Equilibrium Spin Configurations Research Articles

Низкотемпературный антиферромагнетизм в модели Изинга с конкурирующими взаимодействиями

The paper presents a numerical analysis of equilibrium state and spin configuration of square lattice Ising model with competing interaction. The most detailed description is given for case of ferromagnetic interaction of the first-order neighbours and antiferromagnetic coupling of the second-order neighbours. The numerical method is based on Metropolis algorithm. It uses 128×128 lattice with periodic boundary conditions. At first, the simulation results show that the system is in saturation state at low temperatures, and it turns into paramagnetic state at the Curie point. The competing second-order interaction makes possible the domain structure realization. This state is metastable, because its energy is higher than saturation energy. The domains are small at low temperature, and their size increases when temperature is growing until the single domain occupies the whole simulation area. In addition, the antiferromagnetic coupling of the second-order neighbours reduces the Curie temperature of the system. If it is large enough, the lattice has no saturation state. It turns directly from the domain state into paramagnetic phase. There are no extra phases when the system is antiferromagnetic in main order, and only the Neel temperature shift realizes here.

Introduction to antiferromagnetic magnons

The elementary spin excitations in strongly magnetic materials are collective spin deviations, or spin waves, whose quanta are called magnons. Interest in the experimental and theoretical investigation of magnons attracted many groups worldwide about 4–6 decades ago and then waned for some time. In recent years, with the advent of the field of spintronics, the area of magnonics has gained renewed attention. New phenomena have been discovered experimentally, and others have been predicted theoretically. In this tutorial, we briefly review the basic concepts of magnons in antiferromagnetic (AF) materials. Initially, we present a semiclassical view of the equilibrium spin configurations and of the antiferromagnetic resonance in AF materials with two types of magnetic anisotropy, easy-axis and easy-plane. Then, we present a quantum theory of magnons for these materials and apply the results to two important AF insulators, MnF2 and NiO. Finally, we introduce the concept of antiferromagnetic magnonic spin current that plays a key role in several phenomena in antiferromagnetic spintronics.

Slow spin dynamics and self-sustained clusters in sparsely connected systems.

To identify emerging microscopic structures in low-temperature spin glasses, we study self-sustained clusters (SSC) in spin models defined on sparse random graphs. A message-passing algorithm is developed to determine the probability of individual spins to belong to SSC. We then compare the predicted SSC associations with the dynamical properties of spins obtained from numerical simulations and show that SSC association identifies individual slow-evolving spins. Studies of Erdos-Renyi (ER) and random regular (RR) graphs show that spins belonging to SSC are more stable with respect to spin-flip fluctuations, as suggested by the analysis of fully connected models. Further analyses show that SSC association outperforms local fields in predicting the spin dynamics, specifically the group of slow- and fast-evolving spins in RR graphs, for a wide temperature range close to the spin-glass transition. This insight gives rise to a powerful approach for predicting individual spin dynamics from a single snapshot of an equilibrium spin configuration, namely from limited static information. It also implies that single-sample SSC association carries more information than local fields in describing the state of individual spins, when little information can be extracted from the system's topology.

Intrinsic spin-orbit torque in an antiferromagnet with a weakly noncollinear spin configuration

An antiferromagnet is a promising material for spin-orbit torque generation. Earlier studies of the spin-orbit torque in an antiferromagnet are limited to collinear spin configurations. We calculate the spin-orbit torque in an antiferromagnet whose spin ordering is weakly noncollinear. Such noncollinearity may be induced spontaneously during the magnetization dynamics even when the equilibrium spin configuration is perfectly collinear. It is shown that deviation from perfect collinearity can modify properties of the spin-orbit torque since noncollinearity generates extra Berry phase contributions to the spin-orbit torque, which are forbidden for collinear spin configurations. In sufficiently clean antiferromagnets, this modification can be significant. We estimate this effect to be of relevance for fast antiferromagnetic domain wall motion.

