Large magneto-optical effects and magnetic anisotropy energy in two-dimensional ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$

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.

Figure 1

(a) Crystalline structure of bulk rhombohedral ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ with the red dashed rectangle indicating one monolayer (ML) ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$, and (b) the corresponding Brillouin zone with its irreducible wedge indicated by blue lines. (c) Top and (d) side views of trilayer (TL) ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ structure in ABC stacking. In (d), the magenta dashed rectangle indicates a bilayer (BL) unit cell. (e) Two-dimensional hexagonal Brillouin zone for ML, BL, and TL ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ with its irreducible wedge indicated by blue lines.

Figure 2

Four considered intralayer magnetic configurations: (a) FM, (b) AF-$\mathrm{N}\acute{\mathrm{e}}\mathrm{el}$, (c) AF-zigzag, and (d) AF-stripe types. Only magnetic Cr atoms are shown with red (blue) balls indicating up (down) spins. Intralayer exchange coupling parameters $J_1$, $J_2$, and $J_3$ are indicated by magenta, green, and orange arrows, respectively. (e) Interlayer AF configuration with the interlayer exchange coupling parameter $J_{z1}$ indicated by blue lines.

Figure 3

Relativistic band structures of (a) bulk and (b) ML ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ in the FM state with out-of-plane magnetization. Horizontal dashed lines denote the top of the valence band.

Figure 4

Scalar-relativistic site-, orbital-, and spin-projected DOS of ferromagnetic bulk ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$. The top of the valence band is at 0 eV.

Figure 5

Scalar-relativistic site-, orbital-, and spin-projected DOS of FM ML ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$.

Figure 6

Real (a) [(e)] diagonal and (b) [(f)] off-diagonal, imaginary (c) [(g)] diagonal components and (d) [(h)] off-diagonal components of the optical conductivity tensor of bulk (ML) ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ in the FM state with out-of-plane magnetization. All the spectra have been convoluted with a Lorentzian of 0.2 eV to simulate the finite electron lifetime effects.

Figure 7

Relativistic band structure of ferromagnetic monolayer ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ with out-of-plane magnetization. Horizontal dashed lines denote the top of the valence band. The symmetry of the band states at the $\mathrm{\Gamma }$-point is labeled according to the irreducible representation of the $C_{3i}$ double point group. The principal interband transitions and the corresponding peaks in ${\sigma }_{xy}$ in Fig. 6 are indicated by green arrows.

Figure 8

Kerr rotation (${\theta }_K$) and ellipticity (${\varepsilon }_K$) spectra for (a) bulk, (b) ML, (c) BL, and (d) TL ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ in the FM state with out-of-plane magnetization. Green diamonds denote the ${\theta }_K$ value from the recent experiments [16].

Figure 9

Faraday rotation (${\theta }_F$) and ellipticity (${\varepsilon }_F$) spectra for (a) ML, (b) BL, and (c) TL ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ in the FM state with out-of-plane magnetization.