The London theory of the crossing-vortex lattice in highly anisotropic layered superconductors
A novel description of Josephson vortices (JVs) crossed by the pancake vortices (PVs) is proposed on the basis of the anisotropic London theory. The field distribution of a JV and its energy have been calculated for both dense ($a<λ_J$) and dilute ($a>λ_J$) PV lattices with distance $a$ between PVs, and the nonlinear JV core size $λ_J$. It is shown that the ``shifted'' PV lattice (PVs displaced mainly along JVs in the crossing vortex lattice structure), formed in high out-of-plane magnetic fields transforms into the PV lattice ``trapped'' by the JV sublattice at a certain field, lower than $Φ_0/γ^2s^2$, where $Φ_0$ is the flux quantum, $γ$ is the anisotropy parameter and $s$ is the distance between CuO$_2$ planes. With further decreasing $B_z$, the free energy of the crossing vortex lattice structure (PV and JV sublattices coexist separately) can exceed the free energy of the tilted lattice (common PV-JV vortex structure) in the case of $γs<λ_{ab}$ with the in-plane penetration depth $λ_{ab}$ if the low ($B_x<γΦ_0/λ_{ab}^2$) or high ($B_x\gtrsim Φ_0/γs^2$) in-plane magnetic field is applied. It means that the crossing vortex structure is realized in the intermediate field orientations, while the tilted vortex lattice can exist if the magnetic field is aligned near the $c$-axis and the $ab$-plane as well. In the intermediate in-plane fields $γΦ_0/λ_{ab}^2\lesssim B_x \lesssim Φ_0/γs^2$, the crossing vortex structure with the ``trapped'' PV sublattice seems to settle in until the lock-in transition occurs since this structure has the lower energy with respect to the tilted vortex structure in the magnetic field ${\vec H}$ oriented near the $ab$-plane.