Magnetic properties of 2D nano-islands II: Ising spin model with out-of-plane magnetic field

An Ising effective field theory model is presented to calculate the magnetic properties of 2D nano-islands on a nonmagnetic substrate, subject to an externally out-of-plane applied magnetic field. The system Hamiltonian contains nearest neighbor exchange interactions, single-atom magnetic anisotropies, and the Zeeman term. The calculations yield, in particular, the single site spin correlations, the magnetizations, and the isothermal susceptibilities, for the core and periphery domains of the nano-island. The choice of a spin S=1 for the atoms of the system permits the analysis of local spin fluctuations via the single site spin correlations. We investigate in this respect the effects due to the different magnetocrystalline anisotropies and reduced dimensionalities, for the core and periphery domains, and in particular the critical influence of the applied magnetic field. Detailed theoretical results are presented for the square and hexagonal lattice symmetries, with numerical applications for the 2D monolayer Co nano-islands on a Pt substrate. It is shown that the remarkable differences between the magnetic properties of the core and periphery domains in zero field are washed out when an out-of-plane field is applied. The applied field also provokes critical discontinuities for the spin correlations and magnetization reversals, for the core and periphery domains, which are especially evident for the hexagonal lattice nano-island in the range of fields of interest. The discontinuities and magnetization reversals occur over elementary temperature widths, and shift to lower temperatures with increasing field. The field-dependant isothermal susceptibilities show new features very different from those for the susceptibilities in zero field. The present Ising model does not show any blocking temperature transition to superparamagnetism.     

An effective-field theory study of hexagonal Ising nanowire: Thermal and magnetic properties

By means of the effective-field theory （EFT） with correlations, the thermodynamic and magnetic quantities （such as magnetization, susceptibility, internal energy, specific heat, free energy, hysteresis curves, and compensation behaviors） of the spin-l/2 hexagonal Ising nanowire （HIN） system with core/shell structure have been presented. The hysteresis curves are obtained for different values of the system parameters, in both ferromagnetic and antiferromagnetic cases. It has been shown that the system only undergoes a second-order phase transition. Moreover, from the thermal variations of the total magnetization, the five compensation types can be found under certain conditions, namely the Q-, R-, S-, P-, and N-types.     

An effective-field study of Ising metamagnet in an external field

The phase diagrams and magnetization curves of a two-sublattice Ising metamagnet at finite temperature with longitudinal crystal field H are investigated by the use of an effective-field theory (EFT) with correlations. In addition to the second-order transition lines, the first-order transition lines are also presented, since a method to calculate the Gibbs free energy numerically at finite temperature within EFT is found in this work. The results show that there is no fourth-order critical point or reentrant phenomenon in the phase diagrams given by using EFT as found by using mean-field theory (MFT).     

Ground-state magnetic properties of the spin-2 transverse Ising model

The ground-state magnetic properties of the spin-2 transverse Ising model with a longitudinal crystal field are studied within the framework of mean-field theory (MFT) and effective-field theory (EFT), respectively. The phase diagrams and magnetization curves are examined in detail. It is found that the system exhibits a tricritical behavior in the ground-state phase diagrams. Some interesting phenomena have been found, especially the first-order phase transition from one ordered phase to the other ordered phase, which is due to the high spin. The spin correlation has important effect on the magnetic properties of the system. We also find that the ground-state phase diagrams of the spin-2 transverse Ising model are very different from those of the spin-3/2 transverse Ising model.     

A quadrangular transverse Ising nanowire with an antiferromagnetic spin configuration

The phase diagrams and the temperature dependences of magnetizations in a transverse Ising nanowire with an antiferromagnetic spin configuration are investigated by the use of the effective-field theory with correlations (EFT) and the core–shell concept. Many characteristic and unexpected behaviors are found for them, especially for thermal variation of total magnetization m_T. The reentrant phenomenon induced by a transverse field in the core, the appearance of a compensation point, the non-monotonic variation with a compensation point, the reentrant phenomena with a compensation point and the existence of both a broad maximum and a compensation point have been found in the thermal variations of m_T.     

