SUB-THz ELECTROMECHANICAL RESONATORS AND METHODS OF USE

Abstract

Provided herein are electromechanical resonators including a substrate; a Lamb wave resonator over the substrate; and a millimeter-wave dual-rail resonator coupled to the Lamb wave resonator. Also provided herein are electromechanical resonators further including an air gap formed between the dual-rail resonator and the Lamb wave resonator; wherein at least a portion of the dual-rail resonator is suspend over the Lamb wave resonator in the air gap.

Description

[0001]Attorney Docket No.047162-7389WO1 (02512) TITLE OF THE INVENTION SUB-THz ELECTROMECHANICAL RESONATORS AND METHODS OF USE CROSS-REFERENCE TO RELATED APPLICATIONS The present application claims priority under 35 U.S.C. § 119(e) to U.S. Provisional Patent Application No.63 / 551,897, filed February 9, 2024, which application is incorporated herein by reference in its entirety. STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT This invention was made with government support under W911NF-18-1-0020 awarded by Army Research Office, 1640959 and 1941583 awarded by National Science Foundation, and DE-SC0019406 awarded by Department of Energy. The government has certain rights in the invention. BACKGROUND OF THE INVENTION Modern communication systems critically rely on high-quality electromechanical resonators as band-selecting filters. As their operating frequency directly determines the communication speed, it is appealing to use electromechanical resonators of higher and higher frequencies in wireless communications. The fifth-generation (5G) communication systems are being commercially deployed today throughout the world, with assigned bands consisting mainly of sub-6 GHz bands and prototypes being developed up to 60 GHz. Looking ahead, the future sixth-generation (6G) communication systems will focus on sub-THz and THz bands, with the Federal Communications Commission (FCC) having created experimental licenses for the use of frequencies between 95 GHz and 3 THz. The ultrafast data rate and greater capacity that come with the 6G networks will continually revolutionize RF communication technologies. Thus, the development of sub-THz electromechanical resonators (95GHz and above) serves as a vital part of this endeavor. Micromechanical resonators in the sub-THz regime also provide great opportunities in studying the quantum motion of micromechanical structures, quantum entanglement of massive objects, and long-distance, hybrid quantum networks. Since the 1 GHz barrier of detecting - 1 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) nanomechanical motion was surpassed in 2003, current mechanical resonators in the microwave GHz frequency regime are routinely refrigerated to their mechanical ground state in milli-Kevin environments. According to the Bose-Einstein distribution, a higher frequency millimeter-wave resonator in the sub-THz regime can maintain quantum ground state with a temperature on the order of Kelvin, making them appealing elements for accessing quantum mechanical motion. However, advancing a mechanical resonator into the sub-THz regime is a non-trivial task. An approach to address high-frequency mechanical motion is using ultrafast laser pulses through an optical pump-probe scheme, which significantly limits the scaling of the systems for device applications. Electromechanical actuation and readout, on the other hand, are desirable for integrated device architectures. Considering the typical acoustic velocities within solids being on the order of thousands of meters per second, the acoustic wavelength for sub-THz phonons is only tens of nanometers, which raises great challenges in effective transduction and nanofabrication. Accordingly, there is a need in the art for articles and methods that improve on existing resonators by exhibiting increased operating frequencies. The present invention addresses this need. SUMMARY In one aspect, an electromechanical resonator includes a substrate; a Lamb wave resonator over the substrate; and a millimeter-wave dual-rail resonator coupled to the Lamb wave resonator. In some embodiments, the electromechanical resonator further includes a patterned insulating layer between the Lamb wave resonator and the substrate. In some embodiments, the patterned insulating layer includes at least one open section that is devoid of insulating material between the Lamb wave resonator and the substrate. In some embodiments, a portion of the Lamb wave resonator is suspended over the substrate in the open section. In some embodiments, the substrate comprises a semiconducting material. In some embodiments, the semiconducting material comprises silicon (Si) or sapphire. In some embodiments, the patterned insulating layer comprises a sacrificial material. In some embodiments, the sacrificial material comprises silicon dioxide (SiO2). In some embodiments, the Lamb wave resonator comprises a material having high piezoelectric coupling strength. In some embodiments, the Lamb wave resonator comprises lithium niobate (LN), lithium tantalate (LT), or scandium aluminum nitride (ScAlN). - 2 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) In some embodiments, the Lamb wave resonator has a thickness of less than or equal to 300nm. In some embodiments, the Lamb wave resonator has a thickness of less than or equal to 100nm. In some embodiments, the dual-rail resonator comprises metal electrodes on the Lamb wave resonator. In some embodiments, the metal electrodes comprise superconductors. In some embodiments, the metal electrodes comprise gold. In some embodiments, the metal electrodes comprise two central electrodes on the suspended portion of the Lamb wave resonator and two peripheral electrodes on outer portions of the Lamb wave resonator. In some embodiments, the dual-rail resonator comprises a patterned metal between the Lamb wave resonator and the substrate. In some embodiments, the patterned metal comprises two peripheral portions and two central rail portions. In some embodiments, the patterned metal comprises an array of dual rails. In some embodiments, the patterned metal comprises a superconductor. In some embodiments, the patterned metal comprises gold. In some embodiments, the substrate comprises a semiconductor material. In some embodiments, the substrate is high resistivity silicon (HR-Si). In some embodiments, the Lamb wave resonator includes a central portion and a peripheral portion on either side of the central portion. In some embodiments, the Lamb wave resonator comprises lithium niobate (LN). In another aspect, an electromechanical resonator includes a substrate; a Lamb wave resonator over the substrate; a millimeter-wave dual-rail resonator over the Lamb wave resonator; and an air gap formed between the dual-rail resonator and the Lamb wave resonator; wherein at least a portion of the dual-rail resonator is suspend over the Lamb wave resonator in the air gap. In some embodiments, the air gap is from 5 nm to 1 micron between the Lamb wave resonator and the dual-rail resonator. In some embodiments, the air gap is from 5 nm to 100 nm between the Lamb wave resonator and the dual-rail resonator. In some embodiments, the air gap is from 0 to 100 nm between the Lamb wave resonator and the dual-rail resonator. In some embodiments, the millimeter-wave dual-rail resonator can operate at a frequency up to 3 THz. In some embodiments, the millimeter-wave dual-rail resonator can operate at a frequency down to 60 GHz for fifth-generation (5G) communication systems. In some embodiments, the millimeter-wave dual-rail resonator can operate at a frequency down to 71 GHz for 5G communication systems. In some embodiments, the millimeter-wave dual-rail resonator further includes a - 3 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) patterned insulating layer between the Lamb wave resonator and the substrate. In some embodiments, the patterned insulating layer includes at least one open section that is devoid of insulating material between the Lamb wave resonator and the substrate. In some embodiments, a portion of the Lamb wave resonator is suspended over the substrate in the open section. In some embodiments, the air gap is configured to reduce acoustic loss. BRIEF DESCRIPTION OF THE DRAWINGS FIGS.1A-E show images illustrating a dual-rail coupled Lamb wave resonator. (A) False color SEM image of the device. The mechanical resonator is embedded between two signal lines GS1G and GS2G to form a dual-rail coupled Lamb wave resonator (DRCLR). (B) The distributed DRCLR model. The Lamb wave resonator is modeled as distributed conductance Gm in parallel with the distributed mutual capacitance CM . (C) (left) The differential mode (DR+−) has opposite voltages on line S1and line S2. Thus an electric wall is formed between the lines represented by the dashed blue line. (right) The common mode (DR++) has identical voltages on line S1 and line S2. A magnetic wall is formed between the lines represented by the dashed green line. (D) Zoom- in of the suspended structure (not to scale). The red and black curves on the electrodes represent the voltages Vs1and Vs2on line S1and S2respectively when the DRR is on resonance. The red curve on the cross-section illustrates the displacement of the mechanical modes, with cyan arrows marking the displacement direction. (E) Cross-section of the mechanical resonator (not to scale). (F) Simulation of the horizontal electric field with a voltage applied between the electrodes (film thickness not to scale). (G) Simulation of the mechanical displacement fields for A1, A5 and A21 modes, with color surfaces representing the displacement amplitude and cyan arrows marking the displacement direction. FIG.2 shows graphs illustrating sub-THz electromechanics. (Top) Amplitude of the reflection spectrum for a device of length 155 µm. (Bottom) Phase plot of the reflection spectrum. The inset shows the optical image of the measured device. Device parameters: s = 2 µm, g = 2 µm, he = 200 nm (see Methods for definitions). (Middle) Smith chart representations of the reflection spectra for devices of length 80 µm, 155 µm, and 245 µm. FIGS.3A-B show graphs illustrating DRR-enhanced electromechanics and figure-of- merit extraction. (A) Reflection spectra for devices of length 110 µm, 155 µm, and 245 µm. Device parameters: s = 2 µm, g = 2 µm, he = 200 nm. (B) (top) Mechanical resonant frequency - 4 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) and mode order relationship; (middle-top) Quality factor and mode order relationship; (middle- bottom) f Q product and mode order relationship; (bottom) Electromechanical coupling coefficient and mode order relationship. FIGS.4A-D show graphs and images illustrating thickness and orientation variations. (A) The measured reflection spectra for devices of thickness 365 nm and 270 nm. Device parameters: s1 = 2 µm, s2 = 1 µm, g = 2 µm, he= 300 nm (see Methods for definitions). (B) Schematic for devices of thickness 365 nm and 270 nm. (C) The smith chart representations of the measured reflection spectra for devices of different orientations. Device parameters: s1 = 4 µm, s2 = 2 µm, g = 6 µm, he= 300 nm. (D) Optical image of a typical device array. FIG.5 shows process flow for fabricating sub-THz electromechanical resonators. FIGS.6A-B show reflection measurement setups for (A) V band (50 - 75 GHz) and (B) W band (75 - 110 GHz). FIGS.7A-C show a dual-rail coupled Lamb wave resonator. (A) Optical image of a typical DUT. (B) Schematic of the suspended structure. Shown in the dashed rectangle is a ∆z section of the suspended lines, where CM is the distributed mutual capacitance and Gm is the distributed mechanical conductance. (C) (top) The differential mode (DR+−) has opposite voltages on line S1and S2, thus it can read out the embedded mechanical resonator; (bottom) The common mode (DR++) has identical voltages on line S1 and S2, thus it does not see the embedded mechanical resonator. FIGS.8A-B show network analysis of the DRCLR. (A) Schematic of the DUT, which is probed at port 1, open-circuited at port 2 and 4, and short-circuited at port 3. (B) Schematic of the applied common-mode current sources ic1, ic2and differential-mode current sources id1, id2. FIG.9 shows an image illustrating DR+−voltage distribution. FIGS.10A-B show a comparison between the DRCLR and a modified Butterworth-Van Dyke (MBVD) model. (A) The DRCLR model. The mechanical component is viewed as distributed elements embedded between the lines. (B) The MBVD model. The mechanical component is viewed as lumped elements. FIG.11 shows a graph illustrating the measured and fitted reflection spectrum for a 155 µm-long DUT. FIGS.12A-E show the operation principle of IDEAL resonator based on thin film LN. A dual-rail transmission line forms a mm-wave tank resonator which acts as an impedance - 5 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) transformer to interface with Lamb waves in the LN plate. (A) Differential signal of the excited mm-wave and (B) common mode signal of the mm-wave. (C) Equivalent circuit model describing the coupling of differential signal to the A-mode. Vertical dashed line marks an “electrical wall.” (D) The common signal does not couple to the A-modes. The vertical dashed line indicates a “magnetic wall.” (E) Simulated three lowest order asymmetric Lamb modes. FIG.13 shows reflection S11 spectrum of three IDEAL devices spanning 10-110 GHz. The TFLN has a thickness of 365nm. The spectral are taken with 3 sets of equipment covering separate bands (0-40GHz, 45-75GHz, 75-110GHz). With the calibration procedure, the spectral can be seamlessly concatenated between different bands. The admittance can be calculated through Y / Y0 = (1-S11) / (1+S11). FIGS.14A-D show mode number dependence of (A) resonant frequency, (B) unloaded quality factor, (C) frequency-quality factor (fQ) product, and (D) electroacoustic coupling strength ^^^^^2^^^. Data are obtained from z-cut resonators. FIG.15 shows Smith chart plots of W-band resonances recorded from IDEAL devices of varying tank resonator lengths. The center frequency of 110µm tank resonance matches that of the A17 mode where the highest signal extinction is obtained. FIGS.16A-D show (A-B) Device image and layout. (C) Distributed-element model and (D) lumped-element model of IDEAL resonator suite. Here, the large FSR limit and spurious modes are suppressed. FIG.17 shows admittance spectroscopy in the low frequency regime (0-40 GHz) showing resonances and antiresonances. The relevant circuit parameters for each resonance can be extracted by fitting the measured admittance using a lumped element equivalent circuit based on multi-resonance modified BVD model shown in FIGS.16A-D. FIG.18 shows the thickness dependence of A-mode acoustic resonances. FIG.19 shows device orientation dependence of the A-mode resonators in the W-band. The device performance exhibits a high level of in-plane isotropy with z-cut films. FIG.20 shows intrinsic ^^^^^^^2^ of various LN wafer cut for Lamb wave propagation at a selected angle. The highest overall ^^^^^^^2^ value is obtained with 115o y-cut films. The device specific ^^^^^2^^^is smaller than the intrinsic ^^^^^^2^^value here due to imperfect electric and acoustic field overlap. FIG.21 shows phononic cavity design for enhancing acoustic confinement in LN and - 6 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) reducing mechanical loading of metal electrodes. FIGS.22A-C show schematic of S11 measurement setup in the (A) W band and (B) V- band. (C) Snapshot of the measurement setup. FIG.23 shows proposed setup for V-band measurement with high precision network measurement and device de-embedding. FIG.24 shows process flow for LN-on-Gold (LNOG) IDEAL devices. FIGS.25A-B show LNOI-IDEAL device cross-sectional and top-view layouts. (A) Top views of different design variances. (B) Cross-sectional views of different design variances. FIGS.26A-B show device layout for LNOG-IDEAL devices. (A) Top views of different design variances. (B) Cross-sectional views of different design variances. FIGS.27A-C show schematics and simulated results of LN electro-mechanical resonators. (A) Gold electrodes that touch LN membrane. (B) Gold electrodes that are suspended on LN membrane. (C) Simulated results for Q of the system versus Q of gold electrodes. FIGS.28A-C show images of a flip-chip device and a schematic of fabrication thereof. (A) Schematic of a flip-chip device. (B) Image under microscope in which the electrode chip is on top. (C) Fabrication flow. FIGS.29A-C show plots of (A) logarithmic graph of reflection spectrum of two devices, (B) the system mechanical Q in A3 mode versus Q of gold electrodes and Q of TFLN in simulation, and (C) comparison of experimental Q versus frequency in the offloading case and the loaded case. FIGS.30A-C show images illustrating suspended Lamb-wave resonators with varying thicknesses. (A) Image of the prepared lithium niobate-on-insulator chip featuring areas with multiple thickness levels: 67, 107, 165, 230, and 300 nm. The variation in colors in the central region is due to the light reflection of a fluorescent lamp, highlighting the differing thickness levels across the chip. Inset shows the incremental thickness of LN film stair steps. (B) Image of the chip with fabricated devices. Five regions labeled (i) to (v) are marked. (C) Crosssectional schematics of the suspended LWRs with varying LN thicknesses: 67nm (i), 107nm (ii), 165nm (iii), 230nm (iv), 300nm (v). FIGS.31A-C show images and graphs illustrating electrical responses of nanomechanical resonators. (A) Illustration of TS mode displacement for varying film thicknesses, all with similar frequencies around 150 GHz, with mode orders indicated. (B) - 7 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) Reflection spectra for devices with thicknesses of 67 nm, 107 nm, 165 nm, 230 nm, and 300 nm, plotted over 110-220 GHz. Mode orders are labeled adjacent to respective resonances. Curves are shifted vertically for clarity. (C) Smith chart representations of the spectra for devices with thicknesses of 300nm (i), 230nm (ii), and 165nm (iii). Mode orders are labeled adjacent to respective resonances. The origin is marked by a black dot. FIG.32 shows plots illustrating study of Qs in LN resonators. Panel (a) shows extracted Qs of mechanical resonances across different mode orders for devices of varying thickness. Each curve represents a single device. Panel (b) shows scatter plots displaying