Abstract
The energetics and atomic structure associated with the localized hole formed near an Al-atom dopant in α-quartz are calculated using a variational, self-consistent implementation of the Perdew–Zunger self-interaction correction with complex optimal orbitals. This system has become an important test problem for theoretical methodology since generalized gradient approximation energy functionals, as well as commonly used hybrid functionals, fail to produce a sufficiently localized hole due to the self-interaction error inherent in practical implementations of Kohn–Sham density functional theory. The self-interaction corrected calculations are found to give accurate results for the energy of the defect state with respect to both valence and conduction band edges as well as the experimentally determined atomic structure where only a single Al–O bond is lengthened by 11%. The HSE hybrid functional, as well as the PW91 generalized gradient approximation functional, however, gives too small an energy gap between the defect state and the valence band edge, overly delocalized spin density and lengthening of more than one Al–O bond.
Highlights
While Kohn–Sham density functional theory (KS-DFT) has been extremely successful and is widely used in studies of molecules and condensed matter [1, 2], there are several shortcomings of this approach of which the description of localized electronic states in semiconductors and insulators is problematic [3]
The properties of an Al-doped crystal were calculated by replacing one of the Si-atoms with an Al-atom and minimizing the energy given by the EsSIC functional with respect to both orbitals and atomic structure
The resulting atomic structure and spin density are shown in figure 1
Summary
While Kohn–Sham density functional theory (KS-DFT) has been extremely successful and is widely used in studies of molecules and condensed matter [1, 2], there are several shortcomings of this approach of which the description of localized electronic states in semiconductors and insulators is problematic [3]. An experimental estimate of the defect energy level has been obtained from optical absorption measurements where an electron is excited from a lone pair of an O-atom adjacent to the Al-atom into the localized defect state, a light-induced transfer of the hole between oxygen atoms of the AlO4 tetrahedron. The band gap of α-quartz is estimated experimentally to be 9 eV so the defect level is much closer to the valence band edge than the conduction band edge. Both the energetics as well as atomic structure related to the Al substitutional defect in α-quartz are, well established from experimental measurements
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