Scaling behavior of the crossover to short-stack regimes of Josephson vortex lattices in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ stacks

We report the systematic investigations of the oscillation of the Josephson vortex (JV) flow resistance in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ micro-fabricated junctions with various geometries and superconducting anisotropy parameters. As the applied magnetic field parallel to the $ab$-plane is increased, oscillation with a period corresponding to a $\phi_0/2$ par atomic Josephson junction changes to oscillation with a doubled period. This crossover is scaled by both the junction length and the anisotropy parameter, indicating that the bulk inductive coupling that favors the triangular JV lattice is replaced with the surface deformation energy as the dominant interaction for a JV lattice. These results suggest that the in-phase square JV lattice is pronounced at a higher magnetic field in a smaller and more anisotropic sample.