DFT-based Theoretical Calculation of Nb- and W-doped Anatase TiO$_{2}$

The structure and electronic states in the Nb-doped TiO$_{2}$ (TNO) and W-doped TiO$_{2 }$(TWO) in anatase phase were investigated from the first-principle using DFT-based band structure method. In addition to the cases where the dopant substituting a Ti atom, cells containing a dopant (M$_{Ti}$; M = Nb, W) and an oxygen vacancy (V$_{O})$ were calculated in order to clarify the role of the oxygen vacancy in the system. Furthermore, cells containing two dopants and an oxygen vacancy (2M$_{Ti}$--V$_{O})$, and cells with a dopant and two oxygen vacancies (M$_{Ti}$--2V$_{O})$ were calculated. Energetically stable structures were found among the sampled 2W$_{Ti}$--V$_{O}$ and W$_{Ti}$--2V$_{O}$ cells, while the corresponding structures in TNO did not show any significant energy stabilization. Impurity states were found in the stable 2W$_{Ti}$--V$_{O}$ and W$_{Ti}$--2V$_{O}$ structures, and an approach of the two W$_{Ti}$ atoms was observed in the former. The present results rationalize the lower electronic conductivity of TWO than that of TNO, and suggest possible formation of complex structures consisting of the W$_{Ti}$ dopants and the oxygen vacancies.