A quantum simulation approach for a three-dimensional Ising spin model—Comparison to mean field theory

We use a recently developed quantum simulation approach to study the properties of a three-dimensional Ising model consisting of S = 1/2 quantum spins localized at the sites of a simple cubic lattice. We assume nearest-neighbor interaction between spins with an exchange interaction that can be either ferromagnetic or antiferromagnetic. It is found that the computational method quickly converges towards the expected equilibrium spin configurations. The resulting spontaneous magnetization curves corresponding to the two types of magnetic interactions under consideration were found to be almost identical to the ones obtained via quantum mean field theory at all temperatures. The derived total energies, total free energies, magnetic entropies and specific heats per mole of spins show no sizeable differences from known theoretical values. Furthermore, the results of the simulations for two different 3D Ising systems containing 4×4×4 and 20×20×20 spins localized at the sites of a simple cubic lattice were found to be almost identical to each other. This finding suggests that the self-consistent algorithm approach of the current simulation method allows one to obtain the physical bulk properties of a large magnetic system by relying on simulations of a much smaller spin system sample. Therefore, the method presently considered appears to be not only very accurate as gauged by comparison to mean field theory, but also able to greatly increase the speed of simulations.

Spectrum of noncollinear metastable configurations of a finite-size discrete planar spin chain with a collinear ferromagnetic ground state

A theoretical method is used to find, in an accurate and systematic way, all the possible equilibrium magnetic states of a finite-size one-dimensional chain of classical planar spins, interacting via ferromagnetic nearest-neighbor symmetric exchange and subjected to a uniaxial anisotropy which favors the orientation of the spins along an easy axis within the plane perpendicular to the chain. For a chain with $N$ spins, the energy is a function of $N$ absolute orientation angles $({\ensuremath{\theta}}_{i}$, $i=1,...,N)$ and, owing to the absence of energy competition, the ground state is collinear ferromagnetic and doubly degenerate (either ${\ensuremath{\theta}}_{i}=0$ $\ensuremath{\forall}i$, or ${\ensuremath{\theta}}_{i}=\ensuremath{\pi}$ $\ensuremath{\forall}i$). The ground state corresponds to a global minimum of the energy in the $N$-dimensional space, while metastable states correspond to local minima. As a function of $N$, we have numerically determined the minimal value ${\ensuremath{\gamma}}_{\mathrm{min}}$ to be assumed by the ratio $\ensuremath{\gamma}$ between the anisotropy and the exchange constant for the first noncollinear metastable state to appear. For $N\ensuremath{\le}13$, all the noncollinear equilibrium spin configurations of the system were calculated with great accuracy at two different values, $\ensuremath{\gamma}=0.18$ and 0.21. A stability analysis was subsequently performed, showing that metastable noncollinear states can appear in the model even in the absence of energy competition. The various kinds of metastable states can be classified, in order of increasing energy, on the basis of the increasing number of $\ensuremath{\pi}$ walls (including both domains and antidomains) which can be inserted into the discrete finite chain with open ends when $N$ and/or $\ensuremath{\gamma}$ are increased. The effects of open boundary conditions and finite size on the Peierls-Nabarro barrier (i.e., the difference between the energy of an unstable onsite-centered domain wall and the energy of a stable intersite-centered domain wall) were also investigated.

Critical behavior of the geometrical spin clusters and interfaces in the two-dimensionalthermalized bond Ising model

The fractal dimensions and the percolation exponents of geometrical spin clusters of likesign at criticality are obtained numerically for an Ising model with temperaturedependent annealed bond dilution, also known as the thermalized bond Ising model(TBIM), in two dimensions. For this purpose, a modified Wolff single-clusterMonte Carlo simulation is used to generate equilibrium spin configurations onsquare lattices in the critical region. A tie-breaking rule is employed to identifynonintersecting spin cluster boundaries along the edges of the dual lattice. Thevalues obtained for the fractal dimensions of the spanning geometrical clustersDc, and theirinterfaces DI, are in perfect agreement with those reported for the standard two-dimensional ferromagneticIsing model. Furthermore, the variance of the winding angles results in a diffusivityκ = 3 for the two-dimensional thermalized bond Ising model, thus placing it in the universalityclass of the regular Ising model. A finite-size scaling analysis of the largest geometricalclusters results in a reliable estimation of the critical percolation exponents forthe geometrical clusters in the limit of an infinite lattice size. The percolationexponents thus obtained are also found to be consistent with those reported for theregular Ising model. These consistencies are explained in terms of the Fisherrenormalization relations, which express the thermodynamic critical exponentsof systems with annealed bond dilution in terms of those of the regular modelsystem.

