ISSN 0021-3640, JETP Letters, 2018, Vol. 108, No. 5, pp. 312–317. © Pleiades Publishing, Inc., 2018. Original Russian Text © A.V. Sadovnikov, A.A. Grachev, S.A. Odintsov, A.A. Martyshkin, V.A. Gubanov, S.E. Sheshukova, S.A. Nikitov, 2018, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2018, Vol. 108, No. 5, pp. 332–338.
CONDENSED MATTER
Neuromorphic Calculations Using Lateral Arrays of Magnetic Microstructures with Broken Translational Symmetry A. V. Sadovnikova, b, *, A. A. Gracheva, S. A. Odintsova, A. A. Martyshkina, V. A. Gubanova, S. E. Sheshukovaa, and S. A. Nikitova, b, c a Saratov
State University, Saratov, 410012 Russia Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Moscow, 125009 Russia c Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow region, 141700 Russia *e-mail:
[email protected]
b Kotel’nikov
Received July 13, 2018; in final form, August 10, 2018
A possibility to control the characteristics of spin waves in a lateral array of magnetic microstructures with broken translational symmetry is demonstrated. The regimes of spatial and frequency selection of the spinwave signal are studied by Brillouin light scattering and by numerical simulations. The micromagnetic simulation is used to study the effect of geometric parameters on the characteristics of dipole-coupled spin waves. The specific features of the coupling between the transverse modes propagating in the system with broken translational symmetry are revealed. The results can be applied to develop multiplexers, power dividers, couplers, and the ultrahigh frequency signal processing circuits using the neuromorphic principles, which are based on the lateral arrays of magnetic microstructures. DOI: 10.1134/S0021364018170113
The studies of spin waves (SWs) propagating in magnetic waveguide micro- and nanostructures attract considerable current interest [1–5]. The main attention is focused on the usage of SWs as information-carrying signals, since this allows implementing various signal processing devices based on the principles of magnonics [6, 7]. It is well known that the characteristics of SWs in magnetic materials are determined by the dipole and exchange interactions [8–10] and can vary within a wide range with change in the parameters of the material, for example, in the magnitude and direction of the bias field. At the same time, the patterning of thin magnetic films allows controlling the SW characteristics by employing both the geometrical effects arising at the waveguide-type propagation of SWs [11, 12] and the shape anisotropy of magnetic structures manifesting itself in change in local characteristics such as internal magnetic fields [1, 13, 14]. One of the methods allowing one to control the SW characteristics is the usage of translational symmetry breaking in waveguide-type magnetic arrays formed, e.g., by creating a bend (or turn) in a magnonic waveguide [1, 13, 14]. Such approach has provided an opportunity to develop a diversity of functional magnonic devices for the multiplexing and demultiplexing of signals in the frequency domain [15, 16], spatial and frequency separation of signals [17, 18], and linear and nonlinear switching in magnetic waveguide arrays
[19]. At the same time, one of the main current problems in the development of functional units based on spin-wave arrays is a demand of working up a design for interconnections supporting the efficient SW transmission in magnonic networks [20]. The latter are systems of connected micro- and nanoarrays with a complicated topology. Note that the formation of controllable links in magnonic networks with vertical and lateral topologies and nonlinear SW propagation modes ensure the possibility of implementing signal processing devices employing neuron-like (neuromorphic) principles [21]. Such devices are mainly applied in pattern recognition [22]. In [23–27], it is shown that dipole coupling effects for spin waves in lateral strips and multilayered structures lead to the periodic power transfer between strips/layers. In such case, the coupling induced by spin waves can be used to implement the regimes of spatial and frequency SW selection [18] and to create interconnections within magnonic networks [17, 26, 28]. It is important to take into account the intermode coupling for SWs propagating along parallel magnetic strips because the actually used methods for the excitation of spin-wave signals involve the multimode SW propagation regimes [12, 29–31]. In [17], it is shown that the energy of SWs propagating in the multimode propagation regime cannot be completely transferred from one microstrip to another. This is a drawback of such structures. Our
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Fig. 1. (Color online) (а) Layout of the structure under study. (b, c) Spatial distributions of the (b) intensity and (c) mx component of the ac magnetization at 5.114 GHz. (c, e) Spatial distribution of the BLS signal at (c) 5.091 and (e) 5.114 GHz. The insets of panels (d) and (e) demonstrate the integral intensity of the BLS signal I BLS ( y) in sections (circles) S1 and (squares) S2 versus the longitudinal coordinate y.
