Electrical tuning of the magnetic properties of 2D magnets: the case of ${\mathrm{C}\mathrm{r}}_2{\mathrm{G}\mathrm{e}}_2{\mathrm{T}\mathrm{e}}_6$

Motivated by growing interest in atomically-thin van der Waals magnetic materials, we present an {\it ab initio} theoretical study of the dependence of their magnetic properties on the electron/hole density $\rho$ induced via the electrical field effect. By focusing on the case of monolayer ${\mathrm{C}\mathrm{r}}_2{\mathrm{G}\mathrm{e}}_2{\mathrm{T}\mathrm{e}}_6$ (a prototypical 2D Ising ferromagnet) and employing a hybrid functional, we first study the dependence of the gap and effective mass on the carrier concentration $\rho$. We then investigate the robustness of magnetism by studying the dependencies of the exchange couplings and magneto-crystalline anisotropy energy (MAE) on $\rho$. In agreement with experimental results, we find that magnetism displays a bipolar electrically-tunable character, which is, however, much more robust for hole ($\rho >0$) rather than electron ($\rho <0$) doping. Indeed, the MAE vanishes for an electron density $\rho \approx -7.5\times {10}^{13}\mathrm{e}\times {\mathrm{c}\mathrm{m}}^{-2}$, signalling the failure of a localized description based on a Heisenberg-type anisotropic spin Hamiltonian. This is in agreement with the rapid increase of the coupling between fourth-neighbor atoms with increasing electron density.