Qs at approximately 63 GHz (i) and 168 GHz (ii) for devices of different thicknesses. Each color represents a group of multiple devices with the same thickness. FIG.33 shows a process flow for fabricating nanomechanical resonators with multiple thickness levels. FIG. 34 shows a graph illustrating the relationship between ^^^^2^^^^⁄ ^^^^2^^^^0 and ^^^^⁄ ^^^^^^^^ for oddand even mode orders. DETAILED DESCRIPTION OF THE INVENTION Definitions Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. Although any methods and materials similar or equivalent to those described herein can be used in the practice or testing of the present invention, the preferred methods and materials are described. The articles “a” and “an” are used herein to refer to one or to more than one (i.e., to at least one) of the grammatical object of the article. By way of example, “an element” means one element or more than one element. “About” as used herein when referring to a measurable value such as an amount, a temporal duration, and the like, is meant to encompass variations of ±20% or ±10%, more preferably ±5%, even more preferably ±1%, and still more preferably ±0.1% from the specified value, as such variations are appropriate to perform the disclosed methods. Ranges: throughout this disclosure, various aspects of the invention can be presented in a range format. It should be understood that the description in range format is merely for - 8 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) convenience and brevity and should not be construed as an inflexible limitation on the scope of the invention. Accordingly, the description of a range should be considered to have specifically disclosed all the possible subranges as well as individual numerical values within that range. For example, description of a range such as from 1 to 6 should be considered to have specifically disclosed subranges such as from 1 to 3, from 1 to 4, from 1 to 5, from 2 to 4, from 2 to 6, from 3 to 6 etc., as well as individual numbers within that range, for example, 1, 2, 2.7, 3, 4, 5, 5.3, and 6. This applies regardless of the breadth of the range. Detailed Description Provided herein are electromechanical resonators with high operating frequencies. Referring to FIG.1A, in various aspects and embodiments, an electromechanical resonator 100 includes a substrate 110, a Lamb wave resonator 120 over the substrate 110, and a millimeter- wave dual-rail resonator 130 positioned relative to the Lamb wave resonator 120. The substrate 110 includes any suitable semiconducting material, such as, but not limited to, silicon (Si) or sapphire. The Lamb wave resonator 120 includes any suitable material having high piezoelectric coupling strength, such as, but not limited to, lithium niobate (LN), lithium tantalate (LT), or scandium aluminum nitride (ScAlN). The dual-rail resonator 130 includes any suitable high conductivity metal(s), such as, but not limited to, gold, aluminum, copper, silver, superconductor material, or a combination thereof. For example, in certain embodiments, electromechanical resonator 100 includes a Si or high resistivity Si (HR-Si) substrate 110 with a LN Lamb wave resonator 120 coupled to a gold electrode dual-rail resonator 130. Although described herein primarily with respect to Si substrates with LN Lamb wave resonators and gold electrode dual-rail resonators, as will be appreciated by those skilled in the art, the disclosure is not so limited and expressly includes any other combination of suitable materials. For example, in some cases, the electrode materials can include any high conductivity metal(s) such as, but not limited to, gold, aluminum, copper, silver, superconducting electrodes, combinations thereof, and / or the like. In some cases, the Lamb wave resonator / membrane can include material such as, for example, LN, lithium tantalate, scandium aluminum nitride, combinations thereof, and / or the like. Any and all combinations of various electrode materials and Lamb wave resonator materials are explicitly contemplated herein. The dual-rail resonator 130 includes two transmission lines, or ports, with each port - 9 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) having any suitable number and / or configuration of traces. For example, still referring to FIG. 1A, in some embodiments, the dual-rail resonator 130 includes a first port 131 and a second port 133, with the first port 131 having a single first trace 132 extending therefrom and the second port 133 having a single second trace 134 extending therefrom. Alternatively, in some embodiments, at least one of the ports includes two or more traces. For example, referring to FIG.4B, in some embodiments, the first port 131 includes a single first trace 132, and the second port 133 includes two second traces 134. In some such embodiments, the two second traces 134 are symmetrical and function the same. Turning to FIG.26, in another example, the first port 131 includes two first traces 132 and the second port 133 includes two second traces 134. In some such embodiments, the first traces 132 and the second traces 134 extend from the respective ports in an alternating pattern. As will be appreciated by those skilled in the art, although described herein primarily with respect to one or two traces extending form each port, the disclosure is not so limited and may include any other suitable number or combination of traces extending from each port. When positioned relative to the Lamb wave resonator 120, the dual-rail resonator 130 may be above or below the Lamb wave resonator 120, with respect to the substrate 110. Additionally, the dual-rail resonator 130 can be in contact with or separated from the Lamb wave resonator 120. For example, in some embodiments, as illustrated in FIGS.1D and 27A, the dual-rail resonator 130 is coupled to the top of (i.e., in contact with and above) the Lamb wave resonator 120. In such embodiments, the electromechanical resonator 100 includes a patterned insulating layer 140 between the Lamb wave resonator 120 and the substrate 110. The patterned insulating layer 140 includes any suitable insulating sacrificial material, such as, but not limited to, silicon dioxide (SiO2). As used herein, the term “sacrificial material” refers to any material which may be selectively removed (e.g., by etching) to form a released structure. The term “released structure,” as used herein, refers to an electromechanical resonator 100 where a portion of the patterned insulating layer 140 has been removed to form at least one open section 150 that is devoid of insulating material 140. In some embodiments, the open section 150 is formed between the Lamb wave resonator 120 and the substrate 110, such that the Lamb wave resonator 120 is suspended over the substrate 110 in the open section 150. For example, in certain embodiments, the Lamb wave resonator 120 includes a central suspended portion 121 and a peripheral portion 123 on either side of the central suspended portion 121. - 10 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) Referring to FIG.27B, in some embodiments, the electrodes, or traces, of the dual-rail resonator 130 are suspended over (i.e., separate from and above) the Lamb wave resonator 120. In contrast to embodiments where the electrodes are directly touching the Lamb wave resonator 120 (FIG.27A), which are referred to herein as “loaded,” the embodiments including the electrodes suspended over the Lamb wave resonator 120 are considered “offloaded.” The offloaded electromechanical resonator 100 can include any suitable combination of materials described herein, such as, but not limited to, a Si substrate 110; a SiO2patterned insulating layer 140, an LN Lamb wave resonator 120, and a gold dual-rail resonator 130. The offloaded device includes any suitable air gap 2800 between the electrodes of the dual-rail resonator 130 and the Lamb wave resonator 120. For example, the device may include an air gap 2800 of up to 1 micron, between 5 nm and 1 micron, up to 500 nm, between 5 and 500 nm, up to 100 nm, between 5 nm and 100 nm, between 10 nm and 100 nm, or any suitable combination, sub- combination, range, or sub-range thereof. Still referring to FIGS.27A-C, the displacement of the A3 eigenmode is shown in FIGS. 27A(i) and B(i), where magnitude is mapped from blue to red. FIGS.27A(ii) and B(ii) show the stored mechanical energy distribution at resonance (magnitude is mapped from blue to orange). FIG.27C shows simulated results for Q of the system versus Q of gold electrodes. The blue line (bottom line) plots the loaded case, the red line (middle line) plots the offloading case with a 50nm air gap, and the purple line (top line) plots the offloading case with a 1350nm air gap. All simulation results demonstrated in FIGS.27A-C are based on the A3 mode of 300nm LN resonator (18GHz). The offloading of the mechanical resonator according to one or more of the embodiments disclosed herein reduces the acoustic loss channel and / or significantly improves the Q of the electro-mechanical platform, as compared to devices where electrodes directly contact the LN membrane. For example, as illustrated in FIGS.29A-C, the Qs of the offloaded devices were 2 to 3 times the Qs of the devices where electrodes directly contact the LN membrane. Moreover, offloading of the mechanical resonator is universally applicable to enhancing the Q-factor in materials with varying loss properties. For example, as shown in FIG.29B, the intrinsic Q of the LN thin film and the loss of gold electrodes can be varied, which displays that the off-loaded platform maintains its competitive advantage on the system mechanical Q over the loaded case. Turning to FIGS.26A-B, in some embodiments, the dual-rail resonator 130 is positioned - 11 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) between the substrate 110 and the Lamb wave resonator 120. In such embodiments, the electromechanical resonator 100 does not include the patterned insulating layer 140 as described above. Instead, the Lamb wave resonator 120 is directly supported by the patterned material of the dual-rail resonator 130. This configuration is also referred to herein as LNOG, or lithium niobate on gold. As will be appreciated by those skilled in the art, the Lamb wave resonator 120 on top of the dual-rail resonator 130 may include the same configuration as when the dual-rail resonator 130 is on top, or the configuration of the Lamb wave resonator 120 may be varied. For example, in some embodiments, the Lamb wave resonator 120 may extend over the central electrodes (traces) and outer portions of the dual-rail resonator 130, or may include only a central portion over the electrodes of the dual-rail resonator 130. The Lamb wave resonator 120 according to any of the embodiments disclosed herein includes any suitable thickness 160 (FIGS.1D and 26B) for generating the operating frequencies disclosed herein. Suitable thicknesses of the Lamb wave resonator include, but are not limited to, up to 1 µm, up to 900 nm, up to 800 nm, up to 700 nm, up to 600 nm, up to 500 nm, up to 400 nm, up to 300 nm, up to 200 nm, up to 100 nm, between 100 nm and 1 µm, or any suitable combination, sub-combination, range, or sub-range thereof. In some embodiments, such as, for example, as illustrated in FIG.1D, the Lamb wave resonator 120 includes a uniform thickness 160. Alternatively, in some embodiments, the Lamb wave resonator 120 includes variations in thickness 160 within the electromechanical resonator 100. For example, in some embodiments, as illustrated in FIGS.21, 25B, and 26B, the Lamb wave resonator 120 includes a raised section 2100 with an increased thickness 160 as compared to the rest of the Lamb wave resonator 120. In some embodiments, the raised section 2100 is centered between the electrodes of the dual-rail resonator 130. Furthermore, the electrodes of the dual-rail resonator 130 may be positioned relative to variations in thickness 160 of the Lamb wave resonator 120. For example, the electrodes of the dual-rail resonator may be positioned on either side of a raised section 2100 of the Lamb wave resonator 120 (e.g., on or below thinner sections abutting the raised section). Also provided herein are devices and / or chips including one or more of the electromechanical resonators according to any of the embodiments disclosed. For example, in some embodiments, as illustrated in FIGS.4D, 19A, and 30A-B, a device 400 includes an array of the electromechanical resonators 100. The array may include identical electromechanical resonators 100 or any suitable combination of different electromechanical resonators 100. For - 12 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) example, the array may include loaded electromechanical resonators 100, offloaded electromechanical resonators 100, electromechanical resonators 100 having differing numbers of electrodes, electromechanical resonators 100 having the dual-rail resonator 130 on the Lamb wave resonator 120, electromechanical resonators 100 having the dual-rail resonator 130 between the Lamb wave resonator 120 and the substrate 110, electromechanical resonators 100 with varying thickness in the Lamb wave resonator 120, electromechanical resonators 100 with varying thickness between Lamb wave resonators 120, or any combination thereof. Turning to FIGS.30A-B, in some embodiments, for example, the device 400 includes an array of electromechanical resonators 100 divided into different sections 3100. In some embodiments, the electromechanical resonators 100 in one section 3100 are different from the electromechanical resonators 100 in at least one other section 3100. In some embodiments, the electromechanical resonators 100 in each section 3100 are different from the electromechanical resonators 100 in the other sections 3100. The differences between electromechanical resonators 100 may include any of the differences disclosed herein. In some embodiments, as illustrated in FIG.30C, the electromechanical resonators 100 in each section 3000 include a different thickness of the Lamb wave resonators 120 as compared to the electromechanical resonators 100 in each other section 3000. The thickness of the Lamb wave resonators 120 in each section 3000 includes any suitable thickness for generating a desired operating frequency. In some embodiments, the thicknesses of the Lamb wave resonators 120 are distributed in successive ranges across the different sections 3000. In some embodiments, the thickness of the Lamb wave resonators 120 increases or decreases between sections 3000 in a stepwise manner. For example, in some embodiments, the device 400 includes five sections, with a first section 3010 having the smallest thickness of the Lamb wave resonators 120, a second section 3020 having a thickness of the Lamb wave resonators 120 greater than that of the first section 3010, a third section 3030 having a thickness of the Lamb wave resonators 120 greater than that of the second section 3020, a fourth section 3040 having a thickness of the Lamb wave resonators 120 greater than that of the third section 3030, and a fifth section 3050 having a thickness of the Lamb wave resonators 120 greater than that of the fourth section 3040. In some embodiments, the thicknesses of the Lamb wave resonators 120 are distributed between 50 nm and 300 nm across the different sections 3000. In some embodiments, the first section 3010 includes a Lamb wave resonator 120 thickness of between 50 and 100 nm, the second section - 13 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) 3020 includes a Lamb wave resonator 120 thickness of between 101 and 150 nm, the third section 3030 includes a Lamb wave resonator 120 thickness of between 151 and 200 nm, the fourth section 3040 includes a Lamb wave resonator 120 thickness of between 201 and 250 nm, and the fifth section 3050 includes a Lamb wave resonator 120 thickness of between 251 and 300 nm. For example, in some embodiments, the first section 3010 includes a Lamb wave resonator 120 thickness of 67 nm, the second section 3020 includes a Lamb wave resonator 120 thickness of 107 nm, the third section 3030 includes a Lamb wave resonator 120 thickness of 165 nm, the fourth section 3040 includes a Lamb wave resonator 120 thickness of 230 nm, and the fifth section 3050 includes a Lamb wave resonator 120 thickness of 300 nm. When coupled as described herein, the Lamb wave resonator and the dual-rail resonator, and device incorporating the same, provide sub-THz and THz operating frequencies, which provide increased data rate and capacity as compared to existing resonators. For example, in some embodiments, the resonators disclosed herein can provide operating frequencies of at least 40 GHz, at least 50 GHz, at least 60 GHz, at least 70 GHz, at least 80 GHz, at least 90 GHz, at least 100 GHz, at least 110 GHz, at least 220 GHz, or any suitable combination, sub- combination, range, or sub-range thereof. In some cases, the resonators disclosed herein can provide operating frequencies between 60 GHz and 3 THz, between 70 GHz and 3 THz, between 80 GHz and 3 THz, between 90GHz and 3THz, between 100 GHz and 3 THz, between 110GHz and 3THz, between 220 GHz and 3THz, or any suitable combination, sub-combination, range, or sub-range thereof. In some embodiments, the resonators disclosed herein can provide operating frequencies of down to 60 GHz for 5G communications and / or at 71 GHz for supporting 6G communications (and consequently 5G, 4G, etc.,). Without wishing to be bound by theory, it is believed that the extremely high operation frequency of the sub-THz electromechanical resonators disclosed herein make them an excellent candidate for wireless communication technologies, since a higher carrier frequency directly translates to faster data transfer rates. At above 95 GHz, the sub-THz electromechanical resonators disclosed herein have broken into the 6G communication spectrum and will remove a major bottleneck for future deployment of 6G networks by serving as front-end channel filters, etc. Additionally, from a quantum science and technology perspective, with such an extremely high resonant frequency above 100 GHz, the sub-THz resonators disclosed herein are more resilient to thermal fluctuations than GHz resonators. They can reach the quantum ground state at - 14 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) more accessible Kelvin temperatures compared to the stringent milli-Kelvin environment, which is only achievable