Ginzburg-Landau Equation Research Articles

Stability and local bifurcations of single-mode equilibrium states of the Ginzburg-Landau variational equation

One of the versions of the generalized variational Ginzburg-Landau equation is considered, supplemented by periodic boundary conditions. For such a boundary value problem, the question of existence, stability, and local bifurcations of single-mode equilibrium states is studied. It is shown that in the case of a nearly critical threefold zero eigenvalue, in the problem of stability of single-mode spatially inhomogeneous equilibrium states, subcritical bifurcations of two-dimensional invariant tori filled with spatially inhomogeneous equilibrium states are realized. The analysis of the stated problem is based on such methods of the theory of infinite-dimensional dynamical systems as the theory of invariant manifolds and the apparatus of normal forms. Asymptotic formulas are obtained for the solutions that form invariant tori.

PyTDGL: Time-dependent Ginzburg-Landau in Python

Time-dependent Ginzburg-Landau (TDGL) theory is a phenomenological model for the dynamics of superconducting systems. Due to its simplicity in comparison to microscopic theories and its effectiveness in describing the observed properties of the superconducting state, TDGL is widely used to interpret or explain measurements of superconducting devices. Here, we introduce pyTDGL, a Python package that solves a generalized TDGL model for superconducting thin films of arbitrary geometry, enabling simulations of vortex and phase dynamics in mesoscopic superconducting devices. pyTDGL can model the nonlinear magnetic response and dynamics of multiply connected films, films with multiple current bias terminals, and films with a spatially inhomogeneous critical temperature. We demonstrate these capabilities by modeling quasi-equilibrium vortex distributions in irregularly shaped films, and the dynamics and current-voltage-field characteristics of nanoscale superconducting quantum interference devices (nanoSQUIDs). Program summaryProgram Title: pyTDGLCPC Library link to program files:https://doi.org/10.17632/t6z7szt9bj.1Developer's repository link:http://www.github.com/loganbvh/py-tdglCode Ocean capsule:https://codeocean.com/capsule/2460583Licensing provisions:MIT LicenseProgramming language: PythonNature of problem:pyTDGL solves a generalized time-dependent Ginzburg-Landau (TDGL) equation for two-dimensional superconductors of arbitrary geometry, enabling simulations of vortex and phase slip dynamics in thin film superconducting devices.Solution method:: The package uses a finite volume adaptive Euler method to solve a coupled TDGL and Poisson equation in two dimensions.

Emergence of a Non-Van der Waals Magnetic Phase in a Van der Waals Ferromagnet.

Manipulation of long-range order in 2D van der Waals (vdW) magnetic materials (e.g., CrI3 , CrSiTe3 ,etc.), exfoliated in few-atomic layer, can be achieved via application of electric field, mechanical-constraint, interface engineering, or even by chemical substitution/doping. Usually, active surface oxidation due to the exposure in the ambient condition and hydrolysis in the presence of water/moisture causes degradation in magnetic nanosheets that, in turn, affects the nanoelectronic /spintronic device performance. Counterintuitively, the current study reveals that exposure to the air at ambient atmosphere results in advent of a stable nonlayered secondary ferromagnetic phase in the form of Cr2 Te3 (TC2 ≈160K) in the parent vdW magnetic semiconductor Cr2 Ge2 Te6 (TC1 ≈69K). The coexistence of the two ferromagnetic phases in the time elapsed bulk crystal is confirmed through systematic investigation of crystal structure along with detailed dc/ac magnetic susceptibility, specific heat, and magneto-transport measurement. To capture the concurrence of the two ferromagnetic phases in a single material, Ginzburg-Landau theory with two independent order parameters (as magnetization) with a coupling term can be introduced. In contrast to the rather common poor environmental stability of the vdW magnets, the results open possibilities of finding air-stable novel materials having multiple magnetic phases.

On the existence of Meissner solutions to the Ginzburg-Landau system in two dimensions

In this paper we establish the existence of the stable Meissner solutions of the Ginzburg-Landau equations for superconductivity in R2.

