Far-infrared Studies of the Two-Gap Behavior in Epitaxial MgB$_{2}$ Films

Far-infrared transmission and reflectivity measurements have been carried out for a series of pure and carbon-doped epitaxial MgB$_{2}$ films. While the carbon-doped film exhibits the typical characteristics for a dirty BCS superconductor in the T$_{S}$ /T$_{N}$ and R$_{S}$ /R$_{N}$ ratios, the pure MgB$_{2}$ films can only be understood knowing the multi-gap nature of the superconducting state in MgB$_{2} $. As a function of increasing T$_{c}$, the fraction of the Cooper pairs having the larger gap increases. Both gaps appear to follow the BCS temperature dependence. However, the two gaps exhibit different behavior when a magnetic field is applied along c-axis. While the smaller gap can be suppressed by a relatively small field, the larger gap can persist up to 10 T. These infrared measurements indicate that the two superfluids coexist quite independently in the superconducting state of MgB$_{2}$.