with dilution refrigerators. Such relaxation in the refrigeration requirements makes the presently disclosed sub-THz resonators perfect candidates for quantum physics studies and facilitates the construction of large-scale quantum networks leveraging mechanical resonators. Also provided herein, in various aspects and embodiments, are methods of forming the electromechanical resonators. In some embodiments, as illustrated in FIG.5, forming the electromechanical resonator 100 with the dual-rail resonator 130 on the Lamb wave resonator 120 includes forming a layer of insulating material 140 (e.g., SiO2) on a substrate material 110 (e.g., Si), and forming a layer of Lamb wave resonator material 120 (e.g., LN) over the layer of insulating material 140. This may be done by any suitable method, such as, but not limited to, smart-cut techniques. In certain embodiments, the method includes applying an annealing process to suppress any microscopic defects introduced during the smart-cut technique. Next, a two-step dry etching process is used to provide a desired Lamb wave resonator 120 thickness. In some embodiments, the dry etching process includes covering the Lamb wave resonator material 120 outside a release window with an HSQ mask 510, then applying a first Ar-ion mill etch to the Lamb wave resonator material 120 in the release window. The HSQ mask 510 is removed after the Ar-ion mill etch and the whole wafer is exposed to a second Argon etch to target Lamb wave resonator material 120 thickness, providing fully etched Lamb wave resonator material 120 in the release window. The dual-rail resonator material 130 is then deposited through a standard lift-off process. In some embodiments, the dual-rail resonator material 130 is deposited without applying a metal adhesion layer, such as Cr or Ti. Finally, the resonator is released with a timed buffered oxide etch in BOE. Turning to FIG.24, in some embodiments, forming the electromechanical resonator 100 with the dual-rail resonator 130 positioned between the substrate 110 and the Lamb wave resonator 120 includes initially preparing two separate wafers or dies. The first wafer includes a layer of insulating material (e.g., SiO2) on a substrate material (e.g., Si) and a layer of Lamb wave resonator material (e.g., LN) over the layer of insulating material. The second wafer includes a coating of dual-rail resonator material (e.g., Au) over a substrate (e.g., HR-Si). Next, the dual-rail resonator material is patterned to form a tank resonator and annealed in vacuum to reduce RF losses. The two wafers are then joined by a plasma-activated direct-flip chip bonding - 15 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) technique, after which the substrate handle of the first wafer is removed by DRIE, which stops at the buried insulator layer. The remining insulator layer is then selectively removed in BOE, leaving a continuous layer of Lamb wave resonator material on the surface of the wafer with more than 99.9% of its surface area is supported by the dual-rail resonator ground planes and electrodes. Finally, the fabrication process outlined above for the Lamb wave resonator material is applied to form the Lamb wave Resonator on the dual-rail resonator. In some embodiments, the resonator is formed using a flip-chip fabrication process. For example, FIG.28A shows an example schematic of a flip-chip device where the scale has been intentionally modified for visualization. FIG.28B shows an image of the device under microscope (the electrode chip is on top). In some embodiments, as illustrated in FIG.28C, the LN is first etched by Ar ion milling (FIG.28C(i)). As further illustrated in FIG.28C, gold is then deposited on the bonding area (FIG.28C(ii)) and the SiO2 buffer layer is removed in BOE (FIG.28C(iii)). Following removal of the SiO2buffer layer, gold and Ni are deposited on the corresponding bonding area and sapphire is etched around 150 nm (FIG.28C(iv)). Next, 200nm gold electrodes are deposited (FIG.28C(v)),nickel is removed (FIG.28C(vi)), and the two chips are bonded (FIG.28C(vii)).In some embodiments, after bonding of the chips the handle (e.g., Si and / or SiO2) is removed. Referring to FIG.33, in some embodiments, These different thicknesses are achieved through multiple blanket etching cycles, with Si chips placed on top of the completed LN regions between cycles to prevent further etching. For device fabrication, a gold electrode pattern is defined through electron-beam lithography (EBL), with polymethyl methacrylate (PMMA) as the resist, followed by a liftoff procedure. Next, the release window pattern is defined using hydrogen silsesquioxane (HSQ) resist and then argon-ion-milled to remove all LN within the designated area. Finally, the mechanical resonators are suspended by etching away the silicon dioxide beneath the LN beam with buffered oxide etchant (BOE). Those skilled in the art will recognize, or be able to ascertain using no more than routine experimentation, numerous equivalents to the specific procedures, embodiments, claims, and examples described herein. Such equivalents were considered to be within the scope of this invention and covered by the claims appended hereto. For example, it should be understood, that modifications in reaction conditions, including but not limited to reaction times, reaction size / volume, and experimental reagents, such as solvents, catalysts, pressures, atmospheric - 16 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) conditions, e.g., nitrogen atmosphere, and reducing / oxidizing agents, with art-recognized alternatives and using no more than routine experimentation, are within the scope of the present application. It is to be understood that wherever values and ranges are provided herein, all values and ranges encompassed by these values and ranges, are meant to be encompassed within the scope of the present invention. Moreover, all values that fall within these ranges, as well as the upper or lower limits of a range of values, are also contemplated by the present application. The following examples further illustrate aspects of the present invention. However, they are in no way a limitation of the teachings or disclosure of the present invention as set forth herein. EXAMPLES EXAMPLE 1 Modern communication systems critically rely on high-quality electromechanical resonators as band-selecting filters. As their operating frequency directly determines the communication speed, it is appealing to use electromechanical resonators of higher and higher frequencies in wireless communications. The fifth-generation (5G) communication systems are being commercially deployed today throughout the world, with assigned bands consisting mainly of sub-6 GHz bands and prototypes being developed up to 60 GHz. Looking ahead, the future sixth-generation (6G) communication systems will focus on sub-THz and THz bands, with the Federal Communications Commission (FCC) having created experimental licenses for the use of frequencies between 95 GHz and 3 THz. The ultrafast data rate and greater capacity that come with the 6G networks will continually revolutionize RF communication technologies. Thus, the development of sub-THz electromechanical resonators (95GHz and above) serves as a vital part of this endeavor. Micromechanical resonators in the sub-THz regime also provide great opportunities in studying the quantum motion of micromechanical structures, quantum entanglement of massive objects, and long-distance, hybrid quantum networks. Since 1 GHz barrier of detecting nanomechanical motion was surpassed in 2003, nowadays mechanical resonators in the microwave GHz frequency regime are routinely refrigerated to their mechanical ground state in milli-Kevin environments. According to the Bose-Einstein distribution, a higher frequency millimeter-wave resonator in the sub-THz regime can maintain quantum ground state with a - 17 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) temperature on the order of Kelvin, making them appealing elements for accessing quantum mechanical motion. However, advancing a mechanical resonator into the sub-THz regime is a non-trivial task. An approach to address high-frequency mechanical motion is using ultrafast laser pulses through an optical pump-probe scheme, which can significantly limit the scaling of the systems for device applications. Electromechanical actuation and readout, on the other hand, can be desirable for integrated device architectures. Considering the typical acoustic velocities within solids being on the order of thousands of meters per second, the acoustic wavelength for sub-THz phonons is only tens of nanometers, which can raise great challenges in effective transduction and nanofabrication. The newly emerged thin-film lithium niobate (TFLN) platform can be a favorable candidate to address the actuation challenge owing to its excellent piezoelectric properties and low phonon loss. To circumvent the stringent requirements on the fabrication resolution in the fundamental-tone actuation scheme, great efforts have been made in developing electromechanical overtones up to 60 GHz. Yet the inverse-square decrease of the electromechanical coupling with respect to the mode order largely limits further advance of electromechanical overtones into the sub-THz regime. In some implementations, a novel millimeter-wave dual-rail resonator (DRR) can be implemented directly on a suspended lithium niobate resonator for efficient actuation and detection of sub-THz Lamb wave overtones. Serving as a tank circuit, the DRR can greatly aid the electromechanical transduction by providing on- chip impedance matching to the mechanical modes. Together with a well-calibrated reflection measurement setup that mitigates perturbation to fragile sub-THz signals, electromechanical oscillations beyond 100 GHz can be achieved, leaping into the sub-THz regime. Based on the significance of the DRR enhancement shown in the experimental results and the achieved high signal fidelity, such a DRR-enhanced electromechanical transduction scheme can be further utilized to scale electromechanical frequencies further beyond the microwave W band. The false color SEM image of the complete resonator suite, namely the dual-rail coupled Lamb wave resonator (DRCLR), is shown in FIG.1A. To mitigate the phonon loss, the Lamb wave resonator can be suspended by chemically removing the silicon dioxide (SiO2) beneath the lithium niobate (LN) film. The DRR can be formed on top of the Lamb wave resonator and can comprise two coupled transmission lines GS1G and GS2G, short-circuited at the end of the transmission line GS2G and probed at the start of the transmission line GS1G. The DRR can - 18 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) feature a more uniformly distributed electric field distribution between the lines than a typical quarter-wavelength resonator, making it advantageous to provide efficient piezoelectric coupling to the distributed Lamb wave resonator. Such a distributive characteristic of the Lamb wave resonator originates from the device dimension being comparable to the sub-THz signal wavelength. Therefore, different from the lumped element model where the mechanical resonator is modeled as lumped RLC components, the DRCLR model, as shown in FIG.1B, can treat the mechanical resonator as distributed conductance between the coupled lines, defined as^^^^^^^^ = 1 / (^^^^^^^^ +1 ^^^^^^^^^^^^^^^^ + ^^^^^^^^^^^^^^^^), with Rm, Cm, and Lm representing the distributed motionalresistance, respectively, and ω representing the signal angular frequency. frequency f and quality factor Q can be expressed as ^^^^ =1⁄ (2^^^^�^^^^^^^^^^^^^^^^) and ^^^^ = 1⁄ (2^^^^^^^^^^^^^^^^^^^^^^^^) respectively. The embedded Lamb wave resonator canmeasuring the reflection (Γ) spectrum of the DRCLR with vector network . The input impedance Zin can be derived from the measured reflection, following Γ = (Zin− Z0) / (Zin+ Z0), where Z0= 50 Ω, is the impedance of a standard transmission line. By viewing the dual rails as a four-port network, as shown in FIG.1B, the boundary conditions of the DRCLR, where port 2 and port 4 are open-circuited, and port 3 is short- circuited. In the theoretical derivation of the input impedance Zin seen at port 1, the propagating signals on the dual rails can be treated as a superposition of the differential mode (DR+−) and the common mode (DR++), each with corresponding transmission line parameters. FIG.1C, illustrates their differences in terms of distributed capacitance and conductance. The DR+−mode has opposite voltages across the two signal lines S1and S2. The Lamb wave resonator can be read out efficiently by contributing distributed motional conductance Gmin parallel with the mutual capacitance CM . The electromechanical coupling coefficient is defined as the ratio of themotional capacitance to the mutual capacitance, ^^^^2 = ^^^^^^^^⁄ ^^^^ ++^^^^ . While for the DR mode, thevoltages on both signal lines S1 and S2 are identical, therefore the Lamb wave resonator does not contribute distributed conductance between the signal lines. Generally speaking, the DRR resonant condition Im(Zin) = 0 requires contributions from both DR+−and DR++modes. Note that the DR++mode has a significant higher characteristic impedance (Zc = 327 Ω for device shown in FIG.2) than that of the DR+−mode (Zd= 42 Ω). As additional advantages of this configuration, the DR+−mode supported by the dual rails provides a more uniform and larger differential voltage distribution between the signal lines than singly ended resonator design such as quarter- - 19 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) lambda resonators, making it ideal to couple to the distributed Lamb wave resonator. FIG.1D highlights the suspended device structure as a zoom-in of FIG.1A. Plotted at the cross-section of the suspended structure is the displacement of a typical anti-symmetric Lamb wave mode (A mode), which can be efficiently excited via a horizontal electric field between the deposited electrodes (FIG.1F), thanks to the large e51 value of the LN piezoelectric coupling tensor. To further elucidate the mechanical overtones utilized, plotted in FIG.1G are the finite-element- method (FEM) simulated displacement fields of the mechanical modes for mode order n =1, 5, and 21, with the color surface plots representing the displacement amplitude and the cyan arrows marking the displacement direction. For the n-th order mechanical overtone with an acoustic wavelength of λnin the thickness direction, its resonant condition is when the film thickness h can be written as nλn = 2h, or in terms of frequency f, nv = 2hf , where v is the acoustic velocity. Due to the reduced or even vanished effective overlap between the applied electric field and the mechanical piezoelectric field of high-order overtones, only odd-order overtones can be excited and their electromechanical coupling coefficient decreases quadratically with respect to the mode order (K2∝ 1 / n2). Large electromechanical coupling and significant DRR-enhancement in can be emphasized in the DRCLR design, as both are important to overcome the challenge of the diminishing transduction efficiency at sub-THz frequencies. The flexibility of tuning the DRR frequency via device length can aid in augmenting mechanical features in desired bands. For W- band Lamb wave readout (75 - 110 GHz), a device length of 155 µm can be chosen with its reflection spectrum shown in FIG.2 (top) (optical image in FIG.2 (bottom)). In the full W-band span, three prominent Lamb modes A17, A19, and A21 respectively at 84 GHz, 94 GHz, and 104 GHz can be resolved. To compare with devices without the DRR enhancement in the W band, FIG.2 (middle) shows plots of the smith chart representations of the reflection spectra for device lengths of 80 µm and 245 µm. The minor and even invisible resonant circles of these devices showcase the significance of DRR enhancement in the sub-THz mechanical mode actuation and detection. A systematic demonstration of the DRR behavior is shown in FIG.3A, where reflection spectra obtained from three separate measurement setups respectively covering the microwave X-Ka, V, and W bands are combined. The broadband reflection spectrum spans from 10 GHz to 110 GHz. The calibration procedure ensures accurate stitching of the measured spectrum from - 20 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) each band, except for a spectral gap between 43 GHz and 50 GHz. As the DRR length increases from 110 µm to 245 µm, the dual-rail resonance shifts to lower frequencies as expected. As a tank circuit, the DRR effectively mediates impedance mismatch to the Lamb wave resonators and facilitates the electromechanical transduction, as seen by the higher extinction of the mechanical modes in the reflection spectra. With this broadband measurement, the aforementioned linear relationship between the mechanical resonant frequency f and the mode order n is experimentally demonstrated in FIG.3B (top). To assess the mechanical damping, frequency-quality factor (fQ) product can be used as a figure of merit for the Lamb wave system. Benefiting from the non-degrading mechanical Q shown in FIG.3B (middle-top) as the mechanical frequency increases, the highest fQ product around 2.5 × 1013at the 21stmode (FIG. 3B (middle-bottom)) can be achieved, setting a record for the TFLN. Yet, the increasing trend of the f Q product suggests opportunities for further scaling up device frequencies. Consistent with the theoretically predicted inverse-square dependence of the electromechanical coupling coefficient on the mode order in unloaded resonators, 1 / n2.2relationship can be approximately extracted from the measured reflection spectra, as shown in FIG.3B (bottom). Another way that aids efficient electromechanical transduction in the sub-THz regime is to utilize a lower-order overtone to preserve a high electromechanical coupling coefficient, as described by the aforementioned inverse-square relationship between K2and n. Given the resonant condition of the Lamb wave thickness modes, a thinner film yields a larger mechanical free spectral range (FSR), thus resulting in a faster scaling up to the sub-THz regime with a lower mode order. Shown in FIG.4A are the reflection spectra for a 365 nm-thick device and a 270 nm-thick device. The 365 nm-thick device scales up to the sub-THz regime with a mode order of 21 (K2= 4.3 × 10−4); while the 270 nm-thick device achieves that with a mode