This paper provides a tutorial overview over recent vigorous efforts to develop computing systems based on spin waves instead of charges and voltages. Spin-wave computing can be considered a subfield of spintronics, which uses magnetic excitations for computation and memory applications. The Tutorial combines backgrounds in spin-wave and device physics as well as circuit engineering to create synergies between the physics and electrical engineering communities to advance the field toward practical spin-wave circuits. After an introduction to magnetic interactions and spin-wave physics, the basic aspects of spin-wave computing and individual spin-wave devices are reviewed. The focus is on spin-wave majority gates as they are the most prominently pursued device concept. Subsequently, we discuss the current status and the challenges to combine spin-wave gates and obtain circuits and ultimately computing systems, considering essential aspects such as gate interconnection, logic level restoration, input-output consistency, and fan-out achievement. We argue that spin-wave circuits need to be embedded in conventional complementary metal-oxide-semiconductor (CMOS) circuits to obtain complete functional hybrid computing systems. The state of the art of benchmarking such hybrid spin-wave-CMOS systems is reviewed, and the current challenges to realize such systems are discussed. The benchmark indicates that hybrid spin-wave-CMOS systems promise ultralow-power operation and may ultimately outperform conventional CMOS circuits in terms of the power-delay-area product. Current challenges to achieve this goal include low-power signal restoration in spin-wave circuits as well as efficient spin-wave transducers.

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Featuring low heat dissipation, devices based on spin-wave logic gates promise to comply with increasing future requirements in information processing. In this work, we present the experimental realization of a majority gate based on the interference of spin waves in an Yttrium-Iron-Garnet-based waveguiding structure. This logic device features a three-input combiner with the logic information encoded in the phase of the spin waves. We show that the phase of the output signal represents the majority of the phase of the input signals. A switching time of about 10 ns in the prototype device provides evidence for the ability of sub-nanosecond data processing in future down-scaled devices.The scaling of conventional CMOS-based nanoelectronics is expected to become increasingly intrinsically limited in the next decade. Therefore, novel beyond-CMOS devices are being actively developed as a complement to expand functionally in future nanoelectronic technology nodes 1 . In particular, the field of magnonics 2-7 (see also reviews 8-12) which utilizes the fundamental excitations of a magnetic system -spin waves 13 and their quanta -magnons 14 as data carriers, provides promising approaches to overcome crucial limitations of CMOS since they may provide ultralow power operation as well as nonvolatility 9,12,15 . Magnonic devices are especially amenable to building majority gates 7,16-19 with excellent scaling potential leading to an improved circuit efficiency. Hence, majority gates can be considered to be key devices in a novel approach to circuit design with strongly improved area and power scaling behavior 20 .Spin waves cover characteristic frequencies in the GHz regime and their wavelength can easily be reduced down to the nanometer range 12,21 . Furthermore, their dispersion relation is highly versatile depending on material parameters as well as magnetization and field configuration 8 making them usable in a wide range of devices [2][3][4][5][6][7]10,[22][23][24][25] . In this context, majority gates are of special interest since a simple spin-wave combiner substitutes several tens of transistors, and three majority gates suffice for creating a full-adder 26 . Multi-frequency operation allows for parallel data processing 27 .In this work, we present the experimental realization and investigation of a prototype of a spin-wave majority gate, whose functionality and performance on the microscopic scale have been investigated in numerical simulations 7,17 . The investigated majority gate has three

We develop an all electrical experiment to perform the broadband phase-resolved spectroscopy of propagating spin waves in micrometer sized thin magnetic stripes. The magnetostatic surface spin waves are excited and detected by scaled down to 125 nm wide inductive antennas, which award ultra broadband wavevector capability. The wavevector selection can be done by applying an excitation frequency above the ferromagnetic resonance. Wavevector demultiplexing is done at the spin wave detector thanks to the rotation of the spin wave phase upon propagation. A simple model accounts for the main features of the apparatus transfer functions. Our approach opens an avenue for the all electrical study of wavevector-dependent spin wave properties including dispersion spectra or non-reciprocal propagation.Spin wave based computing 1 -a paradigm-shifting technology that uses the interference of spin waves-offers potential for significant power and area reduction per computing throughput with respect to complementary metaloxide-semiconductor (CMOS) transistor technology. Efficient solutions for spin wave routing 2,3 , spin wave emission 4 , amplification 1 , and spin wave combination 5,6 have been developed. However, these solutions often rely on materials that are difficult to integrate 7 into a CMOS environment. Moreover, they are often demonstrated only for long wavelength (≥ 1 µm) spin waves, for which the low group velocity limits the speed of computation and communication. Efficient methods to generate and detect spin waves with short wavelengths are still lacking. Inductive methods have commonly been employed for long wavelength spin waves as has Brillouin light scattering spectroscopy 8 , which is however diffraction limited and requires complex procedure to retrieve phase information 9 . Alternative spin wave generation and detection methods based on magneto-elastic coupling in surfaceacoustic wave devices 10 are still under development and raise questions regarding their high frequency capability 11 . Longto-short wavelength conversion can be done in magnonic crystals 12 for a geometrically limited discrete set of wavevectors at the expense of high conversion loss. A better conversion efficiency can be obtained by periodically folded coplanar antennas 13,14 but this at the expense of any flexibility in the generated wavevevector. Overall, none of the above methods has so far demonstrated the combination of phase resolution, broad frequency coverage, high sensitivity, and large wavevector (short wavelength) capability.In this work, we demonstrate that the use of deep submicron inductive antennas can circumvent these limitations and allow for the generation and detection of spin waves with ultra-wide frequency band capability and broad wavevector capability up to 15 − 20 rad/µm. We illustrate our method on micrometer-sized Permalloy stripes and describe its behavior within an analytic framework. Our method complements the advantages of Brillouin Light scattering in a compact all electrical device in which the phase resol...