Spin-wave Switching in the Side-Coupled Magnonic ... - IEEE Xplore

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMAG.2017.2709540, IEEE Transactions on Magnetics BQ-06 1

Spin-wave Switching in the Side-Coupled Magnonic Stripes Alexandr V. Sadovnikov1,2, Member, IEEE, Sergey A. Odintsov1, Evgeniy N. Beginin1, Member, IEEE,

Svetlana E. Sheshukova1, Member, IEEE, Yurii P. Sharaevskii1, Sergey A. Nikitov1,2, Member, IEEE Laboratory “Metamaterials”, Saratov State University, Saratov, 410012, Russia Kotel’nikov Institute of Radio Engineering and Electronics of Russian Academy of Science, Moscow, 125009, Russia 1

2

The study of the spin waves coupling in the magnonic coupler, formed by the laterally coupled magnonic stripes, is performed by the means of finite-element and micromagnetic simulation. The efficiency of the spin-wave coupling is improved by the geometry design and choose of the appropriate magnetization direction inside each magnonic stripe. The proposed side-coupled magnetic stripes enable the improvement of performance of the advanced integrated magnonic devices and offer a range of further opportunities in planar magnonics. Index Terms—magnonic coupler, spin waves, magnetic stripe.

I. INTRODUCTION The extensive development of the fabrication technique of magnonic micro and nanosized elements over the last decade leads to the fabrication of functional devices for planar magnonic applications [1-7], which are based on the unique features of the magnetic materials. The new paradigm of the signal processing, based on propagating spin waves, lies in the basis of the beyond CMOS computing architecture [3-5]. It was shown, that the directional coupler for magnonic application can be fabricated using the side-coupling magnetic stripes and/or magnonic crystals [8-10]. The adjacent magnetic stripes can be proposed as an alternative approach to the frequency and wavenumber filtering techniques as the frequency filtering elements in the magnonic networks [6, 10]. The coupling of magnetostatic surface waves [11, 12] propagating in a “sandwich” geometry where two magnetic films are located in parallel, leads to the spin-wave power exchange between the films [13-15]. In the Refs.[16,17], the planar topology is addressed as a convenient way to fabricate the magnonic directional coupler due to the tunable frequency filtering and frequency demultiplexing characteristics of the coupled waveguiding structures. Nevertheless, the control of spin waves propagation and coupling in adjacent magnonic waveguides was not addressed before. It is known, that three types of magnetostatic spin waves in thin magnetic films is typically considered [11,12] – forward volume magnetostatic waves (FVMSV), backward volume magnetostatic waves (BVMSW) and surface magnetostatic waves (MSSW). The question about the spin-wave coupling in adjacen magnonic stripes in each of the geometry is still open. Moreover, the recent works on spin-wave multiplexing [18,19] demonstrated the splitting of spin waves in the Y-shaped magnonic waveguide. It is important to optimize the geometry design of the side-coupled stripes in order to provide the enchanced functionality of the Y-shaped beam splitter or multiplexer [19]. Therefore, the problem of the optimization of both the geometry and static magnetic properties is of great importance nowadays. In this letter, we report on the study of the spin waves coupling in the laterally coupled magnonic stripes, which are formed as an adjacent spin-wave waveguides with the gap between them. The dispersion of eigenmodes and transverse

modes profiles were calculated with the finite element method (FEM). Transmission and coupling of spin wave modes were numerically studied using micromagnetic simulations to show the functionality of the operational regimes of the proposed magnonic coupler. The geometry design optimization reveals the parameters of the coupler (magnonic stripe’s width and gap between the stripes), which are enable to effective spinwave coupling. The different directions of magnetic field orientation are considered. The proposed side-coupled magnetic stripes enable the improvement of performance of the advanced integrated magnonic devices and offer a range of further opportunities in planar magnonics. II. METHODOLOGY We use the modification of the finite-element method (FEM) [20,21] to model the electrodynamic problem of electromagnetic wave propagation and coupling in the adjacent magnetic stripes. We solve the full system of Maxwell's equations in the 2D geometry [20]. Next in this letter we consider the magnetic stripes with the geometry defined in Ref.[17], which is shown to be the appropriate candidate as a directional two channel magnonic coupler. The investigated structure consists of two side coupled magnonic stripes S1 and S2 of ferrimagnetic yttrium iron garnet (Y3Fe5O12, YIG) film with saturation magnetization of M0 = 1750/(4π) G, located on a tg = 100 μm-thick gadolinium gallium garnet (Gd3Ga5O12) substrate. Figure 1(a) shows the schematic of the computational area of the structure. The width of the waveguide w is varied from 100 μm to 600 μm. The efficient coupling between spin waves in the adjacent magnonic stripes can be achieved only in the case, if the propagating spin waves in each stripe have almost equal propagation constants. Thus we consider only the case of the stripes of the equal width. The gap d between the stripes is varied from 10 μm to 60 μm. We use the boundary condition in the form of the perfect electric conductor (PEC) in the bottom and top edges and perfect magnetic conductor (PMC) in the left and right edges of the computational domain [20]. The distance between the top of YIG stripes and the top edge of computational area was l1 = 145 μm. l2 = 45 μm, thickness of waveguides ty = 10 μm. The distance between the magnetic stripe and PMC is q = w, thus q is varied with the stripes’ width. The structure is infinite in the direction of the

