The interaction of phase-slip centers in superconducting filaments

Theoretical step structures for ideal homogeneous filaments based on the interactions implicit in the simple phase-slip-center model of Skocpol, Beasley, and Tinkham are displayed explicitly.Research supported in part by NSF and ONR.Senior U.S. Scientist awardee of Alexander von Humboldt Foundation.     

Instabilities in the array of phase-slip centers in superconducting filaments

The dynamics of formation and destruction of phase-slip centers (PSC) in the resistive state of a narrow superconducting channel is considered. Using the principle of minimum entropy production, stability is studied of a periodic array of PSCs with respect to their displacements from the equilibrium positions. It is shown that the periodic array of PSCs is stable only if the period is long enough. By means of numerical integration of the dynamic equations, we have considered the behavior of PSCs over times longer than the order parameter relaxation time. More specifically, we studied formation and destruction of localized and nonlocalized PSCs after variations of the current through the channel, as well as the stability of an array of PSCs as a function of its structure period. Results of the numerical integration agree well with the theoretical analysis. Moreover, we have observed numerically a new phenomenon, self-sustained oscillations in the system of PSCs. This effect consists in formation and destruction of a PSC periodically in time at certain values of the current.     

Structure and properties of the dissipative phase-slip state in narrow superconducting filaments with and without inhomogeneities

Using the generalized time-dependent Ginzburg-Landau equations for dirty superconductors nearT _c we study the oscillatory phase-slip solutions in quasione-dimensional superconductors at arbitrary pair-breaking parameter and calculate their properties. For small pair breaking (low temperature), a static approximation can be used, which allows analytic evaluation of most quantities and connects the theory to simple models. The influence of two types of short weak regions that pin the phase slips is considered. They tend to decrease the differential resistance and influence its temperature dependence.     

Theory of periodically driven,current-carrying superconducting filaments. II. Stability limits of the homogeneous state. Periodic and chaotic phase-slip states

Using the generalized time-dependent Ginzburg-Landau equations for dirty superconductors in local equilibrium, we determine the stability limits of the homogeneous superconducting state in high-frequency, current-carrying filaments. For weak pair-breaking by inelastic scattering, the instability sets in when the gap is depressed by the current to the universal ratio (2/3)^1/2. In the unstable region (and beyond it) there exist stable ac phase-slip solutions. In appropriate ranges of frequency and temperature, they exhibit complex time behavior with period-doubling and intermittency-type transition to chaos.     

The structure of phase-slip centers and the resistive state of narrow superconducting channels

The resistive state of narrow superconducting channels is considered on the basis of dynamic equations of the microscopic theory of superconductivity. The analytical solution for the order parameter is obtained in the immediate vicinity of a phase-slip center. The current-voltage characteristic of the resistive state is calculated.     

Interaction between phase-slip centers in superconducting indium microbridges

We have studied the electric behavior of phase-slip centers in superconducting indium microbridges of 0.9 µm thickness, 3–4 µm width, and 70–130 µm length. Our measurements were performed in the temperature range 3–10 mK below the critical temperature T_c resulting in relatively large values (about 10–20 µm) of the quasiparticle diffusion length. The temperature dependent Ginzburg-Landau coherence length (T) has been determined for all samples from the measured temperature dependence of the critical current density. The fabrication of two notches in the indium bridge has been found to localize reliably the nucleation site of the first two phase-slip centers. Our measurements quantitatively confirmed the dc interaction predicted by the model of Kadin, Smith, and Skocpol. In addition, for temperatures very close toT _c , an ac interaction has been observed which disappears abruptly for decreasing temperature. No voltage coupling has been detected.     

Current-induced relaxation of charge imbalance in superconducting phase-slip centers

In superconductors with a relatively long inelastic scattering time, relaxation of charge imbalance generated in phase-slip centers will be dominated by elastic scattering in the presence of a supercurrent. The spatial dependence of the electrochemical potential is calculated. Important qualitative differences occur from the usual model of phase-slip centers, based on relaxation by inelastic scattering.Research supported by the Stichting voor Fundamenteel Onderzoek der Materie FOM.     

