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Hybrid elastic and spin waves hold promises for energy-efficient and versatile generation and detection of magnetic signals, with potentially long coherence times. Here we report on the combined elastic and magnetic dynamics in a one-dimensional magneto-mechanical crystal composed of an array of magnetic nanowires. Phononic and magnonic modes are impulsively excited by an optical ultrafast trigger and their decay is monitored by time resolved Magneto Optical Kerr Effect, with complementary Brillouin Light Scattering measurements and micromagnetic simulations. The strength and degree of mixing of coherent and dissipative coupling of the quasi-particles is determined quantitatively.

We present a combined Brillouin light scattering and micromagnetic simulation investigation of the magnetic-field dependent spin-wave spectra in a hybrid structure made of permalloy (NiFe) artificial spin-ice (ASI) systems, composed of stadium-shaped nanoislands, deposited on the top of an unpatterned permalloy film with a nonmagnetic spacer layer. The thermal spin-wave spectra were recorded by Brillouin light scattering (BLS) as a function of the magnetic field applied along the symmetry direction of the ASI sample. Magneto-optic Kerr effect magnetometry was used to measure the hysteresis loops in the same orientation as the BLS measurements. The frequency and intensity of several spin-wave modes detected by BLS were measured as a function of the applied magnetic field. Micromagnetic simulations enabled us to identify the modes in terms of their frequency and spatial symmetry and to extract information about the existence and strength of the dynamic coupling, relevant only to a few modes of a given hybrid system. Using this approach, we suggest a way to understand if dynamic coupling between ASI and film modes is present or not, with interesting implications for the development of future three-dimensional magnonic applications and devices.

We present a combined experimental and numerical study of the spin-wave dispersion in NiFe artificial spin-ice (ASI) system consisting of an array of stadium-shaped nanoislands deposited on the top of a continuous NiFe film with nonmagnetic spacer layers of varying thickness. The spin-wave dispersion, measured by wavevector resolved Brillouin light scattering spectroscopy in the Damon-Eshbach configuration, consists of a rich number of modes, with either stationary or propagating character. We find that the lowest frequency mode displays a bandwidth of approximately 0.5 GHz, which is independent of the presence of the film underneath. On the contrary, the BLS intensity of some of the detected modes strongly depends on the presence of the extended thin-film underlayer. Micromagnetic simulations unveil the details of the dynamic coupling between ASI lattice and film underlayer. Interestingly, the ASI lattice facilitates dynamics of the film either specific wavelengths or intensity modulation peculiar to the modes of the ASI elements imprinted in the film. Our results demonstrate that propagating spin waves can be modulated at the nanometer length scale by harnessing the dynamic mode coupling in the vertical, i.e., the out-of-plane direction of suitably designed magnonic structures.

Magnonics is a field of science that addresses the physical properties of spin waves and utilizes them for data processing. Scalability down to atomic dimensions, operations in the GHz-to-THz frequency range, utilization of nonlinear and nonreciprocal phenomena, and compatibility with CMOS are just a few of many advantages offered by magnons. Although magnonics is still primarily positioned in the academic domain, the scientific and technological challenges of the field are being extensively investigated, and many proof-of-concept prototypes have already been realized in laboratories. This roadmap is a product of the collective work of many authors that covers versatile spin-wave computing approaches, conceptual building blocks, and underlying physical phenomena. In particular, the roadmap discusses the computation operations with Boolean digital data, unconventional approaches like neuromorphic computing, and the progress towards magnon-based quantum computing. The article is organized as a collection of sub-sections grouped into seven large thematic sections. Each sub-section is prepared by one or a group of authors and concludes with a brief description of the current challenges and the outlook of the further development of the research directions., Comment: 74 pages, 57 figures, 500 references

In this paper we introduce the notion of coalgebra symmetry for discrete systems. With this concept we prove that all discrete radially symmetric systems in standard form are quasi-integrable and that all variational discrete quasi-radially symmetric systems in standard form are Poincar\'e--Lyapunov--Nekhoroshev maps of order $N-2$, where $N$ are the degrees of freedom of the system. We also discuss the integrability properties of several vector systems which are generalisations of well-known one degree of freedom discrete integrable systems, including two $N$ degrees of freedom autonomous discrete Painlev\'e I equations and an $N$ degrees of freedom McMillan map., Comment: 38 pages, 4 figures

