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

Spin-orbit coupling and magnetic interactions in Si(111):{C,Si,Sn,Pb}