  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. 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 3x3 phases, using mainly the Hartree-Fock
approximation. For U larger than the bandwidth we find a non-collinear
antiferromagnetic 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 3x3 CDW phases can be stabilized by two
different physical mechanisms. One is the inter-site repulsion V, that together
with electron-phonon coupling can lower the energy of a charge modulation. The
other is a novel 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.
