Time-dependent Ginzburg Research Articles

Dynamics of vortex–antivortex creation and annihilation in current-driven mesoscopic superconducting squares

In this paper, based on the time-dependent Ginzburg–Landau theory, we study the dynamics of vortex–antivortex (V–Av) pairs in a mesoscopic superconducting square with a small hole under applied bias currents. For the sample with a centered hole, a V–Av pair can nucleate at the hole edges and moves in opposite directions perpendicular to applied constant DC drive. The influence of the external magnetic field on the (anti)vortex velocity and the lifetime of V–Av pairs is mainly investigated. Different modes in the dynamical process of the V–Av collision and annihilation are identified. Moreover, in the case when the hole is displaced from the center of the square, the V–Av dynamics behaves quite differently from the symmetric case due to the shift of the V–Av creation point.

Theoretical prediction of the minimum height for negligible demagnetization field in a superconducting sample

The limit below which the magnetization curve of a mesoscopic superconducting prism of square cross-section and finite height approximates the characteristic bi-dimensional limit, e.g. long prism, which with both the magnetic induction and the superconducting electron density are invariant along the direction where the magnetic external field is applied, was numerically found. Using the three- and two-dimensional time-dependent Ginzburg–Landau models, we have determined an analytical dependence of this height limit as a function of the lateral size of the sample.

Superconducting properties of a mesoscopic parallelepiped with anisotropic surface conditions

We consider a mesoscopic superconducting parallelepiped with different boundary conditions on different parts of the surface, xy, xz and yz surface planes. This is realized by considering different values of the de Gennes extrapolation length b on different surfaces of the sample. Our investigation was carried out by solving the three-dimensional (3D) time dependent Ginzburg–Landau (TDGL) equations. We studied the local magnetic field, order parameter, and both the magnetization and vorticity curves as functions of the external applied magnetic field for different values of b on the surfaces of the sample. We show that this surface anisotropy has very strong influence on the vortex configurations and the magnetization as a function of the external applied magnetic field, both experimentally accessible.

Asymptotic global fast dynamics and attractor fractal dimension of the TDGL-equations of superconductivity

In this paper, we study a model system of equations of the time dependent Ginzburg–Landau equations of superconductivity in a Lorentz gauge, in scale of Hilbert spaces \(E^{\alpha }\) with initial data in \(E^{\beta }\) satisfying \(3\alpha + \beta \ge \frac{N}{2}\), where \(N=2,3\) is such that the spatial domain of the equations Open image in new window. We show in the asymptotic dynamics of the equations, well-posedness of the dynamical system for a global exponential attractor \({\mathcal {U}}\subset E^{\alpha }\) compact in \(E^{\beta }\) if \(\alpha >\beta \), uniform differentiability of orbits on the attractor in \(E^{0}\cong L^{2}\), and the existence of an explicit finite bounding estimate on the fractal dimension of the attractor yielding that its Hausdorff dimension is as well finite. Uniform boundedness in \((0,\infty )\times \Omega \) of solutions in \(E^{\frac{1}{2}}\cong H^{1}(\Omega )\) is in addition investigated.

A new approach for numerical simulation of the time-dependent Ginzburg–Landau equations

We introduce a new approach for finite element simulations of the time-dependent Ginzburg–Landau equations (TDGL) in a general curved polygon, possibly with reentrant corners. Specifically, we reformulate the TDGL into an equivalent system of equations by decomposing the magnetic potential to the sum of its divergence-free and curl-free parts, respectively. Numerical simulations of vortex dynamics show that, in a domain with reentrant corners, the new approach is much more stable and accurate than the traditional approaches of solving the TDGL directly (under either the temporal gauge or the Lorentz gauge); in a convex domain, the new approach gives comparably accurate solutions as the traditional approaches.

