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 _x O₂ 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 ≈ 4μ _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 ≈ 4μ _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.