LOW-LOSS AND WIDE-BAND ACOUSTIC DELAY LINES USING ALUMINUM NITRIDE THIN FILMS

Abstract

A piezoelectric thin film (PTF) is located above a carrier substrate. The PTF can be an aluminum nitride thin film adapted to propagate an acoustic wave in at least one of a first mode excited by an electric field oriented at least partially in a longitudinal direction along a length of the PTF or a second mode excited by the electric field oriented in a thickness direction of the PTF. A first interdigitated transducer (IDT) is disposed on a first end of the PTF and converts a first electromagnetic signal, traveling in the longitudinal direction, into the acoustic wave. A second IDT is disposed on a second end of the PTF with a gap between the second IDT and the first IDT. The second IDT is to convert the acoustic wave into a second electromagnetic signal, and the gap determines a time delay of the acoustic wave.

Claims

1. An apparatus comprising: a piezoelectric thin film located above a carrier substrate, wherein the piezoelectric thin film is one of a c-axis aluminum nitride (AlN) or scandium aluminum nitride (ScAlN) thin film adapted to propagate an acoustic wave in at least one of: a first mode excited by an electric field oriented at least partially in a longitudinal direction along a length of the piezoelectric thin film; or a second mode excited by the electric field oriented in a thickness direction of the piezoelectric thin film; a first interdigitated transducer (IDT) disposed on a first end of the piezoelectric thin film, the first IDT to convert a first electromagnetic signal, traveling in the longitudinal direction, into the acoustic wave; a second IDT disposed on a second end of the piezoelectric thin film with a gap between the second IDT and the first IDT, the second IDT to convert the acoustic wave into a second electromagnetic signal, and the gap to determine a time delay of the acoustic wave before output of the second electromagnetic signal; and a floating bottom electrode disposed beneath the piezoelectric thin film.

2. The apparatus of claim 1, wherein the first mode is a first-order antisymmetric (A1) mode.

3. The apparatus of claim 1, wherein the second mode is one of a fundamental symmetric (S0) mode or a first-order symmetric (S1) mode.

4. The apparatus of claim 1, wherein the piezoelectric thin film is suspended above the carrier substrate.

5. The apparatus of claim 1, wherein the piezoelectric thin film is disposed on a high-acoustic impedance layer interposed between the piezoelectric thin film and the carrier substrate, the high-acoustic impedance layer comprising one of silicon (Si), sapphire, fused silica, quartz, silicon carbide (SiC), diamond, aluminum nitride (AlN), aluminum oxide (Al.sub.2O.sub.3), tungsten, molybdenum, platinum, or combinations thereof.

6. The apparatus of claim 1, wherein the piezoelectric thin film is disposed on a combination of a plurality of high-acoustic impedance layers and a plurality of low-acoustic impedance layers interposed between the piezoelectric thin film and the carrier substrate.

7. The apparatus of claim 6, wherein respective ones of the plurality of the low-acoustic impedance layers are alternately disposed on respective ones of the plurality of high-acoustic impedance layers.

8. The apparatus of claim 1, further comprising: a waveguide inside of which is disposed the piezoelectric thin film, the first IDT, and the second IDT; a first port coupled to the first IDT, the first port to receive the first electromagnetic signal; and a second port coupled to the second IDT, the second port to output the second electromagnetic signal.

9. The apparatus of claim 8, wherein the first IDT comprises at least a transducer unit cell comprising: a ground line coupled to the first port; a back half of a first floating top electrode, wherein a center of the first floating top electrode comprises a first reflection center; a front half of a second floating top electrode, wherein a center of the second floating top electrode comprises a second reflection center; a first transduction electrode coupled to the ground line of the first port; a signal line coupled to the first port; and a second transduction electrode coupled to the signal line and disposed between the first floating top electrode and the first transduction electrode, wherein a transduction center is located between the first transduction electrode and the second transduction electrode.

10. The apparatus of claim 9, wherein the first floating top electrode and the second floating top electrode are embedded reflectors.

11. The apparatus of claim 9, wherein the first reflection center is located at a first end of the transducer unit cell at a first distance away from the transduction center, the second reflection center is located at a second end of the transducer unit cell at a second distance away from the transduction center, the second distance is different from the first distance, and wherein the first transduction electrode is located between the transduction center and the second reflection center.

12. The apparatus of claim 9, wherein the first reflection center is located at a first distance from the transduction center on a first side of the transduction center and the second reflection center is located at a second distance different than the first distance from the transduction center on a second side of the transduction center, the second side being opposite from the first side such that a first plurality of components of the acoustic wave propagating toward the second reflection center interferes constructively and a second plurality of components of the acoustic wave propagating toward the first reflection center interferes destructively.

13. The apparatus of claim 1, wherein the first IDT and the second IDT are thickness-field-excited single-phase unidirectional transducers (TFE-SPUDTs).

14. The apparatus of claim 1, wherein the first IDT and the second IDT are bi-directional transducers.

15. A full-duplex radio comprising: an antenna to transmit a first radio frequency (RF) signal in a first frequency range and receive a second RF signal at the first frequency range; transmit (TX) chain circuitry coupled to the antenna; receive (RX) chain circuitry coupled to the antenna, wherein the RX chain circuitry receives the second RF signal and a reflected portion of the first RF signal; a directional coupler in the TX chain circuitry, the directional coupler to direct a portion of the first RF signal to the RX chain circuitry; and a plurality of acoustic delay lines (ADLs) coupled between the TX chain circuitry and the RX chain circuitry to provide a signal delay, wherein the portion of the first RF signal experiences the signal delay and destructively interferes with the reflected portion of the first RF signal, and wherein each ADL of the plurality of ADLs comprises: a piezoelectric thin film located above a carrier substrate, wherein the piezoelectric thin film is one of a c-axis aluminum nitride (AlN) or scandium aluminum nitride (ScAlN) thin film adapted to propagate an acoustic wave; a first interdigitated transducer (IDT) disposed on a first end of the piezoelectric thin film, the first IDT to convert a first electromagnetic signal, traveling in a longitudinal direction along a length of the piezoelectric thin film, into the acoustic wave; and a second IDT disposed on a second end of the piezoelectric thin film with a gap between the second IDT and the first IDT, the second IDT to convert the acoustic wave into a second electromagnetic signal, and the gap to determine a time delay of the acoustic wave before output of the second electromagnetic signal.

16. The full-duplex radio of claim 15, wherein the acoustic wave is propagated in at least one of: a first mode excited by an electric field oriented at least partially in a longitudinal direction along a length of the piezoelectric thin film; or a second mode excited by the electric field oriented in a thickness direction of the piezoelectric thin film.

17. The full-duplex radio of claim 16, wherein: the first mode is a first-order antisymmetric (A1) mode; and the second mode is one of a fundamental symmetric (S0) mode or a first-order symmetric (S1) mode.

18. The full-duplex radio of claim 15, further comprising: a waveguide inside of which is disposed the piezoelectric thin film, the first IDT, and the second IDT; a first port coupled to the first IDT, the first port to receive the first electromagnetic signal; and a second port coupled to the second IDT, the second port to output the second electromagnetic signal.

19. The full-duplex radio of claim 18, wherein the first IDT comprises at least a transducer unit cell comprising: a ground line coupled to the first port; a first portion of a first floating top electrode, wherein a center of the first floating top electrode comprises a first reflection center; a second portion of a second floating top electrode, wherein a center of the second floating top electrode comprises a second reflection center; a first transduction electrode coupled to the ground line of the first port; a signal line coupled to the first port; and a second transduction electrode coupled to the signal line and disposed between the first floating top electrode and the first transduction electrode, wherein a transduction center is located between the first transduction electrode and the second transduction electrode.

20. A method comprising: converting, by a first interdigitated transducer (IDT) disposed on a first end of a piezoelectric thin film, a first electromagnetic signal traveling in a longitudinal direction along a length of the piezoelectric thin film into an acoustic wave, wherein the piezoelectric thin film is one of a c-axis aluminum nitride (AlN) or scandium aluminum nitride (ScAlN) thin film located above a carrier substrate; propagating the acoustic wave in at least one of: a first mode excited by an electric field oriented at least partially in the longitudinal direction; or a second mode excited by the electric field oriented in a thickness direction of the piezoelectric thin film; converting, by a second IDT disposed on a second end of the piezoelectric thin film, the acoustic wave into a second electromagnetic signal after a delay determined by a gap between the first IDT and the second IDT; and outputting the second electromagnetic signal.

21. The method of claim 20, wherein: the first mode is a first-order antisymmetric (A1) mode; and the second mode is one of a fundamental symmetric (S0) mode or a first-order symmetric (S1) mode.

22. The method of claim 20, further comprising applying a voltage potential across a signal line coupled to the first IDT to generate the electric field.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0005] A more particular description of the disclosure briefly described above will be rendered by reference to the appended drawings. Understanding that these drawings only provide information concerning typical embodiments and are not therefore to be considered limiting of its scope, the disclosure will be described and explained with additional specificity and detail through the use of the accompanying drawings.

[0006] FIG. 1A is a schematic illustration of an acoustic delay line (ADL) with an air gap according to one embodiment.

[0007] FIG. 1B is a schematic illustration of an ADL device with a high-acoustic impedance layer according to one embodiment.

[0008] FIG. 1C is a schematic illustration of an ADL device with a set of high-acoustic impedance layers and a set of low-acoustic impedance layers according to one embodiment.

[0009] FIG. 2A is a top view of an ADL according to one embodiment.

[0010] FIG. 2B is a top view of a distributed acoustic reflector transducer (DART) unit cell according to one embodiment.

[0011] FIG. 2C is a top view of an electrode width controlled (EWC) unit cell according to one embodiment.

[0012] FIG. 3A is a schematic diagram of a Lamb mode ADL with a pair of thickness-field-excited single-phase unidirectional transducers (TFE-SPUDT) transducers on a suspended AlN thin film according to one embodiment.

[0013] FIG. 3B is a top view of a TFE-SPUDT unit cell of one of the TFE-SPUDTs transducers (FIG. 3A) according to one embodiment.

[0014] FIG. 3C is a cross-sectional view of a TFE-SPUDT cell, showing the configuration of top electrodes (including the floating electrodes, the signal electrode, and the ground electrode) and a floating bottom electrode according to one embodiment.

[0015] FIG. 4 is a top view of a bi-directional ADL according to one embodiment.

[0016] FIG. 5A is a side-view of a single-phase unidirectional transducer (SPUDT) with an effective electric field and a corresponding strain curve and displacement curve according to one embodiment.

[0017] FIG. 5B is a side-view of the SPUDT to illustrate a reflection center RC according to one embodiment.

[0018] FIG. 5C is a side-view of the SPUDT to illustrate unidirectionality according to one embodiment.

