Abstract
We present the solution of the full magnetic problem arising from the absence of magnetic translation symmetry in two dimensions due to an extended magnetic surface step on the surface boundary of an insulating magnetic substrate. The calculation concerns in particular the spin fluctuation dynamics of a magnetic atomic step in the surface of a ferromagnetic simple cubic lattice, the spin order being in the direction normal to surface boundary. Only exchange interactions are considered between the spins in the model. The theoretical approach determines the evanescent spin fluctuation field in the two dimensional plane normal to the direction of the step edge. This field arises owing to the absence of magnetic translation symmetry in this plane, and is completely independent of the form of the surface defect, underlying the general character of the calculation. We show the existence of optical localised magnon modes propagating along the step, their fields being evanescent in the plane normal to the step direction.
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