Patent Publication Number: US-11381220-B2

Title: Lateral bulk acoustic wave filter

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is a continuation of and claims priority to U.S. patent application Ser. No. 16/127,171, filed on Sep. 10, 2018. The disclosure of the prior application is considered part of and are incorporated by reference in the disclosure of this application. 
    
    
     BACKGROUND 
     Technical Field 
     This specification relates to thin film radio-frequency acoustic wave filters. 
     Background 
     Radio-frequency (“RF”) components, such as resonators and filters, based on microacoustic and thin-film technology are widely used in radio applications such as: mobile phones, wireless networks, satellite positioning, etc. Their advantages over their lumped-element, ceramic, and electromagnetic counterparts include small size and mass-production capability. 
     SUMMARY 
     This specification describes technologies for Lateral Bulk Acoustic Wave (“LBAW”) filters. In the present disclosure, LBAW filters include an input and an output resonator formed from a piezoelectric layer sandwiched between two pairs of electrodes. LBAW filters are formed from a piezoelectric layer sandwiched between two pairs of electrodes. One electrode from each pair is located on the top surface of the piezoelectric layer, and forms an input or an output of the LBAW. The input and output electrodes are separated by a gap. Each pair also has a counter electrode located on the bottom surface of the piezoelectric layer. By applying an alternating voltage across the piezoelectric layer at the input resonator, a mechanical resonance is formed in the piezoelectric layer below the input electrode. The piezoelectric layer thickness and the gap between electrodes can be designed such that this mechanical resonance is coupled across the gap to the output resonator. The frequency range at which such coupling occurs determines the achievable bandwidth (or width of passband) for the LBAW filter. 
     In general, one innovative aspect of the subject matter described in this specification can be embodied in an acoustic wave filter device that includes a piezoelectric layer, an input electrode and an output electrode located on a top surface of the piezoelectric layer and physically separated from one another, and a counter electrode having a top surface connected to a bottom surface of the piezoelectric layer. The input and output electrodes each include a base and at least one extension extending from the base. The at least one extension of the input electrode extending alongside and in a generally opposite direction to and separated by a gap width from an adjacent extension of the at least one extensions of the output electrode. Thickness of the piezoelectric layer and the gap width between adjacent extensions of the input and output electrodes are such that application of a radio frequency voltage between the input electrode and the counter electrode will create symmetric and antisymmetric acoustic thickness-extensional resonance modes in the piezoelectric layer. The at least one extension of the input or output electrodes has a width that changes from a first end of the at least one extension to a second end. 
     The foregoing and other embodiments can each optionally include one or more of the following features, alone or in combination. 
     The first end of the extension may be proximal to the base of the corresponding electrode, and the second end of the extension may be distal from the base of the corresponding electrode. The at least one extension of one of the input or output electrodes may have a width that decreases from the first end of the at least one extension to the second end. In some examples, the at least one extension of the other of the input or output electrodes has a width that increases from the first end of the at least one extension to the second end. The at least one extension of the input electrode can increase in width from the first end to the second end of the least one extension of the input electrode, and the at least one extension of the output electrode can decrease from the first end to the second end of the least one extension of the output electrode. The at least one extension of one of the input or output electrodes can have a width that increases from a first end of the at least one extension to a second end. 
     In some embodiments, the gap width between extensions is constant. In some embodiments, the width of the at least one extension of the input or output electrodes changes continuously from a first end of the at least one extension to a second end. 
     In some embodiments, the input and output electrodes each have a comb structure with interdigitated extensions. In some examples, the input and output electrodes have at least 10 extensions. 
     The acoustic wave filter can further include a Bragg reflector in contact with a bottom surface of the counter electrode. 
     In some embodiments, the piezoelectric layer thickness is at least 0.5 μm. The width of each extension may increase between 20% and 250% from a first end of at least one extension to a second end. The width of each extension may increase by between 25% and 100% from the first end to the second end. In some embodiments, the width of each extension increases from 4 μm to 5 μm from the first end to the second end. 
     The acoustic wave filter can be a laterally coupled bulk acoustic wave filter. 
     One innovative aspect of the subject matter described in this specification can be embodied in an acoustic wave filter device that includes a piezoelectric layer, an input electrode and an output electrode located on a top surface of the piezoelectric layer and physically separated from one another, and a counter electrode having a top surface connected to a bottom surface of the piezoelectric layer. The input and output electrodes each include a base and at least one extension extending from the base. The at least one extension of the input electrode extending alongside and in a generally opposite direction to and separated by a gap width from an adjacent extension of the at least one extensions of the output electrode. Thickness of the piezoelectric layer and the gap width between adjacent extensions of the input and output electrodes are such that application of a radio frequency voltage between the input electrode and the counter electrode will create symmetric and antisymmetric acoustic thickness-extensional resonance modes in the piezoelectric layer. The gap width between adjacent extensions of the input and output electrodes changes between the base of the input electrode and the base of the output electrode. 
     The foregoing and other embodiments can each optionally include one or more of the following features, alone or in combination. 
     In some embodiments, the gap width between adjacent extensions of the input and output electrodes increases between the base of the input electrode and the base of the output electrode. In some examples, the gap width between adjacent extensions of the input and output electrodes increases from 2 μm to 3 μm. 
     In some embodiments, the gap width between adjacent extensions of the input and output electrodes decreases between the base of the input electrode and the base of the output electrode. 
     The input and output electrodes each can have a comb structure with interdigitated extensions. 
     The subject matter described in this specification can be implemented in particular embodiments so as to realize one or more of the following advantages. LBAW filters described herein can be simpler to fabricate because they use only a single piezoelectric layer as compared to two in vertically stacked bulk acoustic wave (BAW) coupled resonator filters. LBAW filters can also operate at higher frequencies as surface acoustic wave (SAW) filters as their operation is determined more by piezoelectric layer thickness than IDT electrode dimensions. In some embodiments, LBAW filters can also achieve a wider bandwidth than BAW filters. LBAW filters can perform as filters with a single lithographic patterning step as compared to multiple (close to 10) lithographic steps used in BAW and can operate without reflectors needed in SAW, and thus is smaller in size. 
     In some embodiments, LBAW filters can be designed to suppress side bands. In some embodiments, LBAW filters can be designed to reduce ripple in the passband. 
     The details of one or more embodiments of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a schematic perspective view of a solidly-mounted LBAW filter. 
         FIG. 1B  is a schematic perspective view of a self-supported LBAW filter. 
         FIG. 1C  is a schematic planar view of an IDT electrode structure. 
         FIGS. 2A-B  are schematic diagrams of two types of propagating plate wave modes in LBAW piezo layer. 
         FIG. 3  is a dispersion curves for an exemplary LBAW. 
         FIG. 4A  is a schematic diagram of two resonant modes in an LBAW. 
         FIG. 4B  is an illustrative transmission response of an LBAW as a function of frequency. 
         FIG. 5  is an experimental transmission curve of an LBAW as a function of frequency. 
         FIG. 6  is a schematic illustration of the different acoustic modes formed under multiple electrode extensions. 
         FIG. 7  is an experimental plot of insertion loss versus frequency for an exemplary LBAW. 
         FIG. 8A  is a schematic top view of a section of a fan-shaped LBAW electrode with tapered extensions. 
         FIG. 8B  is a schematic top view of a fan-shaped LBAW electrode with tapered extensions. 
         FIG. 9A  is a schematic top view of a section of a fan-shaped LBAW electrode with tapered gap width. 
         FIG. 9B  is a schematic top view of a fan-shaped LBAW electrode with tapered gap width. 
         FIG. 10A  is a plot of insertion loss versus frequency for two LBAW filters with electrode extensions of constant width and spacing. 
         FIG. 10B  is a plot of insertion loss versus frequency for the combination of the filters in  FIG. 10A . 
         FIG. 11  is a plot of insertion loss versus frequency comparing the performance of an LBAW with electrode extensions of constant width and spacing to an LBAW with a fan shaped electrode with tapered extensions. 
         FIG. 12  is a plot of insertion loss versus frequency comparing the performance of LBAW filters with electrode extensions of constant width and spacing to an LBAW with a fan shaped electrode. 
     
