Transfer-matrix approach to multiband josephson junctions

We study the influence of multiple bands on the properties of Josephson junctions. In particular, we focus on the two gap superconductor magnesium diboride. We present a formalism to describe tunneling at a point contact between two $\mathrm{Mg}{\mathrm{B}}_{2}$ electrodes generalizing the transfer-matrix approach to multiple bands. A simple model is presented to determine the effective hopping amplitudes between the different energy bands as a function of the misorientation angle of the electrodes. We calculate the critical current and the current-voltage characteristics for $N\text{\ensuremath{-}}I\text{\ensuremath{-}}S$ and $S\text{\ensuremath{-}}I\text{\ensuremath{-}}S$ contacts with different orientation for junctions with both high and low transparency. We find that interband tunneling processes become increasingly important with increasing misorientation angle. This is reflected in certain features in the differential tunneling conductance in both the tunneling limit as well as for multiple Andreev reflections.