Normal forms, resonance and bifurcation analysis via the Carleman linearization

AbstractEquivalence between the normal form (NF) and the Carleman linearization of a nonlinear dynamic system is established. The near-identity nonlinear coordinate transformation which brings a system to its NF is shown to be the similarity transformation bringing a Carleman system to a Jordan canonical form. It is shown that the steady-state multiplicity is given by the nullity of the first Carleman matrix with noncomplete nullspace. The stability of the limit cycles at Hopf bifurcation is determined by the direction of the iω-generalized eigenvector of the first odd order Carleman matrix which has a noncomplete iω-eigenspace. The coefficients of the resonant terms that are retained in the NF are explicitly determined

Mechanism for nonequilibrium symmetry breaking and pattern formation in magnetic films

Magnetic thin films exhibit a strong variation in properties depending on their degree of disorder. Recent coherent x-ray speckle experiments on magnetic films have measured the loss of correlation between configurations at opposite fields and at the same field, upon repeated field cycling. We perform finite temperature numerical simulations on these systems that provide a comprehensive explanation for the experimental results. The simulations demonstrate, in accordance with experiments, that the memory of configurations increases with film disorder. We find that non-trivial microscopic differences exist between the zero field spin configuration obtained by starting from a large positive field and the zero field configuration starting at a large negative field. This seemingly paradoxical beahvior is due to the nature of the vector spin dynamics and is also seen in the experiments. For low disorder, there is an instability which causes the spontaneous growth of line-like domains at a critical field, also in accord with experiments. It is this unstable growth, which is highly sensitive to thermal noise, that is responsible for the small correlation between patterns under repeated cycling. The domain patterns, hysteresis loops, and memory properties of our simulated systems match remarkably well with the real experimental systems.Comment: 12 pages, 10 figures Added comparison of results with cond-mat/0412461 and some more discussio

Thermally induced error: density limit for magnetic data storage

Magnetic data storage is pervasive in the preservation of digital information and the rapid pace of computer development requires ever more capacity. Increasing the storage density for magnetic hard disk drives requires a reduced bit size, previously thought to be limited by the thermal stability of the constituent magnetic grains. The limiting storage density in magnetic recording is investigated treating the writing of bits as a thermodynamic process. A 'thermal writability' factor is introduced and it is shown that storage densities will be limited to 15 to 20 TBit/in^2 unless technology can move beyond the currently available write field magnitudes.Comment: Improved manuscript for readabilit

Domain Dynamics of Magnetic Films with Perpendicular Anisotropy

We study the magnetic properties of nanoscale magnetic films with large perpendicular anisotropy comparing polarization microscopy measurements on Co_28Pt_72 alloy samples based on the magneto-optical Kerr effect with Monte Carlo simulations of a corresponding micromagnetic model. We focus on the understanding of the dynamics especially the temperature and field dependence of the magnetisation reversal process. The experimental and simulational results for hysteresis, the reversal mechanism, domain configurations during the reversal, and the time dependence of the magnetisation are in very good qualitative agreement. The results for the field and temperature dependence of the domain wall velocity suggest that for thin films the hysteresis can be described as a depinning transition of the domain walls rounded by thermal activation for finite temperatures.Comment: 7 pages Latex, Postscript figures included, accepted for publication in Phys.Rev.B, also availible at: http://www.thp.Uni-Duisburg.DE/Publikationen/Publist_Us_R.htm

Monte Carlo simulation with time step quantification in terms of Langevin dynamics

For the description of thermally activated dynamics in systems of classical magnetic moments numerical methods are desirable. We consider a simple model for isolated magnetic particles in a uniform field with an oblique angle to the easy axis of the particles. For this model, a comparison of the Monte Carlo method with Langevin dynamics yields new insight in the interpretation of the Monte Carlo process, leading to the implementation of a new algorithm where the Monte Carlo step is time-quantified. The numeric results for the characteristic time of the magnetisation reversal are in excellent agreement with asymptotic solutions which itself are in agreement with the exact numerical results obtained from the Fokker-Planck equation for the Neel-Brown model.Comment: 5 pages, Revtex, 4 Figures include

