Charge-density waves and surface Mott insulators for adlayer structures on semiconductors: Extended Hubbard modeling

Motivated by the recent experimental evidence of commensurate surface charge-density waves (CDW) in Pb/Ge(111) and Sn/Ge(111) $\sqrt{3}$-adlayer structures, as well as by the insulating states found on K/Si(111):B and SiC(0001), we have investigated the role of electron-electron interactions, and also of electron-phonon coupling, on the narrow surface-state band originating from the outer dangling-bond orbitals of the surface. We model the $\sqrt{3}$ dangling-bond lattice by an extended two-dimensional Hubbard model at half filling on a triangular lattice. The hopping integrals are calculated by fitting first-principle results for the surface band. We include an on-site Hubbard repulsion U and a nearest-neighbor Coulomb interaction V, plus a long-ranged Coulomb tail. The electron-phonon interaction is treated in the deformation potential approximation. We have explored the phase diagram of this model including the possibility of commensurate $3\times 3$ phases, using mainly the Hartree-Fock approximation. For U larger than the bandwidth we find a noncollinear antiferromagnetic spin-density wave (SDW) insulator, possibly corresponding to the situation on the SiC and K/Si surfaces. For U comparable or smaller, a rich phase diagram arises, with several phases involving combinations of charge and spin-density-waves (SDW), with or without a net magnetization. We find that insulating, or partly metallic $3\times 3$ CDW phases can be stabilized by two different physical mechanisms. One is the intersite repulsion V, which together with electron-phonon coupling can lower the energy of a charge modulation. The other is a magnetically-induced Fermi-surface nesting, stabilizing a net cell magnetization of 1/3, plus a collinear SDW, plus an associated weak CDW. Comparison with available experimental evidence, and also with first-principle calculations is made.