Thermal fluctuations in 1D superconducting samples

For a system that obeys diffusive evolution, an additional term due to thermal fluctuations should be added. We present an intuitive derivation of the distribution of this term. The formalism developed to deal with fluctuations can be incorporated into two models that are widely used in the dynamics of superconductors: the time-dependent Ginzburg–Landau model and the Kramer–Watts-Tobin model. We review three problems that can be studied with these models: supercurrent flow around a ring close to Tc; fluctuations of the supercurrent in a filament that bridges between two bulk superconductors with order parameters at a given phase difference; and the Kibble–Zurek mechanism, intended to describe phase transitions that are governed by kinetics rather than by thermodynamics, in the case of a superconducting loop.

Vortex States in Nanosized Superconducting Strips with Weak Links Under an External Magnetic Field

The time-dependent Ginzburg–Landau equations have been solved numerically by a finite element analysis for the nanosized superconducting strips with one central weak link. Anisotropy is included through the spatially dependent anisotropy coefficient ζ in different layers of the sample. For given applied magnetic fields, we have simulated the dynamical behavior of the penetrating magnetic vortices into the superconductor. Our results show the energy barrier for vortices to enter a weak link is smaller than that for vortices to enter superconducting layers. The final distribution of vortices is determined by the competing interactions of vortices with Meissner currents and the weak link boundaries.

Three-Dimensional Phase Field Simulations of Hysteresis and Butterfly Loops by the Finite Volume Method **Supported by the Research Starting Funds for Imported Talents of Ningxia University under Grant No BQD2012011.

Three-dimensional simulations of ferroelectric hysteresis and butterfly loops are carried out based on solving the time dependent Ginzburg–Landau equations using a finite volume method. The influence of externally mechanical loadings with a tensile strain and a compressive strain on the hysteresis and butterfly loops is studied numerically. Different from the traditional finite element and finite difference methods, the finite volume method is applicable to simulate the ferroelectric phase transitions and properties of ferroelectric materials even for more realistic and physical problems.

Comparison of flux motion in type-II superconductors including pinning centers with the shapes of nano-rods and nano-particles by using 3D-TDGL simulation

Time-dependent Ginzburg–Landau (TDGL) equations are very useful method for simulation of the motion of flux quanta in type-II superconductors. We constructed the 3D-TDGL simulator and succeeded to simulate the motion of flux quanta in 3-dimension. We carried out the 3D-TDGL simulation to compare two superconductors which included only pinning centers with the shape of nano-rods and only nano-particle-like pinning centers in the viewpoint of the flux motion. As a result, a motion of “single-kink” caused the whole motion of a flux quantum in the superconductor including only the nano-rods. On the other hand, in the superconductor including the nano-particles, the flux quanta were pinned by the nano-particles in the various magnetic field applied angles. As the result, no “single-kink” occurred in the superconductor including the nano-particles. Therefore, the nano-particle-like pinning centers are effective shape to trap flux quanta for various magnetic field applied angles.

Perfectly Ordered Patterns via Corner-Induced Heterogeneous Nucleation of Self-Assembling Block Copolymers Confined in Hexagonal Potential Wells

The ordering dynamics of cylinder-forming diblock copolymer/homopolymer blends confined in hexagonal potential wells is systematically investigated using time-dependent Ginzburg–Landau (TDGL) theory. It is demonstrated that a high-efficient method to obtain large-scale ordered hexagonal patterns is to utilize corner-induced heterogeneous nucleation processes, in which nucleation events with controlled positions and orientations are triggered exclusively at the six corners of the confining hexagonal wells. Subsequent growth of the six domains originated from the corners leads to the formation of perfectly ordered patterns occupying the entire hexagonal well. The heterogeneous nucleation rate is regulated by the homopolymer concentration as well as the surface potential of the confining walls. Defect-free hexagonal patterns are obtained in hexagons with a diagonal size containing up to 61 cylinders (about 2 μm). The robustness of the method is examined by studying the tolerance window of the size-commensura...

