High temperature behaviour of spin waves in Fe 3-xMn xSi

Previous neutron scattering experiments on single crystal of Sendust alloy (73.5 at. % Fe, 17 at. % Al, and 9.5 at. % of Si) revealed extremely high spin wave damping at room temperature. The aim of present studies was to test if the spin wave damping originates from lattice disorder or comes from mutual spin waves interactions or other excitations that would be temperature dependent. The spin wave dispersion relation and damping were studied in temperature range from 8 to 295 K. No regular changes of both spin wave damping and stiffness constant (which characterises dispersion relation) with temperature were found. Thus, within this temperature range, spin waves interactions with dynamical excitations of all kinds must be negligible, and their damping is most likely produced by lattice disorder alone. It is of interest to note that the inelastic background intensity below the spin wave peaks increases with temperature and for energy transfers higher than 30 meV this increase is larger than that of spin-wave peak intensity.

We discuss magnetic superlaticces composed of ferromagnetic films with either ferromagnetic or antiferr* magnetic coupling at the interfaces. For ferromagnetic coupling, the low temperature behavior of the magnetization as a function of film thickness depends critically on the strength of the interface coupling constant. For antiferromagnetic coupling we calculate the spin wave excitation spectrum and relate this to a macroscopic ferrimagnet. In this paper we examine the excitation spectrum [I] and thermodynamic properties [2] of a variety of superlattices composed of different ferromagnetic films. The coupling at the interface between the two ferromagnetic films may be ferromagnetic or antiferromagnetic in nature. As we will see the interface coupling will play a critical role in properties of the superlattice. We first consider the low temperature magnetization of a superlattice composed of alternating ferromagnetic films which are coupled ferromagnetically at the interface. In a bulk, uniform sample, the magnetization should decrease according to the Bloch T~~~ law, i.e. M (T) / M (0) = 1 B T ~ / ~ . Except for very thin samples, this result is independent of sample size [3]. The situation in a superlattice can be quite different. As usual, the low temperature magnetization depends on the spin wave spectrum. The spin waves in the superlattice will differ substantially from those. in bulk samples of the individual materials in two respects: 1) the periodicity of the superlattice introduces gaps into the spin wave spectrum. These gaps allow the spin wave frequencies to differ considerably from the w = D I ~ ~ dispersion law for bulk materials. Thus the magnetization can also deviate from the T3I2 Blochs law; 2 ) the interface exchange interaction can be quite different from the exchange constants in each of the consistuent films. For small wavevectors, q, perpendicular to the layers, the dispersion relation will be 2 approximately w = Deffq where Deff depends on the the exchange constants in materials A and B (JA and J e ) , the magnitude of the spins in materials A and B (SA and Sg) and the interface exchange constant ( J I ) . For very thin ferrmagnetic layers, the interface exchange constant can play a significant role in determining Deff. While D,E does not change the 312 power law, it governs the size of the coefficient, B, in the Bloch B T ~ ~ ~ terms, with a larger Des leading to a smaller coefficient B. In our work, we diagonalized the superlattice Hamiltonian, found the net spin deviation for each excitation and then did a thermal average to find the change in magnetization as a function of temperature. The details will be presented elsewhere [4]. We note that care must be taken in the thermal average to avoid nonphysical singularities [3]. Figure 1 shows M ( T ) as a function of layering pa6 tern. The parameters are JA = 1.42 x 10-l6 ergs, JB = 2.76 x lo-'' ergs, SA = 312, SB = 512. We consider superlattices with n layers of material A and n layers of material B in a unit cell (n / n system). In the case of strong interfacial coupling, the magnetization decreases more rapidly as the number of layers in a unit cell is increased. For weak interfacial coupling the behavior is opposite. To understand this, we note that M (T) tends toward the bulk behavior of the maFig. 1. structures. T Low temperature magnetization for different 'work supported by ARO Grant # DAAL03-88-K-0061. 'permanent address: ICMAB-C.S.I.C., Marti i Fra.nau6s S/N Barcelona-08028, Spain. B. M. is grateful for support from C.I.R.I.T. and C.S.I.C. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19888767 C8 1692 JOURNAL DE PHYSIQUE terial with the lower Curie temperature as the number of layers in a unit cell is increased. In the weak coupling limit, Deff is softer for thinner layers (where the interface coupling is important) and then increases for thicker layers (nearing bulk behavior). This leads to the more rapid decrease in magnetization with temperature seen for smaller n in figure 2. The opposite occurs in the strong coupling limit. Fig. 2. Dispersion relation for spin wave propagating perpendicular to the layers of the superlattice. t = T/(Tc of Fe). As mentioned earlier, the band gaps introduced by cells. We then diagonalize a 272 x 2n matrix in order to find the frequency eigenvalues. We obtain an approximate temperature dependence for the spin wave frequencies by replacing Si with its thermal averaged values obtained from the mean field calculation 151. In figure 2, we present the results for the dispersion relation for spin waves propagating perpendicular to the layers of the superlattice. Only the lowest two modes are shown, each for three temperatures. The most interesting feature of this figure is that one mode decreases in frequency as the temperature is raised while the other mode increases in frequency. Normally one would expect that an increase in temperature should result in a lowering of the the frequency of the spin wave modes since the effective fields in the torque equations are reduced. The explantation for the increase in frequency with temperature has to do with the similarity of the superlattice to a macroscopic ferrimagnet. The dispersion relation for a bcc frrimagnet in the small q limit is given by the periodic structure should allow deviations from the T3/2 We ha,,e this deviation to be rather In the superlattice the temperature dependences of Sup small. For example in a 20120 strong coupling configand Sdown are given by their For uration the exponent of is while for a 20/20 the Fe/Gd superlattice considered here, the coefficient weak coupling configuration the exponent is found to in front of the q2 term actually increases with temperbe Stronger interface coupling generally leads to ature. This then results in the increase in frequency larger exponents. of the superlattice mode. To be sure that the analysis here is appropriate, we checked that in this particular We now turn to magnetic superlattices which have spin wave mode the spins in each different material esantiferromagnetic coupling at the interfaces. The static configurations in this system have been invessentially move together so that the mode does indeed look like that of a ferrimagnet. tigated previously [5, 61. It was found that a number of different phases were possible depending on the layering pattern, the temperature, and the applied field. The spin waves in this structure are also of interest. [I] Albuquerque, E. L., Fulco, P., Sarmento, E. and First it is necessary to know the excitation spectrum Tilley, D. R., Solid State Commun. 58 (1986) 41. in order to obtain the proper low temperature mag[2] Fishman, F., Schwabl, F. and Schwenk, D., Phys. netization. Second, spin wave excitations play a key Lett. A 121 (1987) 192. role in some dynamic macrosc opic measurements such [3] Klein, M. J. and*Smith, R. S., Phys. Rev. 81 as infrzcred absorption. Although we have calculated (1951). the spin waves for a number of different structures [4] Martinez, B. and Camley, R. E. (to be published). and phases, we concentrate here on one system, the [5] Camley, R. E. and Tilley, D. R., Phys. Rev. B 37 4 Fe/4 Gd structure in the aligned state. (1988) 3413. The spin waves can be calculated by formulating the [6] Webb, D. J., Walmsley, R. G., Parvin, K., Dickequations of motion for spins in each layer, i, and using inson, P. H., Geballe, T. H., White, R. M., Phys. Bloch's theorem to relate the spins in neighboring unit Rev. B 32 (1985) 4467.