    
    
     Like reference numbers and designations in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
       FIGS. 1A, 1C  show an example of an LBAW filter (or resonator)  100  with input  150  and output  170  electrodes that have an interdigitated geometry (also called “interdigital transducer” or “IDT” LBAW). LBAW filter  100  includes a piezoelectric (“piezo”) layer  110 , having a thickness d, an IDT electrode structure  102  located on the top surface of the piezo layer, and a bottom counter electrode  120  located on the bottom surface of the piezo layer. IDT electrode structure (“IDT”)  102  includes two comb-shaped electrodes,  150  and  170 , of conductive material, e.g., metal or polysilicon. IDT electrodes  150  and  170  have parallel extensions  150   a  and  170   a , respectively, that provide the “tines” or “teeth” or “fingers” of the “comb.” Electrode  150  and counter electrode  120  form an input resonator with piezo layer  110 . Electrode  170  and counter electrode  120  form an output resonator with piezo layer  110 . 
     Acoustic vibrations are created in piezo layer  110  by applying an oscillating (or alternating) input voltage across IDT electrode  150  and bottom counter electrode  120  at an input port  160 . The applied voltage is transformed into a mechanical (e.g., acoustic) vibration via the piezoelectric effect. Under resonance conditions (e.g., with certain acoustic resonance modes, as detailed further below), this vibration can create a standing wave under input electrode  150  and an evanescent wave (with exponentially decaying amplitude) in the gap region  190 . With appropriate selection of vibration frequencies and gap width G, the standing wave can be coupled mechanically across gap  190  from the piezo region under electrode  150  to piezo region under electrode  170  by the evanescent wave and create a similar standing wave in piezo layer  110  under electrode  170 . The standing wave under electrode  170  results in an output signal voltage with the same frequency at an output port  180  via the reverse piezoelectric effect. The frequency range at which this coupling occurs in mechanical resonance with strong piezoelectric coupling forms the passband (or bandwidth) of LBAW filter  100 . In some example, the frequency range is between 1.8 and 1.95 GHz. As discussed further below, the thicknesses and geometries, and spacing of the various layers of LBAW  100  can be tuned to change the RF response and passband of the filter. 
     A reflecting structure  130  can serve to isolate the vibration in piezo layer  110  from an underlying substrate  140  and to prevent acoustic leakage. The reflecting structure can be a stack of thin layers, for example, a Bragg reflector composed of alternating high and low acoustic impedance (“Z ac ”) material layers. The thickness of these layers can be designed such that the frequencies with and near the passband of LBAW filter are reflected back into piezo layer  110  and all other frequencies pass through the mirror. 
     In some embodiments, LBAW  100  is does not directly overlie substrate  140  (as shown in  FIG. 1A ), but is self-supported, as shown in  FIG. 1B . In such an arrangement, substrate  140  and mirror  130  are replaced by an air gap, with portions of piezo that extend laterally past the region in which LBAW  100  is fabricated being supported by substrate  140 . 
     In some embodiments, as shown in  FIG. 1C , extensions  150   a  and  170   a  are rectangular and have a width W, length L, and are spaced by gap width G. Each electrode  150  and  170  has one or more extensions  150   a  and  170   a  respectively. The total number of electrode extensions is designated as K. 
     Although  FIG. 1C  shows rectangular interdigital electrodes  150 / 170  with parallel extensions  150   a / 170   a  of same geometry and spacing G, other electrode geometries are also contemplated. Design considerations include the gap between electrodes, the length of the electrode, and the number, if any, and shape of electrode extensions. The gap can be used to control coupling between the input and output electrodes. Longer electrodes can also increase coupling. The number of extensions K can be used to control the bandwidth and/or to increase coupling while conserving the area taken up by the electrodes. In some embodiments, the electrodes are composed of rectangular strips, with two or more extensions (e.g., K≥2). For example, each extension can be a rectangular strip. In some embodiments, the electrodes are concentric circles or spirals having a common axis. 
