Inter-band scattering effects in the Chevrel phase superconductors

Recent scanning tunneling spectroscopy on the Chevrel phases SnMo$_6$S$_8$ and PbMo$_6$S$_8$ by Petrovi\'{c} et al.~(PRL \textbf{106}, 017003 (2011)) demonstrates clear signatures of multi-band superconductivity. In contrast with MgB$_2$, where there is extremely weak scattering between the $\sigma$ and $\pi$ bands, the inter-band scattering rate is relatively high compared to the intra-band scattering rate in the Chevrel phases. We calculate the quasi-classical Green's function in the presence of strong inter-band scattering and derive comparable results to the measured density of states. We make predictions for proposed measurements on the vortex cores.