Phenomenological theory of surface effects in ferroelectrics and its relationship with transverse Ising model

The relationship between the transverse field Ising model and the Landau phenomenological theory for ferroelectrics is analyzed, and the Landau free energy expression for ferroelectrics having surfaces is derived. It is pointed out that the traditional expression in which the surface integral has only a term of the square polarization is valid only for special cases, in general a term of the polarization to the four should be included as well. By use of the newly derived free energy expression, the thickness-dependence of the spontaneous polarization and Curie temperature of ferroelectric films is calculated; thereby some experimental results incompatible with the traditional phenomenological theory are successfully explained.     

Pattern Formation in Axially Symmetric Landau–Lifshitz–Gilbert–Slonczewski Equations

The Landau–Lifshitz–Gilbert–Slonczewski equation describes magnetization dynamics in the presence of an applied field and a spin-polarized current. In the case of axial symmetry and with focus on one space dimension, we investigate the emergence of space–time patterns in the form of wavetrains and coherent structures, whose local wavenumber varies in space. A major part of this study concerns existence and stability of wavetrains and of front- and domain wall-type coherent structures whose profiles asymptote to wavetrains or the constant up-/down-magnetizations. For certain polarization, the Slonczewski term can be removed which allows for a more complete characterization, including soliton-type solutions. Decisive for the solution structure is the polarization parameter as well as size of anisotropy compared with the difference of field intensity and current intensity normalized by the damping.     

Coherent Reflection of Electromagnetic Plane Waves from Infinite Rough Slabs

We adopt the prospect of an observer interested to optimise the signal-to-noise ratio in the reception of the backward radiation coming from a surface illuminated by an electromagnetic wave with a wavelength chosen to minimize the diffuse scattering so that he has just to point his receiver in the direction of the coherent reflection. Then, to analyse the coherent reflection for harmonic plane waves impinging on a dielectric infinite film deposited on a metallic substrate we develop a formalism generalizing the customary angular spectrum representation used to tackle this kind of problem. This new approach whose efficiency is proved in the easier situation of a dielectric film endowed with an impedance, is used to get the coherent reflection from a structured 1D-dielectric film illuminated by TE and TM electromagnetic plane waves when the rough amplitude h is small enough to justify 0(h ²) approximations. The Idemen technique is used to get the boundary conditions needed to tackle these scattering problems.     

Local deformations of branched projective structures: Schiffer variations and the Teichmüller map

For certain K3 surfaces, there are two constructions of mirror symmetry that appear very different. The first, known as BHK mirror symmetry, comes from the Landau–Ginzburg model for the K3 surface; the other, known as LPK3 mirror symmetry, is based on a lattice polarization of the K3 surface in the sense of Dolgachev’s definition. There is a large class of K3 surfaces for which both versions of mirror symmetry apply. In this class we consider the K3 surfaces admitting a certain purely non-symplectic automorphism of order 4, 8, or 12, and we complete the proof that these two formulations of mirror symmetry agree for this class of K3 surfaces.

     

Existence of Partially Regular Solutions for Landau–Lifshitz Equations in ℝ3

Abstract

We establish the existence of partially regular weak solutions for the Landau–Lifshitz equation in three space dimensions for smooth initial data of finite Dirichlet energy. The construction is based on Ginzburg–Landau approximation. The new key ingredient is a nonlocal representation formula for the penalty term that permits us to take advantage of the special trilinear structure of the limiting nonlinearity.     

Solving an inverse problem for Maxwell’s equations numerically with Laguerre functions

An inverse problem is solved by an optimization method using Laguerre functions. Numerical simulations are carried out for one-dimensional Maxwell’s equations in the wave and diffusion approximations. The spatial distributions of permittivity and conductivity in a medium are determined from a known solution at a certain point. A Laguerre harmonics function is minimized. The minimization is performed by the conjugate gradient method. The results of determining permittivity and conductivity are presented. The influence of the shape and spectrum of a source of electromagnetic waves on the solution accuracy of the inverse problem is investigated. The solutions with broadband and harmonic sources of electromagnetic waves are compared in accuracy.     

Phase field simulations in ferroelectric materials

The hindering of domain wall movement by defects in ferroelectric materials is closely connected to electric fatigue. A movable domain wall in a ferroelectric material in most cases is modelled as a singular surface which allows the use of configurational forces. In contrast, the present approach treats the polarization as an order parameter, extending the total energy by a phase separation energy and a domain wall energy. The polarization then no longer has a discontinuity at the domain wall but is a continuous vector field (phase field). As an example, a numerical simulation of domain evolution under stress free boundary conditions is presented. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

АГ-Convergence Result for Thin Curved Films Bonded to a Fixed Substrate with a Noninterpenetration Constraint

The behavior of a thin curved hyperelastic film bonded to a fixed substrate is described by an energy composed of a nonlinearly hyperelastic energy term and a debonding interfacial energy term. The author computes the F-limit of this energy under a noninterpenetration constraint that prohibits penetration of the film into the substrate without excluding contact between them.     