Ferroelectric Nanopillar Field-Effect Transistors: Quantum Transport Simulations Based on a Three-Dimensional Phase Field

A ferroelectric field-effect transistor (FEFET) with a cylindrical ferroelectric (FE) nanopillar embedded in the gate oxide stack is proposed and investigated. The device utilizes the multiple polar topological states, which can be stabilized in a nanoscale regime. To correctly simulate the proposed FEFET, a full three-dimensional phase-field-based quantum transport model is developed where nonequilibrium Green's function, Poisson, and time-dependent Ginzburg-Landau equations are self-consistently solved. Using the in-house tool, we theoretically demonstrate that the nanopillar FEFET exhibits multibit operation of six-level states with a reasonable memory window (MW) and sufficiently large difference in the current between the states. We also investigate the short-channel effect on the proposed FEFET and assess its scalability. MW linearly increases as the gate length decreases, because the polarization bound charges of the FE nanopillar induce the quasi-bound-states on the channel region and thus degrade the gate controllability.

Boundary superconductivity in the BCS Model

We consider the linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the one-dimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. In particular, the Cooper-pair wave function localizes near the boundary, an effect that cannot be modeled by effective Neumann boundary conditions on the order parameter as often imposed in Ginzburg–Landau theory. We also show that the relative shift in critical temperature vanishes if the coupling constant either goes to zero or to infinity.

Superconductors with a Topological Gap

I review a new superconductivity mechanism in which the gap is opened through a topological mechanism and not through the Landau mechanism of spontaneous symmetry breaking. As a consequence, the low-energy effective theory which describes these new superconductors is not the Landau–Ginzburg theory, formulated in terms of a local-order parameter, but a topological-field theory formulated in terms of emerging gauge fields. This new mechanism is realized as global superconductivty in Josephson junction arrays and in thin superconducting films with thicknesses comparable to the superconducting coherence length, which exhibits emergent granularity.

Zavisimost' dielektricheskoy pronitsaemosti i elektrokaloricheskogo effekta ot razmera granuly segnetoelektricheskoy keramiki

We consider the problem of determining the permittivity and the electrocaloric effect in the model of a ferroelectric ceramics grain. We assume that a grain consists of a spherical ferroelectric core coated with a dielectric shell and placed into a dielectric matrix. The transition layer thickness is assumed small as compared to the grain size. The dependence of the polarization on the electric field in the core is given by the nonlinear Ginzburg–Landau equation. The polarization reversal is induced by a change in the electric field that is considered uniform at large distance from the grain. The electrostriction effect in the core–shell–matrix three-phase system produces an elastic field described by linear equations. To take into account the effect of domain walls on the physical characteristics of the ceramics in the given model, we propose that the Kittel–Mitsui–Furuichi approach be used. The proposed computational algorithm makes it possible to refine the dependence of the number of domains on the spherical grain size. The electrocaloric effect in the grain is represented by the combination of the primary and secondary effects that appear due to ordering of dipole moments of the ferroelectric with the perovskite structure; by way of example, we consider the barium titanate ceramics. For this material, we report on the results of calculations of the dependences of the permittivity and individual electrocaloric effect components on the grain size.

Ginzburg–Landau surface energy of multiband superconductors: derivation and application to selected systems

We determine the energy of an interface between a multiband superconducting and a normal half-space, in presence of an applied magnetic field, based on a multiband Ginzburg–Landau (GL) approach. We obtain that the multiband surface energy is fully determined by the critical temperature, electronic densities of states, and superconducting gap functions associated with the different band condensates. This furthermore yields an expression for the thermodynamic critical magnetic field, in presence of an arbitrary number of contributing bands. Subsequently, we investigate the sign of the surface energy as a function of material parameters, through numerical solution of the GL equations. Here, we consider two distinct cases: (i) standard multiband superconductors with attractive interactions, and (ii) a three-band superconductor with a chiral ground state with phase frustration, arising from repulsive interband interactions. Furthermore, we apply this approach to several prime examples of multiband superconductors, such as metallic hydrogen and MgB2, based on microscopic parameters obtained from first-principles calculations.