Yttrium iron garnet (Y <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> Fe <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</inf> O <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">12</inf> , YIG) is well-known material with an extremely small magnetic damping, $\alpha = \sim 10 ^{-5}$ in bulk, which is two orders of magnitude smaller than that in ferromagnetic metals. The growing demands for YIG-based spintronics have led to the development of YIG thin films with a nanometer thickness range [1, 2]. Recently, magnonics has been attracted considerable interests for the transmission, storage and processing of the information using propagating spin waves [3]. To miniaturize the magnonic devices, it is necessary to reduce the thickness of YIG films for a shorter wavelength. For high quality YIG nanometer films, it has been reported that the YIG thin films by pulsed laser deposition (PLD) show the relatively low damping constant of $2.3 \times 10 ^{-4}$ for 20 nm thickness [1]. From the view point of a broad utility and industry, the sputtering growth is better than PLD. In this study we investigate the spin wave propagation in sputter-deposited YIG nanometer films, and characterize the YIG thickness dependence of the several parameters, such as magnetic damping constant, spin wave group velocity and nonreciprocity. The YIG thin films were grown on 0.5-mm-thick single crystal gallium gadolinium garget (GGG) substrates with (111) orientation by RF magnetron sputtering. During the deposition, the substrate was kept at room temperature, the argon pressure and sputtering power was 0.06 Pa and 150 W, respectively. The films were annealed at 900°C for 8 h in the air. We varied YIG film thickness from 20 nm to 50 nm. Using electron-beam lithography and Ar ion-milling technique, the films were patterned into a circular shape with $10- \mu \mathrm {m} -$diameter for ferromagnetic resonance (FMR) measurement and a rectangular shape with $50- \mu \mathrm {m} -$width for the spin wave measurement. After patterning the YIG films, an insulating layer of 30 nm SiO <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> was deposited on the entire surface. Finally, microwave antennas were deposited for the FMR and spin wave measurement. First, we evaluate the magnetic damping constant $\alpha $ from FMR measurement using a vector network analyzer. Figure 1 (a) shows the FMR linewidth as a function of resonance frequency. $\alpha $ was extracted from the slope of the linear fits to the data. As shown in Fig. 1(b), the $\alpha $ value decreases with increasing the thickness and we obtained lowest value of $\alpha = 1.3 \times 10 ^{-3}$ in 50-nm-thick YIG, which is slightly larger than the reported values for sputter-deposited YIG thin films [2]. While we need further optimization of the sputtering condition and/or annealing process to reduce $\alpha $, it should be noted that the obtained $\alpha $ in this study is significantly smaller than that of ferromagnetic metallic films with similar thickness. Second, the propagating spin wave spectroscopy was performed under the in-plane magnetic field to excite the magnetostatic surface spin wave (MSSW) mode. Figure 2(a) shows the group velocity of spin wave estimated from the oscillation period in transmission spectra. It was about 1.1 km/s in 50-nm-thick YIG waveguide under 14 mT. We found that the spin wave group velocity decreases with increasing the magnetic field and decreasing the film thickness. The group velocity $v_{g}$ can be calculated from the spin wave dispersion $\omega (k)$ as defined $v_{g} = d \omega ( k)/ dk$, which nicely reproduces the experimental results. By comparing the signal intensity, the nonreciprocity defined as $A_{12}/ A_{21}$ was also estimated, where $A_{12}$ and $A_{21}$ denote the signal intensity of $S_{12}$ and $S_{21}$, respectively. The unity value indicates the reciprocal characteristics. In our experiment, the nonreciprocity is mainly caused by x- and z-components of microwave magnetic field [4]. As shown in Fig. 2(b), the nonreciprocity increases with increasing magnetic field and we obtained the largest nonreciprocity of 0.1 at 85 mT. This This nonreciprocity is much more significant than that in ferromagnetic metals [4], which is attributed to the smaller saturation magnetization of YIG than that in ferromagnetic metals.

