Two-Dimensional Brillouin Zones in Thin Magnetic Films and a Mapping of the Regions of Existence of Magnetic Surface States
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We consider the model of a thin ferromagnetic film, described by a standard Heisenberg Hamiltonian involving interactions between nearest neighbours. The surface inhomogeneities are introduced by means of "surface parameters" describing strictly the conditions at the surfaces. We consider the surface orientations SC(111) and FCC(111), for which we diagonalize the Hamiltonian by standard procedure obtaining the characteristic equation for the permitted eigenvalues (and eigenvectors perpendicular to the surface) versus the two-dimensional wave vector in the plane of the reciprocal lattice. It is found that the existence of a surface state (at fixed surface conditions) requires an appropriate propagation direction of the spin wave in the plane of the film. This requirement is visualized graphically by plotting the existence regions of the surface states in the two-dimensional Brillouin zone. These regions are studied as to their form and size versus the surface parameters. For the two surface orientations studied, the existence regions are found to be identical and hence they are not sensitive to the differences in cubic structure between SC and FCC.
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