  By performing systematic $ab$ $initio$ density functional calculations, here
we study two relativity-induced properties of atomically thin ferromagnetic
(FM) Cr$_2$Ge$_2$Te$_6$ films [monolayer (ML), bilayer (BL) and trilayer (TL)
as well as bulk], namely, magnetic anisotropy energy (MAE) and magneto-optical
(MO) effects. Competing contributions of both magneto-crystalline anisotropy
energy (C-MAE) and magnetic dipolar anisotropy energy (D-MAE) to the MAE, are
computed. Calculated MAEs of these materials are large, being in the order of
$\sim$0.1 meV/Cr. Interestingly, we find that the out-of-plane magnetic
anisotropy is preferred in all the systems except the ML where an in-plane
magnetization is favored because here the D-MAE is larger than the C-MAE.
Crucially, this explains why long-range FM order was observed in all the
few-layer Cr$_2$Ge$_2$Te$_6$ except the ML because the out-of-plane magnetic
anisotropy would open a spin-wave gap and thus suppress magnetic fluctuations
so that long-range FM order could be stabilized at finite temperature. In the
visible frequency range, large Kerr rotations up to $\sim$1.0 deg in these
materials are predicted and they are comparable to that observed in famous MO
materials such as PtMnSb and Y$_3$Fe$_5$O$_{12}$. Moreover, they are $\sim$100
times larger than that of 3$d$ transition metal MLs deposited on Au surfaces.
Faraday rotation angles in these 2D materials are also large, being up to
$\sim$120 deg/$\mu$m, and are thus comparable to the best-known MO
semiconductor Bi$_3$Fe$_5$O$_{12}$. These findings thus suggest that with large
MAE and MO effects, atomically thin Cr$_2$Ge$_2$Te$_6$ films would have
potential applications in novel magnetic, MO and spintronic nanodevices.
