Nonlinear Spin Wave Effects in the System of Lateral Magnonic

ISSN 0021-3640, JETP Letters, 2018, Vol. 107, No. 1, pp. 25–29. © Pleiades Publishing, Inc., 2018. Original Russian Text © A.V. Sadovnikov, S.A. Odintsov, E.N. Beginin, A.A. Grachev, V.A. Gubanov, S.E. Sheshukova, Yu.P. Sharaevskii, S.A. Nikitov, 2018, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2018, Vol. 107, No. 1, pp. 29–34.

CONDENSED MATTER

Nonlinear Spin Wave Effects in the System of Lateral Magnonic Structures A. V. Sadovnikova, *, S. A. Odintsovb, E. N. Begininb, A. A. Grachevb, V. A. Gubanovb, S. E. Sheshukovab, Yu. P. Sharaevskiib, and S. A. Nikitova a

Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Moscow, 125009 Russia b Saratov State University, Saratov, 410012 Russia *e-mail: [email protected] Received November 7, 2017

The spin-wave dynamics in the system of lateral magnetic microstructures have been studied experimentally and numerically. Regimes of the propagation of coupled spin waves have been investigated by Brillouin spectroscopy measurement and numerical experiment. A phenomenological model has been proposed to describe the properties of spin waves in a lateral structure. It has been shown that the length of the coupling between spin waves can be governed by varying the power of the signal. The results can be used to fabricate magnetic waveguides of spin-wave demultiplexers, power splitters, and microwave couplers based on the lateral system. DOI: 10.1134/S0021364018010113

nonic switching, where the spatial scales of the intensity distribution of spin waves along lateral microwaveguides depend on the amplitude of the spin wave. The use of nonlinear effects in thin magnetic films [14, 15] resulted in the production of a new class of spin-wave devices [16–18] such as nonlinear phase converters and filters [19, 20], switching devices based on magnonic crystals [21], and devices for the generation of coherent spin waves owing to the interaction of the spin subsystem of yttrium iron garnet (YIG) with spin currents [22, 23]. In this work, the properties of nonlinear spin waves propagating in the system of two lateral microwaveguides with the dipole–dipole coupling are studied experimentally and numerically. We demonstrate the possibility of implementing nonlinear splitters based on such microwaveguides and determine necessary conditions for the appearance of the switching mode. We compare the numerical and experimental results for the characteristics of spin waves propagating in microwaveguides at various input microwave powers. Figure 1а shows the layout of the lateral system of waveguides W1 and W2 . The waveguides were made of a 10-μm-thick YIG [Y3Fe5O12 (111)] film with the saturation magnetization M 0 = 139 G and the width of the ferromagnetic resonance ΔH = 0.54 Oe measured at a frequency of 9.7 GHz. Such YIG microwaveguides with a width of w = 200 μm were placed at a distance of d = 40 μm on a 500-μm-thick gallium gadolinium garnet [Gd3Ga5O12, (111)] film. The microwaveguide W1 has a wedge shape on one of the ends,

Along with the development of information signal processing systems based on semiconductor technologies with charge transfer (electrons, holes), studies directed to the production of similar systems based on the transfer of magnetic moments or spins of electrons without charge transfer are actively developed and constitute the field of magnonics [1–3]. The use of spin waves in magnetically ordered materials and structures based on them allows the processing of information signals in spatially distributed systems with a high degree of time parallelism. In particular, spin-wave interference in these systems makes it possible to apply wave methods of data processing and Boolean and non-Boolean computational devices and to perform logically reversible and parallel computations [4]. The development of topologies of planar coupled magnetic micro- and nanostructures—magnonic networks—which play a key role when creating functional devices of a new generation, is a topical problem for the implementation of such systems [5, 6]. It is known that the modulation of the intensity of spin waves propagating along a system of lateral magnetic microwaveguides [5] is due to the effects of dipole coupling between spin waves [7, 8]. The use of lateral microstructures [9] allows fabricating interconnection elements in planar topologies of magnonic networks [10, 11]. It was shown that the spatial period of transfer of a spin-wave signal between parallel structures can be governed by varying the magnitude [5] and direction [12] of the external magnetic field and by using ferroelectric loads [13]. At the same time, an unsolved problem is the fabrication of all-mag25

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Heff = − δF is the effective magnetic field, and F is δM the free energy of the magnetic. In order to suppress the reflection of a signal from the boundaries of the calculation region ( x = 0 and x = 8 mm in Fig. 1b) in the calculation, we introduced regions (0 < x < 0.5 mm and 7.5 mm < x < 8 mm) where the coefficient α increases in a geometric progression. In the first stage of the numerical simulation, an input signal at the left boundary of the system at x = 0.5 mm was specified in the form bz (t ) = b0 tanh(t ) sin(2πft ) with the frequency f and the initial amplitude b0 = 0.01 mT increasing smoothly after switching on. The resulting distributions of the dynamic magnetization component mz are shown in the upper panels of Figs. 1b–1d for various signal frequencies. The lower panels show the distributions of