Ferromagnet-dielectric heterostructures may possess interfacial magnetic anisotropy, which varies with the electric field created in the dielectric interlayer. Such a voltage-controlled magnetic anisotropy (VCMA) represents an efficient tool for the excitation of spin waves without the use of microwave magnetic fields. In ferromagnets with a significant magnetoelastic coupling between spins and strains, however, the magnetization precession in the spin wave induces elastic vibrations. Here we report micromagnetoelastic simulations of the coupled spin and acoustic dynamics generated in the thick Fe film brought into contact with an MgO nanolayer subjected to a microwave voltage. It is found that the electrically induced periodic modulation of VCMA gives rise to the excitation of both spin and elastic waves in the magnetostrictive Fe film. Remarkably, the magnetoelastic coupling makes it possible to generate propagating spin waves having frequencies well below the ferromagnetic resonance frequency. Furthermore, when the voltage frequency approaches a value ${\ensuremath{\nu}}^{*}$ at which the dispersion relations of pure spin and transverse elastic waves cross, the excitation of traveling magnetoelastic waves with a specific hybrid dispersion occurs in the Fe film. To explain the variety of observed phenomena, we propose a generalized dispersion relation for magnetoelastic waves, which accounts for a possible difference between the wave numbers of their magnetic and elastic components. The analysis of the simulation data also reveals that frequency dependencies of the amplitudes of magnetic oscillations and elastic vibrations display significant anomalies around the crossing-point frequency ${\ensuremath{\nu}}^{*}$, which originate from the electrically induced magnetoacoustic resonance.

