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 Theta 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 lambda, but larger than the coherence length xi. 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 Theta<20-30degrees of the exponential drop of the local critical current density J(b)(Theta) 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(omega) 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(omega) has peaks at the "washboard" ac frequency omega(0)=2piv/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, nhomega/2e with n=1.2,....