No description available.

There are two main tools for determining the stability of nonlinear partial differential equations (PDEs): Lyapunov Theory and linearization. The former has the advantage of providing stability results for nonlinear equations directly, while the latter considers the stability of linear equations and then further justification is needed to show the linear stability implies local stability of the nonlinear equation. Linearization has the advantage of investigating stability on a simpler equation; however, the justification can be difficult to prove. Both Lyapunov Theory and linearization are applied to the Landau--Lifshitz equation, a nonlinear PDE that describes the behaviour of magnetization inside a magnetic object. It is known that the Landau-Lifshitz equation has an infinite number of stable equilibrium points. We present a control that forces the system from one equilibrium to another. This is proved using Lyapunov Theory. The linear Landau--Lifshitz equation is also investigated because it provides insight to the nonlinear equation. The linear model is shown to be well--posed and its eigenvalue problem is solved. The resulting eigenvalues suggest an appropriate control for the nonlinear Landau--Lifshitz equation. Mathematically, the control causes the initial equilibrium to no longer be an equilibrium and the second point to be an asymptotically stable equilibrium point. This implies the magnetization has moved to the second equilibrium and hence the control objective is successfully achieved. The existence of multiple stable equilibria is closely related to hysteresis. This is a phenomenon that is often characterized by a looping behaviour; however, the existence of a loop is not sufficient to identify hysteretic systems. A more precise definition is required, which is presented, and applied to the Landau--Lifshitz equation (both linear and nonlinear) to establish the presence of hysteresis.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-12-22T12:37:32Z No. of bitstreams: 1 josielcarlosdesouzagomes.pdf: 11267373 bytes, checksum: 393a01f57f4f5afaaf46890b84f4a7ac (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-12-22T12:42:22Z (GMT) No. of bitstreams: 1 josielcarlosdesouzagomes.pdf: 11267373 bytes, checksum: 393a01f57f4f5afaaf46890b84f4a7ac (MD5) / Made available in DSpace on 2016-12-22T12:42:22Z (GMT). No. of bitstreams: 1 josielcarlosdesouzagomes.pdf: 11267373 bytes, checksum: 393a01f57f4f5afaaf46890b84f4a7ac (MD5) Previous issue date: 2015-02-26 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / A nanotecnologia é uma área de estudo promissora e que nos mostra resultados bastante surpreendentes. Amostras magnéticas (Cobalto e liga de Permalloy (Ni81Fe19), por exemplo) em escala nanométrica, têm como aplicabilidade importante a gravação magnética devido à crescente demanda por meios de gravação cada vez mais rápidos e de alta capacidade de armazenamento. Para determinados tamanhos de nanofios, observa-se a presença de domínios magnéticos e paredes de domínios do tipo vórtice ou transversal que podem ser transportadas para diferentes regiões sem deformação. Pode-se usar tais paredes como bit de informação mas, para isso, precisa-se conhecer com detalhes o comportamento dessas paredes em diversas situações. Neste presente trabalho utilizamos simulações numéricas para estudar o comportamento da magnetização em nanofios retangulares (nanofitas) de Permalloy-79, que apresentam parede de domínio transversal entre domínios “head-to-head”. Utilizamos nestas simulações um modelo no qual os momentos magnéticos interagem através da interação de troca e a interação dipolar. Embora a maioria dos trabalhos encontrados utilizem campo magnético para mover a parede, optamos por aplicar corrente de spin-polarizado na direção do nanofio devido ao fato de ser mais prático de ser produzido. A dinâmica do sistema é regida pelas equações de Landau-Lifshitz-Gilbert e a atuação da corrente é introduzida nessas equações. Fizemos uma abordagem teórica na qual pode-se mostrar como esta equação de Landau-Lifshitz-Gilbert para aplicação de corrente foi obtida. A integração da equação de Landau-Lifshitz-Gilbert é feita utilizando o método de Runge-Kutta e de Predição-Correção. Baseado nessas teorias, escrevemos um programa na linguagem Fortran-90 para realizar as simulações. Em nossos resultados observamos o comportamento da velocidade da parede de domínio em função do tempo e da densidade de corrente. Comparamos estes resultados com a bibliografia. / Nanotechnology is a promising field of study and show us pretty amazing results. Magnetic samples (Cobalt and alloy Permalloy (81NiFe19), for example) at the nanometer scale, have as important applicability the magnetic recording due to the growing demand for recording media ever faster and high storage capacity. For certain sizes of nanowires, it is observed the presence of magnetic domains and vortex domain walls or transverse domain wall which can be transported to different regions without deformation. It can use such walls as bit of information, but for that it is necessary to know in detail the behavior of these walls in various situations. In this work we used numerical simulations to study the behavior of the magnetization in rectangular nanowires (nanostrip) of Permalloy-79, which have transverse domain wall between domains "head-to-head."We used in these simulations a model in which the magnetic moments interact through the exchange interaction and the dipolar interaction. Although most studies found use magnetic field to move the wall, we decided to apply spin-polarized current toward the nanowire due the fact that it is more practical to be produced. The dynamics of the system is governed by the equations of Landau-Lifshitz-Gilbert and the current performance is introduced in these equations. We made a theoretical approach in which you can show how this equation of Landau-Lifshitz-Gilbert for applying current was obtained. The integration of the equation of Landau-Lifshitz-Gilbert is done using the Runge-Kutta and Prediction-Correction methods. Based on these theories, we wrote a program in Fortran-90 language to perform the simulations. In our results we observed the behavior of the domain wall velocity as a function of time and current density. We compare these results with the literature.