We investigated non-reciprocal spin wave (SW) dispersion relations by using Brillouin Light Scattering (BLS) in inversion symmetry breaking heterostructures. The non-reciprocal SW dispersion relation is a fingerprint of the existence of asymmetric exchange coupling, Dzyaloshinskii-Moriya Interaction (DMI). The quantification of DMI is important, because it is an essential ingredient of the domain wall motion and skyrmion based logic devices. We obtained DMI energy densities of various NM1/ferromagnetic(Co, CoFeB)/NM2 structures, where NM1,2 are non-magnetic heavy metals (Pt, Ir, Ta) or insulators (MgO, AlOx). We revealed that DMI is proportional to the inverse of ferromagnetic layer thickness, which strong evidence of interface nature of DMI in the heterostructure. In order to exclude the possible source of non-reciprocal SW dispersions, we carefully examined DMI with three independent measurement methods, field strength dependence, SW wave vector magnitude dependence, and SW propagating angle dependences, and found all three measurements gave the same results. It implies our measurement results are more reliable compared with other methods. We also found that sign and strength are sensitive function of materials, interface qualities, layer orders, and annealing conditions. Since the strong spin-orbit coupling (SOC) between ferromagnetic and the heavy metal layers is a source DMI, we also investigated SOC related quantities such as perpendicular magnetic anisotropy, magneto-optical Kerr effect, and spin pumpings. And we found that the relation between SOC and DMI is not the same with other SOC-related physical quantities.

We develop a generic all-inductive procedure to measure the dispersion relation of spin waves in a magnetic stripe. Our method works even if several spin wave branches coexist in the investigated frequency interval, provided that the branches possess sufficiently different group velocities. We first measure the microwave scattering matrix of a network composed of distant antennas inductively coupled to the spin wave bath of a magnetic stripe. The antenna-to-antenna coupling that is independent from the applied magnetic field are suppressed by an appropriate calibration to get the complex spin wave transmission coefficient versus frequency. A comparison with the expected contribution of when a single spin wave mode exists argues for the existence of several spin wave branches in the magnetic conduit (Fig. 1). After a mathematical transformation to the time-domain to get the transmission impulse response, the different spin wave branches are viewed as wavepackets that reach successively the receiving antenna after different travel times (Fig. 2) determined by their group velocities. In analogy with time-of flight spectroscopy, the wavepackets are then separated by time-gating. The time-gated responses are used to recalculate the contribution of each spin wave branch to the frequency domain scattering matrix. The dispersion relation of each branch stems from the absolute phase of the time-gated transmission parameter. The spin wave wavevector can be determined unambiguously if the results for several propagation distances are combined, so as to get the dispersion relations of each band, and in the end the full band structure of the spin waves [1].   Measuring the dispersion relations of spin wave bands using time-of-flight spectroscopy

We have investigated optically excited magnetoelastic waves by phase-resolved spin-wave tomography (PSWaT). PSWaT reconstructs the dispersion relation of spin waves together with their phase information by using time-resolved magneto-optical imaging for spin-wave propagation followed by an analysis based on the convolution theorem and a complex Fourier transform. In PSWaT spectra for a Bi-doped garnet film, we found a 180°-phase shift of magnetoelastic waves at around the crossing of the dispersion relations of spin and elastic waves. The result is explained by a coupling between spin waves and elastic waves through the magnetoelastic interaction. We also propose an efficient way for the phase manipulation of magnetoelastic waves by rotating the orientation of magnetization less than 10°.

We study spin excitations in thin magnetic films in the Heisenberg model with magnetic dipole and exchange interactions by the spin operator diagram technique and make comparison of their parameters with characteristics of spin waves in thick films. Dispersion relations of spin waves in thin magnetic films (in two-dimensional magnetic monolayer and bilayer lattices) and the spin-wave resonance spectrum in $N$-layer structures are found. For thick magnetic films, spin excitations are determined by simultaneous solution of the generalized Landau-Lifshitz equations and the equation for the magnetostatic potential. Generalized Landau-Lifshitz equations are derived from first principles and have the integral (pseudodifferential) form. It is found that dispersion relations of spin waves in monolayers and in bilayers differ from dispersion relations of spin waves in continuous thick magnetic films. For normal magnetized ferromagnetic films, the spin-wave damping is calculated in the one-loop approximation for a diagram expansion of the Green functions at low temperature. In thick magnetic films, the magnetic dipole interaction makes a major contribution to the relaxation of long-wavelength spin waves. Thin films have a region of the low relaxation of long-wavelength spin waves. In thin magnetic films, four-spin-wave processes take place and the exchange interaction makes a major contribution to the damping. It is found that the damping of spin waves propagating in a magnetic monolayer is proportional to the quadratic dependence on the temperature and is very low for spin waves with small wave vectors.

