IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,
vol. 61, no. 5,
May
2014
881
Correspondence Design and Simulation of CoupledResonator Filters Using Periodically Slotted Electrodes on FBARs Jiansong Liu, Tatsuya Omori, Member, IEEE, Changjun Ahn, Member, IEEE, and Ken-ya Hashimoto, Fellow, IEEE Abstract—This paper discusses the use of periodically slotted top electrodes in the film bulk acoustic resonator (FBAR) structure for the realization of wideband coupled-resonator filters (CRFs), where evanescent modes in the periodic structure are used for the coupling between adjacent electrodes. First, wave propagation in this structure is investigated. Finite element analysis is performed for the Mo/ZnO/Mo structure. The result suggests that lateral wave propagation is controlled by the Bragg reflection, and that transverse modes can be suppressed when the structure is properly designed. Next, a CRF is designed. It is shown that wideband CRFs are realizable when the period, thickness, and width of slotted electrodes are properly set.
I. Introduction
R
adio-frequency (RF) BAW filters categorized as film bulk acoustic resonators (FBARs) and solidly mounted resonators (SMRs) are widely used as front-end filters and duplexers for mobile phones [1], [2]. The filter function is realized by interconnecting multiple RF BAW resonators in the ladder or lattice topologies [3]. Ladder-type filters offer a flat passband and steep transition bands with the unbalanced termination for the input and output ports; however, achievable out-of-band rejection is usually limited [3]. On the other hand, latticetype filters exhibit good out-of-band rejection far from the passband, but the cut-off characteristic is gradual [4]. It should be noted that lattice-type filters operate only with balanced termination, and do not fit into the current RF front-end, for which the antenna port is usually unbalanced. Coupled-resonator filters (CRFs) [5], [6] are known to offer low insertion loss in the passband and good out-ofband rejection far from the passband, and flatness of the passband and steepness of transition bands are enhanced when multiple CRFs are cascaded. Furthermore, the cascade connection can also be used to realize the balancedto-unbalanced conversion and/or impedance conversion functions [6].
Manuscript received June 12, 2013; accepted February 10, 2014. This work was partially supported by the “Funding Program for World-Leading Innovative R&D on Science and Technology” from the Japan Society for the Promotion of Science. The authors are with the Graduate School of Engineering, Chiba University, Chiba-shi, Japan (e-mail: [email protected]). DOI http://dx.doi.org/10.1109/TUFFC.2014.2978 0885–3010/$25.00
A conventional CRF is realized by sandwiching the multilayer mechanism between two stacked resonators to weaken the acoustic coupling between the resonators [5], [6]; however, the coupling is very delicate and hard to control in fabrication. Later, a single, unique coupling layer was developed to simplify the fabrication process [7]–[9]. Laterally coupled BAW filters fabricated on a thin-film acoustic mirror were reported in which the lateral evanescent field caused by the cut-off of the thickness resonance is responsible for the coupling between closely spaced narrow resonators [10]–[13]. Application of this design is not simple for structures exhibiting the so called type-II dispersion because the resonance frequency for the coupling region is required to be lower than that of the resonator region [14]. Recently, the authors discussed the influence of surface gratings placed on the top electrode of the FBAR structure, and demonstrated that its Bragg reflection allows us to control the lateral propagation of spurious Lamb modes with minor influence on the main thickness resonance [15]. The objective of this paper is to discuss the possibility of realizing wideband CRFs using lateral coupling controlled by the Bragg reflection. Fig. 1 shows the device configuration discussed in this paper, in which narrow FBAR sections are aligned periodically and are electrically connected alternately. The acoustic coupling between adjacent resonators is caused by the evanescent mode in the periodic structure, and the coupling strength is controlled by the slot period and width. The structure shown in Fig. 1 seems equivalent to that used in Lamb wave resonators (LWRs) [16], [17]. Because LWRs employ resonances of laterally propagating Lamb waves, their resonance frequencies are mostly determined by the electrode period. In contrast, the resonance frequencies are mostly determined by the film thickness in the present case. First, wave propagation in the structure shown in Fig. 1 is analyzed using the finite element method (FEM). The Mo/ZnO/Mo structure is used as an example. It is shown that, because of the Bragg reflection, propagation of transverse modes can be forbidden above the main resonance frequency provided that the stopband is designed to cover the frequency range in which the transverse modes occur. The location and width of the stopband are adjustable by the slot period and width. Next, a CRF is designed using the principle given in [18]. It is demonstrated that a wideband CRF is realizable through proper design. II. Simulation Setup Fig. 2 shows the device structure used for the analysis, where p, w, hs, Np, and L are the period, width, thickness,
© 2014 IEEE
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IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,
vol. 61, no. 5,
May
2014
Fig. 3. In-phase excitation. Fig. 1. Coupled-resonator filter configuration using slotted top electrodes.