order of only 15 (K2= 1.1 × 10−3). However, the faster scaling up in frequency resorting to thinner films comes at the cost of compromising the rigidity of the suspended structure. Such a trade-off should be well considered in designing sub-THz electromechanical transducers. Also from the device application perspective, the strong anisotropy of LN may impact the resonator frequency reproducibility due to fabrication variations in the wafer preparation and alignment processes. Thankfully, for anti-symmetric Lamb wave devices fabricated with z-cut LN films, their performances do not show orientation dependence due to the rotational symmetry of the involved piezoelectric properties. This result can be illustrated with the intuitive smith - 21 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) chart representations as shown in FIG.4C, where electromechanical devices of different in- plane orientations are measured. The consistency of the device’s performances, reflected by the similarities of the resonant circles shown in the smith charts, marks the robustness for electromechanical applications that utilize anti-symmetric Lamb wave resonators on a z-cut LN film. The slight differences in the reflection spectra of different devices are likely due to the systematic fabrication error. In conclusion, using a DRR-coupled Lamb-wave electromechanical system, efficient electromechanical transduction into the sub-THz regime can be demonstrated. Such sub-THz phonon control abilities will not only open up new possibilities for mechanical resonators to be used in the RF front-end for future broadband wireless communication systems, but also greatly facilitate the development of quantum phononics studies where the DRR can be in the form of high Q superconducting resonators for interfacing with high-frequency circuit quantum electrodynamics (cQED) systems. Methods 1. Nanofabrication. A 600 nm lithium niobate-on-insulator (LNOI) film can be used. First, the etching mask can be defined with hydrogen silsesquioxane (HSQ) resist patterned to cover the film except for the release window. Then the film is Argon-milled until the LN thickness in the release window is below 200 nm. Afterwards, the etching mask (HSQ) is removed and the film is etched for a second time to reach the target thickness of the mechanical resonator. Next, the gold electrodes are deposited through liftoff using polymethyl methacrylate (PMMA) resist. Finally, the mechanical resonators are released by soaking the chip in buffered oxide etchant (BOE) and removing the SiO2beneath the LN. Device parameters: For the DRR shown in FIG.2, s can be defined as the width for both signal lines S1 and S2, g as the gap between the signal lines, and he as the gold thickness. For the more generalized DRR design shown in FIGS.4A-D, s1 can be defined as the width of signal line S1, s2as the width of signal line S2, g as the gap between signal lines S1 and S2, and he as the gold thickness. 2. Reflection measurement. The reflection spectra of the device under test (DUT) was measured between 10 GHz and 110 GHz, except for a small spectral gap between 43 GHz and 50 GHz. The 10 GHz – 43 GHz spectra was taken with a commercial VNA (ShockLine MS46122B). The 50 GHz – 75 GHz and - 22 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) 75 GHz – 110 GHz spectra were taken with a V-band and W-band reflection measurement setup respectively. For the V-band and W-band setups, frequency multipliers, frequency down- converters and directional couplers can be utilized to perform reflection measurements. Such network analyzers using the off-wafer SOL calibration technique with an impedance standard substrate (ISS) can be calibrated. A shunt capacitor and a shunt conductor in the DRCLR model can be added to account for the difference in the dielectric constant (both real and imaginary parts) between the TFLN platform and the ISS. Supporting Information FIG.5 illustrates the process flow for the dual-rail coupled Lamb wave resonator (DRCLR) devices. First, the etching mask on a 600 nm lithium niobate-on-insulator (LNOI) film with hydrogen silsesquioxane (HSQ) resist to cover the film except for the release windows can be defined. Then the film is Argon (Ar) milled until the lithium niobate (LN) thickness in the release window is below 200 nm. Next, the etching mask (HSQ) is removed and the film is flood etched to reach the target thickness of the mechanical resonator. Afterwards, the gold (Au) electrodes are deposited through liftoff using polymethyl methacrylate (PMMA) resist. Finally, the mechanical resonators are released by soaking the chip in buffered oxide etchant (BOE) and removing the silicon dioxide (SiO2) beneath the LN. II. REFLECTION MEASUREMENT SETUP AND CALIBRATION V-band setup: Shown in FIG.6A is the V-band (50 - 75 GHz) reflection measurement setup. The signal for probing the device under test (DUT) is generated by a 4X multiplier with input RF1 (f1) and coupled to the DUT via a directional coupler. The reflected signal (f1) from the DUT then passes the directional coupler and is received as the RF signal input to a V-band harmonic mixer. The LO of such a mixer is supplied by RF2 (f2). To demodulate the V-band RF signal (4f1), IF signal (δf = f2 - f1) can be used as the reference by mixing RF1 and RF2 signals, and demodulate with a lock-in amplifier (Zurich HF2LI 50 MHz) the IF signal (4δf = 4f2- 4f1) from the V-band mixer at the 4thharmonics of the reference signal. The RF1 frequency f1 can be swept in the range of 12.5 - 18.75 GHz so that the 4X frequency-multiplication of the signal covers 50 - 75 GHz, i.e., the V band. A frequency difference of δf = 10 MHz between RF2 and RF1 can be used. W-band setup: FIG.6B shows the W-band (75 - 110 GHz) reflection measurement setup. The signal for - 23 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) probing the DUT is generated by a 6X multiplier (6f1) with input RF1 (f1) and coupled to the−DUT via a directional coupler. The reflected signal from the DUT (6f1) then passes the directional coupler and is received as the RF signal for the W-band mixer (6f1). For the LO of the W-band mixer, it is obtained by 6X frequency-multiplication (6f2) of the signal from RF2 (f2). To demodulate the W-band RF signal, the IF signal (δf = f2 - f1) can be used as the reference by mixing RF1 and RF2 signals, and demodulate with a lock-in amplifier (Zurich HF2LI 50 MHz) the IF signal (6δf = 6f2− 6f1) from the W-band mixer at the 6thharmonics of the reference signal. The RF1 frequency f1 in the range of 12.5 - 18.34 GHz can be swept so that the 6X frequency- multiplication of covers 75 - 110 GHz, i.e., the W band. A frequency difference of δf = 6 MHz between RF2 and RF1 can be used. Off-wafer calibration: The one-port network SOL (short-open-load) calibration is performed by utilizing an impedance standard substrate (ISS) supplied by FormFactor, Inc (part number: 104-783). The reflection spectral for short, open and load conditions are recorded to compute the error terms in the network, which include the directivity term ED, the reflection tracking term ER and the source match term ES. Such error parameters are then utilized to calibrate the reflection spectrum of the DUT, with the relation ΓΓ^^^^−^^^^^^^^^^^^=^^^^^^^^(Γ^^^^−^^^^^^^^)+^^^^^^^^, where ΓArepresents the calibrated ΓMrepresents the measured reflection spectrum. A shunt capacitor and a shunt conductor in the DRCLR model can be added to account for the difference in the dielectric constant (both real and imaginary parts) between the thin-film lithium niobate (TFLN) platform and the ISS. SNR comparison: As can be seen in the measured reflection spectra in FIGS.3A-B, the V-band spectra has lower signal-to-noise ratio (SNR) than the W-band spectra. This is due to the fact that the V- band harmonic mixer (passive instrument) has much larger conversion loss (~ 20 - 30 dB) than the W-band mixer (active instrument). Nonetheless, since the focus is within the sub-THz regime, the current V-band setup serves well to bridge the lower-frequency X-Ka spectrum and the W-band spectrum. III. NETWORK ANALYSIS OF THE DRCLR In the DRCLR design, a millimeter-wave dual-rail resonator (DRR) is utilized to aid the - 24 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) impedance matching for the Lamb wave resonator. The modeling of such coupled transmission lines differs from the commonly-used lumped Butterworth-Van-Duke (MBVD) model. In the following, the theoretical network analysis of the DRCLR system is performed. A. Electromechanical coupling via the differential mode Shown in FIG.7A is an optical image of a typical DRCLR, which comprises two coupled transmission lines GS1G and GS2G, forming a DRR, and a mechanical resonator beam beneath. Since the aim includes frequencies beyond 100 GHz and the device dimensions are comparable to sub-THz wavelengths, the mechanical resonator can be modeled as distributed motional conductance Gm, in parallel with the mutual capacitance CMbetween the two transmission lines, as highlighted in FIG.7B. The motional conductance can be expressed as 1 G^^^^= 1 where Rm, Cm and Lm represent capacitance and inductance respectively and ω represents the signal angular frequency. To derive the input impedance of the DRCLR, the propagating signals on the dual rails can be treated as a superposition of the differential mode (DR+−) and the common mode (DR++) with different sets of transmission line parameters. FIGS.7C-D illustrate that the electromechanical coupling is achieved via the differential mode. For the common mode, there exists a magnetic wall between the lines, thus it does not see the embedded mechanical resonator; while for the differential mode, there exists an electrical wall between the lines, and the mechanical resonator contributes motional conductance Gm to the distributed parameters of the lines. The common mode distributed resistance, inductance, conductance and capacitance can be denoted by Rc, Lc, Gcand Cc, and those of the differential mode by Rd, Ld, Gd and Cd. And the distributed parameters for the two modes can be expressed as ^^^^^^^^ = ^^^^^^^^ (S3)^^^^^^^^ = ^^^^^^^^ + 2^^^^^^^^ (S4)^^^^^^^^ = ^^^^^^^^^^^^ (S5)^^^^^^^^ = ^^^^^^^^^^^^ + 2^^^^^^^^, (S6)- 25 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) where Cg represents the line-to-ground capacitance, and Gcd, Gdd represent the dielectric losses of the system. B. Input impedance and reflection of the DRCLR FIGS.8A-B provide a more detailed circuit schematic of DRR. The device at the start of GS1G (port 1) can be probed and terminate GS2G with a short-circuit condition (port 3). For port 2 and port 4, the open-circuit condition is applied. For the DR+−and DR++modes, the propagation constants can be written as ^^^^^^^^ =�(^^^^^^^^ + ^^^^^^^^^^^^^^^^)(^^^^^^^^ + ^^^^^^^^^^^^^^^^) (S7)and the characteristic ^^^^^^^ + ^^^^^^^^^^^^^^^^ ^ ^^^^^^^^ =�^^^^ ^^^^^^^^^^^^(S9) ^^^^ ^^^^ ^^^^^^^^ =�^^^^ +(S10)^^^^ ^^^^^^^^^^^^^^^^The network analysis of the DRR can be performed as follows. First, the system can be viewed as a four-port network, with each port terminated with the open-circuit condition (ideal current sources), as shown in FIG.8B. To solve for the open-circuit impedance matrix, it can be assumed that the common mode is driven by current sources ic1 and ic2, and the differential mode is driven by id1and id2. The currents at port 1-4 can be written as The written as - 26 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) with L By impedance seen at port 1 can be solved, The where C. Dual-rail resonance As shown in FIG.3A, the dual-rail resonance, serving as RF tanks, mediates impedance to the mechanical resonator. For simplicity of the analysis, no loss of the lines can be assumed, thus the propagation constants for the common and differential modes can be written as ^^^^^^^^=^^^^^^^^^^^^ = ^^^^^^^^�^^^^^^^^^^^^^^^^, ^^^^^^^^ = ^^^^^^^^^^^^ = ^^^^^^^^�^^^^^^^^^^^^^^^^ where ω is the signal angular frequency. The resonantcondition for the DRR can be expressed as Im(Zin) = 0. Given the input impedance of the DUT in Eqn. S14, the resonant condition can be arrived - 27 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) To frequency can be designed to be within the W band, which, according to Eqn. S16, can be tuned by varying the device length L. Since the Lamb wave resonator is coupled to the mutual capacitance CM as distributed conductance, it is natural for us to examine the differential voltage distribution along the lines. The on-resonance differential voltage distribution across the lines can be written as where I1represents the port 1 current, and x Benefiting from its unique termination conditions and large impedance ratio Zc / Zd, the DRR features a smaller footprint, and a larger and more uniform differential voltage distribution along the lines (FIG.9 and Table S1) compared to commonly-used quarter-wavelength (λ / 4) resonators in band filter applications and quantum electrodynamics studies. Such favorable characteristics of the DRR make it ideal to couple to the distributed Lamb wave resonator. TABLE S1 - Comparison between the DRR (same device parameters as in FIG.2) and the λ / 4 resonator. DRR λ / 4 resonatorCharacteristic Zd= 42 Ω Zλ / 4= 84 Ω impedance Zc = 327 Ω footprint 0.13λ 0.25λ max(|Vd|) / |I1| 104 Ω 84 Ω Voltage Relatively uniform Non-uniform uniformity The large ratio Zc / Zdof DRR yields a smaller footprint, a larger and more uniform voltage distribution between the lines, compared to a λ / 4 resonator with the same layout. D. Comparison between the DRCLR model and the MBVD model Shown in FIGS.10A-B is a detailed comparison between the DRCLR model and the MBVD model. The most significant difference between these two models is whether the mechanical component in the electromechanical system is treated as distributed or lumped. In the DRCLR model, the Lamb wave resonator contributes distributed conductance in parallel with the mutual capacitance CM between the electrodes; while for the MBVD model, the Lamb wave - 28 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) resonator is treated as a lumped element in parallel with the lumped electrode capacitance C0. For a more direct comparison about the formalization of the mechanical conductance between the two models, Eqn. S2 can be used to write the expressions of the mechanical conductance for the DRCLR model and MBVD model respectively, where , can terms of the RLC components resonator, and allows the mechanical properties from the measured reflection spectrum to be properly extracted when such a coupling occurs. While the MBVD model fits well when the mechanical component can be viewed as a lumped element, i.e., in low-frequency bands, however, it doesn’t offer the correct interpretation for the dual-rail resonance, as elaborated in the following section. TABLE S2 – Parameters for the DLCLR model. Parameter Description Cm Distributed motional capacitance Lm Distributed motional inductance - 29 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) Rm Distributed motional resistance Cd Distributed differential mode capacitance Gd Distributed differential mode conductance LdDistributed differential mode inductance Rd Distributed differential mode inductance Cc Distributed common mode capacitance Gc Distributed common mode conductance Lc Distributed common mode inductance Rc Distributed common mode inductance CfShunt capacitance for calibration correction GfShunt conductance for calibration correction TABLE S3 – Parameters for the MBVD model. Parameter Description Cmt Motional capacitance Ltm Motional inductance Rmt Motional resistance C0 Differential mode capacitance G0 Differential mode conductance Cf Shunt capacitance for calibration correction Gf Shunt conductance for calibration correction E. Simulations and fittings The mechanical quality factors (Q), frequency-quality factor (fQ) products and electromechanical coupling coefficients (K2) shown in FIGS.3A-B are extracted through the fittings to the reflection measurement results, which were taken with a commercial VNA operating between 10 GHz and 43 GHz, the V-band (50 - 75 GHz) and W-band (75 - 110 GHz) setups aforementioned in section II. For the V-band and W-band fittings, the DRCLR model described above can be utilized, as it correctly interprets the DRR-enhanced mechanical responses; while for the low-frequency band (10 – 43 GHz), the data can be fitted using the lumped MBVD model, as the device dimension is relatively small compared to the wavelength at these frequencies and the mechanical resonator can be viewed as a lumped element. In the DRCLR model, a total of thirteen parameters need to be determined to describe the reflection response, which are listed in Table S2. In the MBVD model, the number of parameters needed to describe the system is reduced, which are listed in Table S3. For both models, shunt capacitance Cf and shunt conductance Gf can be utilized to account for the dielectric constant difference between the TFLN and the ISS in the off-wafer calibration process. They can be - 30 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) determined via fittings to the measurement results of test structures. For the extraction of other parameters, to avoid the overfitting problem which may occur due to too many parameters, as is the case in the DRCLR model, parts of these parameters can be determined via finite element method (FEM) simulations. The parameter descriptions are listed in Table S2 and S3. As shown in FIG.11 is the measured and fitted reflection spectrum of a 155 µm-long device. The V-band and W-band data are fitted using the DRCLR model, and the lower frequency (10 – 43 GHz) data are fitted using the MBVD model. The fitted parameters relating to the mechanical properties are shown in Table S4, together with the mechanical resonant frequency f, quality factor Q, and electromechanical coupling coefficient K2extracted using Eqn. S20 and Eqn. S21, based on the fitted parameters. The fittings agree well with the experimental results and permit accurate modeling of the mechanical overtones as they scale up in the mode order and frequency. TABLE S4 – The values of the fitted parameters and extracted mechanical resonance frequencies (f ), quality factors (Q) and electromechanical coupling coefficients (K2). Mechanical mode C’m / fF L’m / nH R’m / Ω C0 / fF f / GHz Q K2 / % A3 0.55 207 193 17.3 14.9 101 3.2 A5 0.22 192 170 16.9 24.8 175 1.3 A7 0.12 171 169 18.0 34.6 221 