Atomically thin ferromagnetic (FM) films were recently prepared by mechanical exfoliation of bulk FM semiconductor ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$. They provide a platform to explore novel two-dimensional (2D) magnetic phenomena, and they offer exciting prospects for new technologies. By performing systematic ab initio density functional calculations, here we study two relativity-induced properties of these 2D materials (monolayer, bilayer, and trilayer as well as bulk), namely magnetic anisotropy energy (MAE) and magneto-optical (MO) effects. Competing contributions of both magnetocrystalline anisotropy energy (C-MAE) and magnetic dipolar anisotropy energy (D-MAE) to the MAE are computed. The calculated MAEs of these materials are large, being on the order of $\sim 0.1$ meV/Cr. Interestingly, we find that out-of-plane magnetic anisotropy is preferred in all the systems except the monolayer, where 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 ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ except the monolayer 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 $\sim 2.2^{\circ }$ in these materials are predicted, and they are comparable to that observed in famous MO materials such as PtMnSb and ${\mathrm{Y}}_3{\mathrm{Fe}}_5{\mathrm{O}}_{12}$. Moreover, they are $\sim 100$ times larger than that of $3d$ transition metal monolayers deposited on Au surfaces. Faraday rotation angles in these 2D materials are also large, being up to $\sim {120}^{\circ }/\mu \mathrm{m}$, and they are thus comparable to the best-known MO semiconductor ${\mathrm{Bi}}_3{\mathrm{Fe}}_5{\mathrm{O}}_{12}$. These findings thus suggest that with large MAE and MO effects, atomically thin ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ films would have potential applications in novel magnetic, MO, and spintronic nanodevices.

Superlattice provides a new approach to enrich the class of materials with novel properties. Here, we report the structural and electronic properties of superlattices made with alternate stacking of two-dimensional hexagonal germanene (or silicene) and a MoS₂ monolayer using the first principles approach. The results are compared with those of graphene/MoS₂ superlattice. The distortions of the geometry of germanene, silicene, and MoS₂ layers due to the formation of the superlattices are all relatively small, resulting from the relatively weak interactions between the stacking layers. Our results show that both the germanene/MoS₂ and silicene/MoS₂ superlattices are manifestly metallic, with the linear bands around the Dirac points of the pristine germanene and silicene seem to be preserved. However, small band gaps are opened up at the Dirac points for both the superlattices due to the symmetry breaking in the germanene and silicene layers caused by the introduction of the MoS₂ sheets. Moreover, charge transfer happened mainly within the germanene (or silicene) and the MoS₂ layers (intra-layer transfer), as well as some part of the intermediate regions between the germanene (or silicene) and the MoS₂ layers (inter-layer transfer), suggesting more than just the van der Waals interactions between the stacking sheets in the superlattices.