Dynamics in mesoscopic superconducting rings: relaxation process and vortex-antivortex pairs

Abstract

Abstract. We investigate the behavior of a mesoscopic 1D ring or 2D torus in an external magnetic field by simulating the Time Dependant Ginzburg Landau (TDGL) equations with periodic boundary conditions. In 1D, we analyze the stability and the different possible evolutions for the phase-slip phenomena starting from a metastable state. We used those results to observe the dynamics of a vortex-antivortex pair dissociation in 2D. IntroductionNonequilibrium phenomena in superconductors are a challenging area for both experimental and theoretical research. They are also crucial for the development of applications as they are the key to the appearance of resistive states in superconducting samples and to the possible use of vortex dynamics. In 1D, the resistive phase-slip process foreseen by W. A. Little [1] has been described quantitatively by the LAMH theory developed by J. S. Langer and V. Ambegaokar [2] and extended by D. E. McCumber and N. B. Halperin [3,4]. The LAMH theory describes thermally activated phase-slips (TPS). It evaluates the resistance of a 1D superconductor when driven out of thermodynamical equilibrium by a voltage or current source. Since then, this theory has been accepted to a large extent in rather good agreements with experiments (for a more complete review of theoretical and experimental works, see [5]).More recently, simulations were carried out on different versions of the time dependant Ginzburg Landau (TDGL) equations in order to investigate the dynamics of the process. In particular, the case of the 1D ring has raised interest since it exhibits multiple metastable states which can be reached by phase-slip processes. In [6], the ring is submitted to an electromotive force which constantly accelerates the superconducting electrons. Another approach has been made in [7], where the magnetic field is gradually increased.In the present work, we consider a superconducting ring of thickness d, radius R and length L as represented on figure 1. For simplicity, we consider d << ξ << λ ef f , R ≈ ξ and R << λ ef f , where ξ is the coherence length and λ ef f is the Pearl [8] penetration depth. The first two conditions allow us to treat the ring as one dimensional and the last two conditions account for the mesoscopic size of the ring. We submit this 1D ring to a perpendicular constant magnetic field H, which determines the properties (superfluid density, current etc.) of the stable state. Our approach is different because we observe a relaxation process from the initial state to a more stable one, without any external parameter influencing the dynamics.We also simulated vortex-antivortex dynamics in 2D, starting from a phase-slip line situation. Indeed, the vortex-antivortex pair dissociation is analog to the phase-slip in 1D.

Multiple phase slips phenomena in mesoscopic superconducting rings

“…The phase of the order parameter develops a sharper sinusoidal form, until the minimum and the maximum disconnect, at the same moment when the amplitude vanishes and in the same region, as discussed in Ref. 19. Afterward, it relaxes to a sawtooth pattern corresponding to the new state (15) so that both amplitude and phase are consistent with the solution (17).…”

Electromagnetic response during quench dynamics to the superconducting state: Time-dependent Ginzburg-Landau analysis

“…Changes in the winding number occur via phase slip processes at which the amplitude of the order parameter is driven locally to zero and the phase difference across the region with locally zero order parameter changes by a multiple of 2π. [20][21][22][23]25,26 In the deterministic dynamics studied here phase slips occur when the local amplitude of the current is greater than an energy barrier determined by the local magnitude of the order parameter. (Note that the random initial conditions mean that this magnitude will be different on different sites and that the random currents implied by the random phases will lead to different order parameter magnitudes at intermediate times even if the initial condition is a space independent order parameter magnitude).…”

“…For E-field pulses that are stronger, or applied earlier, phase slips may occur. The stability analysis performed in 17,20,22,23 reveals that within the deterministic TDGL dynamics for fully established phase stiffness the critical value for the vector potential is A c = ∆ 0 /(2 √ 3ξ) ≈ 0.091/ξ. In Fig.…”