Second-harmonic generation in ferroelectric thin films on metal substrates for isotropic and cubic symmetries

The Landau–Ginzburg phenomenological theory of ferroelectrics and Landau–Khalatnikov equations of motion are used to model the interaction of electromagnetic radiation and a ferroelectric thin film on a metal substrate. The analysis starts with the minimization of a Gibb’s free energy functional that accounts for the free energy cost of the boundaries of the thin film through a gradient term that is non-zero when the polarization in the film varies. The minimization procedure leads to an Euler–Lagrange equation that can be solved to find the equilibrium polarization in the film. Next the nonlinear dynamical equations that describe the response to an incident electromagnetic field are solved by a perturbation expansion about the equilibrium polarization. The second-harmonic generation term from the expansion is selected. We then go on to calculate a reflection coefficient for the harmonic generation term and investigate how the finite thickness of the film influences the reflection coefficient. In Landau theory the symmetry of the crystal making up the film in the paraelectric phase is reflected in the free energy expression. The analysis here is done for an isotropic symmetry that is often used as an approximation to the actual crystal symmetry and which is mathematically easier to handle. However it is also indicated how the free energy expression can be changed so that it is appropriate for cubic symmetry, and a discussion of how the second harmonic term can be calculated for this case is given. The theory presented is relevant to experimental studies using far-infrared or terahertz reflection measurements because the ferroelectric film is resonant in the far infrared and terahertz ranges.     

A finite element phase field model for relaxor ferroelectrics

A mechanically coupled phase field model is presented for the domain evolution and mesoscopic response of relaxor ferroelectrics. In the model the spontaneous polarization is treated as order parameter. The model is derived from thermodynamic analysis including the material force theory. Random field theory is adopted to take the disorder of relaxor ferroelectrics into account. Results show that the model is capable of reproducing relaxor features, such as domain miniaturization, small remnant polarization and large piezoelectric response. Dependence of these features on the random field strength is discussed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Polarization of vacuum by an external magnetic field in the (2+1)-dimensional quantum electrodynamics with a nonzero fermion density

The Green's function of the Dirac equation with an external stationary homogeneous magnetic field in the (2+1)-dimensional quantum electrodynamics (QED ₂₊₁) with a nonzero fermion density is constructed. An expression for the polarization operator in an external stationary homogenous magnetic field with a nonzero chemical potential is derived in the one-loopQED ₂₊₁ approximation. The contribution of the induced Chern—Simons term to the polarization operator and the effective Lagrangian for the fermion density corresponding to the occupation of n relativistic Landau levels in an external magnetic field are calculated. An expression of the induced Chern—Simons term in a magnetic field for the case of a finite temperature and a nonzero chemical potential is obtained. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 132–151, October, 2000.     

自由能展开式中点群的不可约表示的基函数

一个群的基函数的选择并非唯一，不同的基函数对应不同的表示矩阵。即使相同的表示矩阵，基函数也可以有不同的选择。在相变的宏观唯象理论中， 自由能展开式可由同一个表示矩阵的基函数来构造，因而给出不可约表示的基函数表就非常有意义。现有文献给出的32点群不可约表示的基函数只写到二次幂，而且有些文献中的同一个不可约表示所选取的不同的基函数却对应不同的表示矩阵，这样在构造群变换不变式时，就会出错。该文将基函数表写至三次幂, 这有助于准确、迅速地写出到六次幂群变换不变的自由能表达式。由新的基函数表发现十八种点群有三次幂的群操作不变式, 高温相属这些点群铁电体, 发生的本征铁电相变为一级相变。     

Nonlinear Mathieu equation and coupled resonance mechanism

The method of multiple-scales is used to determine a third-order solution for a cubic nonlinear Mathieu equation. The perturbation solutions are imposed on the so-called solvability conditions. Solvability conditions in the non-resonance case yield the standard Landau equation. Several types of a parametric Landau equation are derived in the neighborhood of five different resonance cases. These parametric Landau equations contain a parametric complex conjugate term or a parametric second-order complex conjugate term or a parametric complex conjugate term as well as a parametric second-order term. Necessary and sufficient conditions for stability are performed in each resonance case. Stability criteria correspond to each parametric Landau equation and are derived by linear perturbation. Stability criteria for the non-trivial steady-state response are discussed. The analysis leads to simultaneous resonance. Transition curves are performed in each case. Numerical calculations are made for some transition curves to illustrate the coupled resonance regions, where the induced stability tongues within the instability tongues are observed. The amplitude of the periodic coefficient of Mathieu equation plays a dual role in the stability criteria for nonlinear Mathieu equation.     