Fluctuations and Dissipation of Coherent Magnetization

A quantum mechanical model is used to derive a generalized Landau-Lifshitz equation for a magnetic moment, including fluctuations and dissipation. The model reproduces the Gilbert-Brown form of the equation in the classical limit. The magnetic moment is linearly coupled to a reservoir of bosonic degrees of freedom. Use of generalized coherent states makes the semiclassical limit more transparent within a path-integral formulation. A general fluctuation-dissipation theorem is derived. The magnitude of the magnetic moment also fluctuates beyond the Gaussian approximation. We discuss how the approximate stochastic description of the thermal field follows from our result. As an example, we go beyond the linear-response method and show how the thermal fluctuations become anisotropy-dependent even in the uniaxial case.Comment: 22 page

Magnetic relaxation in finite two-dimensional nanoparticle ensembles

We study the slow phase of thermally activated magnetic relaxation in finite two-dimensional ensembles of dipolar interacting ferromagnetic nanoparticles whose easy axes of magnetization are perpendicular to the distribution plane. We develop a method to numerically simulate the magnetic relaxation for the case that the smallest heights of the potential barriers between the equilibrium directions of the nanoparticle magnetic moments are much larger than the thermal energy. Within this framework, we analyze in detail the role that the correlations of the nanoparticle magnetic moments and the finite size of the nanoparticle ensemble play in magnetic relaxation.Comment: 21 pages, 4 figure

Effects of boundary conditions on magnetization switching in kinetic Ising models of nanoscale ferromagnets

Magnetization switching in highly anisotropic single-domain ferromagnets has been previously shown to be qualitatively described by the droplet theory of metastable decay and simulations of two-dimensional kinetic Ising systems with periodic boundary conditions. In this article we consider the effects of boundary conditions on the switching phenomena. A rich range of behaviors is predicted by droplet theory: the specific mechanism by which switching occurs depends on the structure of the boundary, the particle size, the temperature, and the strength of the applied field. The theory predicts the existence of a peak in the switching field as a function of system size in both systems with periodic boundary conditions and in systems with boundaries. The size of the peak is strongly dependent on the boundary effects. It is generally reduced by open boundary conditions, and in some cases it disappears if the boundaries are too favorable towards nucleation. However, we also demonstrate conditions under which the peak remains discernible. This peak arises as a purely dynamic effect and is not related to the possible existence of multiple domains. We illustrate the predictions of droplet theory by Monte Carlo simulations of two-dimensional Ising systems with various system shapes and boundary conditions.Comment: RevTex, 48 pages, 13 figure

Analytical and computational study of magnetization switching in kinetic Ising systems with demagnetizing fields

An important aspect of real ferromagnetic particles is the demagnetizing field resulting from magnetostatic dipole-dipole interaction, which causes large particles to break up into domains. Sufficiently small particles, however, remain single-domain in equilibrium. This makes such small particles of particular interest as materials for high-density magnetic recording media. In this paper we use analytic arguments and Monte Carlo simulations to study the effect of the demagnetizing field on the dynamics of magnetization switching in two-dimensional, single-domain, kinetic Ising systems. For systems in the ``Stochastic Region,'' where magnetization switching is on average effected by the nucleation and growth of fewer than two well-defined critical droplets, the simulation results can be explained by the dynamics of a simple model in which the free energy is a function only of magnetization. In the ``Multi-Droplet Region,'' a generalization of Avrami's Law involving a magnetization-dependent effective magnetic field gives good agreement with our simulations.Comment: 29 pages, REVTeX 3.0, 10 figures, 2 more figures by request. Submitted Phys. Rev.

Influence of demagnetization in remanence curves of magnetic thin films

Remanent magnetization curves of perpendicular magnetic thin films are simulated and measured. The simulations are used to investigate the theoretical influence of the strong demagnetizing field present in these films. Conclusions are drawn from this on how remanence curves should be measured and how they should be corrected for the demagnetizing influence. The experimental part consists of measurements on Fe‐Alumite, Co‐Pt–based multilayers, and Co‐Cr. In addition the latter material is also artificially patterned into microstrips in order to investigate the influence of demagnetization on remanence curves experimentally