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

We study the magnetic properties of the adatom systems on a semiconductor surface Si(111):{C,Si,Sn,Pb}- root 3 x root 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. The Hartree-Fock simulations for the unit cell with nine correlated orbitals put forward insulating ground states with the noncollinear 120 degrees-Neel (for C, Si, Sn monolayer coverages) and 120 degrees -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 in the systems with Sn and Pb adatoms are revealed to be very strong and compatible with the isotropic exchange couplings. To simulate the excited magnetic states we solve the constructed spin models by means of the Monte Carlo method, where 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 at high magnetic fields in the adatom sp electron systems is discussed.