AC response of Abrikosov vortices in layered superconductors

Abstract

In a layered high-T c superconductor an Abrikosov vortex line perpendicular to the layers can be considered as a stack of coupled two-dimensional «pancake» vortices (2DV) with cores residing inside the superconducting layers. When the in-plane penetration length λ ab =λ is smaller than the Josephson length λ J =λ c d/λ (d: layer period) the interaction between 2DV is via a magnetic pair force which is complicated but exactly known. We calculate linear and nonlinear ac responses for this model including randomly distributed pinning centers. In detail, the thermally averaged shift S n (t) of a 2DV in layer n under the influence of a finite external ac current in the frequency range up to 1 GHz is computed from coupled Fokker-Planck equations for the distribution functions of 2DV positions. In the adiabatic approximation, the Fokker-Planck equations can be cast into a nonlinear differential equation for a continuous shift function S(z,t) using a gradient expansion. The constitutive equation for S(z,t) is solved perturbatively for the linear ac response and for the mixing current at frequency 2ω 1 -ω 2 from two external currents at frequencies ω 1 and ω 2 ≃2ω 1 due to a third order process. This pseudo-harmonic mixing probes the nonlinearity of the intervortex force and its interplay with pinning. Explicit results for the film geometry are given for the surface impedance Z s and the mixing current amplitude δj (3) as functions of external frequencies, crossover frequency ω c ∞n i dω D (measuring the pinning), and diffusion frequency ω D of free 2DV