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 $\ensuremath{\vartheta}$ 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 $\ensuremath{\lambda},$ but larger than the coherence length $\ensuremath{\xi}.$ 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 $\ensuremath{\vartheta}l20\char21{}30\ifmmode^\circ\else\textdegree\fi{}$ of the exponential drop of the local critical current density ${J}_{b}(\ensuremath{\vartheta})$ 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 $Hg{H}_{c1}$ 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 $R(\ensuremath{\omega})$ 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 $R(\ensuremath{\omega})$ has peaks at the ``washboard'' ac frequency ${\ensuremath{\omega}}_{0}=2\ensuremath{\pi}v/a,$ where a is the intervortex spacing. This resonance can cause peaks and portions with negative differential conductivity on the averaged dc voltage-current $(V\ensuremath{-}I)$ 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 $I\ensuremath{-}V$ curves at the Josephson voltages, $n\ensuremath{\Elzxh}\ensuremath{\omega}/2e$ with $n=1.2,\dots{}.$