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Utilizing the first-principles calculations together with the group theoryanalysis, we systematically investigate the magnetocrystalline anisotropyenergy, magneto-optical effect, and anomalous transport properties (includinganomalous Hall, Nernst, and thermal Hall effects) of monolayer and bilayerFe$_n$GeTe$_2$ ($n$ = 3, 4, 5). The monolayer Fe$_n$GeTe$_2$ ($n$ = 3, 4, 5)exhibits the out-of-plane, in-plane, and in-plane ferromagnetic orders withconsiderable magnetocrystalline anisotropy energies of -3.17, 4.42, and 0.58meV/f.u., respectively. Ferromagnetic order is predicted in bilayerFe$_4$GeTe$_2$ while antiferromagnetic order prefers in bilayer Fe$_3$GeTe$_2$and Fe$_5$GeTe$_2$. The group theory analysis reveals that in addition tomonolayer ferromagnetic Fe$_n$GeTe$_2$ ($n$ = 3, 4, 5), the magneto-optical andanomalous transport phenomena surprisingly exist in bilayer antiferromagneticFe$_5$GeTe$_2$, which is much rare in realistic collinear antiferromagnets. Ifspin magnetic moments of monolayer and bilayer Fe$_n$GeTe$_2$ are reorientedfrom the in-plane to out-of-plane direction, the magneto-optical and anomaloustransport properties enhance significantly, presenting strong magneticanisotropy. We also demonstrate that the anomalous Hall effect decreases withthe temperature increases. The gigantic anomalous Nernst and thermal Halleffects are found in monolayer and bilayer ferromagnetic Fe$_n$GeTe$_2$, andthe largest anomalous Nernst and thermal Hall conductivities, respectively, of-3.31 A/Km and 0.22 W/Km at 130 K are observed in bilayer ferromagneticFe$_4$GeTe$_2$. Especially, bilayer antiferromagnetic Fe$_5$GeTe$_2$ exhibitslarge zero-temperature anomalous Hall conductivity of 2.63 e$^2$/h as well asanomalous Nernst and thermal Hall conductivities of 2.76 A/Km and 0.10 W/Km at130 K, respectively.

Strong magneto-optical and anomalous transport manifestations in two-dimensional van der Waals magnets Fe$_n$GeTe$_2$ ($n$ = 3, 4, 5)