L’objectif de cette thèse est d’explorer analytiquement et numériquement la dynamique d’aimantation de nanoparticules magnétiques. Nous montrons qu’il est possible de contrôler efficacement le retournement d’aimantation d’une nanoparticule à l’aide d’une excitation autorésonante. Cette étude révèle que l’amortissement de Gilbert et la température altèrent l’efficacité de ce procédé, tandis que les interactions dipolaires peuvent le faciliter. Les propriétés stationnaires d’une monocouche de nanoparticules sont également étudiées en reproduisant numériquement des courbes ZFC. Nous observons ainsi qu’un désordre structurel ne modifie pas la température de blocage. Enfin, nous étudions le comportement d’un ensemble de nanoparticules en interaction dans un fluide à l’aide de simulations de dynamique moléculaire. Nous retrouvons les configurations à l’équilibre en forme de chaînes et d’anneaux, puis nous examinons les aspects dynamiques en mettant en évidence l’existence d’ondes de spins. / The goal of this thesis is to explore analytically and numerically the magnetization dynamics in magnetic nanoparticles. Firstly, we study the Néel dynamics of fixed. We show that one can efficiently control the magnetization reversal of a nanoparticle by using a chirped excitation (autoresonance). This study reveals that the Gilbert damping and the temperature alter the efficiency of the reversal, while dipolar interactions can improve it. The stationary properties of a monolayer of nanoparticles are then examined by computing ZFC curves with a Monte Carlo method. We observe that structural disorder has no effect on the blocking temperature. Finally, we investigate the behavior of an ensemble of interacting nanoparticles moving in a fluid with a molecular dynamics approach. Our numerical simulations reproduce the usual chain and ring-like equilibrium configurations. We then study the dynamics of these structures and show the existence of super-spin waves.

No description available.

Understanding magnetization reversal is very important in designing high density and high data transfer rate recording media. This research has been motivated by interest in developing new nonvolatile data storage solutions as magnetic random access memories - MRAMs. This dissertation is intended to provide a theoretical analysis of static and dynamic magnetization switching of magnetic systems within the framework of critical curve (CC). Based on the time scale involved, a quasi-static or dynamic CC approach is used. The static magnetization switching can be elegantly described using the concept of critical curves. The critical curves of simple uncoupled films used in MRAM are discussed. We propose a new sensitive method for CC determination of 2D magnetic systems. This method is validated experimentally by measuring experimental critical curves of a series of Co/SiO2 multilayers systems. The dynamics switching is studied using the Landau-Lifshitz-Gilbert (LLG) equation of motion. The switching diagram so-called dynamic critical curve of Stonerlike particles subject to short magnetic field pulses is presented, giving useful information for optimizing field pulse parameters in order to make ultrafast and stable switching possible. For the first time, the dynamic critical curves (dCCs) for synthetic antiferromagnet (SAF) structures are introduced in this work. Comparing with CC, which are currently used for studying the switching in toggle MRAM, dCCs show the consistent switching and bring more useful information on the speed of magnetization reversal. Based on dCCs, better understanding of the switching diagram of toggle MRAM following toggle writing scheme can be achieved. The dynamic switching triggered by spin torque transfer in spin-torque MRAM cell has been also derived in this dissertation. We have studied the magnetization's dynamics properties as a function of applied current pulse amplitude, shape, and also as a function of the Gilbert damping constant. The great important result has been obtained is that, the boundary between switching/non-switching regions is not smooth but having a seashell spiral fringes. The influence of thermal fluctuation on the switching behavior is also discussed in this work.

We solved the Landau-Lifshitz equations numerically to examine the time development of a system of magnetic particles. Constant or periodical external magnetic field has been applied. First, the system has been studied without dissipation. Local energy excitations (breathers) and chaotic transients have been found. The behaviour of the system and the final configurations can strongly depend on the initial conditions, and the strength of the external field at an earlier time. We observed some sudden switching between two remarkably different states. Series of bifurcations have been found. When a weak Gilbert-damping has been taken into account, interesting behaviour has been found even in the case of one particle as well: bifurcation series and period multiplication leading to chaos. For a system of antiferromagnetically coupled particles, highly nontrivial hysteresis loops have been produced. The dynamics of the magnetization reversal has been investigated and the characteristic time-scale of the reversal has been estimated. For more particles, the energy spectrum and the magnetization of the system exhibits fractal characteristics for increasing system size. Finally, energy have been pumped into the system in addition to the dissipation. For constant field, complicated phase diagrams have been produced. For microwave field, it has been found that the chaotic behaviour crucially depends on the parity of the number of the particles.

Cette thèse est consacrée à l'étude de quelques problèmes mathématiques issus du micromagnétisme. Le but est d'analyser le comportement des modèles en fonction de différents paramètres physiques, dont les fines variations sont parfois difficilement mesurables. Nous adoptons des approches numériques, asymptotiques ou d'homogénéisation. Les modèles considérés reposent sur l'utilisation de l'équation de Landau-Lifshitz-Gilbert (LLG) décrivant l'évolution du champ d'aimantation dans un matériau ferromagnétique. Nous rappelons d'abord quelques notions importantes en ferromagnétisme. Ensuite, nous menons une étude numérique d'un modèle de la dynamique d'aimantation avec effets d'inertie. Nous proposons un schéma aux différences finies semi-implicite qui respecte de façon intrinsèque les propriétés du modèle continu. Des simulations numériques sont réalisées pour cerner l'effet du paramètre d'inertie. Ces simulations montrent aussi la performance du schéma et confirment l'ordre de convergence obtenu théoriquement. Nous étudions ensuite un modèle de la dynamique de l'aimantation avec amortissement non local. La sensibilité de la dynamique d'aimantation au paramètre d'amortissement est étudiée en donnant le problème limite pour de petites et de grandes valeurs du paramètre. Enfin, nous étudions l'homogénéisation de l'équation LLG dans deux types de matériau, à savoir les composites présentant un fort contraste des propriétés magnétiques et les matériaux périodiquement perforés avec énergie d'anisotropie de surface. Des modèles homogénéisés sont d'abord obtenus formellement puis une dérivation rigoureuse est établie en se basant principalement sur les concepts de la convergence à double échelle et de la convergence à double échelle en surface. Pour traiter les non-linéarités, nous introduisons une nouvelle méthode basée sur le couplage d'un opérateur de dilatation calibré sur les contrastes d'échelle et d'un outil de réduction de dimension, par construction de grilles emboitées adaptées à la géométrie du domaine microscopique. / This thesis is devoted to the study of some mathematical problems arising in micromagnetism. The models considered here are based on the Landau-Lifshitz-Gilbert equation (LLG) describing the evolution of the magnetization field in a ferromagnetic material. Our aim is the analysis of the behavior of the models regarding the slight variations of some physical parameters. We first recall some important notions about ferromagnetism. Then, we carry out a numerical study of a model of magnetization dynamics with inertial effects. We propose a semi-implicit finite difference scheme which intrinsically respects the properties of the continuous model. Numerical simulations are provided for emphasizing the effect of the inertia parameter. These simulations also show the performance of the scheme and confirm the order of convergence obtained theoretically. We then study a model of magnetization dynamics with a non-local damping. The sensitivity of the magnetization dynamics to the damping coefficient is studied by giving the limiting problem for small and large values of the parameter. Finally, we study the homogenization of the LLG equation in two types of structures, namely a composite material with strongly contrasted magnetic properties, and a periodically perforated material with surface anisotropy energy. The homogenized models are first obtained formally. The rigorous derivation is then performed using mainly the concepts of two-scale convergence, two-scale convergence on surfaces together with a new homogenization procedure for handling with the nonlinear terms. More precisely, an appropriate dilation operator is applied in a embedded cells network, the network being constrained by the microscopic geometry.