Fig. 1. (Color online) (а) Layout of the considered structure. (b–d) (Upper panels) Calculated spatial distribution mz ( x, y) of the component of the dynamic magnetization and (lower panels) the intensity of the spin wave I ( x ) for the frequency of the input signal (b) f1 = 5.19 GHz, (c) f2 = 5.25 GHz, and (d) f3 = 5.36 GHz.

which is connected to a feeding section with a 30-μmwide gold microstrip antenna (P0 in Fig. 1а). The length of the coupling region is Lc = 6 mm. The distance between YIG strips near the ports Z1 and Z 2 in Fig. 1а increased linearly to b = 300 μm. The structure was placed in an external magnetic field with the magnitude H 0 = 0.12 T directed along the y axis in order to excite a surface magnetostatic wave [24, 25]. To demonstrate the operating regimes of the structure, we performed the numerical simulation with the application of the Dorman–Prince method [26] to the solution of the Landau–Lifshitz–Gilbert equation [27–31]

the quantity I ( x) = mx2 + mz2 on the axis of each microwaveguide. It is seen that the intensity of the spin-wave signal is redistributed between the ports Z1 and Z 2 depending on the frequency in the input section of the structure: the signal can leave the waveguide W1 (Fig. 1с) or W2 (Fig. 1d) or the power of the signal is divided equally between the output sections of the waveguides (Fig. 1b). Further, the spectral power density of the output signal P1,2( f ) in the regions Z1 and Z2 shown in Fig. 1а and by the solid line in Figs. 1b–1d was calculated with the input signal in the form bz (t ) = b0sinc(2πfct ) with the frequency fc = 10 GHz. Then, the dynamic magnetization mz ( x, y, t ) near the output sections P1,2 was recorded with a step of Δt = 75 fs for the time T = 500 ns. As a result, the frequency dependences of the dynamic magnetizations P1( f ) and P2( f ) at the outputs of the first and second waveguides, respectively, were obtained by means of the Fourier transform (Fig. 2а). It is seen that the signal leaves the port Z 2 in the frequency range 5.23 GHz < f < 5.3 GHz marked in yellow in Fig. 2а, whereas the power of the spin wave beyond this frequency range is localized near the port Z1 . To plot the dispersion characteristic of spin waves at the numerical simulation, we calculated the quantity D(k x , f ) = 1 N

N

∑|Θ [m ( x, y , t)]| , 2

2

i =1

z

(2)

i

(1)

where Θ2 is the two-dimensional Fourier transform operator, yi is the ith cell, and N = 256 is the number of cells along the width of the YIG waveguide.

Here, M is the magnetization vector, γ = 28 GHz/T is the gyromagnetic ratio, α = 10−5 is the damping parameter phenomenologically introduced by Gilbert,

The two-dimensional distribution D(k x , f ) is shown in Fig. 2c by color grades. In the ( f , k x ) plane, it is possible to indicate a set of points corresponding to maxima of D(k x , f ), which represents the dispersion

∂M = γ[H × M] + α ⎡M × ∂M ⎤ . eff ∂t M 0 ⎣⎢ ∂t ⎦⎥

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characteristics of the symmetric and antisymmetric spin-wave modes marked in Fig. 2c as ks and kas , respectively. Using the dispersion characteristic, one can estimate the length L of the coupling between spin waves, which is the distance in which the power is completely transferred from W1 to W2 . The variation of the coupling length L( f ) = π/Δk( f ) = π/|ks ( f ) − kas ( f )| with the variation of the frequency is demonstrated in Figs. 1b–1d. Experimental data for ke ( f ) = ϕ( f )/h , where ϕ( f ) is the frequency dependence of the phase incursion measured by an Agilent E8362C PNA microwave network analyzer and h = 10 mm is the distance between the microstrip antennas located near the ports P0 and Z1 , are shown by circles in Fig. 2с. It is seen that the results of the numerical simulation are in good agreement with the experimental data; the measured wavenumber was ke ( f ) ≈ (ks ( f ) + kas ( f ))/2 . The spatial dynamics of spin waves in the fabricated system of lateral structures was studied by Brillouin spectroscopy based on the inelastic scattering of light on coherently excited magnons [32]. The intensity of the Brillouin spectroscopy signal is proportional to the square of the dynamic magnetization I BLS( x, y) ∼ |mz2( x, y)| . The frequency dependence of the Brillouin spectroscopy signal S1,2 in near the ports P0 and Z1 shown in Fig. 2b qualitatively corresponds to the numerical simulation results (Fig. 2a). To reveal nonlinear propagation modes of spin waves in lateral structures, we obtained spatial intensity distribution maps I BLS( x, y) for the frequency of the input signal f0 = 5.21 GHz and powers of the input signal P0 = (Fig. 3а) –5 and (Fig. 3b) 26 dBm. Scanning was performed in the 5 × 0.5 mm region. It is seen that the Brillouin spectroscopy signal at the output of the microwaveguide W2 increases with the power of the input signal. To quantitatively estimate the power distribution between the output sections of both microwaveguides, we plotted the input-power dependence of the coefficient T0 = 10 log(S2 /S1) (green squares in Fig. 4a), according to which the power is divided equally between the output sections W1 and W2 (i.e., T0 = 0 dB) at P0 = 20 dBm. To describe nonlinear propagation modes and coupling between spin waves in lateral structures, we used a phenomenological model based on the coupled Ginzburg–Landau equations, which are used, e.g., in optics when studying the wave dynamics of dissipative space–time solitons in media with nonlinear damping [33, 34]. An explicit form of two coupled Ginzburg– Landau equations [35] can be obtained from the Landau–Lifshitz–Gilbert equation for the dynamic magnetization taking into account Kerr nonlinearity, i.e., a decrease in the saturation magnetization at an increase in the angle of deviation of the magnetization vector from the equilibrium state: M ≈ JETP LETTERS