Verification of a new quantum simulation approach through its application to two-dimensional Ising lattices

A new quantum simulation approach has been applied in the present work to the two-dimensional (2D) ferromagnetic and antiferromagnetic Ising lattices to calculate their magnetic structures, magnetizations, free energies and specific heats in the absence of an external magnetic field. Surprisingly, no size effects could be observed in our simulations performed for the Ising lattices of different sizes. Most importantly, our calculated spontaneous thermally averaged spins for the two kinds of systems are exactly same as those evaluated with quantum mean field theory, and the magnetic structures simulated at all chosen temperatures are perfectly ferromagnetic or antiferromagnetic, verifying the correctness and applicability of our quantum model and computational algorithm. On the other hand, if the classical Monte Carlo (CMC) method is applied to the ferromagnetic 2D Ising lattice with S=1, it is able to generate correct magnetization well consistent with Onsager's theory; but in the case of S=1/2, the computational results of CMC are incomparable to those predicted with the quantum mean field theory, giving rise to very much reduced magnetization and considerably underestimated Curie temperature. The difficulty met by the CMC method is mainly caused by its improperly calculated exchange energy of the randomly selected spin in every simulation step, especially immediately below the transition temperature, where the thermal averages of spins are much less than 1/2, however they are assigned to ±1/2 by CMC to evaluate the exchange energies of the spins, such improper manipulation is obviously impossible to lead the code to converge to the right equilibrium states of the spin systems.     

Some characteristic properties of initial susceptibility in a Ising nanotube

Initial susceptibility and magnetization of a cylindrical nanotube described by the Ising model are investigated by the use of the effective field theory with correlations (EFT), since the phase diagrams of the system have been examined in the previous work of Kaneyoshi (2010) [8] using the two theoretical frameworks of the mean field theory and the EFT. The effects of the two exchange couplings at the surface shell and in the core to the initial susceptibility are clarified. Some characteristic phenomena are found in the thermal variations, depending on the ratios of the physical parameters in the surface shell and the core. It is also discussed whether the Neel hyperbola in the paramagnetic region is valid for a nano-scaled ferrimagnetic system.     

Magnetic Properties of Transverse Ising Model under a Time Oscillating Longitudinal Field

A transverse Ising spin system, in the presence of time-dependentlongitudinal field, is studied by the effective-field theory (EFT). Theeffective-field equations of motion of the average magnetization are givenfor the simple cubic lattice (Z = 6) and the honeycomb lattice (Z = 3).The Liapunov exponent λ is calculated for discussing the stability of the magnetization and it is used to determine the phase boundary. Thedynamic phase transition diagrams in h₀/ ZJ -Γ/ZJ plane and in h₀/ZJ-T/ZJ plane have been drawn, and there is no dynamical tricritical point on the dynamic phase transition boundary. The effect of the thermal fluctuations upon the dynamic phase boundary has been discussed.     

Magnetization and magnetic susceptibility of the Ising ferromagnetic/antiferromagnetic superlattice

A linear cluster mean-field approximation is used to study the magnetic properties of the Ising ferromagnetic/antiferromagnetic superlattice, which is composed of a spin-1/2 ferromagnetic monolayer and a spin-1 antiferromagnetic monolayer with a single-ion anisotropy alternatively. By using the transfer matrix method, we calculate the magnetization and the initial magnetic susceptibility as functions of temperature for different interlayer coupling, single-ion anisotropy. We summarize the changing behaviors of the spin structure in ferromagnetic and antiferromagnetic layers and the characteristics of the corresponding magnetic susceptibilities, give the transition temperature as a function of the interlayer exchange coupling for different single-ion anisotropy, and analyze the features of the magnetization and the magnetic susceptibility.     

Multicanonical sampling of the space of states of ℋ(2, <Emphasis Type="Italic">n</Emphasis>)-vector models

Problems of temperature behavior of specific heat are solved by the entropy simulation method for Ising models on a simple square lattice and a square spin ice (SSI) lattice with nearest neighbor interaction, models of hexagonal lattices with short-range (SR) dipole interaction, as well as with long-range (LR) dipole interaction and free boundary conditions, and models of spin quasilattices with finite interaction radius. It is established that systems of a finite number of Ising spins with LR dipole interaction can have unusual thermodynamic properties characterized by several specific-heat peaks in the absence of an external magnetic field. For a parallel multicanonical sampling method, optimal schemes are found empirically for partitioning the space of states into energy bands for Ising and SSI models, methods of concatenation and renormalization of histograms are discussed, and a flatness criterion of histograms is proposed. It is established that there is no phase transition in a model with nearest neighbor interaction on a hexagonal lattice, while the temperature behavior of specific heat exhibits singularity in the same model, in case of LR interaction. A spin quasilattice is found that exhibits a nonzero value of residual entropy.     