0.69 Mechanical mode Cm / (pF / m) Lm / (pH·m) Rm / (Ω·m) CM / (pF / m) f / GHz Q K2 / % A11 0.20 43.6 0.0542 84.9 54.4 275 0.24 A13 0.12 51.4 0.0748 84.9 64.3 278 0.14 A15 0.10 44.3 0.1014 84.9 74.2 204 0.12 A17 0.092 39.3 0.1017 101 83.8 203 0.089 A19 0.065 44.3 0.1168 101 93.9 223 0.065 A21 0.040 58.1 0.1594 101 103.9 238 0.040 EXAMPLE 2 Discussed herein is an IDEAL mm-wave resonator architecture to achieve COFFEE TA3 goals. IDEAL, standing for “Integrated Differential Electro-Acoustic Lamb wave” resonator, is the first electromechanical resonators in the mm- wave W-band (75-110 GHz). Based on thin film lithium niobate (LN), IDEAL offers strong electro-acoustic coupling, low mm-wave loss, and high potential for wafer-scale production. With this device architecture, quality factor beyond 200 up to 110 GHz can be routinely achieved. Judging from high signal fidelity in the - 31 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) W-band, the IDEAL platform can support resonators and filters in the full mm-wave spectrum (30-300 GHz) and above, thus presents unprecedented opportunities for 5G / 6G communications, advanced Radar and electronic warfare system. IDEAL achieves high transduction efficiency because the microwave energy is preferentially coupled to the asymmetric Lamb mode which exhibits higher acoustic confinement and lower damping than other form of acoustic waves. The construction of IDEAL resonator is illustrated in FIG.12A. At the core is a pair of head-to-head dual-rail transmission lines which support common (++) and differential (+-) hybridized mm-wave modes. The excited differential mm-wave mode couples to the asymmetric lamb waves while the common mm-wave does not. In essence, two energy transfer processes take place in the IDEAL resonator. The first step involves energy transfer from input waveguide to the differential mm-wave resonator, and the second step stores the energy further to the acoustic resonance. Thus, the differential mm-wave mode functions as a RF tank and efficiently transforms the large impedance mismatch between acoustic resonator and the input waveguide. Table 1. Proposed TA3 COFFEE Resonator Metrics TA3 goals Goals Yale SOA Phase 1 Phase 2 Center frequency 0.1-110 GHz 50 GHz 50 GHz Quality factor 278@64 GHz, 238 250 500 @104GHz Energy cpl factor 0.25%@64GHz, 5% 10% (^^^^^^^^) 0.1%@104GHz ^^^^ Max dim (wafer 0.4mm 0.3 mm 0.3 mm thickness) Among COFFEE’s four TA3 metrics, IDEAL devices already satisfy the center frequency and device dimension requirements. The major challenges addressed herein are the quality factor (Q) and energy coupling factor (a.k.a. ^^^^^2^^^) metrics. Presently, the highest f·Q = 2.5x1013is reached at 104 GHz with A21 overtone mode. If this f·Q could be retained at 50 - 32 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) GHz, the Phase 2 quality factor metrics (Q > 500) would be realized. In practice, as the overtone number goes down, the acoustic confinement becomes weaker, and retaining such high f·Q value is quite challenging. Analyses suggest that IDEAL is not yet reaching fundamental loss mechanisms such as Akhiezer effect. The main loss channels are attributed to the gold-LN interfaces, the dielectric and conductive losses. Therefore, the eMech & eMag co-design aims to achieve further enhanced acoustic energy confinement and optimized gold electrodes layout for simultaneous high transduction efficiency and low metal-induced loss. To boost ^^^^^2^^^at 50 GHz from the current value of 0.25% to 5% and beyond, the IDEAL devices can be operated at the much lower overtone A3-mode, which requires aggressive scaling of the LN thickness down to 110nm. A two material configurations, namely LN- on-insulator (LNOI) and LN-on-gold (LNOG), may be implemented to realize ultrathin, high ^^^^^2^^^resonators at 50 GHz. The LNOI resonator is suspended with gold electrodes deposited atop the LNOG device is supported directly by gold electrodes patterned on a high purity Specially oriented LN wafer-cut with high intrinsic piezoelectric coefficient are employed for the implementation of both device architectures. There are pros and cons for each of the LNOI and LNOG IDEAL device configurations. LNOI is a more mature material platform, however, the device consists of suspended structures which introduces anchoring loss and has more demanding packaging requirements. LNOG, on the other hand, is a new material platform and require more process development. It provides higher structure rigidity, potentially lower mm-wave loss, therefore better scaling capability to support thin film resonators below 100nm. These complementary features make LNOG an attractive alternative to LNOI devices. B. Technical Approach 1. IDEAL device design The IDEAL resonator is schematically illustrated in FIG.12A. A mm-wave tank resonator is formed by a dual-rail gold transmission line (GS1S2G) with signal traces S1 and S2 patterned on a released LN beam and ground electrodes placed in the unleased regime across the airgaps. The tank resonator can be probed at the start of GS1G and the GS2G can be terminated with a short. The dual-rail transmission line supports two eigen modes, the differential (asymmetric) mode and the common (symmetric) mode with their voltage profiles respectively shown in FIGS.12A-B. For the differential mode, the voltages on the two signal lines are - 33 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) opposite and an electric wall is formed between the lines represented by the dash blue line in FIG.12C; while for the common mode, the voltages are the same, and a magnetic wall is formed between the lines represented by the dashed green line in FIG.12D. The embedded acoustic resonator between the signal lines can be read out when the dual rail resonator is excited to the differential mode. This tank resonator mimics the function of a quarter lambda impedance transformer and provides efficient interface to the high impedance LN acoustic resonator. Compared to a regular quarter lambda resonator, the dual rail tank resonator has a relatively uniform asymmetric voltage distribution along the lines, thus enables a stronger coupling to the distributed mechanical modes. Considering the large dielectric constant of LN, the length of the tank resonator is only 280 µm long to support the fundamental mm-wave mode at 50 GHz (140 µm at 100 GHz). For the acoustic modes, this proposal focuses on the asymmetric Lamb modes (A-mode), which feature high electro-mechanical coupling, high Q and high-power handling capability. For illustration purpose the simulation of the first three A-modes (A1, A3, A5) can be shown in FIG. 12E. The higher order modes are also simulated but their displacement profiles are too fine to be visualized within this plot. The wavelength of n-th order A- mode is set by the resonant condition ^^^^^^^^n= 2^^^^ where ^^^^ is the film thickness. The electroacoustic coupling (^^^^^^2^^) scales with ^^^^-2, thus as the mode order increases, the mechanical excitation decreases quadratically and the mechanical displacement becomes more challenging to detect. Nevertheless, with the IDEAL devices patterned from 365nm-thick LN thin films, A1 through A21 modes can be observed spanning micro / mm-wave C, X, K, V, W bands, up to the instrumentation limit of 110 GHz. FIG.13 shows the measured reflection spectral for 3 different IDEAL devices of same thickness and but with varying transmission line length L of 110, 155, and 245 µm. All the A-mode resonances appear at projected frequencies with free spectra range of 9.88 GHz which is set by the shear acoustic velocity through: Δ^^^^ = ^^^^ / ^^^^. The mm-wave tank resonance shows up as a broad background low-Q resonance. As the device length decreases, the tank resonance shifts from V- band to W-band. When L = 245 µm, the V-band tank resonance leads to enhanced extinction observed for A11 and A13 mode. The dual-rail resonance pushes to W band for device with L = 110µm, and the detection of A17, A19 and A21 modes from 80 to 110 GHz is enhanced. The advantage of IDEAL design can be better revealed when all the A17, A19, and A21 resonances are shown in a single smith chart (FIG.15). The resonance enhanced detection is - 34 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) clearly manifested in device with L = 110µm which is specifically designed to target W-band. A similar plot can be also generated in the V-band. 2. IDEAL device modeling and extraction of motional parameters The IDEAL device can be modeled with a lumped-element or a distributed-element equivalent circuit. In the mm- wave band, the distributed element circuit is a more precise model because the electromechanical coupling strength varies along dual-rail transmission line however the system analysis is more involved. A simple lumped-element model can be also exploited to capture the essential features of the IDEAL device reflection and transmission spectrum. FIGS. 16C and D respectively show these two circuit models where the R0, L0, C0are associated with the tank resonator and Rm, Lm, Cm are associated with the motional branch. The value of motional elements is related to the unloaded mechanical Q and electromechanical coupling efficiency ^^^^^2^^^through: ^^^^ ^^^^21 8^^^^ 2 0^^^^ ^^^^21 ^^^^= ^^^^2^^^^ ^^^^2^^^^, ^^^^^^^^=^^^^^^^^2, ^^^^^^^^= ^^^^^^^ These values can Experimentally, they are extracted from broadband fitting to the measured admittance or reflection spectrum. At low frequencies when the electromechanical coupling strength is large, ^^^^ can be calculated from the maximum and minimum of the admittance spectrum which in turn is converted from the measured reflection spectrum S11 through ^^^^ / ^^^^0 = (1 - ^^^^11) / (1 + ^^^^11). FIG.17 shows an example of calculated admittance spectrum for a device operating in the low frequency below 40 GHz. The fitted motional branch circuit parameters Rm, Lm, Cm are labeled at each resonance and used to extract the respective Q and ^^^^^2^^^. A more reliable approach is to apply the equivalent circuit model and directly fit the S11 spectrum over a broadband. FIG.14D shows the extracted ^^^^^2^^^values for all nine A-modes over the full Ka, V, W bands from a device with L =155 µm (FIG.13, red curve). The mode dependence of ^^^^^2^^^is empirically described by ^^^^^2^^^= 42.3 / ^^^^2.2, closely following the theoretical prediction. 3. Resonator center frequency: the impact of LN thickness, gold thickness, and resonator orientation. The impact of LN thickness. The resonant frequency of IDEAL devices is pre-dominantly determined by the thickness, which presents an important knob for adjusting center frequency. For a given film thickness, different overtone modes are separated by a free spectral range ^^^^^^^^^^^^ = - 35 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) 6GHz � ^−13. ^^^^^^^^^^^� . This relation is clearly reflected in FIG.18, where four modes with FSR = 9.88 GHz in a 365nm-thick device and three modes with FSR=13.75 GHz in a thinner 270nm-thick device within the entire W-band can be measured. For a selected mode, the center frequency shifts linearly as the film thickness is fine-tuned,^^^^^^^^^^^^−^^^^^^^^^^^^^^^^=^^^^ , or 50MHz / nm tuning rate for a device of 1µm-thick operating at 50GHz, of center frequency for each 1nm thickness change. In the actual device distribution in the whole wafer can be mapped and the film thickness can be adjusted to the target value by ion-mill. The ion-mill rate is tuned to 10nm per 30m so that precision control of film thickness can be achieved iteratively. Resonator frequency “fine” tuning by on-chip dc electrodes. The center frequency can be tuned by applying an in-plane dc-electric field through additional electrodes patterned on chip. The tuning rate is measured to be on the order of 0.5 ppm / V for a pair of electrodes separated by 500 µm, and the applied voltage is in the range ±130V before thin film breakdown. This electrical tuning function can be exploited for compensating environmental effects such as temperature drift in practical filter applications. Impact of gold thickness. In the high overtone resonators, the gold thickness (typically 200- 300nm) and location have negligible impact on the resonant frequency. Gold participates in the device mostly electromagnetically rather than electromechanically. This is because gold has very high acoustic impedance compared to LN and the acoustic energy is mostly confined within LN. However, for lower overtone modes, the acoustic confinement becomes weaker and gold electrodes start to influence resonant frequency and quality factor. In order to minimize the impact of gold electrodes, a photonic cavity may be created to achieve better acoustic confinement within LN. Details are presented in Section 5. Impact of device orientations. Lithium niobate is a highly anisotropic material. Its piezoelectric coupling constant ^^^^^2^^^strongly depends on the acoustic mode, crystal orientation, electrical field profile, and wave propagation direction (FIG.20). There is a trade-off between attaining highest electromechanical coupling efficiency and achieving orientation-independent device fabrication. The z-cut films have reasonably high electro-mechanical coupling (> 40% for A1 mode, 59% theoretical limit) and high insensitivity to device orientation in plane. FIG.19 shows a set of devices of identical geometry but rotated in the xy plane at incremental angles. The A17, A19, and A21 modes are recorded at identical frequencies for all device orientations. - 36 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) To meet DARPA’s ^^^^^2^^^metrics, in addition to z-cut films, resonators with LN wafer-cut of other crystal orientations, such as 115-128oY-cut films may be used. In these films, the device performance is highly orientation-dependent in-plane therefore a similar approach developed for z-cut film may be used to systematically vary the resonator in-plane orientation for optimized performances. 4. LN wafer cut selection and approaches for boosting ^^^^^^^^^^^^ A specific LN wafer-cut is used herein to maximize piezoelectric coupling. In the past, z- cut and x-cut wafers were used for their commercial availability. A LNOI foundry can also be used to smart-cut specialty wafers sourced from a 3rd-party as well as bonding on non- conventional substrates. These resources may be used to fabricate strong piezoelectric LNOI wafers on low loss substrates. FIG.20 presents the simulated intrinsic electromechanical coupling (^^^^^^2^^) of A1 LNOI Lamb mode resonator for different wafer-cut orientations. For a selected wafer-cut, ^^^^^^2^^is plotted against the propagating angle of Lamb wave resonator. The z- cut film has clear overall advantages over x-cut and y- cut films for its large electromechanical coupling (~60%) and insensitivity to device in-plane orientation. The largest electro- mechanical coupling, however, happens with a 93% coupling efficiency in x-propagating Lamb wave of 122oy-cut film. In this film, devices fabricated within ±20oall have ^^^^^^2^^>80%. The more commercially viable 128oy-cut film only has a slightly lower conversion efficiency of 91% therefore may be selected for device demonstration. z-cut film may also be used for fabrication runs and for ultrathin devices targeting A1 mode. It should be noted that ^^^^^^2^^is inherently a material property calculated without mechanical loading (i.e. infinitesimal electrodes) and assuming perfect electro-acoustic overlap�^^^^ = ∫^^^^^^^^^^^^^^^^^^^^^^^^⁄�∫^^^^^2^^^^^^^^^^^ ∫ ^^^^^2 2^^^^^^^^^^^ = 1�. A practical devices ^^^^^^^^ is geometry and modaldependent. For example, for A1 mode shown in FIG.13, ^^^^^2^^^= 42% can be inferred, which is less than ^^^^^^2^^= 59% predicted in the z-cut films. Nevertheless, the model shown in FIG.20 accurately describes the angle dependence in z-cut films (FIG.19). In the phase 1 estimates provided in this proposal, it was assumed that s = 0.72 based on the experimental data and those of references. It is believed that this value may be pushed to beyond 90% in ultrathin films and phonon-engineered structures. 4. Approaches for resonator Q improvement - 37 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) In FIG.13D, it can be found that the f·Q of IDEAL resonators increases as the mode number goes up. The highest fQ is ~ 2.5x1013Hz recorded at the 21stmode. Although this value is a record for among all the piezoelectric materials, the intrinsic material limited Q is likely much higher because a saturation behavior normally expected in thin films was not observed. Indeed, fQ beyond 8x1013Hz was demonstrated in bulk LN resonator. Akhiezer damping, prominent for MHz resonators, predicts fQ > 1014Hz at mm-wave frequencies, therefore should not be a limiting factor either. The non-saturation behavior of fQ suggests that there are plenty opportunities to further boost the quality factor by materials and device engineering. The following techniques may be used for further Q improvement with IDEAL resonators: (1) Reduce the dielectric loss and conduction losses. During the smart-cut process, microscopic defects can be introduced into the LN crystal and impact the mechanical loss. These defects can be mitigated by applying an annealing process established previously in the lab. A similar annealing process can be also applied to the metal film prior to the bonding. (2) Reduce substrate induced losses. The doped or unintentionally doped silicon substrate can introduce mm-wave losses. The LNOI and LNOG chips may be prepared with high- resistivity, high purity silicon substrates, which exhibit loss tangent (tan ^^^^) less than 2x10-4in mm-wave bands. The loss arising from thermal oxide can be also reduced by suitable design of release structure in LNOI. (3) Phonon cavity design for improved lateral acoustic confinement and reduced electrode loading. Simple rectangular slabs have primarily been used to support Lamb modes. With thin film LN, the lamb modes are very well confined along the thickness direction, and the lateral confinement is often weaker and leads to mechanical loading due to electrodes. This mechanical loading is partially lifted by the large acoustic impedance of gold which however is not infinite. A phononic cavity may be used to further trap the acoustic energy in between the gold electrodes. FIG.21 shows a simple cavity design and the simulated trapped acoustic mode within a two-step etched trapezoidal LN structure. Under the electrodes, the mechanical motion is minimized at the gold / LN interfaces, leading to significant reduced damping. This configuration also improves the electromechanical coupling due to improved e-field uniformity and better overlap with the acoustic mode. (4) Control of electrodes. The use of integrated tank resonators reduces the amount of metal deposited on the resonator therefore contributes to the reduction of the acoustic loss in the - 38 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) resonator platform. While the layout options for the LNOI resonator are limited, there is more flexibility in arranging the electrodes in the LNOG architecture. For example, thicker and narrower electrodes can be used to raise the acoustic impedance of gold electrodes without increasing their electrical impedance. 