Time-dependent Ginzburg-Landau Equation Research Articles

Electric-field-induced crystallization of Hf0.5Zr0.5O2 thin film based on phase-field modeling

Ferroelectricity in crystalline hafnium oxide has attracted considerable attention because of its potential application for memory devices. A recent breakthrough involves electric-field-induced crystallization, allowing HfO2-based materials to avoid high-temperature crystallization, which is unexpected in the back-end-of-line process. However, due to the lack of clarity in understanding the mechanisms during the crystallization process, we aim to employ theoretical methods for simulation, to guide experimental endeavors. In this work, we extended our phase-field model by coupling the crystallization model and time-dependent Ginzburg-Landau equation to analyze the crystalline properties and the polarization evolution of Hf0.5Zr0.5O2 thin film under applying an electric field periodic pulse. Through this approach, we found a wake-up effect during the process of crystallization and a transformation from orthorhombic nano-domains to the stripe domain. Furthermore, we have proposed an innovative artificial neural synapse concept based on the continuous polarization variation under applied electric field pulses. Our research lays the theoretical groundwork for the advancement of electric-field-induced crystallization in the hafnium oxide system.

Roundabout-like defect as a guide of phase-slip lines in a superconducting thin film at low temperature

AbstractIn this work, we examined the kinematic vortex state in a low-temperature, mesoscopic superconducting thin film in presence of an external applied current $$\varvec{J}$$ J , at zero magnetic field. We analyzed the voltage-current ($$\varvec{V-I}$$ V - I ), resistivity-current ($$\varvec{R-I}$$ R - I ) curves, as well as the vortex-anti-vortex velocity $$\varvec{V_m}$$ V m and the superconducting-electron density (norm of the superconduting order parameter $$\varvec{\psi }$$ ψ ) as an applied current function. Additionally, we have calculated the voltage as a function of characteristic time ($$\varvec{V-t}$$ V - t ). The sample presents a roundabout shaped pinning center fabricated with a lower low-temperature superconducting material, which allows for control over the vortex dynamics within the sample. To study this system, we have solved the generalized time-dependent Ginzburg-Landau equations for a single-condensate system. Our results demonstrate that the kinematic vortices enter the sample through regions where the superconductivity is depleted, revealing an intriguing and novel behavior of the phase-slip lines in the defect. We found that when a higher $$\varvec{J}$$ J is applied and the defect is not centered, the point where the vortex anti-vortex pairs enter the film moves away from the defect and are located in a point tangent to the outer circle of the defect. Besides, our results show a way to control the dynamics of the vortices, critical currents, and points of annihilation or creation of the cinematic vortices by including defects in the sample, important in the design and development of devices applied in industry and engineering.

Steering of Vortices by Magnetic Field Tilting in Open Superconductor Nanotubes.

In planar superconductor thin films, the places of nucleation and arrangements of moving vortices are determined by structural defects. However, various applications of superconductors require reconfigurable steering of fluxons, which is hard to realize with geometrically predefined vortex pinning landscapes. Here, on the basis of the time-dependent Ginzburg-Landau equation, we present an approach for the steering of vortex chains and vortex jets in superconductor nanotubes containing a slit. The idea is based on the tilting of the magnetic field B at an angle α in the plane perpendicular to the axis of a nanotube carrying an azimuthal transport current. Namely, while at α=0∘, vortices move paraxially in opposite directions within each half-tube; an increase in α displaces the areas with the close-to-maximum normal component |Bn| to the close(opposite)-to-slit regions, giving rise to descending (ascending) branches in the induced-voltage frequency spectrum fU(α). At lower B values, upon reaching the critical angle αc, the close-to-slit vortex chains disappear, yielding fU of the nf1 type (n≥1: an integer; f1: the vortex nucleation frequency). At higher B values, fU is largely blurry because of multifurcations of vortex trajectories, leading to the coexistence of a vortex jet with two vortex chains at α=90∘. In addition to prospects for the tuning of GHz-frequency spectra and the steering of vortices as information bits, our findings lay the foundation for on-demand tuning of vortex arrangements in 3D superconductor membranes in tilted magnetic fields.

A generalized scalar auxiliary variable method for the time-dependent Ginzburg-Landau equations

A generalized scalar auxiliary variable method for the time-dependent Ginzburg-Landau equations

A convex splitting method for the time-dependent Ginzburg-Landau equation

A convex splitting method for the time-dependent Ginzburg-Landau equation

Fully discrete scheme for a time-dependent Ginzburg-Landau equation in macromolecular microsphere composite hydrogels

In this article, we propose and analyze a fully discrete scheme for the time-dependent Ginzburg-Landau equation, which describes the phase field evolution of macromolecular microsphere composite hydrogels. This fully discrete scheme is a combination of the mass lumped finite element approximation for spatial discretization and the first order backward Euler scheme for temporal discretization. Besides, the convex-concave decomposition is adopted technically in the treatment of energy functional, where the concave part is treated explicitly and the convex part is treated implicitly. Moreover, we obtain the unique solvability and unconditional energy stability of the scheme. Finally, several numerical experiments are presented to illustrate the convergence order and effectiveness of the presented full-discrete scheme.