III. Resonance Characteristics number of the slotted top electrodes, and total length of the structure, respectively, and hp and he designate the thicknesses of the piezoelectric layer and bottom electrode. The metallization ratio m is defined as w/p. In the following analysis, ZnO and Mo are chosen as the piezoelectric layer and electrode material, respectively. The FEM analysis was carried out using Ansys Multiphysics 13.0 (ANSYS Inc., Canonsburg, PA), reading an Ansys parametric design language (APDL) script into the batch mode. Parametric input, data evaluation, and calculation flow control were performed by Matlab 7.7 (The MathWorks Inc., Natick, MA). In the simulation, harmonic analysis in the frequency domain was performed. The absorption mechanism was adapted at both ends to decrease the reflection and mode conversion there. The absorption mechanism is composed of a set of damping layers with exponentially increasing material viscous loss v = 1/Q(n), where Q(n) = Qstart(Qend/Qstart)[(n−1)/(N−1)], n = 1, 2, 3, …, N. N is the number of damping layers. In the simulation, the Qstart and Qend were set to 1000 and 20. With appropriate design, wave reflection and mode conversion are not allowed even within the damping mechanism. The detailed design can be seen in [19]. The sinusoidal voltage V with the frequency f was applied to the slotted top electrodes while the bottom electrode was grounded. The electric potentials of all the nodes of the individual electrodes were coupled to keep the voltage at the electrode constant for a given time. After the FEM analysis, the Ansys post-processor gives total charge q on the slotted top electrodes. Then, the admittance Y is given by 2πjfq/V.
Fig. 2. Slotted top electrodes on the FBAR structure.
First, wave propagation in the structure with in-phase excitation shown in Fig. 3 is analyzed. Fig. 4 shows, for comparison, calculated admittance Y of the Mo/ZnO/Mo structure without the slots. In this calculation, hp and he were set at 1.52 μm and 0.1 μm, respectively. The main resonance giving |Y |−1 ~ 0 is seen at fr = 1.637 GHz, and the anti-resonance giving |Y | ~ 0 is seen at fa = 1.714 GHz. A series of spurious resonances is seen at frequencies above fr. As explained in [15], they are caused by the Lamb mode (S1) propagation, which occurs at frequencies above the main resonance in the Mo/ZnO/Mo structure. Fig. 5 shows the calculated admittance Y when hs, hp, he, p, Np, and m were set as 0.1 μm, 1.52 μm, 0.1 μm, 3.125π μm, 20, and 0.7, respectively. Because of the decreased mass loading, fr and fa slightly increase to 1.689 GHz and 1.742 GHz, respectively. From the figure, it is seen that spurious resonances are completely suppressed and only a clean thickness resonance is visible. The Ansys post-processor also gives out-of-plane displacements uz(x) at the top surface of piezoelectric film. Then, their spectrum Uz in the wavenumber (βx) domain is readily obtained by applying the fast Fourier transform (FFT) to uz(x). The dispersion relation of the Lamb modes is given by calculating |Uz| as a function of f and assigning |Uz| to brightness at a point (βx, f).
Fig. 4. Calculated admittance of the Mo/ZnO/Mo structure without slots.
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Fig. 7. Anti-phase excitation.
IV. CRF Design and Discussion Let us express the electric characteristics of the CRF structure shown in Fig. 1 as the following admittance matrix: Fig. 5. Calculated admittance of the Mo/ZnO/Mo FBAR structure with slotted top electrodes.
Fig. 6 shows the dispersion curves for this case. Because of the structural periodicity, the identical dispersion curves appear periodically with a period of 2π/p = 0.64 rad/μm. It is clear that the S1 mode branch has partially disappeared because of the Bragg reflection. A spurious free resonance is obtainable when the stopband covers the frequency range where the lateral mode resonances occur. Note that the lateral wavenumber β is π/p at the stopband edges, and the stopband width increases with hg. Thus, p should be set so that the stopband is located in the frequency range where the spurious resonances occur, and hg should be set so that the stopband fully covers the frequency range. These results indicate that the slotted top electrode is effective for controlling propagation of lateral modes, similar to the grating placed on the top electrode [15].
