We develop a systematic approach by which to derive boundary conditions at an interface between two ferromagnetic materials in the continuous medium approximation. The approach treats the interface as a two-sublattice material, although the final equations connect magnetizations outside of the interface and therefore do not explicitly depend on its structure. Instead, the boundary conditions are defined in terms of some average properties of the interface, which may also have a finite thickness. In addition to the interface anisotropy and symmetric exchange coupling, this approach allows us to take into account coupling resulting from inversion symmetry breaking in the vicinity of the interface, such as the Dzyaloshinskii-Moriya antisymmetric exchange interaction. In the case of negligible interface anisotropy and Dzyaloshinskii-Moriya exchange parameters, the derived boundary conditions represent a generalization of those proposed earlier by Barnaś and Mills and are therefore named "generalized Barnaś-Mills boundary conditions". We demonstrate how one could use the boundary conditions to extract parameters of the interface via fitting of appropriate experimental data. The developed theory could be applied to modeling of both linear and non-linear spin waves, including exchange, dipole-exchange, magnetostatic, and retarded modes, as well as to calculations of nonuniform equilibrium micromagnetic configurations near the interface, with a direct impact on the research in magnonics and micromagnetism. * Corresponding author: V.V.Kruglyak@exeter.ac.uk 2
We report a theoretical study of the spin-wave band spectrum of magnonic crystals formed by stacking thin-film magnetic layers, with general assumptions about the properties of the interfaces between the layers. The use of the Barnaś-Mills magnetization boundary conditions has enabled us to systematically trace the origin of the magnonic band gaps in the spin-wave spectrum of such systems. We find that the band gaps are a ubiquitous attribute of a weakened interlayer coupling and a finite interface anisotropy (pinning). The band gaps in such systems represent a legacy of the discrete spinwave spectrum of the individual magnetic layers periodically stacked to form the magnonic crystal rather than result from Bragg scattering. At the same time, magnonic crystals with band gaps due to the Bragg scattering can be described by natural boundary conditions (i.e. those maintaining continuity of the magnetization direction across the whole sample). We generalize our conclusions to systems beyond thin-film magnonic crystals and propose magnonic crystals based on the ideas of graded-index magnonics and those formed by Fano resonances as a possible way to circumvent the discovered issues.
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