Noncollinear magnetic configurations and substrate-mediated interactions in Mn trimers on the GaN (000$\bar{1}$) surface
Collinear and noncollinear calculations based on density functional theory are carried out to elucidate the magnetic ordering of Mn trimers on a GaN ($000\bar{1}$) substrate. These trimers had previously been observed in $3a \times 3a$ surface reconstructions through Mn deposition onto the N-polar face of wurtzite GaN($000\bar{1}$). In this work, we start off by studying the effect of spin orbit coupling for the case of monomers and dimers of Mn atoms on top of a GaN surface. Based on an effective spin Hamiltonian, we estimate the magnetic anisotropy energy (MAE) for those cases and found that it is four orders of magnitude weaker than the exchange magnetic coupling between Mn adatoms. In the Mn trimer case, the magnetic ground state has the Mn spins in-plane with the GaN surface in which the relative spin orientation within each trimer is noncollinear due to the competition between the two antiferromagnetic interactions that affect each Mn spin in the trimer, which leads to the found energy minimum with 120 degree angles between the spins. By exploring the nature and fundamental mechanisms for the magnetic interaction among the Mn trimers, we find that the surface states of the substrate play a key role, involving a Ruderman-Kittel-Kasuya-Yosida (RKKY)-type interaction. We report on an electron-mediated long-distance exchange coupling between localized magnetic moments on a GaN($000\bar{1}$) surface.