5. Measurement setup and precision calibration in the mm-wave bands Mm-wave measurement setup. The home-built V-band and W-band S11 measurement system can achieve calibrated reflection measurement between 40 GHz and 110GHz with better than 6 µHz resolution. The setup is based on frequency up / down-conversion scheme sourced by a Zurich Instruments ultrahigh frequency lock-in amplifier (UHFLI). In the W-band, the probe signal is generated by a 6X multiplier with input RF1 swept from 12.5GHz to 18.3GHz. The probe signal is then coupled to the DUT via a direction coupler. The reflected signal from the DUT is taken from the through port of the directional coupler and then passed on to a W-band mixer for frequency down-conversion to RF band. The LO of this W-band mixer is derived by 6X multiplication of a second source RF2 which is detuned from RF1 by a fixed frequency ^^^^ = 80 MHz. The output of the W-band mixer is demodulated at 6-th harmonics of the reference signal ^^^^. The complete measurement setup shares the same high precision clock source therefore maintain phase coherence and can be kept stable over extended measurement time. The V-band setup is almost identical to the W-band except that 4x multipliers can be used and all the WR10 waveguides are replaced with WR15 waveguides. FIGS.22A-C show the schematic setup for W-band (A), V-band (B) and a photo of the combined experiment setup (C). For the COFFEE setup, V-band setup may be modified to lift the reference plane to the probe input and output and therefore allow full scattering matrix measurement. Precision calibration of scattering matrix. The electromechanical coupling scales inversely with respect to n2. Thus, at high frequencies, as the mode order increases, the mechanical excitation will decrease quadratically and the mechanical displacement will be more challenging to detect. In addition, as the measurement frequency goes higher, a perturbation to the measurement setup can cause a significant phase and amplitude change to the signals. Thus, a thorough calibration of the measurement network is essential to effectively read out the weak mechanical mode. The one-port network SOL calibration is performed by utilizing an impedance standard substrate (ISS) supplied by FormFactor, Inc. The reflection spectrums for short, open and load conditions are measured to calculate the error parameters in the network shown in FIG. - 39 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) 23 when the left-signal path is switched on. Such error parameters will be further used to calibrate the reflection spectrum for the DUT. To compensate for the dielectric constant difference between the ISS and the TFLN chip, test structures are used to de-embed the feedthrough capacitance and conductance. The one-path two-port calibration is performed by utilizing the through, reflect and line (TRL) conditions fabricated on the same chip as the DUT. The reflection spectrums for these conditions are measured to calculate the error parameters for the one-path two-port network. Such error parameters can be further used to calibrate the reflection and through spectrum for the DUT. 6. Process flow, resonator uniformity, and manufacturability Process flow of LNOI-IDEAL resonators. The LNOI processing is an established process and the film thickness may be scaled from 300nm down to 100nm. The major steps are shown in FIG.5. • Step 1: Start from commercial LN- on-SiO2(2µm)-on-silicon wafers prepared by smart-cut techniques. During smart-cut, microscopic defects can be introduced into the LN crystal and impact the mechanical loss. These defects can be suppressed by applying an annealing process established previously. • Step 2-4: To reach a specific LN thickness, apply a two-step dry etching process. With HSQ resist covering the LN film except for the release windows, a first Ar-ion mill etch the LN film in the release window. The HSQ mask is then removed to expose the whole wafer to a second Argon etch to a target LN thickness. The LN in the release window should be fully etched at this point. • Step 5: The gold electrodes are then deposited through standard lift-off process. In order to retain high acoustic impedance contrast between gold and LN, don’t apply metal adhesion layer such as Cr or Ti. • Step 6: The resonator is finally released with a timed buffered oxide etch in BOE. Process flow of LNOG IDEAL resonators. The proposed process flow for LN-on-Gold resonators is illustrated in FIG.24 and elaborated below. • Step 1: The fabrication process starts with the preparation of two separate wafers (or dies to save material cost): one standard commercial LNOI wafer, and one gold-coated high resistivity silicon wafer. The LNOI wafer is thinned to a target thickness of LN. • Step 2: The gold film on HR-Si is patterned to form tank resonator and annealed in vacuum - 40 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) to reduce RF losses. • Step 3: The two wafers are then joined by a plasma-activated direct-flip chip bonding technique. This bonding process is particularly suitable for LN and routinely utilized to join LNOI with other type of wafers, see reference. • Step 4: The silicon handle of LNOI wafer is removed by DRIE. The DRIE stops at the buried oxide layer (BOX). • Step 5: The remaining BOX is then selectively removed in BOE. At this point, LN film is continuous on the surface of the wafer and more than 99.9% of its surface area is supported by the gold ground planes and electrodes. The air pockets have very small footprint therefore should not impact the fabrication flow and yield. • Step 6: Standard LNOI fabrication process outlined earlier will be then applied to form the LN resonator. Resonator uniformity. One of the advantages for IDEAL resonators is that their center frequency is set by the thickness. Its variance dependance on film thickness is expressed by^^^^^^^^^^^^^^^^^^^^= −^^^^^^^^. T ^^^^ o maintain less than 1% frequency difference, the thickness variation of LN than less than 1% or 1nm across the die. This level of control is challenging but achievable using the timed wafer thinning process. In this fabrication, the film thickness is pre- screened by mapping the LN thickness using a high resolution ellipsometer. Manufacturability. IDEAL resonators operate in air and at room temperature, and standard non- vacuum MEMS packaging can be applied for ambient operations. The fabrication process at Yale starts from 4-inch wafers and all the fabrication tools are 4-inch compatible. To save materials cost, chips in 10mm x 10mm dies can be processed. The yield is close to 100% if a critical point dryer (CPD) is used after structural release. The CPD process is not required if a thicker buried oxide layer is utilized. The critical dimension of the devices is on the order microns. Although ebeam lithography can be used for fast turn-around in the university fab, all the structures can be defined with a stepper and the complete device processing can be scaled up in a MEMS foundry with high level of manufacturability. 7. Power handling Currently, for LNOI devices operating in the W-band, structural damage or noticeable resonance shift up to 20dBm input power was not observed, which is the highest power the - 41 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) instrumentation can provide. A tiny amount of static deflection of the LN membrane under microscope was observed through the change of interference pattern. This is likely due to the heating of the device which induces static stress in the membrane. The power handling in LNOG devices is believed to be significantly higher due to the use of high conductivity gold as structural support. 8. Proposed device designs and layouts LNOI-IDEAL device design and layout. LNOI designs have the tank resonator patterned on top of the LN thin film. The electrical field is predominately lateral because the film thickness (d) is much smaller than typical lateral dimension (W0~ 10-15µm). This configuration also suppresses unwanted spurious modes near the target A-mode. In most of the current devices, thanks to the tank resonator, the metal trace on the resonator can be minimized and meanwhile provide highly uniform electrical field in the film. Because ^^^^^2^^^scales as ^^^^ / ^^^^S^^^^E^^^^2, to achieve 5% energy conversion efficiency the overtone number of a 50 GHz resonator can be decreased from current value of ^^^^ = 11 to 3, by using very thin LN film (d ~ 110nm) of high piezoelectric coefficient e, such as 108oY-cut LN (FIG.20 for detailed analysis). The gold electrodes can also be made thinner to avoid fracturing the film and reduce the mechanical loading. Phononic cavity engineering (FIG.21) may be applied to enhance acoustic confinement, reduce the acoustic / electric losses, and enlarge electro-acoustic coupling. Several cross-sectional and top view layouts are shown in FIGS.25A-B. Critical dimensions, such as film thickness, electrodes size and spacing can be found in Table 2. LNOG-IDEAL device design and layout. LNOG designs place LN thin film on top of the tank resonator through a flip-chip bonding process. Hence, it enables more flexibility on electrode layout and the gold films can be made much thicker to reduce conduction losses and increase the acoustic impedance. The complete chip is oxide-free on high resistivity silicon substrate therefore the dielectric loss is also reduced. It is also possible to use T- shaped gold electrodes to form narrow contacts without increasing electrical impedance. The high structural rigidity of the LNOG design also permits the fabrication ultrathin LN thin film resonators down to 40nm-thick. This will allow the operation of A1 mode at 50GHz (See Table 2, column 2). In this configuration, more than 40% electromechanical coupling (59% simulated) can be realized in conveniently available z-cut films. Without wishing to be bound by theory, it is believed that the metal-induced loss and - 42 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) acoustic radiation losses through the anchors and substrates can be minimized by controlling the placement of electrodes and by optimally distributing the acoustic impedances across the resonator structure. Additionally, the loss in the electro / acoustic domain may be reduced by 1) selection of annealing conditions of the bonded LN films to suppress defects in LN, for example by controlling the temperature ramping profile, gaseous environment of annealing, and etc.2) selection of deposition condition of gold to ensure high conductivity and high acoustic impedance.3) configuration of the bonding process to minimize the acoustic and mm-wave loss arising from bonding interfaces. EXAMPLE 3 High-performance electromechanical resonators are in great demand in communication industries. In terms of communication speed, the use of millimeter-wave spectra can offer unprecedented opportunities for next-generation networks, enabling high-bandwidth applications. Among the materials being explored, thin-film lithium niobate resonator stands out as a promising candidate due to its strong piezoelectric properties and low acoustic loss. However, in nearly all existing lithium niobate (LN) electromechanical devices, the configuration is such that the electrodes are in direct contact with the mechanical resonator. This can give rise to undesirable mass-loading effect, introducing spurious modes and additional damping to the resonator. A novel electromechanical platform can include electrodes which are separated from the mechanical resonator with a flip-chip bonding technique. By offloading the nanomembrane, increased quality factor of these resonators can be achieved. Electromechanical resonators can be pivotal components in modern-day communication technologies. The ongoing deployment of 5G networks has significantly heightened the need for high-performance electromechanical resonators. Frequency-wise, millimeter-wave frequencies can offer a broader bandwidth for high-data-rate communications. Yet, the large insertion loss at these frequencies can present challenges for scaling electromechanical resonators to operate within this range. As a result, high quality factor (Q) acoustic configurations can be in great demand. In some cases, thin-film lithium niobate (TFLN) can exhibit excellent piezoelectric properties, low acoustic loss and CMOS compatibility. LN related devices can be a distinct electro-mechanical platform to scale up to millimeter-wave frequency. In some embodiments, - 43 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) asymmetrical lamb wave modes of a suspended membrane on z-cut TFLN can be utilized. In some implementations, the asymmetrical lamb wave models can be effectively excited via a horizontal electric field through an e51 element of a piezoelectric coupling tensor. In some instances, sub-terahertz electromechanics can showcase the compatibility of such a platform to millimeter-wave applications. However, for most existing electromechanical configurations, the metal electrodes are directly in contact with the vibrating material (reflected in FIG.27A). Therefore, acoustic waves unavoidably propagate into electrodes (displacement field shown in FIG.27A(i)), and the electrodes can serve as a dissipation channel of the mechanical mode as shown in FIG.27A(ii). In this scenario, mechanical damping of electrodes can contribute to the overall decay of stored energy in the LN mechanical resonator. To circumvent this problem, FIG.27B outlines a platform where an air gap (or at least an open section) separates the electrodes and the membrane. In this case, most acoustic energy are confined in a LN membrane due to the significant mechanical impedance mismatch between the solid and the air. This can be visualized by the displacement field and the energy distribution plotted in FIG.27B(i) and FIG. 27B(ii). In contrast to the previous case (as shown in FIG.27A(i) and FIG.27A(ii)), the off- loaded electrode configuration neither bring spurious modes nor loss channels to the mechanical system. 1. Simulations of Mechanical Responses with Different Electrode Damping Coefficients To further illustrate the advantages of the off-loaded configuration, simulations of the mechanical responses with different electrode damping coefficients, using a finite-element method (FEM) in COMSOL can be used. As for the damping coefficient of the LN thin film, a realistic value of 2×10−4 was set, based on the extracted data from a LN acoustic delay line. In some implementations, the damping coefficient of gold electrodes can be varied from 0.001 to 0.02 and plotted the corresponding A3 mode quality factor (as shown in FIG.29C), for three different configurations: 1. electrodes in contact with the membrane; 2.50nm air gap (or at least an open section) between electrodes and membrane; 3.1350nm air gap between electrodes and membrane. Thanks to the air gap that separates the LN membrane from the electrode loss channel, the offloaded system Q can be much higher than the loaded case, close to the intrinsic value of LN. In some cases, however, in the loaded case, although the Q increases slightly with the decrement of gold damping, it is still largely limited by the metal. - 44 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) 2. Fabrication Process for Resonator Configuration with Separation Between Membrane and Electrodes To achieve a separation between the membrane electrodes, a flip-chip bonding technique can be implemented. The complete resonator configuration is shown in FIG.28A, where a sapphire chip with metal electrodes is bonded to a TFLN chip with suspended membranes, with a gold-gold bonding technique. The fabrication process is illustrated in FIG.28C. The initial LN wafer for processing is LN(300 nm) / SiO2(5 μm) / Si substrate. For the definition of release windows, the LN layer can be patterned by electron-beam lithography using FOX16 as the resist, and etched by Ar ion milling (as shown in FIG.28C(i)). An adhesion layer of 5nm chromium and 40nm gold can then deposited using a lift-off technique (as shown in FIG.28C(ii)). The membrane chip can be finally released in buffered oxide etchant (BOE) by isotropically removing the SiO2 layer beneath the beam (as shown in FIG.28C(iii)). As for the electrode chip, the chip can include a bare sapphire wafer and a transparent material favorable for flip-chip alignment. The chromium / gold adhesion layer can be deposited first and then protected by nickel. In order to grow thicker electrodes, the electrode area can be etched using Cl2 and BCl3 plasma with Ni as a hard mask (as shown in FIG.28C(iv)). Then, 200nm gold electrodes can be deposited with a lift-off technique and nickel can be removed by piranha solution before bonding (FIG.28C(v)(vi)). For the final step, a flip-chip bonder is used to bond the LN membrane chip and sapphire electrode chip. As a result, a tight bonding between these two chips can be achieved and a well-defined air gap between the membrane and electrode can be obtained (as shown in FIG.28C(vii)). The optical microscope image of a completed flip-chip bonded device can be shown in FIG.28B). 