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Fig. 2. (Color online) (а) Frequency spectrum of the spinwave signal in the output section of the microwaveguides W1 and W2 . (b) Frequency dependence of the Brillouin spectroscopy signal for the microwaveguides W1 and W2 . (c) Calculated dispersion characteristics of spin waves for the (upper dash-dotted line) symmetric and (lower dashdotted line) antisymmetric modes in comparison with the experimental data marked by circles.

M 0(1 − (mx2 + mz2 /(2M 02 )) = M 0(1 − Φ 2 /2),

where

The input power of the spin Φ= + wave can be estimated as P0 ≈ |Φ 0 |2M 02v g wt , where v g is the group velocity and Φ 0 is the initial amplitude of magnetostatic surface waves, for the numerical integration of the system of two coupled Ginzburg–Landau equations:

(mx2

mz2 )/M 02 .

⎧i d Φ1 = k Φ + χΦ + (ζ − iν )|Φ |2Φ − iν Φ , 1 2 2 1 1 1 1 ⎪ dx (3) ⎨ ⎪i d Φ 2 = k Φ 2 + χΦ1 + (ζ − i ν2 )|Φ 2 |2Φ1 − iν1Φ 2. ⎩ dx Here, Φ1( x) and Φ 2( x) are the amplitudes of the spin wave in the waveguides W1 and W2, respectively; k = k( f ) is the wavenumber of the spin wave propagating in a single microwaveguide; χ = χ( f ) ≈ |ks ( f ) − kas ( f )| is the coupling constant of spin waves;

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20 log Fig. 4. (Color online) (а) Coefficient T versus (lower scale) the amplitude of the input signal Φ 0 and (upper scale) the power of the input microwave signal P0 and the nonlinear phase incursion Δφ(Φ 0 ) versus the amplitude of the input signal. (b) Intensity of the spin wave in the output sections of the microwaveguides W1 and W2 at various amplitudes of the input signal.

Fig. 3. (Color online) Spatial distribution of the Brillouin spectroscopy signal I BLS ( x, y) at the input powers P0 = (а) –5 and (b) 26 dBm. (c–f) Calculated intensity of the spin wave at the input signal amplitudes Φ 0 = (c) 0.02, (d) 0.45, (e) 0.5, and (f) 0.53.

(2 fH + fM ) exp(2kt ) − fM is the nonlinear ζ = ∂k2 fM t ∂Φ Φ=0 coefficient, where fH = γH 0 and fM = 4πγM 0 ; and 2 are ν1 = 1 ∂ω ΔH and ν2 = − 12 ζ ΔH ∂ω ∂ ω2 v g ∂H 2 v g 2 ∂H ∂k Φ=0 the linear and nonlinear damping of the spin wave, respectively, where ω = 2πf is the circular frequency. To calculate nonlinear propagation modes of the spin-wave signal in the case of the excitation of the waveguide W1 at the frequency f0 , we used the nonlinear parameter ζ = 1.77 × 105 m–1 and the linear and nonlinear damping parameters ν1 = 13.03 m–1 and ν2 = 50.09 m–1. The results of calculation at an increase in the amplitude Φ 0 of the input signal demonstrate that the intensity of the spin wave in the output section of the waveguide W2 can increase because of an increase in the length of coupling between spin waves (Figs. 3d–3e). A further increase in the amplitude results in the nonlinear switching effect at which the signal intensity is concentrated in the output section of the waveguide W1 (Fig. 3f). A

