Time-dependent Ginzburg-Landau Equation Research Articles

Sudden generation of anomalous anti-vortex states in a two-component superconductor

We studied the influences of the inclusion of different geometrical defects (circle, triangle, and square) with different Ginzburg–Landau parameters [Formula: see text] on the vortex state of a mesoscopic superconducting square immersed in an external applied magnetic field. We calculated the magnetization, vorticity, and density of Cooper pairs for this system, solving the time-dependent Ginzburg–Landau equations. We found a novel and interesting behavior of the vorticity [Formula: see text] at low magnetic fields: a spontaneous generation of anti-vortices due to the breaking inversion symmetry.

A Theoretical Study of Multidomain Ferroelectric Switching Dynamics With a Physics-Based SPICE Circuit Model for Phase-Field Simulations

In this paper, the multi-domain nature of ferroelectric (FE) polarization switching dynamics in a metal-ferroelectric-metal (MFM) capacitor is explored through a physics-based phase field approach, where the three-dimensional time-dependent Ginzburg-Landau (TDGL) equation and Poisson's equation are self-consistently solved with the SPICE simulator. Systematically calibrated based on the experimental measurements, the model well captures transient negative capacitance in pulse switching dynamics, with domain interaction and viscosity being the key parameters. It is found that the influence of pulse amplitudes on voltage transient behaviors can be attributed to the fact that the FE free energy profile strongly depends on how the domains are interacted. This finding has an important implication on the charge-boost induced by stabilization of negative capacitance in an FE + dielectric (DE) stack since the so-called capacitance matching needs to be designed at a specific operation voltage or frequency. In addition, we extract the domain viscosity dynamics during polarization switching according to the experimental measurements. For the first time, a physics-based circuit-compatible SPICE model for multi-domain phase field simulations is established to reveal the effect of domain interaction on the FE energy profile and microscopic domain evolution.

Theory for the response of a superconducting kinetic inductance detector to an electromagnetic wave packet

Voltage pulses generated by local heating due to the irradiation of an electromagnetic (EM) wave packet in a superconducting current-biased kinetic inductance detector are theoretically investigated on the basis of our previous theory [J. Phys. Conf. Ser. 1293, 012050]. The growth and decay of a hot spot created by an EM wave packet is described with the heat diffusion equation coupled with the time-dependent Ginzburg-Landau (TDGL) equation. Variation in the kinetic inductance of the current-biased stripline in the detector induced by a hot spot is explicitly derived. We clarify the characteristic feature in the shape of the voltage pulses.

Multi-Domain Negative Capacitance Effects in Metal-Ferroelectric-Insulator-Semiconductor/Metal Stacks: A Phase-field Simulation Based Study

We analyze the ferroelectric domain-wall induced negative capacitance (NC) effect in Metal-FE-Insulator-Metal (MFIM) and Metal-FE-Insulator-Semiconductor (MFIS) stacks through phase-field simulations by self-consistently solving time-dependent Ginzburg Landau equation, Poisson’s equation and semiconductor charge equations. Considering Hf0.5Zr0.5O2 as the ferroelectric material, we study 180° ferroelectric domain formation in MFIM and MFIS stacks and their polarization switching characteristics. Our analysis signifies that the applied voltage-induced polarization switching via soft domain-wall displacement exhibits non-hysteretic characteristics. In addition, the change in domain-wall energy, due to domain-wall displacement, exhibits a long-range interaction and thus, leads to a non-homogeneous effective local negative permittivity in the ferroelectric. Such effects yield an average negative effective permittivity that further provides an enhanced charge response in the MFIM stack, compared to Metal-Insulator-Metal. Furthermore, we show that the domain-wall induced negative effective permittivity is not an intrinsic property of the ferroelectric material and therefore, is dependent on its thickness, the gradient energy coefficient and the in-plane permittivity of the underlying insulator. Similar to the MFIM stack, MFIS stack also exhibits an enhanced charge/capacitance response compared to Metal-Oxide-Semiconductor (MOS) capacitor. Simultaneously, the multi-domain state of the ferroelectric induces non-homogeneous potential in the underlying insulator and semiconductor layer. At low applied voltages, such non-homogeneity leads to the co-existence of electrons and holes in an undoped semiconductor. In addition, we show that with the ferroelectric layer being in the 180° multi-domain state, the minimum potential at the ferroelectric-dielectric interface and hence, the minimum surface potential in the semiconductor, does not exceed the applied voltage (in-spite of the local differential amplification and charge enhancement).