In this work, we study experimentally by broadband ferromagnetic resonance measurements, the dependence of the spin-wave excitation spectra on the magnetic applied field in CoFeB meander-shaped films. Two different orientations of the external magnetic field were explored, namely parallel or perpendicular to the lattice cores. The interpretation of the field dependence of the frequency and spatial profiles of major spin-wave modes were obtained by micromagnetic simulations. We show that the vertical segments lead to the easy-axis type of magnetic anisotropy and support the in-phase and out-of-phase spin-wave precession amplitude in the vertical segments. The latter could potentially be used for the design of tunable metasurfaces or in magnetic memories based on meandering 3D magnetic films., Comment: This project has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 801055 "Spin Wave Computing for Ultimately-Scaled Hybrid Low-Power Electronics" CHIRON

In this work, we investigate the spin-wave propagation in three-dimensional nanoscale CoFeB/Ta/NiFe meander structures fabricated on a structured SiO2/Si substrate. The magnonic band structure has been experimentally determined by wavevector-resolved Brillouin light scattering (BLS) spectroscopy and a set of stationary modes interposed by two dispersive modes of Bloch type have been identified. The results could be understood by micromagnetic and finite element simulations of the mode distributions in both real space and the frequency domain. The dispersive modes periodically oscillate in frequency over the Brillouin zones and correspond to modes, whose spatial distributions extend over the entire sample and are either localized exclusively in the CoFeB layer or the entire CoFeB/Ta/NiFe magnetic bilayer. Stationary modes are mainly concentrated in the vertical segments of the CoFeB and NiFe layers and show negligible amplitudes in the horizon-tal segments. The findings are compared with those of single-layer CoFeB meander structures with the same geometry parameters, which reveals the influence of the dipolar coupling between the two ferromagnetic layers on the magnonic band structure., Comment: 10 pages, 4 figures. This project has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 801055 "Spin Wave Computing for Ultimately-Scaled Hybrid Low-Power Electronics" - CHIRON

Nineteen classical superintegrable systems in two-dimensional non-Euclidean spaces are shown to possess hidden symmetries leading to their linearization. They are the two Perlick systems [A. Ballesteros, A. Enciso, F.J. Herranz and O. Ragnisco, Class. Quantum Grav. 25, 165005 (2008)], the Taub-NUT system [A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco, and D. Riglioni, SIGMA 7, 048 (2011)], and all the seventeen superintegrable systems for the four types of Darboux spaces as determined in [E.G. Kalnins, J.M. Kress, W. Miller, P. Winternitz, J. Math. Phys. 44, 5811--5848 (2003)].

We discretise the Lax pair for the reduced Nahm systems and prove its equivalence with the Kahan-Hirota-Kimura discretisation procedure. We show that these Lax pairs guarantee the integrability of the discrete reduced Nahm systems providing an invariant. Also, we show with an example that Nahm systems cannot solve the general problem of characterisation of the integrability for Kahan-Hirota-Kimura discretisations., Comment: 14 pages

We compactify and regularize the space of initial values of a planar map with a quartic invariant and use this construction to prove its integrability in the sense of algebraic entropy. The system turns out to have certain unusual properties, including a sequence of points of indeterminacy in $\mathbb P^1\cross \mathbb P^1$. These indeterminacy points are shown to lie on a singular fibre of the mapping to a corresponding QRT system and provide the existence of a one-parameter family of special solutions., Comment: 10 pages, 1 figures

In this paper we present a class of four-dimensional bi-rational maps with two invariants satisfying certain constraints on degrees. We discuss the integrability properties of these maps from the point of view of degree growth and Liouville integrability., Comment: 26 pages. arXiv admin note: text overlap with arXiv:1808.04942

