Time-dependent Ginzburg-Landau Theory Research Articles

Measurement of the gap relaxation time of superconducting NbTi strips on a sapphire substrate

We report on the study of the transient voltage response of superconducting NbTi strips to an over-critical current pulse ($${I}> {I}_\mathrm{c}$$, where $${I}_\mathrm{c}$$ is the pair-breaking current). In this experiment, a localized normal spot appeared for a current amplitude larger than the critical current. The induced metastable superconducting state was identified as either a hotspot or phase-slip center. These two dissipative modes share the feature of the voltage response occurring after a delay time $${t}_\mathrm{d}$$, a solution of the time-dependent Ginzburg–Landau (TDGL) theory developed by M. Tinkham. The gap relaxation time was subsequently deduced from fitting the experimental data with the TDGL theory. An agreement was found by choosing an effective gap relaxation time $$\tau _{\Delta }= 4.75 \, \hbox {ns}$$ for a thickness of 50 nm. Assuming the proportionality to sample thickness, this indicates a thermal relaxation time of 96 ps/nm for a NbTi film sputtered at room temperature on polished crystalline $$\hbox {Al}_{{2}} \hbox {O}_{{3}}$$. If we assume that the electron and the phonon specific heats have the same ratio as for pure Nb, then it results in a phonon heat escape time $$\tau _\mathrm{es}/{d} = 32 \, \hbox {ps/nm}$$.

Modeling the Critical Current of Polycrystalline Superconducting Films in High Magnetic Fields

We have performed simulations using time-dependent Ginzburg–Landau theory on a two-dimensional (2-D) polycrystalline system, where grain boundaries are modeled as narrow regions with a locally reduced critical temperature ( T c ). For the small system sizes investigated, we find that the critical current density ( J c ) is not sensitive to changes in grain size until the grain size is sufficiently small that it limits the average superparticle density in the system through the proximity effect. Furthermore, once T c in the boundary regions is sufficiently low relative to the surrounding superconductor that grain boundary regions act as preferred channels for flux flow, further reductions in the boundary T c only weakly reduce J c across the superconductor.

Generation of Terahertz oscillations by thin superconducting film in fluctuation regime

Explicit analytical expressions for conductivity of a superconducting film above and below critical temperature in an arbitrary electric field are derived in the frameworks of the time dependent Ginzburg-Landau theory. It is confirmed that slightly below critical temperature the differential conductivity of superconducting film can become negative for small enough values of electric field. This fact may cause generation of electromagnetic oscillations if the superconducting film is appropriately coupled of with a resonator. Their maximal frequency is proportional to the value of critical temperature of superconducting transition. The obtained results can stimulate the development of Terahertz generators on the basis of high temperature superconducting films.

Hybrid patterns from directed self-assembly of diblock copolymers by chemical patterns.

The surface affinity is a critical factor for controlling the formation of monolayer nanostructures in block copolymer thin films. In general, strong surface affinity tends to induce the formation of domains with low spontaneous curvature. Abiding by this principle, we propose a facile chemoepitaxial scheme for producing large-scale ordered hybrid patterns by the directed self-assembly of diblock copolymers. The guiding chemical pattern is designed as periodic stripes with alternately changing surface affinities. As a consequence, two different geometries of domains are formed on the stripes with different affinities. The self-assembly process of block copolymers guided by the stripe patterns is investigated using cell dynamics simulations based on time-dependent Ginzburg-Landau theory, and the kinetic stability diagram is estimated. Hybrid patterns are successfully achieved with both cylinder-forming and sphere-forming diblock copolymers. In the cylinder-forming system, the major hybrid pattern exhibiting a considerable stability window is composed of parallel cylinders and perforated lamellae, while it is composed of monolayer spheres and parallel cylinders in the other system. Encouragingly, the chemoepitaxial method is valid till the period of the guiding pattern is a large multiple of the domain spacing. The chemoepitaxial scheme demonstrated in this work serves as a nice supplement to the graphoepitaxial one proposed in our previous work.

Thermally activated flux flow, vortex-glass phase transition and the mixed-state Hall effect in 112-type iron pnictide superconductors

The transport properties in the mixed state of high-quality Ca0.8La0.2Fe0.98Co0.02As2 single crystal, a newly discovered 112-type iron pnictide superconductor, are comprehensively studied by magneto-resistivity measurement. The field-dependent activation energy, U0, is derived in the framework of thermally activated flux flow (TAFF) theory, yielding a power law dependence U0~Hα with a crossover at a magnetic field around 2 T in both H⊥ab and H//ab, which is ascribed to the different pinning mechanisms. Moreover, we have clearly observed the vortex phase transition from vortex-glass to vortex-liquid according to the vortex-glass model, and vortex phase diagrams are constructed for both H⊥ab and H//ab. Finally, the results of mixed-state Hall effect show that no sign reversal of transverse resistivity ρxy(H) is detected, indicating that the Hall component arising from the vortex flow is also negative based on the time dependent Ginzburg-Landau (TDGL) theory. Meanwhile, the transverse resistivity ρxy(H) and the longitudinal resistivity ρxx(H) follow the relation |ρxy(H)|=Aρxx(H)β well with an exponent β~2.0, which is in line with the results in theories or experiments previously reported on some high-Tc cuprates.