Dispersion relations of spin waves in ferromagnetic metals with multiple bands are obtained by the method of normal mode within the random phase approximation and a certain approximation for the Coulomb interaction. It is found that spin wave spectra consist of one acoustical branch, some optical branches and other branches due to inter-band transitions. The inter-band transitions have an important effect on the dispersion relation of an acoustical spin wave at larger momentum. The coefficient of the square of momentum in the dispersion relation of the acoustical spin wave is found to be the sum of D B and D H which are derived from the intra-atomic Coulomb and inter-atomic exchange interactions, respectively. For nickel, the value of D B is estimated as about 0.1 eVÅ 2 by using the model of two overlapping bands, and it is concluded that \(D_{B}{\lesssim}D_{H}\).

We investigate the magnonic band structure of in-plane magnetized two-dimensional magnonic crystals composed of cobalt dots embedded into a permalloy antidot lattice. Our analysis is based on the results of numerical calculations carried out by the plane wave method. The complex magnonic band structure found in square-lattice magnonic crystals is explained on the basis of the spin wave dispersion relations calculated in the empty lattice model. We show that four principal effects influence the formation of a magnonic band structure in planar two-dimensional bi-component magnonic crystals: a folding effect, Bragg scattering, hybridization between various spin wave modes, and a demagnetizing field. While the first two effects are found for other types of waves in periodic composites, the third one exists in an anisotropic medium and the last one is specific to spin waves propagating in magnonic crystals with magnetization in the film plane. The strong anisotropy in the dispersion relation of spin waves in thin ferromagnetic films results in the crossing and anti-crossing of the fast, Damon–Eshbach-like mode with a number of other spin waves folded to the first Brillouin zone. The demagnetizing field can induce the formation of channels for spin waves which are propagating perpendicular to the external magnetic field direction, but this property exists only in the limiting range of the thicknesses and the lattice constants of the bi-component magnonic crystals. Based on the model analysis we propose a modification of the magnonic crystal structure by changing its thickness, lattice constant and aspect ratio along the direction of the applied magnetic field to significantly modify the magnonic band structure and obtain partial magnonic band gaps.

To know the properties of a particle or a wave, one should measure how its energy changes with its momentum. The relation between them is called the dispersion relation, which encodes essential information of the kinetics. In a magnet, the wave motion of atomic spins serves as an elementary excitation, called a spin wave, and behaves like a fictitious particle. Although the dispersion relation of spin waves governs many of the magnetic properties, observation of their entire dispersion is one of the challenges today. Spin waves whose dispersion is dominated by magnetostatic interaction are called pure-magnetostatic waves, which are still missing despite of their practical importance. Here, we report observation of the band dispersion relation of pure-magnetostatic waves by developing a table-top all-optical spectroscopy named spin-wave tomography. The result unmasks characteristics of pure-magnetostatic waves. We also demonstrate time-resolved measurements, which reveal coherent energy transfer between spin waves and lattice vibrations.

