Orbital-dependent two-band superconductivity in MgB2

We show that a two-band model with ${\bf k}$-dependent superconducting gaps well describes the transmission and optical conductivity measured for MgB$_2$ thin films. It is also shown that the two-band anisotropic model consistently describes the specific-heat jump and thermodynamic critical magnetic field $H_c$. A single-gap anisotropic model is shown to be insufficient to understand consistently optical and thermodynamic behaviors. In our model, the pairing symmetry in each band has an anisotropic characteristic which is determined almost uniquely; the superconducting gap in the $\sigma$-band has anisotropy in the ab-plane and the gap in the $\pi$-band has a prolate form exhibiting anisotropy in the c-direction. The anisotropy in the $\sigma$-band produces rather small effects on the physical properties compared to the anisotropy in the $\pi$-band.