Effect of pinning on the vortex motion in superconducting strip

We study the effect of different sizes and shapes of pinning centers on the vortex motion in superconducting strip in the presence of external current and magnetic field using the time-dependent Ginzburg–Landau theory. The size of pinning centers obviously affects the threshold current corresponding to the appearance of phase-slip line. The larger size results in a considerable decrease of the threshold current and a larger output voltage. For the same size, the different shapes of pinning centers lead to different behaviors of vortices. The dynamic behavior of vortices is dependent on the minimum distance dm between pinning center and the sample surface and the maximum length lm of the pinning center paralleling with the sample edge. However, the obvious changes are observed in the dynamic of vortex motion in superconductor with different pinning centers.

Time-Dependent Ginzburg–Landau Simulations of the Critical Current in Superconducting Films and Junctions in Magnetic Fields

Understanding the magnetic field dependence of the critical current density (Jc) of superconductors is of considerable interest for optimizing their use in high field applications. Using time-dependent Ginzburg-Landau theory, we have completed simulations of the average electric field generated in thin film systems subject to transport currents in applied magnetic fields, and compared them to thin film systems containing narrow junctions of reduced critical temperature (Tc). For thin films in contact with insulating surfaces, Jc approaches the depairing current density at applied magnetic fields below the initial vortex penetration field and remains nonzero until close to Tinkham's parallel critical field. For thin films in contact with highly metallic surfaces, Jc was found to decrease to zero with decreasing film width. Adding a junction region to the film was found to broaden the transition to the normal state at all applied magnetic fields and reduce Jc of the film at zero field.

Defect Patterns from Controlled Heterogeneous Nucleations by Polygonal Confinements.

Defects are often observed in crystalline structures. To regulate the formation or annihilation of defects presents an interesting question. In this work, we propose a method to fabricate defect patterns composed of regularly distributed steady "programmed defects", which is proceeded via the heterogeneous nucleation of a hexagonal pattern from a homogeneous state. The nucleation process occurring in a model system of AB-diblock/C-homopolymer blends under polygonal confinement is modeled by the time-dependent Ginzburg-Landau theory and is simulated by the cell dynamics simulations. Specifically, we demonstrate the validity of this method by means of three polygonal confinements including square, pentagon, and octagon, which have mismatched angles with the hexagonal lattice. Each corner or side of the polygons induces a nucleation event separately. Two nucleated domain grains by two neighboring corners or sides exhibit incommensurate orientations, and thus their merging leads to a radial line of clustered defects in the form of five-seven pairs. As a result, these radial lines constitute a radial pattern of defects, and their number is equal to the side number of the polygon. The distance of five-seven defect pairs is dictated by the incommensurate angle between two neighboring grains, which is similar to that of defects in hard crystals. This method can be extended to fabricate diverse defect patterns by programming the nucleation agents beyond simple polygonal confinements.

Phase transition of a diblock copolymer and homopolymer hybrid system induced by different properties of nanorods**Project supported by the National Natural Science Foundation of China (Grant No. 21373131), the Provincial Natural Science Foundation of Shanxi, China (Grant No. 2015011004), and the Research Foundation for Excellent Talents of Shanxi Provincial Department of Human Resources and Social Security, China.

We investigated phase transitions in a diblock copolymer–homopolymer hybrid system blended with nanorods (NRs) by using the time-dependent Ginzburg–Landau theory. We systematically studied the effects of the number, length and infiltration properties of the NRs on the self-assembly of the composites and the phase transitions occurring in the material. An analysis of the phase diagram was carried out to obtain the formation conditions of sea island structure nanorodbased aggregate, sea island structure nanorod-based dispersion, lamellar structure nanorod-based multilayer arrangement and nanowire structure. Further analysis of the evolution of the domain sizes and the distribution of the nanorod angle microphase structure was performed. Our simulation provides theoretical guidance for the preparation of ordered nanowire structures and a reference to improve the function of a polymer nanocomposite material.