A statistical-mechanical study of evolution of robustness: An approach from two-temperature models

Recently we proposed a simple statistical-mechanical model of interacting spins for studying evolutional process in biological system. In our model, a configuration of spins and their interactions J represent a phenotype and genotype, respectively. The phenotype dynamics are given by a stochastic process with temperature TS under a fixed Hamiltonian with J. Meanwhile, the evolution of J is also stochastic with temperature TJ and follows mutations introduced into J and selection based on a fitness given by equilibrium spin configurations of a given set of target spins. Using Monte Carlo simulations, it is found that frustration around the target spins is strongly suppressed for the interactions J evolved in the intermediate TS temperature region. The evolved J's give the funnel-like dynamics and show robustness to mutations. This model can be regarded as an extension of partial annealing model, in which the interactions between spins are dynamical variables with characteristic time scale widely separated from that of the spins. We also discuss the partially annealed Sherrington-Kirkpatrick model by particularly paying attention to the frustration.

Funnel Landscape and Mutational Robustness as a Result of Evolution under Thermal Noise

Using a statistical-mechanical model of spins, the evolution of phenotype dynamics is studied. Configurations of spins and their interaction J represent the phenotype and genotype, respectively. The fitness for selection of J is given by the equilibrium spin configurations determined by a Hamiltonian with J under thermal noise. The genotype J evolves through mutational changes under selection pressure to raise its fitness value. From Monte Carlo simulations we find that the frustration around the target spins disappears for J evolved under temperature beyond a certain threshold. The evolved Js give the funnel-like dynamics, which is robust to noise and also to mutation.

Surface spin dynamics of antiferromagnetically coupled frustrated triangular films

Results are presented for spin-wave dispersions in geometrically frustrated stacked triangular antiferromagnets with a thin film or semi-infinite geometry having either zero, easy-plane, or easy-axis anisotropy. Surface effects on the equilibrium spin configurations and excitation spectrum are investigated for the case of antiferromagnetically coupled films, serving to extend previous results on ferromagnetically coupled layers [E. Meloche et al., Phys. Rev. B 74, 094424 (2006)]. An operator equation of motion formalism is applied to systems which are quasi-one and quasi-two dimensional in character. In contrast to the case of ferromagnetically coupled films the new results show surface modes that are well separated in frequency from bulk excitations. Magnetic excitations in thin films with an even or an odd number of layers show qualitatively different behavior. These results are relevant for a wide variety of stacked triangular antiferromagnetics materials.

Ferromagnetic nanorings

Ferromagnetic metal rings of nanometre range widths and thicknesses exhibitfundamentally new spin states, switching behaviour and spin dynamics, whichcan be precisely controlled via geometry, material composition and applied field.Following the discovery of the ‘onion state’, which mediates the switching toand between vortex states, a range of fascinating phenomena has been found inthese structures. In this overview of our work on ring elements, we first show howthe geometric parameters of ring elements determine the exact equilibrium spinconfiguration of the domain walls of rings in the onion state, and we show howsuch behaviour can be understood as the result of the competition between theexchange and magnetostatic energy terms. Electron transport provides an extremelysensitive probe of the presence, spatial location and motion of domain walls, whichdetermine the magnetic state in individual rings, while magneto-optical measurementswith high spatial resolution can be used to probe the switching behaviour of ringstructures with very high sensitivity. We illustrate how the ring geometry hasbeen used for the study of a wide variety of magnetic phenomena, including thedisplacement of domain walls by electric currents, magnetoresistance, the strength of thepinning potential introduced by nanometre size constrictions, the effect of thermalexcitations on the equilibrium state and the stochastic nature of switching events.