similar situation appears in other physical systems, e.g., in the problem of the coupling between waves propagating in optical splitters [36]. To reveal propagation modes of the spin-wave signal at parameters corresponding to experimental data, we plotted a color map (see Fig. 4b) of the wave intensity near the output sections Z1,2 (at x = 6.4 mm in Fig. 3f). It is seen that there are two characteristic amplitudes of spin waves. The first amplitude is Ath1 at which T = 0 dB and the signal power is divided equally between W1 and W2 near the output section (Fig. 3d). The second amplitude is Φ th2 at which the maximum possible intensity of spin waves occurs in the output section of the waveguide W2 (Fig. 3e). Figure 4a shows the numerically calculated amplitude dependences of the (solid line) coefficient T and (dashed line) nonlinear phase incursion Δφ(Φ 0 ) [35] at the output of the waveguide W1. The results of the numerical simulation qualitatively coincide with the experimental data. A further increase in the power of the input signal P0 leads to a significant thermal heating of the YIG film in the region of the input microstrip antenna, which determines the range of the initial amplitude of the signal in the numerical simulation. Agreement of the results of the numerical simulation with the experimental data in a certain power range of the input microwave signal can be a criterion of the correctness of the presented model. To summarize, propagation modes of coupled spin waves in the system of lateral magnetic microwaveguides have been experimentally studied. Using the developed numerical model, we have revealed mechaJETP LETTERS

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nisms responsible for the characteristic propagation modes of spin waves at different intensities and frequencies. The results of the numerical simulation are in good agreement with the experimental data. It has been shown that the frequency-selective properties of the proposed structure can be controlled by the input power, which can be used to produce spin-wave demultiplexers, power splitters, and microwave couplers based on the lateral system of magnetic waveguides. This work was supported by the Russian Science Foundation (project no. 16-19-10283) and by the Council of the President of the Russian Federation for Support of Young Scientists and Leading Scientific Schools (project nos. SP-313.2015.5 and MK5837.2016.9). REFERENCES 1. S. A. Nikitov, Ph. Tailhades, and C. S. Tsai, J. Magn. Magn. Mater 236, 320 (2001). 2. V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. 43, 264001 (2010). 3. International Technology Roadmap for Semiconductors. http://www.itrs2.net/itrs-reports.html. Accessed April 1, 2017. 4. D. Sander, S. O. Valenzuela, D. Makarov, C. H. Marrows, E. E. Fullerton, P. Fischer, J. McCord, P. Vavassori, S. Mangin, P. Pirro, B. Hillebrands, A. D. Kent, T. Jungwirth, O. Gutfleisch, C. G. Kim, and A. Berger, J. Phys. D: Appl. Phys. 50, 363001 (2017). 5. A. V. Sadovnikov, E. N. Beginin, S. E. Sheshukova, D. V. Romanenko, Y. P. Sharaevsky, and S. A. Nikitov, Appl. Phys. Lett. 107, 202405 (2015). 6. Yu. P. Sharaevsky, A. V. Sadovnikov, E. N. Beginin, M. A. Morozova, S. E. Sheshukova, A. Yu. Sharaevskaya, S. V. Grishin, D. V. Romanenko, and S. A. Nikitov, in Spin Wave Confinement: Propagating Waves, Ed. by S. Demokritov, 2nd ed. (Pan Stanford, Singapore, 2017), Chap. 2. 7. G. Gubbiotti, S. Tacchi, G. Carlotti, P. Vavassori, N. Singh, S. Goolaup, A. O. Adeyeye, A. Stashkevich, and M. Kostylev, Phys. Rev. B 72, 224413 (2005). 8. M. P. Kostylev, G. Gubbiotti, J.-G. Hu, G. Carlotti, T. Ono, and R. L. Stamps, Phys. Rev. B 76, 054422 (2007). 9. A. V. Sadovnikov, S. A. Odintsov, E. N. Beginin, S. E. Sheshukova, Y. P. Sharaevskii, and S. A. Nikitov, IEEE Trans. Magn. 53, 2801804 (2017). 10. C. S. Davies, A. Francies, A. V. Sadovnikov, S. V. Chertopalov, M. T. Bryan, S. V. Grishin, D. A. Allwood, Y. P. Sharaevskii, S. A. Nikitov, and V. V. Kruglyak, Phys. Rev. 92, 020408 (2015). 11. A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015). 12. A. V. Sadovnikov, A. A. Grachev, S. A. Odintsov, S. E. Sheshukova, Y. P. Sharaevskii, and S. A. Nikitov, IEEE Magn. Lett. PP (2017). 13. A. V. Sadovnikov, A. A. Grachev, E. N. Beginin, S. E. Sheshukova, Y. P. Sharaevskii, and S. A. Nikitov, Phys. Rev. Appl. 7, 014013 (2017). JETP LETTERS

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Translated by R. Tyapaev