Time-dependent London approach: Dissipation due to out-of-core normal excitations by moving vortices

The dissipative currents due to normal excitations are included in the London description. The resulting time dependent London equations are solved for a moving vortex and a moving vortex lattice. It is shown that the field distribution of a moving vortex looses it cylindrical symmetry, it experiences contraction which is stronger in the direction of the motion, than in the direction normal to the velocity $\bm v$. The London contribution of normal currents to dissipation is small relative to the Bardeen-Stephen core dissipation at small velocities, but approaches the latter at high velocities, where this contribution is no longer proportional to $v^2$. To minimize the London contribution to dissipation, the vortex lattice orients as to have one of the unit cell vectors along the velocity, the effect seen in experiments and predicted within the time-dependent Ginzburg-Landau theory.

Soliton-induced critical current oscillations in two-band superconducting bridges

Using time-dependent Ginzburg-Landau theory we find oscillations of critical current density $j_c$ as a function of the length $L$ of the bridge formed from two-band superconductor. We explain this effect by appearance of the phase solitons in the bridge at $j<j_c$, those number changes with change of $L$. In case of sufficiently strong interband coupling oscillations of $j_c$ disappear.

Spin Seebeck effect in a simple ferromagnet near Tc: a Ginzburg–Landau approach

A time-dependent Ginzburg–Landau theory is used to examine the longitudinal spin Seebeck effect in a simple ferromagnet in the vicinity of the Curie temperature Tc. It is shown analytically that the spin Seebeck effect is proportional to the magnetization near Tc, whose result is in line with the previous numerical finding. It is argued that the present result can be tested experimentally using a simple magnetic system such as EuO/Pt or EuS/Pt.

Weak links array generated by a continuous wave laser in a superconducting sample

Vortex state and magnetic response in a disk with an array of weakly-superconducting (or normal regions) created in the sample increasing the temperature T locally by using a continuous wave laser (CWL) that emits a beam with a controlled heat output, beam duration and intensity are studied. The time-dependent Ginzburg–Landau theory is used to describe the interaction of the light with the superconductor, where the defects are include through the spatially-dependent temperature. Vortices penetrating into the weak-superconducting regions which are generated considering that T > Tc (Tc is the critical temperature of the sample) where the CWL is pointing. Qualitative changes are observed in the dynamics of the superconducting condensate in the presence of pinning with different shapes and sizes which are possible to be modified in the same superconducting sample without any structural changes that could include metals, permanent magnets and implantation of heavy ions.

Potential of energy harvesting in barium titanate based laminates from room temperature to cryogenic/high temperatures: measurements and linking phase field and finite element simulations

This paper studies the energy harvesting characteristics of piezoelectric laminates consisting of barium titanate (BaTiO3) and copper (Cu) from room temperature to cryogenic/high temperatures both experimentally and numerically. First, the output voltages of the piezoelectric BaTiO3/Cu laminates were measured from room temperature to a cryogenic temperature (77 K). The output power was evaluated for various values of load resistance. The results showed that the maximum output power density is approximately 2240 nW cm−3. The output voltages of the BaTiO3/Cu laminates were also measured from room temperature to a higher temperature (333 K). To discuss the output voltages of the BaTiO3/Cu laminates due to temperature changes, phase field and finite element simulations were combined. A phase field model for grain growth was used to generate grain structures. The phase field model was then employed for BaTiO3 polycrystals, coupled with the time-dependent Ginzburg–Landau theory and the oxygen vacancies diffusion, to calculate the temperature-dependent piezoelectric coefficient and permittivity. Using these properties, the output voltages of the BaTiO3/Cu laminates from room temperature to both 77 K and 333 K were analyzed by three dimensional finite element methods, and the results are presented for several grain sizes and oxygen vacancy densities. It was found that electricity in the BaTiO3 ceramic layer is generated not only through the piezoelectric effect caused by a thermally induced bending stress but also by the temperature dependence of the BaTiO3 piezoelectric coefficient and permittivity.

The Influence of Pinning Centers in the Magnetization of the Mesoscopic Superconductors

In this work, we study the mesoscopic superconducting samples with pinning centers using the time-dependent Ginzburg-Landau theory (TDGL). The pinning centers are introduced via the phenomenological function f(r) Sorensen et al. (Physica C 533, 40–43 2017). We calculated the magnetization curves M(H) for some distances d from the boundary of the sample to the position of the pinning center. From this investigation arises the relation between the first magnetic field H p and the distance d. It shows that the pinning centers located close to the boundary of the sample decrease H p , and also the existence of two regimes of the penetration of the vortices. The magnetization curves revel the existence of ruddle of jumps for low magnetic fields for small distances d, indicating a complex vortex penetration.

