Confined magnetoelastic waves in thin waveguides

Abstract

In recent years, the coupling between the elastic and magnetic domain in magnetostrictive materials has gained strong interest [1]. The magnetoelastic coupling in these materials results in elastic response originating from magnetic excitations and vice versa. At microwave frequencies, this mutual interaction leads to coupling between elastic and magnetic waves, forming magnetoelastic waves. Whereas plane magnetoelastic waves are already studied in bulk materials some decades ago [2-4], the magnetoelastic waves in thin magnetostrictive films gained attention only recently [5,6]. By contrast, many recent spin-wave-based information processing applications employ nanoscale waveguides for information transfer and computation [7]. At these nanoscales, confinement of both the elastic and magnetic waves leads to multiple modes each of which with a specific mode profile and consequently also to confined magnetoelastic waves. The shape of the magnetoelastic field depends on the displacement profile and the shape of the magnetoelastic body force depends on the magnetization profile. As a consequence, it is expected that every mode experiences different magnetoelastic coupling strength. This presents a rather complex multiphysics problem where analytical methods are hard or even impossible, and thus numerical approaches are preferred. Therefore, in this work, we present the development of a novel mumax3 extension that allows the numerical calculation of magnetoelastic waves in scaled waveguides [8]. Furthermore, we present results of magnetoelastic waves in a nanoscale waveguide with static magnetization along the wave propagation direction.The studied system consists of a thin CoFeB waveguide with width of 200 nm and thickness of 20 nm. A weak external field of 5 mT is applied in the longitudinal direction of the waveguide to initiate the static magnetization along this direction. The resonant modes in the structure are numerically found by applying a 5 ps excitation pulses and performing a two-dimensional fast Fourier transform. By comparing the numerical result with the analytical uncoupled dispersion relations of confined elastic and magnetic waves, all modes were identified. The uncoupled resonant waves of this structure consist of confined magnetic spin waves, antisymmetric and symmetric lamb waves and out-of-plane shear elastic waves. Fig. 1a shows the numerically obtained dispersion relations and Fig. 1b shows the overlay with the analytically calculated dispersion relations for uncoupled elastic and magnetic waves.The magnetoelastic coupling leads to the formation of anti-crossings near the reciprocal point where the uncoupled elastic and magnetic dispersion relations would intersect with each other. These anti-crossings identify regions of strong interaction between the elastic and magnetic domain and correspond to pure magnetoelastic waves. The magnetoelastic waves are shown to have different behavior as compared to the pure elastic or magnetic waves. The magnetoelastic wave group velocity is rather unaffected in the quasi-magnetic or quasi-elastic regime whereas it strongly differs close to the anti-crossing. Furthermore, the magnetization and displacement value are shown to take both high values close to the anti-crossing. However, when approaching the quasi-elastic regime there is a very strong decay of the magnetization component of the magnetoelastic wave and vice versa for the displacement component and the quasi-magnetic regime.Lastly, careful analysis of the dispersion relations also shows reciprocal points with weak or even without anti-crossing. The frequency gap formed by the anti-crossing is a good measure of the magnetoelastic coupling strength and thus provides much information. The variation in amplitude of this frequency gap is attributed to the wave confinement and their mode profiles. Every elastic and magnetic mode has its own mode profile and thus also its own magnetoelastic body force and field. Considering these mode dependent body forces and fields together with the mode’s overlap integral allows to identify the strength of the coupling between the different elastic and magnetic waves. The strongest coupling is shown to exist between the odd magnetic modes and symmetric lamb modes and even magnetic modes and anti-symmetric lamb modes.This work has been partially funded by the European Union’s Horizon 2020 research and innovation programme within the FET-OPEN project CHIRON under grant agreement No. 801055. FV acknowledge the financial support from Fonds voor Wetenschappelijk Onderzoek (FWO) under the grant No. 1S05719N. **