A passive GHz frequency-division multiplexer/demultiplexer based on anisotropic magnon transport in magnetic nanosheets

F. Heussner, G. Talmelli, M. Geilen, B. Heinz, T. Brächer, T. Meyera, F. Ciubotaru, C. Adelmann, K. Yamamoto, A. A. Serga, B. Hillebrands and P. Pirro 1 Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany 2 imec, 3001 Leuven, Belgium 3 KU Leuven, Departement Materiaalkunde, SIEM, 3001 Leuven, Belgium 4 Graduate School Materials Science in Mainz, 55128 Mainz, Germany 5 Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan a Current affiliation: THATec Innovation GmbH, 68165 Mannheim, Germany ★ e-mail: heussner@rhrk.uni-kl.de

Their typical frequencies in the range from GHz to THz together with the associated wavelengths on the micro-and nanometre scale [15,16,17] allows for their utilization in microchips and nanostructured devices while operating at the same frequency as many current and emerging techniques, e.g. Bluetooth, WLAN and 5G mobile networks. This makes a conversion between different frequencies redundant when alternating between data transmission via air and data processing in devices. Furthermore, the enormous tunability of spin-wave properties by various parameters such as the external magnetic field or the local magnetization [18] can be a significant advantage compared to the utilization of, e.g., surface acoustic waves.
One exceptional property utilized in the following work is the anisotropic propagation of spin waves in suitable magnetic media, which can lead to the creation of narrow spin-wave beams and caustics [8,19,20,21,22,23,24,25] (see Methods). Their sub-wavelength transverse aperture [23] is an outstanding reason for their potential use in nanostructured networks, especially when considering the beam formation of high-wavevector magnons [20]. The spin-wave beams can be radiated from point-like sources into unpatterned magnetic films [21,22,23,24] and they are formed due to the non-collinearity of the wavevector and the group velocity vector. Furthermore, their propagation direction can be versatilely controlled by, e.g., frequency [8,19,21,24,25], magnetic field strength [19] and direction of the local magnetization [22,23].
In this work, we experimentally demonstrate how these spin-wave beams can be used to realize a controllable information transport in two-dimensional magnetic films with thicknesses of a few tens of nanometers, so called nanosheets. The guidance of the signal is not achieved by geometrical structuring [9,10] or magnetically induced channels [26] but by the inherent anisotropy of the magnon system. Utilizing this anisotropic signal propagation allows for the realization of passive elements without any additional energy consumption since the magnon system inherently collimates and steers the energy transport. As just one of the many possible applications of this concept, we present the realization of a passive demultiplexer exploiting the frequencydependence of the beam direction. By inverting the geometry a multiplexer can be realised equally well. The discussed two-dimensional signal transport could be used in various logic circuits, for example to distribute information in artificial neural networks [4]. Finally, the presented concept could be transferred to other systems which can show a pronounced anisotropy of energy propagation as well like, e.g., surface acoustic waves [27] or photons in hyperbolic materials [28].
The experimental realization of the frequency-division demultiplexer is based on 30-nm-thin films of the magnetic alloy CoFeB. Being based on our previous studies [8], we have designed a prototype by micromagnetic modelling. Subsequently, the sample has been fabricated and studied by detecting the spin-wave intensity using micro-focused Brillouin light scattering spectroscopy (µBLS) [29]. After investigating the frequency-dependence of the spin-wave beam directions and comparing it to theoretical calculations, we verify the full functionality of the demultiplexer by twodimensional measurements of the spin-wave intensity. Figure 1a,b presents the simulated intensity distribution inside of the designed structure for spin waves of frequency 1 = 11.2 GHz and 2 = 13.8 GHz. The dashed arrows mark the beam directions that are predicted by the developed theoretical model (see Fig. 1c and Methods). The functional part of the device comprises the unpatterned area in the centre of the structure in which the two- propagation at 1 = 11.2 GHz. The signal is guided from the input waveguide into output 1 by a spin-wave beam. b, Spin-wave distribution at 2 = 13.8 GHz. The signal is channelled into output 2 due to the changed direction of the spin-wave beams in the unstructured film area. c, Isofrequency curves of the studied system. The directions of the group velocity vectors, which are perpendicular to the curve, vary with spin-wave carrier frequency. d, Fabricated sample according to the developed layout of the demultiplexer. dimensional energy transport takes place. Input and output waveguides, which can serve as interfaces to adjacent magnonic building blocks in larger magnonic networks, are connected to this area. To provide an input signal, magnon excitation is simulated by applying localized alternating magnetic fields hrf inside the input waveguide. The field distribution is chosen in accordance with the experiment in which these fields are created by microwave currents flowing through a microstrip antenna placed across the waveguide. The spin waves propagate towards the unpatterned area and, in agreement with previous observations [8,21,22,23], two symmetric spin-wave beams are emitted from