     Piezo layer  110  can be formed from various piezoelectric materials. Exemplary materials include ZnO, AlN, CdS, PZT, LiNbO 3 , LiTaO 3 , quartz, KNN, BST, GaN, Sc alloyed AlN, or the aforementioned materials doped or alloyed with an additional element. Doping can be used to improve or tailor electromechanical properties of piezo layer  110 . As detailed further below, piezo layer thickness d is selected such that thickness-extensional modes near the frequencies of the desired bandwidth of the LBAW filter are produced in the piezo layer. In some embodiments, piezo layer thickness d is 20% to 50% of λ z , or 30% to 45% of λ z , where λ z  is the wavelength of the piezoelectric vibration in the thickness direction. In some embodiments, d is 1500 nm to 2500 nm, or 1800 to 2200 nm. 
     Thin film IDT  102  can be composed of various materials. In some embodiments, IDT electrodes  150  and  170  are metal. For example, the electrode material can include Al, Mo, Pt, Cu, Au, Ag, Ti, W, Ir, Ru, or multilayers of metals and/or metals doped with additional materials, e.g., AlSi, AlSiCu, polysilicon, etc. Doping can be used to improve or tailor IDT electric or mechanical properties. 
     Although  FIG. 1A  shows a single common counter electrode  120 , filter  100  can include separate electrodes for the input and output resonators. Various materials are suitable for the counter electrode(s) (e.g., electrode  120 ). For example, the electrodes can include a metal, such as Al, Mo, Pt, Cu, Au, Ag, Ti, W, Ir, Ru, or multilayers of metals and/or metals doped with additional materials, e.g., AlSi, AlSiCu, etc. Doping can be used to improve or tailor IDT electric or mechanical properties. For example, the electrodes can be Ti+Mo, Ti+W, AlN+Mo, or Al+W. The electrodes can be multilayered. The electrodes can have a special thin seed layer deposited below the electrode. 
     Reflecting structure  130  can be composed of alternating layers of different materials. For example, reflecting structure  130  can include alternating layers of two of: tungsten (W), SiO 2 , silicon (Si), and carbon (C). For example, layers of high acoustic impedance include W, Mo, Ir, Al 2 O 3 , diamond, Pt, AlN, Si 3 N 4 . Layers of low acoustic impedance can include SiO 2 , glass, Al, Ti, C, polymers, or porous materials. Layer of Si provides an intermediate acoustic impedance. Various materials are suitable for the substrate  140 , such as Si or SiO 2  or glass, sapphire, quartz. Substrate  140  materials can have high electrical resistivity. The substrate can have a thickness appropriate for RF applications, such as integration into mobile phone platforms. For example, the substrate can have a thickness less than 500 microns, or less than 200 microns. For example, Si wafers can be purchased with a thickness of 675 μm and thinned down to a achieve desired device thickness, e.g., for mobile platforms. 
     Modeling of the acoustic response of LBAW  100  can provide guidance on how to tune the design parameters for individual elements of the structure to achieve desired bandpass properties. For example, LBAW  100  can be designed to have resonance modes at specific frequencies. In general, the geometry of various LBAW  100  components can be selected to achieve various acoustic properties. LBAW  100  properties can depend on the combination of these geometries, which may not be independent of one another. 
     In piezoelectric layer  110 , different bulk acoustic vibration modes can arise at different excitation frequencies f of input voltage (e.g., at port  160 ). Acoustic vibrations in piezo layer  110  can propagate laterally as Lamb waves (or plate waves), wherein particle motion lies in the plane that contains the direction of wave propagation and the plate normal (e.g., the z-axis in  FIG. 1A ). Two such modes are shown in  FIGS. 2A-2B . Referring to  FIG. 2A , a thickness-extensional (TE or longitudinal) bulk mode  200  has particle displacement  210  perpendicular to the propagation direction (in the z-direction). Referring to  FIG. 2B , a second order thickness-shear (TS2) bulk mode  220  has particle displacement  230  parallel to the propagation direction (in the y-direction). For both modes, the lowest frequency at which resonance in the thickness direction can arise is when the thickness d of piezo layer  110  is equal to an integer number of half wavelengths λ z  (disregarding the thickness of electrodes  150 / 170 ); in other words, when 
               d   =       N   ⁢     λ   z       2       ,         
where N is an integer that indicates the order of the resonance. For the TE1 mode,
 
