Magnetization boundary conditions at a ferromagnetic interface of finite thickness

Abstract

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

Magnetic interfaces as sources of coherent spin waves

“…At the same time, we showed in Ref. [17] that interfaces of finite thickness could be described using boundary conditions of Barnaś-Mills form [18,19], which are fully compatible with the method described here.…”

“…By integrating the linearized Landau-Lifshitz equation without the damping term over infinitely small vicinities of the interface in each medium [1,16,17], we obtain the following pair of boundary conditions for the dynamic magnetization:…”

“…Considering only the waves propagating away from the interface, i.e., μ A ∝ exp(−ik A z) and μ B ∝ exp(+ik B z), and neglecting the interface thickness compared to the wavelength of the spin waves [17], we obtain…”

Influence of magnetic surface anisotropy on spin wave reflection from the edge of ferromagnetic film

“…Here, extension of the analytical model with properly defined boundary conditions will be required. 56,58 The micromagnetic simulations were conducted for the value of the damping parameter α = 0.0005, which is close to the value of thick 26 and also very thin films (tens of nm thick) of YIG as demonstrated experimentally recently. 59 We have performed additional simulations for slightly smaller and larger damping to check the influence of damping on GH shift.…”

Propagation of Spin Waves Through an Interface Between Ferromagnetic and Antiferromagnetic Materials

“…The tasks of this work are: firstto find the abridged general form of the boundary conditions between FM and two-sublattice AFM taking into account the fact that the interface is a composite material with finite thickness δ which is much less than the length of the spin wave λ sw including only energies of uniform and non-uniform exchange between all sublattices [1,2]; and secondto derive the coefficients of transmission and reflection of surface evanescent spin wave through the FM/AFM interface [19]. The normal to the interface of magnets n is parallel to the y-axis.…”