Fig. 6. Dispersion curves for the Mo/ZnO/Mo structure with slotted top electrodes when hs, hp, he, p, Np, and m were set as 0.1 μm, 1.52 μm, 0.1 μm, 3.125π μm, 20, and 0.7, respectively.
( II ) = (YY 1
11
2
12
)( )
Y12 V1 , (1) Y 22 V 2
where Vn and In are the voltage and current applied to the electric port n. Because of the structural symmetry, Y22 = Y11 in this case. When the CRF is terminated by the impedance G0, the transfer function S21 is given by
S 21 = −
2G 0Y12 . (2) (Y11 + G 0)(Y 22 + G 0)2 − Y122
When V2 = V1, all FBAR sections vibrate in phase, whereas adjacent FBAR sections vibrate in anti-phase when V2 = −V1. Let us define Yi = Y11 + Y12 and Ya = Y11 − Y12, which correspond to the input admittances for these inphase and anti-phase resonance modes, respectively. The design principles for the CRF [18] are that 1) the resonance frequency for Yi (or Ya) coincides with the anti-resonance frequency for Ya (or Yi), and 2) the shunt capacitance C0 is set at approximately G0/ωr, where ωr is the resonance frequency. The simulation setup shown in Fig. 3 gives Yi, whereas that shown in Fig. 7 gives Ya.
Fig. 8. Admittance characteristic of the designed coupled-resonator filter.
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2014
Fig. 11. Coupled-resonator filter configuration using slotted top and bottom electrodes.
Fig. 9. Calculated transfer function S21 of the designed coupled-resonator filter.
Fig. 8 shows Yi and Ya of a CRF designed under this principle. In this calculation, hs, hp, he, p, Np, and m were set as 0.09 μm, 1.52 μm, 0.09 μm, 2π μm, 30, and 0.52, respectively. The circuit impedance was set at ωrC0. It can be seen that the anti-resonance frequency of Yi coincides with the resonance frequency of Ya. Fig. 9 shows the transfer function S21 of the designed CRF. It is seen that a relatively flat and wide passband is realized. Relatively large spurious peaks are seen at 1.73 GHz and 1.86 GHz. They are due to spurious resonances in Ya, which can also be seen in Fig. 8. Achieved out-of-band rejection is somewhat limited at frequencies far from the passband. The main reasons are the electrostatic coupling between adjacent electrodes and the influence of spurious resonances far from the main resonance. Fig. 10 shows the calculated dispersion for this case. The target stopband can be seen from 1.819 to 1.905 GHz. As indicated in the figure, the resonance frequencies f ri of Yi and f ra of Ya correspond to the cut-off for the thickness
extension vibration with β = 0 and the lower stopband edge with β = π/p, respectively. Another stopband can be seen from 1.68 to 1.73 GHz. This may be due to the Bragg reflection of the TS2 mode, and causes a relatively large spurious peak in Ya at 1.73 GHz (see Fig. 8). It is expected that CRFs are also realizable when both the top and bottom electrodes are slotted as shown in Fig. 11. Electrical isolation between electric ports can be used to realize the balanced-to-unbalanced and/or impedance conversion functions [6]. However, because the layer on which the piezoelectric layer is grown critically influences the quality of the piezoelectric layer at the deposition, nonuniform bottom electrodes may not fit into the current FBAR/SMR fabrication process.
V. Conclusion This paper investigated the influence of the slotted top electrodes placed on the FBAR structure for the realization of CRFs. First, it was shown that, similar to a surface grating, the slotted top electrodes cause Bragg reflection and can be used to suppress transverse modes. Then it was shown that the slotted top electrodes can be used to design wideband CRFs. A CRF was designed and its performance was demonstrated. References
Fig. 10. Dispersion curves for the Mo/ZnO/Mo structure with slotted top electrodes when hs, hp, he, p, Np, and m were set as 0.09 μm, 1.52 μm, 0.09 μm, 2π μm, 30, and 0.52, respectively.
