We construct the tight-binding model Hamiltonians of transition metals on the basis of $s$, $d$, and $p_z$ orbitals by employing a two-center approximation based on Slater-Koster tables. We systematically investigate the electronic structures of $3d$ and $5d$ transition-metal monolayers, and the material-specific parameters are extracted by fitting the first-principles calculations. We find that, compared with bulk systems, the local environments of atoms in these monolayers have lower symmetries, making $d$ orbitals split into more energy levels. The energy splittings are comparable with the $d$-band width and thus have significant influence on the band structure. Moreover, the lattice constants of these monolayer systems are remarkably smaller than those of corresponding bulk systems, indicating a dramatic change of hopping energies, band widths, and magnetisms. By further invoking the spin degree of freedom, we find that most $3d$ transition metals exhibit magnetizations that are closely related to their lattice constants. By fitting spin-minority and spin-majority bands separately, we find that the spin-majority electron exhibits a smaller $d$-state radius and therefore is more localized. In contrast, most $5d$ monolayers are nonmagnetic while exhibiting strong spin-orbit coupling, whose strength increases approximately linearly as the atomic number increases.

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• Electronic structure
• Spin-orbit coupling
• First-principles calculations