Time-dependent Ginzburg-Landau Equation Research Articles

Reversible ratchet effects in a narrow superconducting ring

We study the ratchet effect in a narrow pinning-free superconductive ring based on time-dependent Ginzburg-Landau (TDGL) equations. Voltage responses to external dc an ac currents at various magnetic fields are studied. Due to asymmetric barriers for flux penetration and flux exit in the ring-shaped superconductor, the critical current above which the flux-flow state is reached, as well as the critical current for the transition to the normal state, are different for the two directions of applied current. These effects cooperatively cause ratchet signal reversal at high magnetic fields, which has not been reported to date in a pinning-free system. The ratchet signal found here is larger than those induced by asymmetric pinning potentials. Our results also demonstrate the feasibility of using mesoscopic superconductors to employ superconducting diode effect in versatile superconducting devices.

Strain engineering of ferroelectric negative capacitance in PbZr(1-x)TixO3 thin films

Negative capacitance (NC) in ferroelectric materials has been proposed to overcome the fundamental limitation on the energy efficiency of electronics. Understanding the impact of mechanical strain on the NC of ferroelectrics is of scientific significance and technological importance because the strain-polarization coupling could substantially influence the multifunctional properties of epitaxial ferroelectric thin films. Based on time-dependent Ginzburg-Landau equation and Kirchhoff's law, we reveal that a tensile strain decreases the magnitude and duration time of the transient NC of PbZr(1-x)TixO3 (PZT) thin films while a compressive strain increases it in a resistor-ferroelectric capacitor circuit. In particular, the tensile strain makes the transient NC completely disappear near the morphotropic phase boundary (MPB), whereas the compressive strain can maintain it up to a temperature higher than Curie temperature. The detailed energy analysis reveals that the change of transient NC with strain and composition is attributed to the change of the negative curvature of energy landscape, which is the origin of NC in ferroelectric materials. The work gives considerable new insight into the mechanical control of ferroelectric transient NC, and offers guidance in searching for large transient NC in the widely used PZT thin films through strain and composition engineering.

A phase field approach to study of transformation induced micro–cracking in a martensitic phase transformation

AbstractIn this study, a coupled phase field (PF) method for the simulation of crack propagation and martensitic phase transformations is developed. In order to investigate the crack field and martensitic microstructure evolution the concept of the thermodynamic driving force, interfacial energy, the elastic energy, and the kinetic of phase field equations are introduced (time dependent Ginzburg Landau equation) [1]. The weak form and an algorithm for the solution of corresponding equations are implemented in the finite element program (FEAP). Since the phase transformation can form during the application of high amount of stresses, the influence of microcrack propagation on the formation of the martensitic phase has been studied. The crack tip produces high amount of concentrated stresses, which lead to a change in the distribution of the martensitic phases and it can also deviate the crack direction [2].

До кінетики руйнування надпровідного стану нелінійного копланарного хвилеводу на основі плівки високотемпературного надпровідника

Subject and Purpose. The mechanism of destruction of the S-state of a nonlinear high-temperature superconductor as part of a coplanar waveguide has not been properly elucidated as the effect of avalanche-type transition to a highly dissipative state, which was experimentally detected by the authors, takes place. The present work is concerned with the development of an appropriate approach describing kinetics of destruction of the S-state of a nonlinear high-temperature superconductor in a coplanar waveguide with allowances made for an inhomogeneous distribution of the microwave current in the superconducting film strip. Methods and Methodology. Use of I.B. and O.G. Vendiks’ reasoning [2] is made on kinetics of the destruction of the superconducting state of a wide film when a direct current governed by the time-dependent Ginzburg-Landau equation is applied. Keeping unchanged their idea as to the S–N boundary forming in the film strip with the boundary movement to the middle of the strip, the S–N boundary motion equation is obtained for a coplanar waveguide, proceeding in doing this from the motion equation of magnetic flux vortices under certain restrictions specified. Results. The time of S-state destruction has been numerically estimated: 1) for a wide superconducting film of YBa2Cu3O7–d composition, the destruction is by the direct current and 2) for a coplanar waveguide based on the same film, the destruction is by the microwave current. When the superconductivity is small (I / I c ³ 1), the destruction time values in both cases are close to each other within the order of magnitude. Conclusion. It is for the first time that the S-state destruction time in a coplanar waveguide has been expressed in terms of the microwave current distribution in the waveguide. It has been shown that this characteristic linearly depends on the ratio between the critical current and the microwave current amplitude in contrast to a quadratic dependence obtained for a superconducting strip with a direct current.

