Donor-vacancy complexes in Ge: Cluster and supercell calculations

We present a comprehensive spin-density functional modeling study of the structural and electronic properties of donor-vacancy complexes ($\mathrm{P}V$, $\mathrm{As}V$, $\mathrm{Sb}V$, and $\mathrm{Bi}V$) in Ge crystals. Special attention is paid to spurious results which are related to the choice of the boundary conditions (supercell-cluster approach), the resulting band-gap width, and the choice of the points to sample the Brillouin zone. The underestimated energy gap, resulting from the periodic conditions together with the local-density approximation to the exchange-correlation energy, leads to defect-related gap states that are strongly coupled to crystalline states within the center of the zone. This is shown to produce a strong effect even on relative energies. Our results indicate that in all $E$ centers the donor atom occupies a nearly substitutional site, as opposed to the split-vacancy form adopted by the $\mathrm{Sn}V$ complex in Si. The $E$ centers can occur in four charge states, from positive to double negative, and produce occupancy levels at $E(0∕+)={E}_{v}+0.1\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, $E(\ensuremath{-}∕0)={E}_{v}+0.3\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, and $E(=∕\ensuremath{-})={E}_{c}\ensuremath{-}0.3\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$.

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