London theory of the crossing vortex lattice in highly anisotropic layered superconductors

A description of Josephson vortices (JV’s) crossed by the pancake vortices (PV’s) is proposed on the basis of the anisotropic London equations. The field distribution of a JV and its energy have been calculated for both dense $(a<\mathrm{}{\lambda }_J)$ and dilute $(a>\mathrm{}{\lambda }_J)$ PV lattices with distance a between PV’s and the nonlinear JV core size ${\lambda }_J.$ It is shown that the “shifted” PV lattice (PV’s displaced mainly along JV’s in the crossing-vortex lattice structure), formed in high out-of-plane magnetic fields $B_z>{\Phi }_0/{\gamma }^2{s}^2$ [A. E. Koshelev, Phys. Rev. Lett. 83, 187 (1999)], transforms into the PV lattice “trapped” by the JV sublattice at a certain field, lower than ${\Phi }_0/{\gamma }^2{s}^2,$ where ${\Phi }_0$ is the flux quantum, $\gamma$ is the anisotropy parameter, and s is the distance between ${\mathrm{CuO}}_2$ 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 $\gamma s<\mathrm{}{\lambda }_{\mathrm{ab}}$ with the in-plane penetration depth ${\lambda }_{\mathrm{ab}}$ if the low ${(B}_x<\gamma {\Phi }_0/{\lambda }_{\mathrm{ab}}^2)$ or high ${(B}_x\gtrsim {\Phi }_0/\gamma s^2)$ 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 $\mathrm{ab}$ plane as well. In the intermediate in-plane fields $\gamma {\Phi }_0/{\lambda }_{\mathrm{ab}}^2\lesssim B_x\lesssim {\Phi }_0/\gamma s^2,$ 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 $\mathbf{H}$ oriented near the $\mathrm{ab}$ plane. The recent experimental results concerning the vortex-lattice melting transition and transitions in the vortex-solid phase in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ single crystals are discussed in the context of the presented theoretical model.