The electronic structure and vacancy formation energy of rutile with ordered oxygen vacancies were calculated using the density functional theory with on-site Coulomb corrections between Ti 3d orbital and O 2p orbital (LDA+Ud+Up). The calculated band gaps were about 3eV, using LDA+Ud+Up, and a hybrid functional proposed by Heyd–Scuseria–Ernzerhog. The ordered oxygen vacancies were introduced along the [001] direction within a 3x3x4 super-cell of rutile TiO2-x that consisted of 72 Ti and 136 O atoms. Biaxial strain was induced in the rutile TiO2 along the x- and y-directions up to ±5%. The lowest formation energy of ordered oxygen vacancies was found in 5% compressive strain and deemed as a thermodynamically favorable structure.

Oxygen Vacancy Chain Formation in TiO2 under External Strain for Resistive Switching Memory. Yoo, D.S., Ahn, K., Cho, S.B., Lee, M., Chung, Y.C.: Japanese Journal of Applied Physics, 2012, 51[6], 06FG14