  We investigate the electronic and magnetic structures of two-dimensional
transition metal tri-chalcogenide CrSiTe$_{3}$ and CrGeTe$_{3}$ materials by
carrying out first-principles calculations. The single-layer CrSiTe$_3$ and
CrGeTe$_3$ are found to be a ferromagnetic insulator, where the presence of the
strong $dp\sigma$-hybridization of Cr $e_{\mathrm{g}}$-Te $p$ plays a crucial
role for the ferromagnetic coupling between Cr ions. We observe that the
bandgaps and the interlayer magnetic order vary notably depending on the
magnitude of on-site Coulomb interaction $U$ for Cr $d$ electrons. The bandgaps
are 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 the
Fermi 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 the
electronic and magnetic properties of the $ABX_3$-type transition metal
tri-chalcogenides in general.