Microscopic mechanism of thermomagnetic instabilities in type-II superconducting thin films under AC magnetic fields

Thermomagnetic instability is one of the significant challenges for the application of superconducting devices. In this paper, the microscopic mechanism of thermomagnetic instability in superconducting films subjected to a transient AC magnetic field is numerically investigated by coupling the generalized time dependent Ginzburg–Landau equations and the heat diffusion equation. The influences of magnetic field ramp rate, ambient temperature, and nanometer-sized artificial pinning on the vortex matter are considered in our simulations. It has been found that vortex alignment and repulsion play significant roles in the branching of the penetration trajectories of the magnetic flux. Under fast ramping magnetic fields, the increase in the temperature and instability in the vortex matter are more significant. However, the rising temperature and jump size in the magnetization weaken as the ambient temperature increases. Pronounced hysteresis in the vortex dynamics has been found in the film subjected to AC magnetic fields. As the AC cycle proceeds, the vortex penetration process gets more unstable. We have also found that the nanometer sized pinning strongly modulates the penetration of vortices and the vortex matter is highly correlated with the lattice structure of the pinning sites. Our results provide new insights into vortex dynamics and give a mesoscopic understanding on the channeling and branching in the vortex penetration paths in superconductors under AC magnetic fields.

Critical properties of superconducting nano-sized sample placed on the top of a dipole

We deal with the nanoscale superconductivity. Vortex configurations of superconducting 3D nano disk and square placed on the top of a dipole are studied by phenomenological Ginzburg-Landau (GL) formalism. Gibbs free energy is observed to identify the stability range of different vortex state. Effect of the confinement to the vortex flow is investigated by time dependent GL equation. Amount of flux is determined by the potential difference and only giant vortex state are observed for the dipole. In the presence of transport current and magnetic field, voltage-current characteristics of nano-sized disk and square are obtained. Depairing current of the sample are studied with applied field. Maximum dissipation-less current of disk/square has shown the Quantum oscillation.

Explicit Integrators Based on a Bipartite Lattice and a Pair of Affine Transformations to Solve Quantum Equations with Gauge Fields

An explicit integrator is proposed to solve the time-dependent Ginzburg–Landau equation and the time-dependent Gross–Pitaevskii equation, which describe quantum dynamics with gauge fields. We const...

The heat dissipations and vortices motion in the open superconducting square tube

In this paper, we study the vortex dynamics of the open superconducting square tubes in the presence of external magnetic field by the time-dependent Ginzburg-Landau (TDGL) equations coupling with heat dissipation equation. The influences of geometry and the environment temperature are considered. It is found that the size and location of the hole have a significant effect on the vortex dynamics and configuration in the magnetic field increasing with time. However, they are more preferring to enter the samples with square hole due to the much larger current density at the inner corner, while the vortices can penetrate into the samples with circle hole along special directions such as the horizontal axes from the inner surface. When the thermal effect and the energy dissipation are considered, the energy dissipation caused by the vortex motion leads to a more concentrated distribution of the vortices and a local temperature rise which can attract the following vortices. And when the sample is in the constant magnetic field, the thermal effect also has an obvious effect on the vortex configuration and causes a large jump in number of vortices under the relatively high field.

Finite strain phase-field microelasticity theory for modeling microstructural evolution

Implementing the phase-field model at finite strains is usually considered unattainable through the spectral method, largely because of the nonlinearity in the transformation-induced microelasticity. Here we present a phase-field microelasticity (PFM) theory at finite strains with a representation in the reference configuration, allowing the spectral method to be readily incorporated. Following the spirit of Khachaturyan’s PFM theory at small strains, the elastic energy is formulated as a functional of microstructure (order parameters) solely, which should automatically satisfy the mechanical equilibrium. Thermodynamic consistency of the current theory under multiplicative decomposition of the total deformation gradient (into elastic and inelastic parts) and in conjunction with hyperelasticity and the time-dependent Ginzburg-Landau equation is shown rigorously. The new theory is first applied to the classical Eshelby’s inclusion problem, where shear-dilation coupling due to geometric nonlinearity is shown and a convergence study between small strain and finite strain theories is also carried out. The effects of geometric nonlinearity on the co-evolution of micromechanics and microstructure is further studied through modeling the growth of {101¯2}〈1¯011〉 deformation twins in magnesium. The simulation results suggest significant differences in terms of the shape of and the stress field around the deformation twin. In particular, the current finite strain PFM theory predicts a deviation of the twin boundary plane from the theoretical K1 plane, which is not captured in the small strain theory nor in the crystallographic theory. A parametric study further reveals that the observed deviation is caused by the tip effect of the finite-sized twin plate when the aspect ratio is relatively small. The symmetry of the stress field distribution around the twin tip is also found to be drastically different between the small strain and finite strain based phase-field modeling. The sharp twin tip observed in experiments is also shown to be likely related to the anisotropy in twin/matrix interface mobility.