[0019] FIG. 5D is a side-view of the SPUDT to illustrate multiple reflections of acoustic waves according to one embodiment.

[0020] FIG. 6A is a schematic diagram illustrating a COMSOL model for calculating the mechanical reflection coefficient from a step-up discontinuity caused by metallization according to one embodiment.

[0021] FIG. 6B is a graph illustrating a magnitude of y-axis displacement u.sub.y at 160 MHz according to one embodiment.

[0022] FIG. 7 is a schematic diagram of a SPUDT formed by N transducer unit cells according to one embodiment.

[0023] FIG. 8 is a schematic diagram illustrating a sectional Mason's model for a single unit cell according to one embodiment.

[0024] FIG. 9A is a schematic diagram illustrating a finite element method (FEM) model built in COMSOL to simulate the response of the ADLs to validate the circuit model of FIG. 8 according to one embodiment.

[0025] FIG. 9B is a schematic diagram illustrating a magnitude of a displacement along the y-axis at the center frequency according to one embodiment.

[0026] FIGS. 10A-10E are a set of graphs illustrating the simulated performance of the AlN Lamb mode ADL with TFE-SPUDT according to one embodiment.

[0027] FIGS. 11A-11B are optical microscope images of a fabricated aluminum nitride (AlN) ADL according to one embodiment.

[0028] FIGS. 12A-12D are a set of graphs illustrating the measured S-parameters of the fabricated AlN ADLs (L.sub.g=0.1 mm, =46.4 m) with different N (20-40) according to one embodiment.

[0029] FIGS. 13A-13D are a set of graphs illustrating the measured S-parameters of AlN ADLs (N=20, =46.4 m) with different L.sub.g (0.1-0.8 mm) according to one embodiment.

[0030] FIGS. 14A-14C are a set of graphs illustrating the measured S-parameters of the ADLs (N=20, L.sub.g=0.1 mm) with different (32.0-46.4 m) or center frequencies according to one embodiment.

[0031] FIG. 15 is a schematic diagram of a full-duplex transceiver with an acoustic delay synthesizer to attain self-interference cancellation (SIC) according to one embodiment.

[0032] FIG. 16 is a flow diagram of a method of the operation of an ADL according to one embodiment.

DETAILED DESCRIPTION

[0033] By way of introduction, the present disclosure relates to acoustic delay lines (ADLs) with interdigital transducers (IDTs) on a piezoelectric thin film, an apparatus, and a full-duplex radio that include disclosed ADLs. Aluminum nitride (AlN) thin films can be excellent platforms for implementing low-loss ADLs based on unidirectional transducers. These AlN thin films can also successfully be implemented with bi-directional transducers.

[0034] Chip-scale radio frequency (RF) acoustic delay lines (ADLs) may promise a wide range of applications from matched filtering and frequency synthesis to nonreciprocal components and time-domain equalization. One thin film lithium niobate (LNbO.sub.3)-based acoustic delay line shows significant advances in attaining low insertion loss (IL) and a wide range of delays over a broad bandwidth (BW). However, integration of LiNbO.sub.3 thin films requires layer transfer processes that may not be readily scaled to beyond 6-inch sizes, consequently prohibiting more seamless integration of delay-enabled signal processing with Complementary metal-oxide-semiconductor (CMOS) materials.

[0035] Aluminum nitride (AlN), on the other hand, can be reactively sputtered, thus holding the prospect of monolithic integration with CMOS and more intimate coalescence of RF acoustics and active circuitry. Although AlN devices based on various vibration modes (longitudinal mode, thickness mode, and two-dimensional modes) may feature notably lower electromechanical coupling (k.sup.2), AlN ADLs can offer application-worthy delay performance due to the lack of fundamental investigations on its insertion loss versus fractional bandwidth (IL-FBW) design space. Moreover, the some AlN ADLs adopt bi-directional transducers, which may lead to a minimum 6-dB IL and large ripples in the group delay.

[0036] Design tradeoffs allowed in the AlN thin films of ADLs are discussed herein and a set of low-loss AlN Lamb mode ADLs are disclosed with delays ranging from 105 ns to 165 ns with a 3 dB FBW of 4.5%, a minimum IL of 5.9 dB, and center frequencies ranging from 175 MHz to 255 MHz is subsequently demonstrated. The performance can be enabled by the unidirectional transducers proposed herein, namely the thickness-field-excited single-phase unidirectional transducers (TFE-SPUDT). The significantly lower IL demonstrated herein is expected to open new horizons for hybridized signal processing based on AlN and CMOS.

[0037] Aspects of the present disclosure address the above challenges among others by using low-loss and wide-band acoustic delay lines (ADLs). The ADLs include a piezoelectric thin film located above a carrier substrate. A first interdigitated transducer (IDT) may be disposed at a first end of the thin film and a second IDT may be disposed at a second end of the piezoelectric thin film. The first IDT is to convert an input electromagnetic signal (e.g., an RF signal traveling in a longitudinal direction along a length of the piezoelectric thin film) into an acoustic wave. The second IDT is to convert the acoustic wave into an output electromagnetic signal, which can be delayed in time compared to the first electromagnetic signal. In some embodiments, the IDTs are unidirectional. In other embodiments, the IDTs are bi-directional.

[0038] In some embodiments, the piezoelectric thin film is suspended above the carrier substrate. In other embodiments, the piezoelectric thin film is disposed on a high-acoustic impedance layer interposed between the piezoelectric thin film and the carrier substrate. In still further embodiments, the high-acoustic impedance layer includes at least one of silicon (Si), sapphire, fused silica, quartz, silicon carbide (SiC), diamond, aluminum nitride (AlN), aluminum oxide (Al.sub.2O.sub.3), tungsten, molybdenus, platinum, or combinations thereof. In some embodiments, the piezoelectric thin film is disposed on a Bragg reflector interposed between the piezoelectric thin film and the carrier substrate. In some embodiments the Bragg reflector includes a set of alternating high-acoustic impedance layers and low-acoustic impedance layers. The low-acoustic impedance carrier may be at least one of silicon nitride (Si.sub.3N.sub.4) or silicon dioxide (SiO.sub.2). In some embodiments, interfaces between the high-acoustic impedance layers and low-acoustic impedance layers can reflect the acoustic waves, and can lead to multiple reflections from the alternating layers. In further embodiments, acoustic energy can be confined in a layer (e.g., the piezoelectric thin film) above the Bragg reflector, which may prevent or minimize energy leakage into the carrier substrate via the multiple reflections.

[0039] In various embodiments, the acoustic wave travels within the piezoelectric thin film in at least one of a fundamental symmetrical (S0) mode, a first-order symmetrical (S1) mode, or a first-order antisymmetric (A1). In some embodiments, the modes are excited by at least one of a longitudinal-direction (e.g., along a length of the piezoelectric thin film) component of an electric field or a thickness-direction component of the electric field. In some embodiments, the electric fields are induced by incoming electromagnetic signal(s) (e.g., RF signal(s)). In some embodiments, the orientation of the induced electric field is determined by the configuration of electrodes of the IDTs in relation to the piezoelectric thin film. In some embodiments, the electric field is generated by a voltage potential that is applied between a signal bus line and a ground bus line. In various embodiments, the piezoelectric thin film includes one of a reactively sputtered c-axis aluminum nitride (AlN) or scandium aluminum nitride (ScAlN).

[0040] FIGS. 1A-1C are schematic illustrations of a cross-sectional view of ADL devices 100 according to one embodiment. FIG. 1A is a schematic illustration of an ADL device 110 with an air gap 106 according to one embodiment. The ADL device 110 includes a piezoelectric thin film 102 suspended above a carrier substrate 104. An air gap 106 is located between the carrier substrate 104 (e.g., carrier wafer) and the piezoelectric thin film 102 (e.g., piezoelectric layer). Electrodes and/or reflectors are located above the piezoelectric thin film 102, and are represented generally by 108. The electrodes and/or reflectors 108 can be physically and electrically coupled to the piezoelectric thin film 102.

[0041] FIG. 1B is a schematic illustration of an ADL device 120 with a high-acoustic impedance layer 112 according to one embodiment. The ADL device 120 is similar to the ADL device 110 except that the air gap is replaced by a high-acoustic impedance layer 112. In other words, the piezoelectric thin film 102 is located on the high-acoustic impedance layer 112. Illustrated is an ADL mock-up including single-phased unidirectional transducers (SPUDTs) disposed on top of an AlN thin film The high-acoustic impedance layer is located between the piezoelectric thin film 102 and the carrier substrate 104. In various embodiments, the high-acoustic impedance layer can be composed of one of silicon (Si), sapphire, fused silica, quartz, silicon carbide (SiC), diamond, aluminum nitride (AlN), aluminum oxide (Al.sub.2O.sub.3), tungsten, molybdenus, platinum, combinations thereof, or the like. Electrodes and/or reflectors 108 are located on top of the piezoelectric thin film 102. Electrodes and/or reflectors are located above the piezoelectric thin film 102, and are represented generally by 108. The electrodes and/or reflectors 108 can be physically and electrically coupled to the piezoelectric thin film 102.

[0042] FIG. 1C is a schematic illustration of an ADL device 130 with a set of high-acoustic impedance layers 112 and a set of low-acoustic impedance layers 114 according to one embodiment. The ADL device 130 is similar to the ADL device 110 except that the air gap is replaced by the set of high-acoustic impedance layers 112 and set of low-acoustic impedance layers 114. The piezoelectric thin film 102 is located on a combination of high-acoustic impedance layers 112 and a combination of low-acoustic impedance layers 114. In some embodiments, the high-acoustic impedance layers 112 and the low-acoustic impedance layers 114 form a stack in an alternating pattern and that stack is interposed between the piezoelectric thin film 102 and the carrier substrate 104. For example, respective ones of the low-acoustic impedance layers 114 can be alternately disposed on respective ones of the high-acoustic impedance layers 112.

[0043] The ADL device 130 illustrates a further embodiment in which the piezoelectric thin film 102 is disposed on a Bragg reflector which is composed of multiple alternating layers of high-acoustic impedance layers 112 and low-acoustic impedance layers 114. In some embodiments, each of the high-acoustic impedance layers 112 and the low-acoustic impedance layers 114 have the same thickness. In other embodiments, the high-acoustic impedance layers 112 can have a different thickness than the low-acoustic impedance layers 114. The low-acoustic impedance carrier of the low-acoustic impedance layers can be at least one of silicon nitride (Si.sub.3N.sub.4), silicon dioxide (SiO.sub.2), benzocyclobutene (BCB), or other suitable polymers. The Bragg reflector can be disposed between the carrier substrate 104 (e.g., carrier wafer) and the piezoelectric thin film 102 (e.g., piezoelectric layer). Electrodes and/or reflectors 108 can be located on top of or above the piezoelectric layer. In some embodiments, interfaces between the high-acoustic impedance layers 112 and the low-acoustic impedance layers 114 can reflect the acoustic waves, and can lead to multiple reflections from the alternating layers. In further embodiments, acoustic energy can be confined in a layer above the Bragg reflector, and can prevent energy leakage into the carrier substrate. In some embodiments, high impedance devices, such as ADL device 120) can provide better power handling. Further, air gap devices, such as the ADL device 110, can provide higher quadrature (Q) values compared to devices that have no air gap.