Hysteresis losses in a superconducting plate

On the basis of theoretical analysis expressions have been derived suitable for calculating the hysteresis losses in the presence of variable and constant components of both external magnetic field and transport current in a superconducting layer. The calculation scheme can be applied to any arbitrary shape of the field variation cycle, provided the variable components of the field and current are proportional. It has been assumed that the critical current density is dependent only on the external field. For the more common forms of this relationship the results have been expressed in an analytical form. Application of the results is illustrated by calculating the losses in a superconducting solenoid under variable current in the winding and the external field.     

Charge imbalance waves and nonequilibrium dynamics near a superconducting phase-slip center

Using a generalized two-fluid picture to describe a quasi-one-dimensional superconductor near T _c,we provide a heuristic derivation for a set of equations governing the temporal and spatial evolution of the charge imbalance (or branch imbalance) in the quasiparticles. We show that these equations are isomorphic to those that describe a simple electrical transmission line, so that charge imbalance waves may propagate in the superconductor in analogy with electrical signals that propagate down the transmission line. We propose as a model for a phase-slip center in a superconducting filament a localized Josephson oscillator coupled to the transmission line. Applying standard transmission-line theory to solve the problem, we show that the Josephson oscillations in the center generate charge imbalance waves that propagate out to a frequency-dependent distance of the order of the quasiparticle diffusion length _Q ^*= (D Q ^*)^1/2 before they damp out. The time-averaged behavior of the model reduces to the earlier model of Skocpol, Beasley, and Tinkham. A novel consequence of the model is a prediction of intrinsic hysteresis in the dc current-voltage relation. The model also provides a convenient framework for dealing with ac effects in phase-slip centers, including resonance and synchronization in systems of closely spaced phase-slip centers and microbridges.Research supported in part by NSF and ONR.     

Phase-slip centers in long superconducting filaments

The behavior of phase slip centers and their interactions via exchange of ac and dc quasiparticles in long superconducting filaments are studied with samples fabricated by using an electron beam lithography technique. The data show that the ac decay length of quasiparticles is shorter than the dc decay length in indium at temperatures nearT _c . The interactions among phase slip centers in the long filament are interpreted on the basis of critical current variations due to quasiparticle exchange, from which a dc decay length 5.0 µm for indium atT/T _c =0.992 has been obtained. The results indicate that a phase slip center can be used as a quasiparticle detector to obtain information about other phase slip centers and their mutual interactions.Supported in part by the National Science Foundation.     

Dynamic phase-slip centers and the resistive state of a strong-nonequilibrium superconductor

The model of dynamic phase-slip centers is considered for the resistive state of a strong-nonequilibrium, quasi-one-dimensional superconductor. It is shown that at a certain type of nonequilibrium, the factor in the first time-derivative of the order parameter in the time-dependent Ginzburg-Landau equation can be made small. In this case the dynamic phase-slip centers in a two-dimensional space-time x; t have a structure quite similar to that of vortex lines in the mixed state of a type II superconductor in ordinary space.     

Behavior of a phase-slip center at a tunable weak link in superconducting Sn whiskers

We investigated cross-type specimens of two Sn whiskers with a metallic connection at their crossover. Applying a control current to one of the whiskers yields a tunable weak link (TWL) in the other whisker at the crossing point. The properties of the voltage-current characteristic generated by a phase-slip center developing at the TWL depend on the weakness of superconductivity in this region. The measurements show that the different behavior of phase-slip centers in Sn microbridges and Sn whiskers observed in former experiments may be a consequence of the different grade of homogeneity of these specimens. Finally, the weakening mechanism responsible for the weakening of superconductivity in the TWL is discussed.     

