Theory of dual-tunable thin-film multiferroic magnonic crystal Aleksei A. Nikitin, Andrey A. Nikitin, Alexander V. Kondrashov, Alexey B. Ustinov, Boris A. Kalinikos, and Erkki Lähderanta Citation: Journal of Applied Physics 122, 153903 (2017); View online: https://doi.org/10.1063/1.5000806 View Table of Contents: http://aip.scitation.org/toc/jap/122/15 Published by the American Institute of Physics Articles you may be interested in A switchable spin-wave signal splitter for magnonic networks Applied Physics Letters 111, 122401 (2017); 10.1063/1.4987007 A microwave field-driven transistor-like skyrmionic device with the microwave current-assisted skyrmion creation Journal of Applied Physics 122, 153901 (2017); 10.1063/1.4999013 Temperature induced phase transformations and negative electrocaloric effect in (Pb,La)(Zr,Sn,Ti)O3 antiferroelectric single crystal Journal of Applied Physics 122, 154101 (2017); 10.1063/1.4986849 Voltage tunable plasmon propagation in dual gated bilayer graphene Journal of Applied Physics 122, 153101 (2017); 10.1063/1.5007713 Nonlinear terahertz metamaterials with active electrical control Applied Physics Letters 111, 121101 (2017); 10.1063/1.4990671 Controlled creation of nanometric skyrmions using external magnetic fields Applied Physics Letters 111, 092403 (2017); 10.1063/1.4993855 JOURNAL OF APPLIED PHYSICS 122, 153903 (2017) Theory of dual-tunable thin-film multiferroic magnonic crystal Aleksei A. Nikitin,1,2 Andrey A. Nikitin,1,2 Alexander V. Kondrashov,1,2 Alexey B. Ustinov,1,2 €hderanta2 Boris A. Kalinikos,1 and Erkki La 1 Department of Physical Electronics and Technology, St. Petersburg Electrotechnical University, St. Petersburg 197376, Russia 2 Laboratory of Physics, Lappeenranta University of Technology, Lappeenranta 53850, Finland (Received 18 August 2017; accepted 5 October 2017; published online 19 October 2017) A theory has been developed for the waveguiding characteristics of dual-tunable multiferroic magnonic crystals (MCs). The crystals are constructed as periodically width-modulated microwave transmission slot-lines placed in between thin ferrite and ferroelectric films. Dispersion characteristics of the spin-electromagnetic waves (SEWs) in the investigated periodic waveguiding structure were derived using the method of approximate boundary conditions and the coupled-mode approach. The transmission-loss characteristics (TLCs) were calculated by the transfer-matrix method. The results show that the TLCs of the structures consist of pass-bands and stop-bands. The stop-bands are due to Bragg reflections in the structure. The magnetic and electric fields control the stop-band frequencies. The ferroelectric film polarization produced with the application of control voltage to the slot-line electrodes reduces its dielectric permittivity and provides up-shift of the stop-band frequencies. The most effective electric tuning is achieved in the area of the maximum hybridization of SEWs. As a result, the investigated multiferroic MCs combine the advantages of thin-film planar topology and dual tunability of magnonic band-gaps. Published by AIP Publishing. https://doi.org/10.1063/1.5000806 I. INTRODUCTION Nowadays, increasing interest to study composite materials for their possible microwave applications is evident.1 Artificial multiferroic structures are promising candidates for microwave devices due to a combination of advantages of ferrite and ferroelectric materials. The development of frequency-agile materials for microwave devices has led to the appearance of multilayered multiferroic structures.2–6 An interaction between the ferromagnetic and ferroelectric phases is realized through electrodynamic coupling of spin waves (SWs) and electromagnetic waves (EMWs). This interaction leads to a formation of spin-electromagnetic waves (SEWs).2 The frequency spectrum of these waves is dually controllable by both electric and magnetic fields. Electric tuning of the SEW spectrum is due to a dependence of ferroelectric dielectric permittivity on the bias electric field, whereas magnetic tuning is provided by a dependence of magnetic permeability on the bias magnetic field. In addition, devices based on multilayered multiferroic structures demonstrate small insertion losses3 and small power consumption.6 Owing to the dual tunability, multiferroic materials have been used to develop microwave devices. Such devices added the advantages of electric tuning to spin-wave devices.7–15 Among such devices, there are a delay line7 