  The atomic structure, energy of formation, and electronic states of vacancies
in H-passivated Ge nanocrystals are studied by density functional theory (DFT)
methods. The competition between quantum self-purification and the free surface
relaxations is investigated. The free surfaces of crystals smaller than 2 nm
distort the Jahn-Teller relaxation and enhance the reconstruction bonds. This
increases the energy splitting of the quantum states and reduces the energy of
formation to as low as 1 eV per defect in the smallest nanocrystals. In
crystals larger than 2 nm the observed symmetry of the Jahn-Teller distortion
matches the symmetry expected for bulk Ge crystals. Near the nanocrystal's
surface the vacancy is found to have an energy of formation no larger than 0.5
to 1.4 eV per defect, but a vacancy more than 0.7 nm inside the surface has an
energy of formation that is the same as in bulk Ge. No evidence of the
self-purification effect is observed; the dominant effect is the free surface
relaxations, which allow for the enhanced reconstruction. From the evidence in
this paper, it is predicted that for moderate sized Ge nanocrystals a vacancy
inside the crystal will behave bulk-like and not interact strongly with the
surface, except when it is within 0.7 nm of the surface.
