  The vortex structure is studied in light of MgB$_2$ theoretically based on a
two-band superconducting model by means of Bogoliubov-de Gennes framework. The
field dependence of the electronic specific heat coefficient $\gamma (H)$ is
focused. The exponent $\alpha$ in $\gamma (H)\propto H^{\alpha}$ is shown to
become smaller by adjusting the gap ratio of the two gaps on the major and
minor bands. The observed extremely small value $\alpha\sim 0.23$ could be
explained reasonably well in this two-band model with the gap ratio $\sim 0.3$.
