  We study the magnetic properties of the adatom systems on a semiconductor
surface Si(111):\{C,Si,Sn,Pb\} - ($\sqrt{3} \times \sqrt{3}$). On the basis of
all-electron density functional theory calculations we construct effective
low-energy models taking into account spin-orbit coupling and electronic
correlations. In the ground state the surface nanostructures are found to be
insulators with the non-collinear 120$^{\circ}$ N\'eel (for C, Si, Sn monolayer
coverages) and 120$^{\circ}$ row-wise (for Pb adatom) antiferromagnetic
orderings. The corresponding spin Hamiltonians with anisotropic exchange
interactions are derived by means of the superexchange theory and the
calculated Dzyaloshinskii-Moriya interactions are revealed to be very strong
and compatible with the isotropic exchange couplings in the systems with Sn and
Pb adatoms. To simulate the excited magnetic states we solve the constructed
spin models by means of the Monte Carlo method. At low temperatures and zero
magnetic field we observe complex spin spiral patterns in Sn/Si(111) and
Pb/Si(111). On this basis the formation of antiferromagnetic skyrmion lattice
states in adatom $sp$ electron systems in strong magnetic fields is discussed.
