Interplay of ballistic and diffusive superconductivity in the vortex core in the model two-band system MgB$_2$

A revived interest in multi-band superconductivity has emerged due to the unexpected and interesting simultaneous presence of diffusive and ballistic bands in the superconductor MgB$_2$. Motivated by recent experimental data on the vortex state in MgB$_2$ obtained by scanning tunneling spectroscopy, we theoretically study the intriguing effects of superconductivity in a diffusive band (`$\pi$ band') induced by superconductivity in a ballistic band (`$\sigma $ band'). We apply a unique model that has been developed recently [1] for describing such a system, based on coupled Eilenberger and Usadel equations. Results are presented for the spatial variation of the order parameter, the current density, and the vortex core spectrum in the two bands. A particularly interesting result emerging from our studies is the possibility of additional bound states near the gap edge in the `strong' $\sigma$ band, which arise from hybridization with the `weak' $\pi$ band. The development of such gap-edge bound states is examined for various sets of physical parameters that are relevant for MgB$_2$. We will also discuss the induced Kramer-Pesch effect in the $\pi$ band and magnetic-field dependence of vortex core size. [1] K. Tanaka, D. F. Agterberg, J. Kopu, M. Eschrig, Phys. Rev. B {\bf 73}, 220501(R) (2006).