Spin fluctuations dynamics on the insulating antiferromagnetic stepped surface model

We present a numerical method to calculate the spin fluctuations dynamics on the stepped surface. The model discussed here consists in an extended antiferromagnetic surface steps on the surface boundary of an insulating antiferromagnetic substrate. The stepped surface is formed by two straight steps dropped randomly and the spins moments of the steps and the substrate are considered as local with no electronic effects. The calculations concern in particular the energies of localized spin-wave modes near the stepped surface and employ the matching procedure in the random-phase approximation and mean field approximation. Only nearest neighbors exchange interactions are considered between the spins in the model. The theoretical formalism presented here solves for the three dimensional evanescent crystal spin field in the bulk and the surface domains around the steps, that arises owing to the breakdown of magnetic translation symmetry of the system. The results are used to calculate the spin modes energies associated with the steps and surface terraces. We show the presence of acoustic and optical localized spin-wave modes propagating along the surface and the steps, their spin fluctuations fields are also described as evanescent in the plane normal to the stepped surface and depend on the nature of the exchange parameters near the steps.