  We present first-principles calculations of Schottky barrier heights (SBHs)
at interfaces relevant for silicon-based merged-element transmon qubit devices.
Focusing on Al(111)/Si(111) and CoSi$_2$(111)/Si(111), we consider various
possible interfacial structures, for which we study the relaxations of the
atoms near the interface, calculate the formation energies and Schottky barrier
heights, and provide estimates of the Josephson critical currents based on the
WKB tunneling formalism as implemented in the Simmons/Tsu-Esaki model. We find
that the formation energies and SBHs are very similar for all Al(111)/Si(111)
structures, yet vary significantly for the CoSi$_2$(111)/Si(111) structures. We
attribute this to the more covalent character of bonding at CoSi$_2$/Si, which
leads to configurations with distinct atomic and electronic structure. Our
estimated Josephson critical currents, which govern the behavior of
merged-element transmons, provide insight into the trends as a function of
Schottky-barrier height. We show that desirable qubit frequencies of 4-5 GHz
can be obtained with a Si barrier thickness of about 5-10 nm, and demonstrate
that the critical current density as a function of Schottky barrier height can
be modeled based on the tunneling probability for a rectangular barrier.
