Time-dependent Ginzburg-Landau Research Articles

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.

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.

Ultrafast Switching from the Charge Density Wave Phase to a Metastable Metallic State in 1T-TiSe_{2}.

The ultrafast electronic structures of the charge density wave material 1T-TiSe_{2} were investigated by high-resolution time- and angle-resolved photoemission spectroscopy. We found that the quasiparticle populations drove ultrafast electronic phase transitions in 1T-TiSe_{2} within 100fs after photoexcitation, and a metastable metallic state, which was significantly different from the equilibrium normal phase, was evidenced far below the charge density wave transition temperature. Detailed time- and pump-fluence-dependent experiments revealed that the photoinduced metastable metallic state was a result of the halted motion of the atoms through the coherent electron-phonon coupling process, and the lifetime of this state was prolonged to picoseconds with the highest pump fluence used in this study. Ultrafast electronic dynamics were well captured by the time-dependent Ginzburg-Landau model. Our work demonstrates a mechanism for realizing novel electronic states by photoinducing coherent motion of atoms in the lattice.

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.

An elastoplastic phase-field study of the precipitation behaviors of Mg17Al12 phase in Mg-Al-Based alloys: Part I. Modeling and the role of plasticity

An elastoplastic phase-field model for typical diffusional transformations was developed, where plastic deformation was considered through a modified time-dependent Ginzburg-Landau dynamic equation. The model was applied to investigate the precipitation in Mg-Al-based alloys, with a focus on studying the effects of elastoplastic deformation associated with phase transformation on the morphology and growth kinetics of β-Mg17Al12 precipitates (with Burgers orientation). The elastoplastic modeling results show that the precipitates adopt a lath-shaped morphology with lozenge ends, consistent with experimental observations. Although the plastic strain accumulated in the transformation front has a moderate impact on the morphology of the precipitate, it exhibits a clear dual effect on the growth of the precipitate along [112̅0]α. Specifically, plasticity promotes the growth of the precipitate during the initial stage, owing to its stress relaxation effect, and induces a solute poor zone of atomic Al with a greater concentration gradient at the transformation front. However, the plastic strain tends to retard the growth of the precipitate afterward due to its limited ability to release the increasing compressive strain between the two phases. These results shed light on understanding the role of plasticity in general diffusional transformations and provide guidelines for tailoring the microstructure of alloys.

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.

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.

Simulation of martensitic transformation in porous alloys based on phase-field method

Porous materials, characterized by the presence of interconnected pores, have been found to exhibit unique properties that differ from their bulk counterparts. One area of interest is the influence of pores on the martensitic transformation in shape memory alloys (SMAs), which directly affects the material's shape memory effect and mechanical properties. The martensitic transformation is accompanied by the formation of different martensitic variants, which determine the overall morphology, distribution, and self-accommodation effect of the transformed regions. Previous experimental studies have shown that the presence of pores, particularly at the metal-air interface, can significantly impact the martensitic variant structure, leading to its thinning. This thinning effect has been found to enhance the damping performance of the alloy. Experimental observations have indicated that there is no relief of martensitic variants around the metal-air interface, and there is non-transformed regions present. These observations suggest that the metal-air interface in porous materials is not a free surface and plays a crucial role in influencing the martensitic transformation. To further investigate the effect of martensitic variant self-accommodation on different constrained interfaces in porous materials, a three-dimensional phase-field model based on the Time Dependent Ginzburg Landau (TDGL) function is proposed in this study. The phase-field model allows for a comprehensive understanding of the evolution of martensitic variants and their interaction with the constrained interfaces. Remarkably, the simulation results align well with the experimental findings, demonstrating the presence of fine martensitic variants near the metal-air interface. The simulations under different interface constraint conditions reveal that increasing the specific surface area of porous materials is an effective strategy to obtain a more refined martensitic variant structure. The system's total energy is minimized by reducing the strain energy, which leads to the formation of a greater number of fine martensitic variants. This finding suggests that controlling the specific surface area of porous materials can be a promising approach to tailor the mechanical properties of SMAs for specific applications. In conclusion, the presence of metal-air interface in porous materials significantly influences the evolution of the martensitic transformation in SMAs. Experimental observations have shown that the introduction of pore can modify the martensitic variant structure, resulting in improved damping performance. The proposed phase-field model successfully captures the behavior of martensitic variants near constrained interfaces. The simulation results emphasize the importance of specific surface area in obtaining fine martensitic variant structures. These findings contribute to a deeper understanding of the role of porous materials in shaping the properties of SMAs and provide valuable insights for their design and application in various fields.

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.

Driven responses of periodically patterned superconducting films

We simulate the motion of a commensurate vortex lattice in a periodic lattice of artificial circular pinning sites having different diameters, pinning strengths, and spacings using the time-dependent Ginzburg-Landau formalism. Above some critical DC current density ${J}_{\mathrm{c}}$, the vortices depin, and the resulting steady-state motion then induces an oscillatory electric field $E(t)$ with a defect ``hopping'' frequency ${f}_{0}$, which depends on the applied current density and the pinning landscape characteristics. The frequency generated can be locked to an applied AC current density over some range of frequencies, which depends on the amplitude of the DC as well as the AC current densities. Both synchronous and asynchronous collective hopping behaviors are studied as a function of the supercell size of the simulated system and the (asymptotic) synchronization threshold current densities determined.

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.

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.

Experimental study of phase separation in dynamically asymmetric unentangled polymer blend.

We examined a phase separation process of a dynamically asymmetric blend of unentangled polyisoprene (PI) and poly(4-ethylstyrene) (PC2St) exhibiting the upper critical solution temperature. PI having the type-A dipole was the dielectrically active fast component, whereas PC2St was the dielectrically inert slow component whose dynamics can be detected by rheological measurements. To precisely model the phase separation process, it is important to estimate the composition dependence of the mobility, which is needed to describe the phase separation dynamics. For that purpose, we conducted dielectric and rheological measurements to determine the friction coefficient of each component in a homogeneous state sufficiently above the phase separation temperature. The temperature dependence of the friction coefficient of each component was reasonably expressed by the Williams-Landel-Ferry equation. Extrapolating this dependence obtained for blends of various compositions to the test temperature T* below the phase separation temperature, we were able to estimate the friction coefficient of the chain at T* as a function of the composition. This friction coefficient was then used to determine the mobility Λ defined for the material fluxes at T*. The time-dependent Ginzburg-Landau (TDGL) equation incorporating this Λ well described the experimentally observed phase separation dynamics. In particular, the 2D TDGL simulation with this Λ qualitatively captured the phase-separated structure observed with the optical microscope as well as broad dielectric mode distribution of the blend at T*.

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.

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.

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.