Time-dependent Ginzburg-Landau Theory Research Articles

Dynamics of Vortex Matter in 2D Gapless Superconducting Materials with Impurities.

The recent interest in using ultrafast single-photon detectors in research and commercial applications has garnered significant attention from the scientific community. The dynamic event in the detection process consists of a photon causing local destruction of the order parameter, and then the applied current dissipates heat, bringing the material even more out of the superconducting state and then spiking a voltage peak in a measurement device. We investigated the role of superconducting and thermal parameters of the Generalized Time-Dependent Ginzburg-Landau (GTDGL) theory within the event of the first vortex penetration and the thermal dissipation in superconductors near the critical temperature. Moreover, in the vicinity of the Bogomolny point of the static Ginzburg-Landau theory, where κ ≈ 1/√2 and T → Tc, it has been found severely different vortex profiles. We point out that within the GTDGL theory, the presence of impurities (usual or paramagnetic) influences flux penetration and enhances the dissipation of heat, causing anomalous vortex configurations.

Characteristic Times for Gap Relaxation and Heat Escape in Nanothin NbTi Superconducting Filaments: Thickness Dependence and Effect of Substrate.

We measured the temporal voltage response of NbTi superconducting filaments with varied nanoscale thicknesses to step current pulses that induce non-equilibrium superconducting states governed by a hot spot mechanism. Such detected voltage emerges after a delay time td, which is intimately connected to the gap relaxation and heat escape times. By employing time-dependent Ginzburg-Landau theory to link the delay time to the applied current, we determined that the gap relaxation time depends linearly on film thickness, aligning with the acoustic mismatch theory for phonon transmission at the superconductor-substrate interface. We thereby find a gap relaxation time of 104 ps per nm of thickness for NbTi films on polished sapphire. We further show that interfacial interaction with the substrate significantly impacts the gap relaxation time, with observed values of 9 ns on SiOx, 6.8 ns on fused silica, and 5.2 ns on sapphire for a 50 nm thick NbTi strip at T=5.75 K. These insights are valuable for optimizing superconducting sensing technologies, particularly the single-photon detectors that operate in the transient regime of nanothin superconducting bridges and filaments.

Angle-dependent Hall resistivity and longitudinal resistivity of type-II superconductor

We use time-dependent Ginzburg–Landau theory in a three-dimensional model, including thermal noise, to analyze angle-dependent Hall resistivity and longitudinal resistivity of type-II superconductor. The Hall resistivity and longitudinal resistivity are calculated as functions of temperature, magnetic field and the angle θ between the magnetic field and the ab-plane in the vortex-liquid regime. Our theoretical calculations within a self-consistent fluctuation approximation for MgB2 and HgBa2CaCu2O6 materials are in good agreements with the experimental findings for both below and above the critical temperature Tc. We observe that when the field angle decreases, the transition temperature increases and the magnitude of longitudinal resistivity decreases, which is qualitatively comparable to decrease of the perpendicular field component. However, when the magnetic field direction approaches the layer surface, it shows a clear different effect from that of a perpendicular field with the same normal component.

Stable implicit numerical algorithm of time-dependent Ginzburg–Landau theory coupled with thermal effect for vortex behaviors in hybrid superconductor systems

Stable implicit numerical algorithm of time-dependent Ginzburg–Landau theory coupled with thermal effect for vortex behaviors in hybrid superconductor systems

A quantitative theory and atomistic simulation study on the soft-sphere crystal-melt interfacial properties. I. Kinetic coefficients.

By employing non-equilibrium molecular dynamics (NEMD) simulations and time-dependent Ginzburg-Landau (TDGL) theory for solidification kinetics [Cryst. Growth Des. 20, 7862 (2020)], we predict the kinetic coefficients of FCC(100) crystal-melt interface (CMI) of soft-spheres modeled with an inverse-sixth-power repulsive potential. The collective dynamics of the local interfacial liquid phase at the equilibrium FCC(100) CMIs are calculated based on a recently proposed algorithm [J. Chem. Phys. 157, 084 709 (2022)] and are employed as the resulting parameter that eliminates the discrepancy between the predictions of the kinetic coefficient using the NEMD simulations and the TDGL solidification theory. A speedup of the two modes of the interfacial liquid collective dynamics (at wavenumbers equal to the principal and the secondary reciprocal lattice vector of the grown crystal) is observed. With the insights provided by the quantitative predictive theory, the variation of the solidification kinetic coefficient along the crystal-melt coexistence boundary is discussed. The combined methodology (simulation and theory) presented in this study could be further applied to investigate the role of the inter-atomic potential (e.g., softness parameter s = 1/n of the inverse-power repulsive potential) in the kinetic coefficient.