               d   =       λ   z     2       .         
As discussed further below, the width W of the electrodes and the gap G between electrodes can be designed such that TE1 mode standing waves with certain lateral wavelengths λ ∥  are formed that can couple through their evanescent tails across gap G to create two mechanical resonant modes.
 
     Acoustic properties of an LBAW resonator  100  can be described with dispersion curves. Referring to  FIG. 3 , an example dispersion curve for an LBAW  100  shows the lateral wave number k ∥  of the vibration, where 
                 k               =       2   ⁢   π       λ                   ,         
as a function of frequency f. The first-order longitudinal (thickness extensional, TE1) vibration mode, in which the combined thickness of the piezoelectric layer d and the thickness of electrode(s)  150  or  170  contains approximately half a wavelength of the bulk vibration, λ z /2, and the second-order thickness shear (TS2) mode, in which the bulk vibration is perpendicular to the thickness direction (z-axis in  FIG. 2B ) and approximately one acoustic wavelength A is contained in the combined piezoelectric layer thickness d and the thickness of electrode(s)  150  or  170 , are denoted in the figure. The TE1 mode is the darker portion of each dispersion curve, and TS2 mode is the lighter region of each dispersion curve. The top curve (“no electrode”) represents the dispersion properties of the piezoelectric layer under the gap  190 . The bottom curve (“electrode”) represents the dispersion properties of the piezoelectric layer under electrodes  150 / 170 , also known as the active region. More specifically, where the “electrode” curve intersects k=0, the TE1 mode has approximately λ z /2 contained in the combined thickness of the electrodes  150  or  170  and the piezoelectric layer. This is approximate because the wave can extend into the Bragg reflector. “No Electrode” curve intersection with k=0 lines shows the modes where approximately λz/2 is contained in the combined thickness of the bottom electrode only and the piezolayer. This type of dispersion, in which the TE1 mode has increasing k ∥  with increasing frequency f, is called Type 1. The difference in intersect k ∥ =0 frequencies between electrode and non-electrode areas determined the hard limits for the achievable bandwidth of the filter. The gap width G, electrode width W, and number of extensions K can be used to vary the coupling strength within the limits set by the dispersion difference.
 
