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Large magneto-optical effect and magnetic anisotropy energy in two-dimensional metallic ferromagnet Fe$_3$GeTe$_2$

Few layers Fe$_3$GeTe$_2$ is currently the only atomically thin ferromagnetic metal, and thus has drawn huge attention in the field of two-dimensional (2D) magnetism. In this paper, we perform a systematic first principle study on the electronic structure, magnetic anisotropy energy (MAE), and magneto-optical (MO) effects in monolayer (ML), bilayer (BL) and trilayer (TL) as well as bulk Fe$_3$GeTe$_2$. All the considered structures of Fe$_3$GeTe$_2$ are predicted to have large MAE of order $\sim$3.0 meV/f.u., being larger than reported 2D ferromagnetic semiconductors Cr$_2$Ge$_2$Te$_6$ and CrI$_3$ and also being comparable to that of FePt which has the largest MAE among the transition metal alloys. This large MAE thus stabilizes the long range ferromagnetic order down to atomically thin layers and also suggests promising applications of 2D Fe$_3$GeTe$_2$ in high density data storage. Furthermore, the calculated magneto-optical spectra show large magnetic circular dichroism, thus resulting in large MO Kerr rotation and Faraday rotation angles. In visible frequency range, Kerr rotation angles up to $\sim$1.0$^\circ$ for TL Fe$_3$GeTe$_2$ are found. Such values are larger than famous MO transition metal alloy MnBi. Also, large Faraday rotation angles are predicted for all considered Fe$_3$GeTe$_2$ structures. In particular, ML Fe$_3$GeTe$_2$ has a Faraday rotation angle of -156$^\circ$/$μm$, which is three times larger than famous MO oxide Bi$_3$Fe$_5$O$_{12}$. These important findings are analysed in terms of the calculated orbital-decomposed density of states and dipole selection rule derived from the group theory. Our findings thus suggest that few layers and bulk Fe$_3$GeTe$_2$ are promising MO materials and could be widely applied to nano MO devices in the future.