[0044] FIG. 2A is a top view of an ADL 200 according to one embodiment. Illustrated is a mock-up of the ADL 200 including SPUDTs disposed on top of a suspended AlN thin film. In one embodiment, the ADL 200 includes a suspended thin film 202 made of AlN. The suspended thin film 202 may constitute a propagation medium for the SH0 acoustic waves. In other embodiments, the suspended thin film may be ScAlN, or other suitable piezoelectric thin film. For the following discussion, c-axis reactively sputtered AlN is used as an illustrative example. Acoustic modes can be excited by electric fields oriented in a longitudinal direction, e.g., along a direction of propagation of the acoustic wave, or in other words along the x-axis in FIG. 2A, or in the thickness direction, e.g., normal to a direction of propagation of the acoustic wave, or in other words along the y-axis in FIG. 2A.

[0045] In some embodiments the AlN thin film can be adapted to propagate an acoustic wave in at least one of a first mode excited by an electric field oriented at least partially in the longitudinal direction along a length of the piezoelectric thin film or in a second mode excited by the electric field oriented in the thickness direction of the piezoelectric thin film. The first mode can include at least a first-order antisymmetric (A1) mode. The second mode can include at least a fundamental symmetric (S0) mode or a first-order shear-horizontal (SH1) mode. In various embodiments, the AlN thin film can be taken to be between 30 nm and 100 m. For illustrative purposes herein, the thickness of the AlN thin film can be chosen to be 800 nm. It should be noted that various dimensions depicted in FIG. 2A are solely for illustrative purposes, and should not be considered to be restrictive.

[0046] With continued reference to FIG. 2A, two sets of interdigital transducers (IDTs) 216a and 216b may include metal electrodes interconnected by bus lines 218 and may be disposed on top of the AlN thin film 202. The IDTs 216a and 216b may be composed of at least one of gold (Au), aluminum (Al), molybdenum (Mo), platinum (Pt), or any other suitable conductive material. In one embodiment, the IDTs 216a and 216b may be unidirectional. In other embodiments, the IDTs 216a and 216b may be bi-directional. Either set of IDTs 216a and 216b can serve as the transmitting transducer (input port), while the other IDT serves as the receiving transducer (output port). In the depicted embodiment, the IDTs 216a and 216b are separated by a gap length 201 L.sub.G that may set the time delay experienced by an electrical signal traversing from an input port 220 to an output port 222. In some cases, the gap length 201 can be between 0 m and several centimeters. In some cases, the gap length 201 may be larger. Each IDT 216a and 216b can be formed by cascading N identical transducer unit cells (which are described in more detail in reference to FIGS. 2B-2C). In some embodiments, the number N of transducer unit cells may range from 1 to 20. The number N of identical transducer cells may be as large as required for a given application. In some embodiments, the transducer unit cell may be a distributed acoustic reflector transducer (DART) unit cell (such as DART unit cell 230b of FIG. 2B). In other embodiments, the transducer unit cell may be an electrode width controlled (EWC) unit cell (such as transducer unit cell 230c of FIG. 2C).

[0047] FIG. 2B is a top view of a DART unit cell 230b according to one embodiment. The DART unit cell 230b is one example of a SPUDT. The DART unit cell 230b includes a signal electrode 226b coupled to an upper bus line 228b and a ground electrode 224b coupled to a ground bus line 218b. The signal electrode 226b and the ground electrode 224b can be collectively referred to as transduction electrodes herein. The transduction electrodes have a width of .sub.0/8. The DART unit cell 230b also includes wider electrodes that are reflectors 232a and 232b (e.g., acoustic reflectors). A width of the reflectors 232a and 232b of the DART unit cell 230b is 3.sub.0/8.

[0048] While an ADL, such as the ADL 200, itself can be treated as an electrical device with two ports, the individual IDTs, as well as each included transducer unit cell (such as the DART unit cell 230b), can be analyzed as a three-port network that effectively has one electrical port 234b and two acoustic ports, including a forward (FWD) acoustic port 236b and a backward (BWD) acoustic port 238b. The two acoustic ports 236b and 238b effectively represent the two propagation directions (e.g., +x-axis and x-axis) into the acoustic medium.

[0049] In some cases, a transducer unit cell, and thus the corresponding IDT, can be a bi-directional transducer with no directionality. Such a transducer can emit the same amount of power towards both acoustic ports. Thus, in an ADL formed by bi-directional transducers and an acoustic media, only half of the acoustic power available at the input transducer is sent towards the output transducer, while the other half may get lost. By reciprocity, the bi-directional output transducer may only convert half of the incident acoustic power to the electric domain. Consequently, ADLs formed by bi-directional transducers may suffer from an intrinsic minimum IL of 6 dB.

[0050] In order to mitigate the acoustic power loss due to bi-directionality, unidirectional transducers, such as single-phased unidirectional transducers (SPUDTs), such as the DART unit cell 230b can be employed, although this may come at the cost of reduced bandwidth. The operation principle of SPUDTs can be explained from the analysis of the transduction and reflection centers founded in their electrode layouts. A transduction center (TC) is a reference plane at which the acoustic waves launched towards both longitudinal directions (e.g., the +x and x directions) have the same amplitude and phase. Similarly, a reflection center (RC) is a reference plane at which the wave reflections from both longitudinal directions (e.g., the +x and x directions) are equal.

[0051] In some IDTs, such as bi-directional IDTs, the TCs and RCs can be evenly distributed along the transducer. Alternatively, in SPUDTs, TC 240b can be arranged asymmetrically with respect to the RCs 242a and 242b, in a way such that the launched acoustic waves, through both transduction and reflection, interfere constructively (illustrated by acoustic wave 203b) towards one of the acoustic ports, while the waves launched towards the opposite acoustic port interfere destructively (illustrated by acoustic wave 205b), thus leading to the unidirectionality. For simplicity, and by way of example, the former port will be referred to as the FWD acoustic port 236b and the latter port will be referred to as the BWD acoustic port 238b in the present disclosure. However, it should be noted that in other embodiments, the former port can be referred to as the BWD acoustic port and the latter port can be referred to as the FWD acoustic port. In some cases, the TC can be placed closer to the nearest RC towards the BWD acoustic port than to the nearest RC towards the FWD acoustic port. The difference between these distances may be .sub.0/4 to produce the mentioned constructive (destructive) interaction at the FWD (BWD) port.

[0052] FIG. 2C is a top view of an EWC unit cell 230c according to one embodiment. The EWC unit cell 230c is another example of a SPUDT. The EWC unit cell 230c includes a signal electrode 226c coupled to an upper bus line 228c and a ground electrode 224c coupled to a ground bus line 218c. The signal electrode 226c and the ground electrode 224c can be collectively referred to as transduction electrodes herein. The transduction electrodes have a width of .sub.0/8. The EWC unit cell 230c also includes wider electrodes that are reflectors 232c and 232d (e.g., acoustic reflectors). A width of the reflectors 232c and 232d of the EWC unit cell 230c is .sub.0/4.

[0053] The EWC unit cell 230c can also be analyzed as a three-port network that effectively has one electrical port 234b and two acoustic ports, including an FWD acoustic port 236c and a BWD acoustic port 238c. The two acoustic ports 236c and 238c effectively represent the two propagation directions (e.g., +x axis and x axis) into the acoustic medium.

[0054] Similar to the DART unit cell 230b, in the case of the EWC unit cell 230c, the TC 240c can be arranged asymmetrically with respect to the RCs 242c and 242c, in a way such that the launched acoustic waves, through both transduction and reflection, interfere constructively (illustrated by acoustic wave 203c) towards one of the acoustic ports, while the waves launched towards the opposite acoustic port interfere destructively (illustrated by acoustic wave 205c), thus leading to the unidirectionality. For simplicity, and by way of example, the former port will be referred to as the FWD acoustic port 236c and the latter port will be referred to as the BWD acoustic port 238c in the present disclosure. However, it should be noted that in other embodiments, the former port can be referred to as the BWD acoustic port and the latter port can be referred to as the FWD acoustic port. In some cases, the TC can be placed closer to the nearest RC towards the BWD acoustic port than to the nearest RC towards the FWD acoustic port. The difference between these distances may be .sub.0/4 to produce the mentioned constructive (destructive) interaction at the FWD (BWD) port.

[0055] With reference to FIGS. 2A-2C, each transducer unit cell can contain two types of electrodes: ground electrodes 232 and 224 that are connected to a lower bus line 218 and signal electrodes 226 that are connected to an upper bus line 228. In some cases, the thickness of the electrodes (including the ground electrodes 224 and 232 as well as the signal electrodes 226) may be between 5 nm and 10 m. In some cases, the thickness of the electrodes may be larger. When a voltage is applied between the lower bus line 218 and the upper bus line 228 (e.g., from an electromagnetic signal, for example, an RF signal), electric fields (e.g., E-fields) may be generated between the signal electrodes 226 and the ground electrodes 224 along the propagation direction (e.g., the x-axis). In some embodiments, the electric fields may be induced by one or more incoming electromagnetic signals. Further, the orientation of the induced electric fields may be determined by the configuration of the electrodes 224 and 226. Through the inverse piezoelectric effect, the E-fields can subsequently launch fundamental shear-horizontal strain and stress waves (SH0) in the xy-plane towards both the +x and x directions. By reciprocity, the shear stress and/or strain in the xy-plane associated with an acoustic wave propagating through the receiving IDT 216 can generate a voltage difference across the corresponding input electrical port 220. The efficiency of the conversion between electrical and acoustic energy can be maximum at a center frequency, f.sub.0, at which an acoustic wave is phase-delayed by 360 after traveling through a transducer unit cell. The value of f.sub.0 can be determined by the length of the unit cells .sub.0 as

[00001]$\begin{array}{cc}{f}_0=\frac{v_t}{{}_0},& \left(1\right)\end{array}$

where v.sub.t is the average phase velocity of the acoustic wave in the transducer unit cell. In some cases, the length of the transducer unit cell can range between 0.1 m to 100 m. An average phase velocity of the acoustic wave can be calculated as a weighted average between a phase velocity v.sub. of the un-metallized AlN film and a phase velocity v.sub.m, of the metallized film which can be expressed as

v.sub.t=v.sub.m+(1)v.sub.,(2)

where is the metallization ratio of the transducer unit cell. The dependence of f.sub.0 on the thickness of the piezoelectric thin film can be neglected in some embodiments.

