Theoretical study on the magnetic moments formation in Ta-doped anatase TiO₂

We present a theoretical study on Ti-vacancy induced ferromagnetism in Ta-doped anatase TiO₂. Experimental study of Ti_1−xTa_xO₂ thin film has shown that Ti-vacancies (assisted by Ta doping) induce the formation of localized magnetic moment around it, then, the observed ferromagnetism is caused by the alignment of localized magnetic moments through Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. In this study, we focus on the formation of the localized magnetic moments in this system. We hypothesize that on a unit cell, Ti-vacancy has caused four electrons from the surrounding oxygen atoms to become unpaired. These unpaired electrons then arrange themselves into a configuration with a non-zero net magnetic moment. To examine our hypothesis, we construct a Hamiltonian of the four unpaired electrons, incorporating the Coulomb intra- and inter-orbital interactions, in matrix form. Using a set of chosen parameter values, we diagonalize the Hamiltonian to get the eigenstates and eigenvalues, then, with the resulting eigenstates, we calculate the magnetic moment, μ , by obtaining the expectation value of the square of total spin operator. Our calculation results show that in the ground state, provided that the ratio of parameters satisfies some criterion, μ ≈ 4 μ_B , corresponding to the four electron spins being almost perfectly aligned, can be achieved. Further, as long as we keep the Coulomb intra-orbital interaction between 0.5 and 1 eV, we find that μ ≈ 4 μ_B is robust up to far above room temperature. Our results demonstrate that Ti vacancies in anatase TiO₂ can form very stable localized magnetic moments. (paper)