Vortices in a wedge made of a type-I superconductor

Using the time-dependent Ginzburg–Landau approach, we analyze vortex states and vortex dynamics in type-I superconductor films with a thickness gradient in one direction. In the thinnest part of the structures under consideration, the equilibrium states manifest the typical type-II vortex patterns with only singly-quantized vortices. At the same time, in the regions with larger thickness the singly-quantized and giant vortices coexist, in a qualitative agreement with our scanning Hall probe microscopy measurements on relatively thick Pb films. In the presence of an external current applied perpendicularly to the thickness gradient direction, the singly-quantized vortices, which enter the wedge through its thinnest edge, merge into giant vortices when propagating to the thicker parts of the structure. Remarkably, the results of our simulations imply that at moderate external current densities a regime is possible where the winding number of giant vortices, formed as a result of vortex coalescence, takes preferentially (or even exclusively) the values given by positive integer powers of two.

Superconducting slab in an antisymmetric magnetic field: Vortex–antivortex dynamics

When a type II superconductor with slab geometry is subjected to a magnetic field which is antisymmetric with respect to the middle of the slab the induced vortex matter consists of vortex–antivortex pairs or double kinks. These double kinks and their role in the generation of a considerable asymmetry in the critical current of the slab are addressed here both numerically, in the framework of time dependent Ginzburg Landau model, and semi-analytically, using the concept of surface energy barriers for flux entry and flux exit.

Effect of Dzyaloshinsky–Moriya interaction on magnetic vortex: A real-space phase-field study

A real-space phase field model is developed to investigate the effect of Dzyaloshinsky–Moriya (DM) interaction on magnetic vortex in ultrathin ferromagnetic film. Based on the time-dependent Ginzburg–Landau (TDGL) equation, the phase field model takes into account the DM interaction of magnetizations, which includes the coupling between the magnetization and magnetization gradient. The governing equations in the phase field model are solved simultaneously by means of a nonlinear finite elements method, which can be employed to simulate magnetization vortex without periodic boundary condition. The simulation results demonstrate that the DM interaction has significant influence on the structure of the magnetization vortex. It is found that the DM interaction induces an out-of-plane magnetization on the edge of the vortex and enlarges the size of the vortex core. The magnitude of the out-of-plane magnetization at the vortex core and the edge increases with the increase of DM constant. Besides, the handedness of magnetization vortex also changes when the sign of DM constant changes.

Phase-field model coupling cracks and dislocations at finite strain

A new 3D continuum model is developed to simultaneously describe cracks and dislocations in materials undergoing finite strain deformations. Significant advances are made to account for fracturing in a cubic symmetrical system and to correctly model the microscopic dislocation properties into a Time-Dependent Ginzburg–Landau formalism. In addition, the interaction between dislocations and free borders (crack lips, surfaces…) is naturally described. As an example, the model is used to investigate the effects of plasticity on the delamination and buckling of thin films deposited on substrates. It may also be helpful in any other situation where cracks and dislocations take place and in which finite strain effects must be considered.

Stable large-scale solver for Ginzburg–Landau equations for superconductors

Understanding the interaction of vortices with inclusions in type-II superconductors is a major outstanding challenge both for fundamental science and energy applications. At application-relevant scales, the long-range interactions between a dense configuration of vortices and the dependence of their behavior on external parameters, such as temperature and an applied magnetic field, are all important to the net response of the superconductor. Capturing these features, in general, precludes analytical description of vortex dynamics and has also made numerical simulation prohibitively expensive. Here we report on a highly optimized iterative implicit solver for the time-dependent Ginzburg–Landau equations suitable for investigations of type-II superconductors on massively parallel architectures. Its main purpose is to study vortex dynamics in disordered or geometrically confined mesoscopic systems. In this work, we present the discretization and time integration scheme in detail for two types of boundary conditions. We describe the necessary conditions for a stable and physically accurate integration of the equations of motion. Using an inclusion pattern generator, we can simulate complex pinning landscapes and the effect of geometric confinement. We show that our algorithm, implemented on a GPU, can provide static and dynamic solutions of the Ginzburg–Landau equations for mesoscopically large systems over thousands of time steps in a matter of hours. Using our formulation, studying scientifically-relevant problems is a computationally reasonable task.

