Hybridized Abrikosov Flux-lines and Pancake Vortices in Two-band Superconductors with Mixed Dimensionality

We study electronic structure and thermodynamic properties of a two-band superconductor, in which one band is ballistic and quasi-two dimensional (2D), and the other is diffusive and three dimensional (3D). 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 properties of the ``strong'' 2D ballistic band. In particular, the effects of Coulomb interactions in the diffusive band and unusual Kramer-Pesch effect are examined. Furthermore, based on a circular-cell approximation within the quasiclassical theory of superconductivity, we explore the effects of magnetic field on vortex structure in such a two-band superconductor. We discuss hybridization of Abrikosov flux lines in the 3D diffusive band with pancake vortices in the 2D ballistic band.