Time-dependent Ginzburg-Landau Equation Research Articles

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.

Magnon-Fluxon Interaction in Coupled Superconductor/Ferromagnet Hybrid Periodic Structures

We study the magnon propagation in a micrometer-size ferromagnetic waveguide affected by the proximity to a superconductor that hosts a vortex formation. By solving the time-dependent Ginzburg-Landau equations of superconductivity, we obtain the equilibrium state of the vortex configuration and the associated stray fields that act on the magnetic dynamics. Spin wave dynamics in the presence of the superconductors are inferred from the Landau-Lifschitz equation. For a quantitative comparison, the simulations are performed under the experimental conditions as reported by Dobrovolskiy et al. (Nature Physics, 15, 477 (2019)). We found that the presence of the vortex lattice leads to the formation of a Bloch-like band structure in the magnon spectrum. The width and the number of allowed bands are found to depend mainly on the in-plane component of the vortex field along the waveguide. A shift to lower energies of the lowest-order-allowed bands occurs due to the confinement of the magnonic modes above the locations of the superconductor vortices. We also studied the role of the films thickness. At least two GHz-frequency backward-volume magnetostatic spin waves modes are observed at the top and bottom surfaces (in addition to these magnetostatic surface modes, the vertical confinement can also give rise to longer-wavelength and low-frequency perpendicular standing spin waves). Effects of the vortex fields are found to depend strongly on the type of the magnon modes. It is found that the effect of the superconducting vortices of the spin waves can be emulated well by the stray fields of air-separated magnetic nanocubes of appropriate dimensions.

High-frequency diode effect in superconducting Nb3Sn microbridges

The superconducting diode effect has recently been reported in a variety of systems and different symmetry-breaking mechanisms have been examined. However, the frequency range of these potentially important devices still remains obscure. We investigated superconducting microbridges of ${\mathrm{Nb}}_{3}\mathrm{Sn}$ in out-of-plane magnetic fields; optimum magnetic fields of $\ensuremath{\sim}$10 mT generate $\ensuremath{\sim}10%$ diode efficiency, while higher fields of $\ensuremath{\sim}$15--20 mT quench the effect. The diode changes its polarity with magnetic field reversal. We documented superconductive diode rectification at frequencies up to 100 kHz, the highest reported as of today. Interestingly, the bridge resistance during diode operation reaches a value that is a factor of two smaller than in its normal state, which is compatible with the vortex-caused mechanism of resistivity. This is confirmed by finite-element modeling based on time-dependent Ginzburg-Landau equations. To explain experimental findings, no assumption of lattice thermal inequilibrium has been required. Dissimilar edges of the superconductor strip can be responsible for the inversion symmetry breaking by the vortex penetration barrier; visual evidence of this opportunity was revealed by scanning electron microscopy. Estimates are in favor of a much higher (GHz) range of frequencies for this type of diode.

Effect of mesoscopic defect on vortex dynamics of prestrained type-II superconducting film

In this paper, we numerically investigate the vortex dynamics of type-II superconducting films containing mesoscopic defects based on the time-dependent Ginzburg-Landau (TDGL) equation modified by linear deformation theory. In this study, we also take into account the effect of prestrain on the vortex properties of superconducting thin films. The effect of defects on the wave function, flux motion, and energy in specific regions of the superconducting film is analyzed by solving the TDGL equation under an applied magnetic field. The results show that the presence or absence of defects in the presence of prestrain affects the vortex flow in superconducting thin films significantly. It influences both the quantity of vortices and the time it takes for them to develop. A section of the energy fluctuation of the thin film is also investigated. The findings demonstrate that mesoscopic flaws significantly affect the development of vortices in superconductors when linear elastic deformation is taken into account. When analyzing the superconductivity and electromagnetic characteristics of superconductors, mesoscopic flaws must be taken into consideration.

An efficient iterative method for dynamical Ginzburg-Landau equations

In this paper, we propose a new finite element approach to simulate the time-dependent Ginzburg-Landau equations under the temporal gauge, and design an efficient preconditioner for the Newton iteration of the resulting discrete system. The new approach solves the magnetic potential in H(curl) space by the lowest order of the second kind Nédélec element. This approach offers a simple way to deal with the boundary condition, and leads to a stable and reliable performance when dealing with the superconductor with reentrant corners. The comparison in numerical simulations verifies the efficiency of the proposed preconditioner, which can significantly speed up the simulation in large-scale computations.