     In some embodiments, LBAW  100  can be designed to produce Type 1 dispersion. For example, piezo layer  110  materials can be selected in which Type 1 dispersion can occur. For example, ZnO can be used. In another example, appropriate design of acoustic Bragg reflector  130  can help achieve Type 1 dispersion. For example, using Aluminum nitride (“AIN”) for piezo layer  110  can typically produce a Type 2 dispersion, where TE1 mode behaves non-monotonically having initially decreasing k 81   with increasing frequency f, and then increasing k ∥  with increasing frequency f, (roughly similar to what is described in the dispersion curves of in  FIG. 3  but with TE1 and TS2 interchanged). However, in some embodiments, with an appropriate design of the reflecting structure  130  (e.g., acoustic Bragg reflectors), the LBAW  100  can use AIN in piezo layer  100  and still achieve a Type 1 dispersion. See for example Fattinger et al. “Optimization of acoustic dispersion for high performance thin film BAW resonators,” Proc. IEEE International Ultrasonics Symposium, 2005, pp. 1175-1178. 
     In  FIG. 3 , positive values of k∥ denote real wave numbers (propagating waves) and negative k∥ values correspond to imaginary wave numbers (evanescent waves). For a resonance to arise, the acoustic energy must be trapped inside the LBAW resonator structure. In the thickness (z-axis) direction, isolation from the substrate (using reflecting structure  130 ) can be used for energy trapping. In the lateral direction, energy trapping can occur when an evanescent wave forms outside the electrode region (e.g., on the “no electrode” curve). To get resonant coupling between the two resonators (e.g., electrodes  150 / 170  and  120 ) of an LBAW, standing waves of a TE1 mode form in the active regions of the piezo layer (under the electrodes), and evanescent waves form in the “no electrode” region. In other words, k∥ is positive for the TE1 “electrode” curve and negative for the TE1 “no electrode” curve. According to  FIG. 3 , this occurs in the labeled “trapping range” frequency range. Energy trapping can be easier to realize in Type I dispersion. Without wishing to be bound by theory, with monotonically increasing dispersion curves as the thick TE1 lines in  FIG. 3 , for the “Electrode”, at a single frequency in the trapping range there is either a single imaginary wave number available or above the trapping range a single real wave number. The former means that the TE1 does not propagate outside the electrode and the latter that the TE1 can couple to a propagating wave outside the electrode and thus “leak”. The Type 2 dispersion can be described by similar curves but with the TE1 and TS2 curves interchanged. The fact that the curve in Type 2 is non-monotonic means that at a given frequency there may be several real wavenumbers. Having several wavenumbers for a frequency means propagating waves are available outside the electrode, which can cause a “leak”. 
       FIGS. 4A-4B  illustrate the relationship between standing wave resonance modes and the LBAW bandgap. Referring to  FIG. 4A , a portion of LBAW  100  includes two adjacent electrodes  401  and  402  with width W (e.g., corresponding to extensions  150   a  and  170   a  of respective electrodes  150  and  170  of  FIG. 1A ). The bandpass frequency response of LBAW  100  is formed by two (or more) laterally standing resonance modes,  410  and  420 , arising in the structure. Lateral standing wave resonance can arise when plate waves are reflected from edges of the adjacent electrodes  401  and  402 . In the even mode resonance  410 , the piezo layer under both electrodes  150  and  170  vibrates in-phase, whereas in the odd mode resonance  420 , the phases are opposite. The even lateral standing wave resonance can arise when the total width of the structure is roughly equal to half of the lateral wavelength λ ∥  of the mode: 
                 λ     e   ⁢   v   ⁢   e   ⁢   n       2     =         λ               2     ≈       2   ·   W     +     G   .               
In the limit of infinitely small gap width G, λ even  approaches the total width from below. As shown in  FIG. 4A , λ even  gets smaller when G gets larger and gets larger when G gets larger. In case of small gap (e.g., zero-gap) λ even  gets close to 4W and in case of a large gap λ even  gets close to 2W. The odd lateral standing wave resonance can arise when the width of the electrode is roughly equal to half of the lateral wavelength λ ∥  of the mode:
 
     
       
         
           
             
               
                 
                   λ 
                   
                     o 
                     ⁢ 
                     d 
                     ⁢ 
                     d 
                   
                 
                 2 
               
               = 
               
                 
                   
                     λ 
                     
                        
                        
                     
                   
                   2 
                 
                 ≈ 
                 W 
               
             
             . 
           
         
       
     
     Referring to  FIG. 4B , the even  410  and odd  420  modes are shown as transmission peaks as a function of input frequency f for an LBAW with Type 1 dispersion. For Type 1 dispersion, the even mode  410  has a longer wavelength and is lower in frequency than the shorter-wavelength odd mode  420 . The frequency difference  430  between the modes determines the achievable bandwidth of LBAW filter  100 , and depends on the acoustic properties of the structure and on the dimensions of IDT resonator  102 . Acoustic coupling strength can be defined in terms of the (resonance) frequency difference between even (symmetric) and odd (antisymmetric) resonances: 
                   f   asymm     -     f   symm         f   0       ,         
where f symm  and f asymm  are the symmetric and antisymmetric eigenfrequencies, respectively, and f 0 =(f symm +f asymm )/2 is the center frequency between the two modes.
 
     In some embodiments, increasing the number of extensions (e.g., 150a and  170   a ) in each electrode (e.g.,  150  and  170 ) can increase the frequency difference between the even and odd mode in the LBAW, and thus increase the bandwidth. This effect can result from the fact that the lateral wavelength of the odd mode can depend on the periodicity of the electrode structure (e.g., width W), while the even mode can depend on the entire width of the structure (e.g., adding up all widths W and gaps G). For example, if the total number of electrode extensions K, the electrode width is W, and the gap width is G, the wavelength λ ∥  of the lateral acoustic wave at the even mode resonance frequency approaches or is slightly shorter than: 
                 λ     e   ⁢   v   ⁢   e   ⁢   n       2     ≈       K   ·   W     +     K   ·     G   .               
The odd lateral standing wave resonance in this structure, however, approaches or is slightly larger than:
 
                   λ     o   ⁢   d   ⁢   d       2     ≈   W     .         
Additionally, or alternatively, in some embodiments, the total width of the structure K·W+K·G can be such that the highest-order mode trapped in the structure is the desired odd mode resonance. For example, K can be 31, W can be 3 μm, and G can be 2 μm.
 