Multi-band analysis on physical properties of superconducting FeSe films

The origins of superconductivity and pairing symmetry of order parameters are still controversial problems for FeSe thin films up to date. Under the Neumann boundary conditions, the electromagnetic properties of this system are investigated using the two-band Ginzburg–Landau theory. We calculate the temperature dependence of upper critical field in arbitrary direction and critical supercurrent density through the FeSe film. It is revealed that the normalized upper critical field is independent of the film thickness and all of our theoretical results are in accordance with the experimental data. These thus strongly indicate the existence of two-gap s-wave superconductivity in this material.

Grain boundary- and triple junction-induced martensitic transformations: A phase-field study of effects of grain boundary width and energy

An original thermodynamically consistent large strains-based general multiphase phase-field (PF) approach of Ginzburg–Landau type is developed for studying the grain boundary (GB)- and triple junction (TJ)-induced martensitic transformations (MTs) in polycrystalline materials at the nanoscale considering the structural stresses within the interfaces. In this general PF approach, N+1 order parameters are used for describing the austenite (A)↔martensite (M) transformations and N(>1) martensitic variants and other M order parameters are used to describe M(>1) grains in the polycrystalline samples. Neglecting the kinetics of the GBs and TJs and assuming a uniform temperature of the sample, the system of coupled mechanics and Ginzburg–Landau equations for the order parameters related to MTs is derived. Distinct energies for the GBs in austenitic and martensitic phases due to their structural rearrangement are considered by using variable energy for the GB regions as a function of the order parameter related to the A↔M transformation and misorientations. The strong effects of the austenitic GB width (size-effect), the difference of GB energies of two phases, and applied strains on heterogeneous nucleation of the phases and their subsequent growth are explored in bicrystals with a symmetric planar tilt GB during the forward and reverse transformations. A rich plot for the temperatures of transformations between A, premartensite (the intermediate states between A and M), and M in a bicrystal is plotted for varying austenitic GB width exhibiting a strong GB size effect. The TJ energy and the energy and width of the adjacent GBs are also shown to significantly influence the nucleation and microstructures using the tricrystals having three symmetric planar tilt GBs meeting at 120° dihedral angles. The elastic and structural stresses across the interfaces are plotted, which is essential for understanding the role of GBs and TJs in material failure. The plausible reasons for the nonintuitive behaviour of the GBs on martensite nucleation observed in experiments and atomistic studies are explained.

FerroX: A GPU-accelerated, 3D phase-field simulation framework for modeling ferroelectric devices

We present a massively parallel, 3D phase-field simulation framework for modeling ferroelectric materials based scalable logic devices. This code package, FerroX, self-consistently solves the time-dependent Ginzburg Landau (TDGL) equation for ferroelectric polarization, Poisson's equation for electric potential, and charge equation for carrier densities in semiconductor regions. The algorithm is implemented using the AMReX software framework [1], which provides effective scalability on manycore and GPU-based supercomputing architectures. We demonstrate the performance of the algorithm with excellent scaling results on NERSC multicore and GPU systems, with a significant (15×) speedup on the GPU using a node-by-node comparison. We further demonstrate the applicability of the code in simulations of ferroelectric domain-wall induced negative capacitance (NC) effect in Metal-Ferroelectric-Insulator-Metal (MFIM) and Metal-Ferroelectric-Insulator-Semiconductor-Metal (MFISM) devices. The charge (Q) v.s. voltage (V) responses for these 3D structures clearly indicate stabilized negative capacitance with multidomain formation, which is corroborated by amplification of the voltage at the interface between the ferroelectric and dielectric layers.

