Nonlinear dynamics of vortices in easy flow channels along grain boundaries in superconductors

A theory of nonlinear dynamics of mixed Abrikosov vortices with Josephson cores (AJ vortices) on low-angle grain boundaries (GB) in superconductors is proposed. As the misorientation angle ϑ increases, vortices on low-angle GBs evolve from the Abrikosov vortices with normal cores to intermediate AJ vortices with Josephson cores, whose length l along GB is smaller that the London penetration depth λ, but larger than the coherence length ξ. Dynamics and pinning of the AJ vortex structures determine the in-field current transport through GB and the microwave response of polycrystal in the crucial misorientation range ϑ<20–30° of the exponential drop of the local critical current density J_b(ϑ) through GB. An exact solution for an overdamped periodic AJ vortex structure driven along GB by an arbitrary time-dependent transport current in a dc magnetic field H>H_c1 is obtained. It is shown that the dynamics of the AJ vortex chain is parametrized by solutions of two coupled first-order nonlinear differential equations which describe self-consistently the time dependence of the vortex velocity and the AJ core length. Exact formulas for the dc flux flow resistivity R_f(H), and the nonlinear voltage-current characteristics are obtained. Dynamics of the AJ vortex chain driven by superimposed ac and dc currents is considered, and general expressions for a linear complex resistivity R(ω) and dissipation of the ac field are obtained. A flux flow resonance is shown to occur at large dc vortex velocities v for which the imaginary part of R(ω) has peaks at the “washboard” ac frequency ω₀=2πv/a, where a is the intervortex spacing. This resonance can cause peaks and portions with negative differential conductivity on the averaged dc voltage-current (V-I) characteristics. ac currents of large amplitude cause generation of higher voltage harmonics and phase locking effects which manifest themselves in steps on the averaged dc I-V curves at the Josephson voltages, nħω/2e with n=1.2,….