3. Reflection Spectra and Q Results As shown in FIG.28A, a ground-signal-ground (GSG) probe can be used to interface with the electrodes on the sapphire chip. The mechanical mode can be piezoelectrically coupled to the microwave field and can be read out through a reflection measurement using a vector network analyzer. The reflection spectra (S11) of two devices are plotted in FIG.29A. A modified Butterworth-Van Dyke (MBVD) model can be used to fit the experimental data, represented by solid lines in the insets of FIG.29A. The measured membrane has a thickness of 300 nm. In the reflection spectrum, three major resonances with a well-defined FSR are observed, corresponding to the fundamental, third-order and fifth-order A modes. The highest Q - 45 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) obtained is of the A3 mode, with a value of 934 at 18GHz. The Qs of A1 and A3 modes are 608 and 649 respectively. All of them are around 2∼3 times of the Qs of the devices where electrodes directly contact the LN membrane, as illustrated in FIG.29C. This demonstrates that by offloading the mechanical resonator and thus reducing the acoustic loss channel, the Q of the electro-mechanical platform can be significantly improved. Moreover, this technique for enhancing the Q-factor can be universally applicable to materials with varying loss properties. As shown in FIG.29B, the intrinsic Q of the LN thin film and the loss of gold electrodes can be varied, which can display that the off-loaded platform maintains its competitive advantage on the system mechanical Q over the loaded case. The experimental data are also marked in this figure. As for the TFLN chip after releasing, there is around 1.3 μm buckling of 300nm LN film towards silicon substrate, which is measured by a 3D optical profilometer and the buckling magnitude is consistent with optical interference patterns on the suspended LN structure in FIG.27B. Therefore, the estimated air gap between electrodes and the suspended LN membrane is around 1.35 μm, including the height offset between bonding layers and electrodes. As a result of the larger air gap, the electromechanical coupling coefficient (^^^^^2^^^) are lower than expected, which are 0.02% and 0.01% for the A3 and A5 modes respectively. However, based on FEM simulations, if the air gap is 50nm or smaller, the ^^^^^2^^^of the off-loaded case is close to the loaded case. Importantly, such a small air gap allows for the complete elimination of electrode loss channels. Consequently, the off-loaded configuration can offer both high-Q and high ^^^^^2^^^features. EXAMPLE 4 Advancing electromechanical resonators towards terahertz frequencies opens vast bandwidths for phononic signal processing. In quantum phononics, mechanical resonators at these frequencies can remain in their quantum ground state even at kelvin temperatures, obviating the need for millikelvin cooling typically required for gigahertz resonators. However, electrical actuation and detection of mechanical motion at such high frequencies present significant challenges, primarily due to the need for device miniaturization to support acoustic waves with nanometer-scale wavelengths. One effective strategy is to aggressively thin down piezoelectric thin films, ideally to a thickness on the order of the acoustic wavelength, which is in the tens of nanometers. In this work, we systematically reduce the thickness of lithium niobate from 300 nm to 67 nm through several stages, and fabricate suspended Lamb-wave resonators at - 46 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) each thickness level. These resonators achieve resonant frequencies as high as 220 GHz, doubling the previous record and approaching the terahertz frequency threshold. While ultrathin films exhibit a clear advantage in frequency gains, they also experience increased acoustic losses. Our results suggest that future advances in terahertz nanomechanics will critically depend on mitigating surface defects in sub-100 nm thin films. INTRODUCTION Located within the under-explored spectral region that bridges the gap between conventional microwave and infrared optical domains, the terahertz (THz) frequency range (∼0.3 – 10 THz) presents a landscape rich with opportunities. Terahertz waves are favored in sensing and spectroscopy fields, thanks to their non-ionizing nature, their sensitivity to vibrational and rotational modes of a variety of molecules, and their exceptional ability to penetrate non-polar and non-conductive materials. This has led to their extensive use in multiple sectors, including industrial and medical applications, as well as basic sciences such as astronomy, condensed matter physics, and biochemistry. However, the lack of reliable sources, detectors, and control components has always been a bottleneck for THz technologies. Devices like electromechanical resonators, capable of interchanging energy between the electrical and mechanical domains, could play a key role in these systems. THz electromechanical resonators could offer coherent manipulation of THz phonons, thereby serving as excellent inspection tools. Other than classical applications, from a quantum science perspective, nanomechanical resonators have emerged as promising platforms for fundamental quantum studies. THz mechanical resonators, with an ultrahigh resonant frequency, can be directly refrigerated to their mechanical ground state with a temperature of several kelvins. In comparison, gigahertz (GHz) resonators would necessitate stringent millikelvin temperatures or the employment of sideband cooling techniques. Recently, sub-THz electromechanical resonators operating near 110 GHz have been demonstrated on thin-film lithium niobate (TFLN) platform, leveraging its excellent piezoelectric properties. However, advancing mechanical resonators towards the THz regime still presents significant challenges. At 300 GHz, the acoustic wavelength of z-cut lithium niobate (LN) thickness-shear (TS) modes is around 10 nm, which creates substantial difficulties in terms of efficient transduction. One effective strategy involves thinning down piezoelectric thin films to scale up the high-order thickness (T) modes, while still preserving electromechanical coupling - 47 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) K2even at hundreds of GHz. In this Example, we systematically reduce the LN thickness from 300 nm to 67 nm to achieve multi-thickness levels. By fabricating suspended Lamb-wave resonators (LWRs) at each thickness level, we are able to study the relationship between mechanical quality factors (Qs) and thin film thicknesses. We demonstrate resonant frequencies up to 220 GHz, doubling our previous record and nearing the THz frequency threshold. However, thinner sub-100 nm films show increased acoustic losses likely due to surface defects. RESULTS 220 GHz nanomechanics FIG.30A displays the optical image of lithium niobate-on- insulator (LNOI) chip as prepared, featuring incremental thickness steps: 67, 107, 165, 230, and 300 nm (see Methods). FIG.30B illustrates the fabricated chip, consisting of LWRs across different thickness levels (see Methods for fabrication details). In these images, optical interference leads to the perceived color difference among the films with different thicknesses. FIG.30C shows the cross-sectional schematics of the LWRs. In this configuration, the resonators are suspended to minimize acoustic losses through the substrate. The electric field between the electrodes is mostly in the horizontal direction, facilitating coupling to the TS modes supported in the suspended z-cut LN through the large piezoelectric coupling element ^^^^51. The TS mode displacement schematics of varying film thicknesses are shown in FIG.31A. Clearly, to achieve a frequency around 150 GHz, thicker films necessitate an increased mode order (n). This escalation in mode order is accompanied by a reduction in the electromechanical coupling coefficient K2, which is inversely proportional to the square of the mode order (∝ 1 / n2). FIG.31B displays the reflection spectra of devices, each with a length of 110 µm and varying thicknesses of 67, 107, 165, 230, and 300 nm. The data were collected using two vector network analyzers (VNA) that cover the D-band and G-band, respectively (see Methods). Here, the spectral data are concatenated at 140 GHz. Precise calibration methods are employed to minimize the stitching of spectra across different bands. Mechanical resonances around 220 GHz are observed in FIG.31B, including T35, T27, T19, and T13 modes for films of 300 nm, 230 nm, 165 nm, and 107 nm thicknesses, respectively. Additionally, the broadband tank resonance shifts rightward with decreasing film thickness due to a reduced effective dielectric constant. Typically, even resonances are not effectively excited because the overlap between their strain fields and the applied electric field is mostly cancelled out. However, we - 48 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) observed an intriguing phenomenon where, as film thickness decreases, resonances with even mode numbers begin to appear. For instance, in the 165 nm-thick device, modes T10, T12, T14, T16, and T18 can be observed, although their responses are significantly weaker than those of the odd resonances. As the LN thickness decreases to 107 nm, the extinctions of the even resonances approach those of the odd resonances. In the case of the 67 nm-thick film, the odd resonances diminish significantly, and the even resonances are predominantly excited, likely due to the increased etch damage which inadvertently modifies the surface ferroelectric properties. Additionally, we plot in FIG.31C the Smith chart representations for films of 300 nm, 230 nm, and 165 nm thickness. Even with lower mode orders, the resonance circles of the 165 nm films are generally smaller than those of thicker films, indicating a lower Q for thinner films. This phenomenon is also evident in FIG.31B, where thinner films exhibit wider mechanical linewidths. Mechanical Q and piezoelectric film thickness The fitted intrinsic mechanical Qs (with a Lorentzian model) for various mode orders are shown as scattered points in FIG.32, with each color indicating different device thicknesses. Markers of the same color denote data from the same device as depicted in FIG.31B. Due to the low signal-to-noise ratio of some resonances and uncertainty in fitting, we have omitted them from the plot. For relatively thick resonators, the Q values exhibit considerable variation across different frequencies, possibly linked to the varying energy participation ratio of the lossy gold electrodes at these frequencies. Further investigation is needed to better understand this phenomenon. Furthermore, we observe that resonators with greater thickness typically exhibit higher Q factors compared to thinner ones. This is more evident in FIG.32 (panel b), which statistically presents the Q values of various devices with different thicknesses at approximately 63 GHz (FIG.32 (panel b(i))) and 168 GHz (FIG.32 (panel b(ii))). The data clearly show that a larger film thickness is associated with a higher Q. This negative correlation between the Q and the resonator’s surface-to-volume ratio likely originates from the surface-related losses, including the surface damage induced during the plasma etch. As a result, the low-Q performance of sub-100 nm films poses challenges in demonstrating THz mechanical resonances. Nonetheless, efforts towards minimizing surface defects and improving the film quality of sub-100 nm films are promising for unlocking significant potential and broad applications in THz nanomechanics. - 49 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) Lamb-wave resonator’s advantage for advancing THz nanomechanics Over the years, a wide range of micromechanical / nanomechanical resonators has been developed, tailored for various applications such as sensing, signal processing, communications, and quantum information studies. These types include beam resonators, thin-film bulk acoustic resonators (FBAR), surface acoustic wave (SAW) resonators, optomechanical crystals (OMC), and LWR. From a frequency scaling perspective, advances in fabrication techniques, such as more precise lithography and etching processes, improved material platforms, and innovative design approaches, have consistently driven mechanical frequencies upward. This progression is evident in the evolution from the leading resonators operating below 1 GHz two decades ago to the recent achievement of our current 220 GHz electromechanical resonators setting a new frequency record. Unlike many other resonator types where frequencies are constrained by the horizontal dimensions due to current lithographic limits, LWR frequencies largely depend on the thickness of the plate. This relaxes the precision requirement in fabrication and enables frequency scaling utilizing thinner films. DISCUSSION In our experiments, we observed that as film thickness decreases, resonances with even mode numbers begin to appear. This observation is atypical, as even resonances are generally not efficiently excited due to the cancellation of the overlap between their strain fields and the applied electric field. This phenomenon can be modeled by considering a specific thickness of piezoelectrically inactive layers, resulting from surface damage induced during the plasma etch. These non- piezoelectric layers do not contribute to electromechanical excitation but still participate in the resonator’s vibration. Consequently, the electromechanical coupling becomes non-zero for even mode orders. Below, we describe in detail the modeling of piezoelectrically inactive layers. The electromechanical coupling coefficient K2can be written as ∫�⃗ ( ^) ∙ ^^^^ ∙ ^^^^(^^^^)2 �^^^^ ^^^ ^^^^^^^^�where ^�^^⃗^ represents the piezoelectric coupling tensor and elasticity tensor respectively, and ^^^^ is the dielectric constant. In - 50 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) this study of z-cut thickness-shear resonators, we simplify the analysis by focusing on the primary displacement in the x-direction ^^^^^^^^, which corresponds to a shear strain ^^^^^^^^^^^^. This shear strain is coupled to the horizontal electric field ^^^^^^^^through the ^^^^51piezoelectric coupling element. For a resonator of thickness h, the resonant condition is written as ^^^^^^^^^^^^ = 2ℎ, where n isthe mode order, and ^^^^^^^^is the wavelength (FIG.34 inset). The shear displacement ^^^^^^^^of the n-th mode can be estimated as ^^^^^^^^^^^^^^^^ ^^^^ cos�ℎ�, with ^^^^^^^^representing the amplitude. We then have the 1 ^^^^^^^^ ^^^^^^^^ ^^^^^^^^^^^^ ^^^^^^^ ^^^ ^^^^^^^^^ ^^^^^^^^^^^^^ =2 �+ ^^^^^^^^^^^^^^^^� = −2ℎsin�ℎ�(S2)Considering a does not contribute to piezoelectric coupling, the electromechanical coupling for the n-th mode ^^^^^2^^^can be written as 2 ^^^^2�2^^^^ℎ−^^^^ ^^^^ ∫ ^^^^^ ^^^^^^^^� ^^^^251 0 ^^^^^^^ Where ^^^^^2^^^0is defined as 4^^^^2^^^^2^2^^^^2 =51^^^^^^^When the ^^^^^2^^^0 [(−1)^^^^ − 1]2, indicating that ^^^^2 is nonzero only for odd n. However, electromechanical begins to emerge for even modes. In FIG.34, we plot the relationship between ^^^^2^^^^⁄ ^^^^2^^^^0 and ^^^^⁄ ^^^^^^^^ for both odd and even mode orders. As the thickness t- 51 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) increases to 0.5^^^^^^^^, ^^^^^2^^^decreases for odd n but increases for even n. When t ranges from 0.5^^^^^^^^to ^^^^^^^^, the trend in ^^^^^2^^^reverses. This explains why, in thinner films, even modes tend to be more efficiently excited: thinner films are subject to more aggressive etching, which may result in comparatively piezoelectrically inactive layers. CONCLUSIONS We demonstrate mechanical resonance frequencies up to 220 GHz based on TFLN. By systematically thinning down LN through several stages and fabricating LWRs at each thickness level, we are able to study the correlation between mechanical Qs and the resonator’s surface-to- volume ratio. This paves the way for further optimizing sub-100 nm film qualities for improved device performance in THz nanomechanics. Additionally, we traced the evolution of frequency scaling in mechanical resonators, highlighting the advantages of LWRs in progressing towards THz frequencies. METHODS Nanofabrication. Initially, an LNOI chip with multiple thickness stages from 67 to 300 nm is prepared. These different thicknesses are achieved through multiple blanket etching cycles, with Si chips placed on top of the completed LN regions between cycles to prevent further etching. For device fabrication, a gold electrode pattern is defined through electron-beam lithography (EBL), with polymethyl methacrylate (PMMA) as the resist, followed by a liftoff procedure. Next, the release window pattern is defined using hydrogen silsesquioxane (HSQ) resist and then argon-ion-milled to remove all LN within the designated area. Finally, the mechanical resonators are suspended by etching away the silicon dioxide beneath the LN beam with buffered oxide etchant (BOE). In FIG.33, we illustrate the fabrication process flow for the nanomechanical devices of various thicknesses. Initially, a lithium niobate-on-insulator (LNOI) chip undergoes blanket etching through multiple cycles. Between each cycle, Si chips are positioned atop the completed lithium niobate (LN) regions to prevent further etching. This produces an LNOI chip with multiple thickness stages from 67 to 300 nm. For device fabrication, gold electrodes are first defined using a liftoff procedure. Subsequently, the release window pattern is defined using hydrogen silsesquioxane (HSQ) resist, followed by argon-ion milling to remove all LN within the designated area. The process concludes with the suspension of mechanical resonators by - 52 - 55056855.3 Attorney Docket No.047162-7389WO1 (02512) etching away the silicon dioxide (SiO2) beneath the LN beam using buffered oxide etchant (BOE). Calibration. Measurements of the scattering parameters are conducted using a Keysight E8361C network analyzer with frequency extenders (N5260 millimeter heads for the W Band, WR6.5- VNAX for the D band and V05VNA2-T / R-A for the G band). For each band, a tier-1 calibration is initially performed using off-wafer impedance standards, applying the line-reflect-reflect- match (LRRM) method. Subsequently, the tier-2 calibration is performed with on-chip impedance standards, utilizing the through-reflect-line (TRL) calibration technique. The disclosures of each and every patent, patent application, and publication cited herein are hereby incorporated herein by reference in their entirety. While this invention has been disclosed with reference to specific embodiments, it is apparent that other embodiments and variations of this invention may be devised by others skilled in the art without departing from the true spirit and scope of the invention. The appended claims are intended to be construed to include all such embodiments and equivalent variations. - 53 - 55056855.3

