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_dd/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 curreut 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 equatious 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 2ω₁ - ω₂ from two external currents at frequencies ω₁ and ω₂ ≈ 2ω₁ 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 Z₀ and the rnixing current amplitude δj₍₃₎ as functions of external frequencies, crossover frequency ω_c α n_idω_D (measuring the pinning), and diffusion frequency ω_D of free 2DV.