We present a detailed analytical derivation of the spin wave (SW) dispersion relation in magnetic nanotubes with magnetization along the azimuthal direction. The obtained formula can be used to calculate the dispersion relation for any longitudinal and azimuthal mode. The obtained dispersion is asymmetric for all azimuthal modes traveling along the axial direction. As reported in our recent publication [Phys. Rev. Lett. 117, 227203 (2016)], the asymmetry is a curvature-induced effect originating from the dipole-dipole interaction. Here, we discuss the asymmetry of the dispersion for azimuthal modes by analyzing the SW asymmetry $\mathrm{\ensuremath{\Delta}}f({k}_{z})={f}_{n}({k}_{z})\ensuremath{-}{f}_{n}(\ensuremath{-}{k}_{z})$, where ${f}_{n}({k}_{z})$ is the eigenfrequency of a magnon with a longitudinal and azimuthal wave vectors, ${k}_{z}$ and $n$, respectively; and the dependence of the maximum asymmetry with the nanotube radius $R$. The analytical results are in perfect agreement with micromagnetic simulations. Furthermore, we show that the dispersion relation simplifies to the thin-film dispersion relation with in-plane magnetization when analyzing the three limiting cases: (i) ${k}_{z}=0$, (ii) ${k}_{z}\ensuremath{\gg}1/R$, and (iii) ${k}_{z}\ensuremath{\ll}1/R$. In the first case, for the zeroth-order modes the thin-film Kittel formula is obtained. For modes with higher order the dispersion relation for the Backward-Volume geometry is recovered. In the second case, for the zeroth-order mode the exchange dominated dispersion relation for SW in Damon-Esbach configuration is obtained. For the case ${k}_{z}\ensuremath{\ll}1/R$, we find that the dispersion relation can be reduced to a formula similar to the Kalinikos-Slavin [J. Phys. C: Sol. State Phys. 19, 7013 (1986)] type.

Nonreciprocal properties of the magnetostatic spin waves propagating in thin ferromagnetic films are potentially useful for design miniaturized microwave isolators and circulators, essential elements in microwave technology. Here, we study theoretically surface magnetostatic spin waves propagating perpendicularly to the external magnetic field in thin ferromagnetic films with periodic pattern, i.e., in one dimensional magnonic crystals. We identified influence of a periodic pattern on a surface localization of the spin wave amplitude. In particular we showed, that the surface character vanishes at wavenumbers related to the symmetry points of the reciprocal space, making distribution of the amplitude across the film thickness symmetric. Moreover, we show, that the asymmetric amplitude distribution can be enhanced by modification one of the surfaces or surroundings of the ferromagnetic film. This modification introduces also nonreciprocal dispersion relations for spin waves. Despite the nonreciprocity of the dispersion relation, the surface spin waves form magnonic band gaps in the magnonic spectra. However, the band gaps are shifted to higher frequencies and wave vectors far from the border of the Brillouin zone in the case of nonreciprocal dispersion. The potential applications of nonreciprocal properties of spin waves are discussed.

We present an experimental study of high-wave-vector spin waves in 8 monolayer (ML) thick hexagonal closed-packed (hcp) Co films performed by spin-polarized electron energy loss spectroscopy (SPEELS). Using inelastic electron scattering, we were able to follow the spin wave dispersion up to the surface Brillouin zone boundary $(\overline{K})$, i.e., up to a wave-vector transfer of $1.64\phantom{\rule{0.3em}{0ex}}{\mathrm{\AA{}}}^{\ensuremath{-}1}$. The spin wave dispersion was found to agree surprisingly well with the dispersion relation of a surface spin wave calculated by a nearest-neighbor Heisenberg model. From this description, we obtain a value for the product of the exchange coupling constant $(J)$ and the magnetic moment $(S)$ of $JS=14.8\ifmmode\pm\else\textpm\fi{}1\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$. This value, within the error bar, is identical to our results obtained on thin fcc Co films on Cu(001). We also find that the spin wave features measured by SPEELS at high-wave-vectors are strongly broadened. This is in agreement with expectations from nonadiabatic theoretical descriptions in which the broadening is ascribed to a strong damping of these high-wave-vectors spin waves by Stoner excitations. Similar to the observations in previously studied systems, we also observe a strong dependence of the measured spin wave intensities on the kinetic energy of the incident electrons $({E}_{\mathrm{kin}})$. Highest spin wave intensities were found for low kinetic energies $({E}_{\mathrm{kin}}<10\phantom{\rule{0.3em}{0ex}}\mathrm{eV})$.

We present a semi-classical theory, based on the torque equation of motion for the magnetization, to investigate the spin wave spectra in magnetic superlattices, whose constituents are alternating ferromagnetic and antiferromagnetic layers. We consider the presence of an external magnetic field applied in the plane of the layers and parallel to the easy axis of the structure, supposed to be in the z-direction. By using a transfer-matrix approach, to simplify the algebra which is otherwise quite complex, we explicitly obtain the analytical expression for the spin wave's dispersion relation. For numerical purpose, we confine our discussion to uniaxial antiferromagnetic fluorides (e.g. MnF2), while the ferromagnetic material is considered to be EuS. We illustrate our theoretical results numerically, and compare them with previous works done by using a microscopic Heisenberg model.