Tilted and crossing vortex chains in layered superconductors

In the presence of the Josephson vortex lattice in layered superconductors, a small c-axis magnetic field penetrates in the form of vortex chains. In general, the structure of a single chain is determined by the ratio of the London [$\lambda$] and Josephson [$\lambda_{J}$] lengths, $\alpha= \lambda/\lambda_{J}$. The chain is composed of tilted vortices at large $\alpha$'s (tilted chain) and at small $\alpha$'s it consists of a crossing array of Josephson vortices and pancake-vortex stacks (crossing chain). We study chain structures at intermediate $\alpha$'s and found two types of phase transitions. For $\alpha\lesssim 0.6$ the ground state is given by the crossing chain in a wide range of pancake separations $a\gtrsim [2-3]\lambda_J$. However, due to attractive coupling between deformed pancake stacks, the equilibrium separation can not exceed some maximum value depending on the in-plane field and $\alpha$. The first phase transition takes place with decreasing pancake-stack separation $a$ at $a=[1-2]\lambda_{J}$, and rather wide range of the ratio $\alpha$, $0.4 \lesssim \alpha\lesssim 0.65$. With decreasing $a$, the crossing chain goes through intermediate strongly-deformed configurations and smoothly transforms into a tilted chain via a second-order phase transition. Another phase transition occurs at very small densities of pancake vortices, $a\sim [20-30]\lambda_J$, and only when $\alpha$ exceeds a certain critical value $\sim 0.5$. In this case a small c-axis field penetrates in the form of kinks. However, at very small concentration of kinks, the kinked chains are replaced with strongly deformed crossing chains via a first-order phase transition. This transition is accompanied by a very large jump in the pancake density.