based on the yttrium iron garnet film and single-crystal lead magnesium niobate-lead titanate bilayer, the tunable microwave resonators with wide frequency tuning range8 and changeable quality factor,9 the ferromagnetic resonance phase shifters,10–12 and a multiband filter.13 Besides, an increased interest to investigate one-dimensional and two-dimensional 0021-8979/2017/122(15)/153903/6/$30.00 magnonic crystals (MCs) was observed (see, e.g., Refs. 16–19 and literature therein). Typical MCs are made from magnetic materials with a spatial periodic modulation of their physical properties or geometry. The periodicity of the magnetic film structures results in the appearance of the band-gaps in the spin-wave spectrum. It modifies the dispersion of spin waves in the vicinity of the band-gaps. Similar phenomena were observed for EMWs in photonic crystals20,21 and for matter waves of a Bose-Einstein condensate.22,23 Various linear and nonlinear microwave phenomena in the MCs are caused by peculiarities of the SW dispersion near the magnonic band-gaps.24–31 The magnonic crystals were used to construct various microwave devices such as phase shifters,32 spin-wave logic devices,33 magnetic field sensors,34 microwave oscillators,35 and others. Note that the epitaxial yttrium-iron garnet (YIG) films were successfully used in MCs due to small magnetic losses.36 It is clear that a combination of multiferroic and MC features is promising for the development of a new class of microwave devices. Recent advances in this field include the development of periodic ferrite-ferroelectric structures. In particular, the periodic structure composed of a thin-film MC and a ferroelectric slab was fabricated and studied.37 After that, a number of theoretical and experimental works were carried out.38–44 In these works, rather thick ferroelectric layers (of thickness more than 100 lm) were used in order to provide an effective hybridization of microwave SWs with EMWs. As a result, a relatively high control voltage (up to 1000 V) was needed for an effective electric tuning of SEW dispersion. 122, 153903-1 Published by AIP Publishing. 153903-2 Nikitin et al. In order to reduce the control voltage, the so-called allthin-film multiferroic structures were suggested.5,11,12,45,46 However, the problem of relatively high control voltage for multiferroic MCs still remained. To solve this problem, in this work, we suggest a thin-film MC based on a slot transmission line. As it was shown, the thin-film regular structures based on a slot transmission line demonstrate an effective electric tunability under voltage less than 100 V.5 Therefore, these structures could be useful for thin-film multiferroic MCs with a relatively low control voltage. The purpose of the present work is twofold: (i) to develop a general theory for waveguiding characteristics of thin-film MCs based on a slot transmission line and (ii) using this theory to find ways to enhance the electric tuning range for reduction of the control voltage. This paper is organized as follows. Section II describes the topology of the thin-film MCs under investigation. Section III is devoted to the theoretical model. The dispersion characteristics of SEWs and the transmission-loss characteristics (TLCs) of the periodical structure are investigated in Sec. IV. Section V provides a summary and conclusions. II. TOPOLOGY OF THE MAGNONIC CRYSTAL The thin-film multiferroic MC based on a slot-line having a periodic modulation of the slot width is shown in Fig. 1. It is composed of several layers enumerated with index j: a sapphire substrate (j ¼ 1), a polycrystalline ferroelectric film of the barium strontium titanate (j ¼ 2), an epitaxial YIG film with saturation magnetization M0 (j ¼ 3), and a gadolinium gallium garnet substrate (j ¼ 4). We denote the thickness of the layers as dj and their relative permittivities as ej . We assume that a SEW propagates along the slot-line in the multiferroic structure, i.e., along the x-axis. The structure is magnetized to saturation tangentially along the z-axis. The transmission slot-line is assumed to be formed as a narrow slot between two infinitely thin and perfectly conducting metal electrodes placed between the ferrite (j ¼ 3) and ferroelectric (j ¼ 2) films, as is shown in Fig. 1. These electrodes provide two functions, namely, waveguiding of J. Appl. Phys. 122, 153903 (2017) SEWs and electric biasing of the ferroelectric film. The thinfilm multiferroic MC is formed by segments of the slot-line. The