of short-range attractions and long-range dipolar repulsions. The theory predicts temperature and domain-size scaling as a function of the relative repulsion strength eta, the ratio of the repulsive to the attractive coupling constant in the system's Hamiltonian. As eta decreases, the domain size explodes exponentially and the melting temperature for a system of ordered stripes increases. Our findings shed new light on the phase diagram and critical excitations for the dipolar Ising ferromagnet or lattice gas and their continuum analogs. We show that the features described by the scaling theory are insensitive to details like cutoffs for the dipolar interactions and, therefore, should be widely applicable. Our corresponding states analysis explains the experimentally observed stripe melting upon compression in a Langmuir monolayer. A phenomenological extension of the analytic scaling theory describes how the system's behavior is modified in the presence of magnetization or density fluctuations. Fluctuations are found to suppress domain size and the stripe melting temperature. In regimes where fluctuations are important, we predict that domain size will decrease with increasing temperature.

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.