Optical conductivity of granular aluminum films near the Mott metal-to-insulator transition

We report measurements of the energy gap of granular aluminum films by THz spectroscopy. We find that as the grains progressively decouple, the coupling ratio $2\Delta(0)/k_{B}T_{c}$ increases above the BCS weak coupling ratio $3.53$, and reaches values consistent with an approach to BCS-BEC crossover for the high resistivity samples, expected from the short coherence length. The Mattis-Bardeen theory describes remarkably well the behavior of $\sigma_{1,s}/\sigma_{1,n}$ for all samples up to very high normal state resistivities.