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z-axes. The computational domain is covered with the triangular mesh with the refinement of the element size in the area of YIG stripes (see Fig.1(b)).

in Fig.1(c),(d)) and an anti-symmetric (solid curves in Fig.1(c),(d)) modes. The former corresponds to the case when the amplitudes of the magnetic potentials in both magnetic stripes have the same phase, and in the anti-symmetric mode they are out of phase. We demonstrate also the mode profile of the individual stripe with the dashed curves in Fig. 1(c),(d). It is clearly seen that the overlap integral of mode profiles is decreased as the frequency is increased and reaches almost zero value at the wavenumber of 475 cm-1. We show the possibility of the tuning of the electrodynamic characteristics with the variation of the design of the proposed coupled magnonic stripes. The main result of FEM simulation is the dispersion of symmetric and anti-symmetric modes of electromagnetic waves, which are propagated in the adjacent stripes. Moreover, the guided power may be interchanged between the side-coupled magnetic stripes in periodic manner with the period equal to the coupling length: L   k s  k as , where k s and k as are the wavenumbers of the symmetrical and anti-symmetrical modes, respectively. Figure 2(a) shows the dispersion characteristics of the symmetrical and antisymmetrical modes for the parameters of w = 200 μm and d = 20 μm. It should be noted that the cut-off frequencies of anti-symmetric mode f coas is lower than that for the symmetric modes f cos . Moreover, the value of f cos is almost equal to the frequency of ferromagnetic resonance f p of the in-plane magnetized film [12]. This is due to the fact, that at the wavenumbers k  1 tY the dispersion of the symmetric and anti-symmetric modes almost coincide with the dispersion of the first and second transverse modes [24,25] of YIG stripe of width w2  2 w  400 μm, respectively. If the distance

Fig. 1. (a) The schematic sketch of the computational domain and the geometry design of the magnonic coupler. (b) The results of the mesh generation for the FEM simulation. (c) and (d) shows the distribution of Ex component at different frequencies for anti-symmetric modes (solid blue curves), symmetric modes (black dash-dotted curves) and eigenmodes of individual magnetic stripe (red dashed line) at H0=600 Oe. Vertical dotted lines show the left and right boundaries of adjacent waveguides.

III. RESULTS AND DISCUSSION First, we consider the MSSW geometry[22] in the case, when the uniform static magnetic field H 0  600 Oe is directed along positive x-direction. In this case, the permeability tensor of YIG is written in the form: 0  1 0  ˆ  0  i a  , 0 i a   where

 f  

fH  fH  fM   f 2 f H2

f

2

, a  f  

fM f f H2

 f2

,

f H   H 0 , f M  4 M 0 and γ = 2.8 MHz/Oe is the gyromagnetic ratio for YIG. The relative permittivity of YIG is   14 . Figures 1 (c) and (d) show the distribution of Ex component of electromagnetic field across the x-direction at the frequencies of 3.4 GHz and 3.8 GHz, respectively. These profiles correspond to the center of the each YIG film. It is known [23, 10], that the spectrum of eigenmodes of two magnonic stripes consists of a symmetric (dash-dotted curves

between the stripes d is increased the values of f cos and f coas are decreased. Frequency dependence of the coupling length for different values of parameter d is depicted in Fig. 2(b). Thus, the value of L is monotonically increased with the increase of frequency. If the stripe’s width is increased, the well-pronounced minima in L  f  at the frequency of f = 3.33 GHz is observed (see Fig. 2(c)). It is worth to note, that the variation of the width w has the more impact on the value of L, rather than the variation of d (see Fig.2(d)). In order to demonstrate the functionality of the proposed magnonic coupler, we present the results of the micromagnetic simulation [26] of spin-wave coupling in the adjacent stripes with the parameters: w = 200 μm and d = 40 μm. We take into account, that the effective magnetic field and magnetization configurations are not uniform inside each magnetic stripe (see Fig.1 in Ref.[17]). Moreover, the lateral confinement of coupled waveguides leads not only to the reduction of internal magnetic field but also to the asymmetrical profile of the internal field distribution inside S1 and S2with respect to the center of each stripe. The spin waves were excited in the vicinity of the left edge of S1 by a local magnetic field of harmonic temporal profile at frequency of f=3.4 GHz (see Fig.2(e)) and f=3.5 GHz (see Fig.2(f)). The periodic power exchange between the stripes is observed. It should be noted, that the results of coupling length calculation are in a good



agreement with FEM simulation (Fig.3(b)): L=1.1 mm at f=3.4 GHz and L=1.8 mm at f=3.4 GHz.