Spin-flop transition in uniaxial antiferromagnets: Magnetic phases, reorientation effects, and multidomain states

The classical spin-flop is the field-driven first-order reorientation transition in easy-axis antiferromagnets. A comprehensive phenomenological theory of easy-axis antiferromagnets displaying spin-flops is developed. It is shown how the hierarchy of magnetic coupling strengths in these antiferromagnets causes a strongly pronounced two-scale character in their magnetic phase structure. In contrast to the major part of the magnetic phase diagram, these antiferromagnets near the spin-flop region are described by an effective model akin to uniaxial ferromagnets. For a consistent theoretical description both higher-order anisotropy contributions and dipolar stray-fields have to be taken into account near the spin-flop. In particular, thermodynamically stable multidomain states exist in the spin-flop region, owing to the phase coexistence at this first-order transition. For this region, equilibrium spin-configurations and parameters of the multidomain states are derived as functions of the external magnetic field. The components of the magnetic susceptibility tensor are calculated for homogeneous and multidomain states in the vicinity of the spin-flop. The remarkable anomalies in these measurable quantities provide an efficient method to investigate magnetic states and to determine materials parameters in bulk and confined antiferromagnets, as well as in nanoscale synthetic antiferromagnets. The method is demonstrated for experimental data on the magnetic properties near the spin-flop region in the orthorhombic layered antiferromagnet (C_2H_5NH_3)_2CuCl_4.

Observation of spinor dynamics in optically trapped 87Rb Bose-Einstein condensates.

We measure spin mixing of F=1 and F=2 spinor condensates of 87Rb atoms confined in an optical trap. We determine the spin mixing time to be typically less than 600 ms and observe spin population oscillations. The equilibrium spin configuration in the F=1 manifold is measured for different magnetic fields and found to show ferromagnetic behavior for low field gradients. An F=2 condensate is created by microwave excitation from the F=1 manifold, and this spin-2 condensate is observed to decay exponentially with time constant 250 ms. Despite the short lifetime in the F=2 manifold, spin mixing of the condensate is observed within 50 ms.

Magnetic states and reorientation transitions in antiferromagnetic superlattices

Equilibrium spin configurations and their stability limits have been calculated for models of magnetic superlattices with a finite number of thin ferromagnetic layers coupled antiferromagnetically through spacers. Depending on values of applied magnetic field and uniaxial anisotropy, the system assumes collinear (antiferromagnetic, ferromagnetic, various ``ferrimagnetic'') phases, or spatially inhomogeneous (symmetric spin-flop phase and asymmetric, canted and twisted, phases) via series of field induced continuous and discontinuous transitions. Contrary to semi-infinite systems a surface phase transition, so-called ``surface spin flop,'' does not occur in the models with a finite number of layers. It is shown that ``discrete jumps'' observed in some Fe/Cr superlattices and interpreted as ``surface spin flop'' transition are first-order ``volume'' transitions between different canted phases. Depending on the system these collinear and canted phases can co-exist as metastable states in broad ranges of the magnetic fields, which may cause severe hysteresis. The results explain magnetization processes in recent experiments on antiferromagnetic Fe/Cr and Co/Ru superlattices.

Dynamic micromagnetic simulations of susceptibility spectra in thin films with nonuniform magnetization distributions

The high-frequency response of magnetic systems to a weak rf magnetic field can be characterized by the dynamic susceptibility tensor. The calculation of this quantity requires a precise knowledge of the magnetization configuration inside the sample. The aim of this paper is to demonstrate the ability of dynamic micromagnetic simulations for investigating the high-frequency susceptibility spectra of thin films with nonuniform magnetization distributions. As an illustrative example, the case of ferromagnetic films supporting a weak-stripe-domain structure is addressed. The zero-field susceptibility spectra over the frequency range 0.1-40 GHz are computed using a two-dimensional dynamic micromagnetic model. These spectra exhibit multiple resonances whose number and resonance frequencies depend on the equilibrium spin configuration and on the rf exciting field orientation. A modal analysis permits us to correlate these magnetic excitations with spin regions of the stripe domain structure. The influence of the uniaxial perpendicular anisotropy, the film thickness and an in-plane static magnetic field applied along the stripe direction on the main features of spectra (number of resonances, resonance frequencies, intensities, and linewidths) is then investigated. Finally, these theoretical results are discussed in light of available experimental data.