Photogalvanic transport in fluctuating Ising superconductors

In a two-dimensional noncentrosymmetric Ising superconductor in the fluctuating regime under the action of a uniform external electromagnetic field there emerge two contributions to the photogalvanic effect due to the trigonal warping of the valleys. The first contribution stems from the current of the electron gas in its normal state, while the second contribution is of Aslamazov-Larkin nature: it originates from the presence of fluctuating Cooper pairs when the ambient temperature approaches (from above) the temperature of superconducting transition in the sample. The way to lift the valley degeneracy is the application of a weak out-of-plane external magnetic field producing a Zeeman effect. The Boltzmann equations approach for the electron gas in the normal state and the time-dependent Ginzburg-Landau equations for the fluctuating Cooper pairs allow for the study of the photogalvanic current in two-dimensional transition metal dichalcogenide Ising superconductors.

Vortices in a Ginzburg-Landau theory of superconductors with nematic order

In this work we explore the interplay between superconductivity and nematicity in the framework of a Ginzburg-Landau theory with a nematic order parameter coupled to the superconductor order parameter. In particular, we focus on the study of the vortex-vortex interaction in order to determine the way nematicity affects its attractive or repulsive character. To do so, we use a dynamical method based on the solutions of the time-dependent Ginzburg-Landau equations in a bulk superconductor. An important contribution of our work is the implementation of a pseudospectral method to solve the dynamics, known to be highly efficient and of very high order in comparison to the usual finite-differences and -elements methods. The coupling between the superconductor and the (real) nematic order parameters is represented by two terms in the free energy: a biquadratic term and a coupling of the nematic order parameter to the covariant derivatives of the superconductor order parameter. Our results show that there is a competing effect: while the former independently of its competitive or cooperative character generates an attractive vortex-vortex interaction, the latter always generates a repulsive interaction.

Anti-Poiseuille flow: Increased vortex velocity at superconductor edges

Using the time-dependent Ginzburg Landau equations we study vortex motion driven by an applied current in two dimensional superconductors in the presence of a physical boundary. At smaller sourced currents the vortex lattice moves as a whole, with each vortex moving at the same velocity. At larger sourced current, vortex motion is organized into channels, with vortices in channels nearer to the sample edges moving faster than those farther away from sample edges, opposite to the Poiseuille flow of basic hydrodynamics where the velocity is lowest at the boundaries. At intermediate currents, a stick-slip motion of the vortex lattice occurs in which vortices in the channel at the boundary break free from the Abrikosov lattice, accelerate, move past their neighbors and then slow down and reattach to the vortex lattice at which point the stick-slip process starts over. These effects could be observed experimentally, e.g. using fast scanning microscopy techniques.

Topological transitions in ac/dc-driven superconductor nanotubes

Extending of nanostructures into the third dimension has become a major research avenue in condensed-matter physics, because of geometry- and topology-induced phenomena. In this regard, superconductor 3D nanoarchitectures feature magnetic field inhomogeneity, non-trivial topology of Meissner currents and complex dynamics of topological defects. Here, we investigate theoretically topological transitions in the dynamics of vortices and slips of the phase of the order parameter in open superconductor nanotubes under a modulated transport current. Relying upon the time-dependent Ginzburg–Landau equation, we reveal two distinct voltage regimes when (i) a dominant part of the tube is in either the normal or superconducting state and (ii) a complex interplay between vortices, phase-slip regions and screening currents determines a rich FFT voltage spectrum. Our findings unveil novel dynamical states in superconductor open nanotubes, such as paraxial and azimuthal phase-slip regions, their branching and coexistence with vortices, and allow for control of these states by superimposed dc and ac current stimuli.