[0056] FIG. 3A is a schematic diagram of a Lamb mode ADL 300 with a pair of TFE-SPUDT transducers on a suspended AlN thin film according to one embodiment. The schematic of an example fundamental symmetric (S0) mode AlN ADL 300 is depicted in FIG. 3A. In the depicted embodiment, the S0 mode AlN ADL 300 includes a suspended AlN thin film 302, sandwiched by IDTs 316a and 316b on top of float bottom electrodes 304. In some embodiments, the suspended AlN thin film can be approximately 1 m, the IDTs can be aluminum with a thickness of approximately 150 nm, and the floating bottom electrodes can be platinum with a thickness of approximately 100 nm. However, it should be noted that in other embodiments, an AlN ADL can be designed with different parameter values, which can in some cases affect the propagation of modes, the time delay generated by the AlN ADL, and the like. For example, some suitable ranges (though not an exhaustive list) are listed in Table 1. In some embodiments, the piezoelectric thin film is one of a c-axis aluminum nitride (AlN) or scandium aluminum nitride (ScAlN) thin film adapted to propagate an acoustic wave within the piezoelectric thin film in a first-order antisymmetric (A1) mode excited by an electric field oriented at least partially in a longitudinal direction along a length of the piezoelectric thin film. These AlN thin films may also propagate the acoustic wave in a fundamental symmetric (S0) mode or a first-order symmetric (S1) mode excited by the electric field oriented in a thickness direction of the piezoelectric thin film. In some embodiments ScAlN (e.g., AlN doped with Sc), may offer higher electromechanical coupling (e.g., k.sup.2).

[0057] In some embodiments, the piezoelectric thin film 302 can be suspended above a carrier substrate. In other embodiments, the piezoelectric thin film 302 can be disposed on a high-acoustic impedance layer interposed between the piezoelectric thin film and the carrier substrate and the high-acoustic impedance layer may composed of at least one of silicon (Si), sapphire, fused silica, quartz, silicon carbide (SiC), diamond, aluminum nitride (AlN), or aluminum oxide (Al.sub.2O.sub.3). In other embodiments, the piezoelectric thin film 302 can be disposed on a Bragg reflector interposed between the piezoelectric thin film and the carrier substrate and the Bragg reflector can be composed of a number of alternating layers including a first layer with a first acoustic impedance (e.g., a high-acoustic impedance layer) and a second layer with a second acoustic impedance (e.g., a low-acoustic impedance layer). The second acoustic impedance may be lower than the first acoustic impedance. A pair of IDTs 316a and 316b can be placed on the two longitudinal ends of the acoustic waveguide, and may serve as the input and output ports. Each port is coupled to a ground line and a signal line. As seen in FIG. 3A, the pair of IDTs 316a and 316b are TFE-SPUDTs that may be composed of cascaded transducer unit cells. The piezoelectric thin film as well as the IDTs 316a and 316b can be located inside of the waveguide and are designed as unidirectional IDTs. A gap between the first IDT 316a and the second IDT 316b determines a time delay of the acoustic wave prior to converting the acoustic wave back into an electromagnetic signal and outputting the electromagnetic signal.

[0058] FIG. 3B is a top view of a TFE-SPUDT unit cell 330b of one of the TFE-SPUDTs transducers (FIG. 3A) according to one embodiment. In FIG. 3B, the relevant dimensions, the reflection centers 342a and 342b, and the transduction center 340b are labeled.

[0059] FIG. 3C is a cross-sectional view of a TFE-SPUDT cell 330c, showing the configuration of top electrodes (including the floating electrodes 304, the signal electrode 326b, and the ground electrode 324f) and a floating bottom electrode 332 according to one embodiment. Each TFE-SPUDT unit cell can include two transduction electrodes 324f and 326b (/8 wide), and a back half of a floating reflector 304a (of width /4) and a front half of a floating reflector 304b of width /4) that function as an embedded reflector. An embedded reflector refers to a reflector (e.g., of acoustic waves) that is arranged (e.g., located) between adjacent transduction electrodes, such as a ground electrode and a signal electrode. The floating reflectors 304a and 304b can be floating top electrodes. In each cell, the transduction center 340b may be arranged non-symmetrically (leading to the unidirectionality of the IDTs 316a and 316b) to the reflectors on the opposite sides with different spacings of 3/8 (e.g., a first distance) and 5/8 (e.g., a second distance). The reflection center 342a can be located in a center of the floating reflector 304a and the reflection center 342b can be located in a center of the floating reflector 304b (as the transduction electrodes of width /8 are non-reflective). The transduction center 340b can be in the center of between the two transduction electrodes 326b and 324f since the reflector is floating. The bottom electrode 332 may only be placed in the transducer sections. In operation, due to the non-symmetry, the acoustic wave reflected from the reflector 304a on the left can constructively interfere with the acoustic wave that propagates toward the right, or the forward direction (FWD). On the contrary, the wave reflected from the reflector 304b on the right can destructively interfere with the wave launched towards the left, or the backward direction (BWD). Given an adequate number of cells and reflectivity per cell, the energy would be mostly launched towards FWD. Thus, unidirectional transducers for low-loss ADLs can be achieved. In some embodiments, the width of the floating electrodes 304a and 304b is equal to or less than four times the width of the transduction electrodes 324f and 326b.

[0060] FIG. 4 is a top view of a bi-directional ADL 400 according to one embodiment. Illustrated is a mock-up of the bi-direction ADL 400 including bi-directional IDTs disposed on top of a suspended AlN thin film. The suspended thin film 402 may constitute a propagation medium for the S0 acoustic waves. In other embodiments, the suspended thin film can be another suitable piezoelectric thin film. Similar to the ADL 200 of FIG. 2, propagation modes may be excited by electric fields oriented in a longitudinal direction, in a thickness direction or a combination thereof. In particular, the longitudinal direction is defined to be along a direction of propagation of the acoustic wave along the x-axis in FIG. 4. The thickness direction is defined to be normal to a direction of propagation of the acoustic wave, or in other words along the y-axis in FIG. 4.

[0061] In some embodiments the AlN thin film can be adapted to propagate an acoustic wave in at least one of a first mode excited by an electric field oriented at least partially in the longitudinal direction along a length of the piezoelectric thin film or in a second mode excited by the electric field oriented in the thickness direction of the piezoelectric thin film. The first mode can include at least a first-order antisymmetric (A1) mode. The second mode can include at least one of a fundamental symmetric (S0) mode or a first-order symmetric (S1) mode. With continued reference to FIG. 4, two sets of interdigital transducers (IDTs) 416a and 416b may include metal electrodes interconnected by bus lines 418 and may be disposed on top of the LiNbO.sub.3 thin film 402. The IDTs 416a and 416b may be composed of at least one of gold (Au), aluminum (Al), molybdenum (Mo), platinum (Pt), or any other suitable conductive material. In some embodiments, the IDTs 416a and 416b may be bi-directional transducers, as depicted in FIG. 4A. Either set of IDTs 416a and 416b can serve as the transmitting transducer (input port), while the other IDT serves as the receiving transducer (output port). In the depicted embodiment, the IDTs 416a and 416b are separated by a gap length 401 L.sub.G that may set the time delay experienced by an electrical signal traversing from an input port 420 to an output port 422. In some cases, the gap length 401 can be between 0 m and several centimeters. In some cases, the gap length 401 may be larger. Each IDT 416a and 416b can be formed by cascading N identical transducer unit cells. In some embodiments, the number N of transducer unit cells may range from 1 to 20. The number N of identical transducer cells may be as large as required for a given application.

[0062] A transduction center (TC) is a reference plane at which the acoustic waves launched towards both longitudinal directions (e.g., the +x and x directions) have the same amplitude and phase. Similarly, a reflection center (RC) is a reference plane at which the wave reflections from both longitudinal directions (e.g., the +x and x directions) are equal.

[0063] In some IDTs, such as bi-directional IDTs, the TCs and RCs can be evenly distributed along the transducer. Alternatively, in SPUDTs, TC can be arranged asymmetrically with respect to the RCs, in a way such that the launched acoustic waves, through both transduction and reflection, interfere constructively towards one of the acoustic ports, while the waves launched towards the opposite acoustic port interfere destructively, thus leading to the unidirectionality.

[0064] With reference to FIG. 4, each transducer unit cell can contain two types of electrodes: ground electrodes 432 and 424 that are connected to a lower bus line 418 and signal electrodes 426 that are connected to an upper bus line 428. In some cases, the thickness of the electrodes (including the ground electrodes 424 and 432 as well as the signal electrodes 426) may be between 5 nm and 10 m. In some cases, the thickness of the electrodes may be larger. When a voltage is applied between the lower bus line 418 and the upper bus line 428 (e.g., from an electromagnetic signal, for example, an RF signal), electric fields (e.g., E-fields) may be generated between the signal electrodes 426 and the ground electrodes 424 along the propagation direction (e.g., the x-axis). In some embodiments, the electric fields may be induced by one or more incoming electromagnetic signals. Further, the orientation of the induced electric fields may be determined by the configuration of the electrodes 424 and 426. Through the inverse piezoelectric effect, the E-fields can subsequently launch fundamental shear-horizontal strain and stress waves (SH0) in the xy-plane towards both the +x and x directions. By reciprocity, the shear stress and/or strain in the xy-plane associated with an acoustic wave propagating through the receiving IDT 416 can generate a voltage difference across the corresponding input electrical port 420. The efficiency of the conversion between electrical and acoustic energy can be maximum at a center frequency, f.sub.0, at which an acoustic wave is phase-delayed by 360 after traveling through a transducer unit cell.

[0065] FIG. 5A is a side-view of a SPUDT 530 with an effective electric field 507 and a corresponding strain 509 curve and displacement 511 curve according to one embodiment. Although not all components of the SPUDT 530 are shown, the SPUDT 530 is similar to the DART unit cell 230b and/or the EWC unit cell 230c, as noted by similar reference numbers.

[0066] To locate the TCs 240b and 240c of the transduction unit cells 230b and 230c of FIGS. 2B-2C, respectively it should first be noted that shear-horizontal waves can be generated through piezoelectricity in the areas with x-polarized electric fields, such as the electric field 507. These areas may be the gaps between the signal electrode 526 and the adjacent ground electrodes 524 and 532 on either side. In adjacent gap areas, the x-polarized electric fields induced by the electrodes may have opposite signs, as seen in FIG. 5A. Therefore, in a transduction unit cell (such as the DART unit cell 230b), a center of the signal electrode 526 may be approximately an axis of anti-symmetry for a generated xy-plane strain .sub.xy 509. Since .sub.xy 509 is the derivative of a y-axis displacement u.sub.y 511 with respect to x, u.sub.y 511 is symmetric with respect to the center of the signal electrode 526. Thus, this point can be considered the TC for the displacement wave u.sub.y(x, t) 511. The same approximation can be adopted for EWC unit cells. FIG. 5A shows that the induced strain 509 and displacement 511 may be respectively antisymmetric and symmetric with respect to the center of the signal electrode 526.