Evolution of coherent collective modes through consecutive charge-density-wave transitions in the(PO2)4(WO3)12monophosphate tungsten bronze

All optical femtosecond relaxation dynamics in a single crystal of mono-phosphate tungsten bronze (PO$_{2}$)$_{4}$(WO$_{3}$)$_{2m}$ with alternate stacking m=6 of WO$_{3}$ layers was studied through the three consequent charge density wave (CDW) transitions. Several transient coherent collective modes associated to the different CDW transitions were observed and analyzed in the framework of the time dependent Ginzburg-Landau theory. Remarkably, the interference of the modes leads to an apparent rectification effect in the transient reflectivity response. A saturation of the coherent-mode amplitudes with increasing pump fluence well below the CDWs destruction threshold fluence indicates a decoupling of the electronic and lattice parts of the order parameter under strong optical drive.

Phase-Slip Phenomena in Proximitized NbN/NiCu Superconducting Nanostripes

Ultrathin superconducting/ferromagnet (S/F) nanostripes are very interesting systems to investigate both the physics involved on nanoscale size and as light sensitive elements for superconducting single photon detectors. The electrical transport properties of NbN(8 nm)/NiCu(10 nm) nanostripes are presented down to a temperature of 4.2 K. A number of voltage steps were observed on the current-voltage characteristics, and they were investigated at different temperatures. A possible explanation in terms of active phase-slip phenomena has been proposed on the basis of the time-dependent Ginzburg-Landau theory, leading to an estimation of the inelastic electron-phonon relaxation time around 0.8 ps. The latter value was found to be in good agreement with a relaxation time, independently measured by femtosecond transient optical reflectivity experiments performed on the same bilayer.

Measurements of the Delay Time Between a Critical Current Pulse and the First Resistive Response in Superconducting Niobium Strips

We have measured in superconducting niobium filaments the delay time $\text{t}_{d}$ that separates the initiation of a pulse of overcritical current $(\text{I} > \text{I}_{c})$ from its first resistive response. The experiments were performed at various temperatures, typically 4, 5, and 6 K, well below the critical temperature T $_{c}$ , for delays in the range 0.3 ns $ t $_{d}$ $ 80 ns, divided into two parts for technical reasons. The data t $_{d}$ (T/T $_{c}$ , I/I $_{c}$ ) were analyzed through a time-dependent Ginzburg–Landau theory leaving the gap relaxation time $\tau _{d}$ as an unknown parameter. In a restricted range of current amplitudes (I/I $_{c}$ $ 1.15), a fit is obtained by choosing $\tau _{d}$ between 1.65 and 1.75 ns, which we interpret as a film cooling time of 23 ps per nm thickness, almost independently of the temperature.

Flexoelectricity, strain gradients, and singularities in ferroelectric nanostructures

The effect of flexoelectricity on the formation and evolution of domain structures in ferroelectric materials is developed by integrating strain gradient theory into a finite element phase field model. Length scales associated with elastic strain gradients and the corresponding polarization gradient across a domain wall are integrated into the time-dependent Ginzburg–Landau theory and numerical simulated. Theoretical relations of a shear strain gradient along an electrode/dielectric interface are first solved and verified numerically using the finite element model. A singularity in the strain gradient–induced polarization is shown to occur as the elastic strain gradient length scale approaches a flexoelectric length scale. The theory and finite element modeling is then extended to quantify strain gradient electromechanics near 180° and 90° tetragonal phase domain structures. It is shown that the strain gradient length scale [Formula: see text] strongly influences changes in strain across the domain walls but has a negligible effect on the polarization. Strain gradient effects become negligible when [Formula: see text], where [Formula: see text] is the polarization domain wall length scale.

The position-dependent vortex dynamics in the asymmetric superconducting ring

We study the position-dependence of vortex motion around asymmetric mesoscopic superconducting ring for the external current flowing from inner boundaries to outer boundaries based on time-dependent Ginzburg-Landau theory. The inner hole position can have a great impact on not only the vortex configuration but also the current-voltage (I-V) characteristics. Different from the vortex rotation in the symmetric structure, we demonstrate that vortices enter/exit from outer boundaries periodically and the formation of curved vortex channel strongly depend on the inner hole position. As the inner hole is close enough to the outer boundaries, vortices get deformed even at low applied current. Flux-flow state (i.e., slow-moving Abrikosov vortices) and phase-slip state (i.e., fast-moving vortices) coexist during a multiharmonic voltage oscillation. In this way, the vortex motion and critical current of the sample can be manipulated by the hole position. At the critical current corresponding to the abrupt jump in I-V curve, vortex motion becomes unstable and the vortices are trapped in the hole for the symmetric ring, while the vortices disappear at the outer boundaries for the asymmetric ring.