Ac Hall Effect and Photon Drag of Superconducting Condensates.

We suggest a theoretical description of the photogalvanic phenomena arising in superconducting condensates in the field of electromagnetic wave. The ac Hall effect and photon drag are shown to originate from the second-order nonlinear response of superconducting carriers caused by the suppression of their concentration due to the combined influence of the electron-hole asymmetry and charge imbalance generated by the incident electromagnetic wave. Starting from the time-dependent Ginzburg-Landau theory with the complex relaxation constant, we develop a phenomenological description of these phenomena and investigate the resulting behavior of the dc supercurrent and second harmonic induced by microwave radiation incident on a superconductor surface.

Asymmetric fracture behavior in ferroelectric materials induced by flexoelectric effect

Ferroelectric materials are widely used in actuators, exciters, and memory devices due to their excellent electromechanical properties. However, the instinctive brittleness of ferroelectric materials makes them easy to fracture under external load. Since giant strain gradient can be easily generated near the crack tip, the flexoelectric effect is indispensable in the research of fracture properties of ferroelectric materials. With the combination of time-dependent Ginzburg–Landau theory and phase-field model, the electromechanical behavior of PbTiO3 in the vicinity of the crack tip is determined in this work. The simulation results demonstrate that the domain structure near the crack tip becomes asymmetric with the flexoelectric effect. The polarization switching-induced toughening, which is characterized by the J-integral, depends on the direction of the crack relative to the original polarization orientation. Furthermore, the longitude flexoelectric coefficient f11 has more significant impact on the fracture toughness than that of the transverse flexoelectric coefficient f12 and the shear flexoelectric coefficient f44. The results of the present work suggest that the flexoelectric effect must be considered in the reliable design of ferroelectric devices.

Achieving optimal magnetic flux expulsion of a Nb3Sn superconducting radio-frequency cavity via spatial temperature gradient

The trapped magnetic flux leads to residual resistance, thereby reducing the quality factor of the superconducting cavity and affecting its power consumption. It has been experimentally shown that the trapped magnetic flux of the superconducting radio-frequency cavity can be expelled during cooldown under a spatial temperature gradient. In the present paper, we simulate the magnetic flux expulsion behavior during the cooling process of a Nb3Sn superconducting cavity using time-dependent Ginzburg-Landau (TDGL) theory. In order to achieve optimal magnetic flux expulsion, the effect of grain size, grain boundary strength |ψ|GB, and cooling time on flux expulsion ratio ε for different temperature gradients are studied theoretically. The results show that ε first increases rapidly and then varies slightly to a steady value with increasing temperature gradient, which is in good agreement with the experimental results. In general, for larger |ψ|GB, ε decreases with increasing grain size. While for smaller |ψ|GB, ε first increases to the maximum then decreases with increasing grain size. Moreover, It is found that a larger cooling time is more beneficial for the expulsion of the flux vortices from the sample. The results of this paper The vortex dynamics behavior can give an insight into the mechanism of magnetic flux expulsion, which can provide a theoretical basis for further enhancing the performance of superconducting cavities.

Detection and Measurement of Picoseconds-Pulsed Laser Energy Using a NbTiN Superconducting Filament

We investigate non-equilibrium states created by a laser beam incident on a superconducting NbTiN filament subject to an electrical pulse at 4 K. In absence of the laser excitation, when the amplitude of the current pulse applied to the filament exceeds the critical current value, we monitored the delay time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t_{d}$</tex-math></inline-formula> that marks the collapse of the superconducting phase which is then followed by a voltage rise. We linked the delay time to the applied current using the time-dependent Ginzburg-Landau (TDGL) theory, which enabled us to deduce the cooling (or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">heat-removal</i> ) time from the fit to the experimental data. Subsequently, we exposed the filament biased with a current pulse close to its critical value to a focused laser beam, inducing a normal state in the impact region of the laser beam. We showed that the energy of the incident beam and the incurred delay time are related to each other by a simple expression, that enables direct measurement of incident beam energy by temporal monitoring of the transport response. This method can be extended for usage in single-photon detection regime, and be used for accurate calibration of an arbitrary light source.

Comparing energy dissipation mechanisms within the vortex dynamics of gap and gapless nano-sized superconductors

The presence of magnetic fields and/or transport currents can cause penetration of vortices in superconductors. Their motion leads to dissipation and resistive state arises, which in turn strongly affects the performance of superconducting devices such as single-photon and single-electron detectors. Therefore, an understanding of the dissipation mechanisms in mesoscopic superconductors is not only of fundamental value but also very important for further technological advances. In the present work, we analyzed the contributions and interplay of the dissipative mechanisms due to the locally induced electric field and an intrinsic relaxation of the superconducting order parameter, Ψ, in mesoscopic samples by using the time-dependent Ginzburg–Landau theory. Although often neglected, we show that the dissipated energy due to relaxation of Ψ must be taken into account for an adequate description of the total dissipated energy. The local increase of the temperature due to vortex motion and its diffusion in the sample were also analyzed, where the joint effect of thermal relaxation and vortex dynamics plays an important role for the dissipative properties presented by the superconducting systems.