segments of narrow and wide slots (w1 and w2 ) are hereinafter referred to as segment I and segment II. The lengths of each segments are l1 and l2 , respectively. Thus, the period of the MC is K ¼ l1 þ l2 . The ferrite and ferroelectric layers are assumed to be relatively thin (on the order of unity of micrometers). Other dielectric layers are assumed to be relatively thick (on the order of hundreds of micrometers). Running ahead, we note that during our simulations, the following parameters were varied: the YIG film thickness d3 , the BST film thickness d2 , the barium strontium titanate film permittivity e2 , the external magnetic field H as well as geometrical parameters of the MC such as the slot widths w1 and w2, the period K, and the number of the periods N. All the rest parameters were fixed. These were d1 ¼ d4 ¼500 lm, e1 ¼ 10, e4 ¼ 12, e3 ¼ 14, and M0 ¼ 1750 G. III. THEORETICAL MODEL OF THE THIN-FILM MULTIFERROIC MAGNONIC CRYSTALS The development of the theoretical model was carried out in several stages. At the first stage, the dispersion relations for the SEWs propagating in the multiferroic MCs were found according to the coupled-mode approach.47 This approach supposes that forward and backward waves are propagating independently and that a waveguide parameter variation (in our case, it is the slot width) provides a coupling between them. In this case, a dispersion equation has the following form: cos ðK KÞ ¼ cosðk1 l1 Þ cosðk2 l2 Þ k12 þ k22 sin ðk1 l1 Þ sin ðk2 l2 Þ; 2 k1 k2 (1) where K is the Bloch wave vector, and k1 and k2 are the wave numbers of the SEWs in the segments I and II. The wave numbers in Eq. (1) were numerically calculated according to the SEW dispersion relation based on the approximate boundary conditions method described in detail in Ref. 5. Note that Eq. (1) takes into account two aspects of the wave process in the investigated MCs. First, the SEWs are damped waves for the frequencies where the Bloch wave vector has a complex value. In the microwave range, this phenomenon causes the Bragg gaps. Second, the roots of the Eq. (1) are real at the frequencies outside the band-gaps. Such solutions are responsible for propagating waves. At the second stage, TLCs of the periodic multiferroic structure were obtained according to the transfer-matrix method.48 This method allows one to calculate transmission characteristics of a finite-length periodic structure taking into account SEW insertion losses. IV. RESULTS AND DISCUSSION FIG. 1. The thin-film multiferroic MC based on a slot transmission line with modulation of the slot width. The dispersion characteristics of the spin-electromagnetic waves and transmission characteristics of the thin-film multiferroic magnonic crystals were calculated and analyzed by using the theoretical model discussed in Sec. III. 153903-3 Nikitin et al. The corresponding calculations were carried out for the typical parameters of the experimental multiferroic structures based on ferrite and ferroelectric films, as was outlined in Sec. II. Figure 2 shows the dispersion characteristics for the regular thin-film multiferroic structures with geometry presented in Fig. 2, i.e., without width modulation. The calculations were carried out for the different slot widths w ¼ 25 lm (solid lines) and w ¼ 90 lm (dashed lines) with the following parameters: d3 ¼ 13.6 lm, H ¼ 1350 Oe, d2 ¼ 2 lm, and e2 ¼ 1500. For a comparison, Fig. 2 also shows the dispersion branches for the fundamental electromagnetic mode of the individual slot with the ferroelectric film on the dielectric substrate and for the surface spin wave mode of the free-standing magnetic film (dotted lines). One can clearly see a hybridization of the two fundamental modes appearing due to their electrodynamic interaction. It becomes more pronounced near the point where the dispersion branches of the pure EMW and the pure SW cross each other. As is clear from Fig. 2, a reduction of the slot width w shifts the SEW dispersion characteristic to the higher wave numbers. Consequently, the SEWs at a fixed frequency accumulate the different phase shifts in different segments of the periodic structure. The band-gaps appear at the frequencies where this phase shift is a multiple of p. The SEW dispersion characteristic and the TLC of the thin-film multiferroic MC of Fig. 1 are shown in Figs. 3(a) and 3(b), respectively. The MC was formed by a series connection of two slot-line segments investigated before. The period and the number of the periods were taken to be