Motivated by growing interest in atomically thin van der Waals magnetic materials, we present an ab initio theoretical study of the dependence of their magnetic properties on the electron/hole density

ρ

induced via the electrical field effect. By focusing on the case of monolayer

{Cr}_{2} {Ge}_{2} {Te}_{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

ρ

. We then investigate the robustness of magnetism by studying the dependencies of the exchange couplings and magneto crystalline anisotropy energy (MAE) on

ρ

. In agreement with experimental results, we find that magnetism displays a bipolar electrically-tunable character, which is, however, much more robust for hole (

ρ > 0

) rather than electron (

ρ < 0

) doping. Indeed, the MAE vanishes for an electron density

ρ \approx - 7.5 \times 10^{13} e \times {cm}^{- 2}

, signaling 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.

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