Fig. 2. The calculation results of characteristics of MSSW in the adjacent magnonic stripes. (a) Dispersion characteristics of symmetric (blue curve) and anti-symmetric (red curve) mode at w = 200 μm and d = 20 μm; (b) frequency dependence of coupling length at various gap between the stripes; (c) frequency dependence of coupling length at various width of each stripe; (d) coupling length as a function of stripes’ width (red curve) and distance between the stripes (blue curve) at f = 3.33 GHz. Results of the micromagnetic simulation of spin-wave coupling in the adjacent stripes with the parameters: w = 200 μm and d = 40 μm at frequency of f = 3.4 GHz (e) and f = 3.5 GHz (f). The dynamic magnetization component my is depicted. All shown data are presented for H0=600 Oe.

Second, we consider the case of the “anti-parallel” magnetization of the both magnetic stripes. This can be achieved in experiment by e.g. the current bus on the top of each magnetic stripe. The current direction in each bus should be anti-parallel. This situation corresponds to the opposite direction of the internal static magnetic field inside each stripe. We suppose, that H 0 || x in the stripe S1 and H 0 ||- x in the stripe S2. This leads to the intersection of the dispersion curves for the symmetric k0s and anti-symmetric k0as modes at the frequency of f i  3.275 GHz (see Fig. 3(a)). This results in the well-marked increase of the L in the vicinity of f i . Figure 3(b) shows that the increase of stripe’s width w leads to the shift of f i to the lower frequencies. Figure 3(a) demonstrates also that, when the wavenumber is increased the dispersion of the modes in the “anti-parallel” case coincides with the “parallel” case. It is worth to note, that we obtain the same results, if we consider the case, when the magnetic field is directed along the positive x-direction in S2 and along the negative x-direction in S1.

Fig. 3. (a) The comparison of the dispersion characteristics of MSSW in the adjacent stripes at w = 200 μm and d = 40 μm in the case of “parallel” and “anti-parallel” magnetization of each stripe. (b) The frequency dependence of the coupling length at various width of magnetic stripes and fixed distance between them d = 40 μm.

Next, we consider the case of BVMSW and FVMSW geometry for the parameters w = 200 μm and d = 20 μm. The dispersion of symmetric and anti-symmetric modes for BVMSW and FVMSW is shown in Fig. 4(a) and (b), respectively. For the case of BVMSV, the permeability tensor of YIG is written in the form of [11,12]:

i a 0     ˆ   i a  0  ,  0 0 1  whereas, for case of FVMSV, the permeability tensor of YIG is written in the form of [11,12]:   ˆ   0  i a

0 i a  1 0  . 0  

Figures 4 (c) and (d) show the frequency dependence of coupling length for both cases. It is seen, that the coupling length is decreased with the frequency increase and is approximately two times larger than that for the MSSW geometry in the same wavenumber range. The using the FVMSW to fabrication of spin-wave coupler with d=40 m is not appropriate due to the well-pronounced growth of the value of L even at the smallest wavenumbers (see Fig. 4(e)).



ACKNOWLEDGMENT Micromagnetic numerical simulation is supported by the Grant from Russian Science Foundation (No. 16-19-10283). Finite element numerical calculations and anti-parallel paradigm of magnetization in adjacent channels were supported by the Grant from Russian Science Foundation (No. 14-19-00760). The research leading to these results has received funding from the Russian Foundation for Basic Research (Project Nos. 16-37-00217, 16-02-00789) and the Grant and Scholarship of the President of the Russian Federation (No. МК-5837.2016.9, SP-313.2015.5). REFERENCES

Fig. 4. Dispersion of symmetric (blue curve) and anti-symmetric (red curve) modes of BVMSW(a) and FVMSW(b) in the side-coupled magnonic stripes with the parameters w = 200 μm and d = 20 μm. Frequency dependence of the coupling length at various gap between the stripes for BVMSW(c) and FVMSW(d) at w = 200 μm. Coupling length as a function of d for all three types of dipolar waves (e).

Therefore, the minimum appropriate value of the gap between the stripes should be d=20 m. It should be noted also, that for the case of FVMSW we use the value of internal field H int  H 0  600 Oe, whereas in experiment the value of H 0   4 M 0  600  Oe should be used [12]. It is worth to

note also, that the MSSW configuration is typically used in the dual-tunable magnonic coupler [27]. Thus the functionality of this dual-tunable coupler can be improved by employing the “anti-parallel” magnetization in the MSSW geometry, which leads to the intensive increase of the coupling length at the frequencies near the frequency of the ferromagnetic resonance. Exploration of this challenges is however beyond the scope of this paper. It should be noted that during the publication of our manuscript the results of numerical simulation in nanoscale directional coupler has been reported in Ref.[28]. IV. CONCLUSIONS We have shown, that the efficient control of spin waves coupling in the laterally coupled magnonic stripes is possible by the variation of both the geometry and the magnetization inside each stripe. Both the dispersion of eigenmodes and spatial distribution of the magnetization demonstrate the spinwave power exchange between the side-coupled stripes. We have shown that for the practical application the in-plane magnetization of the stripes is preferable rather that the out-ofplane magnetization. Moreover, we have demonstrated, that the increased coupling efficiency can be achieved in the case of anti-parallel magnetization of each adjacent magnonic stripe. Our results are relevant to the development of the advanced integrated magnonic couplers and offer a range of further opportunities in planar magnonics for high frequency signal processing.

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