Magnetic polaron and antiferro-ferromagnetic transition in doped bilayer CrI$_3$

Gate-induced magnetic switching in bilayer CrI$_3$ has opened new ways for the design of novel low-power magnetic memories based on van der Waals heterostructures. The proposed switching mechanism seems to be fully dominated by electrostatic doping. Here we explain, by first-principle calculations, the ferromagnetic transition in doped bilayer CrI$_3$. For the case of a very small electron doping, our calculations predict the formation of magnetic polarons (ferrons, fluctuons) where the electron is self-locked in a ferromagnetic droplet in an antiferromagnetic insulating matrix. The self-trapping of holes is impossible, at least, within our approximation.