The dynamics of current-driven vortex in two-band superconductor with s+d wave pairing

Two-band mesoscopic superconductors are known to exhibit specific vortex states. In this paper, finite element method (FEM) is used to numerically solve the time-dependent Ginzburg-Landau (TDGL) equation with s+d wave coupling for a 3D mesoscopic superconducting strip. We discuss the effect of external magnetic field H and applied current I on s+d wave coupling at the temperatures Ts<T<Td, where Ts and Td are the critical temperatures of s- and d-bands, respectively. The obtained results show that this coupling is enhanced by the applied current and reflected by the vortex dynamics in two bands. A small amount of Cooper pairs in the s-band form a series of different vortex and antivortex states. Mutual attraction of vortex and antivortex induces simultaneous motion of s- and d-vortices at applied current I. This work provides theoretical background for the design of mesoscopic superconducting devices.

Vortex state in a superconducting mesoscopic irregular octagon

Our study sample is a superconducting bi-dimensional octagon with different boundary conditions immersed in a magnetic external field [Formula: see text]. The boundary conditions are simulated by considering different values of the de Gennes extrapolation length [Formula: see text] on different surfaces of the sample. Our investigation was carried out by solving the time-dependent Ginzburg–Landau equations. We analyzed the superconducting electron density and the magnetization curves as functions of [Formula: see text] and temperature [Formula: see text] in field-cooling and in zero-field-cooling processes for different values of [Formula: see text] and size of the sample. We found a strong dependence of the critical fields on [Formula: see text] and size of the sample.

Intrinsic 90° charged domain wall and its effects on ferroelectric properties

The domain is an important characteristic of ferroelectric materials, and the type of domain can have significant effects on the performance of ferroelectric materials. In particular, the unique wake-up and fatigue effects of Hf0.5Zr0.5O2 (HZO) films severely restrict improvements in terms of film performance. Recently, using integrated differential phase contrast (iDPC) scanning transmission electron microscopy (STEM) imaging, we observed the existence of intrinsic tail-to-tail 90° domain structures, charged domain walls (CDWs), and polarization relaxation on the TiN/HZO/TiN interfaces of HZO films. Based on this experimental investigation, we establish a phase-field model of the tail-to-tail 90°CDWs based on the time-dependent Ginzburg–Landau equation. In our model, the tail-to-tail 90°CDW affects the mesoscopic domain switching through a built-in electric field, which, in turn, affects the macroscopic ferroelectric properties of the thin film. The following conclusions are drawn: (a) the CDWs that are distributed horizontally or vertically along the thickness of the film cause a A-wake-up effect (antiferroelectric-like wake-up) with double hysteresis loops or a S-wake-up effect (spindle-like wake-up), with thick portions in the middle and both ends points of the hysteresis loop; (b) the S-wake-up caused by the CDWs distributed along the thickness of the film performs badly in fatigue tests and is easier to break down. The model proposed herein reveals the mechanism of the wake-up effect in hafnium oxide-based ferroelectric films. It provides a theoretical basis for the development of a new type of hafnium oxide-based ferroelectric memory system.

Topological defects in open superconducting nanotubes after gradual and abrupt switching of the transport current and magnetic field

We analyze dynamics of the order parameter in superconductor open nanotubes under a strong transport current in an external homogeneous magnetic field using the time-dependent Ginzburg-Landau equation. Near the critical transport current, the dissipation processes are driven by vortex and phase-slip dynamics. A transition between the vortex and phase-slip regimes is found to depend on the external magnetic field only weakly if the magnetic field and/or transport current are switched on gradually. In the case of an abrupt switch-on of the magnetic field or transport current, the system can be triggered to the stable phase-slip regime within a certain window of parameters. Finally, a hysteresis effect in the current-voltage characteristics is predicted in superconductor open nanotubes.

Antiferromagnetic spin Seebeck effect across the spin-flop transition: A stochastic Ginzburg-Landau simulation

We investigate the antiferromagnetic spin Seebeck effect across the spin-flop transition in a numerical simulation based on the time-dependent Ginzburg-Landau equation for a bilayer of a uniaxial insulating antiferromagnet and an adjacent metal. By directly simulating the rate of change of the conduction-electron spin density $\mathbit{s}$ in the adjacent metal layer, we demonstrate that a sign reversal of the antiferromagnetic spin Seebeck effect across the spin-flop transition occurs when the interfacial coupling of $\mathbit{s}$ to the staggered magnetization $\mathbit{n}$ of the antiferromagnet dominates, whereas no sign reversal appears when the interfacial coupling of $\mathbit{s}$ to the magnetization $\mathbit{m}$ dominates. Moreover, we show that the sign reversal is influenced by the degree of spin dephasing in the metal layer. Our result indicates that the sign reversal is not a generic property of a simple uniaxial antiferromagnet, but controlled by microscopic details of the exchange coupling at the interface and the spin dephasing in the metal layer.