Inverse Faraday Effect in Superconductors with a Finite Gap in the Excitation Spectrum

The inverse Faraday effect (generation of a time-independent magnetic moment under the action of a circularly polarized electromagnetic wave) in mesoscopic superconducting samples with a finite gap in the excitation spectrum is analytically described. Within the modified time-dependent Ginzburg–Landau theory (Kramer–Watts-Tobin equations) for thin superconducting disks, it is shown that the temperature dependence of the optically induced magnetic moment is nonmonotonic in a wide range of parameters and contains a maximum. This maximum is due to the dephasing between the spatial oscillations of the magnitude and the phase of the order parameter, which arises with a decrease in the temperature and, correspondingly, in the characteristic relaxation time of perturbations in the superconducting condensate.

Vortex dynamics in NbTi films at high frequency and high DC magnetic fields

We report on the characterization of NbTi films at sim 11 GHz and in DC magnetic fields up to 4 T, performed by means of the coplanar waveguide resonator technique, providing quantitative information about the penetration depth, the complex impedance, and the vortex-motion-induced complex resistivity. This kind of characterization is essential for the development of radiofrequency cavity technology. To access the vortex-pinning parameters, the complex impedance was analyzed within the formalism of the Campbell penetration depth. Measurements in this frequency range allowed us to determine the complete set of vortex-pinning parameters and the flux flow resistivity, both analyzed and discussed in the framework of high-frequency vortex dynamics models. The analysis also benefits from the comparison with results obtained by a dielectric-loaded resonator technique on similar samples and by other ancillary structural and electromagnetic characterization techniques that provide us with a comprehensive picture of the material. It turns out that the normalized flux flow resistivity follows remarkably well the trend predicted by the time dependent Ginzburg-Landau theory, while the pinning constant exhibits a decreasing trend with the field which points to a collective pinning regime.

Inverse Faraday Effect in Superconductors with a Finite Gap in the Excitation Spectrum

The inverse Faraday effect (generation of a time-independent magnetic moment under the action of a circularly polarized electromagnetic wave) in mesoscopic superconducting samples with a finite gap in the excitation spectrum is analytically described. Within the modified time-dependent Ginzburg–Landau theory (Kramer–Watts-Tobin equations) for thin superconducting disks, it is shown that the temperature dependence of the optically induced magnetic moment is nonmonotonic in a wide range of parameters and contains a maximum. This maximum is due to the dephasing between the spatial oscillations of the magnitude and the phase of the order parameter, which arises with a decrease in the temperature and, correspondingly, in the characteristic relaxation time of perturbations in the superconducting condensate.

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.

Atomistic insights into sluggish crystal growth in an undercooled CoNiCrFe multi-principal element alloy

Crystal growth in undercooled CoNi, CoNiCr and CoNiCrFe equi-atomic alloys was investigated by molecular-dynamics simulation to show atomistic insights into sluggish crystal growth in the undercooled CoNiCrFe multi-principal element alloy (MPEA). The simulation results showed that the growth velocity decreases gradually and the crystallization kinetics became sluggish as the increasing number of principal elements, being qualitatively consistent with the experiments. Independent of the number of principal elements, the activation controlled crystal growth mechanism always played a dominative role. The sluggish crystal growth phenomenon in the undercooled CoNiCrFe MPEA was found to be due to the increased growth activation energy and the decreased ultimate growth velocity. By converting the atomic displacement arose at the process of crystallization into timescales, the contribution of each element to the growth activation energy was quantified for the first time. Consequently, the important role of the Cr element in increasing the growth activation energy was revealed, being qualitatively consistent with experimental findings that the adding amount of Cr element played a dominate role in sluggish crystal growth. An implementation of the time dependent Ginzburg-Landau theory for solidification kinetics indicated that the weakened density relaxation capacity of the melt led to a significant decrease of the ultimate growth velocity as the increasing number of principal elements.

Phase field simulation of martensitic transformation in Ti–24Nb–4Zr–8Sn alloy

A phase field model for cubic to orthorhombic martensitic transformation (MT) at the nanoscale in a β titanium (Ti) alloy Ti–24Nb–4Zr–8Sn (in wt.%) is investigated by finite element simulation. The approach is based on phase field theory, time-dependent Ginzburg-Landau theory, and mechanical equilibrium equations. Partial differential equations (PDEs) were solved using the commercial software COMSOL Multiphysics. The morphology of the product phase exhibits plate-like or needle-like shapes that reduce the elastic strain energy of the system. The simulation result for random initial order parameters is in agreement with previous experimental observations. The final volume fractions of two different orthorhombic martensitic variants are not dependent on the initial conditions.

