  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$_2$.