We compute the algebraic entropy of a class of integrable Volterra-like five-point differential-difference equations recently classified using the generalised symmetry method. We show that, when applicable, the results of the algebraic entropy agrees with the result of the generalised symmetry method, as all the equations in this class have vanishing entropy., Comment: 21 pages

We have studied both experimentally and theoretically the reprogrammable spin wave band structure in Permalloy(10nm)/Cu(5nm)/Permalloy(30nm) nanowire arrays of width w=280 nm and inter-wire separation in the range from 80 to 280 nm. We found that, depending on the inter-wire separation, the anti-parallel configuration, where the magnetizations of the two Permalloy layers point in opposite directions, is stabilized over specific magnetic field ranges thus enabling us to directly compare the band structure with that of the parallel alignment. We show that collective spin waves of the Bloch type propagate through the arrays with different magnonic bandwidths as a consequence of the interplay between the intra- and inter-nanowire dynamic dipolar interactions. A detailed understanding, e.g. whether they have a stationary or propagating character, is achieved by considering the phase relation (in-phase or out-of-phase) between the dynamic magnetizations in the two ferromagnetic layers and their average value. This work opens the path to magnetic field-controlled reconfigurable layered magnonic crystals that can be used for future nanoscale magnon spintronic devices.

In this paper we construct the autonomous quad-equations which admit as symmetries the five-point differential-difference equations belonging to known lists found by Garifullin, Yamilov and Levi. The obtained equations are classified up to autonomous point transformations and some simple non-autonomous transformations. We discuss our results in the framework of the known literature. There are among them a few new examples of both sine-Gordon and Liouville type equations., Comment: 27 pages

In this paper we present an algorithm to find the discrete Lagrangian for an autonomous recurrence relation of arbitrary even order $2k$ with $k>1$. The method is based on the existence of a set of differential operators called annihilation operators which can be used to convert a functional equation into a system of linear partial differential equations. This completely solves the inverse problem of the calculus of variations in this setting., Comment: 29 pages

In this letter we give fourth-order autonomous recurrence relations with two invariants, whose degree growth is cubic or exponential. These examples contradict the common belief that maps with sufficiently many invariants can have at most quadratic growth. Cubic growth may reflect the existence of non-elliptic fibrations of invariants, whereas we conjecture that the exponentially growing cases lack the necessary conditions for the applicability of the discrete Liouville theorem., Comment: 16 pages, 2 figures

Using Brillouin light scattering spectroscopy and dynamical matrix method calculations, we study collective spin waves in dense arrays of periodically double-side width-modulated Permalloy nanowires. Width modulation is achieved by creating a sequence of triangular notches on the two parallel nanowire sides, with a periodicity of p=1000 nm, and tunable relative displacement (D) of the notches sequence on the two lateral sides. Both symmetric (D=0) and asymmetric (D= 250 and 500 nm) width-modulated nanowires were investigated. We have found that the detected modes have Bloch-type character and belong to a doublet deriving from the splitting of the modes characteristics of the nanowire with homogeneous width. Interestingly, the amplitude of the magnonic band, the frequency difference of the doublet, as well as their relative scattering intensity, can be efficiently controlled by increasing D rather than having single- or symmetric (D=0) double-side width-modulation.

Reversed structures of artificial spin-ice systems, where elongated holes with elliptical shape (antidots) are arranged into a square array with two orthogonal sublattices, are referred to as anti-squared spin-ice. Using Brillouin light scattering spectroscopy and plane wave method calculations, we investigate the spin wave propagation perpendicular to the applied field direction for two 20 nm thick Permalloy nanostructures which differ by the presence of single and double elliptical antidots. For the spin waves propagation along the principal antidot lattice axis, the spectrum consists of flat bands separated by several frequency gaps which are the effect of spin wave amplitude confinement in the regions between antidots. Contrarily, for propagation direction at 45 degrees with respect to the antidot symmetry axis, straight and narrow channels of propagation are formed, leading to broadening of bands and closing of the magnonics gaps. Interestingly, in this case, extra magnonic band gaps occur due to the additional periodicity along this direction. The width and the position of these gaps depend on the presence of single or double antidots. In this context, we discuss possibilities for the tuning of spin wave spectra in anti-squared spin ice structures.

