Magnetic states of graphene proximitized Kitaev materials

Single-layer $\ensuremath{\alpha}$-ruthenium trichloride $(\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}₃)$ has been proposed as a potential quantum spin liquid. Graphene/${\mathrm{RuCl}}₃$ heterobilayers have been extensively studied with a focus on the large interlayer electron transfer that dopes both materials. Here we examine the interplay between the competing magnetic state of ${\mathrm{RuCl}}₃$ layer and graphene electronic properties. We perform self-consistent Hartree-Fock calculations on a Hubbard-Kanamori model of the $4{d}⁵\phantom{\rule{4pt}{0ex}}{t}_2g$ electrons of $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}₃$ and confirm that out-of-plane ferromagnetic and zigzag antiferromagnetic states are energetically competitive. We show that the influence of hybridization between graphene and ${\mathrm{RuCl}}₃$ bands is strongly sensitive to the magnetic configuration of ${\mathrm{RuCl}}₃$ and the relative orientations of the two layers. We argue that strong hybridization leads to graphene magnetoresistance and that it may tilt the balance between closely competing magnetic states. Our analysis can be applied to any van der Waals heterobilayer system with weak interlayer hybridization and allows for arbitrary lattice constant mismatch and relative orientation.