     In some embodiments, the number of electrode extensions K is between 2 and 200, or between 10 and 60. In some embodiments, the length L of electrode extensions can be between 50 μm and 2000 μm, or between 70 μm and 500 μm. 
     In some embodiments, the gap G is selected to allow coupling of the evanescent tails of standing waves formed under electrodes  150  and  170 . For example, the gap G between electrode extensions can be 0.1 μm and 10 μm, or between 2 μm and 5 μm. 
     In some embodiments, the topology of the electrodes  150  and  170  can be designed such that the gap width G provides good enough coupling between electrode extensions to create a single even mode  410  across the entire width of the structure. For example, the gap width G can be 2%-300%, or 10%-100% of the evanescent acoustic wave&#39;s decay length, i.e. the length at which amplitude A=A 0 ·e −1  of the original amplitude A o , in the gap at the desired even resonance mode. The gap width G can be optimized. Decreasing the gap to a too small width (1) can eventually pull the even and odd modes too far from each other creating a dip in the passband, (2) can lead to reduced coupling coefficient for the odd mode, or (3) can increase capacitive feedthrough from finger to finger causing poor out of band attenuation. 
     In some embodiments, the gap width G can be defined with respect to piezo layer thickness d. For example, G can be designed to be 10% to 300% of d, or 25% to 150% of d. 
     In some embodiments, the width of electrode extensions W can be between 0.1 μm and 30 μm, or between 2 μm and 5 μm. In some embodiments, W can be designed such that the wavelength λ ∥  of the lateral acoustic wave at the desired odd mode resonance frequency λ odd  is obtained. 
     In some embodiments, the electrode width W is designed such that multiple half-wavelengths cannot fit within the electrode width. For example, W can be designed to be smaller than the lateral acoustic wave&#39;s wavelength λ ∥  at the desired odd resonance mode, e.g., where λ ∥ =λ odd . 
     In some embodiments, the thicknesses of various LBAW  100  components can be selected to achieve various acoustic properties and may be interdependent. For example, the piezo layer  110  thickness d (minimum and maximum value) can first be determined with respect to the acoustic wavelength in the piezo material (λ) at the operation frequency f. In some embodiments, thicknesses (min and max) of the other LBAW  100  layers can be selected based on the choice of piezo thickness d. For example, the combined thickness of the electrodes (including the counter electrode  120 ) and the piezoelectric layer can be selected to be approximately half a wavelength of the mode that is being used, for example longitudinal bulk wave for the thickness extensional mode. Fundamental modes with N=1 (the first mode, i.e., first harmonic) can allow for greater coupling, but N&gt;1 modes are also possible. For example, the thickness of electrodes  150  and  170 , bottom electrode  120 , and reflecting structure  130  can be defined as a percentage of piezo layer thickness d. In some embodiments, once all thickness are selected, the geometry of the electrode extensions  150   a  and  170   a , such as number K, width W, and length L, can be tuned to match the LBAW  100  electrical impedance to the system impedance. Without wishing to be bound by theory, impedance matching can help avoid losses and reflections in the system. 
     In some embodiments, thickness of electrodes  150  and  170  is between 1% to 30% of d, or 5% to 25% of d, or 3% to 15% of d. 
     In some embodiments, the thickness of bottom electrode  120  is between 5% to 50% of d, or 10% to 30% of d, or 10% to 20% of d. 
     In some embodiments, where the reflecting structure  130  is a Bragg reflector, the alternative layers of the reflector can be designed such that the required reflectivity of passband wavelengths is obtained. For example, the thickness of each layer can be equal to or less or more than one quarter of the acoustic wavelength λ z  in the thickness direction to reflect the odd and even TE1 resonance modes. In some embodiments, a single layer in the Bragg reflector can be 15% to 80% of d, or 20% to 70% of d. 
     The mass loading of the IDT  102 , determined by the thickness and material of electrodes  150  and  170 , can be designed such that the frequency difference between the k ∥= 0 frequency of the electrode region&#39;s TE1 mode and the outside electrode region&#39;s TS2 mode is small. Without wishing to be bound by any particular theory, when the frequency difference between outside region&#39;s TS2 mode and electrode region&#39;s TE1 mode is small, the trapping range is large. More particularly, the k ∥= 0 frequency of the outside region&#39;s TS2 mode can be 95%-99% of the electrode region&#39;s TE1 cutoff frequency. The frequency difference between the outside region&#39;s TS2 and outside region&#39;s TE1 modes&#39; k ∥= 0 frequencies is designed to be large, e.g. 5%-15%, for example 6.5%-7.5%, of the electrode region&#39;s TE1 mode cutoff frequency. 
     According to certain embodiments of the present invention, the k ∥= 0 frequency of the outside region&#39;s TS2 mode is greater than, or equal to 98%, or between 98% and 99.5%, or is 98.9% of the electrode region&#39;s TE1 cutoff frequency. Similarly, the frequency distance expressed as the frequency difference between electrode region TE1 and outside region TS2 k ∥ =0 frequencies: 
                 electrode   ⁢           ⁢   TE   ⁢           ⁢   1     -     outside   ⁢           ⁢   TS   ⁢           ⁢   2         outside   ⁢           ⁢   TS   ⁢           ⁢   2           
should be small, for example on the order of 1%. As an example, said frequency distance can be between 0.2% and 2.1%, or between 0.5% and 1.8%, or between 0.8% and 1.5%, or for example, 1.1%.
 