Modulational instability in vector exciton-polariton condensates with photonic spin-orbit coupling

The paper demonstrates the existence of modulational instability (MI) in nonlinear exciton-polariton condensates (EPCs). A set of coupled two-dimensional (2D) driven-dissipative Gross-Pitaevskii equations describing the spin-up and spin-down polaritons, containing the photonic spin-orbit (SO) coupling and the rate equations describing the density of incoherent excitonic reservoir density, is reduced to a system of two coupled 2D complex Ginzburg-Landau (2DCGL) equations, with saturable nonlinearities for complex polariton order parameters, thanks to the adiabatic approximation and small density fluctuations approximation. The analytical approach relies on the linear stability analysis of continuous waves (CWs) to derive an expression for the growth rate and conduct a parametric study of MI. The effect of different parameters on the growth rate spectrum is discussed with an emphasis on the photonic SO coupling and the pumping power. The predictions are verified against a direct simulation of the 2DCGL equations, and excellent agreement is found. The emergence of solitonic clusters manifests the evolution of MI. Phase diagrams on a MI spectrum for the CWs are presented against the magnetic field ($B$) and SO coupling strength ($\ensuremath{\sigma}$), based on which the dynamical behaviors of the emerging structures are debated, along with their response to changing $\ensuremath{\sigma}$ and $B$. The results suggest that the photonic SO coupling and the magnetic field constitute efficient tools for nonlinear mode selection and characterization via the MI process and are equally important in any eventual experimental realization of such a process in the studied EPC system.

Landau-Ginzburg theory of charge density wave formation accompanying lattice and electronic long-range ordering

We propose an analytical Landau-Ginzburg theory of the charge density waves coupled with lattice and electronic long-range order parameters. Examples of long-range order include electronic wave function of superconducting Cooper pairs, structural distortions, electric polarization, and magnetization. We formulate the Landau-Ginzburg free energy density as power expansion with respect to the charge density and other long-range order parameters, as well as their spatial gradients, and biquadratic coupling terms. We introduced a biquadratic coupling between the charge density gradient and long-range order parameters, as well as nonlinear higher gradients of the long-range order parameters. The biquadratic gradient coupling is critical to the appearance of different spatially-modulated phases in charge-ordered ferroics and high-temperature superconductors. We derived the thermodynamic conditions for the stability of the spatially-modulated phases, which are the intertwined spatial waves of charge density and lattice/electronic long-range order. The analytical expressions for the energies of different phases, corresponding order parameters, charge density waves amplitudes and modulation periods, obtained in this work, can be employed to guide the comprehensive physical explanation, deconvolution and Bayesian analysis of experimental data on quantum materials ranging from charge-ordered ferroics to high-temperature superconductors.

Nonlinear stability of two dusty magnetic liquids surrounded via a cylindrical surface: impact of mass and heat spread

The current article examines a nonlinear axisymmetric streaming flow obeying the Rivlin–Ericksen viscoelastic model and overloaded by suspended dust particles. The fluids are separated by an infinite vertical cylindrical interface. A uniform axial magnetic field as well as mass and heat transmission (MHT) act everywhere the cylindrical flows. For the sake of simplicity, the viscous potential theory (VPT) is adopted to ease the analysis. The study finds its significance in wastewater treatment, petroleum transport as well as various practical engineering applications. The methodology of the nonlinear approach is conditional primarily on utilizing the linear fundamental equations of motion along with the appropriate nonlinear applicable boundary conditions (BCs). A dimensionless procedure reveals a group of physical dimensionless numerals. The linear stability requirements are estimated by means of the Routh–Hurwitz statement. The application of Taylor’s theory with the multiple time scales provides a Ginzburg–Landau equation, which regulates the nonlinear stability criterion. Therefore, the theoretical nonlinear stability standards are determined. A collection of graphs is drawn throughout the linear as well as the nonlinear approaches. In light of the Homotopy perturbation method (HPM), an estimated uniform solution to the surface displacement is anticipated. This solution is verified by means of a numerical approach. The influence of different natural factors on the stability configuration is addressed. When the density number of the suspended inner dust particles is less than the density number of the suspended outer dust particles, and vice versa, it is found that the structure is reflected to be stable. Furthermore, as the pure outer viscosity of the liquid increases, the stable range contracts, this means that this parameter has a destabilizing effect. Additionally, the magnetic field and the transfer of heat don’t affect the nature of viscoelasticity.

Construction of Blowup Solutions for the Complex Ginzburg-Landau Equation with Critical Parameters

We construct a solution for the Complex Ginzburg-Landau (CGL) equation in a general critical case, which blows up in finite time T T only at one blow-up point. We also give a sharp description of its profile. In the first part, we formally construct a blow-up solution. In the second part we give the rigorous proof. The proof relies on the reduction of the problem to a finite dimensional one, and the use of index theory to conclude. The interpretation of the parameters of the finite dimension problem in terms of the blow-up point and time allows to prove the stability of the constructed solution. We would like to mention that the asymptotic profile of our solution is different from previously known profiles for CGL or for the semilinear heat equation.

Matricials methods applied to the solution of the Ginzburg-Landau equations

In this contribution we propose two numerical methods for the solution of the system of two non-linear coupled differential Ginzburg-Landau equations. These proposals are based firstly on taking a matrix view considering the quasi-linear coupled system, as a second option, considering the computational molecule with its respective restriction of the values and eigen-vectors of the matrix. We clearly and concisely obtained the eigenvalues that lead us to an optimal spatio-temporal convergence solution of said system of equations. We compare the numerical convergence times obtained by these three methods with the known time found by applying the link variable method.

THE THIRD CRITICAL FIELD (HC3) OF SINGLE-CRYSTALLINE K0.73Fe1.68Se2 BY GINZBURG-LANDAU APPROACH

In this paper, two-band Ginzburg-Landau (GL) equations for magnetic superconductors were solved analytically to determine the temperature dependence of surface critical magnetic field (Hc3). By variation method analytically from 1st GL equations and modified to the Changjan & Udomsamuthirun’s temperature dependence model. We found that, our model could find a good agreement with the experimental data of single-crystalline K0.73Fe1.68Se2 superconductors vicinity the critical temperature scenery and the ratio Hc3/Hc2 (where Hc2 is the upper critical field of single-crystalline K0.73Fe1.68Se2) indicated powerful temperature dependence.

Vector pulsating solitons and soliton molecules under higher-order effects in passively mode-locked fiber lasers

We numerically study the transport dynamics of vector solitons in ytterbium-doped fiber lasers based on a coupled Ginzburg-Landau equation with higher-order effects. By adjusting the small signal gain coefficient and intracavity birefringence, the number of soliton molecules can be manipulated, the transmission of synchronous and asynchronous vector solitons and the generation of chaotic pulse sequences can be realized. Synchronous vector solitons can be generated when the intracavity birefringence is relatively low, the stable transmission of multi-soliton molecules and the interaction between soliton molecules can be achieved. By changing the value of the phase delay, periodic pulsating solitons can be produced. Furthermore, the phenomenon of the pulse train from the period-doubling to chaos can also be observed. With the enhancement of the intracavity birefringence, asynchronous vector solitons can also achieve stable transmission in the cavity, which the amplitudes and widths of two components are opposite with the different phases. The proposed model has universal application value and the numerical results have guiding significance for studying polarization tunable lasers and exploring the nonlinear mechanism in fiber lasers.

Optimization of the Hysteresis Effect in Superconductor Nb Open Nanotubes

Superconductor Nb open nanotubes are analyzed under a strong transport current in an external magnetic field using the time‐dependent Ginzburg–Landau equations. The geometric properties of the hysteresis loop in current–voltage characteristics are described for nanotubes of different radii from 190 to 440 nm as a function of the magnetic field. The hysteresis effect appears at a constant temperature as a result of the order parameter dynamics. In nanotubes of radii equal to 240 nm and larger, the area of the hysteresis loop manifests a non‐monotonic behavior as a function of the external magnetic field with a maximum ranging from 15 to 20 mT. Normal conductivity is shown to have a crucial impact on the hysteresis effect in superconductor open nanotubes. Namely, a higher normal conductivity leads to the disappearance of the hysteresis effect. As a rule, the upper and lower branches of the hysteresis loop are provided by the phase‐slip and vortex regimes, correspondingly. Just before the transition of the system from the upper branch to the lower one, the way of nucleation and annihilation of vortices in the region with suppressed superconductivity experiences readjustment.