Nonstoichiometry as a source of magnetism in otherwise nonmagnetic oxides:: Magnetically interacting cation vacancies and their percolation

We discuss the physical conditions required for the creation of collective ferromagnetism in nonmagnetic oxides by intrinsic point defects such as vacancies. We use HfO2 as a case study because of recent pertinent calculations and observations. It was previously noted theoretically that charge-neutral Hf vacancies in HfO2 have partially occupied electronic levels within the band gap, and thus the vacancies carry a nonvanishing local magnetic moment. Such density functional supercell calculations have further shown that two such vacancies interact ferromagnetically if they are separated by up to third-neighbor distance. This suggested to the authors that Hf vacancies could explain the observed collective ferromagnetism in thin HfO2 films. Here we use our previously developed more complete methodology [Phys. Rev. Lett. 96, 107203 (2006)] to inquire if such vacancies can lead to collective ferromagnetism. Applying this methodology to HfO2, we find the following: (i) Hf vacancies appear in a few possible charge states but not all of these have a local magnetic moment. (ii) We calculate the energy required to form such vacancies in HfO2 as a function of the chemical potential and Fermi energy, and from this we compute, as a function of growth temperature and oxygen pressure, the equilibrium concentration of those vacancies that have a nonvanishing local magnetic moment. We find that under the most favorable equilibrium growth conditions the concentration of Hf vacancies with magnetic moment at room temperature does not exceed 6.4x10(15) cm(-3) (fractional composition of x(eq)=2.2x10(-7)%). (iii) Independently, we calculate the minimum Hf vacancy concentration needed to achieve wall-to-wall percolation in the HfO2 lattice, given the range of the magnetic V-Hf-V-Hf interaction (five neighbors) obtained from our supercell calculations. It turns out that the minimum percolation concentration x(perc)=13.5% needed for collective ferromagnetism is eight orders of magnitude higher than the equilibrium vacancy concentration x(eq) in HfO2 under the most favorable growth conditions. We conclude that equilibrium growth cannot lead to ferromagnetism and that ferromagnetism can be established only if one beats the equilibrium Hf vacancy concentration during growth by as much as eight orders of magnitude. This paper presents also an Appendix that gives the Monte Carlo-calculated percolation thresholds of various lattices as a function of the percolation radius of the interaction.