We measure and analyze the chirality of the Dzyaloshinskii-Moriya interaction (DMI) stabilized spin textures in multilayers of Ta/Co$_{20}$Fe$_{60}$B$_{20}$/MgO. The effective DMI is measured experimentally using domain wall motion measurements, both in the presence (using spin orbit torques) and absence of driving currents (using magnetic fields). We observe that the current-induced domain wall motion yields a change in effective DMI magnitude and opposite domain wall chirality when compared to field-induced domain wall motion (without current). We explore this effect, which we refer to as current-induced DMI, by providing possible explanations for its emergence, and explore the possibilty of its manifestation in the framework of recent theoretical predictions of DMI modifications due to spin currents.

Reprogrammability of magnonic band structure in layered Permalloy/Cu/Permalloy nanowires is demonstrated to depend on the relative orientation of the two layers magnetization. By using Brillouin light spectroscopy, we show that when the layers are aligned parallel two dispersive modes, with positive and negative group velocity, are observed while when the magnetic layers are aligned anti-parallel, only one dispersive mode, with positive group velocity, is detected. Our findings are successfully compared and interpreted in terms of a microscopic (Hamiltonian-based) method. An explanation for the observed behavior can be attributed to mode-mixing (or hybridization) effect when the two magnetic layers are aligned anti-parallel. This work opens the path to magnetic field-controlled reconfigurable magnonic crystals with multi-modal frequency transmission characteristics.

In this paper we present a simple, algorithmic test to establish if a Hamiltonian system is maximally superintegrable or not. This test is based on a very simple corollary of a theorem due to Nekhoroshev and on a perturbative technique called multiple scales method. If the outcome is positive, this test can be used to suggest maximal superintegrability, whereas when the outcome is negative it can be used to disprove it. This method can be regarded as a finite dimensional analog of the multiple scales method as a way to produce soliton equations. We use this technique to show that the real counterpart of a mechanical system found by Jules Drach in 1935 is, in general, not maximally superintegrable. We give some hints on how this approach could be applied to classify maximally superintegrable systems by presenting a direct proof of the well-known Bertrand's theorem., Comment: 30 pages, 4 figure

We have experimentally and numerically investigated the dispersion of collective spin waves prop-agating through arrays of longitudinally magnetized nanowires with periodically modulated width. Two nanowire arrays with single-side modulation and different periodicity of modulation were studied and compared to the nanowires with homogeneous width. The spin-wave dispersion, meas-ured up to the third Brillouin zone of the reciprocal space, revealed the presence of two dispersive modes for the width-modulated NWs, whose amplitude of magnonic band depends on the modula-tion periodicity, and a set of nondispersive modes at higher frequency. These findings are different from those observed in homogeneous width NWs where only the lowest mode exhibits sizeable dis-persion. The measured spin-wave dispersion has been satisfactorily reproduced by means of dynam-ical matrix method. Results presented in this work are important in view of the possible realization of frequency tunable magnonic device.

An efficient way for the control of spin wave propagation in a magnetic medium is the use of periodic patterns known as magnonic crystals (MCs). However, the fabrication of MCs especially bi-components, with periodicity on nanoscale, is a challenging task due to the requirement for sharp interfaces. An alternative method to circumvent this problem is to use homogeneous ferromagnetic film with modified periodically surrounding. The structure is also suitable for exploiting nonreciprocal properties of the surface spin waves. In this work, we demonstrate that the magnonic band structure forms in thin permalloy film due to dynamical magnetostatic coupling with Ni stripes near its surface. We show, that the band gap width can be systematically tuned by the changing interlayer thickness between film and stripes. We show also the effect of nonreciprocity, which is seen at the band gap edge shifted from the Brillouin zone boundary and also in nonreciprocal interaction of propagating spin waves in Py film with the standing spin waves in Ni stripes. Our findings open possibility for further investigation and exploitation of the nonreciprocity and band structure in magnonic devices., Comment: 12 pages, 5 figures