Violation of magnetic flux conservation by superconducting nanorings

The behavior of magnetic flux in ring-shaped finite-gap superconductors is explored from the view-point of the flux-conservation theorem, which states that under the variation of an external magnetic field ‘the magnetic flux through the ring remains constant’ (see, e.g., Landau and Lifshitz 1960 Electrodynamics of Continuous Media (New York: Pergamon)). Our results, based on the time-dependent Ginzburg–Landau equations and COMSOL modeling, made it clear that in the general case this theorem is incorrect. While for macroscopic rings the corrections are small, for micro- and nanorings they become rather substantial. The physical reasons behind this effect are discussed. The dependence of flux deviation on ring size, bias temperature and the speed of external flux evolution is explored. The detailed structure of flux distribution inside the ring opening, as well as the electric field distribution inside the ring wire cross section, are revealed. Our results and the developed finite element modeling approach can assist in elucidating various fundamental topics in superconducting nanophysics and in the advancement of nanosize superconducting circuits prior to time-consuming and costly experiments.

Variational Analysis of the Speed of Propagating Fronts in Superconductors

A new approach has taken to determine the speed of the superconducting-normal interface (front) propagating from a stable superconducting phase to the unstable normal phase by means of a variational method. In this process a trial function g(x) is introduced which has a relation with the superconducting order parameter F. Starting from the time-dependent Ginzburg-Landau (TDGL) equations an expression is derived for the front speed in terms of GL parameter κ. The Chittagong Univ. J. Sci. 43(1): 41-50, 2021

Dynamic peculiarities of interphase boundaries and domain walls for non-spin domains

The analysis of magnetization oscillations (de Haas – van Alphen effect; dHvA effect) and features of the Fermi surface shape in three (3D)- and two-dimensional (2D) metals shows that the dynamic phenomena — the dynamics of domain walls and interphase boundaries during diamagnetic phase transitions — are described by a new nonlinear partial differential equation. The equation is a result of the inclusion of the case of multiple extremal cross sections of the Fermi surface in these metals. Our consideration indicates the possibility of the appearance of metastable non-spin domains (Condon domains) and first-order phase transitions to the ordered phase in the regime of dHvA oscillations for the two-frequency case. Sine–Gordon, Klein–Gordon, double sine–Gordon, time-dependent Ginzburg–Landau equations, and the telegraph equation are limiting cases of the derived equation. We show that particular moving kink-soliton solutions of the equation describe traveling wave fronts being moving domain walls and interphase boundaries in the Condon domain phase.

Influence of a nonuniform thermal quench and circular polarized radiation on spontaneous current generation in superconducting rings

We theoretically study the nonequilibrium dynamics of the order parameter of a superconducting ring inhomogeneously quenched through its transition temperature. Numerical simulations based on spectral decomposition of the time-dependent Ginzburg-Landau equation reveal that current-carrying superconducting states can be generated in the ring under certain fast local temperature quench conditions. We also show that illumination of the ring with a circularly polarized electromagnetic radiation during rapid cooling strongly promotes the generation of current-carrying states with rotation directions controlled by the helicity of the radiation field.

Time-Dependent Landau-Ginzburg Equation-Based Ferroelectric Tunnel Junction Modeling With Dynamic Response and Multi-Domain Characteristics

Overcoming the drawbacks of the existing ferroelectric tunnel junction (FTJ) models which ignore the dynamic or multi-domain switching behaviors, we develop a more comprehensive FTJ model by combining the Time-Dependent Landau-Ginzburg (TDLG) equations to solve the multi-domain dynamic switching of ferroelectric layer and the Non-Equilibrium Green Function (NEGF) to solve the tunneling current. The model successfully reproduces the experimental results of our fabricated metal-ferroelectrics-insulator-semiconductor (MFIS) FTJ. This model empowers us to predict both the dynamic and multi-state switching of FTJ, showing its promise for applications in the high-density data storage and analog computing.