       FIG. 5  shows a curve of insertion loss IL versus frequency f for an exemplary LBAW  100 . The curve shows two passbands with peak  510  corresponding to TE1 waves and peak  520  corresponding to TS2 waves. As discussed above, the width of each passband is determined by the frequency difference of the even and odd modes for the respective type of wave. Here, the TS2 modes correspond to sideband  520   a  (also referred to herein as “TS2 passband”), and the TE1 modes correspond to passband  510   a  (also referred to herein as “TE1 passband”). In some embodiments, LBAW  100  is designed to suppress peak  520  corresponding to TS2 modes, while maintaining the properties of peak  510  corresponding to TE1 modes. Without wishing to be bound by any particular theory, TE1 mode operation can be selected because piezo thin film materials have electromechanical coupling that is stronger in the thickness direction. In other words, TE1 longitudinal mode vibrations couple more efficiently to the electrical excitation over the thickness of piezo layer  110 . 
     In some embodiments, LBAW  100  can be designed to have a passband for TE1 modes between 0.5 and 10 GHz, or between 1 and 4 GHz. In some examples, TE1 passband is between 1.8 and 3.7 GHz. The limits of the passband can incorporate design considerations. For example, the dimensions of the device can grow very large or very small. Too large dimensions may take too much space and cause inefficiencies. Too small dimensions can deteriorate performance due to thin and narrow electrodes leading to resistance and losses. In some embodiments, LBAW  100  can be designed to have a TE1 passband width  510   a  of 0.5-15% relative to center frequency, e.g., 10% relative to center frequency, or 5%, or 2%, or 1%. In some embodiments, the insertion loss at the passband is −7 dB to −0.5 dB or −5 dB to −1.5 dB. 
     Varying the width of extensions or their spacing along the length of the extension can be used to reduce the ripple in the LBAW passband, such as the ripple in the passband  510   a  shown in  FIG. 5  These ripples are in part due to intermediate standing modes that form between IDT electrodes.  FIG. 6  is a schematic illustration of some of these modes, where the gray zones represent electrode extensions and the curves represent the amplitude of the acoustic modes forming under the extensions in the LBAW piezo layer. The symmetric  602  and antisymmetric  604  modes can form the frequency boundaries of the passband, while intermediate modes  606  can contribute to the ripple within the passband frequencies. 
     In contrast, as shown in  FIG. 7 , by using extension with widths that vary along their lengths, the passband  710  is much flatter. Without wishing to be bound by theory, varying extension width or gap spacing between extensions can cause these intermediate modes to occur at slightly different frequencies between along the length of the extensions, reducing the ripple effect in the passband. 
     The various embodiments discussed above can be implemented with extensions (e.g.,  150   a  and  170   a ) that are tapered, or gaps widths G that taper, or a combination of both. Referring to  FIG. 8A , a section  800  of interdigitated electrodes  850  and  870  with tapered extensions (e.g., where the width changes from one end to the other) is shown. Electrodes  850  and  870  are physically separated from one another. Referring to  FIG. 8B , a full set  820  of interdigitated (fan shaped) comb electrodes (e.g.,  850  and  870 ) with tapered extensions is shown. Electrode  850  has a base  802  and extensions  806  extending from base  802 , and electrode  870  has a base  804  and extensions  808  extending from base  804 . The extensions from input electrode  850  extend alongside and in generally opposite direction to and separated by a gap width  810  from adjacent extensions of output electrode  870 . Extensions  806  have a smaller width  806   a  in regions distal from base  802 , and a larger width  806   b  in regions proximal to base  802 . Conversely, extensions  808  have a larger width  808   a  in regions distal from base  804 , and a smaller width  808   b  in regions proximal to base  804 . 
     In some embodiments, the width of electrode extensions changes monotonically, e.g., continuously, along the length of the extension between proximal and distal regions. In some embodiments, the width changes in a step-wise manner, or a non-linear manner (e.g., parabolic), or in a randomized manner (e.g., dithered, with increased variation along the finger length). In some embodiments, the width increases or decreases along the length of the extension. In some embodiments, the width of the electrode can change by 10-100% along its length. So the wider end of the extension can be 10-250% wider, e.g., 10-25%, 25-50%, 50-75%, 75-100%, or 100%-250%, wider than the narrower end. In some embodiments, the width of the electrode can change from 10 μm to 15 μm, or 2 μm to 7 μm, or 3 μm to 4 μm or 3.75 μm to 4 μm, depending on the selected bandpass frequency range. 
     In some embodiments, the gap width remains constant between electrodes from base  802  to base  804 . In some embodiments, the gap width is between 0.6 μm and 3 μm, for example 1 μm. 
     Referring to  FIG. 9A , a section  900  of interdigitated electrodes  950  and  970  with tapered gap width is shown. Electrodes  950  and  970  are physically separated from one another. Referring to  FIG. 9B , a full set  920  of interdigitated comb electrodes (e.g.,  950  and  970 ) with tapered gap width is shown. Electrode  950  has a base  902  and extensions  906  extending from base  902 , each with width  906   a . Electrode  970  has a base  904  and extensions  908  extending from base  904 , each with width  908   a . The extensions from input electrode  950  extend alongside and in generally opposite direction to and separated by a gap width from adjacent extensions of output electrode  970 . The gap width between adjacent extensions of the input  950  and output  970  electrodes changes between base  902  of the input electrode and base  904  of the output electrode. Adjacent extensions from electrodes  950  and  970  are separated by a larger gap width  910   a  at a region proximal to base  902  and by a smaller gap width  910   b  at a region distal to base  902 . 