[1] R. Ruby, “Review and comparison of bulk acoustic wave FBAR, SMR technology,” in Proc. IEEE Ultrasonics Symp., 2007, pp. 1029– 1039. [2] R. Ruby and M. Gat, “FBAR filters and oscillators—2012,” in Proc. 2012 Int. Symp. Acoustic Wave Devices for Future Mobile Comunications, pp. 149–152. [3] M. Ylilammi, J. Ella, M. Partanen, and J. Kaitila, “Thin film bulk acoustic wave filter,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 49, no. 4, pp. 535–539, 2002. [4] K. Lakin, “Resonator and filter topologies,” in RF Bulk Acoustic Filters for Communications, K. Hashimoto, Ed., Norwood, MA: Artech House, 2009, ch. 2, pp. 21–49. [5] K. Lakin, “Coupled resonator filters,” in Proc. IEEE Ultrasonics Symp., 2002, vol. 1, pp. 901–908. [6] G. Fattinger, R. Aigner, and W. Nessler, “Coupled bulk acoustic wave resonator filters: Key technology for single-to-balanced RF filters,” in IEEE MTT-S Int. Microwave Symp. Tech. Dig., 2004, vol. 2, pp. 927–929. [7] S. R. Gilbert, P. Nikkel, T. Jamneala, R. Ruby, and J. D. Larson, III, “Improved coupled resonator filter performance using a carbondoped oxide de-coupling layer,” in Proc. IEEE Ultrasonics Symp., 2009, pp. 867–871.
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[8] T. Jamneala, U. B. Koelle, A. Shirakawa, S. R. Gilbert, P. Nikkel, C. Feng, and R. Ruby, “Ultra-miniature coupled resonator filter with single-layer acoustic coupler,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 56, no. 11, pp. 2553–2558, 2009. [9] S. R. Gilbert, R. Ruby, Y.-H. Chow, W. Hii, and S. Lim, “GNSS LNA-filter module using BAW CRF filters,” in Proc. IEEE Ultrasonics Symp., 2010, pp. 99–102. [10] J. Meltaus, T. Pensala, K. Kokkonen, and A. Jansman, “Laterally coupled solidly mounted BAW resonators at 1.9 GHz,” in Proc. IEEE Ultrasonics Symp., 2009, pp. 847–850. [11] J. Meltaus and T. Pensala, “Laterally coupled BAW filters with 5% bandwidth,” in Proc. IEEE Ultrasonics Symp., 2010, pp. 966–969. [12] J. Meltaus, T. Pensala, and K. Kokkonen, “Parametric study of laterally acoustically coupled bulk acoutic wave filters,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 59, no. 12, pp. 2742–2751, 2012. [13] G. Kline, R. S. Ketcham, and K. Lakin, “Low insertion loss filters synthesized with thin film resonators,” in Proc. IEEE Ultrasonics Symp., 1987, pp. 375–380. [14] S. Seto, S. Horiuchi, and K. Yamada, “Use of trapped-energy mode of backward-wave-type thickness vibration for liquid-level sensing,” Jpn. J. Appl. Phys., vol. 49, no. 7S, art. no. 07HC05, 2010.
[15] J. Liu, T. Omori, C. Ahn, and K. Hashimoto, “Impact of surface periodic grating on film bulk acoustic resonator structure to spurious transverse resonances,” J. Appl. Phys., vol. 113, no. 14, art. no. 144507, 2013. [16] V. Yantchev and I. Katardjiev, “Design and fabrication of thin film Lamb wave resonators utilizing Longitudinal wave and interdigital transducers,” in Proc. IEEE Ultrasonics Symp., 2005, pp. 1580–1583. [17] C.-M. Lin, T.-T. Yen, Y.-J. Lai, V. V. Felmetsger, M. A. Hopcroft, J. H. Kuypers, and A. P. Pisano, “Temperature-compensated aluminum nitride Lamb wave resonators,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 57, no. 3, pp. 524–532, 2010. [18] K. Hashimoto, T. Omori, and M. Yamaguchi, “Operation mechanism of double-mode surface acoustic wave filter with pitch-modulated IDTs and reflectors,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 54, no. 10, pp. 2152–2158, 2007. [19] F. Thalmayr, K. Hashimoto, T. Omori, and M. Yamaguchi, “Frequency domain analysis of Lamb wave scattrering and application to film bulk acoutic wave resonators,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 57, no. 7, pp. 1641–1648, 2010.
