Density functional theory
Doping of TiO2 is a very active field, with a particularly large effort expended using density functional theory (DFT) to model doped TiO2; this interest has arisen from the potential for doping to be used in tuning the band gap of TiO2 for photocatalytic applications. Doping is also of importance for modifying the reactivity of an oxide. Finally, dopants can also be unintentionally incorporated into an oxide during processing, giving unexpected electronic properties. To unravel properly how doping impacts on the properties of a metal oxide requires a modelling approach that can describe such systems consistently. Unfortunately, DFT, as used in the majority of studies, is not suitable for application here and in many cases cannot even yield a qualitatively consistent description. In this paper we investigate the doping of bulk rutile TiO2 with trivalent cations, Al, Ga and In, using DFT, DFT corrected for on-site Coulomb interactions (DFT + U, with U on oxygen 2p states) and hybrid DFT (the screened exchange HSE06 exchange correlation functional) in an effort to better understand the performance of DFT in describing such fundamental doping scenarios and to analyse the process of charge compensation with these dopants. With all dopants, DFT delocalizes the oxygen hole polaron that results from substitution of Ti with the lower valence cation. DFT also finds an undistorted geometry and does not produce the characteristic polaron state in the band gap. DFT + U and hybrid DFT both localize the polaron, and this is accompanied by a distortion to the structure around the oxygen hole site. DFT + U and HSE06 both give a polaron state in the band gap. The band gap underestimation present in DFT + U means that the offset of the gap state from both the valence and the conduction band cannot be properly described, while the hybrid DFT offsets should be correct. We have investigated dopant charge compensation by formation of oxygen vacancies. Due to the large number of calculations required, we use DFT + U for these studies. We find that the most stable oxygen vacancy site has either a very small positive formation energy or is negative, so under typical experimental conditions, anion vacancy formation will compensate for the dopant.