In this paper we prove that the trapezoidal $H^{4}$ and the $H^{6}$ families of quad-equations are Darboux integrable systems. This result sheds light on the fact that such equations are linearizable as it was proved using the Algebraic Entropy test [G. Gubbiotti, C. Scimiterna and D. Levi, Algebraic entropy, symmetries and linearization for quad equations consistent on the cube, \emph{J. Nonlinear Math. Phys.}, 23(4):507543, 2016]. We conclude with some suggestions on how first integrals can be used to obtain general solutions., Comment: 34 pages

The classical quantization of the motion of a free particle and that of an harmonic oscillator on a double cone are achieved by a quantization scheme [M.C. Nucci, Theor. Math. Phys. 168 (2011) 994], that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schroedinger equation. The result is different from that given in [K. Kowalski, J.Rembielnski, Ann. Phys. 329 (2013) 146]. A comparison of the different outcomes is provided., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1406.0192 in the Introduction since the authors' method of quantization is described again

Spin-Hall oscillators are promising sources of spin-wave signals for magnonics applications, and can serve as building blocks for magnonic logic in ultralow power computation devices. Here, we analytically and micromagnetically study magnetization dynamics excited in a Spin-Hall oscillator with oblique magnetization when the spin-Hall effect and interfacial Dzyaloshinskii-Moriya interaction act simultaneously. Our key results are (i) excitation of nonreciprocal spin-waves propagating perpendicularly to the in-plane projection of the static magnetization, (ii) skyrmions generation by pure spin-current, (iii) excitation of a new spin-wave mode with a spiral spatial profile originating from a gyrotropic rotation of a dynamical skyrmion. These results demonstrate that Spin-Hall oscillators can be used as generators of magnetic skyrmions and different types of propagating spin-waves for magnetic data storage and signal processing applications.

In the developing field of magnonics,it is very important to achieve tailoring of spin wave propagation by both a proper combination of materials with different magnetic properties and their nanostructuring on the submicrometric scale. We have exploited deep ultra-violet lithography, in combination with tilted shadow deposition technique, to fabricate arrays of closely spaced bi-layer nanowires (NWs), with separation d=100 nm and periodicity a=440 nm, having bottom and top layers made of Permalloy and Iron, respectively. The NWs have either rectangular cross section (bottom and upper layers of equal width) or L-shaped cross section (upper layer of half width). The frequency dispersion of collective spin wave excitations in the above bi-layered NWs arrays has been measured by the Brillouin light scattering technique while sweeping the wave vector perpendicularly to the wire length over three Brillouin zones of the reciprocal space. For the rectangular NWs, the lowest frequency fundamental mode, characterized by a quasi- uniform profile of the amplitude of the dynamic magnetization across the NW width, exhibits a sizeable and periodic frequency dispersion. A similar dispersive mode is also present in L-shaped NWs, but the mode amplitude is concentrated in the thin side of the NWs. The width and the center frequency of the magnonic band associated with the above fundamental modes has been analyzed, showing that both can be tuned by varying the external applied field. Moreover, for the L-shaped NWs it is shown that there is also a second dispersive mode,at higher frequency,characterized by an amplitude concentrated in the thick side of the NW. These experimental results have been quantitatively reproduced by an original numerical model that includes two-dimensional Green's function of the dipole field of the dynamic magnetization and interlayer exchange coupling between the layers., Comment: 17 pages, 5 figures

We report the observation of a Pt layer thickness dependence on the induced interfacial Dzyaloshinskii-Moriya interaction in ultra-thin Pt($d_{\text{Pt}}$)/CoFeB films. Taking advantage of the large spin-orbit coupling of the heavy metal, the interfacial Dzyaloshinskii-Moriya interaction is quantified by Brillouin light scattering measurements of the frequency non-reciprocity of spin-waves in the ferromagnet. The magnitude of the induced Dzyaloshinskii-Moriya coupling is found to saturate to a value $0.45$ mJ$/$m${}^2$ for Pt thicknesses larger than $\sim 2$ nm. The experimental results are explained by analytical calculations based on the 3-site indirect exchange mechanism that predicts a Dzyaloshinskii-Moriya interaction at the interface between a ferromagnetic thin layer and a heavy metal. Our findings open up a way to control and optimize chiral effects in ferromagnetic thin films through the thickness of the heavy metal layer.

