Microscopic picture of the single vacancy in germanium

A complete microscopic picture of the germanium vacancy is presented, and our results are compared with recent measurements. We analyze, through first principles calculations, the structural relaxations, Jahn-Teller distortions, and orbitals for the charge states $(++,+,0,\ensuremath{-},\ensuremath{-}\ensuremath{-}).$ The formation energies for the different charge states are presented, as well as the positions of the $(++/+),$ $(+/0),$ $(0/\ensuremath{-}),$ and $(\ensuremath{-}/\ensuremath{-}\ensuremath{-})$ levels, and we obtain that the vacancy in Ge is not an Anderson negative-$U$ system, as opposed to the silicon vacancy. We propose as an explanation a much smaller electron-lattice coupling for the E mode in germanium than in silicon.