Large enhancement of conductivity in a strongly layered type-II superconductor with an artificial pinning array

Abstract We use the time-dependent Ginzburg–Landau equation to describe a type-II superconductor in a magnetic field in the presence of both strong thermal fluctuations and an artificial pinning array. Thermal fluctuations are represented by the Langevin white noise. The layered structure of the superconductor is taken into accounted with the Lawrence–Doniach model. The self-consistent Gaussian approximation is used to treat the nonlinear interaction term in the time-dependent Ginzburg–Landau equation. In the case of the $\delta $-function model for the pinning centers and the matching field, analytic expressions for the fluctuation electrical and thermoelectric conductivity are obtained. It is found that the fluctuations in electrical and thermoelectric conductivities increase with increasing pinning strength, and when the pinning strength comes near a critical value, the fluctuation conductivity is greatly enhanced. Our result shows that if a pinning array is added to a mixed state superconductor, the original properties of the superconductor are recovered. Physically, in the presence of thermal fluctuations, when the energy scale of the vortex lattice shear fluctuations becomes comparable to the pinning energy scale there is a large enhancement of the fluctuation conductivity in the presence of pinning.

Simulation of dynamics of the order parameter in superconducting nanostructured materials: Effect of the magnetic field renormalization

The effect of the magnetic field renormalization in superconducting nanostructured materials is quantitatively evaluated. For demonstration purposes, three superconducting structures with various geometric shapes and dimensions and functioning in different resistive regimes are considered. Simulation is based on a set of equations including the time-dependent Ginzburg-Landau equation coupled with the Maxwell equations. An impact of the order parameter on the vector and scalar potentials is taken into account. It is shown that for Nb structures having thicknesses (∼200 nm) less than the magnetic field penetration depth (∼300 nm), the effect of the magnetic field renormalization equivocally affects the spatiotemporal distribution of superconducting vortices. For a slab with the thickness of ∼100 nm, the absolute value of the average voltage generated by moving vortices changes by less than 1%. With increasing the thickness of C-shaped structures up to 500 nm, the renormalization effect leads to the growth of the average voltage by up to 10%.

Two-band superconducting square with a central defect: role of the deGennes extrapolation length

We present the influence of the deGennes extrapolation length on the vortex matter in a two band superconducting square with a central defect. This is determined by considering different values of the de Gennes extrapolation length b on the lateral surface of the sample. Our investigation was carried out by solving the two-dimensional time-dependent Ginzburg-Landau equations for a two-band system (2B-TDGL). We studied the density, magnetization, vorticity, and critical fields of the superconducting electrons as functions of the external applied magnetic field for different values of b on the surfaces of the sample and different sizes of the internal square defect. We show an unconventional vortex state due to the repulsive (attractive) short- (long-) range potential between the vortices. Finally, we show that at the limit of the thin film, a two-band system behaves like a three-band system.

Time-dependent Ginzburg-Landau simulations of superconducting vortices in three dimensions

Here we describe the development of a computer algorithm to simulate the Time-Dependent Ginzburg-Landau equation (TDGL) and its application to understand superconducting vortex dynamics in confined geometries. Our initial motivation to get involved in this task was trying to understand better our experimental measurements on the dynamics of superconductors with vortices at high frequencies leading to microwave stimulated superconductivity due to the presence of vortex [A. Lara, et al., Sci. Rep. 5, 9187 (2015)].

Amplitude dynamics of the charge density wave in LaTe3 : Theoretical description of pump-probe experiments

We formulate a dynamical model to describe a photo-induced charge density wave (CDW) quench transition and apply it to recent multi-probe experiments on LaTe$_3$ [A. Zong et al., Nat. Phys. 15, 27 (2019)]. Our approach relies on coupled time-dependent Ginzburg-Landau equations tracking two order parameters that represent the modulations of the electronic density and the ionic positions. We aim at describing the amplitude of the order parameters under the assumption that they are homogeneous in space. This description is supplemented by a three-temperature model, which treats separately the electronic temperature, temperature of the lattice phonons with stronger couplings to the electronic subsystem, and temperature of all other phonons. The broad scope of available data for LaTe$_3$ and similar materials as well as the synergy between different time-resolved spectroscopies allow us to extract model parameters. The resulting calculations are in good agreement with ultra-fast electron diffraction experiments, reproducing qualitative and quantitative features of the CDW amplitude evolution during the initial few picoseconds after photoexcitation.