Despite its historical importance, a perfect gas enclosed by a pistons and in contact with a thermal reservoirs is a system still largely under study. Its thermodynamic properties are not yet well understood when driven under non-equilibrium conditions. In particular, analytic formulas that describe the heat exchanged with the reservoir are rare. In this paper we prove a power series expansions for the heat when both the external force and the reservoir temperature are slowly varying over time but the overall process is not quasi-static. To do so, we use the dynamical equations from [Cerino \emph{et al.}, \textit{Phys. Rev. E}, \textbf{91} 032128] and an uncommon application of the regular perturbation technique., Comment: 8 pages, 3 figures

Several examples of classical superintegrable systems in two-dimensional spac are shown to possess hidden symmetries leading to their linearization. They are those determined 50 years ago in [Phys. Lett. 13, 354 (1965)], and the more recent Tremblay-Turbiner-Winternitz system [J. Phys. A: Math. Theor. 42, 242001 (2009)]. We conjecture that all classical superintegrable systems in two-dimensional space have hidden symmetries that make them linearizable., Comment: 17 pages

The non-equilibrium thermodynamics of a gas inside a piston is a conceptually simple problem where analytic results are rare. For example, it is hard to find in the literature analytic formulas that describe the heat exchanged with the reservoir when the system either relaxes to equilibrium or is compressed over a finite time. In this paper we derive such kind of analytic formulas. To achieve this result, we take the equations derived by Cerino \textit{et al.} [Phys. Rev. E \textbf{91}, 032128] describing the dynamic evolution of a gas-piston system, we cast them in a dimensionless form and we solve the dimensionless equations with the multiple scales expansion method. With the approximated solutions we obtained, we express in a closed form the heat exchanged by the gas-piston system with the reservoir for a large class of relevant non-equilibrium situations., Comment: 23 pages, 5 figures

We discuss the linearization of a non-autonomous nonlinear partial difference equation belonging to the Boll classification of quad-graph equations consistent around the cube. We show that its Lax pair is fake. We present its generalized symmetries which turn out to be non-autonomous and depending on an arbitrary function of the dependent variables defined in two lattice points. These generalized symmetries are differential difference equations which, in some case, admit peculiar B\"acklund transformations., Comment: arXiv admin note: text overlap with arXiv:1311.2406 by other authors

We employed micro-focused Brillouin light scattering to study the amplification of the thermal spin wave eigenmodes by means of a pure spin current, generated by the spin-Hall effect, in a transversely magnetized Pt(4nm)/NiFe(4nm)/SiO2(5nm) layered nanowire with lateral dimensions 500x2750 nm2. The frequency and the cross section of both the center (fundamental) and the edge spin wave modes have been measured as a function of the intensity of the injected dc electric current. The frequency of both modes exhibits a clear redshift while their cross section is greatly enhanced on increasing the intensity of the injected dc. A threshold-like behavior is observed for a value of the injected dc of 2.8 mA. Interestingly an additional mode, localized in the central part of the nanowire, appears at higher frequency on increasing the intensity of the injected dc above the threshold value. Micromagnetic simulations were used to quantitatively reproduce the experimental results and to investigate the complex non-linear dynamics induced by the spin-Hall effect, including the modification of the spatial profile of the spin wave modes and the appearance of the extra mode above the threshold., Comment: 13 pages, 4 figures in journal of applied physics 2015

This letter studies the dynamical behavior of spin-Hall nanoscillators from a micromagnetic point of view. The model parameters have been identified by reproducing recent experimental data quantitatively. Our results indicate that a strongly localized mode is observed for in-plane bias fields such as in the experiments, while predict the excitation of an asymmetric propagating mode for large enough out-of plane bias field similarly to what observed in spin-torque nanocontact oscillators. Our findings show that spin-Hall nanoscillators can find application as spin-wave emitters for magnonic applications where spin waves are used for transmission and processing information on nanoscale., Comment: 14 pages