Micromagnetics is a continuum mechanics theory of magnetic materials widely used in industry and academia. In this thesis we describe a complete numerical method, with a number of novel components, for the computational solution of dynamic micromagnetic problems by solving the Landau-Lifshitz-Gilbert (LLG) equation. In particular we focus on the use of the implicit midpoint rule (IMR), a time integration scheme which conserves several important properties of the LLG equation. We use the finite element method for spatial discretisation, and use nodal quadrature schemes to retain the conservation properties of IMR despite the weak-form approach. We introduce a novel, generally-applicable adaptive time step selection algorithm for the IMR. The resulting scheme selects error-appropriate time steps for a variety of problems, including the semi-discretised LLG equation. We also show that it retains the conservation properties of the fixed step IMR for the LLG equation. We demonstrate how hybrid FEM/BEM magnetostatic calculations can be coupled to the LLG equation in a monolithic manner. This allows the coupled solver to maintain all properties of the standard time integration scheme, in particular stability properties and the energy conservation property of IMR. We also develop a preconditioned Krylov solver for the coupled system which can efficiently solve the monolithic system provided that an effective preconditioner for the LLG sub-problem is available. Finally we investigate the effect of the spatial discretisation on the comparative effectiveness of implicit and explicit time integration schemes (i.e. the stiffness). We find that explicit methods are more efficient for simple problems, but for the fine spatial discretisations required in a number of more complex cases implicit schemes become orders of magnitude more efficient.

Cette thèse est dédiée à l'étude des fluctuations non-linéaires dans les condensats de Bose-Einstein à deux composantes. On présente dans le premier chapitre la dynamique de champ moyen des condensats à deux composantes et les différents phénomènes typiques associés au degré de liberté spinoriel. Dans ce même chapitre, on montre que la dynamique des excitations se sépare en deux modes distincts : un mode dit de densité correspondant au mouvement global des atomes à l'intérieur du condensat et un mode dit de polarisation correspondant à la dynamique relative entre les deux espèces constituant le condensat. Ce calcul est généralisé dans le deuxième chapitre où l'on montre que le mode de polarisation persiste en présence d'un couplage cohérent entre les deux composantes. En particulier on analyse la stabilité modulationnelle du mode en déterminant, à l'aide d'une analyse multi-échelle, la dynamique des excitations non-linéaires. On montre alors que les excitations de polarisation, au contraire des excitations de densité, souffrent d'une instabilité de Benjamin-Feir. Cette instabilité est stabilisée aux grandes impulsions par une résonance onde longue - onde courte. Enfin dans le dernier chapitre, on dérive de façon non-perturbative la dynamique de polarisation proche de la limite de Manakov, dynamique quise révèle être régie par une équation de Landau-Lifshitz sans dissipation. Les équations de Landau-Lifshitz appartiennent à une hiérarchie d'équations intégrables (hiérarchie Ablowitz-Kaup-Newell-Segur) et on étudie les solutions à une phase à l'aide de la méthode d'intégration finite-gap ; on détermine notamment à l'aide de cette méthode un nouveau type de soliton pour les condensats à deux composantes. Finalement, profitant de l'intégrabilité du système, on résout le problème de Riemann à l'aide de la théorie de modulation de Whitham et on montre que les condensats à deux composantes peuvent propager des ondes de raréfaction ainsi que des ondes de choc dispersives ; on décrit notamment la modulation de ces ondes de choc par la propagation d'ondes simples et d'ondes de contact d'invariants de Riemann. / This thesis is devoted to the study of nonlinear fluctuations in two-component Bose-Einstein condensates. In the first chapter we derive the mean field dynamics of two-component condensates and we present the distinctive phenomena associated to the spinorial degree of freedom. In the same chapter, we show that the dynamics of the excitations is divided in two distinct modes: a so-called density mode which corresponds to the global motion of the atoms, and a so-called polarization mode which corresponds to the relative motion between the two species composing the condensate. The computation is generalized in the second chapter in which we demonstrate that the polarization mode remains in presence of a coherent coupling between the two components. In particular we study the modulational stability of the mode and we determine through a multi-scaling analysis the dynamics of non-linear excitations. We show that the excitations of polarization undergo a Benjamin-Feir instability contrary to the density excitations. This instability is then stabilized in the short wavelength regime by a long wave - short wave resonance. Finally in the last chapter, we derive in a non-perturbative way the polarisation dynamics close the Manakov limit.In this limit, the dynamics proves to be governed by a Landau-Lifshitz equation without dissipation. Landau-Lifshitz equations belong to a hierarchy of integrable equations (Ablowitz-Kaup-Newell-Segur hierarchy) and we derive the single-phase solutions thanks to the finite-gap method; in particular we identify a new type of soliton for the two-component Bose-Einstein condensates. Finally, taking advantage of the integrability of the system, we solve the Riemann problem thanks to the Whitham modulation theory and we show that the two-component condensates can propagate rarefaction waves as well as dispersive shockwaves; we describe the modulation of the shockwaves by the propagation of simple waves and contact waves of Riemann invariants.