A Generalization of Mean Field Theory in a Cluster with Many Sites on the Ising Model from the Bogoliubov Inequality: Hexagonal Nanowire and Nanotube

A generalization of mean field theory in a cluster with many sites was obtained for the spin-1/2 Ising model from the Gibbs-Bogoliubov inequality. The expressions for the free energy and the magnetization were obtained. The generalization was applied in a structure of the nanowire and nanotube hexagonal lattices, for clusters of seven sites and six sites, respectively. The results for the magnetization, the free energy, the internal energy, the entropy, the specific heat, and the critical frontiers were obtained. The critical temperature and the compensation temperature in a cylindrical Ising nanowire are investigated, in order to clarify the distinction between the ferromagnetic and ferrimagnetic behaviors when the core-shell exchange coupling takes a different sign. The results were compared with other works.     

Duality Between Spin Networks and the 2D Ising Model

The goal of this paper is to exhibit a deep relation between the partition function of the Ising model on a planar trivalent graph and the generating series of the spin network evaluations on the same graph. We provide respectively a fermionic and a bosonic Gaussian integral formulation for each of these functions and we show that they are the inverse of each other (up to some explicit constants) by exhibiting a supersymmetry relating the two formulations. We investigate three aspects and applications of this duality. First, we propose higher order supersymmetric theories that couple the geometry of the spin networks to the Ising model and for which supersymmetric localization still holds. Secondly, after interpreting the generating function of spin network evaluations as the projection of a coherent state of loop quantum gravity onto the flat connection state, we find the probability distribution induced by that coherent state on the edge spins and study its stationary phase approximation. It is found that the stationary points correspond to the critical values of the couplings of the 2D Ising model, at least for isoradial graphs. Third, we analyze the mapping of the correlations of the Ising model to spin network observables, and describe the phase transition on those observables on the hexagonal lattice. This opens the door to many new possibilities, especially for the study of the coarse-graining and continuum limit of spin networks in the context of quantum gravity.     

An antiferromagnetic transverse Ising nanoisland; unconventional surface effects

The phase diagrams and temperature dependences of magnetizations in a transverse Ising nanosisland with an antiferromagnetic spin configuration are studied by the use of the effective-field theory with correlations (EFT). Some novel features, such as the re-entrant phenomena with two compensation points being free from disorder induced frustration, are obtained for the magnetic properties in the system.     

Comparison of diluted antiferromagnetic Ising models on frustrated lattices in a magnetic field

We study diluted antiferromagnetic Ising models on triangular and kagome lattices in a magnetic field, using the replica-exchange Monte Carlo method. We observe seven and five plateaus in the magnetization curve of the diluted antiferromagnetic Ising model on the triangular and kagome lattices, respectively, when a magnetic field is applied. These observations contrast with the two plateaus observed in the pure model. The origin of multiple plateaus is investigated by considering the spin configuration of triangles in the diluted models. We compare these results with those of a diluted antiferromagnetic Ising model on the three-dimensional pyrochlore lattice in a magnetic field pointing in the [111] direction, sometimes referred to as the “kagome-ice” problem. We discuss the similarity and dissimilarity of the magnetization curves of the “kagome-ice” state and the two-dimensional kagome lattice.     

Compensation temperatures,magnetic susceptibilities and phase diagrams of a mixed ferrimagnetic ternary system on the Bethe lattice

Compensation behaviors, magnetic susceptibilities and the phase diagrams of the ternary system of the type ABC consisting of Ising spins σ = 1/2, S = 3/2 and m = 5/2 in the presence of a single-ion anisotropy are studied on the Bethe lattice within the framework of the exact recursion relations. Both ferromagnetic and antiferromagnetic exchange interactions are considered. The exact expressions for sublattice magnetizations and magnetic susceptibilities are obtained, and then thermal behaviors of the sublattice magnetizations, total magnetization, magnetic sublattice susceptibilities and total susceptibility are investigated. We find that the system only undergoes a second order phase transition for the different and same bilinear nearest-neighbor exchange interaction parameters, but displays compensation behaviors for only different bilinear interaction parameters. We also present the phase diagrams for the different and same bilinear nearest-neighbor exchange interaction parameters. A comparison is made with the other ternary system of the type ABC consisting of different spin values.     

