Non-Stationary Thickness Profiles of Spin Wave Modes Propagating in Obliquely Magnetized Magnetic Films

Abstract

The dynamics of spin waves propagating in thin magnetic films has been studied intensively for several decades, and is relatively well-understood [1,2]. However, investigations have been mainly focused on the cases of either perpendicular or in-plane magnetization, where forward volume, backward volume, and surface magnetostatic (or dipolar) long-wavelength wave modes have been identified [1]. A more general dipole-exchange theory of the spin wave spectrum in magnetic films applicable also to the short-wavelength spin waves has been later developed by the research group of B.A. Kalinikos [2]. The conventional theory led to the understanding that a spin wave in a magnetic film with spin-sink-free surfaces is a non-uniform travelling wave having a stationary distribution of the variable magnetization along the film thickness, and can transfer energy and angular momentum only along the in-plane direction of the spin wave propagation.It turned out, that the situation becomes substantially more complicated in the case when the magnetic film is magnetized obliquely (see Fig.1), and short-wavelength dipole-exchange spin waves are considered. The results of recent experiments [3], performed using Brillouin light-scattering spectroscopy, in combination with the numerical results, based on the theory of dipole-exchange spin-wave spectra [2], have shown two effects: (i) thickness profiles of the propagating spin waves (both in in-plane and obliquely magnetized films) are strongly dependent on the in-plane wavenumber k due to the strong dipolar hybridization between the modes having different thickness profiles, and demonstrate “dipolar pinning” [4] with the increase on the in-plane wavenumber (see Fig.2, top panel); (ii) transverse (thickness) profiles of the spin waves propagating in the film plane in an obliquely magnetized film are non-stationary , i.e. are changing with time in the process of the wave propagation (see Fig.2, bottom panel). This last phenomenon might mean that there could be a transverse (along the film thickness) transfer of spin angular momentum without corresponding transverse transfer of energy. In the end, this issue of transverse spin transport is related to the boundary conditions at the surfaces of the magnetic film. In our simple system Fig. 1, we consider a magnetic film with spin-sink-free surfaces. However, if there is a spin-sink formed by a layer of a heavy metal, such as Pt, present at the film surface, it may be possible to experimentally detect the transfer of spin angular momentum along the film thickness in an obliquely magnetized magnetic film supporting the unidirectional spin wave propagation in the film plane. Further experimental studies are necessary to clarify the issue of possible transverse spin angular momentum transfer in obliquely magnetized magnetic films. **