     For fan shaped electrodes with tapered gap width, the gap width increases or decreases along the length of the gap from one base to the other. In some embodiments, e.g., if the electrodes extension  906 ,  908  have uniform width along their length, the gap width changes continuously between the bases of the two electrodes (e.g.,  902  and  904 ). In some embodiments, the gap width changes in a step-wise manner. In some embodiments, the gap width can have a change between 0.5 μm and 3 μm, for example 2 μm to 3 μm, or 1.25 μm to 1.75 μm, or 0.75 μm to 2 μm, depending on the selected bandpass frequency range. 
     In some embodiments, the extension width remains constant along its length. In some embodiments, the extension width is between 1 μm and 30 μm, for example 4 μm. In embodiments where the gap width G and extension width W are both varied, the ratio G/W is between 0.25 to 5. 
     Having the width of extensions or the width of the gap between the extensions vary along the length of the gap or extensions, respectively, can be used to select the acoustic properties of an LBAW filter (e.g., change the acoustic properties relative to a filter having extension and gaps both of uniform width along their lengths). For example, interdigitated electrodes can be designed to suppress TS2 sidebands, or other parasitic TS1 sidebands that do not fall into the desired LBAW passband frequency. 
     Without wishing to be bound by any particular theory, using interdigitated electrodes that vary in extension widths or spacing along their length can be acoustically similar to having a series of parallel connected LBAW filters with constant but different extension widths and spacing. As such, this theoretical series of different LBAW filters can be selected to have TS2 sidebands at different frequencies. Thus, when the LBAW filters are connected in parallel (simulating the effect of a fan shaped LBAW electrode), the TS2 sidebands average out (or “smear” out) between those frequencies and the TS2 band is generally suppressed. The effect on the TE1 passband is weaker because it is a wider band that is less effected by electrode geometry. Similar methods can be used to suppress parasitic TS1 bands. In some embodiments, TS2 sideband suppression takes precedence over TS1 sideband suppression because TS2 sidebands generally introduce higher insertion losses at frequencies closer to the passband of an LBAW filter. 
     A proof of concept is shown in  FIGS. 10A-10B .  FIG. 10A  is a plot of measured frequency versus insertion loss for two separate LBAW filters. Curve  1002  corresponds to an LBAW filter with regular interdigitated electrodes having a constant gap width of 1.5 μm and constant extension widths of 3 μm. Curve  1004  corresponds to an LBAW filter with regular interdigitated electrodes having a constant gap width of 1.5 μm and constant extension widths of 4 μm. The TS2 peak  1002   a  and parasitic TS1 peak  1002   b  of curve  1002  are located at different frequencies from TS2 peak  1004   a  and parasitic TS1 peak  1004   b  of curve  1004 . 
       FIG. 10B  shows the resulting curves of frequency versus insertion loss from the combination of the two filters in  FIG. 10A . Curve  1006  is synthesized from the measured responses of the filters in  FIG. 10A  by cascading the filters so that the output of the first filter is connected to the input of the second filter. Curve  1008  is a measured response from a filter similar in geometry (e.g., fan shaped) to the one shown in  FIG. 8B , with constant gap width of 1.5 μm and the width of extensions varying from 3 μm to 4 μm along their length. The similarity of curves  1006  and  1008  demonstrates that in-series LBAW filters perform similarly to fan shaped IDT LBAW filters of varying extension width. Moreover, TS2 peaks  1010  and parasitic TS1 peaks  1012  for both curves are significantly lower than for the LBAW filters separately, as shown in  FIG. 10A . In sum, fan shaped LBAW electrodes can be effective for sideband suppression. 
       FIG. 11  shows additional experimental confirmation that fan shaped electrode extensions of varying (tapered) width can be used to suppress sidebands in a plot of frequency versus insertion loss. Measured curve  1104  corresponds to a regular LBAW filter with interdigitated electrodes having a constant gap width of 1.5 μm and constant extension widths of 4 μm. Measured curve  1002  for the fan shaped LBAW electrode corresponds to interdigitated fan shaped electrodes with constant gap width of 1.5 μm and extension widths varying from 3 μm to 4 μm. Curve  1002  has measurably lower insertion loss for the TS2 peak  1106  and parasitic TS1 side peak  1108 . The width and insertion loss of the passband can be optimized by conventional methods. 
     The effect of different electrode geometries can be predicted by simulation. In some embodiments, a continuously varying electrode width can be approximated by dividing the electrode into multiple sections (e.g.,  3  section) and summing the result. 
     Referring to  FIG. 12 , simulated insertion loss versus frequency curves are shown for three regular LBAW electrode geometries versus a fan-shaped LBAW electrode geometry. Curves  1202 ,  1204 , and  1206  correspond to regular LBAW geometries of 2 μm gap width and 10 extensions each with constant extension widths of 4 μm, 4.5 μm, and 5 μm respectively. Curve  1208  corresponds to an LBAW with a fan-shaped IDT electrode of extension widths varying from 4 μm to 5 μm along the length of the extension, with a 2 μm gap width between extensions. Curve  1208  is formed by combining the three other curves to approximately simulate a tapered LBAW structure. As predicted, the lower frequency region  1208   a  of curve  1208  shows a reduction of ripple in the passband between 1920 MHz and 1940 MHz.