Vortex lattice structures in uniaxial superconductors

The structure of the vortex lattice in anisotropic, unaxial superconductors in the domain $H_{c1\mathrm{}}\ll H\ll H_{c2\mathrm{}}$ is considered within the London approach. To first order in the small parameter ${\frac{(L}{\lambda )}}^2$, where $L$ is the average intervortex spacing and $\lambda$ is the average penetration depth, there exists a continuum of different lattices with the same free energy for any direction of the magnetic induction $\mathbf{B}$ with respect to the crystal. It is shown that the degeneracy is removed if terms in the free energy of order ${\frac{(L}{\lambda )}}^4$ are taken into account, yielding both a unique structure and a preferred orientation of the vortex lattice with respect to the direction of $\mathbf{B}$ within the crystal. Parameters of the primitive cell for this structure are obtained and evaluated for known values of the anisotropy of Y${\mathrm{Ba}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_7$. For the particular case of vortices parallel to the Cu-O planes the degeneracy remains exact (within the London approach), which should make this lattice more susceptible to disorder. The magnetization is shown to be almost parallel to the $\hat{c}$ crystal direction for all orientations of the external field ${\mathbf{H}}_0$, except in a narrow domain where ${\mathbf{H}}_0$ is nearly normal to $\hat{\mathbf{c}}$.