Spin and exchange coupling for Ti embedded in a surface dipolar network

We have studied the spin and exchange coupling of Ti atoms on a Cu$_2$N/Cu(100) surface using density functional theory. We find that individual Ti have a spin of 1.0 (i.e., 2 Bohr Magneton) on the Cu$_2$N/Cu(100) surface instead of spin-1/2 as found by Scanning Tunneling Microscope. We suggest an explanation for this difference, a two-stage Kondo effect, which can be verified by experiments. By calculating the exchange coupling for Ti dimers on the Cu$_2$N/Cu(100) surface, we find that the exchange coupling across a `void' of 3.6\AA\ is antiferromagnetic, whereas indirect (superexchange) coupling through a N atom is ferromagnetic. We confirm the existence of superexchange interactions by varying the Ti-N angle in a model trimer calculation. For a square lattice of Ti on Cu$_2$N/Cu(100), we find a novel spin striped phase.