Simulation of domain structures in cracked ferroelectric materials

The domain structure around a crack tip plays a significant role in the fracture behavior of ferroelectrics. A continuum phase field model is used to investigate the microstructure at the crack front. The concept of the Eshelby momentum tensor and configurational forces is then generalized to account for the contributions of the polarization term. Implementation of the generalized configurational force in the Finite Element code enables us to numerically obtain the driving force at the crack tip, which corresponds to the crack-tip energy release rate. Calculations show that additional positive electric fields tend to prohibit crack growth, whereas additional negative electric fields tend to promote crack growth. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vortex dynamics of the full time‐dependent Ginzburg‐Landau equations

In the Ginzburg‐Landau model for superconductivity a large Ginzburg‐Landau parameter κ corresponds to the formation of tight, stable vortices. These vortices are located exactly where an applied magnetic field pierces the superconducting bulk, and each vortex induces a quantized supercurrent about the vortex. The energy of large‐κ solutions blows up near each vortex which brings about difficulties in analysis. Rigorous asymptotic static theory has previously established the existence of a finite number of the vortices, and these vortices are located precisely at the critical points of the renormalized energy (the free energy less the vortex self‐induction energy). A rigorous study of the full time‐dependent Ginzburg‐Landau equations under the classical Lorentz gauge is done under the asymptotic limit κ → ∞. Under slow times the vortices remain pinned to their initial configuration. Under a fast time of order κ the vortices move according to a steepest descent of the renormalized energy. © 2002 John Wiley & Sons, Inc.     

An internal-variable theory of thermo-viscoelastic constitutive relations at finite strain

Based on the nonequilibrium thermodynamic theory, a new thermo-viscoelastic relation at finite strain is proposed. Under the assumption that the specific heat at a fixed strain and fixed internal variables can be regarded as a constant, a new expression for the free energy which decouples the mechanical and the thermal effects is derived. Through an analysis of the mesoscopic deformation mechanism of solid polymers, a set of internal variables is introduced, and an internal-variable constitutive theory in thermo-viscoelasticity at finite strain is formulated. An explicit expression of a thermoviscoelastic constitutive relation is obtained for solid polymers in the case where their molecular network has a randomly oriented distribution function at reference configuration. Moreover, the relationship between the relaxation time and the temperature is also discussed. The viscoelastic constitutive theory proposed in reference is only a linear approximation of the present theory.     

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.     

On a nonlocal phase separation model

An alternative to the Cahn-Hilliard model of phase separation for two-phase systems in a simplified isothermal case is given. The model is derived from a free energy with a nonlocal interacting term and allows reasonable bounds for the concentrations. Using the free energy as Lyapunov functional the asymptotic state of the system is investigated and characterized by a variational principle.     

Semiclassical determination of exponentially small intermode transitions for 1 + 1 spacetime scattering systems

We consider the semiclassical limit of systems of autonomous PDEs in 1 + 1 spacetime dimensions in a scattering regime. We assume the matrix‐valued coefficients are analytic in the space variable, and we further suppose that the corresponding dispersion relation admits real‐valued modes only with one‐dimensional polarization subspaces. Hence a BKW‐type analysis of the solutions is possible. We typically consider time‐dependent solutions to the PDE that are carried asymptotically in the past and as x → ?∞ along one mode only and determine the piece of the solution that is carried for x → +∞ along some other mode in the future. Because of the assumed nondegeneracy of the modes, such transitions between modes are exponentially small in the semiclassical parameter; this is an expression of the Landau‐Zener mechanism. We completely elucidate the spacetime properties of the leading term of this exponentially small wave, when the semiclassical parameter is small, for large values of x and t, when some avoided crossing of finite width takes place between the involved modes. © 2006 Wiley Periodicals, Inc.     

On Landau damping

Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of non-linear echoes; sharp “deflection” estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions.     

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.     

Minimum free energy for an electromagnetic conductor with memory

A closed expression is determined in the frequency domain for the minimum free energy associated with a state of a linear electromagnetic conductor with memory effects, using the fact that this quantity is equal to the maximum recoverable work obtainable from the given state of the material. Another equivalent expression is also derived and applied to evaluate explicit formulae for a discrete spectrum model. In particular, for such a model we present the results corresponding to only one inverse time decay for each of the three kernels of the constitutive equations. These results clearly show the effects of various parameters on the expression for the minimum free energy. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Kinetic Magnetohydrodynamics of Neutron Matter

Equations for the magnetohydrodynamics of neutron matter are derived within a microscopic approach based on the Landau theory of a Fermi liquid. Along with the strong short-distance nuclear interactions, the equations account for the weak long-distance magnetic interactions. Applications of the derived magnetohydrodynamic equations to the theory of shock waves in neutron matter are discussed.     

Quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field

For fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrödinger operator with a constant magnetic field and an axisymmetrical electric potential V. In various, mostly singular settings, asymptotic expansions for the resolvent of the Hamiltonian H _m+H_om+V are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair (H _m, H _om) is established and asymptotic expansions of the scattering matrix are derived as the energy parameter tends to the lowest Landau threshold.