Inelastic Spin-Wave Beam Scattering by Edge-Localized Spin Waves in a Ferromagnetic Thin Film

Spin waves are promising chargeless information carriers for the future, energetically efficient beyond CMOS systems. Among many advantages are the ease of achieving nonlinearity, the variety of possible interactions, and excitation types. Although the rapidly developing magnonic research has already yielded impressive realizations, multimode nonlinear effects, particularly with propagating waves and their nanoscale realizations, are still an open research problem. We theoretically study the dynamic interactions of spin waves confined to the edge of a thin ferromagnetic film with the spin-wave beam incident at this edge. We find inelastically scattered spin-wave beams at frequencies increased and decreased by the frequency of the edge spin-wave relative to the specularly reflected beam. We observe a strong dependence of the angular shift of the inelastic scattered spin-wave beam on the edge-mode frequency, which allows us to propose a magnonic demultiplexing of the signal encoded in spin waves propagating along the edge. Since dynamic magnetostatic interactions, which are ubiquitous in the spin-wave dynamics, are decisive in this process, this indicates the possibility of implementing the presented effects in other configurations and their use in magnonic systems.

Figure 1

(a) Schema of the system under investigation. The red (at the edge), yellow, and blue (beams) colors correspond to SWs at frequencies $\nu$ (ESW), $f$ (incident and reflected bulk SWs), and $f\pm \nu$ (inelastically scattered SWs), respectively. The white color denotes areas where at least two frequencies are mixed. The black arrows represent the group velocities related to the incident and scattered beams, whereas the dashed red arrow represents the wavevector of the incident wave. The white arrow represents the edge SW. (b) Simulated dispersion relation (colormap in the background, where the color intensity corresponds to the averaged-in-space power related to the $x$ component of the wavevector) with a well visible edge-SW band at frequencies downshifted with respect to the continuum of the bulk-SW band (highlighted area). The dispersion of the bulk-SW mode with the wavevector perpendicular to the static magnetization, i.e., the Damon-Eshbach mode, is marked by the green dash-dot line.

Figure 2

(a) The spectrum of SWs with the most prominent frequency peaks corresponding to the edge SWs ($\nu$), second harmonic generated SWs ($2\nu =24\mathrm{GHz}$), SW beam of frequency $f_0=50\mathrm{GHz}$, and the inelastically scattered SW beams $f_0\pm \nu$ at frequencies 38 and 62 GHz. (b) The intensity map of the incident and reflected SW beams at frequency 50 GHz. (c),(d) Intensity maps of SWs at frequencies 38 and 62 GHz, respectively. The angles of propagation of the scattered beams in (c) and (d) differ slightly with respect to each other and with respect to the primary reflected beam presented in (b). The locations of antennas exciting both the SW beam and edge SWs are schematically shown in Fig. 1.

Figure 3

The momentum-space representation of the ISSW. The grayscale color map in the background represents the profile of SWs in $k$ space obtained from simulations. The sharp black regions that are marked by the colored arrows represent the wavevectors of the SW beams. In (a)–(c) we show results obtained for the case from Fig. 2, where the source of the edge SW is placed on the left side of the incident beam spot, whereas in (d)–(f) there are two sources of edge SWs located on both sides of the incident beam spot. In (a) and (c) we show results for $f_0-\nu =38\mathrm{GHz}$, and in (b) and (e) for 50 GHz—specular reflection—whereas in (c) and (f) we show results for $f_0+\nu =62\mathrm{GHz}$. The solid ellipses represent isofrequency contours for corresponding frequencies resulting from the analytical dispersion relation, Eq. (2). The thin-orange arrows correspond to roup velocities obtained analytically for a given wavevector (their length notes the magnitude of ${\mathbf{v}}_g$). The angle $\phi ={\tan }^{-1}(k_x/k_y)$ in (a)–(c) denotes the angle of the phase velocity with respect to the normal to the film edge.

Figure 4

(a) The result of micromagnetic simulations showing the scattering of a 50 GHz bulk-SW beam (yellow color) incident at an angle of ${30}^{\circ }$ on a 12 GHz edge SW (ESW) with $\kappa <0$ (red color). Two scattered beams at 38 GHz (green) and 62 GHz (blue) propagate at significantly different angles to each other, and to the reflected beam. Areas where excitations of the bulk-SW beam and edge SWs take place are schematically marked with gray lines. (b) SW beams at frequencies 39.2 GHz (red color), 38.6 GHz (green color), 38 GHz (blue color), and 37.4 GHz (yellow color) excited at the edge of a ferromagnetic film as a result of the SW beam (at 50 GHz) scattering at edge SWs of frequencies 10.8, 11.4, 12.0, and 12.6 GHz, respectively.

Figure 5

The amplitudes of the $m_z$ component of magnetization of the inelastically scattered SWs at frequencies $f_0\pm \nu$ for different amplitudes of $m_z$ of the edge SW (a) and the bulk-SW beam (b). The results of simulations presented in (a) are obtained for the bulk-SW beam of amplitude $0.93\times {10}^{-3}$, whereas the results presented in (b) are for the edge SW (ESW) of amplitude $6.72\times {10}^{-3}$.

Figure 6

(a)–(c) Dependencies of the $|m_z|$ component of dynamic magnetization detected at $y=100\mathrm{nm}$ distance from the film edge for the reflection and inelastically scattered SWs at 50, 38, and 62 GHz, respectively. Different colours represent results at different angles of incidence of the SW beam (50 GHz) falling at the edge, where edge SWs at frequency 12 GHz propagate along the $+x$ direction ($\kappa >0$). The amplitudes are normalized to the amplitude of the elastically reflected beams at a given angle, i.e., $m_{\mathrm{scattered}}(y=100\mathrm{nm})/m_{\mathrm{reflected}}(y=100\mathrm{nm})$, where $m_{\mathrm{scattered}}(y=100\mathrm{nm})$ and $m_{\mathrm{reflected}}(y=100\mathrm{nm})$ are amplitudes of the scattered and reflected waves at $y=100$. (d) Comparison of the results of micromagnetic simulations (the empty black and red squares) with the analytical model (dash-dot black line) for inelastically scattered waves is done using the function $|m_z|=C\sin (2\phi )\chi [(k_y-K_y)\mathrm{\Delta }]$ with $\mathrm{\Delta }=L=10\mathrm{nm}$ and $C=1.4\times {10}^{-3}$.

Figure 7

The dependencies of $|m_z|$ at $y=100\mathrm{nm}$ distance from the film edge at the specular reflection [(a) 30 GHz] and inelastic scattering SW [(b) 42 GHz] propagating outward from the edge. Different colours represent results at different angles of the SW beam incidence at the edge where the edge SW at frequency 12 GHz propagates. Similarly as in Fig. 6, the amplitudes are normalized to the amplitude of the elastically reflected beams at a given angle. (c) Comparison of the results of micromagnetic simulations (the empty black squares) with the analytical model (dash-dot black line) for inelastically scattered waves using the equation $|m_z|=C\sin (2\phi )\chi [(k_y-K_y)\mathrm{\Delta }]$ with $\mathrm{\Delta }=L=10\mathrm{nm}$ and $C=3.2\times {10}^{-3}$.