Static and dynamic properties of stacked Josephson junctions: Analytic solution

Abstract

Static and dynamic properties of stacked Josephson junctions are studied theoretically. An approximate analytic solution for a stack with arbitrary junction parameters was obtained. The analytic solution is in good agreement with numerical simulations. Characteristic penetration depths, Swihart velocities, the lower critical field, the first integral, and the free energy for a stack of nonidentical junctions were derived and studied for different parameters of the stack. We show that attractive interaction of fluxons in adjacent junctions exists in the dynamic state of the stack, leading to appearance of the ``in-phase'' state with fluxons on top of each other. In a given external magnetic field the Gibbs free energy has a number of local minima corresponding to particular fluxon distributions (modes) in the stack each representing a quasiequilibrium state. For a stack of $N$ junctions each mode would result in $N$ distinct flux-flow branches in the current-voltage characteristic. Taking into account that different modes with equal total number of fluxons are not identical we conclude that the total possible number of flux-flow branches can be much larger than the number of junctions in the stack.