the waveguide opening, which acts as a point-like source exhibiting a broad wavevector spectrum. The anisotropy of the spin-wave dispersion leads to a strong focusing of the energy. The observed frequency dependence of the beam directions is utilized to realize the demultiplexing functionality by properly adding two output waveguides after a certain propagation distance. Their positions are asymmetric with respect to the input, leading to the following consequence: at frequency 1 = 11.2 GHz (Fig. 1a), the upper spin-wave beam, which propagates at an angle of B,1 theo = 77.6° as predicted by the theoretical model, is channelled into output 1 and transmits the information through the device. The lower one is blocked by the edge of the magnetic structure so that the signal cannot reach output 2. In contrast, the beam angles of spin waves of frequency 2 = 13.8 GHz are changed to B,2 theo = 68.9° ( B,2 ′ = 180°− B,2 , resp.) resulting in a sole signal transmission into output 2 (Fig. 1b). Hence, the device separates spin-wave signals of two frequencies into two spatially detached output waveguides, whose transition zones are especially tailored to enable an efficient channelling of the spin waves. The exploited frequency dependence of the beam directions result from the modification of the spinwave isofrequency curve with frequency (see Fig. 1c) and the according variation of the groupvelocity vectors, which are always perpendicular to this curve (see Methods). It should be mentioned that the exact operating frequencies of the device can be controlled by changing, e.g., the external field strength since this shifts the magnon dispersion. Here, the external magnetic field is set to μ 0 ext = 75 mT. reached. The measured data agrees well with the theoretically predicted curve (solid line in Fig.   2b). This is due to the fact that, in addition to the peculiarities of anisotropic magnon propagation [19], our model considers such important parameters like the excited wavevector spectrum and the distance between the observation point and the source. The optimum angles B,1 exp = 77.9° and B,2 exp = 69.6°, leading to maximum intensity in the outputs, are indicated in Fig. 2b together with the shaded acceptance intervals, which result from the assumption that beams are channelled into the outputs if their direction deviates less than 2.5° from the optimum angles. These acceptance intervals in combination with the curve B ( ) can explain the differing widths of the accepted frequency bands of the two outputs. The angle variation is quite large in the region of 1 = 11.2 GHz. Hence, only frequencies within a small range around f1 create beams whose directions lie inside the angular acceptance interval of output 1. In contrast, the slope of the mentioned curve is significantly lower at 2 = 13.8 GHz. Thus, a larger variation of the frequency is permitted around f2 until the occurring beam angles are out of the acceptance interval of output 2.
For a detailed visualisation, the spin-wave intensity was measured in the entire demultiplexer area at the frequencies f1 and f2 by performing two-dimensional µBLS scans. A comparison of these experimental results (Fig. 3) with the results of the micromagnetic modelling (Fig. 1a,b) reveals a remarkably good agreement. This confirms that the whole demultiplexing mechanism works exactly as predicted and is based on the creation of spin-wave beams, their frequency-dependent propagation though the unstructured area, and the channelling of the signals into different outputs.
Furthermore, these measurements can be used to determine the input-output ratio of the device to reveal the influence of the beam formation on signal losses. Considering the transmission from the input waveguide into output 1 at frequency f1 yields a value of 1 exp = (18.1 ± 2.1) dB whereas a ratio of 1 exp = (18.2 ± 1.9) dB is obtained for a signal at frequency f2 channelled into output 2.
For comparison, the ratio theo can be calculated analytically considering the energy splitting and the propagation losses only (see Methods). The resulting values 1 theo = (18.95 ± 1.84) dB and 2 theo = (17.64 ± 1.69) dB show that the major fraction of the losses arise from the intrinsic spinwave damping during their propagation and not due the demultiplexing mechanism based on beam formation and channelling (the splitting is suppressible [8]).
Finally, the scalability and advantages of the presented concept should be discussed. The employed spin-wave beams are created due to the anisotropy of the magnonic system and, for limited length scales, their width is primary determined by the size of the source [23]. In the presented case, the anisotropy is induced by the dipole-dipole-interaction, which is dominant in the low-wavevector range of the magnon spectrum. However, a beam formation is also possible for high-wavevector spin waves whose dispersion characteristics are mainly dominated by the exchange interaction [20]. Hence, the presented concept to guide two-dimensional energy transport by spin-wave beams, formed due to the intrinsic anisotropy of the system, can be scaled down to significantly smaller length scales. Furthermore, the passive character of the signal steering is an important advantage over other concepts requiring energy-consuming external control.
This is the case, e.g., with the previously demonstrated spin-wave time-division multiplexer, which relies on external charge currents [30]. In addition, the technique of time-division multiplexing transmits multiple signals through a single line one after the other. Hence, this does not enable simultaneous data processing in single devices.