K ¼ 1.7 mm and N ¼ 10. The lengths of the segments I and II were equal, i.e., l1 ¼ l2 ¼ K/2. This set of parameters was chosen in order to achieve the second purpose of this work. Indeed, as it was shown in Ref. 5, the effective electric and magnetic tuning ranges for multiferroic structures with slot transmission lines are possible only around the point of effective hybridization of electromagnetic and spin waves. For the considered set of the parameters, the frequency of hybridization is located near 5.74 GHz (see Fig. 2). Therefore, the periodic modulation of the slot FIG. 2. Dispersion characteristics of SEWs in the slot-lines for widths of w ¼ 25 lm (solid lines) and w ¼ 90 lm (dashed lines). Pure EMWs and SW are presented with dotted lines. J. Appl. Phys. 122, 153903 (2017) FIG. 3. Dispersion (a) and transmission-loss (b) characteristics of SEWs of the thin-film multiferroic magnonic crystal. width of the MC was chosen in such a way that the first magnonic band-gap appears near 5.74 GHz. Thus, effective electric tuning of the TLC becomes possible. As can be seen from Fig. 3, the modulation of the slot width leads to an appearance of the band-gaps in the spectrum of the SEWs [see Fig. 3(a)]. These band-gaps cause the dips in the TLC of the investigated structure [see Fig. 3(b)]. The dips have finite depths due to a limited number of the periods. In the considered case, the depth of the first bandgap (denoted by I in Fig. 3) is about -34 dB. The width of the band-gaps reduces with the frequency increasing. This behavior is determined by an efficiency of hybridization between the EMWs and the SWs. Let us now discuss transmission-loss characteristics. Figure 4 demonstrates an influence of the slot-line parameters such as the width of the segments I and II [see Figs. 4(a)] and the period K [see Fig. 4(b)] on the TLCs for thin-film multiferroic MCs. As can be seen from Fig. 4(a), a change in the modulation of the slot-line width affects the first bandgap of the thin-film multiferroic MC. For the first band-gap, an increase in the slot width difference (w1 w2 ) leads to a reduction in the band-gap frequency and to an increase in the band-gap width. These two effects are due to an increase in the wavenumber difference for SEWs propagating in segments I and II (see Fig. 2). Note that the band-gap widths were calculated at a level of 3 dB from the maximum loss. Similar effects can be observed in the case of different periods K [see Fig. 4(b)]. According to Bragg’s diffraction law, the maxima of reflection occur at k ¼ np=K. Therefore, an increase in the length of the period decreases the wavenumbers, which correspond to the band-gaps. Due to this 153903-4 Nikitin et al. FIG. 4. Influence of the slot line width w (a) and period length K (b) on a MC band structure. phenomenon, the band-gaps are shifted to the lower frequencies. In addition, this down-frequency shift determines increasing in the width of the first band-gap. Finally, an influence of the number of periods N on the behavior of the transmission-loss characteristic is analyzed. As follows from the transfer-matrix method, an increase in this parameter increases the losses for a TLC. For the considered structure, the transmission coefficient of the first bandgap is 34 dB for 10 periods, while for 20 periods, this value reaches 68 dB. Also, numerical simulations were carried out for the different thicknesses of the ferroelectric and ferrite films, i.e., d2 and d3 . In particular, Fig. 5(a) shows that a decrease in the ferroelectric film thickness d2 leads to a shift of the band-gap position to higher frequency and brings a change in a width of the first band-gap. This behavior is determined by the decrease in the difference between the SEW dispersion in different segments of the MC. In addition, the higher frequency band-gaps depend weakly from the ferroelectric film thickness. Such a behavior is determined by their frequency positions that are higher than 5.74 GHz. Therefore, the higher frequency band-gaps are located far from the point of effective hybridization between the EMW and the SW (see Fig. 2). In this case, properties of the ferroelectric film play a dominant role in the wave process whereas the influence of the ferroelectric subsystem is negligible. Figure 5(b) demonstrates an influence of the ferrite film thickness d3 on the transmission-loss characteristics. As one can see, the frequencies of the first band-gap depend weakly J. Appl. Phys. 122, 153903 (2017) FIG. 5. Transmission-loss