study demonstrates that the formation of nonidentical lateral arrays with broken translational symmetry provides a tool for overcoming such a problem, at least, partially. This paves a way for developing controllable magnonic devices with the option of spatial and frequency selection of SWs. Experimental and numerical studies reveal the mechanisms underlying the specific features of the intermode coupling of SWs and those responsible for the directed branching of spin-wave signals. We also analyze the effect of the geometrical parameters of lateral microwaveguides on the characteristics of SWs propagating within them and find the characteristics of intermode coupling of SWs in nonidentical lateral microwaveguide arrays. We compare the results of numerical simulations with the corresponding experimental data. In Fig. 1a, we illustrate the layout of the structure under study, which consists of two irregular arrays (G1 and G2 ). Array G1 can be treated as a structure with broken translational symmetry along the z axis. Within the rectangular region Rc bounded by the dashed line, the microstructures form an array of parallel microstrips of length Lc (the length of the coupling region). Such geometry is necessary to implement the spin-wave coupling regime [17]. As a material for microwaveguides, we use yttrium iron garnet [Y3Fe5O12 (111)] (YIG) [32] grown by liquid-phase epitaxy on a gadolinium gallium garnet [Gd3Ga5O12, JETP LETTERS
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(111)] (GGG) substrate. To determine the geometrical sizes of the array under study, we performed a micromagnetic simulation based on the numerical solution of the Landau–Lifshitz–Gilbert equations [33−35]:
∂M = γ[H × M] + α ⎡M × ∂M ⎤ , eff ∂t M 0 ⎣⎢ ∂t ⎦⎥
(1)
where M is the magnetization vector, α = 10−5 is the attenuation parameter, Heff = − δF is the effective δM magnetic field, F is the free energy of the ferromagnetic material, and γ = 2.8 MHz/Oe is the gyromagnetic ratio. To reduce the reflections of signals from the boundaries of the region under study in the numerical simulations, we introduce the series of regions with the attenuation coefficient α decreasing according to a geometric progression. These regions are located at the beginning of section S0 and at the end of sections S1 and S3 . The micromagnetic simulation method makes it possible to numerically solve the problem of the excitation and propagation of spin waves in an irregular magnetic microstructure [1, 36, 37]. Thus, by determining the excitation range and the parameters of the input signal in the waveguide S0 by the numerical solution of Eq. (1), it is possible to find the intensity distribution I ( y, z ) = mx2 + mz2 for the spin wave (Fig. 1b) and that for the x component mx of
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the magnetization (Fig. 1с). Using these distributions, one can find the coupling domain size Rc . In our experiment, we used the single-crystalline YIG film of the thickness t = 10 μm characterized by the saturation magnetization M 0 = 139 G and the width of the ferromagnetic resonance line ΔH = 0.54 G measured at a frequency of 9.7 GHz. The film was patterned by the precision laser cutting facility implementing the optical fiber laser SPLM MiniMarker2 -20А4 used for the formation of the array of magnetic structures. The parameters of the structure under study were chosen according to the micromagnetic simulation results: the widths of YIG waveguides within the Rc region were w1 = 500 μm and w2 = 250 μm and the distance between them was d = 40 μm. In the input segment S0 of array G1, we placed a 50-μm-wide microstrip antenna driven by microwave signals generated by an oscillator. The structure was subjected to the applied magnetic field H 0 = 0.12 T directed along the y axis providing the possibility for excitation of surface magnetostatic spin waves within the S0 region [8, 10]. The steady-state spatial distribution of the ac magnetization was studied by the Brillouin light scattering (BLS). This method is based on inelastic light scattering by coherently excited magnons [29]. The BLS signal intensity is proportional to the ac magnetization squared, I BLS( y, z) ∼ |mz2( y, z)| . In Figs. 1d and 1e, we demonstrate the spatial maps of the BLS intensity I BLS( y, z) at the input signal frequencies f1 = 5.091 GHz and f2 = 5.114 GHz, respectively. The input signal power equals P0 = −10 dB mW, which ensures the linear regime of SW propagation [38]. The experiment was performed using the pulse technique with a pulse duration of 400 ns and a pulse repetition period of 2 μs. The spatial scanning was performed within the 1.875 × 2.75 -mm region. We can see that the surface magnetostatic spin wave propagates along the section S0 and, then, it is transformed [39, 40] to the inverse bulk magnetostatic spin wave propagating along the section S1 and exciting SWs in the section S2 owing to the dipole coupling. At the frequency f1 , the complete energy transfer to S2 is not observed. At the frequency f2 , the second width mode [12, 30] of the inverse bulk magnetostatic SW with the transverse wavenumber kz = 2π/w1 propagates in the section S1 and the energy of the SW is transferred from the section S1 to the section S2 in the longitudinal coordinate range 0.75 < y < 1.75 . This is seen in the inset of Fig. 1e, where we show the dependence of the integral intensity I BLS( y) in each microwaveguide on the longitudinal coordinate y. The micromagnetic simulation allows us to perform the calculations for different widths w1 and w2 of sections S1 and S2, respectively (Fig. 2). Using the
Fig. 2. (Color online) Calculated spatial distributions of (a, c, e) spin-wave intensity I ( y, z) and (b, d, f) x component mx ( y, z) of ac magnetization at different parameters w1 and w2 and frequencies f (indicated in the figure).
