Time-dependent Ginzburg-Landau Research Articles

Electric-field-induced crystallization of Hf0.5Zr0.5O2 thin film based on phase-field modeling

Ferroelectricity in crystalline hafnium oxide has attracted considerable attention because of its potential application for memory devices. A recent breakthrough involves electric-field-induced crystallization, allowing HfO2-based materials to avoid high-temperature crystallization, which is unexpected in the back-end-of-line process. However, due to the lack of clarity in understanding the mechanisms during the crystallization process, we aim to employ theoretical methods for simulation, to guide experimental endeavors. In this work, we extended our phase-field model by coupling the crystallization model and time-dependent Ginzburg-Landau equation to analyze the crystalline properties and the polarization evolution of Hf0.5Zr0.5O2 thin film under applying an electric field periodic pulse. Through this approach, we found a wake-up effect during the process of crystallization and a transformation from orthorhombic nano-domains to the stripe domain. Furthermore, we have proposed an innovative artificial neural synapse concept based on the continuous polarization variation under applied electric field pulses. Our research lays the theoretical groundwork for the advancement of electric-field-induced crystallization in the hafnium oxide system.

Ginzburg–Landau simulations of three-terminal operation of a superconducting nanowire cryotron

Abstract Superconducting nanowire cryotrons (nTrons) are expected to be used as interfaces for super-high-performance hybrid devices in which superconductor and semiconductor circuits are combined. However, nTrons are still under development, and diverse analyses of these devices are needed. Accordingly, we have developed a numerical technique to simulate the three-terminal operation of an nTron by using the finite element method to solve the time-dependent Ginzburg–Landau (TDGL) equation and the heat-diffusion equation. Simulations using this technique offer understanding of the dynamics of the order parameter, the thermal behavior, and the characteristics of three-terminal operation, and the TDGL model reproduces qualitatively the results of nTron experiments conducted at the Massachusetts Institute of Technology. In addition, we investigated how some geometric and physical parameters (the design elements) affect the operation characteristics. The TDGL model has far fewer free parameters compared with the lumped-element electrothermal model commonly used for simulating superconducting devices. Furthermore, the TDGL model provides time-dependent visual information about the superconducting state and the normal state, thereby offering insights into the relationship between nTron geometry and three-terminal operation. These simulation results offer a route to nTron optimization and the development of nTron applications.

A Comparative Computational Study of the Solidification Kinetic Coefficients for the Soft-Sphere BCC-Melt and the FCC-Melt Interfaces

Using the non-equilibrium molecular dynamics (NEMD) simulations and the time-dependent Ginzburg–Landau (TDGL) theory for solidification kinetics, we study the crystal-melt interface (CMI) kinetic coefficients for both the soft-sphere (SS) BCC-melt and the FCC-melt interfaces, modeled with the inverse-power repulsive potential (n=8). The collective dynamics of the interfacial liquids at four equilibrium CMIs are calculated and employed to eliminate the discrepancy between the predictions of the kinetic coefficient using the NEMD simulations and the TDGL solidification theory. The 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) at the equilibrium FCC CMI is observed. The calculated local collective dynamics of the SS BCC CMIs are compared with the previously reported data for the BCC Fe CMIs, validating a hypothesis proposed recently that the density relaxation times of the interfacial liquids at the CMIs are anisotropic and material dependent. With the insights provided by the improved application of the TDGL solidification theory, an attempt has been made to interpret the variation physics of the crystal-structure dependence of the solidification kinetic coefficient.

Roundabout-like defect as a guide of phase-slip lines in a superconducting thin film at low temperature

AbstractIn this work, we examined the kinematic vortex state in a low-temperature, mesoscopic superconducting thin film in presence of an external applied current $$\varvec{J}$$ J , at zero magnetic field. We analyzed the voltage-current ($$\varvec{V-I}$$ V - I ), resistivity-current ($$\varvec{R-I}$$ R - I ) curves, as well as the vortex-anti-vortex velocity $$\varvec{V_m}$$ V m and the superconducting-electron density (norm of the superconduting order parameter $$\varvec{\psi }$$ ψ ) as an applied current function. Additionally, we have calculated the voltage as a function of characteristic time ($$\varvec{V-t}$$ V - t ). The sample presents a roundabout shaped pinning center fabricated with a lower low-temperature superconducting material, which allows for control over the vortex dynamics within the sample. To study this system, we have solved the generalized time-dependent Ginzburg-Landau equations for a single-condensate system. Our results demonstrate that the kinematic vortices enter the sample through regions where the superconductivity is depleted, revealing an intriguing and novel behavior of the phase-slip lines in the defect. We found that when a higher $$\varvec{J}$$ J is applied and the defect is not centered, the point where the vortex anti-vortex pairs enter the film moves away from the defect and are located in a point tangent to the outer circle of the defect. Besides, our results show a way to control the dynamics of the vortices, critical currents, and points of annihilation or creation of the cinematic vortices by including defects in the sample, important in the design and development of devices applied in industry and engineering.