Films courbés minces ferromagnétiques

We consider a thin curved ferromagnetic film not submitted to an external magnetic field. The behavior of the film is described by an energy depending on the magnetization of the film verifying the saturation constraint. The energy is composed of an induced magnetostatic energy and an energy term with density including the exchange energy and the anisotropic energy. We study the behavior of this energy when the thickness of the curved film goes to zero. We show with Γ-convergence arguments that the minimizers of the free energy converge to the minimizers of a local energy depending on a two-dimensional magnetization. To cite this article: H. Zorgati, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Hidden symmetry and the number-theoretic structure of the energy levels of a perturbed harmonic oscillator

The energy levels of a three-dimensional isotropic harmonic oscillator that is perturbed by a quartic potential term possess a remarkable degeneracy structure that originates not from a symmetry group, but from properties of the natural numbers. A comprehensive theory of these energy multiplets, based on properties of the prime numbers, that gives their complete classification into 1-plets, 2-plets, etc., is developed. The number-theoretic origin of this degeneracy is contrasted with that of hidden symmetry. Generalizations to other dimensions are indicated.     

Justification of the Ginzburg–Landau approximation for an instability as it appears for Marangoni convection

The Ginzburg–Landau equation appears as a universal amplitude equation for spatially extended pattern forming systems close to the first instability. It can be derived via multiple scaling analysis for the Marangoni convection problem that is driven by temperature‐dependent surface tension and is the subject of our interest. In this paper, we prove estimates between this formal approximation and true solutions of a scalar pattern forming model problem showing the same spectral picture as the Marangoni convection problem in case of a thin fluid. The new difficulties come from neutral modes touching the imaginary axis for the wave number k = 0 and from identical group velocities at the critical wave number k = k_c and the wave number k = 0. The problem is solved by using the reflection symmetry of the system and by using the fact that the modes concentrate at integer multiples of the critical wave number k = k_c. The paper presents a method that is applicable whenever this kind of instability occurs. Copyright © 2012 John Wiley & Sons, Ltd.     

The Sine-Gordon regime of the Landau–Lifshitz equation with a strong easy-plane anisotropy

It is well-known that the dynamics of biaxial ferromagnets with a strong easy-plane anisotropy is essentially governed by the Sine-Gordon equation. In this paper, we provide a rigorous justification to this observation. More precisely, we show the convergence of the solutions to the Landau–Lifshitz equation for biaxial ferromagnets towards the solutions to the Sine-Gordon equation in the regime of a strong easy-plane anisotropy. Moreover, we establish the sharpness of our convergence result.This result holds for solutions to the Landau–Lifshitz equation in high order Sobolev spaces. We first provide an alternative proof for local well-posedness in this setting by introducing high order energy quantities with better symmetrization properties. We then derive the convergence from the consistency of the Landau–Lifshitz equation with the Sine-Gordon equation by using well-tailored energy estimates. As a by-product, we also obtain a further derivation of the free wave regime of the Landau–Lifshitz equation.     

On domain switching in deformable ferroelectrics, seen as continua with microstructure

We obtain a three-dimensional continuum model for deformable ferroelectric bodies in their polar phase characterized by a spontaneous polarization. This is accomplished by assuming the body as comprised of a continuum with vectorial microstructure: in each point of the body therefore a gross and a fine structure are superposed, the gross structure representing a non linear polarizable elastic body and the vectorial fine structure describing the spontaneous polarization.¶Among the distinctive features of ferroelectric materials, the most interesting is represented by the organization of spontaneous polarization into a domain structure, which minimizes electrostatic energy and which can be modified by the application of electric and deformation fields. This process, called "polarization reversal" or "domain switching", is associated with various hysteresis loops, the most typical being those between spontaneous polarization and electric field (dielectric hysteresis), and between strain and electric field ("butterfly" loop).¶From the balance laws of continua with vectorial microstructure and dissipation inequality we arrive at the evolution equation for the spontaneous polarization which allows for both inertial and dissipative terms and describes domain switching. We postulate a simple interaction mechanism between the spontaneous polarization and the pair electric field, deformation and arrive at, in the quasi-static case, to a minimization problem for a functional which reminds the micromagnetic energy of deformable ferromagnetics.¶For linearized kinematics we study, in the one-dimensional case, stable relative minimizers and give a simple justification for dielectric hysteresis and butterfly loops: under the hypothesis that the domain walls are sharp interfaces, the solutions we find explain the banded twin domains morphology which is typical of many ferroelectrics.     