An improved car-following model accounting for the time-delayed velocity difference and backward looking effect

In order to explore the impacts of the time-delayed velocity difference and backward looking effect on traffic flow, this paper proposes an improved car-following model based on the full velocity difference model (FVDM) by accounting for the time-delayed velocity difference and backward looking effect. The linear stability condition of the proposed model is derived by taking advantage of the linear stability theory. The time-dependent Ginzburg-Landau (TDGL) equation and the modified Korteweg-de Vries (mKdV) equation are established based on the nonlinear theory to describe the evolution of the traffic density waves near the critical stability point. Moreover, the link between the TDGL and mKdV equations is also provided. Finally, the results from both the numerical simulation and the theoretical analysis show that the proposed model can not only strengthen the stability of traffic flow, but also suppress the traffic congestion.

Instabilities of the normal state in current-biased narrow superconducting strips

We study the current-voltage characteristic of narrow superconducting strips in the gapless regime near the critical temperature in the framework of the Ginzburg-Landau model. Our focus is on its instabilities occurring at high current biases. The latter are consequences of dynamical states with periodic phase-slip events in space and time. We analyze their structure and derive the value of the reentrance current at the onset of the instability of the normal state. It is expressed in terms of the kinetic coefficient of the time-dependent Ginzburg-Landau equation and calculated numerically.

Time-dependent Ginzburg–Landau equations for multi-gap superconductors**Project supported by the National Natural Science Foundation of China (Grant Nos. 11564003 and 11865005) and the Natural Science Foundation of Guangxi Province of China (Grant No. 2018GXNSFAA281024).

We numerically solve the time-dependent Ginzburg–Landau equations for two-gap superconductors using the finiteelement technique. The real-time simulation shows that at low magnetic field, the vortices in small-size samples tend to form clusters or other disorder structures. When the sample size is large, stripes appear in the pattern. These results are in good agreement with the previous experimental observations of the intriguing anomalous vortex pattern, providing a reliable theoretical basis for the future applications of multi-gap superconductors.

Random distribution of topological defects in mesoscopic superconducting samples

In the present work we analyze the effect of topological defects at different temperatures in a mesoscopic superconducting sample in the presence of an applied magnetic field H. The time-dependent Ginzburg-Landau equations are solved with the method of link variables. We study the magnetization curves M(H), number of vortices N(H) and Gibbs G(H) free energy of the sample as a applied magnetic field function. We found that the random distribution of the anchor centers for the temperatures used does not cause strong anchor centers for the vortices, so the configuration of fluxoids in the material is symmetrical due to the well-known Beam-Livingston energy barrier.

Linear Stability and Nonlinear Analysis of an Extended Optimal Velocity Model Considering the Speed Limit

In this paper, an extended car-following model is proposed based on an optimal velocity model (OVM), which takes the speed limit into consideration. The model is analyzed by using the linear stability theory and nonlinear analysis method. The linear stability condition shows that the speed limit can enlarge the stable region of traffic flow. By applying the reductive perturbation method, 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. Furthermore, the relation between TDGL and mKdV equations is also given. It is clarified that the speed limit is essentially equivalent to the parameter adjusting of the driver’s sensitivity.

A Positivity-Preserving Second-Order BDF Scheme for the Cahn-Hilliard Equation with Variable Interfacial Parameters

We present and analyze a new second-order finite difference scheme for the Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau (MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes energy potential. This numerical scheme with unconditional energy stability is based on the Backward Differentiation Formula (BDF) method time derivation combining with Douglas-Dupont regularization term. In addition, we present a point-wise bound of the numerical solution for the proposed scheme in the theoretical level. For the convergent analysis, we treat three nonlinear logarithmic terms as a whole and deal with all logarithmic terms directly by using the property that the nonlinear error inner product is always non-negative. Moreover, we present the detailed convergent analysis in $\ell^\infty (0,T; H_h^{-1}) \cap \ell^2 (0,T; H_h^1)$ norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties.