We discuss the physical conditions required for the creation of collective ferromagnetism in nonmagnetic oxides by intrinsic point defects such as vacancies. We use ${\mathrm{HfO}}_2$ as a case study because of recent pertinent calculations and observations. It was previously noted theoretically that charge-neutral Hf vacancies in ${\mathrm{HfO}}_2$ 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 ${\mathrm{HfO}}_2$ 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 ${\mathrm{HfO}}_2$, 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 ${\mathrm{HfO}}_2$ 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 $6.4\times {10}^{15}{\mathrm{cm}}^{-3}$ (fractional composition of $x_{\mathrm{eq}}=2.2\times {10}^{-7}\%$). (iii) Independently, we calculate the minimum Hf vacancy concentration needed to achieve wall-to-wall percolation in the ${\mathrm{HfO}}_2$ lattice, given the range of the magnetic $V_{\mathrm{Hf}}\text{−}{V}_{\mathrm{Hf}}$ interaction (five neighbors) obtained from our supercell calculations. It turns out that the minimum percolation concentration $x_{\mathrm{perc}}=13.5\%$ needed for collective ferromagnetism is eight orders of magnitude higher than the equilibrium vacancy concentration $x_{\mathrm{eq}}$ in ${\mathrm{HfO}}_2$ 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.

• Received 20 January 2007

DOI:https://doi.org/10.1103/PhysRevB.75.184421