vol. 61, no. 5,
May
2014
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Correspondence Design and Simulation of CoupledResonator Filters Using Periodically Slotted Electrodes on FBARs Jiansong Liu, Tatsuya Omori, Member, IEEE, Changjun Ahn, Member, IEEE, and Ken-ya Hashimoto, Fellow, IEEE Abstract—This paper discusses the use of periodically slotted top electrodes in the film bulk acoustic resonator (FBAR) structure for the realization of wideband coupled-resonator filters (CRFs), where evanescent modes in the periodic structure are used for the coupling between adjacent electrodes. First, wave propagation in this structure is investigated. Finite element analysis is performed for the Mo/ZnO/Mo structure. The result suggests that lateral wave propagation is controlled by the Bragg reflection, and that transverse modes can be suppressed when the structure is properly designed. Next, a CRF is designed. It is shown that wideband CRFs are realizable when the period, thickness, and width of slotted electrodes are properly set.
I. Introduction
R
adio-frequency (RF) BAW filters categorized as film bulk acoustic resonators (FBARs) and solidly mounted resonators (SMRs) are widely used as front-end filters and duplexers for mobile phones [1], [2]. The filter function is realized by interconnecting multiple RF BAW resonators in the ladder or lattice topologies [3]. Ladder-type filters offer a flat passband and steep transition bands with the unbalanced termination for the input and output ports; however, achievable out-of-band rejection is usually limited [3]. On the other hand, latticetype filters exhibit good out-of-band rejection far from the passband, but the cut-off characteristic is gradual [4]. It should be noted that lattice-type filters operate only with balanced termination, and do not fit into the current RF front-end, for which the antenna port is usually unbalanced. Coupled-resonator filters (CRFs) [5], [6] are known to offer low insertion loss in the passband and good out-ofband rejection far from the passband, and flatness of the passband and steepness of transition bands are enhanced when multiple CRFs are cascaded. Furthermore, the cascade connection can also be used to realize the balancedto-unbalanced conversion and/or impedance conversion functions [6].
Manuscript received June 12, 2013; accepted February 10, 2014. This work was partially supported by the “Funding Program for World-Leading Innovative R&D on Science and Technology” from the Japan Society for the Promotion of Science. The authors are with the Graduate School of Engineering, Chiba University, Chiba-shi, Japan (e-mail: [email protected]). DOI http://dx.doi.org/10.1109/TUFFC.2014.2978 0885–3010/$25.00
A conventional CRF is realized by sandwiching the multilayer mechanism between two stacked resonators to weaken the acoustic coupling between the resonators [5], [6]; however, the coupling is very delicate and hard to control in fabrication. Later, a single, unique coupling layer was developed to simplify the fabrication process [7]–[9]. Laterally coupled BAW filters fabricated on a thin-film acoustic mirror were reported in which the lateral evanescent field caused by the cut-off of the thickness resonance is responsible for the coupling between closely spaced narrow resonators [10]–[13]. Application of this design is not simple for structures exhibiting the so called type-II dispersion because the resonance frequency for the coupling region is required to be lower than that of the resonator region [14]. Recently, the authors discussed the influence of surface gratings placed on the top electrode of the FBAR structure, and demonstrated that its Bragg reflection allows us to control the lateral propagation of spurious Lamb modes with minor influence on the main thickness resonance [15]. The objective of this paper is to discuss the possibility of realizing wideband CRFs using lateral coupling controlled by the Bragg reflection. Fig. 1 shows the device configuration discussed in this paper, in which narrow FBAR sections are aligned periodically and are electrically connected alternately. The acoustic coupling between adjacent resonators is caused by the evanescent mode in the periodic structure, and the coupling strength is controlled by the slot period and width. The structure shown in Fig. 1 seems equivalent to that used in Lamb wave resonators (LWRs) [16], [17]. Because LWRs employ resonances of laterally propagating Lamb waves, their resonance frequencies are mostly determined by the electrode period. In contrast, the resonance frequencies are mostly determined by the film thickness in the present case. First, wave propagation in the structure shown in Fig. 1 is analyzed using the finite element method (FEM). The Mo/ZnO/Mo structure is used as an example. It is shown that, because of the Bragg reflection, propagation of transverse modes can be forbidden above the main resonance frequency provided that the stopband is designed to cover the frequency range in which the transverse modes occur. The location and width of the stopband are adjustable by the slot period and width. Next, a CRF is designed using the principle given in [18]. It is demonstrated that a wideband CRF is realizable through proper design. II. Simulation Setup Fig. 2 shows the device structure used for the analysis, where p, w, hs, Np, and L are the period, width, thickness,
© 2014 IEEE
882
IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,
vol. 61, no. 5,
May
2014
Fig. 3. In-phase excitation. Fig. 1. Coupled-resonator filter configuration using slotted top electrodes.