Influence of pinning centers of different natures on surrounding vortices

Studies involving vortex dynamics and their interaction with pinning centers are an important issue to reach higher critical currents in superconducting materials. The vortex distribution around arrays of engineered defects, such as blind and through holes, may help to improve the superconducting properties. Thus, in this work, we used the time-dependent Ginzburg-Landau theory to investigate the vortex dynamics in superconductors of mesoscopic dimensions with a large central square defect with three different configurations: (i) a hole which passes through the sample (interface with the vacuum); (ii) a superconducting region with lower critical temperature Tc; and (iii) a region with a more robust superconductivity, i.e., with a higher Tc. Such systems can be envisaged as elementary building blocks of a macroscopic decorated specimen. Therefore, we evaluated the influence of different interfaces on the vortex dynamics and their effects in the field-dependent magnetization and time-dependent induced electric potential variation. The results show that the superheating field is independent from the nature of the defect. However, the currents crowd at the vertices of the through hole producing a lower degradation of the local superconductivity, which may increase the upper critical field. On the other hand, the last type of defect can be used to control the vortex dynamics in the main superconducting region around the defect with more accuracy. While the first two defects are attractive for the vortices, the third type is repulsive, being needed several penetrated vortices in the superconducting matrix to allow penetration of vortices into the higher Tc domain.

Understanding resistance oscillation in the CsV3Sb5 superconductor

A recent demonstration of the periodic oscillation of resistance in the thin film of the ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$ superconductor with a hole in the film suggests that charge-$4e$ and charge-$6e$ Cooper pairs may have condensed in this compound. While exciting, such interpretation calls for a precise determination of the effective area for the passage of Cooper pairs from one end of the lead to the other. Unlike the traditional Little-Parks effect where the rim around the hole is thin, the effective hole area is not obviously defined for the ``thick-rim geometry'' adopted in the experiment. Here, we note that the experiment was conducted in a regime where the superconductivity is strongly fluctuating, which motivates an analysis based on the spacetime formulation of the time-dependent Ginzburg-Landau theory. We argue that under appropriate conditions, the optimal semiclassical path is not the geometrically shortest one, but the one that moves along the edge of the hole and takes advantage of the reduced fluctuations at the boundary of the hole. The condition for the crossover from the geometrically shortest path to the path that sticks to the wall is clarified. In such a scenario, the geometric area of the hole indeed emerges as the effective area for the flux, providing a theoretical justification to the interpretation given by J. Ge et al. (arXiv:2201.10352). The conclusion of our analysis may have implications for similar experiments in other superconductors where the geometry of the device is not obviously that of the Little-Parks experiment employing the thin wall, but of the thick-rim type such as used in the experiment.

Vortex dynamics in the two-dimensional BCS-BEC crossover

The Bardeen–Cooper–Schrieffer (BCS) condensation and Bose–Einstein condensation (BEC) are the two limiting ground states of paired Fermion systems, and the crossover between these two limits has been a source of excitement for both fields of high temperature superconductivity and cold atom superfluidity. For superconductors, ultra-low doping systems like graphene and LixZrNCl successfully approached the crossover starting from the BCS-side. These superconductors offer new opportunities to clarify the nature of charged-particles transport towards the BEC regime. Here we report the study of vortex dynamics within the crossover using their Hall effect as a probe in LixZrNCl. We observed a systematic enhancement of the Hall angle towards the BCS-BEC crossover, which was qualitatively reproduced by the phenomenological time-dependent Ginzburg-Landau (TDGL) theory. LixZrNCl exhibits a band structure free from various electronic instabilities, allowing us to achieve a comprehensive understanding of the vortex Hall effect and thereby propose a global picture of vortex dynamics within the crossover. These results demonstrate that gate-controlled superconductors are ideal platforms towards investigations of unexplored properties in BEC superconductors.

All-optical generation of Abrikosov vortices by the inverse Faraday effect

Within the framework of the time-dependent Ginzburg-Landau theory we show that circularly polarized terahertz or far-infrared radiation induces a dc supercurrent that influences the dynamics of vortex-antivortex pair formation in a mesoscopic superconductor undergoing rapid thermal quenching. The Lorentz force arising from the supercurrents promotes vortex-antivortex separation and allows survival of the vortex polarity defined by the helicity of light. Based on this idea, we propose a two-stage irradiation scheme that provides a powerful method for controlled all-optical generation of Abrikosov vortices.