Combined FEM and phase field method for reliability design of forward degradation in SiC bipolar device

The reliability of 4H-SiC bipolar devices is compromised by the expansion of single Shockley stacking faults (SSFs) during forward-current operation. Because SSF expansion is governed by multiphysical aspects, including electrical, thermal, and stress states, analysis of the mounted structure is important for improving power module design. We propose a practical design method that analyzes the critical condition due to SSF expansion using a combined method with a multiphysical finite element method (FEM) and phase field model based on the time-dependent Ginzburg–Landau equation. In preliminary studies, the thermal deformation of the demonstration module and the variation of threshold current of a bar-shaped SSF were verified from experimental and reference data. Estimating the SSF expansion rate on the constructed response surface under the mutiphysical inputs from FEM, the proposed design method can be used effectively in the design process by changing the various design variables.

Helicity-controlled switching of superconducting states by radiation pulse

We provide a theoretical analysis of the possibility of using circularly polarized radiation to switch between two different quantum states of a superconducting nanoring subjected to the half quantum flux. Numerical modeling, performed in the framework of the time-dependent Ginzburg–Landau equation, reveals the condition for on-demand switching between current-carrying states with different helicities with 100% probability. An experimental demonstration of this effect will be unambiguous evidence of the inverse Faraday effect in superconductors and pave the way for the design of light-operated superconducting devices.

Fast Multigrid Simulations of Pinning in REBCO With Highly Resistive Nanorods

A geometric multigrid method is applied to the frozen-field time-dependent Ginzburg–Landau equations for anisotropic superconductors with spatially varying material properties. This reduces the simulation time for critical current determination in large 3D systems by up to two orders of magnitude compared with a single grid. Critical current density calculations are presented for a model of REBCO tape with nanorod artificial pinning centers. The model consists of an anisotropic superconducting matrix with embedded nanorods which are treated as isotropic highly resistive cylinders. The effects of varying the density, splay, and conductivity of the nanorods are considered. Increasing the resistivity of the nanorods profoundly affects the critical current density in high fields, where current paths are percolative in the vicinity of matrix-pin interfaces and persist well above the upper critical field of the superconducting matrix.

Second-harmonic generation in fluctuating Ising superconductors

In a two-dimensional noncentrosymmetric transition metal dichalcogenide Ising superconductor in the fluctuating regime under the action of a uniform external electromagnetic field, a second-harmonic generation (SHG) effect takes place. There emerge two contributions to this effect, one conventional, which is due to the electron gas in its normal state, and the other one is of the Aslamazov–Larkin nature. Namely, it originates from the presence of fluctuating Cooper pairs in the system when the temperature approaches the temperature of the superconducting transition in the sample from above. Employing a usual approach to Ising superconductors, we lift the valley degeneracy by application of a weak out-of-plane external magnetic field, which produces a Zeeman effect. In calculations, we use the Boltzmann equations approach for the electron gas in the normal state, and the time-dependent Ginzburg–Landau equations for the fluctuating Cooper pairs, and show the emergence of SHG electric current characterized by a temperature-dependent broadening and a redshift.

Fluctuation-mediated inverse Faraday effect in superconducting rings

We study the generation of supercurrent by circularly polarized electromagnetic wave in a superconducting ring in fluctuation regime at T>Tc. Using the time-dependent Ginzburg-Landau equation with a complex relaxation constant we calculate the dc component of the supercurrent, which is the manifestation of the inverse Faraday effect. The observed strong dependence of the current on the radiation frequency and temperature can be additionally enhanced by the application of an external magnetic flux and may represent a distinct feature noticeable in experiments.

A new error analysis of backward Euler Galerkin finite element method for two-dimensional time-dependent Ginzburg–Landau equation

In this paper, the backward Euler Galerkin finite element method (FEM) is investigated for the two dimensional time-dependent Ginzburg–Landau equation. The unconditionally optimal L2 error estimate is obtained without using the boundedness of the numerical solution in L∞ norm, while it is an indispensable requirement in the previous works. A key to the analysis is to deal with the nonlinear term rigorously and skillfully in terms of the boundedness of the numerical solution in L2 norm rather than the L∞ norm. Numerical results are presented to confirm the theoretical findings.

Extracting quantum-geometric effects from Ginzburg-Landau theory in a multiband Hubbard model

We first apply functional-integral approach to a multiband Hubbard model near the critical pairing temperature, and derive a generic effective action that is quartic in the fluctuations of the pairing order parameter. Then we consider time-reversal-symmetric systems with uniform (i.e., at both low-momentum and low-frequency) pairing fluctuations in a unit cell, and derive the corresponding time-dependent Ginzburg-Landau (TDGL) equation. In addition to the conventional intraband contribution that depends on the derivatives of the Bloch bands, we show that the kinetic coefficients of the TDGL equation have a geometric contribution that is controlled by both the quantum-metric tensor of the underlying Bloch states and their band-resolved quantum-metric tensors. Furthermore we show that thermodynamic properties such as London penetration depth, GL coherence length, GL parameter and upper critical magnetic field have an explicit dependence on quantum geometry.

