Josephson flux flow resistance in single crystalline $\mathrm{Bi_2Sr_2CaCu_2O_{8+\delta}}$ in various rectangular geometries

Josephson flux-flow resistance $\rho_c(H,I,\theta)$ has been studied in single crystalline $\mathrm{Bi_2Sr_2CaCu_2O_ {8+\delta}}$ with various rectangular geometries (more than 20 samples) as functions of magnetic field intensity $H$, current $I$ and the magentic field orientation angle $\theta$ with respect to the $\mathrm{CuO_2}$ plane. We observed a clear evidence of the lock-in behavior of the Josephson vortex parallel to the $\mathrm{CuO_2}$ planes. It is found that this lock-in angle $\theta$ strongly depends on the sample dimension, especially, on the length $\ell$ parallel to the magnetic field and is inversly propotional to the $\sqrt{\ell}$. This can be understood by considring the torque energy and energy to create pancake vortices very well. The flux-flow resistance $\rho_c$ shows a characteristic oscillatory behavior as a function of $H$ with a well defined peariodicity of $H=\phi_0/2sw$ in low fields and $H_0=\phi_0/sw$, where $s$ is the layer distance and $w$ is the width of the sample perpendicular to the magnetic field. This oscillatory behavior changes dramatically as a function of $I$ and can be explained by the dynamical ordering of the Josephson vortices in the restricted rectangular dimensions, which impose a potential well to the Josephson vortices.