characteristics for different thicknesses d2 of the ferroelectric film (a) and d3 of the ferrite film (b). on the ferrite film thickness. At the same time, the width of this band-gap is increased. It is because of an increase in the SEW hybridization for thick ferrite films.2 Sufficient reduction in a group velocity with decreasing the ferrite thickness provides higher losses and down-shift in the frequency of the bandgaps. Simultaneously, the narrow band-gaps appeared for the MC based on the thin ferrite film. As shown in Fig. 5(b), the widths of the first band-gap are 38.6 and 19.8 MHz at 20 dB for 20- and 9–lm thick of the ferrite films, respectively. Let us consider now the electric and magnetic tunability of the transmission-loss characteristics. The results of modeling of the electric tuning are shown in Fig. 6(a). The calculations were carried out for the following parameters of the MC: d3 ¼ 13.6 lm, H ¼ 1350 Oe, d2 ¼ 2 lm, K ¼ 1.7 mm, N ¼ 10, w1 ¼ 90 lm, and w2 ¼ 25 lm. An influence of control voltage U applied to the slot-line electrodes was simulated as a reduction of the ferroelectric film permittivity e2 . Note that due to the different widths of the slot-line gaps of the segments I and II (see Fig. 1), the electric field is not the same in different segments and was calculated as E1;2 ¼ U=w1;2 . The relative permittivity of the BST film as a function of the electric field E was calculated by the following formula: e2 ðE1;2 Þ ¼ e2 ð0Þ k E21;2 : (2) The following typical parameters of the barium strontium titanate film were used: e2 ð0Þ ¼ 1500 and k ¼ 0.194 cm2/kV2.5 As can be seen from Fig. 6(a), an increase in the control voltage shifts the band-gaps to the higher frequency. Such 153903-5 Nikitin et al. J. Appl. Phys. 122, 153903 (2017) influence of different geometrical parameters on the waveguiding characteristics and band-gap frequency positions was analyzed. It was found that the investigated structures can provide an excellent signal rejection of more than 30 dB. The optimal rejection efficiency and the required band-gap bandwidth can be obtained by adjusting geometry of a thinfilm magnonic crystal. Furthermore, an enhancement of the electric tuning range and a reduction of the control voltage for the multiferroic MC were achieved due to its unique geometry and physical properties. Thus, application of the bias voltage of 200 V to 2-lm-thick ferroelectric film leads to the shift of the bandgap position by 8.35 MHz. At the same time, the change of the external magnetic field by 10 Oe shifts the band-gaps by 30 MHz. Therefore, the proposed structures are perspective for the development of new microwave devices and investigation of new physical phenomena. ACKNOWLEDGMENTS FIG. 6. Electric (a) and magnetic (b) tuning of the transmission-loss characteristics. The work at SPbETU on numerical modeling was supported by the Russian Science Foundation (Grant No. 14–12-01296-P). The work at SPbETU on Development of Computer Program for Numerical Simulation was supported by the Ministry of Education and Science of the Russian Federation (Project “Goszadanie”) and the Russian Foundation for Basic Research (project No. 16-32-000715 mol_a). The work at LUT was supported in part by the Academy of Finland. 1 behavior of the TLCs can be explained as follows. It is known that a decrease in a ferroelectric permittivity shifts a SEW dispersion characteristic to the lower wavenumbers. In a case of the MC, the values of wavenumbers that correspond to the maxima of reflection (k ¼ np=K) remain constant. Therefore, in the case of e2 decreasing (i.e., increase in the control voltage), this condition will be satisfied in the area of the higher frequency compared to the zero voltage case. In particular, the electric tuning of the TLC for the first band-gap reaches values of 8.35 MHz for the electric voltage of 200 V. Note that the electric tuning is decreased for higher stop-bands due to weak interaction of spin and electromagnetic waves at frequencies higher than 5.8 GHz for the investigated slot-line structure. Turn now to the magnetic tuning. Transmission-loss characteristics were simulated for different values of the external magnetic field H [see Fig. 6(b)]. The following magnetic fields were used: 1340 Oe (solid line), 1345 Oe (dashed line), and 1350 Oe (dotted line). Figure 6 shows that an increase in the external magnetic field leads to the shift of the SW spectrum toward the higher frequencies. V. 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