Nonequilibrium theory of dirty,current-carrying superconductors: phase-slip oscillators in narrow filaments near T c

General equations for the dynamic behavior of dirty superconductors in the Ginzburg-Landau regime T _c-T T _care derived from microscopic theory. In the immediate vicinity of T _ca local equilibrium approximation leads to a simple generalized time-dependent Ginzburg-Landau equation. The oscillatory phase-slip solutions presented previously are discussed in greater detail.Supported by the Science Research Council (England) under grant GR/A65218, and by the Deutsche Forschungsgemeinschaft.     

Thermal expansion in the superconducting state

We present a model for the electronic contribution to the thermal expansion in the superconducting state based on the weak coupling BCS theory. A simple scaling parameter is introduced to accommodate strong coupling. We find satisfactory agreement between the model's predictions and experimental measurements for a number of materials exhibiting qualitatively different expansion behavior.Supported by the Australian Research Grants Committee.     

Hysteresis of critical currents of superconducting bridges in low perpendicular magnetic fields

Hysteresis of critcal currentsI _c of superconducting bridges with In, Nb, and NbN has been studied in low perpendicular magnetic fields. Influences of bridge geometry, small field sweep, trapped flux, and bombardment of argon ions on the hysteresis were made clear. The experimental results suggest that the edge pinning and trapped flux in the bank of bridges are associated with the hysteresis. The peak value ofI _c of NbN bridges, as well as granular Al and In bridges reported before, in decreasing fields agrees with the calculated pair-breaking current. The origin of the hysteresis is discussed.     

The intermediate state in superconducting mercury

Using the magneto-optic technique, an extensive study has been made of the intermediate state in single crystals of superconducting mercury. A regular laminar structure, closely approaching that envisaged by Landau, was realized in the thinnest sample examined (47 Μm). The field dependence of the lamina periodicity displays excellent agreement with the Landau model and the deduced surface energy is in good agreement with the Ginzburg-Landau theory. At applied magnetic fields very close to the critical one, the stable magnetic structure consists of isolated superconducting filaments. A number of observations are reported on their growth and stability.     

De Haas-van Alphen experiments in the superconducting state

Landau quantum oscillations of the magnetization, commonly referred to as the dHvA effect, appear to be a general property of superconductors in the mixed state. This is both intriguing from a fundamental point of view, and fortunate in that it provides a microscopic probe with which to explore the many-body environment of a superconductor. The results of experiments to date can be understood rather well in terms of a general theory of the dHvA effect for an interacting system. We indicate the nature of this and comment on the implications for molecular and oxide superconductors.     

Oxide glass filaments and their superconducting phase by annealing

Bi₄Ca₃Sr₃Cu _x O _y (4334) oxide glasses with different concentrations of the transition metal copper have been prepared by rapid quenching technique from respective melts. These glasses have been characterized from electrical, magnetic, dielectric and other properties. The semiconducting glasses are not superconducting even down to 4·2 K. However, by properly annealing the glasses in air or in oxygen atmosphere they can be converted into the corresponding superconducting phases with superconducting transition temperatures between 70 and 110 K depending on the Cu ion concentrations. The dc conductivity of the glasses could be explained with small polaron adiabatic hopping mechanism. Analysis shows that the correlated barrier hopping (CBH) mechanism is the most appropriate model for explaining the a.c. conductivities of these glasses. Addition of a small amount (about 2 wt%) of B₂O₃ makes it possible to draw glass filaments which could also be converted into the superconducting phase withT _c values lower than those obtained from the respective pure glasses.     

Dynamics of the intermediate state in a superconducting wire

A new approach to the dynamics of the intermediate state in a wire is presented. It represents a generalization of a recent theory for the dynamic resistance of the wire. It is shown that a large dynamic self-inductance also appears. The effects of the wire's environment on the intermediate state dynamics are discussed for the first time, and the classical skin effect is accounted for to all orders. A detailed analysis of the ac voltage response to mixed dc and weak ac current conditions is presented.Supported by CRAD Contributions 3610-609 (N.G.) and 3610-626 (P.R.).