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We investigate the electronic and magnetic structures of two-dimensionaltransition metal tri-chalcogenide CrSiTe$_{3}$ and CrGeTe$_{3}$ materials bycarrying out first-principles calculations. The single-layer CrSiTe$_3$ andCrGeTe$_3$ are found to be a ferromagnetic insulator, where the presence of thestrong $dp\sigma$-hybridization of Cr $e_{\mathrm{g}}$-Te $p$ plays a crucialrole for the ferromagnetic coupling between Cr ions. We observe that thebandgaps and the interlayer magnetic order vary notably depending on themagnitude of on-site Coulomb interaction $U$ for Cr $d$ electrons. The bandgapsare formed between the Cr $e_{\mathrm{g}}$ conduction bands and the Te $p$valence bands for both CrSiTe$_3$ and CrGeTe$_3$ in the majority-spin channel.The dominant Te $p$ antibonding character in the valence bands just below theFermi level is related to the decrease of the bandgap for the increase of $U$.We elucidate the energy band diagram, which may serve to understand theelectronic and magnetic properties of the $ABX_3$-type transition metaltri-chalcogenides in general.

Effect of Coulomb Interactions on the Electronic and Magnetic Properties of Two-Dimensional CrSiTe$_3$ and CrGeTe$_3$ Materials