Non-Equilibrium Coherent Vortex States and Subharmonic Giant Shapiro Steps in Josephson Junction Arrays

Abstract

This is a review of recent work on the dynamic response of Josephson junction arrays driven by dc and ac currents. The arrays are modeled by the resistively shunted Josephson junction model, appropriate for proximity effect junctions, including self-induced magnetic fields as well as disorder. The relevance of the self-induced fields is measured as a function of a parameter κ = λ L /a, with λ L the London penetration depth of the arrays, and a the lattice spacing. The transition from Type II (κ > 1) to Type I (κ < 1) behavior is studied in detail. We compare the results for models with self, self+nearestneighbor, and full inductance matrices. In the κ = ∞ limit, we find that when the initial state has at least one vortex-antivortex pair, after a characteristic transient time these vortices unbind and radiate other vortices. These radiated vortices settle into a parity-broken, time-periodic, axisymmetric coherent vortex state (ACVS), characterized by alternate rows of positive and negative vortices lying along a tilted axis. The ACVS produces subharmonic steps in the current voltage (IV) characteristics, typical of giant Shapiro steps. For finite κ we find that the IV's show subharmonic giant Shapiro steps, even at zero external magnetic field. We find that these subharmonic steps are produced by a whole family of coherent vortex oscillating patterns, with their structure changing as a function of κ. In general, we find that these patterns are due to a break down of translational invariance produced, for example, by disorder or antisymmetric edge-fields. The zero field case results are in good qualitative agreement with experiments in Nb-Au-Nb arrays.

Phase locking of Josephson junctions in a two-dimensional array

“…This behavior is consistent with the assumption that phase locking in two-dimensional arrays of Josephson junctions is described by coherent vortex motion perpendicular to the transport current. [3][4][5][6][7][8][9][10][11][12][13] In some cases, we observed that all junctions of one row, but not the edge junctions, give a voltage signal. The inhomogeneous distribution of dc and ac currents in the array, due to current-induced magnetic fields, 11 may be responsible for this phenomenon.…”

“…Results on the amplitude ⌬I n of a particular Shapiro step of order n versus the applied rf power are obtained experimentally [4][5][6] or by numerical simulations. 7-10, 12 Phillips et al observed in numerical simulations a finite slope of Shapiro steps due to bias current nonuniformity because of the self-induced magnetic field of the bias current. 11 Hagenaars numerically found Shapiro steps showing finite slope due to nonperfect phase locking of all the junctions due to the motion of vortices that do not participate in the coherent motion.…”

Self-field effects in Josephson junction arrays

“…2 Apart from the applications, two-dimensional arrays have been also investigated as an interesting nonlinear system both experimentally and theoretically. 3,4 In the study of such arrays two classes of models have been proposed to describe the dynamics: ͑1͒ Models that neglect the self-field effects. These model are often called ''uniformly frustrated XY ,'' 5-7 because the Hamiltonian for this model is similar to that of a square flat lattice of spins in the magnetic field ͑for that reason it is also called the spin-glass model͒.…”

“…10,11 Domìnguez and Josè have reviewed the topic in Ref. 4. In spite of the simplicity of model ͑1͒, it has been able to explain most of the experimental observations on twodimensional arrays, for example, giant Shapiro steps.…”

Temperature dependence of the low-field magnetization of granular superconductors

“…At low temperatures, intergranular flux penetration is dominant and the magnetic properties of this class of superconductors can be well described by Josephsonjunction array models. [1][2][3][4][5][6][7][8][9] In some of these models 2,7 the presence of the physical grains is taken into account by means of the inductive couplings between junctions and flux penetration after zero-field cooling ͑ZFC͒ takes place exclusively in the intergranular regions because of 2 variations of the gauge-invariant superconducting phase difference of the junctions. 10 However, in order to account for the complete response of the system at higher fields or temperatures, or for different sample histories, intragranular flux penetration must be allowed.…”