Claims

Attorney Docket No.047162-7389WO1 (02512) CLAIMS What is claimed is:

1. An electromechanical resonator, comprising: a substrate; a Lamb wave resonator over the substrate; and a millimeter-wave dual-rail resonator coupled to the Lamb wave resonator.

2. The electromechanical resonator according to claim 1, further comprising a patterned insulating layer between the Lamb wave resonator and the substrate.

3. The electromechanical resonator according to claim 2, wherein the patterned insulating layer includes at least one open section that is devoid of insulating material between the Lamb wave resonator and the substrate.

4. The electromechanical resonator according to claim 3, wherein a portion of the Lamb wave resonator is suspended over the substrate in the open section.

5. The electromechanical resonator according to any one of the preceding claims, wherein the substrate comprises a semiconducting material.

6. The electromechanical resonator according to claim 5, wherein the semiconducting material comprises silicon (Si) or sapphire.

7. The electromechanical resonator according to any one of the preceding claims, wherein the patterned insulating layer comprises a sacrificial material.

8. The electromechanical resonator according to claim 7, wherein the sacrificial material comprises silicon dioxide (SiO2).

9. The electromechanical resonator according to any one of the preceding claims, wherein the Lamb wave resonator comprises a material having high piezoelectric coupling strength. - 54 - 55056855.3Attorney Docket No.047162-7389WO1 (02512) 10. The electromechanical resonator according to any one of the preceding claims, wherein the Lamb wave resonator comprises lithium niobate (LN), lithium tantalate (LT), or scandium aluminum nitride (ScAlN).

11. The electromechanical resonator according to any one of the preceding claims, wherein the Lamb wave resonator has a thickness of less than or equal to 300nm.

12. The electromechanical resonator according to any one of the preceding claims, wherein the Lamb wave resonator has a thickness of less than or equal to 100nm.

13. The electromechanical resonator according to any one of the preceding claims, wherein the dual-rail resonator comprises metal electrodes on the Lamb wave resonator.

14. The electromechanical resonator according to claim 13, wherein the metal electrodes comprise superconductors.

15. The electromechanical resonator according to claim 13 or 14, wherein the metal electrodes comprise gold.

16. The electromechanical resonator according to any one of claims 13-15, wherein the metal electrodes comprise two central electrodes on the suspended portion of the Lamb wave resonator and two peripheral electrodes on outer portions of the Lamb wave resonator.

17. The electromechanical resonator according to claim 1, wherein the dual-rail resonator comprises a patterned metal between the Lamb wave resonator and the substrate.

18. The electromechanical resonator according to claim 17, wherein the patterned metal comprises two peripheral portions and two central rail portions.

19. The electromechanical resonator according to claim 17, wherein the patterned metal comprises an array of dual rails. - 55 - 55056855.3Attorney Docket No.047162-7389WO1 (02512) 20. The electromechanical resonator according to any one of claim 17-19, wherein the patterned metal comprises a superconductor.

21. The electromechanical resonator according to any one of claims 17-20, wherein the patterned metal comprises gold.

22. The electromechanical resonator according to any one of claims 17-21, wherein the substrate comprises a semiconductor material.

23. The electromechanical resonator according to claim 22, wherein the substrate is high resistivity silicon (HR-Si).

24. The electromechanical resonator according to any one of claims 17-23, wherein the Lamb wave resonator includes a central portion and a peripheral portion on either side of the central portion.

25. The electromechanical resonator according to any one of claims 17-24, wherein the Lamb wave resonator comprises lithium niobate (LN).

26. An electromechanical resonator, comprising: a substrate; a Lamb wave resonator over the substrate; a millimeter-wave dual-rail resonator over the Lamb wave resonator; and an air gap formed between the dual-rail resonator and the Lamb wave resonator; wherein at least a portion of the dual-rail resonator is suspend over the Lamb wave resonator in the air gap.

27. The electromechanical resonator according to claim 26, wherein the air gap is from 5 nm to 1 micron between the Lamb wave resonator and the dual-rail resonator.

28. The electromechanical resonator according to claim 26, wherein the air gap is from 5 nm to 100 nm between the Lamb wave resonator and the dual-rail resonator. - 56 - 55056855.3Attorney Docket No.047162-7389WO1 (02512) 29. The electromechanical resonator according to claim 26, wherein the air gap is from 0 to 100 nm between the Lamb wave resonator and the dual-rail resonator.

30. The electromechanical resonator according to any one of claims 26-29, wherein the millimeter-wave dual-rail resonator can operate at a frequency up to 3 THz.

31. The electromechanical resonator according to any one of claims 26-30, wherein the millimeter-wave dual-rail resonator can operate at a frequency down to 60 GHz for fifth- generation (5G) communication systems.

32. The electromechanical resonator according to any one of claims 26-31, wherein the millimeter-wave dual-rail resonator can operate at a frequency down to 71 GHz for 5G communication systems.

33. The electromechanical resonator according to any one of claims 26-32, further comprising a patterned insulating layer between the Lamb wave resonator and the substrate.

34. The electromechanical resonator according to claim 33, wherein the patterned insulating layer includes at least one open section that is devoid of insulating material between the Lamb wave resonator and the substrate.

35. The electromechanical resonator according to claim 34, wherein a portion of the Lamb wave resonator is suspended over the substrate in the open section.

36. The electromechanical resonator according to any one of claims 26-35, wherein the air gap is configured to reduce acoustic loss. - 57 - 55056855.3

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