Large magneto-optical effect and magnetic anisotropy energy in two-dimensional metallic ferromagnet ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$

The recent discovery of long-range magnetic orders in atomically thin semiconductors ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$ and $\mathrm{Cr}{\mathrm{I}}_3$ as well as metal ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$ has opened up exciting opportunities for fundamental physics of two-dimensional (2D) magnetism and also for technological applications based on 2D magnetic materials. To exploit these 2D metallic magnets, the mechanisms that control their physical properties should be well understood. In this paper, based on systematic first-principles density functional theory calculations, we study the magnetic anisotropy energy (MAE) and magneto-optical (MO) effects of ferromagnetic multilayers [mono-, bi-, tri-, tetra-, and pentalayer] and bulk ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$ as well as their connections with the underlying electronic structures of the materials. Firstly, all the considered ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$ structures are found to prefer the out-of-plane magnetization and have gigantic MAEs of $\sim 3.0$ meV/f.u. This gigantic perpendicular anisotropy results from the large magnetocrystalline anisotropy energy (MCE), which is ten times larger than the competing magnetic dipolar anisotropy energy. The giant MCEs are attributed to the large Te $p_{x,y}$ orbital density of states near the Fermi level and also to the topological nodal point just below the Fermi level at the K points in the Brillouin zone. Secondly, 2D and bulk ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$ also exhibit strong MO effects with their Kerr and Faraday rotation angles being $\sim 1.0^{\circ }$ and $\sim 100 \mathrm{deg}/\mu \mathrm{m}$ in the visible-light frequency region, respectively. The strong MO Kerr and Faraday effects are found to result from the large MO conductivity (or strong magnetic circular dichroism) in these ferromagnetic materials. In particular, the calculated MO conductivity spectra are one order of magnitude larger than that of ${\mathrm{Y}}_3{\mathrm{Fe}}_5{\mathrm{O}}_{12}$. The calculated MO conductivity spectra are analysed in terms of the dipole-allowed optical transitions at high symmetry $\mathrm{\Gamma }$, K, and ${\mathrm{K}}^{\prime }$ points, which further indicate that atomically thin ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$ films with odd layer-number might exhibit anomalous ferrovalley Hall effect. All these interesting findings thus suggest that 2D and bulk ferromagnetic ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$ are promising materials for high-density MO and spintronic nanodevices.

Figure 1

Structure and Brillouin zone of bulk and 2D ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$. (a) Side view of the hexagonal bulk structure with two ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$ monolayers per unit cell and (b) the corresponding hexagonal BZ. (c) Top view of one ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$ monolayer whose side view is indicated by the red dashed rectangle in (a). The $\mathrm{\Gamma }$-K-M plane in (b) can be regarded as the corresponding 2D BZ. Note that in (c), the FeII triangular lattice sits right underneath the Te lattice and thus cannot be seen here.

Figure 2

Band structures of ML (left column), BL (middle column) and bulk (right column) ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$. Spin-polarized scalar-relativistic band structures (upper row), relativistic band structures with the magnetization along $z$ axis (middle row) and with an in-plane magnetization along $x$ axis. In [(d)–(i)], the dotted ellipse circles the topological node—gapless: [(d), (f), (g), (i)] and gapped: [(e), (h)]. The Fermi level is at 0 eV.

Figure 3

Scalar-relativistic site-, orbital-, and spin-projected DOS of bulk ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$. The Fermi level is at 0 eV.

Figure 4

Scalar-relativistic site-, orbital-, and spin-projected DOS of monolayer ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$. The Fermi level is at 0 eV.

Figure 5

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] ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$ in the ferromagnetic state with the out-of-plane magnetization. All the spectra have been convoluted with a Lorentzian of 0.1 eV to simulate the finite electron lifetime effects.

Figure 6

Relativistic site- and orbital-projected band structures of bulk ferromagnetic ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$. The Fermi level is at 0 eV. The main interband transitions at the $\mathrm{\Gamma }$ and K point, as well as the corresponding peaks in the ${\sigma }_{xy}^2$ spectrum in Fig. 5 are indicated by red and blue arrows, respectively. Also, the optical transitions in the vicinity of the $\mathrm{\Gamma }$ point are indicated by the magenta arrows, namely, A1 is at (0.04, 0.04, 0) $2\pi /a$ and A2 is at (0.03, 0.03, 0) $2\pi /a$.

Figure 7

Relativistic site- and orbital-projected band structures of ferromagnetic ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$ monolayer. The Fermi level is at 0 eV. The main interband transitions at the $\mathrm{\Gamma }$, K, and ${\mathrm{K}}^{\prime }$ point, as well as the corresponding peaks in the ${\sigma }_{xy}^2$ spectrum in Fig. 5 are indicated by red and blue arrows, respectively. Also, the optical transitions in the vicinity of the $\mathrm{\Gamma }$ point are indicated by the magenta arrows, namely, A1 is at (0.08, $-0.04$, 0) $2\pi /a$ and A2 is at (0.07, $-0.035,0)2\pi /a$.

Figure 8

Kerr rotation (${\theta }_K$) and ellipticity (${\epsilon }_K$) spectra for (a) bulk, (b) monolayer, (c) bilayer, (d) trilayer, (e) tetralayer, and (f) pentalayer ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$ in the ferromagnetic state with the out-of-plane magnetization. In [(b)–(f)], the dotted lines are Kerr rotation and ellipticity spectra calculated without considering the $\mathrm{Si}{\mathrm{O}}_2$ substrate. In (a), the green diamond denotes the experimental ${\theta }_K$ value [3].

Figure 9

Faraday rotation (${\theta }_F$) and ellipticity (${\epsilon }_F$) spectra for (a) bulk, (b) monolayer, (c) bilayer, (d) trilayer, (e) tetralayer, and (f) pentalayer ${\mathrm{Fe}}_3\mathrm{Ge}{\mathrm{Te}}_2$ in the ferromagnetic state with the out-of-plane magnetization.