Electronic structure, spin-orbit interaction, and electron-phonon coupling of triangular adatom lattices on semiconductor substrates

A one-third monolayer of the heavy metals Sn and Pb deposited on semiconductor substrates can lead to a $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ surface reconstruction, constituting an exciting triangular lattice material platform. A long history of experiments identified charge-ordered and magnetic ground states. These discoveries were accompanied by a decades-long debate of whether electron correlations or other effects involving phonons are the driving force of the symmetry-broken states. The most recent discovery of superconductivity in boron-doped Sn/Si(111) with a ${T}_{c}$ between 5 and 9 K led to a renewed excitement. Here we revisit the electronic and phononic properties of Sn and Pb adatom triangular lattices on Si(111) and SiC(0001). For all materials, we compute relativistic band structures using density functional theory (DFT)$+U$, where $U$ is only applied to the substrate atoms in order to adjust the band gap to match the experimental value; as a consequence, some of the resulting tight-binding parameters of the metallic surface band substantially differ compared to previous studies. Remarkably, for Pb/SiC(0001), we predict Rashba spin-orbit coupling as large as 45% of the nearest-neighbor hopping energy. In addition, we compute the phonon spectra and electron-phonon coupling constants for all materials, and for Pb/Si(111) even relativistically, although the inclusion of spin-orbit coupling has surprisingly little effect on the electron-phonon coupling constant. We conclude that the resulting couplings are too weak to account for electron-phonon-mediated superconductivity in any of these materials.