Modeling and calculation of RKKY exchange coupling to explain Ti-vacancy-induced ferromagnetism in Ta-doped TiO₂

We present a theoretical study on Ti-vacancy-induced ferromagnetism in anatase TiO₂. A recent experimental study has revealed room temperature ferromagnetism in Ta-doped anatase TiO₂thin films (Rusydi et al., 2012) [7]. Ta doping assists the formation of Ti vacancies which then induce the formation of localized magnetic moments around the Ti vacancies. As neighboring Ti vacancies are a few unit cells apart, the ferromagnetic order is suspected to be mediated by itinerant electrons. We propose that such an electron-mediated ferromagnetism is driven by Ruderman–Kittel–Kasuya–Yosida (RKKY) exchange interaction. To examine our hypothesis, we construct a tight-binding based model Hamiltonian for the anatase TiO₂ system. We calculate the RKKY exchange coupling constant of TiO₂ as a function of distance between local magnetic moments at various temperatures. We model the system by taking only the layer containing a unit of TiO₂, at which the Ti vacancy is believed to form, as our effective two-dimensional unit cell. Our model incorporates the Hubbard repulsive interactions between electrons occupying Ti d orbitals treated within mean-field approximation. The density of states profile resulting from the model captures the relevant electronic properties of TiO₂, such as the energy gap of ~3.4 eV and the n-type character, which may be a measure of the adequacy of the model. The calculated RKKY coupling constant shows that the ferromagnetic coupling extends up to 3–4 unit cells and enhances slightly as temperature is increased from 0 to 400 K. These results support our hypothesis that the ferromagnetism of this system is driven by RKKY mechanism.