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Magnetism in 2D BN$_{1-x}$O$_x$ and B$_{1-x}$Si$_x$N: polarized itinerant and local electrons

We use density functional theory based first-principles methods to study the magnetism in a 2D hexagonal BN sheet induced by the different concentrations of oxygen and silicon atoms substituting for nitrogen (O$_\mathrm{N}$) and boron (Si$_\mathrm{B}$) respectively. We demonstrate the possible formation of three distinct phases based on the magnetization energy calculated self-consistently for the ferromagnetic (ME$_{\mathrm{FM}}$) and antiferromagnetic (ME$_{\mathrm{AFM}}$) states, i.e. the paramagnetic phase with ME$_{\mathrm{FM}}$=ME$_{\mathrm{AFM}}$, the ferromagnetic phase with ME$_{\mathrm{FM}}$$>$ME$_{\mathrm{AFM}}$ and finally the polarized itinerant electrons with finite ME$_{\mathrm{FM}}$ but zero ME$_{\mathrm{AFM}}$. While the O$_\mathrm{N}$ system was found to exist in all three phases, no tendency towards the formation of the polarized itinerant electrons was observed for the Si$_\mathrm{B}$ system though the existence of the other two phases was ascertained. The different behavior of these two systems is associated with the diverse features in the magnetization energy as a function of the oxygen and silicon concentrations. Finally, the robustness of the polarized itinerant electron phase is also discussed with respect to the O substitute atom distributions and the applied strains to the system.