A phase-field simulation of easily switchable vortex structure for multilevel low-power ferroelectric memory

Ferroelectric materials with multi-directional polarization orders can form a variety of special domain structures, which have a wide range of applications. In this work, lead-free potassium sodium niobate ferroelectric thin films were simulated by solving the Time-Dependent-Landau-Ginzburg (TDLG) equation. Based on vortex topological structure characteristic of perovskite-type ferroelectric, domain types are simplified from 26 to 9 through virtual polarization vectors. By temperature selection, we accomplished a large easily switchable vortex structure with 20% side length. Based on this, as low as 0.04V/nm electric-field pulse was applied to explore vortex domain structures change. Under different low external electric-field pulses, conductance states change in vortex have been calculated and discussed systematically. Free energy computation results reveal that the domain wall (DW) conductance states primarily originate from the competition between electrostatic energy, gradient energy, and elastic energy. Finally, we propose the multilevel data ferroelectric memory through easily switchable vortex topological structures for the first time. Under electric-field pulses, diverse storage states were presented through 0/1 measurement of 8 positions. By achieving an easy-to-distinguish and robust-to-switch data map at the vortex structure, this work might provide a significant idea of vortex structure on low-power ferroelectric memory applications.

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.

Steering of Vortices by Magnetic Field Tilting in Open Superconductor Nanotubes.

In planar superconductor thin films, the places of nucleation and arrangements of moving vortices are determined by structural defects. However, various applications of superconductors require reconfigurable steering of fluxons, which is hard to realize with geometrically predefined vortex pinning landscapes. Here, on the basis of the time-dependent Ginzburg-Landau equation, we present an approach for the steering of vortex chains and vortex jets in superconductor nanotubes containing a slit. The idea is based on the tilting of the magnetic field B at an angle α in the plane perpendicular to the axis of a nanotube carrying an azimuthal transport current. Namely, while at α=0∘, vortices move paraxially in opposite directions within each half-tube; an increase in α displaces the areas with the close-to-maximum normal component |Bn| to the close(opposite)-to-slit regions, giving rise to descending (ascending) branches in the induced-voltage frequency spectrum fU(α). At lower B values, upon reaching the critical angle αc, the close-to-slit vortex chains disappear, yielding fU of the nf1 type (n≥1: an integer; f1: the vortex nucleation frequency). At higher B values, fU is largely blurry because of multifurcations of vortex trajectories, leading to the coexistence of a vortex jet with two vortex chains at α=90∘. In addition to prospects for the tuning of GHz-frequency spectra and the steering of vortices as information bits, our findings lay the foundation for on-demand tuning of vortex arrangements in 3D superconductor membranes in tilted magnetic fields.

A generalized scalar auxiliary variable method for the time-dependent Ginzburg-Landau equations

A generalized scalar auxiliary variable method for the time-dependent Ginzburg-Landau equations

The local rupture of the superconductivity and the curvature of the mesoscopic sample

In this article we revisit a well-known effect in superconductors, which is the penetration of the magnetic field by regions of greater radius of curvature (Tinkham, 2008; De Gennes, 1966). This is exposed in the literature but without a quantitative solution. Here we solve this problem by two distinct numerical simulations. We calculate the superconducting current density using numerical simulations via the London theory in the Meissner state and numerical simulations using the time-dependent Ginzburg–Landau (TDGL) theory for the vortex state. In both simulations, the results obtained are in agreement, as could not be otherwise. We show that in the Meissner state the current density increases much faster in the region where the radius of curvature is larger, thus reaching the critical value first in this location, and therefore local breakdown of superconductivity occurs. In the vortex state, our simulations, using TDGL, show that the vortices penetrate through exactly the same location pointed out by the previous simulation. For this study we worked with a superconducting needle with an ellipsoidal mesoscopic cross section.

Effect of defects on flux expulsion of pure niobium superconducting radio frequency cavities via spatial temperature gradient

Since flux vortices are easily pinned by defects in superconducting radio frequency (SRF) cavities, trapped magnetic flux increases the surface resistance of the SRF cavity. It has been experimentally demonstrated that trapped magnetic flux can be expelled by cooling down a superconducting cavity under a large spatial temperature gradient. In this paper, we use the time-dependent Ginzburg–Landau (TDGL) theory to theoretically investigate the effect of defects on the flux expulsion ability of pure niobium superconducting cavities with an applied spatial temperature gradient. It is shown that the residual flux vortex density displays a notable stepwise decreasing trend with increasing temperature gradient at smaller residual fields, while it shows a stepwise rising with increasing defect size. Both larger defect number and size make vortices difficult to be expelled. Moreover, long cooling time is beneficial for vortices to be expelled, and the cooling time has nearly no significant effect on flux expulsion under conditions of large cooling times. The results in this paper provide some theoretical guidelines for improving the performance of superconducting cavities in practical situations.

