The electronic states of the neutral vacancy in diamond: a quantum mechanical approach

Abstract

The electronic structure of the neutral vacancy in diamond is investigated by using a periodic approach (supercells containing 32, 64 and 128 atoms have been considered), the Hartree–Fock (HF), pure DFT (LDA, PBE), global (B3LYP and PBE0) and range-separated (HSE06) “hybrid” functionals, and a local basis set as implemented in the CRYSTAL14 code. LDA and GGA provide metallic solutions for the three possible spin states (characterized by \(S_z =0,1,2\)), whereas hybrids and HF, thanks to the “exact” exchange, which plays such a crucial role in the present case, provide well-localized defect states in the band gap of the insulating system. As a consequence of the metallic solution, for the \(S_z = 0\) and 1 states, the spin polarization tends to disappear when pure DFT is used, while with HF and hybrids the uncoupled electrons remain strongly localized on the first nearest neighbors, as documented by the spin density maps. The relative stability between the three spin states is \(0 < 1 < 2\) for HF and hybrids, with a small difference between the first two (about 5 \({\text {mE}}_{h}\) in the case of hybrids and HF) and a much larger difference between \(S_z=1\) and 2 (about 50 \({\text {mE}}_{h}\) for hybrids, 28 for HF). As regards PBE and LDA results, the stability order between \(S_z=0\) and 1 is reversed, and \(S_z=2\) is less stable than \(S_z=0\) by only 8–10 \({\text {mE}}_{h}\). The defect crystalline orbitals are classified by symmetry and identified in the band structure with respect to the valence and conducting electrons. The formation energy of the vacancy, evaluated with six different functionals, is close to 7 eV for three of them (7.00 HSE06, 7.05 LDA, 7.11 PBE0), slightly smaller in B3LYP (6.74) and PBE (6.56). Cluster calculations (\({\text {C}}_{35}{\text {H}}_{36}\) and \({\text {C}}_{455}{\text {H}}_{196}\)) are also performed in comparison with the supercell results and for obtaining, with the smaller one, a coupled-cluster estimate of the energy differences between the three spin states that turn out to be close to the ones obtained with hybrids.