Electronic structures of doped anatase ${\mathrm{TiO}}_2:$ ${\mathrm{Ti}}_{1-x}{M}_x{\mathrm{O}}_2$ $(M=\mathrm{Co},$ Mn, Fe, Ni)

We have investigated electronic structures of a room-temperature-diluted magnetic semiconductor: Co-doped anatase ${\mathrm{TiO}}_2.$ We have obtained the half-metallic ground state in the local-spin-density approximation (LSDA) and the insulating ground state in the $\mathrm{LSDA}+U+\mathrm{SO}$ incorporating the spin-orbit interaction. In the stoichiometric case, the low spin state of Co is realized with the substantially large orbital moment. However, in the presence of oxygen vacancies near Co, the spin state of Co becomes intermediate. The ferromagnetisms in the metallic and insulating phases are accounted for by the double-exchange-like and the superexchange mechanisms, respectively. Further, the magnetic ground states are obtained for Mn- and Fe-doped ${\mathrm{TiO}}_2,$ while the paramagnetic ground state is obtained for Ni-doped ${\mathrm{TiO}}_2.$