[0067] FIG. 5B is a side-view of the SPUDT 530 to illustrate an RC according to one embodiment. Following the same symmetry rationale as for the TC, the center of a reflection electrode 532 can be regarded as a reflection center. Due to the symmetry of the electrodes and the law of conservation of power, the reflection coefficients of metal electrodes referred to their centers may be purely imaginary. FIG. 5B shows equal reflection coefficients with respect to to the center of the electrode for incidences from both sides.

[0068] FIG. 5C is a side-view of the SPUDT 530 to illustrate unidirectionality according to one embodiment. As seen in FIG. 5C, in either a DART unit cell or an EWC unit cell, there can be a pair of electrodes 524 and 525 which have a width of .sub.0/8 and are connected to ground and signal, respectively, with a center-to-center distance of .sub.0/4. The acoustic waves respectively reflected by these electrodes can have a phase difference of 1800 at the center frequency f.sub.0 and interfere destructively. Assuming small reflections, their amplitudes can be considered equal, resulting in a perfect theoretical cancellation. In other words, acoustic waves reflected by the two adjacent .sub.0/8 electrodes 524 and 526 can produce an overall substantially zero reflection coefficient. As a result, the .sub.0/8-wide electrodes 524 and 526 with a center-to-center distance of .sub.0/8 can be omitted from the analysis for reflections within the SPUDTs. Different from .sub.0/8-wide electrodes, the wider electrodes (such as the acoustic reflectors, or the wider electrodes 532) may be intended to produce pronounced reflections. In other words, FIG. 5C shows a reflection-less nature of two identical electrodes 524 and 526 separated by a distance of .sub.0/4 at f.sub.0.

[0069] FIG. 5D is a side-view of the SPUDT 530 to illustrate multiple reflections of acoustic waves according to one embodiment. As seen FIGS. 2B-2C, the acoustic emission towards the FWD acoustic port 236 can be a combination of the waves generated at the TC 240 towards the FWD acoustic port 236 and the acoustic waves towards the BWD acoustic port 238 that are reflected from the closest RC on the left. For both DART and EWC designs, the RCs 242a and 242d can be separated from TCs 240b and 240c, respectively, by a first distance of 3.sub.0/8. Assuming a negative imaginary reflection coefficient (with a phase angle of 90 degrees), the reflected acoustic waves can be in phase with the acoustic waves generated at the TC towards the FWD acoustic port 236 at f.sub.0. Note that the acoustic waves sent by the further transduction unit cells on the left may also interfere constructively given the .sub.0 periodicity. The acoustic emission towards BWD acoustic port 238 may be the result of the interference of the waves generated at the TC and their reflection from the closest RC on the right. Due to the second distance of 5.sub.0/8 separation between the TCs 240b and 240 and RCs 242b and 242d, respectively, on the right, the directly transduced waves towards the BWD acoustic port 238 and their reflection from RCs may be out of phase. Hence, the BWD acoustic port 238 may receive less acoustic power than the FWD acoustic port 236 due to the partial cancellation of the directly transduced acoustic waves by the reflection. In some cases, a single reflection may not be sufficient to achieve elimination of transduction towards the BWD acoustic port 238 and unidirectionality towards the FWD acoustic port 236. More transduction unit cells may be required for this purpose. In a multi-cell configuration (e.g., with N transduction unit cells), the RC in each cell can all serve to produce reflection for every TC. Therefore, the interference in both directions combine all the directly transduced waves from all TCs and all the reflections generated by all the RCs. The dynamics in a multi-cell configuration can be analyzed, and it can be shown that a near perfect unidirectionality is thus possible with multiple cells and multi-reflections. In other embodiments, the second distance can be between 40% and 95% greater than the first distance.

[0070] The total reflection illustrated by FIG. 5D is induced by a metal electrode as a combination of two contributions, one electrical (.sub.e) and one mechanical (.sub.m). As discussed above, the directionality of the SPUDT 530 may be based on the reflectivity of the wide electrodes 532 in each unit cell. The reflectivity of each reflector can be quantitatively modeled. The reflection coefficient of an electrode can be considered as the result of two phenomena. First, it can have a mechanical component, .sub.m, caused by the edges of the electrode on the film, along with the change in the acoustic impedance in sections with metal coverage. The change in acoustic impedance can arise from unequal mass density and stiffness of the electrode metal and LiNbO.sub.3. Second, can have an electrical contribution, .sub.e, caused by a constant potential boundary condition created on the top surface of the LiNbO.sub.3 film by the metallization. In other words, .sub.e is the reflection coefficient created by a strip of perfect electric conductor (PEC) of zero thickness. To calculate the total reflection coefficient, the mechanical and electrical reflections can be treated as if they were produced at different locations separated by a distance X.sub.0, as seen in FIG. 5D. By solving the multiple reflections between these two locations and taking a limit X.sub.0.fwdarw.0, the total reflection coefficient can be obtained as

[00002]$\begin{array}{cc}=\frac{{}_e+{}_m}{1+{}_e.Math.{}_m}& \left(3\right)\end{array}$

for small reflections, e.g., .sub.e.sub.m<<1, .sub.e+.sub.m.

[0071] For the reflections that are mechanically-induced by metal electrodes, analytical expressions can be found for SAW devices. For wave propagation in plates, the methods to predict the reflections from mechanical discontinuities can rely on finite element method (FEM) simulations. For an electrode on a thin film, acoustic waves can be reflected as they travel from an un-metalized section to a metalized portion of the LiNbO.sub.3 film (e.g., step-up). Further, acoustic waves can be reflected as they travel from a metalized portion to an un-metalized section (e.g., step-down). As a result of both reflections, an equivalent overall mechanical reflection coefficient .sub.m can be defined for a single electrode.

[0072] FIG. 6A is a schematic diagram illustrating a COMSOL model 600 for calculating the mechanical reflection coefficient from a step-up discontinuity 602 caused by metallization according to one embodiment. The COMSOL model 600 shown in FIG. 6A can be built in COMSOL to evaluate the mechanical reflection from the step-up discontinuity 602 created by an electrode. The model may be composed of sections of the delay medium including a non-metallized region 604 in one end and metallized region 606 in the other. Perfectly matched layer (PML) conditions can be set at both ends of the model to emulate an infinitely long mechanical medium along x and +x. The faces (e.g., surfaces) at y and +y can be modeled as periodic boundaries. Acoustic waves can be excited by a harmonic force applied at the cross section at x=0 and s.sub.1 in FIG. 6A, separated from the discontinuity by a distance L.sub.d. With an excitation force along the y-axis, a first acoustic wave can be propagated in a first direction (e.g., along x) and a second acoustic wave with the same amplitude as the first acoustic wave can be propagated in a second direction opposite to the first direction (e.g., along +x). The first acoustic wave and the second acoustic wave can have opposite phases if the strain .sub.xy is considered as the wave variable. Alternatively, the first acoustic wave and the second acoustic wave can have the same phase if the displacement, u.sub.y is considered as the wave variable.

[0073] FIG. 6B is a graph illustrating a magnitude of y-axis displacement u.sub.y at 160 MHz according to one embodiment. In particular, FIG. 6B shows the solution for the magnitude of u.sub.y. A standing wave can be created between s.sub.1 and the metallization edge as a result of the interference of a(x, t) with the reflected wave b(x, t). A constant amplitude can be observed between s.sub.1 and the PML in the x region, and between the discontinuity and the PML in the +x region. This can indicate a perfect absorption of the acoustic power by the PMLs. The strain field .sub.xy associated with the wave a can be written as

.sub.xy.sup.a(x,t)=Ae.sup.j.sup..sup.xe.sup.jt for x>0

.sub.xy(x,t)=Ae.sup.j.sup..sup.xe.sup.jt for x>0,(4)

where is the angular frequency and .sub.=/v.sub. is the wave-number in the un-metallized region of the thin film. The strain field associated with the reflected wave b can then be obtained as

.sub.xy.sup.b(x,t)=Ae.sup.j.sup..sup.xe.sup.j.sup..sup.x2L.sup.d.sub.sue.sup.jt for x<L.sub.d,(5)

where the subscript su denotes the mechanical reflection coefficient associated with the step-up discontinuity. The stress at the cross-sections s.sub.2 and s.sub.3, separated from s.sub.1 by a distance x (e.g., see FIG. 6A), can be obtained as the superposition of a and b waves at x=x and x=x, respectively:

.sub.xy.sup.s2(t)=A(e.sup.j.sup..sup.x+e.sup.j.sup..sup.xe.sup.j.sup..sup.x2L.sup.d.sub.su)e.sup.jt(6)

.sub.xy.sup.s3(t)=A(e.sup.j.sup..sup.x+e.sup.j.sup..sup.xe.sup.j.sup..sup.x2L.sup.d.sub.su)e.sup.jt.(7)

By taking a limit x.fwdarw.0, the expression below can be obtained:

[00003]$\begin{array}{cc}{}_{\mathrm{su}}=\frac{u_y^b\left(x,t\right)}{u_y^a\left(x,t\right)}=-\frac{{}_{\mathrm{xy}}^b\left(x,t\right)}{{}_{\mathrm{xy}}^a\left(x,t\right)}=e^{-j.Math..Math.{}_{}.Math.2.Math.L_d}.Math.\frac{{}_{x.Math.y}^{s.Math..Math.3}+{}_{x.Math.y}^{s.Math..Math.2}}{{}_{x.Math.y}^{s.Math..Math.3}-{}_{x.Math.y}^{s.Math..Math.2}}& \left(8\right)\end{array}$

where u.sub.y.sup.a and u.sub.y.sup.b are the displacements associated with the incident and reflected waves, respectively. Using this expression, the reflection coefficient .sub.su can then be obtained by evaluating .sub..sup.s2 and .sub..sup.s3 in the COMSOL simulation. As an illustrative example, the procedure described above can be performed for an AlN thin film and four metals that are commonly used as electrodes in microsystems: gold (Au), aluminum (Al), molybdenum (Mo) and platinum (Pt). In all cases, it can be found that .sub.su is substantially constant as a function of frequency up to 500 MHz.