III. Resonance Characteristics number of the slotted top electrodes, and total length of the structure, respectively, and hp and he designate the thicknesses of the piezoelectric layer and bottom electrode. The metallization ratio m is defined as w/p. In the following analysis, ZnO and Mo are chosen as the piezoelectric layer and electrode material, respectively. The FEM analysis was carried out using Ansys Multiphysics 13.0 (ANSYS Inc., Canonsburg, PA), reading an Ansys parametric design language (APDL) script into the batch mode. Parametric input, data evaluation, and calculation flow control were performed by Matlab 7.7 (The MathWorks Inc., Natick, MA). In the simulation, harmonic analysis in the frequency domain was performed. The absorption mechanism was adapted at both ends to decrease the reflection and mode conversion there. The absorption mechanism is composed of a set of damping layers with exponentially increasing material viscous loss v = 1/Q(n), where Q(n) = Qstart(Qend/Qstart)[(n−1)/(N−1)], n = 1, 2, 3, …, N. N is the number of damping layers. In the simulation, the Qstart and Qend were set to 1000 and 20. With appropriate design, wave reflection and mode conversion are not allowed even within the damping mechanism. The detailed design can be seen in [19]. The sinusoidal voltage V with the frequency f was applied to the slotted top electrodes while the bottom electrode was grounded. The electric potentials of all the nodes of the individual electrodes were coupled to keep the voltage at the electrode constant for a given time. After the FEM analysis, the Ansys post-processor gives total charge q on the slotted top electrodes. Then, the admittance Y is given by 2πjfq/V.
Fig. 2. Slotted top electrodes on the FBAR structure.
First, wave propagation in the structure with in-phase excitation shown in Fig. 3 is analyzed. Fig. 4 shows, for comparison, calculated admittance Y of the Mo/ZnO/Mo structure without the slots. In this calculation, hp and he were set at 1.52 μm and 0.1 μm, respectively. The main resonance giving |Y |−1 ~ 0 is seen at fr = 1.637 GHz, and the anti-resonance giving |Y | ~ 0 is seen at fa = 1.714 GHz. A series of spurious resonances is seen at frequencies above fr. As explained in [15], they are caused by the Lamb mode (S1) propagation, which occurs at frequencies above the main resonance in the Mo/ZnO/Mo structure. Fig. 5 shows the calculated admittance Y when hs, hp, he, p, Np, and m were set as 0.1 μm, 1.52 μm, 0.1 μm, 3.125π μm, 20, and 0.7, respectively. Because of the decreased mass loading, fr and fa slightly increase to 1.689 GHz and 1.742 GHz, respectively. From the figure, it is seen that spurious resonances are completely suppressed and only a clean thickness resonance is visible. The Ansys post-processor also gives out-of-plane displacements uz(x) at the top surface of piezoelectric film. Then, their spectrum Uz in the wavenumber (βx) domain is readily obtained by applying the fast Fourier transform (FFT) to uz(x). The dispersion relation of the Lamb modes is given by calculating |Uz| as a function of f and assigning |Uz| to brightness at a point (βx, f).
Fig. 4. Calculated admittance of the Mo/ZnO/Mo structure without slots.
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Fig. 7. Anti-phase excitation.
IV. CRF Design and Discussion Let us express the electric characteristics of the CRF structure shown in Fig. 1 as the following admittance matrix: Fig. 5. Calculated admittance of the Mo/ZnO/Mo FBAR structure with slotted top electrodes.