Two distinct magnetic susceptibility peaks and magnetic reversal events in a cylindrical core/shell spin-1 Ising nanowire

The magnetization and susceptibility of a cylindrical core/shell spin-1 Ising nanowire are investigated within the effective-field theory with correlations for both ferromagnetic and antiferromagnetic exchange interactions between the shell and the core. We find that the nanowire system exhibits two distinct susceptibility peaks and two successive phase transitions; either both of them are second-order transitions or one is a second-order transition and the other is a first-order transition for a small exchange interaction. The susceptibility versus switching field and the hysteresis behavior are investigated for different temperatures. It is found that the magnetization reversal events appear as peaks in the susceptibility versus switching field curve, the positions of which define the coercive field points of the nanowire system; the distance between the two susceptibility peaks decreases with increasing temperature.     

Magnetization reversal in perpendicular exchange-biased multilayers

Co/Pt multilayers with perpendicular magnetic anisotropy exhibit an exchange bias when covered with an IrMn layer. The exchange bias field, which is about 7 mT for 3 Co/Pt bilayer repetitions and a Co layer thickness of 5 Å, can be increased up to 16.5 mT by the insertion of a thin Pt layer at the Co/IrMn interface. The interfacial magnetic anisotropy of the Co/IrMn interface (KSCo/IrMn =-0.09 mJ/m2) favours in-plane magnetization and tends to tilt the Co spins away from the film normal. Dynamical measurements of the magnetization reversal process reveal that both thermally activated spin reversal in the IrMn layer and domain wall nucleation in the Co/Pt multilayer influence the interfacial spin structure and therefore the strength of the perpendicular exchange bias field.     

Effect of the transverse field on the site-diluted Ising model in a longitudinal random field

Phase diagrams and magnetization curves of a diluted spin-3/2 transverse Ising model in a random field on honeycomb lattices are investigated by the use of an effective-field theory with correlations. The tricritical point is found in the system, in contrast to the corresponding spin-1/2 Ising counterpart. The possible reentrant phenomena displayed by the system due to the competition effects that occur for appropriate ranges of the random and transverse fields are investigated.     

Percolation features of cooperative Jahn-Teller systems: Ising EFT framework

Elastic exchange between two nearest Jahn-Teller (JT) centers in two or three dimensional dense crystals, can give an ordered macroscopic distortion known as cooperative JT effect (CJTE). A very diluted JT crystal does not show this effect. In the dynamic JT effect (DJTE), tunneling between different equivalent distorted wells has a pronounced influence on the CJTE. We investigate this phenomenon using a progressive increase in the concentration of these centers in the JT crystals, based on a bond percolation vector spin analogy technique within the framework of effective field theory (EFT). Mean field theory (MFT) was extensively used in previous studies of CJTE; however it neither includes correlation between JT centers in the lattice due to the complexity of the distortion field in the crystal nor the effect of tunneling between wells. We resort to an alternative procedure, by describing a JT center as a pseudo-spin vector \(\vec S\), induced to represent the degenerate JT-distorted states, where two nearest JT centers interact via an elastic exchange described by an Ising type spin interaction. The DJTE is considered to be similar to an elastic transverse field term in the Hamiltonian portraying the effect of tunneling between equivalent wells in the adiabatic potential energy surface (APES). We will be particularly discussing S = 1, S = 3/2 and S = 5/2 spin cases, where 2S + 1 wells in the APES are present and what JT systems they actually represent, with a percolative mechanism applied to the interactions between different JT centers. The different lattices are distinguished by their coordination numbers. Strong tunneling effects can suppress the CJTE and lead to a new state of criticality. Generalizations to higher spin systems will be obtained using a scaling technique. For the relevant distortions, we determine single site correlations, the macroscopic average distortion describing a structural phase transition and the elastic isothermal susceptibility as a function of temperature. The critical bond percolation threshold and the critical tunneling parameter are also obtained.     

Dynamic compensation temperatures in a mixed spin-1 and spin-3/2 Ising system under a time-dependent oscillating magnetic field

We study the existence of dynamic compensation temperatures in the mixed spin-1 and spin-3/2 Ising ferrimagnetic system Hamiltonian with bilinear and crystal-field interactions in the presence of a time-dependent oscillating external magnetic field on a hexagonal lattice. We employ the Glauber transitions rates to construct the mean-field dynamic equations. We investigate the time dependence of an average sublattice magnetizations, the thermal behavior of the dynamic sublattice magnetizations and the total magnetization. From these studies, we find the phases in the system, and characterize the nature (continuous or discontinuous) of transitions as well as obtain the dynamic phase transition (DPT) points and the dynamic compensation temperatures. We also present dynamic phase diagrams, including the compensation temperatures, in the five different planes. A comparison is made with the results of the available mixed spin Ising systems.