  Motivated by the recent experimental evidence of commensurate surface 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 dangling bond orbitals of
the adsorbate. We model the problem by an extended two-dimensional Hubbard
model at half-filling on a triangular lattice. We include an on-site Hubbard
repulsion U and a nearest-neighbor 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 the model including the
possibility of commensurate 3x3 phases, using mainly the Hartree-Fock
approximation. For U larger than the bandwidth we find magnetic insulators,
possibly corresponding to the situation in SiC and in K/Si. For smaller U, the
inter-site repulsion V can stabilize metallic CDW phases, reminiscent of the
3x3 structures of Sn/Ge, and possibly of Pb/Ge.