Fig. 6 shows the dispersion curves for this case. Because of the structural periodicity, the identical dispersion curves appear periodically with a period of 2π/p = 0.64 rad/μm. It is clear that the S1 mode branch has partially disappeared because of the Bragg reflection. A spurious free resonance is obtainable when the stopband covers the frequency range where the lateral mode resonances occur. Note that the lateral wavenumber β is π/p at the stopband edges, and the stopband width increases with hg. Thus, p should be set so that the stopband is located in the frequency range where the spurious resonances occur, and hg should be set so that the stopband fully covers the frequency range. These results indicate that the slotted top electrode is effective for controlling propagation of lateral modes, similar to the grating placed on the top electrode [15].
Fig. 6. Dispersion curves for the Mo/ZnO/Mo structure with slotted top electrodes when hs, hp, he, p, Np, and m were set as 0.1 μm, 1.52 μm, 0.1 μm, 3.125π μm, 20, and 0.7, respectively.
( II ) = (YY 1
11
2
12
)( )
Y12 V1 , (1) Y 22 V 2
where Vn and In are the voltage and current applied to the electric port n. Because of the structural symmetry, Y22 = Y11 in this case. When the CRF is terminated by the impedance G0, the transfer function S21 is given by
S 21 = −
2G 0Y12 . (2) (Y11 + G 0)(Y 22 + G 0)2 − Y122
When V2 = V1, all FBAR sections vibrate in phase, whereas adjacent FBAR sections vibrate in anti-phase when V2 = −V1. Let us define Yi = Y11 + Y12 and Ya = Y11 − Y12, which correspond to the input admittances for these inphase and anti-phase resonance modes, respectively. The design principles for the CRF [18] are that 1) the resonance frequency for Yi (or Ya) coincides with the anti-resonance frequency for Ya (or Yi), and 2) the shunt capacitance C0 is set at approximately G0/ωr, where ωr is the resonance frequency. The simulation setup shown in Fig. 3 gives Yi, whereas that shown in Fig. 7 gives Ya.
Fig. 8. Admittance characteristic of the designed coupled-resonator filter.
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2014
Fig. 11. Coupled-resonator filter configuration using slotted top and bottom electrodes.
Fig. 9. Calculated transfer function S21 of the designed coupled-resonator filter.
Fig. 8 shows Yi and Ya of a CRF designed under this principle. In this calculation, hs, hp, he, p, Np, and m were set as 0.09 μm, 1.52 μm, 0.09 μm, 2π μm, 30, and 0.52, respectively. The circuit impedance was set at ωrC0. It can be seen that the anti-resonance frequency of Yi coincides with the resonance frequency of Ya. Fig. 9 shows the transfer function S21 of the designed CRF. It is seen that a relatively flat and wide passband is realized. Relatively large spurious peaks are seen at 1.73 GHz and 1.86 GHz. They are due to spurious resonances in Ya, which can also be seen in Fig. 8. Achieved out-of-band rejection is somewhat limited at frequencies far from the passband. The main reasons are the electrostatic coupling between adjacent electrodes and the influence of spurious resonances far from the main resonance. Fig. 10 shows the calculated dispersion for this case. The target stopband can be seen from 1.819 to 1.905 GHz. As indicated in the figure, the resonance frequencies f ri of Yi and f ra of Ya correspond to the cut-off for the thickness
extension vibration with β = 0 and the lower stopband edge with β = π/p, respectively. Another stopband can be seen from 1.68 to 1.73 GHz. This may be due to the Bragg reflection of the TS2 mode, and causes a relatively large spurious peak in Ya at 1.73 GHz (see Fig. 8). It is expected that CRFs are also realizable when both the top and bottom electrodes are slotted as shown in Fig. 11. Electrical isolation between electric ports can be used to realize the balanced-to-unbalanced and/or impedance conversion functions [6]. However, because the layer on which the piezoelectric layer is grown critically influences the quality of the piezoelectric layer at the deposition, nonuniform bottom electrodes may not fit into the current FBAR/SMR fabrication process.
V. Conclusion This paper investigated the influence of the slotted top electrodes placed on the FBAR structure for the realization of CRFs. First, it was shown that, similar to a surface grating, the slotted top electrodes cause Bragg reflection and can be used to suppress transverse modes. Then it was shown that the slotted top electrodes can be used to design wideband CRFs. A CRF was designed and its performance was demonstrated. References
Fig. 10. Dispersion curves for the Mo/ZnO/Mo structure with slotted top electrodes when hs, hp, he, p, Np, and m were set as 0.09 μm, 1.52 μm, 0.09 μm, 2π μm, 30, and 0.52, respectively.
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