Size effect of randomly distributed pinning centers on magnetic vortices motion in type-Ⅱ superconductor

We consider the size effect of randomly distributed pinning centers on magnetic vortices motion in a mesoscopic type-Ⅱ superconductor in the present of magnetic field using the time-dependent Ginzburg-Landau (TDGL) equations. The pinning centers distributed randomly in various contents in the system are characterized by a distribution function p(r) which fixes the size and positions of them. We investigate the magnetization curves and the magnetic induction as a function of the external magnetic field along the z-axis. In addition, the density of superconducting electrons and the maximum magnetization values for various sizes of the pinning centers is considered. The size of the pinning centers obviously affects the dynamics of the magnetic vortices in the superconductor. The numerical simulations show that Magnetization values and the stable vortex configuration depend on not only content but also size of the pinning centers. The smaller size R (or w) by which the shape of them is characterized results in an increase (or decrease) of both the lower critical magnetic field H1 and the maximum magnetization value. The present work will show good results into understanding and studying the size effect of pinning centers on dynamics of the magnetic vortices in a mesoscopic type-Ⅱ superconductor.

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.

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.

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.

Universal time-dependent Ginzburg-Landau theory

We study the hydrodynamics of superconductors within the framework of Schwinger-Keldysh effective field theory (EFT). We show that in the vicinity of the superconducting phase transition the most general leading-order EFT satisfying the local Kubo-Martin-Schwinger condition is described by a version of the time-dependent Ginzburg-Landau (TDGL) equations augmented with stochastic terms. This version of TDGL is applicable in the gapless regime independent of any microscopic details. Within this approach, it is possible to include systematically the effects of nonuniform temperature and heat conductivity, as well as explicit or spontaneous breaking of time reversal. We also introduce a thermal version of the Josephson relation and use it to construct an exotic hydrodynamics describing a phase of matter where heat can flow without dissipation.

Optimal Analysis of Non-Uniform Galerkin-Mixed Finite Element Approximations to the Ginzburg–Landau Equations in Superconductivity

This paper is concerned with new error analysis of a lowest-order backward Euler Galerkin-mixed finite element method for the time-dependent Ginzburg–Landau equations. The method is based on a commonly-used nonuniform approximation, in which a linear Lagrange element, the lowest-order Nédélec edge element, and the Raviart–Thomas face element are used for the order parameter , the magnetic field , and the magnetic potential , respectively. This mixed method has been widely used in practical simulations due to its low cost and ease of implementation. In the Ginzburg–Landau model, the order parameter is the most important variable, which indicates the state of the superconductor. An important feature of the method is the inconsistency of the approximation orders. A crucial question is how the first-order approximation of influences the accuracy of . The main purpose of this paper is to establish the second-order accuracy for the order parameter in a spatial direction, although the accuracy for is first order only. Previous analysis only gave the first-order convergence for all three variables due to certain artificial pollution involved in the analysis. Our analysis is based on a nonstandard quasi-projection for and the corresponding more precise estimates, including the -norm. With the quasi-projection, we prove that the lower-order approximation to does not pollute the accuracy of . Our numerical experiments confirm the optimal convergence of . The approach can be extended to many other multiphysics models.

Stability of interlinked neutron vortex and proton flux-tube arrays in a neutron star – III. Proton feedback

ABSTRACT The coupled, time-dependent Gross–Pitaevskii and Ginzburg–Landau equations are solved simultaneously in three dimensions to investigate the equilibrium state and far-from-equilibrium, spin-down dynamics of an interpenetrating neutron superfluid and proton type-II superconductor, as an idealized description of the outer core of a neutron star. The simulations generalize previous calculations without the time-dependent Ginzburg–Landau equation, where proton feedback is absent. If the angle θ between the rotation and magnetic axes does not equal zero, the equilibrium state consists of geometrically complicated neutron vortex and proton flux-tube tangles, as the topological defects pin to one another locally but align with different axes globally. During spin-down, new types of motion are observed. For θ = 0, entire vortices pair rectilinearly with flux tubes and move together while pinned. For θ ≠ 0, vortex segments pair with segments from one or more flux tubes, and the paired segments move together while pinned. The degree to which proton feedback impedes the deceleration of the crust is evaluated as a function of θ and the pinning strength, η. Key geometric properties of vortex-flux-tube tangles, such as filament length, mean curvature, and polarity are analysed. It is found that proton feedback smooths the deceleration of the crust, reduces the rotational glitch sizes, and stabilizes the vortex tangle dynamics. The dimensionless control parameters in the simulations are mutually ordered to match what is expected in a real neutron star, but their central values and dynamic ranges differ from reality by many orders of magnitude due to computational limitations.