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Incorporation of Non-metal Impurities at the Anatase TiO$_2$(001)-(1$\times$4) Surface

We use first-principles calculations to investigate the adsorption and incorporation of non-metal impurities (N, C) at the anatase TiO$_2$(001)-(1$\times$4) reconstructed surface. We analyze in detail the influence of the surface structure and local strain on the impurity binding sites and incorporation pathways and identify important intermediates which facilitate impurity incorporation. We find various subsurface interstitial binding sites and corresponding surface $\rightarrow$ subsurface penetration pathways on the reconstructed surface. This surface also favors the presence of subsurface oxygen-vacancies, to which adsorbed species can migrate to form substitutional impurities. Most notably, we show that the non-exposed oxygen sites just below the surface have a key role in the incorporation of nitrogen and carbon in TiO$_2$(001).