Subwavelength resonant control of the spin-wave phase in thin ferromagnetic films

Abstract

In contemporary developing world demand for small and less energy-consuming devices rapidly increases. To meet this requirement a new generation of technology has to be developed which overcome problems known from the usage of electronics such as frequency limit and heat dissipation. One possible answer promises spin waves (SW) which have a short wavelength and do not suffer from Joule heating as electrons [1]. To successfully use SWs in widespread devices there is a need to find methods that allow handling this type of low energy consuming information carrier.The usage of metasurfaces consisting of arrays of nano-resonators was an innovative concept in modern optics, as made it possible to design a new class of optical systems with properties unheard in classical optics as for example flat lenses [2]. Such devices become available because metasurfaces enable complete control of electromagnetic wave’s phase and amplitude at subwavelength distances. However, such a concept has not been yet widespread in the realm of magnonics, although it would be extremely useful in the design of SW devices [3]. Especially because SW propagation is significantly limited by damping, so requires the usage of technology that works well in the constrain of ultra-short distances.Here, we present extensive theoretical research on the influence of the position and geometry of a ferromagnetic stripe on the reflection and transmission of the SWs from the edge and middle, respectively, of a permalloy film (Fig. 1) [4]. We have focused on how properties of SW change with the resonator width and separation between both elements. Interactions between the stripe and the layer, which depend on the value of separation between the elements and dimensions of the stripe, affect the propagation and reflection of SW. To check the scope of the interactions we use three numerical methods. Namely, finite element method in frequency domain, micromagnetic simulations [5], and finite element modal method. Employing three different methods allowed us to cross-check their results to confirm their correctness.We show that the phase shift as a function of the stripe’s width is characterized by intervals of small and steady growth which are separated by narrow areas of resonances in which phase increases linearly (Fig. 1 inset a). In the areas of resonances there is a visible increase of amplitude and resonances appear periodically in function of a stripe width with a period of about 160 nm. Simulations with increasing separation show (Fig.1 inset b) that resonances also occur with changes of separation but then the function is not periodic. Additionally above certain separation resonances do not appear anymore.The behavior of phase changes is explained with analytical model in which the reflection coefficient consists of two terms of different origins. The first one is responsible for a slow increase of phase with increasing width of the resonator and is a result of phase accumulated in the reflection of the long-wavelength mode. The second term has origin in constructive interference of the short-wavelength mode in the bilayer part of the system, playing a role of the resonator. This term gives significant input only at certain widths of the resonator, which can be predicted from Fabry-Pérot resonance conditions [6]. Intervals of rapid phase changes (Fig. 1 inset a) are caused by the presence of the second term. Because of the properties of the second term, the system used in the simulations can be considered as a realization of the Gires-Tournois type magnonic interferometer [7]. Furthermore, we extend the idea of resonance control of SW phase to the transmission mode.Acquired knowledge allows designing a new class of SW devices based on the Gires-Tournois interferometer concept and manipulations of SW with the usage of magnetic elements smaller than the wavelength of SW. Control of the SW phase can be used for the creation of SW-optical elements such as flat lenses and phase shifters. Especially interesting idea is utilization of proposed interferometer in the system to detect magnetic particles in the vicinity of the bilayer part.The research leading to these results has received funding from the Polish National Science Centre projects No. UMO-2015/17/B/ST3/00118, UMO-2019/33/B/ST5/02013, and UMO-2019/35/D/ST3/03729. The simulations were partially performed at the Poznan Supercomputing and Networking Center (Grant No. 398). **