[0074] The magnitude of .sub.su can be found to be nearly linearly dependent on the metal thickness. The phase of .sub.su is close to 180 for the simulated thickness range. The reflection coefficient of the electrode step-down, .sub.sd, can be found to have the same magnitude but opposite phase as the reflection coefficient of the electrode step-up, (e.g., .sub.sd=.sub.su). The overall mechanical reflection coefficient of an electrode can be found by summing the multiple reflections produced by the step-up and step-down discontinuities. Referencing the reflections to the center of the electrode, the following expression can be obtained

[00004]$\begin{array}{cc}{}_m={}_{\mathrm{su}}.Math.e^{j.Math..Math.}\left(1-e^{-j.Math..Math.2.Math.}.Math.T_{\mathrm{su}}.Math.\underset{n=0}{\overset{}{.Math.}}.Math.{\left({}_{\mathrm{su}}.Math.e^{-j.Math..Math.}\right)}^{2.Math.n}\right)& \left(9\right)\end{array}$

where is the phase retardation for traversing half of the width of a reflector. is be 3/4 for DART and /2 for EWC reflectors. T.sub.su is the transmission coefficient of the step-up discontinuity, given by

T.sub.su=1+.sub.su.(10)

[0075] Introducing T.sub.su to Eq. (9) and simplifying the geometric series, the following result can be obtained

[00005]$\begin{array}{cc}{}_m={}_{\mathrm{su}}.Math.e^{j.Math..Math.}.Math.\frac{1-e^{-j.Math..Math.2.Math.}\left(1-{}_{\mathrm{su}}^2\right)}{1-{}_{\mathrm{su}}^2.Math.e^{-j.Math..Math.2.Math.}}.& \left(11\right)\end{array}$

[0076] The electrical reflection can be calculated in a similar way by considering the change in phase velocity produced by the ground condition set by the reflector electrodes on top of the piezoelectric film. Similar to the approach with the mechanical reflection, a reflection coefficient can be defined as the acoustic wave passes from an un-metalized to a metalized section,

[00006]$\begin{array}{cc}{}_{.Math.0}=\frac{v_0-v_{}}{v_0+v_{}},& \left(12\right)\end{array}$

where v.sub.0 and v.sub. are the phase velocities for a piezoelectric medium with the free and electrically shorted top surfaces, respectively. The reflection coefficient as the acoustic wave passes from a metallized to an un-metallized section is .sub.0=.sub.0. The phase velocities of the SH0 mode can be determined using the finite element method (FEM) in COMSOL. The overall electrical reflection coefficient of an electrode can be obtained following the same procedure as for Eq. (11), as follows:

[00007]$\begin{array}{cc}{}_e={}_{0.Math.}.Math.e^{j.Math..Math.}.Math.\frac{1-e^{-j.Math..Math.2.Math.}\left(1-{}_{0.Math.}^2\right)}{1-{}_{0.Math.}^2.Math.e^{-j.Math..Math.2.Math.}}.& \left(13\right)\end{array}$

[0077] It should be noted that Eq. (13) may not account for non-uniform electric fields created by the uneven charge distribution in an electrode when surrounded by other electrodes in an array or multi-cell configuration. An analytical method to calculate the electrical reflection accounting for this phenomenon can be used. Such a method can assume an array of electrodes with constant width and separation. Since this condition may not be met by the reflectors in SPUDTs, the method may have to be revised before being applied.

[0078] As previously described, multiple unit cells that are spaced by .sub.0 disposed in a cascaded configuration may be required to attain highly unidirectional transduction. In order to be consistent with the framework used for analyzing a single cell, a multi-cell transducer can also be considered with three ports (as described with respect to FIGS. 2B-2C): one electric port that is connected to all the cells for excitation, and two acoustic ports that can be situated at the opposite ends of the multi-cell transducer. To quantitatively measure the directionality of multiple cells, a figure of merit (FoM) dubbed as directionality of transduction can be defined as

[00008]$\begin{array}{cc}D=\frac{P_{\mathrm{FWD}}}{P_{\mathrm{BWD}}},& \left(14\right)\end{array}$

where P.sub.FWD is the power emitted towards the FWD acoustic port and P.sub.BWD is the power emitted towards the BWD acoustic port. When a time-harmonic voltage is applied at the electrical port, a transducer (e.g., with N transducer unit cells) can emit acoustic power towards both acoustic ports. The total emission to each port can be calculated as the superposition of the waves emitted by each TC in the transducer. To determine the power emitted by a single TC in a multi-cell configuration, a voltage source can connected to one TC at a time, while all other TCs are grounded.

[0079] FIG. 7 is a schematic diagram of a SPUDT 716 formed by N transducer unit cells 730 according to one embodiment. Although not all components of the SPUDT 716 are shown, the SPUDT 716 is similar to the IDT 216 of FIG. 2A. The number of transducer unit cells Nis an integer number that can range from 1 to as many as is necessary for a given application. Increasing the number of transducer unit cells (also referred to simply as unit cells herein) can increase the unidirectionality of the SPUDT and can result in a narrower bandwidth. In some embodiments, increasing the unidirectionality can be preferable. In FIG. 7, each rectangle corresponds to a unit cell with marked RC and TC. For the i.sup.th unit cell, the directionality can be calculated by considering its TC and all the RCs at both sides.

[0080] As seen, the transduction center 740 at the unit cell i can have i1 reflectors on its right (FWD) and (Ni+1) reflectors on its left (BWD), with all reflectors being characterized by the same reflection coefficient, . Each RC in the transducer can be denoted by an index k. At the (i1).sup.th RC which is on the immediate right of the i.sup.th TC (i.e., k=i1), an equivalent reflection coefficient .sub.k can be defined, that accounts for all the reflections produced by the unit cells from 1 to k. For k=1, this may simply be .sub.1=. For k=2, the equivalent reflection coefficient must account for the multiple reflections between the RCs of unit cells 1 and 2. At f.sub.0, there may be a 2 phase separation between the RCs, giving:

[00009]$\begin{array}{cc}{}_2^{}=+T^2.Math.{}_1^{}.Math.\underset{n=0}{\overset{}{.Math.}}.Math.{\left({}_1^{}.Math.\right)}^n& \left(15\right)\end{array}$

where T is the transmission coefficient of the RCs, and can be obtained as:

[00010]$\begin{array}{cc}T=\frac{1-{}_{.Math.m}^2}{1-{}_{.Math.m}^2.Math.e^{-j.Math..Math.2.Math.}},& \left(16\right)\end{array}$

where .sub.m=(.sub.su+.sub.0)/(1+.sub.su.sub.0) is the total reflection experienced by a wave traveling from a non-metallized to a metallized section. By substituting Eq. (16) into Eq. (15) and simplifying the geometric series, Eq. (15) can be reduced to

[00011]$\begin{array}{cc}{}_2^{}=+\frac{T.Math.}{1-{}^2}.& \left(17\right)\end{array}$

[0081] This method can be applied to the successive RCs, leading to the recursive definition of .sub.k:

[00012]$\begin{array}{cc}{}_k^{}=+\frac{T.Math.{}_{k-1}^{}}{1-.Math.{}_{k-1}^{}}.& \left(18\right)\end{array}$

[0082] The equivalent reflection coefficients of the RCs on the left of the i.sup.th TC 740 can be obtained in the same way from the right to the left as .sub.Nk+1 (see FIG. 7). Associated to the .sub.k, the equivalent transmission coefficients to each RC can be defined as:

T.sub.k=e.sup.jk{square root over (1|.sub.k|.sup.2)},(19)

where .sub.k is the phase of the transmission coefficient. Then, the calculation of the directionality of unit cell i can be reduced to attending the i.sup.th TC 740 with two overall reflections at the locations of the two most adjacent RCs on the left and right, with reflection coefficients .sub.Ni+1 and .sub.i1 respectively. By solving the multiple reflections for the two waves generated at the TC in the opposite directions, the wave amplitude emitted to the FWD port can be found to be:

[00013]$\begin{array}{cc}{}_i^{F.Math.W.Math.D}=.Math.\frac{e^{-j\left(3.Math./4-{}_{i-1}\right)}.Math.\sqrt{1-{.Math.{}_{i-1}^{}.Math.}^2}.Math.\left(e^{-j.Math..Math./2}+{}_{N-i+1}^{}\right)}{1-{}_{i-1}^{}.Math.{}_{N-i+1}^{}},& \left(20\right)\end{array}$

where is the transduction coefficient. For the wave radiated to the BWD port, the wave amplitude emitted to the BWD port can be found to be:

[00014]$\begin{array}{cc}{}_i^{B.Math.W.Math.D}=.Math.\frac{e^{-j(3.Math./4-\left({}_{N-i+1}\right)}.Math.\sqrt{1-{.Math.{}_{N-i+1}^{}.Math.}^2}.Math.\left(1+e^{-j.Math..Math./2}+{}_{i-1}^{}\right)}{1-{}_{i-1}^{}.Math.{}_{N-i+1}^{}}& \left(21\right)\end{array}$

[0083] Imposing that, from Eq. (18), all the .sub.k may be negative and imaginary, the directionality of the unit cell i can be obtained as

[00015]$\begin{array}{cc}{D}_i=\frac{{.Math.a_i^{F.Math.W.Math.D}.Math.}^2}{{.Math.a_i^{B.Math.W.Math.D}.Math.}^2}=\frac{\left(1+.Math.{}_{i-1}^{}.Math.\right).Math.\left(1+.Math.{}_{N-i+1}^{}.Math.\right)}{\left(1-.Math.{}_{i-1}^{}.Math.\right).Math.\left(1-.Math.{}_{N-i+1}^{}.Math.\right)}.& \left(22\right)\end{array}$

[0084] By evaluating Eq. (18) into Eq. (22), it can be shown that the directionality of each unit cell in a multi-cell configuration has the same value

[00016]$\begin{array}{cc}{D}_i={\left(\frac{1+.Math..Math.}{1-.Math..Math.}\right)}^N& \left(23\right)\end{array}$

which, by linear superposition, may also be the overall directionality of the whole transducer, D. The directionality can further obtained as a composition of two factors, the directionality due to the electrical reflection, D.sub.e, and the directionality due to the mechanical reflection, D.sub.m:

[00017]$\begin{array}{cc}D=D_e.Math.D_m={\left(\frac{1+.Math.{}_e.Math.}{1-.Math.{}_e.Math.}\right)}^N.Math.{\left(\frac{1+.Math.{}_m.Math.}{1-.Math.{}_m.Math.}\right)}^N& \left(24\right)\end{array}$

[0085] The group delay of an ADL (also referred to simply as a delay line herein) employing the abovementioned transducers can be challenging to precisely predict with a closed form expression. This can be due to the complexity introduced by the multiple reflections between the different cells in each transducer. A simplified analysis can be done by disregarding these internal reflections. This can be achieved by considering the transfer function F() from the input port to the center of the ADL. It can be expressed as the superposition of N phase-retarded acoustic waves generated by the transducer unit cells. Assuming lossless propagation, each term in F() can have three phase delays: the one due to the propagation over a distance de from the TC to the right edge of each unit cell, the phase delay from the right edge of each unit cell to the right edge of the entire input transducer, and the phase delay from the right edge of the input transducer to the center of the delay line, over a distance L.sub.G/2. This can be expressed as:

[00018]$\begin{array}{cc}F\left(\right)=\underset{n=1}{\overset{N}{.Math.}}.Math.e^{-j\left({}_t.Math.d_c+{}_t.Math.{}_0\left(n-1\right)+{}_{}.Math.L_G/2\right)}& \left(25\right)\end{array}$

where .sub.t=/v.sub.t is the average wave number within the unit cell. The phase of F() can be calculated at least by using Euler's identity as

[00019]$\begin{array}{cc}.Math.F\left(\right)=-\frac{.Math.L_G}{2.Math.v_{}}-\frac{.Math.d_c}{v_c}-\arctan \left(\frac{\sin \left(N.Math.{}_0.Math./v_t\right)}{\cos \left(N.Math.{}_0.Math./v_t\right)-1}\right)+\arctan \left(\frac{\sin \left({}_0.Math./v_t\right)}{\cos \left({}_0.Math./v_t\right)-1}\right).& \left(26\right)\end{array}$

[0086] By reciprocity and symmetry of the transducers, this can also be equal to the phase shift experienced by a signal from the center of the ADL to the output port. Thus, the total group delay at f.sub.0 can be obtained as

[00020]$\begin{array}{cc}{{}_g\left(f_0\right)=-2.Math.\frac{d.Math..Math.F\left(\right)}{d.Math.}.Math.}_{={}_0}=\frac{L_G}{v_{}}+\frac{2.Math.d_c}{f_0.Math.{}_0}+\frac{N-1}{f_0}.& \left(27\right)\end{array}$

The first term is the delay introduced by the gap L.sub.G between transducers. The second and third terms correspond to the wave propagation within the transducers.

[0087] The overall directionality per unit cell (D/N) can be predicted from Eq. (23) as

[00021]$\begin{array}{cc}D/N\left(\mathrm{dB}\right)=\left(\frac{1+.Math..Math.}{1-.Math..Math.}\right).& \left(28\right)\end{array}$

[0088] It should be noted that, contrary to Eq. (28), the simulated D/N can show a dependence on N for low values of N. This can be explained by fringe effects in the transducer, which can make the transducer unit cells close to the edges present a smaller directionality than those cells located in the middle of the transducer.

[0089] FIG. 8 is a schematic diagram illustrating a sectional Mason's model 800 for a single unit cell according to one embodiment. Each uniform portion of the unit cell is represented by an acoustic transmission-line section. The acoustic impedance, phase velocity, and length of each section are labeled with symbols. In order to predict the response of the described ADLs with intricacies that may have been omitted in the closed-form analysis, an equivalent circuit model can be used. This method, based on Mason's model, can employ a 1D discretization of the ADL by representing each transducer unit cell of the transducers with a sectional equivalent circuit. The schematic of the implemented model for a single transducer unit cell can be found in FIG. 8. Each section with uniform properties can be modeled by a transmission line. The phase velocities for the un-metalized and metalized sections can be calculated in COMSOL. The reflections due to the discontinuities can be modeled by the different characteristic impedances of the sections representing metallized and un-metallized thin film regions, respectively Z.sub.m and Z.sub.. The ratio can be calculated as

[00022]$\begin{array}{cc}\frac{Z_m}{Z_{}}=\frac{1+{}_{.Math.m}}{1-{}_{.Math.m}},& \left(29\right)\end{array}$

where .sub.m is the reflection coefficient for an acoustic wave passing from the un-metallized to the metallized region. The reflection coefficient of an electrode can be approximated as a sum of two reflections at the step-up and step-down discontinuities, assuming small reflections. Given the width of the reflectors, these reflections may be in quadrature for DART and in-phase for EWC. Thus, it can be deduced that

[00023]$\begin{array}{cc}{}_{.Math..Math.m}^{\mathrm{DART}}=\frac{1}{\sqrt{2}}.Math..Math.{}^{\mathrm{DART}}.Math.,& \left(30\right)\\ {}_{.Math..Math.m}^{\mathrm{EWC}}=\frac{1}{2}.Math..Math.{}^{\mathrm{EWC}}.Math.,& \left(31\right)\end{array}$

[0090] The lengths of the transmission line sections are labeled in FIG. 8 for both the DART and EWC designs. The transduction section, which may include the signal electrode 826, can be modeled as a T-shaped network with an ideal transformer 844 connecting to the electrical port. The transformation ratio can be determined by the electromechanical coupling and is given by

r={square root over (2f.sub.0C.sub.sk.sup.2Z.sub.m)},(32)

where C.sub.s represents the static capacitance per transduction unit cell. Note the ratio Z.sub.m/Z.sub. defines the reflections, but the value of Z.sub. (or Z.sub.m) may be irrelevant for the electric response. Hence, Z.sub.=1 can be taken. The angle .sub.m can be obtained as

[00024]$\begin{array}{cc}{}_m=\frac{}{4}.Math.\frac{f.Math.{}_0}{v_m}& \left(33\right)\end{array}$

for both DART and EWC transducers.

[0091] FIG. 9A is a schematic diagram illustrating an FEM model 900 built in COMSOL to simulate the response of the ADLs to validate the circuit model 800 of FIG. 8 according to one embodiment.

[0092] FIG. 9B is a schematic diagram illustrating a magnitude of a displacement along the y-axis at the center frequency according to one embodiment. With all the parameters defined, a model for a complete transducer can be built by concatenating the models of its unit cells. The unit cells can be connected in series in the acoustic domain, and in parallel in the electrical domain to form the electrical port of the transducer. A complete ADL can be simulated by connecting the models of two transducers in the acoustic domain with their FWD ports facing each other. The gap between the transducers can be modeled by an acoustic transmission line with characteristic impedance Z.sub., phase velocity v.sub. and length L.sub.G. To ensure no reflection at the BWD acoustic ports of both transducers, these can be terminated by an impedance Z.sub..

[0093] The IL of a device formed by either DART or EWC transducers of 10 cells can be expressed as

IL (dB)=1.29+5.8.sub.g (s).(34)

[0094] FIGS. 10A-10E are a set of graphs illustrating the simulated performance of the AlN Lamb mode ADL with TFE-SPUDT according to one embodiment. The TFE-SPUDT and the AlN Lamb mode ADL are as described with respect to FIGS. 3A-3C. FIG. 10A illustrates the performance of the cross-sectional mode shape depicting the unidirectionality of the TFE-SPUDT transducers for an ADL with 20 cells. The thickness of the cross-section may be exaggerated. FIG. 10B illustrates the performance for the IL. FIG. 10C illustrates the performance of the zoomed-in IL. FIG. 10D illustrates the performance of the RL. FIG. 10E is the group delay, illustrating their trade-off.

[0095] The simulated performance of AlN ADLs is shown in FIGS. 10B-10D. The simulation can be done with 2D finite element analysis in COMSOL, assuming lossless propagation in the film and perfectly matched layers at the longitudinal ends as described herein. Effects of the clamped ends of the suspended AlN thin film are not included. Different cell numbers from 20 to 40 are investigated with a cell length of 46.4 m and a gap length of 0.1 mm. The displacement mode shape validates the uni-directionality. The TL-FBW tradeoff can be observed, as the 20 cell device can offer an IL of 0.27 dB and a 3 dB FBW of 3.2%, while the 40 cell device can provide a lower IL of 0.1 dB, but a narrower 3 dB FBW of 1.4%. The in-band ripples may be caused by the finite uni-directionality of the transducers and the discontinuity of the bottom electrode. Such low-loss performance may be feasible because an adequate reflectivity of 0.06 per cell is enabled by the notable mechanical reflection of the embedded reflector (e.g., a floating reflector arranged between adjacent transduction electrodes) on the thin-film structure, which can compensate for the small electrical reflection attainable from the moderate k.sup.2 of 2.8% for S0 in the film stack.

[0096] FIGS. 11A-11B are optical microscope images of a fabricated AlN ADL according to one embodiment. FIG. 11A is a zoomed-out view of the fabricated device, and FIG. 1B is a zoomed-in view of the fabricated device. The relevant parameters are shown in Table 1 below. The fabricated devices show no visible warping due to the residue stress. The SPUDTs are also defined with high fidelity in width and excellent uniformity across cells.

TABLE-US-00001 TABLE 1 Sym. Parameter Value Cell length (m) 32.0-46.4 R.sub.e Electrode ratio 0.125 R.sub.r Reflector ratio 0.25 N Number of cells 20-40 L.sub.g Gap length (mm) 0.1-0.8 W.sub.a Aperture width (m) 200 W.sub.d Device width (m) 256 T.sub.AI AI thickness (nm) 150 T.sub.AIN AIN thickness (nm) 1000 T.sub.Pt Pt thickness (nm) 100

[0097] FIGS. 12A-12D are a set of graphs illustrating the measured S-parameters of the fabricated AlN ADLs (L.sub.g=0.1 mm, =46.4 m) with different N (20-40) according to one embodiment. FIG. 12A illustrates the IL. FIG. 12B illustrates the RL. FIG. 12C illustrates the group delay. FIG. 12D illustrates the extracted IL and FBW, illustrating their trade-off. The fabricated ADLs were measured with a vector network analyzer at the 10 dBm power level in air, and then conjugately matched under ideal assumptions in Keysight Advanced Design System (additional loss from matching networks is not included). Devices with the same cell length and gap length, but different cell numbers were measured for investigating the relation between the number of cells and the IL-FBW trade-off An IL of 5.9 dB and a 3 dB FBW of 4.5% are obtained for a 20-cell device while an IL of 5.6 dB and a 3 dB FBW of 2.7% are obtained for a 30-cell device. An IL of 6.4 dB and a 3-dB FBW of 2.3% are obtained for a 40-cell device. The IL-FBW trade-off shown herein significantly surpasses the previously reported results. The IL is larger than the simulated values, which is likely caused by the acoustic propagation loss (PL) in the AlN and transducers, as well as the moderate Qs of transducer static capacitances resulting from the electrical and dielectric loss in the transducers. These loss mechanisms also likely cause the slightly increasing IL for longer ADLs with more cells.

