Access to the full text of the published version may require a subscription. analyse the charge compensation process with these dopants. With all dopants, DFT delocalises the oxygen hole polaron that results from substitution of Ti with the lower valent 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 localise the polaron, which 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 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 2 2 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 that under typical experimental conditions, anion vacancy formation will compensate the dopant.
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