characteristic distributions of the intensity (Figs. 2a, 2c, and 2e) and phase (Figs. 2b, 2d, and 2f) of the spin wave, we can find that the efficient coupling is achieved in the case of laterally arranged magnetic strips with the ratio of widths κ = w2 /w1 = n/m, where n, m = 1, 2, 3 … . To simplify our analysis, we assume further on that w1 = 500 μm and κ is varied by varying w2 . In this case, the phase velocities are close to each other and hence the SW width modes [23] in the microwaveguides S1 and S2 can be coupled. In the parallel magnetic microwaveguides with similar geometry and material parameters, the spin wave intensity distribution results from the interference of the symmetric and antisymmetric modes [17, 41]. In the case of nonidentical arrays, the eigenmode spectrum is transformed. In the spectrum of nonidentical lateral strips, it is important to take into account three eigenmodes Φ1, Φ 2 , and Φ3 with wavenumbers k1 , k2 , and k3 , respectively. The calculations of the electric field components Φ = E y (illustrated in the insets of Figs. 3a– 3c) and the dispersion curves (Figs. 3a–3c) for the Φ1, Φ 2 , and Φ3 modes were performed using the finite eleJETP LETTERS
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Fig. 3. (Color online) (а–c). Dispersion curves for spin waves propagating in lateral arrays consisting of microwaveguides with the thicknesses w1 and w2 (the values are indicated in the figure). The upper insets demonstrate the field distributions of the eigenmodes Φ1, Φ 2 , andΦ3 ; the lower insets illustrate the result of the interference of eigenmodes. (d) Coupling length versus the waveguide width ratio w2 /w1.
ment method [42] by finding the solution of the equation
∇ × (μˆ −1∇ × E) − k 2εE = 0,
(2)
where k = ω/c is the wavenumber in vacuum, ω = 2π/f is the angular frequency, f is the frequency of the electromagnetic wave, and ε = 14 is the effective permittivity of the YIG microwaveguide. In the case of H0 || y , the magnetic permeability tensor can be written as [43]
μ( f ) 0 ıμa ( f ) μˆ = 0 1 0 , −ıμa ( f ) 0 μ( f )
μ( f ) =
fH ( fH + fM ) − f 2 , fH2 − f 2
μa ( f ) =
fM f , − f2
fH2
where fM = γ4πM 0 , fH = γH int (z), and H int (z) is the internal magnetic field in the YIG microwaveguide. Note that the coupling of the spin-wave width modes of different orders can be implemented by changing the width of the microwaveguide S2. In particular, the spatial distribution of the magnetization corresponding to the excitation of one of the lateral YIG microstrips at κ = 1/2 or κ = 2/1 is the superposition of the Φ 2 and Φ3 modes (Figs. 3a and 3c) and at κ = 1/1 is the superposition of the Φ1 and Φ 2 modes JETP LETTERS
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(Fig. 3b). These superpositions coincide with the symmetric and antisymmetric modes of the lateral YIG arrays. In each case, the resulting interference of these modes is illustrated in the lower insets of Figs. 3a–3c. Figure 3d shows the calculated dependence of the characteristic coupling length L = π/|k3 − k2 | (the distance at which the SW power is transferred from one strip to another) on the parameter κ = 0.2, 0.3, 1.0, 2.0, 3.0, 4.0 (squares in Fig. 3d). In the same figure, we show the L values determined in the experiment at κ = 0.5 and κ = 1.0 (circles in Fig. 3d). We can see that the lowest L value is achieved at κ = 0.2 . Note here the unequal coupling lengths at κ = 1/2 and κ = 2/1. At first glance, this is quite unexpected, but they differ by more than a factor of 2 because of the increase both in the longitudinal wavenumber k y and in the difference between wavenumbers of spin-wave modes with the decrease in the waveguide width [29, 30].