A Г-Convergence Result for Thin Curved Films Bonded to a Fixed Substrate with a Noninterpenetration Constraint

Abstract The behavior of a thin curved hyperelastic film bonded to a fixed substrate is described by an energy composed of a nonlinearly hyperelastic energy term and a debonding interfacial energy term. The author computes the Γ-limit of this energy under a noninterpenetration constraint that prohibits penetration of the film into the substrate without excluding contact between them.     

Homogenization of a Ginzburg–Landau model for a nematic liquid crystal with inclusions

We consider a nonlinear homogenization problem for a Ginzburg–Landau functional with a (positive or negative) surface energy term describing a nematic liquid crystal with inclusions. Assuming that inclusions are separated by distances of the same order ɛ as their size, we find a limiting functional as ɛ approaches zero. We generalize the variational method of mesocharacteristics to show that a corresponding homogenized problem for arbitrary, periodic or non-periodic geometries is described by an anisotropic Ginzburg–Landau functional. We obtain computational formulas for material characteristics of an effective medium. As a byproduct of our analysis, we show that the limiting functional is a Γ-limit of a sequence of Ginzburg–Landau functionals. Furthermore, we prove that a cross-term corresponding to interactions between the bulk and the surface energy terms does not appear at the leading order in the homogenized limit.     

Interpretation of parameters in phase field models for ferroelectrics

Ferroelectric ceramics have a complex micro–structure which is responsible for their non–linear behavior on the macroscopic scale. The presented model treats the spontaneous polarization as an order parameter, for which a Ginzburg–Landau type evolution is postulated. Contrary to sharp interface models this allows for the simulation of evolving domain structures. The equations are solved via the Finite Element Method, employing an implicit time integration scheme. Numerical simulations show the physical interpretability of the phase field parameters. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of a Ginzburg–Landau type energy model for smectic C* liquid crystals with defects

This work investigates properties of a smectic C* liquid crystal film containing defects that cause distinctive spiral patterns in the film?s texture. The phenomena are described by a Ginzburg–Landau type model and the investigation provides a detailed analysis of minimal energy configurations for the film?s director field. The study demonstrates the existence of a limiting location for the defects (vortices) so as to minimize a renormalized energy. It is shown that if the degree of the boundary data is positive then the vortices each have degree +1 and that they are located away from the boundary. It is proved that the limit of the energies for a sequence of minimizers minus the sum of the energies around their vortices, as the G–L parameter ε tends to zero, is equal to the renormalized energy for the limiting state.     

Finite-amplitude long-wave instability of Bingham liquid films

The long-wave perturbation method is employed to investigate the weakly nonlinear hydrodynamic stability of a thin Bingham liquid film flowing down a vertical wall. The normal mode approach is first used to compute the linear stability solution for the film flow. The method of multiple scales is then used to obtain the weak nonlinear dynamics of the film flow for stability analysis. It is shown that the necessary condition for the existence of such a solution is governed by the Ginzburg–Landau equation. The modeling results indicate that both the subcritical instability and supercritical stability conditions can possibly occur in a Bingham liquid film flow system. For the film flow in stable states, the larger the value of the yield stress, the higher the stability of the liquid film. However, the flow becomes somewhat unstable in unstable states as the value of the yield stress increases.     

Numerical methods for the modeling of the magnetization vector in multiferroic heterostructures

Multiferroic heterostructures are commonly used to obtain electro-magnetic coupling effects. Thereby, the ferroelectric layer is used to control the magnetization in the ferromagnetic layer. The coupling between the layers is obtained by the mechanical coupling between the layers, which have well-defined interfaces. Within this contribution we use phase field models to define the polarization and magnetization in the ferroelectric and ferromagnetic layers, respectively. A coupling between polarization/magnetization and strains in each layer in combination with coherent deformations at the interface yields an electromagnetic coupling within the entire heterostructure. Numerical formulations for the interpolation of the polarization vector are well-defined in the literature. However, the establishment of a consistent numerical formulation for the ferromagnetic layer, where the length of the magnetization vector has to be constant, remains a difficult task. We propose a new numerical approach for the consistent treatment of the ferromagnetic layer and provide numerical simulations which illustrate the electromagnetic coupling effect. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)