Finite element analysis of coupled phase-field and thermoelasticity equations at large strains for martensitic phase transformations based on implicit and explicit time discretization schemes

In this paper, a staggered, nonlinear finite element procedure is developed to solve the large-strain based coupled system of time dependent Ginzburg-Landau (GL) and thermoelasticity equations for phase transformations at the nanoscale. Geometrical nonlinearities are included based on the total Lagrangian description where the total deformation gradient is defined as the multiplicative decomposition of elastic, transformational and thermal deformation gradient tensors. The principle of virtual work is utilized to obtain the integral form of the Lagrangian equation of motion whose discretized form is solved using the iterative methods which gives the thermal and transformational deformation gradient tensors. Next, the elastic deformation gradient tensor and the first Piola-Kirchhoff stress are calculated and substituted in the GL equation which gives the phase order parameter. The weighted residual method is used to derive the corresponding finite element form which allows for the phase-dependent surface energy BCs. Explicit and different implicit methods belonging to the generalized trapezoidal family are used for time discretization of the GL equation. Various examples of phase transformations are simulated and discussed. The effect of different time integration schemes are also investigated. The current study allows for a proper modeling of various phase field problems at large and small strains.

Effects of Interface Trap on Transient Negative Capacitance Effect: Phase Field Model

Ferroelectric materials have received significant attention as next-generation materials for gates in transistors because of their negative differential capacitance. Emerging transistors, such as the negative capacitance field effect transistor (NCFET) and ferroelectric field-effect transistor (FeFET), are based on the use of ferroelectric materials. In this work, using a multidomain 3D phase field model (based on the time-dependent Ginzburg–Landau equation), we investigate the impact of the interface-trapped charge (Qit) on the transient negative capacitance in a ferroelectric capacitor (i.e., metal/Zr-HfO2/heavily doped Si) in series with a resistor. The simulation results show that the interface trap reinforces the effect of transient negative capacitance.

Calorimetric properties of a superconducting anisotropic square: ZFC process

We analyzed the role of the inclusion of anti-dots on the vortex state and some calorimetric properties of a mesoscopic superconducting square immersed in an external applied magnetic field. We calculated the magnetization, entropy, Gibbs free energy, density of Cooper pairs and specific heat for this system in a zero field cooling process, solving the time-dependent Ginzburg–Landau equations. We found that the critical temperature is non-dependent on the number of anti-dots and dependent slightly on the size of the defects. Oscillations in the entropy and specific heat are found due the temperature dependence of the superconducting characteristics length.

On the crack onset and growth in martensitic micro-structures; a phase-field approach

In this work, a phase-field approach (PFA) is presented to study crack nucleation and propagation in martensitic micro-structures resulted from multi-variant martensitic phase transformations (MPT) within the framework of a finite element method (FEM). To this end, first, a coupled system of the time-dependent Ginzburg-Landau (TDGL) equation and the equilibrium equation is established based on the micro-elasticity theory, which reveals the nucleation and growth of diffusionless martensitic multi-variants forming a twinned martensitic micro-structure. The Helmholtz free energy used in this work consists of a second-degree polynomial of the phase variable, which leads to a nonlinear dependence on the order parameter in the TGDL equation. Thereafter, the nucleation and propagation of a crack is scrutinized in the obtained martensitic specimen, with and without pre-existing crack, according to three types of martensitic embryos. To do so, a damage variable is introduced to the multi-variant MPT model to study the interactions between the martensitic transformation and fracture. The key contributions of this study are not only to shed light on the evolution of the martensitic variants in the micro-structure with three types of pre-existing martensitic embryos, but also to investigate onset and growth of a crack in the martensitic specimen.

Ultra-fast kinematic vortices in mesoscopic superconductors: the effect of the self-field

Within the framework of the generalized time-dependent Ginzburg–Landau equations, we studied the influence of the magnetic self-field induced by the currents inside a superconducting sample driven by an applied transport current. The numerical simulations of the resistive state of the system show that neither material inhomogeneity nor a normal contact smaller than the sample width are required to produce an inhomogeneous current distribution inside the sample, which leads to the emergence of a kinematic vortex–antivortex pair (vortex street) solution. Further, we discuss the behaviors of the kinematic vortex velocity, the annihilation rates of the supercurrent, and the superconducting order parameters alongside the vortex street solution. We prove that these two latter points explain the characteristics of the resistive state of the system. They are the fundamental basis to describe the peak of the current–resistance characteristic curve and the location where the vortex–antivortex pair is formed.

Released power in a vortex-antivortex pairs annihilation process

In this paper, we studied the power dissipation process of a Shubnikov vortex-antivortex pair in a mesoscopic superconducting square sample with a concentric square defect in presence of an oscillatory external magnetic field. The time-dependent Ginzburg-Landau equations and the diffusion equation were numerically solved. The significant result is that the thermal dissipation is associated with a sizeable relaxation of the superconducting electrons, so that the power released in this kind of process might become calculated as a function of the time. Also, we analyzed the effect that the Ginbzurg-Landau κand deformation τparameters have on the magnetization, dissipate power and super-electrons density.