The classical quantization of a family of a quadratic Li\'{e}nard-type equation (Li\'{e}nard II equation) is achieved by a quantization scheme (M.~C. Nucci. {\em Theor. Math. Phys.}, 168:994--1001, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schr\"odinger equation. This method straightforwardly yields the Schr\"odinger equation as given in (A.~Ghose~Choudhury and Partha Guha. {\em J. Phys. A: Math. Theor.}, 46:165202, 2013)., Comment: 13 pages. arXiv admin note: text overlap with arXiv:1307.3803 in the Introduction since the authors' method of quantization is described again

The classical quantization of a Lienard-type nonlinear oscillator is achieved by a quantization scheme (M.C. Nucci. Theor. Math. Phys., 168:997--1004, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schr\"odinger equation. This method straightforwardly yields the correct Schr\"odinger equation in the momentum space (V. Chithiika Ruby, M. Senthilvelan, and M. Lakshmanan. J. Phys. A: Math. Gen., 45:382002, 2012), and sheds light into the apparently remarkable connection with the linear harmonic oscillator., Comment: 18 pages

The application of Brillouin light scattering to the study of the spin-wave spectrum of one- and two-dimensional planar magnonic crystals consisting of arrays of interacting stripes, dots and antidots is reviewed. It is shown that the discrete set of allowed frequencies of an isolated nanoelement becomes a finite-width frequency band for an array of identical interacting elements. It is possible to tune the permitted and forbidden frequency bands, modifying the geometrical or the material magnetic parameters, as well as the external magnetic field. From a technological point of view, the accurate fabrication of planar magnonic crystals and a proper understanding of their magnetic excitation spectrum in the GHz range is oriented to the design of filters and waveguides for microwave communication systems.

Spin wave frequencies are observed in ultra-thin Fe/GaAs(100) films at temperatures where the spontaneous zero field magnetization is zero. The films exhibit good cyrstalline structure, and the effect of magnetic anisotropies is apparent even though no zero field spin wave energy gap exists. An analysis is given in terms of a superparamagnetic model in which the film is treated as a network of non-interacting single domain magnetic islands. A spin wave analysis provides a means to separate measured values of anisotropy parameters from products involving anisotropy and island volume. In this way, a measure of the activation volume associated with superparamagnetic islands is obtained for different Fe film thicknesses. Results suggest that the island lateral area increases with increasing film thickness.

A double-peaked structure was observed in the {\it in-situ} Brillouin Light Scattering (BLS) spectra of a 6 \AA$ $ thick epitaxial Fe/GaAs(001) film for values of an external magnetic field $H$, applied along the hard in plane direction, lower than a critical value $H_c\simeq 0.9$ kOe. This experimental finding is theoretically interpreted in terms of a model which assumes a non-homogeneous magnetic ground state characterized by the presence of perperpendicular up/down stripe domains. For such a ground state, two spin-wave modes, namely an acoustic and an optic mode, can exist. Upon increasing the field the magnetization tilts in the film plane, and for $H \ge H_{c}$ the ground state is homogeneous, thus allowing the existence of just a single spin-wave mode. The frequencies of the two spin-wave modes were calculated and successfully compared with the experimental data. The field dependence of the intensities of the corresponding two peaks that are present in the BLS spectra was also estimated, providing further support to the above-mentioned interpretation., Comment: Shortened version (7 pages). Accepted for publication in Physical Review B

Brillouin light scattering (BLS) measurements were performed for (17-120) Angstrom thick Cu/Ni/Cu/Si(001) films. A monotonic dependence of the frequency of the uniform mode on an in-plane magnetic field H was observed both on increasing and on decreasing H in the range (2-14) kOe, suggesting the absence of a metastable collinear perpendicular ground state. Further investigation by magneto-optical vector magnetometry (MOKE-VM) in an unconventional canted-field geometry provided evidence for a domain structure where the magnetization is canted with respect to the perpendicular to the film. Spin wave calculations confirm the absence of stable collinear configurations., Comment: 6 pages, 3 figures (text, appendix and 1 figure added)