[0098] FIGS. 13A-13D are a set of graphs illustrating the measured S-parameters of AlN ADLs (N=20, =46.4 m) with different L.sub.g (0.1-0.8 mm) according to one embodiment. FIG. 13A is the IL and FIG. 13B is the RL. The IL and RL responses show a 3 dB FBW of 4.5% and a minimum IL of 5.9 dB. FIG. 13C is the group delay. FIG. 13D is the extracted propagation loss and phase velocity of the S0 mode in AlN at 175 MHz. Devices with the same cell number and cell length but different gap lengths were measured for investigating the propagation loss and the phase velocity of the S0 mode in AlN. The extracted group velocity (11560 m/s) is close to the previously reported value and that simulated from the COMSOL finite element analysis (FEA) (10800 m/s). The extracted propagation loss is 1.78 dB/mm for the S0 mode at 175 MHz.

[0099] FIGS. 14A-14C are a set of graphs illustrating the measured S-parameters of the ADLs (N=20, L.sub.g=0.1 mm) with different (32.0-46.4 m) or center frequencies according to one embodiment. FIG. 14A illustrates the IL and FIG. 14B illustrates the RL. The IL and RL responses show a 3 dB FBW around 4.5% at different frequencies. FIG. 14C is the measured group delays of different ADLs in the passbands. Devices with the same cell number and gap length but different cell lengths were measured for investigating the frequency scalability of the AlN ADL. ADLs with center frequencies ranging from 175 MHz to 255 MHz can be demonstrated. The larger IL at higher frequencies is likely caused by the increasing PL with respect to frequency.

[0100] Low-loss Lamb mode ADLs in AlN thin film have been described herein using the proposed TFE-SPUDT. The implemented devices show significantly improved IL-FBW performance trades, which may demonstrate the potentials of integrating AlN ADLs with CMOS for radio frequency signal processing.

[0101] FIG. 15 is a schematic diagram of a full-duplex transceiver 1500 with an acoustic delay synthesizer 1502 to attain self-interference cancellation (SIC) according to one embodiment. In some embodiments, the full-duplex transceiver 1500 may include an antenna 1504, a circulator 1506, a directional coupler 1508, a power amplifier (PA) 1510, a low noise amplifier (LNA) 1512, a receiver 1514, a transmitter 1516, a tunable attenuator 1518, and the acoustic delay synthesizer 1502. In some embodiments, the acoustic delay synthesizer 1505 may include at least one acoustic delay line (ADL), as disclosed in the various embodiments herein.

[0102] ADLs can be useful in implementing full-duplex radios, such as the full-duplex transceiver 1500. One potential challenge for implementing full-duplex radios can include self-interference (SI). Due to the absence of frequency- or time-domain multiplexing, SI can occur when high-power transmitted signals are reflected from antenna packaging or obstacles in the ambiance, and inadvertently received by a highly sensitive receiver, typically after a 0.01-1 s delay. To reduce the SI, e.g., attain SI cancellation (SIC), one approach can be to provide wideband time-domain equalization using true time delays.

[0103] In such a method, a fraction of the transmitted signal is sent into a time-domain equalizer that emulates the channel transfer function of the SI before it is combined with the SI to render cancellation through destructive interference. To accommodate the dynamic in-field conditions, such a system is typically required to provide reconfigurable delays and tunable attenuations. The challenge with such a method is that, although chip-scale tunable attenuation is attainable, miniature delay synthesis over a sufficiently wide bandwidth (BW) and a necessary delay range remains inaccessible. The unavailability of wide-range delay synthesis originates from the fact that the electromagnetic (EM) delay lines in the existing prototypes can hardly provide delays of more than 1 ns on chip scale due to the fast group velocities of EM waves in state-of-the-art slow wave waveguide-related structures. Therefore, EM-based delay synthesis is inadequate for enabling full-duplex in urban environments with dense reflectors (e.g., moving vehicles and buildings). Moreover, the dynamic range of EM-based SIC is also limited. The minimum insertion loss (IL) in the cancellation path is required to be no larger than that in the free space. However, the intrinsically high propagation loss (PL) in the EM delay lines leads to high IL. Moreover, the additional IL from the directional coupler strengthens the requirement of IL, which is challenging for the EM delay lines.

[0104] In some embodiments, the full-duplex transceiver 1500 can also be referred to as a full-duplex radio. A full-duplex radio can transmit and receive signals in the same frequency band simultaneously. The full-duplex transceiver 1500 includes transmit (TX) chain circuitry and receive (RX) chain circuitry. The TX chain circuitry includes at least the directional coupler 1508, the PA 1510, and the transmitter 1516. The RX chain circuitry includes at least the LNA 1512 and the receiver 1514. The TX chain circuitry transmits a first RF signal in a first frequency range via the antenna 1504. The RX chain circuitry receives a second RF signal in the first frequency range via the antenna 1504. The TX chain circuitry can further include the directional coupler 1508, which directs a portion of the first RF signal (e.g., that is transmitted) to the RX chain circuitry. The acoustic delay synthesizer 1502 includes a set of ADLs and is coupled between the TX chain circuitry and the RX chain circuitry in order to provide a signal delay. In other words, the acoustic delay synthesizer 1502 provides a delay to the portion of the first RF signal to the RX chain circuitry such that the first RF signal experiences the signal delay and destructively interferes with a reflected portion of the first RF signal.

[0105] As described above, in an ADL, radio frequency (RF) signals are first converted into the acoustic domain by transducers on one end of the ADL via piezoelectricity. The signals can then propagate as acoustic waves and experience the designed delay before they are turned back into electrical signals by transducers on the other end. In some embodiments, RF ADLs may be realized using surface acoustic waves (SAW) technologies due to their compact sizes and easy fabrication processes. ADLs can be used to enable time delays, filtering, and correlation for improving the signal-to-noise ratios in radar front ends. ADLs can also be used for various sensing applications and construction of nonreciprocal networks. SAW ADLs may not provide sufficiently low IL and wide BW simultaneously for self-interference cancelation (SIC) applications even when custom designed unidirectional transducers are adopted. Such a performance limit can arise from the intrinsic tradeoff between the IL and fractional BW (FBW), which can be fundamentally imposed by the attainable reflectivity of the distributed reflectors and the maximum electromechanical coupling (k.sup.2) of the SAW modes. In addition, the transducer-induced SAW scattering into the substrate may further exacerbate the PL of the SAW and the tradeoff between IL and delay. To work toward an acoustic delay synthesizer, the fundamental performance bounds may be considerably lifted by resorting to a new piezoelectric platform with higher coupling, larger available reflectivity, and better-confined wave guiding at the same time, as will be described in the following embodiments and in more detail with reference to the various figures.

[0106] In some embodiments, longitudinally vibrating modes in thin-film lithium niobate (LiNbO.sub.3), namely, the fundamental shear-horizontal (SH0) mode and fundamental symmetrical (S0) mode, can be utilized in ADL structures for their simultaneously large k.sup.2 and low loss. The large coupling can be harnessed to widen the BW of ADLs, while the confined wave guide within a suspended LiNbO.sub.3 thin film can lower PL and thus also lower IL. Moreover, reflectors on a suspended thin film can provide more substantial reflections in comparison to the same type of reflectors on a SAW structure, which can further improve the tradeoff between IL and BW.

[0107] Such longitudinally vibrating modes can be used for the acoustic delay synthesizer 1502 of the full-duplex transceiver 1500. The acoustic delay synthesizer includes a set of ADLs. Each of the ADLs includes an AlN piezoelectric thin film, a first IDT, and a second IDT. The piezoelectric thin film is located above a carrier substrate. The piezoelectric thin film is adapted to propagate an acoustic wave in at least one of a first mode excited by an electric field oriented at least partially in a longitudinal direction along a length of the piezoelectric thin film or a second mode excited by the electric field oriented in a thickness direction of the piezoelectric thin film. The first IDT is disposed on a first end of the piezoelectric thin film and converts a first electromagnetic signal, which is traveling in the longitudinal direction, into the acoustic wave. The second IDT is disposed on a second end of the piezoelectric thin film. There is a gap between the second IDT and the first IDT. The second IDT converts the acoustic wave into a second electromagnetic signal. In some embodiments, the first mode can be one of an S0 mode, an S1 mode, or an SH0 mode and the second mode can be one of an A1 mode or an SH1 mode.

[0108] FIG. 16 is a flow diagram of a method 1600 of the operation of an ADL according to one embodiment. In one embodiment, the method 1600 is performed by processing logic coupled to or included within an ADL, such as the ADL devices 100-400 as well as other ADL devices described herein. In one embodiment, the piezoelectric thin film of the ADL can be an AlN thin film.

[0109] Referring to FIG. 16, at operation 1602, the method 1600 includes converting, by a first interdigitated transducer (IDT), a first electromagnetic signal into an acoustic wave. The piezoelectric thin film can be an aluminum nitride (AlN) thin film and be located above a carrier substrate. At operation 1604, the method 1600 includes propagating the acoustic wave in at least one of a first mode excited by an electric field oriented at least partially in the longitudinal direction or a second mode excited by the electric field oriented in a thickness direction of the piezoelectric thin film. At operation 1606, the method 1600 includes converting, by a second IDT disposed on a second end of the piezoelectric thin film, the acoustic wave into a second electromagnetic signal after a delay determined by a gap between the first IDT and the second IDT. At operation 1608, the method 1600 includes outputting the second electromagnetic signal.

[0110] In further embodiments, the first mode is a first-order antisymmetric (A1) mode, and the second mode is at least one of a fundamental symmetric (S0) mode or a first-order symmetric (S1) mode. In still further embodiments, a voltage potential can be applied across a signal line coupled to the first IDT to generate the electric field.

[0111] The words example or exemplary are used herein to mean serving as an example, instance, or illustration. Any aspect or design described herein as example or exemplary is not necessarily to be construed as preferred or advantageous over other aspects or designs. Rather, use of the words example or exemplary is intended to present concepts in a concrete fashion. As used in this application, the term or is intended to mean an inclusive or rather than an exclusive or. That is, unless specified otherwise, or clear from context, X includes A or B is intended to mean any of the natural inclusive permutations. That is, if X includes A; X includes B; or X includes both A and B, then X includes A or B is satisfied under any of the foregoing instances. In addition, the articles a and an as used in this application and the appended claims may generally be construed to mean one or more unless specified otherwise or clear from context to be directed to a singular form. Moreover, use of the term an implementation or one implementation or an embodiment or one embodiment or the like throughout is not intended to mean the same implementation or implementation unless described as such. One or more implementations or embodiments described herein may be combined in a particular implementation or embodiment. The terms first, second, third, fourth, etc. as used herein are meant as labels to distinguish among different elements and may not necessarily have an ordinal meaning according to their numerical designation.

[0112] In the foregoing specification, embodiments of the disclosure have been described with reference to specific example embodiments thereof. It will be evident that various modifications can be made thereto without departing from the broader spirit and scope of embodiments of the disclosure as set forth in the following claims. The specification and drawings are, accordingly, to be regarded in an illustrative sense rather than a restrictive sense.