In conclusion, we have developed and experimentally realized a passive frequency-division demultiplexer, which is based on intrinsically steered two-dimensional signal transport in unpatterned magnetic nanosheets. The prototype device enables the spatial separation of spinwave signals of different frequencies. It exploits the frequency-dependence of the direction of narrow spin-wave beams, which are formed in in-plane magnetized films due to the anisotropic

Competing interests
The authors declare no competing interests.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request. in which the occurring energy splitting into two spin-wave beams is taken into account by the factor 1/2. It should be mentioned that the splitting can be suppressed by changing the design of the demultiplexer as shown in [8]. Furthermore, the factor 0.5 has to be introduced since the spinwave intensity is measured instead of the amplitude.

Micro-focused Brillouin light scattering (µBLS) measurements. The spin-wave intensity is
The group velocity of the spin waves differs inside the waveguides ( w ) and in the unpatterned film area ( f ) and can be calculated according to [33,34]. To determine f , the spin-wave wavevector relating to the occurring beam angle is used for the calculations. Furthermore, w is the spin-wave propagation distance in the waveguides whereas f is the corresponding distance in the film area. Finally, the spin-wave lifetime results from the damping parameter α and is approximated by the value ( = 0) at ferromagnetic resonance [34]. by assuming a slightly larger Gilbert damping parameter (α = 0.0048), which is very likely to occur due to, e.g., the structuring process of the device.
Theoretical description of spin-wave beams and caustics. The basis of the developed theoretical model is the anisotropic isofrequency curve (see Fig. 1c), which describes the dependence of the wavevector component ky on the component kx at a particular frequency f. The dispersion calculations are based on [33] taking into account ky || Hext, kx  Hext, µ0Hext = 75 mT and the material parameters of CoFeB. The direction of the anisotropic spin-wave propagation can be derived from the curve since the spin-wave group velocity vector is perpendicular to it at every point. For a precise prediction of the resulting spin-wave beam orientation, the excited wavevector spectrum Ak and the distance between the beam source and the observation point is taken into account as explained in the following. This is the main difference to a previous theoretical approach [19] and our results reveal, that the observed direction of maximum focusing is not always given by the caustic direction (According to [19], caustics occur at the points of the isofrequency curve where its curvature is zero). Due to this reason, we refer to the observed beams resulting from the focused energy transport as (caustic-like) spin-wave beams instead of caustics.
The first step of the developed approach is to include the wavevector spectrum Ak of the beam source in the calculations. It can be assumed that the spin waves propagate trough the input in the form of the first waveguide mode having a sinusoidal shape along the short axis of the waveguide and only one maximum in the centre. If the input waves reach the transition zone, the waveguide mode acts as a source for secondary spin waves propagating into the unpatterned film area and forming beams due to their anisotropic propagation. The confinement of the mode across the waveguide leads to broad spectrum Ak of the respective wavevector component ky, which can be calculated to Ak(ky) = cos(ky weff/2)/( 2 /weff 2 -ky 2 ) by spatial Fourier-transformation of the mode profile (weff is the effective width of the input waveguide. See explanation above.). This spectrum is taken into account by using the following procedure. First, the angle θvg between the group velocity vector and the external field direction is calculated from the isofrequency curve as a function of the wavevector component ky. Second, the wavevector spectrum Ak(ky) is projected onto the curve θvg(ky) by a numerical integration method to determine the amount of excited spin waves propagating into the direction θvg. The result of this procedure is the spin-wave amplitude SW initial ( g ), which reflects the angle distribution of the spin-wave flow under consideration of the initial wavevector spectrum Ak. Maxima of SW initial ( g ) reveal a wave focusing into the respective directions. Their occurrence can be explained by the fact that the anisotropy of the system leads to isofrequency curves with expanded regions where the group velocity direction θvg is (nearly) unchanged for a broad range of wavevectors ky. If the excited spectrum Ak has a significant magnitude in these ranges of ky, an amplitude concentration occurs into the respective directions θvg leading to the maxima of SW initial ( g ). Hence, these maxima indicate the beam creation.
However, a second step is necessary to precisely determine the beam directions. Real beam sources have a certain extent which leads to the fact that the overall spin-wave amplitude at the observation point results from a superposition of spin waves which are generated at different positions within the source. If the observation point is close, the source appears under a large angular range and spin waves from considerably different directions superpose. In this case, the focusing pattern can be significantly influenced. This effect is included in the further calculations by taking into account both the initial angle distribution of the spin-wave amplitudes SW initial ( g ) and the sinusoidal mode profile in the waveguide opening, which represents the source of the beams. Based on these quantities, the final spin-wave amplitude SW final at a given observation point