  Dilute germanium carbides (Ge1-xCx) offer a direct bandgap for compact
silicon photonics, but widely varying results have been reported. This work
uses ab initio simulations with HSE06 hybrid functionals and spin-orbit
coupling to study the Ge1-xCx band structure behavior in the absence of
defects. Contrary to Vegard's law, the conduction band minimum at k=0 is
consistently found to decrease with increasing C content, while L and X valleys
change much more slowly. A vanishing bandgap was observed for all alloys with
x>0.017. Conduction bands deviate from a constant-potential band anticrossing
model except near the center of the Brillouin zone.
