Minimal length of Josephson junctions with stable fluxon bound states

Abstract

Mathematically speaking, the bound states (stationary Josephson vortices) of the magnetic flux in Josephson junctions are presented as solutions of some nonlinear differential equation, whose form and respective boundary conditions depend especially on the geometry of the junction. The generated boundary value problem contains a number of parameters (Filippov et al 1987 Phys. Lett. A 120 47) such as the bias current, the boundary magnetic field, the sizes of junction and inhomogeneities in the inhomogeneous case as well, etc. Depending on the values of both physical and geometrical parameters, the bound states can be stable or unstable with respect to small space–time perturbations. The transition of some concrete vortex from one state to another is defined as a bifurcation of this solution. The bifurcations of the possible bound states in Josephson junctions at change in the main physical parameters—the bias current and the boundary magnetic field, in the one-dimensional case, are well discussed. At the same time, the influence of the geometrical parameters (sizes) of the Josephson junctions on the bifurcations of the vortices has been studied insufficiently so far. The necessity of the above study is conditioned by at least two reasons. First, the sizes of the physical devices utilizing Josephson junctions depend themselves on their sizes and, in particular, on the length of Josephson junctions in the ‘one-dimensional’ case. Hence, from a ‘technological’ point of view it is important to minimize these sizes. In this paper we show numerically that for every nontrivial stable or unstable vortex there exists a minimal length of Josephson junction, for which this state remains stable/unstable. Second, a similar problem arises from a theoretical point of view as well. In the one-dimensional case generally adopted is the division of Josephson junctions to ‘long’ or ‘short’ depending on the inequality λJ >> 1 (held or not) for their lengths, related to the Josephson penetration depth λJ. Our numerical results enable us to make that definition. More strictly, we consider every Josephson junction as ‘long’, in which there exists at least one nontrivial stable distribution of the magnetic flux for fixed values of all the physical and other geometrical parameters. In this way, whether a sample of Josephson junction is ‘long’ or ‘short’ is determined by the specific operating conditions.