Lattice relaxation in many-electron states of the diamond vacancy

Symmetric lattice relaxation around a vacancy in diamond and its effect on many electron states of the defect have been investigated. A molecular approach is used to evaluate accurately electron-electron (e-e) interaction via a semiempirical formalism which is based on a generalized Hubbard Hamiltonian. Coupling of the defect molecule to surrounding bulk is also considered using an improved Stillinger-Weber potential for diamond. Strong dependence of the electronic energy levels to the relaxation size of the nearest neighbor (NN) atoms indicates that in order to obtain quantitative results the effect of lattice relaxation should be considered. Except for the high spin state of the defect (5)A(2), the order of other lowest levels, particularly the ground state of the vacancy E-1 does not change by the relaxation. At 12% outward relaxation, there is a level crossing between (5)A(2) and the excited state of the well-known GR1 transition T-1(2). The reported level crossing confirms the predicted relative energies of these states in the band gap that was speculated by monitoring the temperature dependence of the electron paramagnetic resonance (EPR) signal. By considering the outward relaxation effect, we obtained midgap position for the (5)A(2) state in agreement with the suggestion made by EPR. The position of the low lying T-3(1) level varies from 100 to 400 meV with increasing outward relaxation. When the ion-ion interaction of the NN atoms is included the outward relaxation lowers the energies of all electronic states. The relaxing force is different for investigated electronic states. By considering the interaction of the first and second shell neighbors of the vacancy, the calculated elastic barrier restricts outward relaxation of the vacancy to 12% for the ground and 18% for the (5)A(2) excited state. The calculated equilibrium bond lengths are in very good agreement with ab initio density functional theory and EPR measurement data. Electronic configurations in the unrelaxed and relaxed eigenfunctions of the Hamiltonian are reported. Our results also suggest that there is an outward relaxation if Hund rule is applicable.