Effect of Cr concentration gradient on the intergranular exchange in CoCr(Ta,Pt) thin films

Intergranular exchange in CoCr-alloy polycrystalline thin films utilized for longitudinal recording media is analyzed in terms of details of their chemical composition. Two neighboring interacting grains are modeled by a linear “atomic” chain extending from the center of one grain to the center of another. The composition at each spin site determines the interspin exchange interaction. Thermal fluctuations are included via mean field theory and they yield spacial variations of the saturation magnetization and the crystal anisotropy that differ from the bulk properties. Equilibrium spin configurations for the linear chain are obtained for different Cr concentrations at the grain boundary and for different temperatures. Chromium segregation at the boundary is confirmed to be the mechanism for grain decoupling. An effective exchange parameter, assuming uniformly magnetized grains like those in large-scale micromagnetic simulations, is determined. This parameter decreases significantly with increasing temperature and increases slightly with relative anisotropy axis orientation.

Ferromagnetic resonance spectra in a weak stripe domain structure

The ferromagnetic resonance (FMR) spectra in magnetic films with a nonuniform magnetization configuration are known to exhibit multiple absorption peaks. In this paper, the case of ferromagnetic films supporting a weak stripe domain structure is addressed. A two-dimensional dynamic micromagnetic model is used to investigate the high-frequency response of such a magnetic structure. In a first step, the zero-field susceptibility spectra are computed. The existence of numerous resonances resulting from the excitation of surface and volume modes is predicted. The main features of spectra (number of resonances, resonance frequencies, intensities, and linewidths) strongly depend on the equilibrium spin configuration and on the rf exciting field orientation. In a second step, the susceptibility spectra in the presence of an in-plane static magnetic field applied along the stripe direction are studied in detail. The dispersion relation, frequency versus in-plane magnetic field, is computed for each magnetic excitation. These dispersion curves reveal possible mode couplings and interchange of mode characters with increasing magnetic field. The theoretical in-plane FMR spectra are then deduced from the frequency and magnetic field dependence of the dynamic susceptibility. These theoretical results are discussed in light of available experimental FMR data obtained on thin ferromagnetic films with a moderate perpendicular anisotropy.

Imaging precessional motion of the magnetization vector.

We report on imaging of three-dimensional precessional orbits of the magnetization vector in a magnetic field by means of a time-resolved vectorial Kerr experiment that measures all three components of the magnetization vector with picosecond resolution. Images of the precessional mode taken with submicrometer spatial resolution reveal that the dynamical excitation in this time regime roughly mirrors the symmetry of the underlying equilibrium spin configuration and that its propagation has a non-wavelike character. These results should form the basis for realistic models of the magnetization dynamics in a largely unexplored but technologically increasingly relevant time scale.

Magnetic excitations in a weak-stripe-domain structure: A 2D dynamic micromagnetic approach

The high-frequency susceptibility spectra of ferromagnetic films supporting a weak-stripe-domain structure are computed using a 2D dynamic micromagnetic model that we have developed. The existence of multiple resonances resulting from the excitation of surface and volume modes is predicted. The main features of spectra (number of resonances, resonance frequencies, intensities, and linewidths) strongly depend on the equilibrium spin configuration and on the rf exciting field orientation. These theoretical results are successfully compared with zero-field microwave permeability measurements.

Probing the ferromagnet/antiferromagnet interface with spin waves

A thin ferromagnetic film coupled to an antiferromagnet can exhibit exchange biasing where the hysteresis curve is shifted slightly. A explanation for this shift has been proposed recently by Koon where it was argued that a surface spin flop state in the antiferromagnet exists at the interface. We employ a technique which gives both the equilibrium spin configurations and the spin wave frequencies within one calculation. The spin wave frequencies of both the ferromagnet and the antiferromagnet are substantially influenced by the interface exchange coupling, with a dramatic dip in frequencies where the structure changes from a noncanted state to a canted state.