A convex splitting method for the time-dependent Ginzburg-Landau equation

A convex splitting method for the time-dependent Ginzburg-Landau equation

Fully discrete scheme for a time-dependent Ginzburg-Landau equation in macromolecular microsphere composite hydrogels

In this article, we propose and analyze a fully discrete scheme for the time-dependent Ginzburg-Landau equation, which describes the phase field evolution of macromolecular microsphere composite hydrogels. This fully discrete scheme is a combination of the mass lumped finite element approximation for spatial discretization and the first order backward Euler scheme for temporal discretization. Besides, the convex-concave decomposition is adopted technically in the treatment of energy functional, where the concave part is treated explicitly and the convex part is treated implicitly. Moreover, we obtain the unique solvability and unconditional energy stability of the scheme. Finally, several numerical experiments are presented to illustrate the convergence order and effectiveness of the presented full-discrete scheme.

A phase-field model for ferroelectric materials—Based on the multiphase-field method

A novel phase-field model is developed to investigate the domain structure in ferroelectric materials subjected to an external field. Differing from the time-dependent Ginzburg–Landau (TDGL) theory that describes domain structures based on the spatial and temporal evolution of polarization, our model utilizes the multiphase-field approach to calculate domain structures by minimizing the total energy functional with respect to the multiphase-field order parameter ϕ. The energy functional in the proposed model comprises the general multiphase interfacial energy and the phase-dependent bulk energy, accounting for the contributions from domain walls, and mechanical as well as electric fields. By considering the interplay between the different energy components, the model provides insights into the complex domain structure of ferroelectric materials and their response to external stimuli. This comprehensive approach is validated through simulations of both single-crystal and polycrystalline structures. The approach presented in this study simplifies the computational coefficients required to analyze the domain structure in ferroelectric materials, compared to the TDGL approach, by disregarding the Landau potential. This makes it possible to analyze the domain structure of materials without Landau coefficients. Furthermore, the proposed model serves as a reference for applying the multiphase-field concept to ferroelectric materials.

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.

Fluctuation-mediated inverse Faraday effect in superconducting rings

We study the generation of supercurrent by circularly polarized electromagnetic wave in a superconducting ring in fluctuation regime at T>Tc. Using the time-dependent Ginzburg-Landau equation with a complex relaxation constant we calculate the dc component of the supercurrent, which is the manifestation of the inverse Faraday effect. The observed strong dependence of the current on the radiation frequency and temperature can be additionally enhanced by the application of an external magnetic flux and may represent a distinct feature noticeable in experiments.

A Compact Model for Ferroelectric Capacitors Based on Multidomain Phase-Field Framework

A generalized and fast multidomain phase-field-based compact model for the metal-ferroelectric-metal (MFM) capacitor is presented. Time-dependent Landau–Ginzburg (TDGL) and Poisson’s equations are solved self-consistently to model the polarization dynamics. Additionally, physics-based empirical relationships for voltage-dependent kinetic and gradient energy coefficients are formulated. It is also demonstrated how gradient energy coefficient needs to be modified to accurately capture the physics of the device when a coarse simulation grid is used for fast computation. The developed model is 30 000 times faster than our prior multidomain phase-field model with no degradation in accuracy. Moreover, a further computational speedup has been achieved by decreasing the number of Poisson solver nodes with a slight compromise of accuracy. The model shows a good agreement with the experimental results of both transient characteristics and minor hysteresis loops. The proposed model has the potential to facilitate fast and accurate simulations of large-scale circuits containing ferroelectric capacitors.

Vortex dynamics induced by scanning SQUID susceptometry

We measured the local magnetic response of a niobium thin film by applying a millitesla-scale AC magnetic field using a micron-scale field coil and detecting the response with a micron-scale pickup loop in a scanning superconducting quantum interference device (SQUID) susceptometry measurement. Near the film's critical temperature, we observed a step-like nonlinear and dissipative magnetic response due to the dynamics of a small number of vortex-antivortex pairs induced in the film by the local applied AC field. We modeled the dynamics of the measurement using a combined two-dimensional London-Maxwell and time-dependent Ginzburg-Landau approach, allowing us to construct a detailed real-space picture of the vortex motion causing the observed dissipative response. This work pushes scanning SQUID susceptometry of two-dimensional superconductors beyond the regime of linear response and lays the foundation for microscopic studies of vortex dynamics and pinning in superconducting devices and more exotic materials systems.

Extracting quantum-geometric effects from Ginzburg-Landau theory in a multiband Hubbard model

We first apply functional-integral approach to a multiband Hubbard model near the critical pairing temperature, and derive a generic effective action that is quartic in the fluctuations of the pairing order parameter. Then we consider time-reversal-symmetric systems with uniform (i.e., at both low-momentum and low-frequency) pairing fluctuations in a unit cell, and derive the corresponding time-dependent Ginzburg-Landau (TDGL) equation. In addition to the conventional intraband contribution that depends on the derivatives of the Bloch bands, we show that the kinetic coefficients of the TDGL equation have a geometric contribution that is controlled by both the quantum-metric tensor of the underlying Bloch states and their band-resolved quantum-metric tensors. Furthermore we show that thermodynamic properties such as London penetration depth, GL coherence length, GL parameter and upper critical magnetic field have an explicit dependence on quantum geometry.

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.