A study of Ti${}_n$O${}_{2n-1}$ Magn\'eli phases using Density   Functional Theory
  Defects in the rutile TiO${}_2$ structures have been extensively studied, but the intrinsic defects of the oxygen deficient Ti${}_n$O${}_{2n-1}$ phases have not been given the same amount of consideration. Those structures, known as Magn\'eli phases, are characterized by the presence of ordered planes of oxygen vacancies, also known as shear-planes, and it has been shown that they form conducting channels inside TiO-based memristor devices. Memristors are excellent candidates for a new generation of memory devices in the electronics industry. In this paper we present DFT-based electronic structure calculations for Ti${}_n$O${}_{2n-1}$ Magn\'eli structures using PBESol+U ($0 \leq U \leq 5$ eV) and HSE functionals, showing that intrinsic defects present in these structures are responsible for the appearance of states inside the bandgap, which can act as intrinsic dopants for the enhanced conductivity of TiO${}_2$ memristive devices. 
A. C. M. Padilha, J. M. Os\'orio-Guillen, A. R. Rocha, G. M. Dalpian
cond-mat.mtrl-sci