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Motivated by growing interest in atomically-thin van der Waals magneticmaterials, we present an {\it ab initio} theoretical study of the dependence oftheir magnetic properties on the electron/hole density $\rho$ induced via theelectrical field effect. By focusing on the case of monolayer ${\rm Cr}_2{\rmGe}_2{\rm Te}_6$ (a prototypical 2D Ising ferromagnet) and employing a hybridfunctional, we first study the dependence of the gap and effective mass on thecarrier concentration $\rho$. We then investigate the robustness of magnetismby studying the dependencies of the exchange couplings and magneto-crystallineanisotropy energy (MAE) on $\rho$. In agreement with experimental results, wefind that magnetism displays a bipolar electrically-tunable character, whichis, however, much more robust for hole ($\rho>0$) rather than electron($\rho<0$) doping. Indeed, the MAE vanishes for an electron density$\rho\approx - 7.5 \times 10^{13}~{\rm e} \times {\rm cm}^{-2}$, signalling thefailure of a localized description based on a Heisenberg-type anisotropic spinHamiltonian. This is in agreement with the rapid increase of the couplingbetween fourth-neighbor atoms with increasing electron density.

Electrical tuning of the magnetic properties of 2D magnets: the case of ${\rm Cr}_2{\rm Ge}_2{\rm Te}_6$