Timing Jitter in NbN Superconducting Microstrip Single-Photon Detector

We experimentally study timing jitter of single-photon detection by NbN superconducting strips with width $w$ ranging from 190 nm to $3\phantom{\rule{0.2em}{0ex}}\ensuremath{\mu}\mathrm{m}$. We find that timing jitter of both narrow (190 nm) and micron-wide strips is about 40 ps at currents where internal detection efficiency $\ensuremath{\eta}$ saturates and it is close to our instrumental jitter. We also calculate intrinsic timing jitter in wide strips using the modified time-dependent Ginzburg-Landau equation coupled with a two-temperature model. We find that with increasing width the intrinsic timing jitter increases and the effect is most considerable at currents where a rapid growth of $\ensuremath{\eta}$ changes to saturation. We relate it with complicated vortex and antivortex dynamics, which depends on a photon's absorption site across the strip and its width. The model also predicts that at current close to depairing current the intrinsic timing jitter of a wide strip could be about $\ensuremath{\hbar}/{k}_{B}{T}_{c}$ (${T}_{c}$ is a critical temperature of superconductor), i.e., the same as for a narrow strip.

Enhancement of the Electric Current in High-Tc Superconductor Interacting with an Artificial Pinning Array

The dynamics of the flux flow in the mixed state of high-Tc superconductor interacting with an artificial pinning array is considered. We calculate analytically the current density for the case of when the number of vortices outnumbers that of the pinning centers using the time-dependent Ginzburg-Landau equations in a magnetic field in the presence of strong thermal fluctuations. Our results show that the current density increases considerably with increasing pinning strength. It means that an original property of superconductor is recovered even in strong thermal fluctuations under a magnetic field by adding a pinning array to the mixed state of superconductor.

The Electric Properties of the Medium inside the Crack Effects on the Electric-Field-Induced Domain Evolution in Ferroelectric

Effects of the electric inclusion in ferroelectrics on the stress and electric field concentration and polarization switching are investigated based on a phase field approach containing the time-dependent Ginzburg-Landau equation. To capture a clear physical picture with the simulation, the crack medium in the electric inclusion is taken into account explicitly as a crack fluid medium as water, oil or air in region. The simulations exhibit a macroscopic electric field concentration in the electric inclusion filled with air, and a significant influence on the domain evolution from the micro perspective, while the electric inclusion filled with water have little influence. The numerical calculations indicated that, when the dielectric constant inside the flaw are much smaller than the dielectric constant of the ferroelectric matrix, the electric field inside the crack medium are enhanced much higher than the applied electric loading. The result implies the domain evolution takes place with the minimization of total free energy, which involves the high electric field energy derived from the electric inclusion. Therefore, the crack fluid medium in ferroelectric plays in importation role in the effect on the non-uniform distribution of the stress and electric field.

Dynamic pair-breaking current, critical superfluid velocity, and nonlinear electromagnetic response of nonequilibrium superconductors

We report numerical calculations of a dynamic pairbreaking current density $J_d$ and a critical superfluid velocity $v_d$ in a nonequilibrium superconductor carrying a uniform, large-amplitude ac current density $J(t)=J_a\sin\Omega t$ with $\Omega$ well below the gap frequency $\Omega\ll \Delta_0/\hbar$. The dependencies $J_d(\Omega,T)$ and $v_d(\Omega,T)$ near the critical temperature $T_c$ were calculated from either the full time-dependent nonequilibrium equations for a dirty s-wave superconductor and the time-dependent Ginzburg-Landau (TDGL) equations for a gapped superconductor, taking into account the GL relaxation time of the order parameter $\tau_{GL}$ and the inelastic electron-phonon relaxation time of quasiparticles $\tau_E$. We show that both approaches give similar frequency dependencies of $J_d(\Omega)$ and $v_d(\Omega)$ which gradually increase from their static pairbreaking GL values $J_c$ and $v_c$ at $\Omega\tau_E\ll 1$ to $\sqrt{2}J_c$ and $\sqrt{2}v_c$ at $\Omega\tau_E\gg 1$. Here $J_d$, $v_d$ and a dynamic superheating field at which the Meissner state becomes unstable were calculated in two different regimes of a fixed ac current and a fixed ac superfluid velocity induced by the applied ac magnetic field $H=H_a\sin\Omega t$ in a thin superconducting filament or a type-II superconductor with a large GL parameter. We also calculated a nonlinear electromagnetic response of a nonequilibrium superconducting state, particularly a dynamic kinetic inductance and a dissipative quasiparticle conductivity, taking into account the oscillatory dynamics of superconducting condensate and the kinetics of quasiparticles driven by a strong ac current. It is shown that an ac current density produces multiple harmonics of the electric field, the amplitudes of the higher-order harmonics diminishing as $\tau_E$ increases.

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).

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.

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.