Field dependence of the lock-in transition of Josephson vortex in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ mesoscopic crystals

We have investigated vortex states in magnetic fields close to the $ab$-plane of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi2212) by means of the $c$-axis resistivity $\rho_c$ in Bi2212 mesoscopic crystals fabricated by the FIB technique. When magnetic field $H \leq$ 1T was tilted from the $ab$-plane at $T=60$ K, a sharp drop in $\rho_c$ was observed at an angle $\theta_{th}$ ($\theta=0$ for ${\bf H} \parallel ab$-plane) depending on the field (typically $\theta_{th}$=0.5 deg. for $H=$1T). $\rho_c$ below $\theta_{th}$ does not depend on angle and is almost proportional to $H$, suggesting $\rho_c$ being Josephson vortex (JV) flow resistivity. Above $\theta_{th}$, it is considered that the JV flow is strongly impeded by pancake vortices (PVs), which attract with JVs and do not flow by the $c$-axis current, resulting in $\rho_c \approx 0$. Moreover, the penetration field of PVs $H\sin\theta_{th}$ decreases with increasing $H$. This is considered to be yielded by decrease of the line energy of pancake stacks due to the formation of the crossing lattice (CL). Therefore, the drop in $\rho_c$ corresponds to the transition from the lock-in (LI) state to the CL state. At high magnetic fields, the decrease in $\rho_c$ is broadened and has a shoulder structure. These results suggest that other vortex states between the LI and CL states emerge with increasing magnetic field and have intermediate JV viscosity. The expected vortex structures will be argued.