An efficient fully linearized semi-implicit Galerkin-mixed FEM for the dynamical Ginzburg–Landau equations of superconductivity

The paper focuses on numerical study of the time-dependent Ginzburg–Landau (TDGL) equations under the Lorentz gauge. The proposed method is based on a fully linearized backward Euler scheme in temporal direction, and a mixed finite element method (FEM) in spatial direction, where the magnetic field σ=curlA is introduced as a new variable. The linearized Galerkin-mixed FEM enjoys many advantages over existing methods. In particular, at each time step the scheme only requires the solution of two linear systems for ψ and (σ,A), respectively, with constant coefficient matrices. These two matrices can be pre-assembled at the initial time step and these two linear systems can be solved simultaneously. Moreover, the method provides the same order of optimal accuracy for the density function ψ, the magnetic potential A, the magnetic field σ=curlA, the electric potential divA and the current curlσ. Extensive numerical experiments in both two- and three-dimensional spaces, including complex geometries with defects, are presented to illustrate the accuracy and stability of the scheme. Our numerical results also show that the proposed method provides more realistic predictions for the vortex dynamics of the TDGL equations in nonsmooth domains, while the vortex motion influenced by a defect of the domain is of high interest in the study of superconductors.

Instability in the magnetic field penetration in type II superconductors

Abstract Under the view of the time-dependent Ginzburg–Landau theory we have investigated the penetration of the magnetic field in the type II superconductors. We show that the single vortices, situated along the borderline, between the normal region channel and the superconducting region, can escape to regions still empty of vortices. We show that the origin of this process is the repulsive nature of vortex–vortex interaction, in addition to the non-homogeneous distribution of the vortices along the normal region channel. Using London theory we explain the extra gain of kinetic energy by the vortices situated along this borderline.

The car following model considering traffic jerk

Based on optimal velocity car following model, a new model considering traffic jerk is proposed to describe the jamming transition in traffic flow on a highway. Traffic jerk means the sudden braking and acceleration of vehicles, which has a significant impact on traffic movement. The nature of the model is researched by using linear and nonlinear analysis method. A thermodynamic theory is formulated to describe the phase transition and critical phenomenon in traffic flow. The time-dependent Ginzburg–Landau (TDGL) equation and the modified Korteweg–de Vries (mKdV) equation are derived to describe the traffic flow near the critical point and the traffic jam. In addition, the connection between the TDGL and the mKdV equations are also given.

Geometric effect on the vortex configuration and motion in mesoscopic superconducting cylinders

Based on the time-dependent Ginzburg–Landau equations, we study numerically the vortex configuration and motion in mesoscopic superconducting cylinders. We find that the effects of the geometric symmetry of the system and the noncircular multiply-connected boundaries can significantly influence the steady vortex states and the vortex matter moving. For the square cylindrical loops, the vortices can enter the superconducting region in multiples of 2 and the vortex configuration exhibits the axial symmetry along the square diagonal. Moreover, the vortex dynamics behavior exhibits more complications due to the existed centered hole, which can lead to the vortex entering from different edges and exiting into the hole at the phase transitions.

Effect of pinning on the response of superconducting strips to an external pulsed current

Using the anisotropic time-dependent Ginzburg–Landau theory we study the effect of ordered and disordered pinning on the time response of superconducting strips to an external current that switched on abruptly. The pinning centers result in a considerable delay of the response time of the system to such abrupt switching on of the current, whereas the output voltage is always larger when pinning is present. The resistive state in both cases are characterized either by dynamically stable phase-slip centers/lines or expanding in-time hot-spots, which are the main mechanisms for dissipation in current-carrying superconductors. We find that hot-spots are always initiated by the phase-slip state. However, the range of the applied current for the phase-slip state increases significantly when pinning is introduced. Qualitative changes are observed in the dynamics of the superconducting condensate in the presence of pinning.