The micromagnetic simulation of the dispersion laws demonstrates the existence of the Φ1 and Φ 2 modes in the spectrum of lateral strips. These modes correspond to symmetric and antisymmetric modes in the case of equal widths of the microstructures w1 = w2 = 500 μm (Fig. 4a). If the width of one strip decreases to w2 = 250 μm, the antisymmetric mode in the spectrum is split into two modes owing to the coupling between the second width mode in the strip S1
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Fig. 4. (Color online) (a, b) Dispersion curves for spin waves propagating in (a) identical and (b) different microwaveguides. (c, d) Frequency dependences of the coupling length L at the waveguide width ratios κ = 0.5, 1, 2 for the orientation of the applied magnetic field along the (c) y and (d) z axes.
and the first one in the strip S2 (Fig. 4b). In this case, the dispersion laws obtained by the micromagnetic simulation for Φ1,2,3 modes are in good agreement with those determined by the finite element calculations. Therefore, the frequency dependence of L can be calculated at the width ratios κ = 0.5, 1, 2 for the orientations of the applied magnetic field along the y (Fig. 4с) and z (Fig. 4d) axes. Such orientations correspond to the coupling between the inverse bulk magnetostatic spin waves and between the surface magnetostatic spin waves in the nonidentical lateral structures, respectively. It is important to note that the internal field H int (z) in the sections S1 and S2 changes when the applied magnetic field is directed along the z axis. This is the necessary condition for the excitation of surface magnetostatic spin waves in the section S1 at the excitation of the inverse bulk magnetostatic spin in the section S0. This effect manifests itself owing to the
overlap of dispersion curves for surface magnetostatic SW (in the section S0) and inverse bulk magnetostatic SW (in the section S1) [1, 14]. In Figs 4c and 4d, we can see that the minimum coupling length is achieved at κ = 0.5 . This suggests that the coupling length in nonidentical structures is smaller than that in identical ones ( κ = 1) within the whole frequency range under study. The latter relation is important for the further miniaturization of magnonic networks and functional elements operating on magnonic principles and promising for applications in information signal processing devices. The lateral arrays of magnetic microwaveguides can be used to develop devices with the operating principles involving non-Boolean and fuzzy logic. The specific feature of the latter devices is the signal coding through the use of both the amplitude and phase of spin waves [44–46]. In particular two amplitude levels in combination with two values of the SW phase at the output of the sections S1 and S3 can be used to represent an information signal unit. Thus, we obtain four different combinations each determining a logical value. The alternative combinations of the amplitude and phase can also be used. This is an important advantage of magnonic networks involving lateral microwaveguides useful in applications related to signal processing [46]. To summarize, using the Brillouin light scattering technique and numerical simulations, we have studied the propagation regimes for the coupled spin waves in systems based on nonidentical lateral magnetic microstructures. The possibility to control the intermode coupling of spin waves in the magnetic microstructures with broken translational symmetry has been demonstrated. It has been shown that the coupling length of spin waves can be reduced owing to the increase in the difference between wavenumbers corresponding to the spin-wave modes with the decrease in the width of microwaveguides. The results can be used to develop functional elements for processing of information signals, e.g., demultiplexers, power dividers, and signal couplers for the microwave range, based on the lateral arrays of nonidentical magnetic microwaveguides. Additional Brillouin light scattering studies in the course of revision of the manuscript were supported by the Russian Science Foundation, project no. 18-7900198. The development of the numerical model was supported by the Council of the President of the Russian Federation for Support of Young Scientists and Leading Scientific Schools, project no. MK3650.2018.9. S.A. Nikitov acknowledges the support of the Government of the Russian Federation, state contract no. 074-02-2018-286. REFERENCES 1. A. V. Sadovnikov, C. S. Davies, V. V. Kruglyak, D. V. Romanenko, S. V. Grishin, E. N. Beginin, JETP LETTERS
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