Abrikosov flux-lines in two-band superconductors with mixed dimensionality

We study vortex structure in a two-band superconductor, in which one band is ballistic and quasi-two-dimensional (2D), and the other is diffusive and three-dimensional (3D). A circular cell approximation of the vortex lattice within the quasiclassical theory of superconductivity is applied to a recently developed model appropriate for such a two-band system [Tanaka et al 2006 Phys. Rev. B 73, 220501(R); Tanaka et al 2007 Phys. Rev. B 75, 214512]. We assume that superconductivity in the 3D diffusive band is weak, i.e., mostly induced, as is the case in MgB$_2$. Hybridization with the weak 3D diffusive band has significant and intriguing influence on the electronic structure of the strong 2D ballistic band. In particular, the Coulomb repulsion and the diffusivity in the weak band enhance suppression of the order parameter and enlargement of the vortex core by magnetic field in the strong band, resulting in reduced critical temperature and field. Moreover, increased diffusivity in the weak band can result in an upward curvature of the upper critical field near the transition temperature. A particularly interesting feature found in our model is the appearance of additional bound states at the gap edge in the strong ballistic band, which are absent in the single-band case. Furthermore, coupling with the weak diffusive band leads to reduced band gaps and van Hove singularities of energy bands of the vortex lattice in the strong ballistic band. We find these intriguing features for parameter values appropriate for MgB$_2$.